### (0) Obligation:

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

### (1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

### (2) Obligation:

FIGraph based on JBC Program:
TimesPlusUserDef.main([Ljava/lang/String;)V: Graph of 134 nodes with 0 SCCs.

TimesPlusUserDef.times(II)I: Graph of 47 nodes with 0 SCCs.

TimesPlusUserDef.plus(II)I: Graph of 62 nodes with 0 SCCs.

### (3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 24 rules for P and 38 rules for R.

Combined rules. Obtained 2 rules for P and 8 rules for R.

Filtered ground terms:

1364_1_plus_InvokeMethod(x1, x2, x3, x4, x5, x6) → 1364_1_plus_InvokeMethod(x1, x2, x3, x5, x6)
1333_0_plus_LE(x1, x2, x3, x4) → 1333_0_plus_LE(x2, x3, x4)
Cond_1333_0_plus_LE1(x1, x2, x3, x4, x5) → Cond_1333_0_plus_LE1(x1, x3, x4, x5)
1380_1_plus_InvokeMethod(x1, x2, x3, x4) → 1380_1_plus_InvokeMethod(x1, x3, x4)
Cond_1333_0_plus_LE(x1, x2, x3, x4, x5) → Cond_1333_0_plus_LE(x1, x3, x4, x5)
1537_0_plus_Return(x1, x2, x3) → 1537_0_plus_Return(x2, x3)
1580_0_plus_Return(x1) → 1580_0_plus_Return
1397_0_plus_Return(x1, x2) → 1397_0_plus_Return
1379_0_plus_Return(x1, x2, x3, x4) → 1379_0_plus_Return(x2, x3)
1351_0_plus_Return(x1, x2) → 1351_0_plus_Return

Filtered duplicate args:

1364_1_plus_InvokeMethod(x1, x2, x3, x4, x5) → 1364_1_plus_InvokeMethod(x1, x3, x4, x5)
1333_0_plus_LE(x1, x2, x3) → 1333_0_plus_LE(x1, x3)
Cond_1333_0_plus_LE1(x1, x2, x3, x4) → Cond_1333_0_plus_LE1(x1, x2, x4)
Cond_1333_0_plus_LE(x1, x2, x3, x4) → Cond_1333_0_plus_LE(x1, x2, x4)

Filtered unneeded arguments:

1380_1_plus_InvokeMethod(x1, x2, x3) → 1380_1_plus_InvokeMethod(x1, x2)
1364_1_plus_InvokeMethod(x1, x2, x3, x4) → 1364_1_plus_InvokeMethod(x1, x3, x4)

Combined rules. Obtained 2 rules for P and 8 rules for R.

Finished conversion. Obtained 2 rules for P and 8 rules for R. System has predefined symbols.

Log for SCC 1:

Generated 12 rules for P and 97 rules for R.

Combined rules. Obtained 1 rules for P and 26 rules for R.

Filtered ground terms:

457_0_times_NE(x1, x2, x3, x4) → 457_0_times_NE(x2, x3, x4)
Cond_457_0_times_NE(x1, x2, x3, x4, x5) → Cond_457_0_times_NE(x1, x3, x4, x5)
1580_0_plus_Return(x1, x2) → 1580_0_plus_Return(x2)
Cond_1380_1_plus_InvokeMethod1(x1, x2, x3, x4, x5) → Cond_1380_1_plus_InvokeMethod1(x1, x4, x5)
1397_0_plus_Return(x1, x2) → 1397_0_plus_Return
1380_1_plus_InvokeMethod(x1, x2, x3, x4) → 1380_1_plus_InvokeMethod(x1, x3, x4)
Cond_1380_1_plus_InvokeMethod(x1, x2, x3, x4, x5) → Cond_1380_1_plus_InvokeMethod(x1, x2, x4, x5)
1351_0_plus_Return(x1, x2) → 1351_0_plus_Return
1333_0_plus_LE(x1, x2, x3, x4) → 1333_0_plus_LE(x2, x3, x4)
Cond_1333_0_plus_LE7(x1, x2, x3, x4, x5) → Cond_1333_0_plus_LE7(x1, x3, x4, x5)
1550_0_plus_Return(x1, x2, x3) → 1550_0_plus_Return(x2, x3)
Cond_1333_0_plus_LE6(x1, x2, x3, x4, x5) → Cond_1333_0_plus_LE6(x1, x3, x4, x5)
1507_0_plus_Return(x1, x2, x3) → 1507_0_plus_Return(x2, x3)
Cond_1333_0_plus_LE5(x1, x2, x3, x4, x5) → Cond_1333_0_plus_LE5(x1, x3, x4, x5)
1459_0_plus_Return(x1, x2, x3) → 1459_0_plus_Return(x2, x3)
Cond_1333_0_plus_LE4(x1, x2, x3, x4, x5) → Cond_1333_0_plus_LE4(x1, x3, x4, x5)
1417_0_plus_Return(x1, x2, x3) → 1417_0_plus_Return(x2, x3)
Cond_1333_0_plus_LE3(x1, x2, x3, x4, x5) → Cond_1333_0_plus_LE3(x1, x3, x4, x5)
1387_0_plus_Return(x1, x2, x3) → 1387_0_plus_Return(x2, x3)
Cond_1333_0_plus_LE2(x1, x2, x3, x4, x5) → Cond_1333_0_plus_LE2(x1, x3, x4, x5)
Cond_1333_0_plus_LE1(x1, x2, x3, x4, x5) → Cond_1333_0_plus_LE1(x1, x3, x4, x5)
1537_0_plus_Return(x1, x2, x3, x4) → 1537_0_plus_Return(x2, x3, x4)
Cond_1364_1_plus_InvokeMethod3(x1, x2, x3, x4, x5, x6, x7) → Cond_1364_1_plus_InvokeMethod3(x1, x2, x3, x4, x6, x7)
1379_0_plus_Return(x1, x2, x3, x4) → 1379_0_plus_Return(x2, x3)
1364_1_plus_InvokeMethod(x1, x2, x3, x4, x5, x6) → 1364_1_plus_InvokeMethod(x1, x2, x3, x5, x6)
Cond_1364_1_plus_InvokeMethod2(x1, x2, x3, x4, x5, x6, x7) → Cond_1364_1_plus_InvokeMethod2(x1, x3, x4, x6)
Cond_1364_1_plus_InvokeMethod1(x1, x2, x3, x4, x5, x6, x7) → Cond_1364_1_plus_InvokeMethod1(x1, x2, x3, x4, x6)
Cond_1364_1_plus_InvokeMethod(x1, x2, x3, x4, x5, x6, x7) → Cond_1364_1_plus_InvokeMethod(x1, x2, x3, x4, x6, x7)
Cond_1333_0_plus_LE(x1, x2, x3, x4, x5) → Cond_1333_0_plus_LE(x1, x3, x4, x5)
1522_0_times_Return(x1, x2) → 1522_0_times_Return(x2)
1566_0_times_Return(x1, x2) → 1566_0_times_Return(x2)
1361_0_times_Return(x1, x2) → 1361_0_times_Return
596_0_times_Return(x1, x2, x3, x4) → 596_0_times_Return(x2)

Filtered duplicate args:

754_1_times_InvokeMethod(x1, x2, x3, x4) → 754_1_times_InvokeMethod(x1, x3, x4)
457_0_times_NE(x1, x2, x3) → 457_0_times_NE(x1, x3)
Cond_457_0_times_NE(x1, x2, x3, x4) → Cond_457_0_times_NE(x1, x2, x4)
1333_0_plus_LE(x1, x2, x3) → 1333_0_plus_LE(x1, x3)
Cond_1333_0_plus_LE7(x1, x2, x3, x4) → Cond_1333_0_plus_LE7(x1, x2, x4)
Cond_1333_0_plus_LE6(x1, x2, x3, x4) → Cond_1333_0_plus_LE6(x1, x2, x4)
Cond_1333_0_plus_LE5(x1, x2, x3, x4) → Cond_1333_0_plus_LE5(x1, x2, x4)
Cond_1333_0_plus_LE4(x1, x2, x3, x4) → Cond_1333_0_plus_LE4(x1, x2, x4)
Cond_1333_0_plus_LE3(x1, x2, x3, x4) → Cond_1333_0_plus_LE3(x1, x2, x4)
Cond_1333_0_plus_LE2(x1, x2, x3, x4) → Cond_1333_0_plus_LE2(x1, x2, x4)
Cond_1333_0_plus_LE1(x1, x2, x3, x4) → Cond_1333_0_plus_LE1(x1, x2, x4)
Cond_1364_1_plus_InvokeMethod3(x1, x2, x3, x4, x5, x6) → Cond_1364_1_plus_InvokeMethod3(x1, x2, x4, x5, x6)
1364_1_plus_InvokeMethod(x1, x2, x3, x4, x5) → 1364_1_plus_InvokeMethod(x1, x3, x4, x5)
Cond_1364_1_plus_InvokeMethod2(x1, x2, x3, x4) → Cond_1364_1_plus_InvokeMethod2(x1, x3, x4)
Cond_1364_1_plus_InvokeMethod1(x1, x2, x3, x4, x5) → Cond_1364_1_plus_InvokeMethod1(x1, x2, x4, x5)
Cond_1364_1_plus_InvokeMethod(x1, x2, x3, x4, x5, x6) → Cond_1364_1_plus_InvokeMethod(x1, x2, x4, x5, x6)
Cond_1333_0_plus_LE(x1, x2, x3, x4) → Cond_1333_0_plus_LE(x1, x2, x4)

Filtered unneeded arguments:

Cond_1333_0_plus_LE1(x1, x2, x3) → Cond_1333_0_plus_LE1(x1)
1380_1_plus_InvokeMethod(x1, x2, x3) → 1380_1_plus_InvokeMethod(x1, x2)
Cond_1380_1_plus_InvokeMethod(x1, x2, x3, x4) → Cond_1380_1_plus_InvokeMethod(x1, x2)
Cond_1380_1_plus_InvokeMethod1(x1, x2, x3) → Cond_1380_1_plus_InvokeMethod1(x1)

Combined rules. Obtained 1 rules for P and 26 rules for R.

Finished conversion. Obtained 1 rules for P and 26 rules for R. System has predefined symbols.

