### (0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_22 (Sun Microsystems Inc.) Main-Class: Avg
`public class Avg {	public static void main(String[] args) {		int x, y;		x = args[0].length();		y = args[1].length();		average(x,y);	}	public static int average(int x, int y) {		if (x > 0) {			return average(x-1, y+1);		} else if (y > 2) {			return 1 + average(x+1, y-2);		} else {			return 1;		}	}}`

### (1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

### (2) Obligation:

FIGraph based on JBC Program:
Avg.main([Ljava/lang/String;)V: Graph of 139 nodes with 0 SCCs.

Avg.average(II)I: Graph of 63 nodes with 0 SCCs.

### (3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 28 rules for P and 40 rules for R.

Combined rules. Obtained 2 rules for P and 10 rules for R.

Filtered ground terms:

391_0_average_LE(x1, x2, x3, x4) → 391_0_average_LE(x2, x3, x4)
Cond_391_0_average_LE1(x1, x2, x3, x4, x5) → Cond_391_0_average_LE1(x1, x3, x4, x5)
727_1_average_InvokeMethod(x1, x2, x3, x4) → 727_1_average_InvokeMethod(x1, x4)
Cond_391_0_average_LE(x1, x2, x3, x4, x5) → Cond_391_0_average_LE(x1, x4)
901_0_average_Return(x1, x2, x3) → 901_0_average_Return(x2, x3)
931_0_average_Return(x1) → 931_0_average_Return
857_0_average_Return(x1, x2, x3) → 857_0_average_Return(x2, x3)
784_0_average_Return(x1, x2, x3) → 784_0_average_Return(x2, x3)
699_0_average_Return(x1, x2, x3) → 699_0_average_Return(x2, x3)
631_0_average_Return(x1) → 631_0_average_Return

Filtered duplicate args:

391_0_average_LE(x1, x2, x3) → 391_0_average_LE(x2, x3)
Cond_391_0_average_LE1(x1, x2, x3, x4) → Cond_391_0_average_LE1(x1, x3, x4)

Filtered unneeded arguments:

682_1_average_InvokeMethod(x1, x2, x3, x4, x5) → 682_1_average_InvokeMethod(x1, x4, x5)

Combined rules. Obtained 2 rules for P and 10 rules for R.

Finished conversion. Obtained 2 rules for P and 10 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:

Integer, Boolean

The ITRS R consists of the following rules:
727_1_average_InvokeMethod(699_0_average_Return(1, x1), x1) → 931_0_average_Return
727_1_average_InvokeMethod(784_0_average_Return(1, x1), x1) → 931_0_average_Return
727_1_average_InvokeMethod(857_0_average_Return(1, x1), x1) → 931_0_average_Return
727_1_average_InvokeMethod(901_0_average_Return(1, x1), x1) → 931_0_average_Return
682_1_average_InvokeMethod(631_0_average_Return, 0, x3) → 901_0_average_Return(x0, x1)
682_1_average_InvokeMethod(699_0_average_Return(x0, x1), x0, x1) → 901_0_average_Return(x2, x3)
682_1_average_InvokeMethod(784_0_average_Return(x0, x1), x0, x1) → 901_0_average_Return(x2, x3)
682_1_average_InvokeMethod(857_0_average_Return(x0, x1), x0, x1) → 901_0_average_Return(x2, x3)
682_1_average_InvokeMethod(901_0_average_Return(x0, x1), x0, x1) → 901_0_average_Return(x2, x3)
682_1_average_InvokeMethod(931_0_average_Return, 0, x3) → 901_0_average_Return(x0, x1)

The integer pair graph contains the following rules and edges:
(0): 391_0_AVERAGE_LE(x1[0], 0) → COND_391_0_AVERAGE_LE(x1[0] > 2, x1[0], 0)
(1): COND_391_0_AVERAGE_LE(TRUE, x1[1], 0) → 391_0_AVERAGE_LE(x1[1] - 2, 1)
(2): 391_0_AVERAGE_LE(x1[2], x0[2]) → COND_391_0_AVERAGE_LE1(x1[2] >= 0 && x0[2] > 0, x1[2], x0[2])
(3): COND_391_0_AVERAGE_LE1(TRUE, x1[3], x0[3]) → 391_0_AVERAGE_LE(x1[3] + 1, x0[3] - 1)

(0) -> (1), if ((x1[0] > 2* TRUE)∧(x1[0]* x1[1]))

(1) -> (0), if ((x1[1] - 2* x1[0])∧(1* 0))