### (5) 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:
1380_1_plus_InvokeMethod(1351_0_plus_Return, 0) → 1397_0_plus_Return
1380_1_plus_InvokeMethod(1397_0_plus_Return, x2) → 1580_0_plus_Return
1380_1_plus_InvokeMethod(1580_0_plus_Return, x1) → 1580_0_plus_Return
1364_1_plus_InvokeMethod(1351_0_plus_Return, x1, 0) → 1379_0_plus_Return(x1, x2)
1364_1_plus_InvokeMethod(1379_0_plus_Return(x0, x1), x0, x1) → 1537_0_plus_Return(x0, x3)
1364_1_plus_InvokeMethod(1397_0_plus_Return, x1, 0) → 1537_0_plus_Return(x1, x2)
1364_1_plus_InvokeMethod(1537_0_plus_Return(x0, x1), x0, x1) → 1537_0_plus_Return(x0, x2)
1364_1_plus_InvokeMethod(1580_0_plus_Return, x0, 0) → 1537_0_plus_Return(x0, x1)

The integer pair graph contains the following rules and edges:
(0): 1333_0_PLUS_LE(x0[0], x1[0]) → COND_1333_0_PLUS_LE(x1[0] <= 0 && x0[0] > 0, x0[0], x1[0])
(1): COND_1333_0_PLUS_LE(TRUE, x0[1], x1[1]) → 1333_0_PLUS_LE(x0[1] - 1, x1[1])
(2): 1333_0_PLUS_LE(x0[2], x1[2]) → COND_1333_0_PLUS_LE1(x1[2] > 0, x0[2], x1[2])
(3): COND_1333_0_PLUS_LE1(TRUE, x0[3], x1[3]) → 1333_0_PLUS_LE(x0[3], x1[3] - 1)

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

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

(1) -> (2), if ((x0[1] - 1* x0[2])∧(x1[1]* x1[2]))

(2) -> (3), if ((x1[2] > 0* TRUE)∧(x0[2]* x0[3])∧(x1[2]* x1[3]))

(3) -> (0), if ((x0[3]* x0[0])∧(x1[3] - 1* x1[0]))

(3) -> (2), if ((x0[3]* x0[2])∧(x1[3] - 1* x1[2]))

The set Q consists of the following terms:
1380_1_plus_InvokeMethod(1351_0_plus_Return, 0)
1380_1_plus_InvokeMethod(1397_0_plus_Return, x0)
1380_1_plus_InvokeMethod(1580_0_plus_Return, x0)
1364_1_plus_InvokeMethod(1351_0_plus_Return, x0, 0)
1364_1_plus_InvokeMethod(1379_0_plus_Return(x0, x1), x0, x1)
1364_1_plus_InvokeMethod(1397_0_plus_Return, x0, 0)
1364_1_plus_InvokeMethod(1537_0_plus_Return(x0, x1), x0, x1)
1364_1_plus_InvokeMethod(1580_0_plus_Return, x0, 0)

### (6) 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 1333_0_PLUS_LE(x0, x1) → COND_1333_0_PLUS_LE(&&(<=(x1, 0), >(x0, 0)), x0, x1) the following chains were created:
• We consider the chain 1333_0_PLUS_LE(x0[0], x1[0]) → COND_1333_0_PLUS_LE(&&(<=(x1[0], 0), >(x0[0], 0)), x0[0], x1[0]), COND_1333_0_PLUS_LE(TRUE, x0[1], x1[1]) → 1333_0_PLUS_LE(-(x0[1], 1), x1[1]) which results in the following constraint:

(1)    (&&(<=(x1[0], 0), >(x0[0], 0))=TRUEx0[0]=x0[1]x1[0]=x1[1]1333_0_PLUS_LE(x0[0], x1[0])≥NonInfC∧1333_0_PLUS_LE(x0[0], x1[0])≥COND_1333_0_PLUS_LE(&&(<=(x1[0], 0), >(x0[0], 0)), x0[0], x1[0])∧(UIncreasing(COND_1333_0_PLUS_LE(&&(<=(x1[0], 0), >(x0[0], 0)), x0[0], x1[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)=TRUE1333_0_PLUS_LE(x0[0], x1[0])≥NonInfC∧1333_0_PLUS_LE(x0[0], x1[0])≥COND_1333_0_PLUS_LE(&&(<=(x1[0], 0), >(x0[0], 0)), x0[0], x1[0])∧(UIncreasing(COND_1333_0_PLUS_LE(&&(<=(x1[0], 0), >(x0[0], 0)), x0[0], x1[0])), ≥))

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

(3)    ([-1]x1[0] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1333_0_PLUS_LE(&&(<=(x1[0], 0), >(x0[0], 0)), x0[0], x1[0])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x1[0] ≥ 0∧[(-1)bso_19] ≥ 0)

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

(4)    ([-1]x1[0] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1333_0_PLUS_LE(&&(<=(x1[0], 0), >(x0[0], 0)), x0[0], x1[0])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x1[0] ≥ 0∧[(-1)bso_19] ≥ 0)

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

(5)    ([-1]x1[0] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1333_0_PLUS_LE(&&(<=(x1[0], 0), >(x0[0], 0)), x0[0], x1[0])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x1[0] ≥ 0∧[(-1)bso_19] ≥ 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_1333_0_PLUS_LE(&&(<=(x1[0], 0), >(x0[0], 0)), x0[0], x1[0])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [(-1)bni_18]x1[0] ≥ 0∧[(-1)bso_19] ≥ 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_1333_0_PLUS_LE(&&(<=(x1[0], 0), >(x0[0], 0)), x0[0], x1[0])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [(-1)bni_18]x1[0] ≥ 0∧[(-1)bso_19] ≥ 0)

For Pair COND_1333_0_PLUS_LE(TRUE, x0, x1) → 1333_0_PLUS_LE(-(x0, 1), x1) the following chains were created:
• We consider the chain COND_1333_0_PLUS_LE(TRUE, x0[1], x1[1]) → 1333_0_PLUS_LE(-(x0[1], 1), x1[1]) which results in the following constraint:

(8)    (COND_1333_0_PLUS_LE(TRUE, x0[1], x1[1])≥NonInfC∧COND_1333_0_PLUS_LE(TRUE, x0[1], x1[1])≥1333_0_PLUS_LE(-(x0[1], 1), x1[1])∧(UIncreasing(1333_0_PLUS_LE(-(x0[1], 1), x1[1])), ≥))

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

(9)    ((UIncreasing(1333_0_PLUS_LE(-(x0[1], 1), x1[1])), ≥)∧[(-1)bso_21] ≥ 0)

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

(10)    ((UIncreasing(1333_0_PLUS_LE(-(x0[1], 1), x1[1])), ≥)∧[(-1)bso_21] ≥ 0)

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

(11)    ((UIncreasing(1333_0_PLUS_LE(-(x0[1], 1), x1[1])), ≥)∧[(-1)bso_21] ≥ 0)

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

(12)    ((UIncreasing(1333_0_PLUS_LE(-(x0[1], 1), x1[1])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_21] ≥ 0)

For Pair 1333_0_PLUS_LE(x0, x1) → COND_1333_0_PLUS_LE1(>(x1, 0), x0, x1) the following chains were created:
• We consider the chain 1333_0_PLUS_LE(x0[2], x1[2]) → COND_1333_0_PLUS_LE1(>(x1[2], 0), x0[2], x1[2]), COND_1333_0_PLUS_LE1(TRUE, x0[3], x1[3]) → 1333_0_PLUS_LE(x0[3], -(x1[3], 1)) which results in the following constraint:

(13)    (>(x1[2], 0)=TRUEx0[2]=x0[3]x1[2]=x1[3]1333_0_PLUS_LE(x0[2], x1[2])≥NonInfC∧1333_0_PLUS_LE(x0[2], x1[2])≥COND_1333_0_PLUS_LE1(>(x1[2], 0), x0[2], x1[2])∧(UIncreasing(COND_1333_0_PLUS_LE1(>(x1[2], 0), x0[2], x1[2])), ≥))

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

(14)    (>(x1[2], 0)=TRUE1333_0_PLUS_LE(x0[2], x1[2])≥NonInfC∧1333_0_PLUS_LE(x0[2], x1[2])≥COND_1333_0_PLUS_LE1(>(x1[2], 0), x0[2], x1[2])∧(UIncreasing(COND_1333_0_PLUS_LE1(>(x1[2], 0), x0[2], x1[2])), ≥))

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

(15)    (x1[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1333_0_PLUS_LE1(>(x1[2], 0), x0[2], x1[2])), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]x1[2] ≥ 0∧[(-1)bso_23] ≥ 0)

We simplified constraint (15) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(16)    (x1[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1333_0_PLUS_LE1(>(x1[2], 0), x0[2], x1[2])), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]x1[2] ≥ 0∧[(-1)bso_23] ≥ 0)

We simplified constraint (16) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(17)    (x1[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1333_0_PLUS_LE1(>(x1[2], 0), x0[2], x1[2])), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]x1[2] ≥ 0∧[(-1)bso_23] ≥ 0)

We simplified constraint (17) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(18)    (x1[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1333_0_PLUS_LE1(>(x1[2], 0), x0[2], x1[2])), ≥)∧0 = 0∧[(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]x1[2] ≥ 0∧0 = 0∧[(-1)bso_23] ≥ 0)

We simplified constraint (18) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(19)    (x1[2] ≥ 0 ⇒ (UIncreasing(COND_1333_0_PLUS_LE1(>(x1[2], 0), x0[2], x1[2])), ≥)∧0 = 0∧[(-1)Bound*bni_22] + [bni_22]x1[2] ≥ 0∧0 = 0∧[(-1)bso_23] ≥ 0)

For Pair COND_1333_0_PLUS_LE1(TRUE, x0, x1) → 1333_0_PLUS_LE(x0, -(x1, 1)) the following chains were created:
• We consider the chain COND_1333_0_PLUS_LE1(TRUE, x0[3], x1[3]) → 1333_0_PLUS_LE(x0[3], -(x1[3], 1)) which results in the following constraint:

(20)    (COND_1333_0_PLUS_LE1(TRUE, x0[3], x1[3])≥NonInfC∧COND_1333_0_PLUS_LE1(TRUE, x0[3], x1[3])≥1333_0_PLUS_LE(x0[3], -(x1[3], 1))∧(UIncreasing(1333_0_PLUS_LE(x0[3], -(x1[3], 1))), ≥))

We simplified constraint (20) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(21)    ((UIncreasing(1333_0_PLUS_LE(x0[3], -(x1[3], 1))), ≥)∧[1 + (-1)bso_25] ≥ 0)

We simplified constraint (21) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(22)    ((UIncreasing(1333_0_PLUS_LE(x0[3], -(x1[3], 1))), ≥)∧[1 + (-1)bso_25] ≥ 0)

We simplified constraint (22) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(23)    ((UIncreasing(1333_0_PLUS_LE(x0[3], -(x1[3], 1))), ≥)∧[1 + (-1)bso_25] ≥ 0)

We simplified constraint (23) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(24)    ((UIncreasing(1333_0_PLUS_LE(x0[3], -(x1[3], 1))), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_25] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 1333_0_PLUS_LE(x0, x1) → COND_1333_0_PLUS_LE(&&(<=(x1, 0), >(x0, 0)), x0, x1)
• (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_1333_0_PLUS_LE(&&(<=(x1[0], 0), >(x0[0], 0)), x0[0], x1[0])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [(-1)bni_18]x1[0] ≥ 0∧[(-1)bso_19] ≥ 0)

• COND_1333_0_PLUS_LE(TRUE, x0, x1) → 1333_0_PLUS_LE(-(x0, 1), x1)
• ((UIncreasing(1333_0_PLUS_LE(-(x0[1], 1), x1[1])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_21] ≥ 0)

• 1333_0_PLUS_LE(x0, x1) → COND_1333_0_PLUS_LE1(>(x1, 0), x0, x1)
• (x1[2] ≥ 0 ⇒ (UIncreasing(COND_1333_0_PLUS_LE1(>(x1[2], 0), x0[2], x1[2])), ≥)∧0 = 0∧[(-1)Bound*bni_22] + [bni_22]x1[2] ≥ 0∧0 = 0∧[(-1)bso_23] ≥ 0)