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

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

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

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

The set Q consists of the following terms:
727_1_average_InvokeMethod(699_0_average_Return(1, x0), x0)
727_1_average_InvokeMethod(784_0_average_Return(1, x0), x0)
727_1_average_InvokeMethod(857_0_average_Return(1, x0), x0)
727_1_average_InvokeMethod(901_0_average_Return(1, x0), x0)
682_1_average_InvokeMethod(631_0_average_Return, 0, x0)
682_1_average_InvokeMethod(699_0_average_Return(x0, x1), x0, x1)
682_1_average_InvokeMethod(784_0_average_Return(x0, x1), x0, x1)
682_1_average_InvokeMethod(857_0_average_Return(x0, x1), x0, x1)
682_1_average_InvokeMethod(901_0_average_Return(x0, x1), x0, x1)
682_1_average_InvokeMethod(931_0_average_Return, 0, x0)

### (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 391_0_AVERAGE_LE(x1, 0) → COND_391_0_AVERAGE_LE(>(x1, 2), x1, 0) the following chains were created:
• We consider the chain 391_0_AVERAGE_LE(x1[0], 0) → COND_391_0_AVERAGE_LE(>(x1[0], 2), x1[0], 0), COND_391_0_AVERAGE_LE(TRUE, x1[1], 0) → 391_0_AVERAGE_LE(-(x1[1], 2), 1) which results in the following constraint:

(1)    (>(x1[0], 2)=TRUEx1[0]=x1[1]391_0_AVERAGE_LE(x1[0], 0)≥NonInfC∧391_0_AVERAGE_LE(x1[0], 0)≥COND_391_0_AVERAGE_LE(>(x1[0], 2), x1[0], 0)∧(UIncreasing(COND_391_0_AVERAGE_LE(>(x1[0], 2), x1[0], 0)), ≥))

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

(2)    (>(x1[0], 2)=TRUE391_0_AVERAGE_LE(x1[0], 0)≥NonInfC∧391_0_AVERAGE_LE(x1[0], 0)≥COND_391_0_AVERAGE_LE(>(x1[0], 2), x1[0], 0)∧(UIncreasing(COND_391_0_AVERAGE_LE(>(x1[0], 2), x1[0], 0)), ≥))

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

(3)    (x1[0] + [-3] ≥ 0 ⇒ (UIncreasing(COND_391_0_AVERAGE_LE(>(x1[0], 2), x1[0], 0)), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]x1[0] ≥ 0∧[(-1)bso_21] ≥ 0)

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

(4)    (x1[0] + [-3] ≥ 0 ⇒ (UIncreasing(COND_391_0_AVERAGE_LE(>(x1[0], 2), x1[0], 0)), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]x1[0] ≥ 0∧[(-1)bso_21] ≥ 0)

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

(5)    (x1[0] + [-3] ≥ 0 ⇒ (UIncreasing(COND_391_0_AVERAGE_LE(>(x1[0], 2), x1[0], 0)), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]x1[0] ≥ 0∧[(-1)bso_21] ≥ 0)

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

(6)    (x1[0] ≥ 0 ⇒ (UIncreasing(COND_391_0_AVERAGE_LE(>(x1[0], 2), x1[0], 0)), ≥)∧[(2)bni_20 + (-1)Bound*bni_20] + [bni_20]x1[0] ≥ 0∧[(-1)bso_21] ≥ 0)

For Pair COND_391_0_AVERAGE_LE(TRUE, x1, 0) → 391_0_AVERAGE_LE(-(x1, 2), 1) the following chains were created:
• We consider the chain COND_391_0_AVERAGE_LE(TRUE, x1[1], 0) → 391_0_AVERAGE_LE(-(x1[1], 2), 1) which results in the following constraint:

(7)    (COND_391_0_AVERAGE_LE(TRUE, x1[1], 0)≥NonInfC∧COND_391_0_AVERAGE_LE(TRUE, x1[1], 0)≥391_0_AVERAGE_LE(-(x1[1], 2), 1)∧(UIncreasing(391_0_AVERAGE_LE(-(x1[1], 2), 1)), ≥))

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

(8)    ((UIncreasing(391_0_AVERAGE_LE(-(x1[1], 2), 1)), ≥)∧[(-1)bso_23] ≥ 0)

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

(9)    ((UIncreasing(391_0_AVERAGE_LE(-(x1[1], 2), 1)), ≥)∧[(-1)bso_23] ≥ 0)