• COND_1333_0_PLUS_LE1(TRUE, x0, x1) → 1333_0_PLUS_LE(x0, -(x1, 1))
• ((UIncreasing(1333_0_PLUS_LE(x0[3], -(x1[3], 1))), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_25] ≥ 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(1380_1_plus_InvokeMethod(x1, x2)) = [-1]
POL(1351_0_plus_Return) = [-1]
POL(0) = 0
POL(1397_0_plus_Return) = [-1]
POL(1580_0_plus_Return) = [-1]
POL(1364_1_plus_InvokeMethod(x1, x2, x3)) = [-1]
POL(1379_0_plus_Return(x1, x2)) = [-1]
POL(1537_0_plus_Return(x1, x2)) = [-1]
POL(1333_0_PLUS_LE(x1, x2)) = [-1] + x2
POL(COND_1333_0_PLUS_LE(x1, x2, x3)) = [-1] + x3
POL(&&(x1, x2)) = [-1]
POL(<=(x1, x2)) = [-1]
POL(>(x1, x2)) = [-1]
POL(-(x1, x2)) = x1 + [-1]x2
POL(1) = [1]
POL(COND_1333_0_PLUS_LE1(x1, x2, x3)) = [-1] + x3

The following pairs are in P>:

COND_1333_0_PLUS_LE1(TRUE, x0[3], x1[3]) → 1333_0_PLUS_LE(x0[3], -(x1[3], 1))

The following pairs are in Pbound:

1333_0_PLUS_LE(x0[2], x1[2]) → COND_1333_0_PLUS_LE1(>(x1[2], 0), x0[2], x1[2])

The following pairs are in P:

1333_0_PLUS_LE(x0[0], x1[0]) → COND_1333_0_PLUS_LE(&&(<=(x1[0], 0), >(x0[0], 0)), x0[0], x1[0])
COND_1333_0_PLUS_LE(TRUE, x0[1], x1[1]) → 1333_0_PLUS_LE(-(x0[1], 1), x1[1])
1333_0_PLUS_LE(x0[2], x1[2]) → COND_1333_0_PLUS_LE1(>(x1[2], 0), x0[2], x1[2])

There are no usable rules.

### (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

The ITRS R consists of the following rules:
1380_1_plus_InvokeMethod(1351_0_plus_Return, 0) → 1397_0_plus_Return
1380_1_plus_InvokeMethod(1397_0_plus_Return, x2) → 1580_0_plus_Return
1380_1_plus_InvokeMethod(1580_0_plus_Return, x1) → 1580_0_plus_Return
1364_1_plus_InvokeMethod(1351_0_plus_Return, x1, 0) → 1379_0_plus_Return(x1, x2)
1364_1_plus_InvokeMethod(1379_0_plus_Return(x0, x1), x0, x1) → 1537_0_plus_Return(x0, x3)
1364_1_plus_InvokeMethod(1397_0_plus_Return, x1, 0) → 1537_0_plus_Return(x1, x2)
1364_1_plus_InvokeMethod(1537_0_plus_Return(x0, x1), x0, x1) → 1537_0_plus_Return(x0, x2)
1364_1_plus_InvokeMethod(1580_0_plus_Return, x0, 0) → 1537_0_plus_Return(x0, x1)

The integer pair graph contains the following rules and edges:
(0): 1333_0_PLUS_LE(x0[0], x1[0]) → COND_1333_0_PLUS_LE(x1[0] <= 0 && x0[0] > 0, x0[0], x1[0])
(1): COND_1333_0_PLUS_LE(TRUE, x0[1], x1[1]) → 1333_0_PLUS_LE(x0[1] - 1, x1[1])
(2): 1333_0_PLUS_LE(x0[2], x1[2]) → COND_1333_0_PLUS_LE1(x1[2] > 0, x0[2], x1[2])

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

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

(1) -> (2), if ((x0[1] - 1* x0[2])∧(x1[1]* x1[2]))

The set Q consists of the following terms:
1380_1_plus_InvokeMethod(1351_0_plus_Return, 0)
1380_1_plus_InvokeMethod(1397_0_plus_Return, x0)
1380_1_plus_InvokeMethod(1580_0_plus_Return, x0)
1364_1_plus_InvokeMethod(1351_0_plus_Return, x0, 0)
1364_1_plus_InvokeMethod(1379_0_plus_Return(x0, x1), x0, x1)
1364_1_plus_InvokeMethod(1397_0_plus_Return, x0, 0)
1364_1_plus_InvokeMethod(1537_0_plus_Return(x0, x1), x0, x1)
1364_1_plus_InvokeMethod(1580_0_plus_Return, x0, 0)

### (9) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC 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, Boolean

The ITRS R consists of the following rules:
1380_1_plus_InvokeMethod(1351_0_plus_Return, 0) → 1397_0_plus_Return
1380_1_plus_InvokeMethod(1397_0_plus_Return, x2) → 1580_0_plus_Return
1380_1_plus_InvokeMethod(1580_0_plus_Return, x1) → 1580_0_plus_Return
1364_1_plus_InvokeMethod(1351_0_plus_Return, x1, 0) → 1379_0_plus_Return(x1, x2)
1364_1_plus_InvokeMethod(1379_0_plus_Return(x0, x1), x0, x1) → 1537_0_plus_Return(x0, x3)
1364_1_plus_InvokeMethod(1397_0_plus_Return, x1, 0) → 1537_0_plus_Return(x1, x2)
1364_1_plus_InvokeMethod(1537_0_plus_Return(x0, x1), x0, x1) → 1537_0_plus_Return(x0, x2)
1364_1_plus_InvokeMethod(1580_0_plus_Return, x0, 0) → 1537_0_plus_Return(x0, x1)

The integer pair graph contains the following rules and edges:
(1): COND_1333_0_PLUS_LE(TRUE, x0[1], x1[1]) → 1333_0_PLUS_LE(x0[1] - 1, x1[1])
(0): 1333_0_PLUS_LE(x0[0], x1[0]) → COND_1333_0_PLUS_LE(x1[0] <= 0 && x0[0] > 0, x0[0], x1[0])

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

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

The set Q consists of the following terms:
1380_1_plus_InvokeMethod(1351_0_plus_Return, 0)
1380_1_plus_InvokeMethod(1397_0_plus_Return, x0)
1380_1_plus_InvokeMethod(1580_0_plus_Return, x0)
1364_1_plus_InvokeMethod(1351_0_plus_Return, x0, 0)
1364_1_plus_InvokeMethod(1379_0_plus_Return(x0, x1), x0, x1)
1364_1_plus_InvokeMethod(1397_0_plus_Return, x0, 0)
1364_1_plus_InvokeMethod(1537_0_plus_Return(x0, x1), x0, x1)
1364_1_plus_InvokeMethod(1580_0_plus_Return, x0, 0)

### (11) 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.

### (12) 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, Boolean

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_1333_0_PLUS_LE(TRUE, x0[1], x1[1]) → 1333_0_PLUS_LE(x0[1] - 1, x1[1])
(0): 1333_0_PLUS_LE(x0[0], x1[0]) → COND_1333_0_PLUS_LE(x1[0] <= 0 && x0[0] > 0, x0[0], x1[0])

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

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

The set Q consists of the following terms:
1380_1_plus_InvokeMethod(1351_0_plus_Return, 0)
1380_1_plus_InvokeMethod(1397_0_plus_Return, x0)
1380_1_plus_InvokeMethod(1580_0_plus_Return, x0)
1364_1_plus_InvokeMethod(1351_0_plus_Return, x0, 0)
1364_1_plus_InvokeMethod(1379_0_plus_Return(x0, x1), x0, x1)
1364_1_plus_InvokeMethod(1397_0_plus_Return, x0, 0)
1364_1_plus_InvokeMethod(1537_0_plus_Return(x0, x1), x0, x1)
1364_1_plus_InvokeMethod(1580_0_plus_Return, x0, 0)

### (13) 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 COND_1333_0_PLUS_LE(TRUE, x0[1], x1[1]) → 1333_0_PLUS_LE(-(x0[1], 1), x1[1]) the following chains were created:
• We consider the chain COND_1333_0_PLUS_LE(TRUE, x0[1], x1[1]) → 1333_0_PLUS_LE(-(x0[1], 1), x1[1]) which results in the following constraint:

(1)    (COND_1333_0_PLUS_LE(TRUE, x0[1], x1[1])≥NonInfC∧COND_1333_0_PLUS_LE(TRUE, x0[1], x1[1])≥1333_0_PLUS_LE(-(x0[1], 1), x1[1])∧(UIncreasing(1333_0_PLUS_LE(-(x0[1], 1), x1[1])), ≥))

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

(2)    ((UIncreasing(1333_0_PLUS_LE(-(x0[1], 1), x1[1])), ≥)∧[(-1)bso_11] ≥ 0)

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

(3)    ((UIncreasing(1333_0_PLUS_LE(-(x0[1], 1), x1[1])), ≥)∧[(-1)bso_11] ≥ 0)

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

(4)    ((UIncreasing(1333_0_PLUS_LE(-(x0[1], 1), x1[1])), ≥)∧[(-1)bso_11] ≥ 0)

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

(5)    ((UIncreasing(1333_0_PLUS_LE(-(x0[1], 1), x1[1])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_11] ≥ 0)

For Pair 1333_0_PLUS_LE(x0[0], x1[0]) → COND_1333_0_PLUS_LE(&&(<=(x1[0], 0), >(x0[0], 0)), x0[0], x1[0]) the following chains were created:
• We consider the chain 1333_0_PLUS_LE(x0[0], x1[0]) → COND_1333_0_PLUS_LE(&&(<=(x1[0], 0), >(x0[0], 0)), x0[0], x1[0]), COND_1333_0_PLUS_LE(TRUE, x0[1], x1[1]) → 1333_0_PLUS_LE(-(x0[1], 1), x1[1]) which results in the following constraint:

(6)    (&&(<=(x1[0], 0), >(x0[0], 0))=TRUEx0[0]=x0[1]x1[0]=x1[1]1333_0_PLUS_LE(x0[0], x1[0])≥NonInfC∧1333_0_PLUS_LE(x0[0], x1[0])≥COND_1333_0_PLUS_LE(&&(<=(x1[0], 0), >(x0[0], 0)), x0[0], x1[0])∧(UIncreasing(COND_1333_0_PLUS_LE(&&(<=(x1[0], 0), >(x0[0], 0)), x0[0], x1[0])), ≥))

We simplified constraint (6) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

(7)    (<=(x1[0], 0)=TRUE>(x0[0], 0)=TRUE1333_0_PLUS_LE(x0[0], x1[0])≥NonInfC∧1333_0_PLUS_LE(x0[0], x1[0])≥COND_1333_0_PLUS_LE(&&(<=(x1[0], 0), >(x0[0], 0)), x0[0], x1[0])∧(UIncreasing(COND_1333_0_PLUS_LE(&&(<=(x1[0], 0), >(x0[0], 0)), x0[0], x1[0])), ≥))

We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(8)    ([-1]x1[0] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1333_0_PLUS_LE(&&(<=(x1[0], 0), >(x0[0], 0)), x0[0], x1[0])), ≥)∧[bni_12 + (-1)Bound*bni_12] + [(2)bni_12]x0[0] + [(-1)bni_12]x1[0] ≥ 0∧[2 + (-1)bso_13] ≥ 0)

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

(9)    ([-1]x1[0] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1333_0_PLUS_LE(&&(<=(x1[0], 0), >(x0[0], 0)), x0[0], x1[0])), ≥)∧[bni_12 + (-1)Bound*bni_12] + [(2)bni_12]x0[0] + [(-1)bni_12]x1[0] ≥ 0∧[2 + (-1)bso_13] ≥ 0)

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

(10)    ([-1]x1[0] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1333_0_PLUS_LE(&&(<=(x1[0], 0), >(x0[0], 0)), x0[0], x1[0])), ≥)∧[bni_12 + (-1)Bound*bni_12] + [(2)bni_12]x0[0] + [(-1)bni_12]x1[0] ≥ 0∧[2 + (-1)bso_13] ≥ 0)

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

(11)    (x1[0] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1333_0_PLUS_LE(&&(<=(x1[0], 0), >(x0[0], 0)), x0[0], x1[0])), ≥)∧[bni_12 + (-1)Bound*bni_12] + [(2)bni_12]x0[0] + [bni_12]x1[0] ≥ 0∧[2 + (-1)bso_13] ≥ 0)

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

(12)    (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_1333_0_PLUS_LE(&&(<=(x1[0], 0), >(x0[0], 0)), x0[0], x1[0])), ≥)∧[(3)bni_12 + (-1)Bound*bni_12] + [(2)bni_12]x0[0] + [bni_12]x1[0] ≥ 0∧[2 + (-1)bso_13] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• COND_1333_0_PLUS_LE(TRUE, x0[1], x1[1]) → 1333_0_PLUS_LE(-(x0[1], 1), x1[1])
• ((UIncreasing(1333_0_PLUS_LE(-(x0[1], 1), x1[1])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_11] ≥ 0)

• 1333_0_PLUS_LE(x0[0], x1[0]) → COND_1333_0_PLUS_LE(&&(<=(x1[0], 0), >(x0[0], 0)), x0[0], x1[0])
• (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_1333_0_PLUS_LE(&&(<=(x1[0], 0), >(x0[0], 0)), x0[0], x1[0])), ≥)∧[(3)bni_12 + (-1)Bound*bni_12] + [(2)bni_12]x0[0] + [bni_12]x1[0] ≥ 0∧[2 + (-1)bso_13] ≥ 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(COND_1333_0_PLUS_LE(x1, x2, x3)) = [-1] + [-1]x3 + [2]x2
POL(1333_0_PLUS_LE(x1, x2)) = [1] + [2]x1 + [-1]x2
POL(-(x1, x2)) = x1 + [-1]x2
POL(1) = [1]
POL(&&(x1, x2)) = [-1]
POL(<=(x1, x2)) = [-1]
POL(0) = 0
POL(>(x1, x2)) = [-1]

The following pairs are in P>:

1333_0_PLUS_LE(x0[0], x1[0]) → COND_1333_0_PLUS_LE(&&(<=(x1[0], 0), >(x0[0], 0)), x0[0], x1[0])

The following pairs are in Pbound:

1333_0_PLUS_LE(x0[0], x1[0]) → COND_1333_0_PLUS_LE(&&(<=(x1[0], 0), >(x0[0], 0)), x0[0], x1[0])

The following pairs are in P:

COND_1333_0_PLUS_LE(TRUE, x0[1], x1[1]) → 1333_0_PLUS_LE(-(x0[1], 1), x1[1])

There are no usable rules.

### (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:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_1333_0_PLUS_LE(TRUE, x0[1], x1[1]) → 1333_0_PLUS_LE(x0[1] - 1, x1[1])

The set Q consists of the following terms:
1380_1_plus_InvokeMethod(1351_0_plus_Return, 0)
1380_1_plus_InvokeMethod(1397_0_plus_Return, x0)
1380_1_plus_InvokeMethod(1580_0_plus_Return, x0)
1364_1_plus_InvokeMethod(1351_0_plus_Return, x0, 0)
1364_1_plus_InvokeMethod(1379_0_plus_Return(x0, x1), x0, x1)
1364_1_plus_InvokeMethod(1397_0_plus_Return, x0, 0)
1364_1_plus_InvokeMethod(1537_0_plus_Return(x0, x1), x0, x1)
1364_1_plus_InvokeMethod(1580_0_plus_Return, x0, 0)

### (15) IDependencyGraphProof (EQUIVALENT transformation)

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

### (17) 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:
1380_1_plus_InvokeMethod(1351_0_plus_Return, 0) → 1397_0_plus_Return
1380_1_plus_InvokeMethod(1397_0_plus_Return, x2) → 1580_0_plus_Return
1380_1_plus_InvokeMethod(1580_0_plus_Return, x1) → 1580_0_plus_Return
1364_1_plus_InvokeMethod(1351_0_plus_Return, x1, 0) → 1379_0_plus_Return(x1, x2)
1364_1_plus_InvokeMethod(1379_0_plus_Return(x0, x1), x0, x1) → 1537_0_plus_Return(x0, x3)
1364_1_plus_InvokeMethod(1397_0_plus_Return, x1, 0) → 1537_0_plus_Return(x1, x2)
1364_1_plus_InvokeMethod(1537_0_plus_Return(x0, x1), x0, x1) → 1537_0_plus_Return(x0, x2)
1364_1_plus_InvokeMethod(1580_0_plus_Return, x0, 0) → 1537_0_plus_Return(x0, x1)

The integer pair graph contains the following rules and edges:
(0): 1333_0_PLUS_LE(x0[0], x1[0]) → COND_1333_0_PLUS_LE(x1[0] <= 0 && x0[0] > 0, x0[0], x1[0])
(1): COND_1333_0_PLUS_LE(TRUE, x0[1], x1[1]) → 1333_0_PLUS_LE(x0[1] - 1, x1[1])
(3): COND_1333_0_PLUS_LE1(TRUE, x0[3], x1[3]) → 1333_0_PLUS_LE(x0[3], x1[3] - 1)

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

(3) -> (0), if ((x0[3]* x0[0])∧(x1[3] - 1* x1[0]))

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

The set Q consists of the following terms:
1380_1_plus_InvokeMethod(1351_0_plus_Return, 0)
1380_1_plus_InvokeMethod(1397_0_plus_Return, x0)
1380_1_plus_InvokeMethod(1580_0_plus_Return, x0)
1364_1_plus_InvokeMethod(1351_0_plus_Return, x0, 0)
1364_1_plus_InvokeMethod(1379_0_plus_Return(x0, x1), x0, x1)
1364_1_plus_InvokeMethod(1397_0_plus_Return, x0, 0)
1364_1_plus_InvokeMethod(1537_0_plus_Return(x0, x1), x0, x1)
1364_1_plus_InvokeMethod(1580_0_plus_Return, x0, 0)

### (18) IDependencyGraphProof (EQUIVALENT transformation)

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

### (19) 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, Boolean

The ITRS R consists of the following rules:
1380_1_plus_InvokeMethod(1351_0_plus_Return, 0) → 1397_0_plus_Return
1380_1_plus_InvokeMethod(1397_0_plus_Return, x2) → 1580_0_plus_Return
1380_1_plus_InvokeMethod(1580_0_plus_Return, x1) → 1580_0_plus_Return
1364_1_plus_InvokeMethod(1351_0_plus_Return, x1, 0) → 1379_0_plus_Return(x1, x2)
1364_1_plus_InvokeMethod(1379_0_plus_Return(x0, x1), x0, x1) → 1537_0_plus_Return(x0, x3)
1364_1_plus_InvokeMethod(1397_0_plus_Return, x1, 0) → 1537_0_plus_Return(x1, x2)
1364_1_plus_InvokeMethod(1537_0_plus_Return(x0, x1), x0, x1) → 1537_0_plus_Return(x0, x2)
1364_1_plus_InvokeMethod(1580_0_plus_Return, x0, 0) → 1537_0_plus_Return(x0, x1)

The integer pair graph contains the following rules and edges:
(1): COND_1333_0_PLUS_LE(TRUE, x0[1], x1[1]) → 1333_0_PLUS_LE(x0[1] - 1, x1[1])
(0): 1333_0_PLUS_LE(x0[0], x1[0]) → COND_1333_0_PLUS_LE(x1[0] <= 0 && x0[0] > 0, x0[0], x1[0])

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

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

The set Q consists of the following terms:
1380_1_plus_InvokeMethod(1351_0_plus_Return, 0)
1380_1_plus_InvokeMethod(1397_0_plus_Return, x0)
1380_1_plus_InvokeMethod(1580_0_plus_Return, x0)
1364_1_plus_InvokeMethod(1351_0_plus_Return, x0, 0)
1364_1_plus_InvokeMethod(1379_0_plus_Return(x0, x1), x0, x1)
1364_1_plus_InvokeMethod(1397_0_plus_Return, x0, 0)
1364_1_plus_InvokeMethod(1537_0_plus_Return(x0, x1), x0, x1)
1364_1_plus_InvokeMethod(1580_0_plus_Return, x0, 0)

### (20) 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.

### (21) 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, Boolean

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_1333_0_PLUS_LE(TRUE, x0[1], x1[1]) → 1333_0_PLUS_LE(x0[1] - 1, x1[1])
(0): 1333_0_PLUS_LE(x0[0], x1[0]) → COND_1333_0_PLUS_LE(x1[0] <= 0 && x0[0] > 0, x0[0], x1[0])

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

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

The set Q consists of the following terms:
1380_1_plus_InvokeMethod(1351_0_plus_Return, 0)
1380_1_plus_InvokeMethod(1397_0_plus_Return, x0)
1380_1_plus_InvokeMethod(1580_0_plus_Return, x0)
1364_1_plus_InvokeMethod(1351_0_plus_Return, x0, 0)
1364_1_plus_InvokeMethod(1379_0_plus_Return(x0, x1), x0, x1)
1364_1_plus_InvokeMethod(1397_0_plus_Return, x0, 0)
1364_1_plus_InvokeMethod(1537_0_plus_Return(x0, x1), x0, x1)
1364_1_plus_InvokeMethod(1580_0_plus_Return, x0, 0)

### (22) 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 COND_1333_0_PLUS_LE(TRUE, x0[1], x1[1]) → 1333_0_PLUS_LE(-(x0[1], 1), x1[1]) the following chains were created:
• We consider the chain COND_1333_0_PLUS_LE(TRUE, x0[1], x1[1]) → 1333_0_PLUS_LE(-(x0[1], 1), x1[1]) which results in the following constraint:

(1)    (COND_1333_0_PLUS_LE(TRUE, x0[1], x1[1])≥NonInfC∧COND_1333_0_PLUS_LE(TRUE, x0[1], x1[1])≥1333_0_PLUS_LE(-(x0[1], 1), x1[1])∧(UIncreasing(1333_0_PLUS_LE(-(x0[1], 1), x1[1])), ≥))

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

(2)    ((UIncreasing(1333_0_PLUS_LE(-(x0[1], 1), x1[1])), ≥)∧[1 + (-1)bso_9] ≥ 0)

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

(3)    ((UIncreasing(1333_0_PLUS_LE(-(x0[1], 1), x1[1])), ≥)∧[1 + (-1)bso_9] ≥ 0)

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

(4)    ((UIncreasing(1333_0_PLUS_LE(-(x0[1], 1), x1[1])), ≥)∧[1 + (-1)bso_9] ≥ 0)

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

(5)    ((UIncreasing(1333_0_PLUS_LE(-(x0[1], 1), x1[1])), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_9] ≥ 0)