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

(10)    ((UIncreasing(391_0_AVERAGE_LE(-(x1[1], 2), 1)), ≥)∧[(-1)bso_23] ≥ 0)

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

(11)    ((UIncreasing(391_0_AVERAGE_LE(-(x1[1], 2), 1)), ≥)∧0 = 0∧[(-1)bso_23] ≥ 0)

For Pair 391_0_AVERAGE_LE(x1, x0) → COND_391_0_AVERAGE_LE1(&&(>=(x1, 0), >(x0, 0)), x1, x0) the following chains were created:
• We consider the chain 391_0_AVERAGE_LE(x1[2], x0[2]) → COND_391_0_AVERAGE_LE1(&&(>=(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]), COND_391_0_AVERAGE_LE1(TRUE, x1[3], x0[3]) → 391_0_AVERAGE_LE(+(x1[3], 1), -(x0[3], 1)) which results in the following constraint:

(12)    (&&(>=(x1[2], 0), >(x0[2], 0))=TRUEx1[2]=x1[3]x0[2]=x0[3]391_0_AVERAGE_LE(x1[2], x0[2])≥NonInfC∧391_0_AVERAGE_LE(x1[2], x0[2])≥COND_391_0_AVERAGE_LE1(&&(>=(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])∧(UIncreasing(COND_391_0_AVERAGE_LE1(&&(>=(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥))

We simplified constraint (12) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

(13)    (>=(x1[2], 0)=TRUE>(x0[2], 0)=TRUE391_0_AVERAGE_LE(x1[2], x0[2])≥NonInfC∧391_0_AVERAGE_LE(x1[2], x0[2])≥COND_391_0_AVERAGE_LE1(&&(>=(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])∧(UIncreasing(COND_391_0_AVERAGE_LE1(&&(>=(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥))

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

(14)    (x1[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_391_0_AVERAGE_LE1(&&(>=(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [(2)bni_24]x0[2] + [bni_24]x1[2] ≥ 0∧[(-1)bso_25] ≥ 0)

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

(15)    (x1[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_391_0_AVERAGE_LE1(&&(>=(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [(2)bni_24]x0[2] + [bni_24]x1[2] ≥ 0∧[(-1)bso_25] ≥ 0)

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

(16)    (x1[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_391_0_AVERAGE_LE1(&&(>=(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [(2)bni_24]x0[2] + [bni_24]x1[2] ≥ 0∧[(-1)bso_25] ≥ 0)

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

(17)    (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(COND_391_0_AVERAGE_LE1(&&(>=(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[bni_24 + (-1)Bound*bni_24] + [(2)bni_24]x0[2] + [bni_24]x1[2] ≥ 0∧[(-1)bso_25] ≥ 0)

For Pair COND_391_0_AVERAGE_LE1(TRUE, x1, x0) → 391_0_AVERAGE_LE(+(x1, 1), -(x0, 1)) the following chains were created:
• We consider the chain COND_391_0_AVERAGE_LE1(TRUE, x1[3], x0[3]) → 391_0_AVERAGE_LE(+(x1[3], 1), -(x0[3], 1)) which results in the following constraint:

(18)    (COND_391_0_AVERAGE_LE1(TRUE, x1[3], x0[3])≥NonInfC∧COND_391_0_AVERAGE_LE1(TRUE, x1[3], x0[3])≥391_0_AVERAGE_LE(+(x1[3], 1), -(x0[3], 1))∧(UIncreasing(391_0_AVERAGE_LE(+(x1[3], 1), -(x0[3], 1))), ≥))

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

(19)    ((UIncreasing(391_0_AVERAGE_LE(+(x1[3], 1), -(x0[3], 1))), ≥)∧[1 + (-1)bso_27] ≥ 0)

We simplified constraint (19) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(20)    ((UIncreasing(391_0_AVERAGE_LE(+(x1[3], 1), -(x0[3], 1))), ≥)∧[1 + (-1)bso_27] ≥ 0)

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

(21)    ((UIncreasing(391_0_AVERAGE_LE(+(x1[3], 1), -(x0[3], 1))), ≥)∧[1 + (-1)bso_27] ≥ 0)

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

(22)    ((UIncreasing(391_0_AVERAGE_LE(+(x1[3], 1), -(x0[3], 1))), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_27] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 391_0_AVERAGE_LE(x1, 0) → COND_391_0_AVERAGE_LE(>(x1, 2), x1, 0)
• (x1[0] ≥ 0 ⇒ (UIncreasing(COND_391_0_AVERAGE_LE(>(x1[0], 2), x1[0], 0)), ≥)∧[(2)bni_20 + (-1)Bound*bni_20] + [bni_20]x1[0] ≥ 0∧[(-1)bso_21] ≥ 0)