For Pair 1333_0_PLUS_LE(x0[0], x1[0]) → COND_1333_0_PLUS_LE(&&(<=(x1[0], 0), >(x0[0], 0)), x0[0], x1[0]) the following chains were created:
• We consider the chain 1333_0_PLUS_LE(x0[0], x1[0]) → COND_1333_0_PLUS_LE(&&(<=(x1[0], 0), >(x0[0], 0)), x0[0], x1[0]), COND_1333_0_PLUS_LE(TRUE, x0[1], x1[1]) → 1333_0_PLUS_LE(-(x0[1], 1), x1[1]) which results in the following constraint:

(6)    (&&(<=(x1[0], 0), >(x0[0], 0))=TRUEx0[0]=x0[1]x1[0]=x1[1]1333_0_PLUS_LE(x0[0], x1[0])≥NonInfC∧1333_0_PLUS_LE(x0[0], x1[0])≥COND_1333_0_PLUS_LE(&&(<=(x1[0], 0), >(x0[0], 0)), x0[0], x1[0])∧(UIncreasing(COND_1333_0_PLUS_LE(&&(<=(x1[0], 0), >(x0[0], 0)), x0[0], x1[0])), ≥))

We simplified constraint (6) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

(7)    (<=(x1[0], 0)=TRUE>(x0[0], 0)=TRUE1333_0_PLUS_LE(x0[0], x1[0])≥NonInfC∧1333_0_PLUS_LE(x0[0], x1[0])≥COND_1333_0_PLUS_LE(&&(<=(x1[0], 0), >(x0[0], 0)), x0[0], x1[0])∧(UIncreasing(COND_1333_0_PLUS_LE(&&(<=(x1[0], 0), >(x0[0], 0)), x0[0], x1[0])), ≥))

We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(8)    ([-1]x1[0] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1333_0_PLUS_LE(&&(<=(x1[0], 0), >(x0[0], 0)), x0[0], x1[0])), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] + [bni_10]x0[0] + [(-1)bni_10]x1[0] ≥ 0∧[(-1)bso_11] ≥ 0)

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

(9)    ([-1]x1[0] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1333_0_PLUS_LE(&&(<=(x1[0], 0), >(x0[0], 0)), x0[0], x1[0])), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] + [bni_10]x0[0] + [(-1)bni_10]x1[0] ≥ 0∧[(-1)bso_11] ≥ 0)

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

(10)    ([-1]x1[0] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1333_0_PLUS_LE(&&(<=(x1[0], 0), >(x0[0], 0)), x0[0], x1[0])), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] + [bni_10]x0[0] + [(-1)bni_10]x1[0] ≥ 0∧[(-1)bso_11] ≥ 0)

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

(11)    (x1[0] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1333_0_PLUS_LE(&&(<=(x1[0], 0), >(x0[0], 0)), x0[0], x1[0])), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] + [bni_10]x0[0] + [bni_10]x1[0] ≥ 0∧[(-1)bso_11] ≥ 0)

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

(12)    (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_1333_0_PLUS_LE(&&(<=(x1[0], 0), >(x0[0], 0)), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_10] + [bni_10]x0[0] + [bni_10]x1[0] ≥ 0∧[(-1)bso_11] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• COND_1333_0_PLUS_LE(TRUE, x0[1], x1[1]) → 1333_0_PLUS_LE(-(x0[1], 1), x1[1])
• ((UIncreasing(1333_0_PLUS_LE(-(x0[1], 1), x1[1])), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_9] ≥ 0)

• 1333_0_PLUS_LE(x0[0], x1[0]) → COND_1333_0_PLUS_LE(&&(<=(x1[0], 0), >(x0[0], 0)), x0[0], x1[0])
• (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_1333_0_PLUS_LE(&&(<=(x1[0], 0), >(x0[0], 0)), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_10] + [bni_10]x0[0] + [bni_10]x1[0] ≥ 0∧[(-1)bso_11] ≥ 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(COND_1333_0_PLUS_LE(x1, x2, x3)) = [-1] + [-1]x3 + x2
POL(1333_0_PLUS_LE(x1, x2)) = [-1] + x1 + [-1]x2
POL(-(x1, x2)) = x1 + [-1]x2
POL(1) = [1]
POL(&&(x1, x2)) = [1]
POL(<=(x1, x2)) = [-1]
POL(0) = 0
POL(>(x1, x2)) = [-1]

The following pairs are in P>:

COND_1333_0_PLUS_LE(TRUE, x0[1], x1[1]) → 1333_0_PLUS_LE(-(x0[1], 1), x1[1])

The following pairs are in Pbound:

1333_0_PLUS_LE(x0[0], x1[0]) → COND_1333_0_PLUS_LE(&&(<=(x1[0], 0), >(x0[0], 0)), x0[0], x1[0])

The following pairs are in P:

1333_0_PLUS_LE(x0[0], x1[0]) → COND_1333_0_PLUS_LE(&&(<=(x1[0], 0), >(x0[0], 0)), x0[0], x1[0])

There are no usable rules.

### (24) 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): 1333_0_PLUS_LE(x0[0], x1[0]) → COND_1333_0_PLUS_LE(x1[0] <= 0 && x0[0] > 0, x0[0], x1[0])

The set Q consists of the following terms:
1380_1_plus_InvokeMethod(1351_0_plus_Return, 0)
1380_1_plus_InvokeMethod(1397_0_plus_Return, x0)
1380_1_plus_InvokeMethod(1580_0_plus_Return, x0)
1364_1_plus_InvokeMethod(1351_0_plus_Return, x0, 0)
1364_1_plus_InvokeMethod(1379_0_plus_Return(x0, x1), x0, x1)
1364_1_plus_InvokeMethod(1397_0_plus_Return, x0, 0)
1364_1_plus_InvokeMethod(1537_0_plus_Return(x0, x1), x0, x1)
1364_1_plus_InvokeMethod(1580_0_plus_Return, x0, 0)

### (25) IDependencyGraphProof (EQUIVALENT transformation)

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

### (27) 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_1333_0_PLUS_LE(TRUE, x0[1], x1[1]) → 1333_0_PLUS_LE(x0[1] - 1, x1[1])

The set Q consists of the following terms:
1380_1_plus_InvokeMethod(1351_0_plus_Return, 0)
1380_1_plus_InvokeMethod(1397_0_plus_Return, x0)
1380_1_plus_InvokeMethod(1580_0_plus_Return, x0)
1364_1_plus_InvokeMethod(1351_0_plus_Return, x0, 0)
1364_1_plus_InvokeMethod(1379_0_plus_Return(x0, x1), x0, x1)
1364_1_plus_InvokeMethod(1397_0_plus_Return, x0, 0)
1364_1_plus_InvokeMethod(1537_0_plus_Return(x0, x1), x0, x1)
1364_1_plus_InvokeMethod(1580_0_plus_Return, x0, 0)

### (28) IDependencyGraphProof (EQUIVALENT transformation)

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

### (30) 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, Boolean

The ITRS R consists of the following rules:
457_0_times_NE(x0, 0) → 596_0_times_Return(x0)
754_1_times_InvokeMethod(596_0_times_Return(x0), x0, 0) → 1328_1_times_InvokeMethod(1333_0_plus_LE(0, x0), 0, x0)
754_1_times_InvokeMethod(1361_0_times_Return, 0, x3) → 1328_1_times_InvokeMethod(1333_0_plus_LE(0, 0), 0, 0)
754_1_times_InvokeMethod(1522_0_times_Return(x0), x1, x2) → 1328_1_times_InvokeMethod(1333_0_plus_LE(x0, x1), x0, x1)
754_1_times_InvokeMethod(1566_0_times_Return(x0), 0, x3) → 1328_1_times_InvokeMethod(1333_0_plus_LE(x0, 0), x0, 0)
1328_1_times_InvokeMethod(1351_0_plus_Return, x1, x2) → 1361_0_times_Return
1328_1_times_InvokeMethod(1537_0_plus_Return(x0, x1, x2), x0, x1) → 1522_0_times_Return(x2)
1328_1_times_InvokeMethod(1580_0_plus_Return(x0), x1, x2) → 1566_0_times_Return(x0)
1328_1_times_InvokeMethod(1397_0_plus_Return, x1, x2) → 1566_0_times_Return(1)
1328_1_times_InvokeMethod(1379_0_plus_Return(x0, x1), x0, x1) → 1522_0_times_Return(1)
1333_0_plus_LE(x0, x1) → Cond_1333_0_plus_LE(x1 > 0, x0, x1)
Cond_1333_0_plus_LE(TRUE, x0, x1) → 1364_1_plus_InvokeMethod(1333_0_plus_LE(x0, x1 - 1), x1, x0, x1 - 1)
1364_1_plus_InvokeMethod(1351_0_plus_Return, x2, x1, 0) → 1379_0_plus_Return(x1, x2)
1364_1_plus_InvokeMethod(1537_0_plus_Return(x0, x1, x2), x3, x0, x1) → Cond_1364_1_plus_InvokeMethod(x2 > 0, 1537_0_plus_Return(x0, x1, x2), x3, x0, x1)
Cond_1364_1_plus_InvokeMethod(TRUE, 1537_0_plus_Return(x0, x1, x2), x3, x0, x1) → 1537_0_plus_Return(x0, x3, 1 + x2)
1364_1_plus_InvokeMethod(1580_0_plus_Return(x0), x2, x1, 0) → Cond_1364_1_plus_InvokeMethod1(x0 > 0, 1580_0_plus_Return(x0), x2, x1, 0)
Cond_1364_1_plus_InvokeMethod1(TRUE, 1580_0_plus_Return(x0), x2, x1, 0) → 1537_0_plus_Return(x1, x2, 1 + x0)
1364_1_plus_InvokeMethod(1397_0_plus_Return, x2, x1, 0) → Cond_1364_1_plus_InvokeMethod2(1 > 0, 1397_0_plus_Return, x2, x1, 0)
Cond_1364_1_plus_InvokeMethod2(TRUE, 1397_0_plus_Return, x2, x1, 0) → 1537_0_plus_Return(x1, x2, 1 + 1)
1364_1_plus_InvokeMethod(1379_0_plus_Return(x0, x1), x3, x0, x1) → Cond_1364_1_plus_InvokeMethod3(1 > 0, 1379_0_plus_Return(x0, x1), x3, x0, x1)
Cond_1364_1_plus_InvokeMethod3(TRUE, 1379_0_plus_Return(x0, x1), x3, x0, x1) → 1537_0_plus_Return(x0, x3, 1 + 1)
1333_0_plus_LE(x0, x1) → Cond_1333_0_plus_LE1(x1 <= 0 && x0 <= 0, x0, x1)
Cond_1333_0_plus_LE1(TRUE, x0, x1) → 1351_0_plus_Return
1333_0_plus_LE(x0, x1) → Cond_1333_0_plus_LE2(x1 <= 0 && x0 > 0, x0, x1)
Cond_1333_0_plus_LE2(TRUE, x0, x1) → 1380_1_plus_InvokeMethod(1387_0_plus_Return(x0 - 1, x1), x0 - 1)
1333_0_plus_LE(x0, x1) → Cond_1333_0_plus_LE3(x1 <= 0 && x0 > 0, x0, x1)
Cond_1333_0_plus_LE3(TRUE, x0, x1) → 1380_1_plus_InvokeMethod(1417_0_plus_Return(x0 - 1, x1), x0 - 1)
1333_0_plus_LE(x0, x1) → Cond_1333_0_plus_LE4(x1 <= 0 && x0 > 0, x0, x1)
Cond_1333_0_plus_LE4(TRUE, x0, x1) → 1380_1_plus_InvokeMethod(1459_0_plus_Return(x0 - 1, x1), x0 - 1)
1333_0_plus_LE(x0, x1) → Cond_1333_0_plus_LE5(x1 <= 0 && x0 > 0, x0, x1)
Cond_1333_0_plus_LE5(TRUE, x0, x1) → 1380_1_plus_InvokeMethod(1507_0_plus_Return(x0 - 1, x1), x0 - 1)
1333_0_plus_LE(x0, x1) → Cond_1333_0_plus_LE6(x1 <= 0 && x0 > 0, x0, x1)
Cond_1333_0_plus_LE6(TRUE, x0, x1) → 1380_1_plus_InvokeMethod(1550_0_plus_Return(x0 - 1, x1), x0 - 1)
1333_0_plus_LE(x0, x1) → Cond_1333_0_plus_LE7(x1 <= 0 && x0 > 0, x0, x1)
Cond_1333_0_plus_LE7(TRUE, x0, x1) → 1380_1_plus_InvokeMethod(1333_0_plus_LE(x0 - 1, x1), x0 - 1)
1380_1_plus_InvokeMethod(1351_0_plus_Return, 0) → 1397_0_plus_Return
1380_1_plus_InvokeMethod(1580_0_plus_Return(x0), x2) → Cond_1380_1_plus_InvokeMethod(x0 > 0, 1580_0_plus_Return(x0), x2)
Cond_1380_1_plus_InvokeMethod(TRUE, 1580_0_plus_Return(x0), x2) → 1580_0_plus_Return(1 + x0)
1380_1_plus_InvokeMethod(1397_0_plus_Return, x2) → Cond_1380_1_plus_InvokeMethod1(1 > 0, 1397_0_plus_Return, x2)
Cond_1380_1_plus_InvokeMethod1(TRUE, 1397_0_plus_Return, x2) → 1580_0_plus_Return(1 + 1)