• COND_391_0_AVERAGE_LE(TRUE, x1, 0) → 391_0_AVERAGE_LE(-(x1, 2), 1)
• ((UIncreasing(391_0_AVERAGE_LE(-(x1[1], 2), 1)), ≥)∧0 = 0∧[(-1)bso_23] ≥ 0)

• 391_0_AVERAGE_LE(x1, x0) → COND_391_0_AVERAGE_LE1(&&(>=(x1, 0), >(x0, 0)), x1, x0)
• (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(COND_391_0_AVERAGE_LE1(&&(>=(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[bni_24 + (-1)Bound*bni_24] + [(2)bni_24]x0[2] + [bni_24]x1[2] ≥ 0∧[(-1)bso_25] ≥ 0)

• COND_391_0_AVERAGE_LE1(TRUE, x1, x0) → 391_0_AVERAGE_LE(+(x1, 1), -(x0, 1))
• ((UIncreasing(391_0_AVERAGE_LE(+(x1[3], 1), -(x0[3], 1))), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_27] ≥ 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(727_1_average_InvokeMethod(x1, x2)) = [-1]
POL(699_0_average_Return(x1, x2)) = [-1]
POL(1) = [1]
POL(931_0_average_Return) = [-1]
POL(784_0_average_Return(x1, x2)) = [-1]
POL(857_0_average_Return(x1, x2)) = [-1]
POL(901_0_average_Return(x1, x2)) = [-1]
POL(682_1_average_InvokeMethod(x1, x2, x3)) = [-1]
POL(631_0_average_Return) = [-1]
POL(0) = 0
POL(391_0_AVERAGE_LE(x1, x2)) = [-1] + [2]x2 + x1
POL(COND_391_0_AVERAGE_LE(x1, x2, x3)) = [-1] + x2
POL(>(x1, x2)) = [-1]
POL(2) = [2]
POL(-(x1, x2)) = x1 + [-1]x2
POL(COND_391_0_AVERAGE_LE1(x1, x2, x3)) = [-1] + [2]x3 + x2
POL(&&(x1, x2)) = [-1]
POL(>=(x1, x2)) = [-1]
POL(+(x1, x2)) = x1 + x2

The following pairs are in P>:

COND_391_0_AVERAGE_LE1(TRUE, x1[3], x0[3]) → 391_0_AVERAGE_LE(+(x1[3], 1), -(x0[3], 1))

The following pairs are in Pbound:

391_0_AVERAGE_LE(x1[0], 0) → COND_391_0_AVERAGE_LE(>(x1[0], 2), x1[0], 0)
391_0_AVERAGE_LE(x1[2], x0[2]) → COND_391_0_AVERAGE_LE1(&&(>=(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])

The following pairs are in P:

391_0_AVERAGE_LE(x1[0], 0) → COND_391_0_AVERAGE_LE(>(x1[0], 2), x1[0], 0)
COND_391_0_AVERAGE_LE(TRUE, x1[1], 0) → 391_0_AVERAGE_LE(-(x1[1], 2), 1)
391_0_AVERAGE_LE(x1[2], x0[2]) → COND_391_0_AVERAGE_LE1(&&(>=(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])

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:

Integer, Boolean

The ITRS R consists of the following rules:
727_1_average_InvokeMethod(699_0_average_Return(1, x1), x1) → 931_0_average_Return
727_1_average_InvokeMethod(784_0_average_Return(1, x1), x1) → 931_0_average_Return
727_1_average_InvokeMethod(857_0_average_Return(1, x1), x1) → 931_0_average_Return
727_1_average_InvokeMethod(901_0_average_Return(1, x1), x1) → 931_0_average_Return
682_1_average_InvokeMethod(631_0_average_Return, 0, x3) → 901_0_average_Return(x0, x1)
682_1_average_InvokeMethod(699_0_average_Return(x0, x1), x0, x1) → 901_0_average_Return(x2, x3)
682_1_average_InvokeMethod(784_0_average_Return(x0, x1), x0, x1) → 901_0_average_Return(x2, x3)
682_1_average_InvokeMethod(857_0_average_Return(x0, x1), x0, x1) → 901_0_average_Return(x2, x3)
682_1_average_InvokeMethod(901_0_average_Return(x0, x1), x0, x1) → 901_0_average_Return(x2, x3)
682_1_average_InvokeMethod(931_0_average_Return, 0, x3) → 901_0_average_Return(x0, x1)