The integer pair graph contains the following rules and edges:
(0): 457_0_TIMES_NE(x0[0], x1[0]) → COND_457_0_TIMES_NE(x1[0] > 0, x0[0], x1[0])
(1): COND_457_0_TIMES_NE(TRUE, x0[1], x1[1]) → 457_0_TIMES_NE(x0[1], x1[1] - 1)

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

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

The set Q consists of the following terms:
457_0_times_NE(x0, 0)
754_1_times_InvokeMethod(596_0_times_Return(x0), x0, 0)
754_1_times_InvokeMethod(1361_0_times_Return, 0, x0)
754_1_times_InvokeMethod(1522_0_times_Return(x0), x1, x2)
754_1_times_InvokeMethod(1566_0_times_Return(x0), 0, x1)
1328_1_times_InvokeMethod(1351_0_plus_Return, x0, x1)
1328_1_times_InvokeMethod(1537_0_plus_Return(x0, x1, x2), x0, x1)
1328_1_times_InvokeMethod(1580_0_plus_Return(x0), x1, x2)
1328_1_times_InvokeMethod(1397_0_plus_Return, x0, x1)
1328_1_times_InvokeMethod(1379_0_plus_Return(x0, x1), x0, x1)
1333_0_plus_LE(x0, x1)
Cond_1333_0_plus_LE(TRUE, x0, x1)
1364_1_plus_InvokeMethod(1351_0_plus_Return, x0, x1, 0)
1364_1_plus_InvokeMethod(1537_0_plus_Return(x0, x1, x2), x3, x0, x1)
Cond_1364_1_plus_InvokeMethod(TRUE, 1537_0_plus_Return(x0, x1, x2), x3, x0, x1)
1364_1_plus_InvokeMethod(1580_0_plus_Return(x0), x1, x2, 0)
Cond_1364_1_plus_InvokeMethod1(TRUE, 1580_0_plus_Return(x0), x1, x2, 0)
1364_1_plus_InvokeMethod(1397_0_plus_Return, x0, x1, 0)
Cond_1364_1_plus_InvokeMethod2(TRUE, 1397_0_plus_Return, x0, x1, 0)
1364_1_plus_InvokeMethod(1379_0_plus_Return(x0, x1), x2, x0, x1)
Cond_1364_1_plus_InvokeMethod3(TRUE, 1379_0_plus_Return(x0, x1), x2, x0, x1)
Cond_1333_0_plus_LE1(TRUE, x0, x1)
Cond_1333_0_plus_LE2(TRUE, x0, x1)
Cond_1333_0_plus_LE3(TRUE, x0, x1)
Cond_1333_0_plus_LE4(TRUE, x0, x1)
Cond_1333_0_plus_LE5(TRUE, x0, x1)
Cond_1333_0_plus_LE6(TRUE, x0, x1)
Cond_1333_0_plus_LE7(TRUE, x0, x1)
1380_1_plus_InvokeMethod(1351_0_plus_Return, 0)
1380_1_plus_InvokeMethod(1580_0_plus_Return(x0), x1)
Cond_1380_1_plus_InvokeMethod(TRUE, 1580_0_plus_Return(x0), x1)
1380_1_plus_InvokeMethod(1397_0_plus_Return, x0)
Cond_1380_1_plus_InvokeMethod1(TRUE, 1397_0_plus_Return, x0)

### (31) 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 457_0_TIMES_NE(x0, x1) → COND_457_0_TIMES_NE(>(x1, 0), x0, x1) the following chains were created:
• We consider the chain 457_0_TIMES_NE(x0[0], x1[0]) → COND_457_0_TIMES_NE(>(x1[0], 0), x0[0], x1[0]), COND_457_0_TIMES_NE(TRUE, x0[1], x1[1]) → 457_0_TIMES_NE(x0[1], -(x1[1], 1)) which results in the following constraint:

(1)    (>(x1[0], 0)=TRUEx0[0]=x0[1]x1[0]=x1[1]457_0_TIMES_NE(x0[0], x1[0])≥NonInfC∧457_0_TIMES_NE(x0[0], x1[0])≥COND_457_0_TIMES_NE(>(x1[0], 0), x0[0], x1[0])∧(UIncreasing(COND_457_0_TIMES_NE(>(x1[0], 0), x0[0], x1[0])), ≥))

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

(2)    (>(x1[0], 0)=TRUE457_0_TIMES_NE(x0[0], x1[0])≥NonInfC∧457_0_TIMES_NE(x0[0], x1[0])≥COND_457_0_TIMES_NE(>(x1[0], 0), x0[0], x1[0])∧(UIncreasing(COND_457_0_TIMES_NE(>(x1[0], 0), x0[0], x1[0])), ≥))

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

(3)    (x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_457_0_TIMES_NE(>(x1[0], 0), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_63] + [(2)bni_63]x1[0] ≥ 0∧[(-1)bso_64] ≥ 0)

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

(4)    (x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_457_0_TIMES_NE(>(x1[0], 0), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_63] + [(2)bni_63]x1[0] ≥ 0∧[(-1)bso_64] ≥ 0)

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

(5)    (x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_457_0_TIMES_NE(>(x1[0], 0), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_63] + [(2)bni_63]x1[0] ≥ 0∧[(-1)bso_64] ≥ 0)

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

(6)    (x1[0] ≥ 0 ⇒ (UIncreasing(COND_457_0_TIMES_NE(>(x1[0], 0), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_63 + (2)bni_63] + [(2)bni_63]x1[0] ≥ 0∧[(-1)bso_64] ≥ 0)

For Pair COND_457_0_TIMES_NE(TRUE, x0, x1) → 457_0_TIMES_NE(x0, -(x1, 1)) the following chains were created:
• We consider the chain COND_457_0_TIMES_NE(TRUE, x0[1], x1[1]) → 457_0_TIMES_NE(x0[1], -(x1[1], 1)) which results in the following constraint:

(7)    (COND_457_0_TIMES_NE(TRUE, x0[1], x1[1])≥NonInfC∧COND_457_0_TIMES_NE(TRUE, x0[1], x1[1])≥457_0_TIMES_NE(x0[1], -(x1[1], 1))∧(UIncreasing(457_0_TIMES_NE(x0[1], -(x1[1], 1))), ≥))

We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(8)    ((UIncreasing(457_0_TIMES_NE(x0[1], -(x1[1], 1))), ≥)∧[2 + (-1)bso_66] ≥ 0)

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

(9)    ((UIncreasing(457_0_TIMES_NE(x0[1], -(x1[1], 1))), ≥)∧[2 + (-1)bso_66] ≥ 0)

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

(10)    ((UIncreasing(457_0_TIMES_NE(x0[1], -(x1[1], 1))), ≥)∧[2 + (-1)bso_66] ≥ 0)

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

(11)    ((UIncreasing(457_0_TIMES_NE(x0[1], -(x1[1], 1))), ≥)∧0 = 0∧[2 + (-1)bso_66] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 457_0_TIMES_NE(x0, x1) → COND_457_0_TIMES_NE(>(x1, 0), x0, x1)
• (x1[0] ≥ 0 ⇒ (UIncreasing(COND_457_0_TIMES_NE(>(x1[0], 0), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_63 + (2)bni_63] + [(2)bni_63]x1[0] ≥ 0∧[(-1)bso_64] ≥ 0)

• COND_457_0_TIMES_NE(TRUE, x0, x1) → 457_0_TIMES_NE(x0, -(x1, 1))
• ((UIncreasing(457_0_TIMES_NE(x0[1], -(x1[1], 1))), ≥)∧0 = 0∧[2 + (-1)bso_66] ≥ 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(457_0_times_NE(x1, x2)) = [-1]
POL(0) = 0
POL(596_0_times_Return(x1)) = [-1]
POL(754_1_times_InvokeMethod(x1, x2, x3)) = [-1]
POL(1328_1_times_InvokeMethod(x1, x2, x3)) = [-1]
POL(1333_0_plus_LE(x1, x2)) = [-1] + [-1]x2 + [-1]x1
POL(1361_0_times_Return) = [-1]
POL(1522_0_times_Return(x1)) = [-1]
POL(1566_0_times_Return(x1)) = [-1]
POL(1351_0_plus_Return) = [-1]
POL(1537_0_plus_Return(x1, x2, x3)) = [-1] + [-1]x3 + [-1]x1 + [-1]x2
POL(1580_0_plus_Return(x1)) = x1
POL(1397_0_plus_Return) = [-1]
POL(1) = [1]
POL(1379_0_plus_Return(x1, x2)) = [-1] + [-1]x1 + [-1]x2
POL(Cond_1333_0_plus_LE(x1, x2, x3)) = [-1] + [-1]x3 + [-1]x2
POL(>(x1, x2)) = [-1]
POL(1364_1_plus_InvokeMethod(x1, x2, x3, x4)) = [-1] + [-1]x4 + [-1]x1 + [-1]x2 + [-1]x3
POL(-(x1, x2)) = x1 + [-1]x2
POL(Cond_1364_1_plus_InvokeMethod(x1, x2, x3, x4, x5)) = [-1] + [-1]x5 + [-1]x4 + [-1]x3 + [-1]x2
POL(+(x1, x2)) = x1 + x2
POL(Cond_1364_1_plus_InvokeMethod1(x1, x2, x3, x4, x5)) = [-1] + [-1]x4 + [-1]x3 + [-1]x2
POL(Cond_1364_1_plus_InvokeMethod2(x1, x2, x3, x4, x5)) = [-1] + [-1]x4 + [-1]x3
POL(Cond_1364_1_plus_InvokeMethod3(x1, x2, x3, x4, x5)) = [-1] + [-1]x5 + [-1]x4 + [-1]x3 + [-1]x2
POL(Cond_1333_0_plus_LE1(x1, x2, x3)) = [-1] + [-1]x3 + [-1]x2
POL(&&(x1, x2)) = [-1]
POL(<=(x1, x2)) = [-1]
POL(Cond_1333_0_plus_LE2(x1, x2, x3)) = [-1] + [-1]x3 + [-1]x2
POL(1380_1_plus_InvokeMethod(x1, x2)) = [-1] + [-1]x2 + [-1]x1
POL(1387_0_plus_Return(x1, x2)) = [-1] + [-1]x2 + [-1]x1
POL(Cond_1333_0_plus_LE3(x1, x2, x3)) = [-1] + [-1]x3 + [-1]x2
POL(1417_0_plus_Return(x1, x2)) = [-1] + [-1]x2 + [-1]x1
POL(Cond_1333_0_plus_LE4(x1, x2, x3)) = [-1] + [-1]x3 + [-1]x2
POL(1459_0_plus_Return(x1, x2)) = [-1] + [-1]x2 + [-1]x1
POL(Cond_1333_0_plus_LE5(x1, x2, x3)) = [-1] + [-1]x3 + [-1]x2
POL(1507_0_plus_Return(x1, x2)) = [-1] + [-1]x2 + [-1]x1
POL(Cond_1333_0_plus_LE6(x1, x2, x3)) = [-1] + [-1]x3 + [-1]x2
POL(1550_0_plus_Return(x1, x2)) = [-1] + [-1]x2 + [-1]x1
POL(Cond_1333_0_plus_LE7(x1, x2, x3)) = [-1] + [-1]x3 + [-1]x2
POL(Cond_1380_1_plus_InvokeMethod(x1, x2, x3)) = [-1] + [-1]x3 + [-1]x2
POL(Cond_1380_1_plus_InvokeMethod1(x1, x2, x3)) = [-1] + [-1]x3
POL(457_0_TIMES_NE(x1, x2)) = [2]x2
POL(COND_457_0_TIMES_NE(x1, x2, x3)) = [2]x3

The following pairs are in P>:

COND_457_0_TIMES_NE(TRUE, x0[1], x1[1]) → 457_0_TIMES_NE(x0[1], -(x1[1], 1))

The following pairs are in Pbound:

457_0_TIMES_NE(x0[0], x1[0]) → COND_457_0_TIMES_NE(>(x1[0], 0), x0[0], x1[0])

The following pairs are in P:

457_0_TIMES_NE(x0[0], x1[0]) → COND_457_0_TIMES_NE(>(x1[0], 0), x0[0], x1[0])

There are no usable rules.

### (33) 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, Boolean

The ITRS R consists of the following rules:
457_0_times_NE(x0, 0) → 596_0_times_Return(x0)
754_1_times_InvokeMethod(596_0_times_Return(x0), x0, 0) → 1328_1_times_InvokeMethod(1333_0_plus_LE(0, x0), 0, x0)
754_1_times_InvokeMethod(1361_0_times_Return, 0, x3) → 1328_1_times_InvokeMethod(1333_0_plus_LE(0, 0), 0, 0)
754_1_times_InvokeMethod(1522_0_times_Return(x0), x1, x2) → 1328_1_times_InvokeMethod(1333_0_plus_LE(x0, x1), x0, x1)
754_1_times_InvokeMethod(1566_0_times_Return(x0), 0, x3) → 1328_1_times_InvokeMethod(1333_0_plus_LE(x0, 0), x0, 0)
1328_1_times_InvokeMethod(1351_0_plus_Return, x1, x2) → 1361_0_times_Return
1328_1_times_InvokeMethod(1537_0_plus_Return(x0, x1, x2), x0, x1) → 1522_0_times_Return(x2)
1328_1_times_InvokeMethod(1580_0_plus_Return(x0), x1, x2) → 1566_0_times_Return(x0)
1328_1_times_InvokeMethod(1397_0_plus_Return, x1, x2) → 1566_0_times_Return(1)
1328_1_times_InvokeMethod(1379_0_plus_Return(x0, x1), x0, x1) → 1522_0_times_Return(1)
1333_0_plus_LE(x0, x1) → Cond_1333_0_plus_LE(x1 > 0, x0, x1)
Cond_1333_0_plus_LE(TRUE, x0, x1) → 1364_1_plus_InvokeMethod(1333_0_plus_LE(x0, x1 - 1), x1, x0, x1 - 1)
1364_1_plus_InvokeMethod(1351_0_plus_Return, x2, x1, 0) → 1379_0_plus_Return(x1, x2)
1364_1_plus_InvokeMethod(1537_0_plus_Return(x0, x1, x2), x3, x0, x1) → Cond_1364_1_plus_InvokeMethod(x2 > 0, 1537_0_plus_Return(x0, x1, x2), x3, x0, x1)
Cond_1364_1_plus_InvokeMethod(TRUE, 1537_0_plus_Return(x0, x1, x2), x3, x0, x1) → 1537_0_plus_Return(x0, x3, 1 + x2)
1364_1_plus_InvokeMethod(1580_0_plus_Return(x0), x2, x1, 0) → Cond_1364_1_plus_InvokeMethod1(x0 > 0, 1580_0_plus_Return(x0), x2, x1, 0)
Cond_1364_1_plus_InvokeMethod1(TRUE, 1580_0_plus_Return(x0), x2, x1, 0) → 1537_0_plus_Return(x1, x2, 1 + x0)
1364_1_plus_InvokeMethod(1397_0_plus_Return, x2, x1, 0) → Cond_1364_1_plus_InvokeMethod2(1 > 0, 1397_0_plus_Return, x2, x1, 0)
Cond_1364_1_plus_InvokeMethod2(TRUE, 1397_0_plus_Return, x2, x1, 0) → 1537_0_plus_Return(x1, x2, 1 + 1)
1364_1_plus_InvokeMethod(1379_0_plus_Return(x0, x1), x3, x0, x1) → Cond_1364_1_plus_InvokeMethod3(1 > 0, 1379_0_plus_Return(x0, x1), x3, x0, x1)
Cond_1364_1_plus_InvokeMethod3(TRUE, 1379_0_plus_Return(x0, x1), x3, x0, x1) → 1537_0_plus_Return(x0, x3, 1 + 1)
1333_0_plus_LE(x0, x1) → Cond_1333_0_plus_LE1(x1 <= 0 && x0 <= 0, x0, x1)
Cond_1333_0_plus_LE1(TRUE, x0, x1) → 1351_0_plus_Return
1333_0_plus_LE(x0, x1) → Cond_1333_0_plus_LE2(x1 <= 0 && x0 > 0, x0, x1)
Cond_1333_0_plus_LE2(TRUE, x0, x1) → 1380_1_plus_InvokeMethod(1387_0_plus_Return(x0 - 1, x1), x0 - 1)
1333_0_plus_LE(x0, x1) → Cond_1333_0_plus_LE3(x1 <= 0 && x0 > 0, x0, x1)
Cond_1333_0_plus_LE3(TRUE, x0, x1) → 1380_1_plus_InvokeMethod(1417_0_plus_Return(x0 - 1, x1), x0 - 1)
1333_0_plus_LE(x0, x1) → Cond_1333_0_plus_LE4(x1 <= 0 && x0 > 0, x0, x1)
Cond_1333_0_plus_LE4(TRUE, x0, x1) → 1380_1_plus_InvokeMethod(1459_0_plus_Return(x0 - 1, x1), x0 - 1)
1333_0_plus_LE(x0, x1) → Cond_1333_0_plus_LE5(x1 <= 0 && x0 > 0, x0, x1)
Cond_1333_0_plus_LE5(TRUE, x0, x1) → 1380_1_plus_InvokeMethod(1507_0_plus_Return(x0 - 1, x1), x0 - 1)
1333_0_plus_LE(x0, x1) → Cond_1333_0_plus_LE6(x1 <= 0 && x0 > 0, x0, x1)
Cond_1333_0_plus_LE6(TRUE, x0, x1) → 1380_1_plus_InvokeMethod(1550_0_plus_Return(x0 - 1, x1), x0 - 1)
1333_0_plus_LE(x0, x1) → Cond_1333_0_plus_LE7(x1 <= 0 && x0 > 0, x0, x1)
Cond_1333_0_plus_LE7(TRUE, x0, x1) → 1380_1_plus_InvokeMethod(1333_0_plus_LE(x0 - 1, x1), x0 - 1)
1380_1_plus_InvokeMethod(1351_0_plus_Return, 0) → 1397_0_plus_Return
1380_1_plus_InvokeMethod(1580_0_plus_Return(x0), x2) → Cond_1380_1_plus_InvokeMethod(x0 > 0, 1580_0_plus_Return(x0), x2)
Cond_1380_1_plus_InvokeMethod(TRUE, 1580_0_plus_Return(x0), x2) → 1580_0_plus_Return(1 + x0)
1380_1_plus_InvokeMethod(1397_0_plus_Return, x2) → Cond_1380_1_plus_InvokeMethod1(1 > 0, 1397_0_plus_Return, x2)
Cond_1380_1_plus_InvokeMethod1(TRUE, 1397_0_plus_Return, x2) → 1580_0_plus_Return(1 + 1)

The integer pair graph contains the following rules and edges:
(0): 457_0_TIMES_NE(x0[0], x1[0]) → COND_457_0_TIMES_NE(x1[0] > 0, x0[0], x1[0])

The set Q consists of the following terms:
457_0_times_NE(x0, 0)
754_1_times_InvokeMethod(596_0_times_Return(x0), x0, 0)
754_1_times_InvokeMethod(1361_0_times_Return, 0, x0)
754_1_times_InvokeMethod(1522_0_times_Return(x0), x1, x2)
754_1_times_InvokeMethod(1566_0_times_Return(x0), 0, x1)
1328_1_times_InvokeMethod(1351_0_plus_Return, x0, x1)
1328_1_times_InvokeMethod(1537_0_plus_Return(x0, x1, x2), x0, x1)
1328_1_times_InvokeMethod(1580_0_plus_Return(x0), x1, x2)
1328_1_times_InvokeMethod(1397_0_plus_Return, x0, x1)
1328_1_times_InvokeMethod(1379_0_plus_Return(x0, x1), x0, x1)
1333_0_plus_LE(x0, x1)
Cond_1333_0_plus_LE(TRUE, x0, x1)
1364_1_plus_InvokeMethod(1351_0_plus_Return, x0, x1, 0)
1364_1_plus_InvokeMethod(1537_0_plus_Return(x0, x1, x2), x3, x0, x1)
Cond_1364_1_plus_InvokeMethod(TRUE, 1537_0_plus_Return(x0, x1, x2), x3, x0, x1)
1364_1_plus_InvokeMethod(1580_0_plus_Return(x0), x1, x2, 0)
Cond_1364_1_plus_InvokeMethod1(TRUE, 1580_0_plus_Return(x0), x1, x2, 0)
1364_1_plus_InvokeMethod(1397_0_plus_Return, x0, x1, 0)
Cond_1364_1_plus_InvokeMethod2(TRUE, 1397_0_plus_Return, x0, x1, 0)
1364_1_plus_InvokeMethod(1379_0_plus_Return(x0, x1), x2, x0, x1)
Cond_1364_1_plus_InvokeMethod3(TRUE, 1379_0_plus_Return(x0, x1), x2, x0, x1)
Cond_1333_0_plus_LE1(TRUE, x0, x1)
Cond_1333_0_plus_LE2(TRUE, x0, x1)
Cond_1333_0_plus_LE3(TRUE, x0, x1)
Cond_1333_0_plus_LE4(TRUE, x0, x1)
Cond_1333_0_plus_LE5(TRUE, x0, x1)
Cond_1333_0_plus_LE6(TRUE, x0, x1)
Cond_1333_0_plus_LE7(TRUE, x0, x1)
1380_1_plus_InvokeMethod(1351_0_plus_Return, 0)
1380_1_plus_InvokeMethod(1580_0_plus_Return(x0), x1)
Cond_1380_1_plus_InvokeMethod(TRUE, 1580_0_plus_Return(x0), x1)
1380_1_plus_InvokeMethod(1397_0_plus_Return, x0)
Cond_1380_1_plus_InvokeMethod1(TRUE, 1397_0_plus_Return, x0)

### (34) IDependencyGraphProof (EQUIVALENT transformation)

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

### (36) 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, Boolean

The ITRS R consists of the following rules:
457_0_times_NE(x0, 0) → 596_0_times_Return(x0)
754_1_times_InvokeMethod(596_0_times_Return(x0), x0, 0) → 1328_1_times_InvokeMethod(1333_0_plus_LE(0, x0), 0, x0)
754_1_times_InvokeMethod(1361_0_times_Return, 0, x3) → 1328_1_times_InvokeMethod(1333_0_plus_LE(0, 0), 0, 0)
754_1_times_InvokeMethod(1522_0_times_Return(x0), x1, x2) → 1328_1_times_InvokeMethod(1333_0_plus_LE(x0, x1), x0, x1)
754_1_times_InvokeMethod(1566_0_times_Return(x0), 0, x3) → 1328_1_times_InvokeMethod(1333_0_plus_LE(x0, 0), x0, 0)
1328_1_times_InvokeMethod(1351_0_plus_Return, x1, x2) → 1361_0_times_Return
1328_1_times_InvokeMethod(1537_0_plus_Return(x0, x1, x2), x0, x1) → 1522_0_times_Return(x2)
1328_1_times_InvokeMethod(1580_0_plus_Return(x0), x1, x2) → 1566_0_times_Return(x0)
1328_1_times_InvokeMethod(1397_0_plus_Return, x1, x2) → 1566_0_times_Return(1)
1328_1_times_InvokeMethod(1379_0_plus_Return(x0, x1), x0, x1) → 1522_0_times_Return(1)
1333_0_plus_LE(x0, x1) → Cond_1333_0_plus_LE(x1 > 0, x0, x1)
Cond_1333_0_plus_LE(TRUE, x0, x1) → 1364_1_plus_InvokeMethod(1333_0_plus_LE(x0, x1 - 1), x1, x0, x1 - 1)
1364_1_plus_InvokeMethod(1351_0_plus_Return, x2, x1, 0) → 1379_0_plus_Return(x1, x2)
1364_1_plus_InvokeMethod(1537_0_plus_Return(x0, x1, x2), x3, x0, x1) → Cond_1364_1_plus_InvokeMethod(x2 > 0, 1537_0_plus_Return(x0, x1, x2), x3, x0, x1)
Cond_1364_1_plus_InvokeMethod(TRUE, 1537_0_plus_Return(x0, x1, x2), x3, x0, x1) → 1537_0_plus_Return(x0, x3, 1 + x2)
1364_1_plus_InvokeMethod(1580_0_plus_Return(x0), x2, x1, 0) → Cond_1364_1_plus_InvokeMethod1(x0 > 0, 1580_0_plus_Return(x0), x2, x1, 0)
Cond_1364_1_plus_InvokeMethod1(TRUE, 1580_0_plus_Return(x0), x2, x1, 0) → 1537_0_plus_Return(x1, x2, 1 + x0)
1364_1_plus_InvokeMethod(1397_0_plus_Return, x2, x1, 0) → Cond_1364_1_plus_InvokeMethod2(1 > 0, 1397_0_plus_Return, x2, x1, 0)
Cond_1364_1_plus_InvokeMethod2(TRUE, 1397_0_plus_Return, x2, x1, 0) → 1537_0_plus_Return(x1, x2, 1 + 1)
1364_1_plus_InvokeMethod(1379_0_plus_Return(x0, x1), x3, x0, x1) → Cond_1364_1_plus_InvokeMethod3(1 > 0, 1379_0_plus_Return(x0, x1), x3, x0, x1)
Cond_1364_1_plus_InvokeMethod3(TRUE, 1379_0_plus_Return(x0, x1), x3, x0, x1) → 1537_0_plus_Return(x0, x3, 1 + 1)
1333_0_plus_LE(x0, x1) → Cond_1333_0_plus_LE1(x1 <= 0 && x0 <= 0, x0, x1)
Cond_1333_0_plus_LE1(TRUE, x0, x1) → 1351_0_plus_Return
1333_0_plus_LE(x0, x1) → Cond_1333_0_plus_LE2(x1 <= 0 && x0 > 0, x0, x1)
Cond_1333_0_plus_LE2(TRUE, x0, x1) → 1380_1_plus_InvokeMethod(1387_0_plus_Return(x0 - 1, x1), x0 - 1)
1333_0_plus_LE(x0, x1) → Cond_1333_0_plus_LE3(x1 <= 0 && x0 > 0, x0, x1)
Cond_1333_0_plus_LE3(TRUE, x0, x1) → 1380_1_plus_InvokeMethod(1417_0_plus_Return(x0 - 1, x1), x0 - 1)
1333_0_plus_LE(x0, x1) → Cond_1333_0_plus_LE4(x1 <= 0 && x0 > 0, x0, x1)
Cond_1333_0_plus_LE4(TRUE, x0, x1) → 1380_1_plus_InvokeMethod(1459_0_plus_Return(x0 - 1, x1), x0 - 1)
1333_0_plus_LE(x0, x1) → Cond_1333_0_plus_LE5(x1 <= 0 && x0 > 0, x0, x1)
Cond_1333_0_plus_LE5(TRUE, x0, x1) → 1380_1_plus_InvokeMethod(1507_0_plus_Return(x0 - 1, x1), x0 - 1)
1333_0_plus_LE(x0, x1) → Cond_1333_0_plus_LE6(x1 <= 0 && x0 > 0, x0, x1)
Cond_1333_0_plus_LE6(TRUE, x0, x1) → 1380_1_plus_InvokeMethod(1550_0_plus_Return(x0 - 1, x1), x0 - 1)
1333_0_plus_LE(x0, x1) → Cond_1333_0_plus_LE7(x1 <= 0 && x0 > 0, x0, x1)
Cond_1333_0_plus_LE7(TRUE, x0, x1) → 1380_1_plus_InvokeMethod(1333_0_plus_LE(x0 - 1, x1), x0 - 1)
1380_1_plus_InvokeMethod(1351_0_plus_Return, 0) → 1397_0_plus_Return
1380_1_plus_InvokeMethod(1580_0_plus_Return(x0), x2) → Cond_1380_1_plus_InvokeMethod(x0 > 0, 1580_0_plus_Return(x0), x2)
Cond_1380_1_plus_InvokeMethod(TRUE, 1580_0_plus_Return(x0), x2) → 1580_0_plus_Return(1 + x0)
1380_1_plus_InvokeMethod(1397_0_plus_Return, x2) → Cond_1380_1_plus_InvokeMethod1(1 > 0, 1397_0_plus_Return, x2)
Cond_1380_1_plus_InvokeMethod1(TRUE, 1397_0_plus_Return, x2) → 1580_0_plus_Return(1 + 1)

The integer pair graph contains the following rules and edges:
(1): COND_457_0_TIMES_NE(TRUE, x0[1], x1[1]) → 457_0_TIMES_NE(x0[1], x1[1] - 1)

The set Q consists of the following terms:
457_0_times_NE(x0, 0)
754_1_times_InvokeMethod(596_0_times_Return(x0), x0, 0)
754_1_times_InvokeMethod(1361_0_times_Return, 0, x0)
754_1_times_InvokeMethod(1522_0_times_Return(x0), x1, x2)
754_1_times_InvokeMethod(1566_0_times_Return(x0), 0, x1)
1328_1_times_InvokeMethod(1351_0_plus_Return, x0, x1)
1328_1_times_InvokeMethod(1537_0_plus_Return(x0, x1, x2), x0, x1)
1328_1_times_InvokeMethod(1580_0_plus_Return(x0), x1, x2)
1328_1_times_InvokeMethod(1397_0_plus_Return, x0, x1)
1328_1_times_InvokeMethod(1379_0_plus_Return(x0, x1), x0, x1)
1333_0_plus_LE(x0, x1)
Cond_1333_0_plus_LE(TRUE, x0, x1)
1364_1_plus_InvokeMethod(1351_0_plus_Return, x0, x1, 0)
1364_1_plus_InvokeMethod(1537_0_plus_Return(x0, x1, x2), x3, x0, x1)
Cond_1364_1_plus_InvokeMethod(TRUE, 1537_0_plus_Return(x0, x1, x2), x3, x0, x1)
1364_1_plus_InvokeMethod(1580_0_plus_Return(x0), x1, x2, 0)
Cond_1364_1_plus_InvokeMethod1(TRUE, 1580_0_plus_Return(x0), x1, x2, 0)
1364_1_plus_InvokeMethod(1397_0_plus_Return, x0, x1, 0)
Cond_1364_1_plus_InvokeMethod2(TRUE, 1397_0_plus_Return, x0, x1, 0)
1364_1_plus_InvokeMethod(1379_0_plus_Return(x0, x1), x2, x0, x1)
Cond_1364_1_plus_InvokeMethod3(TRUE, 1379_0_plus_Return(x0, x1), x2, x0, x1)
Cond_1333_0_plus_LE1(TRUE, x0, x1)
Cond_1333_0_plus_LE2(TRUE, x0, x1)
Cond_1333_0_plus_LE3(TRUE, x0, x1)
Cond_1333_0_plus_LE4(TRUE, x0, x1)
Cond_1333_0_plus_LE5(TRUE, x0, x1)
Cond_1333_0_plus_LE6(TRUE, x0, x1)
Cond_1333_0_plus_LE7(TRUE, x0, x1)
1380_1_plus_InvokeMethod(1351_0_plus_Return, 0)
1380_1_plus_InvokeMethod(1580_0_plus_Return(x0), x1)
Cond_1380_1_plus_InvokeMethod(TRUE, 1580_0_plus_Return(x0), x1)
1380_1_plus_InvokeMethod(1397_0_plus_Return, x0)
Cond_1380_1_plus_InvokeMethod1(TRUE, 1397_0_plus_Return, x0)

### (37) IDependencyGraphProof (EQUIVALENT transformation)

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