The integer pair graph contains the following rules and edges:
(0): 391_0_AVERAGE_LE(x1[0], 0) → COND_391_0_AVERAGE_LE(x1[0] > 2, x1[0], 0)
(1): COND_391_0_AVERAGE_LE(TRUE, x1[1], 0) → 391_0_AVERAGE_LE(x1[1] - 2, 1)
(2): 391_0_AVERAGE_LE(x1[2], x0[2]) → COND_391_0_AVERAGE_LE1(x1[2] >= 0 && x0[2] > 0, x1[2], x0[2])

(1) -> (0), if ((x1[1] - 2* x1[0])∧(1* 0))

(0) -> (1), if ((x1[0] > 2* TRUE)∧(x1[0]* x1[1]))

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

The set Q consists of the following terms:
727_1_average_InvokeMethod(699_0_average_Return(1, x0), x0)
727_1_average_InvokeMethod(784_0_average_Return(1, x0), x0)
727_1_average_InvokeMethod(857_0_average_Return(1, x0), x0)
727_1_average_InvokeMethod(901_0_average_Return(1, x0), x0)
682_1_average_InvokeMethod(631_0_average_Return, 0, x0)
682_1_average_InvokeMethod(699_0_average_Return(x0, x1), x0, x1)
682_1_average_InvokeMethod(784_0_average_Return(x0, x1), x0, x1)
682_1_average_InvokeMethod(857_0_average_Return(x0, x1), x0, x1)
682_1_average_InvokeMethod(901_0_average_Return(x0, x1), x0, x1)
682_1_average_InvokeMethod(931_0_average_Return, 0, x0)

### (8) IDependencyGraphProof (EQUIVALENT transformation)

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

### (9) 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:
727_1_average_InvokeMethod(699_0_average_Return(1, x1), x1) → 931_0_average_Return
727_1_average_InvokeMethod(784_0_average_Return(1, x1), x1) → 931_0_average_Return
727_1_average_InvokeMethod(857_0_average_Return(1, x1), x1) → 931_0_average_Return
727_1_average_InvokeMethod(901_0_average_Return(1, x1), x1) → 931_0_average_Return
682_1_average_InvokeMethod(631_0_average_Return, 0, x3) → 901_0_average_Return(x0, x1)
682_1_average_InvokeMethod(699_0_average_Return(x0, x1), x0, x1) → 901_0_average_Return(x2, x3)
682_1_average_InvokeMethod(784_0_average_Return(x0, x1), x0, x1) → 901_0_average_Return(x2, x3)
682_1_average_InvokeMethod(857_0_average_Return(x0, x1), x0, x1) → 901_0_average_Return(x2, x3)
682_1_average_InvokeMethod(901_0_average_Return(x0, x1), x0, x1) → 901_0_average_Return(x2, x3)
682_1_average_InvokeMethod(931_0_average_Return, 0, x3) → 901_0_average_Return(x0, x1)

The integer pair graph contains the following rules and edges:
(1): COND_391_0_AVERAGE_LE(TRUE, x1[1], 0) → 391_0_AVERAGE_LE(x1[1] - 2, 1)
(0): 391_0_AVERAGE_LE(x1[0], 0) → COND_391_0_AVERAGE_LE(x1[0] > 2, x1[0], 0)

(1) -> (0), if ((x1[1] - 2* x1[0])∧(1* 0))

(0) -> (1), if ((x1[0] > 2* TRUE)∧(x1[0]* x1[1]))

The set Q consists of the following terms:
727_1_average_InvokeMethod(699_0_average_Return(1, x0), x0)
727_1_average_InvokeMethod(784_0_average_Return(1, x0), x0)
727_1_average_InvokeMethod(857_0_average_Return(1, x0), x0)
727_1_average_InvokeMethod(901_0_average_Return(1, x0), x0)
682_1_average_InvokeMethod(631_0_average_Return, 0, x0)
682_1_average_InvokeMethod(699_0_average_Return(x0, x1), x0, x1)
682_1_average_InvokeMethod(784_0_average_Return(x0, x1), x0, x1)
682_1_average_InvokeMethod(857_0_average_Return(x0, x1), x0, x1)
682_1_average_InvokeMethod(901_0_average_Return(x0, x1), x0, x1)
682_1_average_InvokeMethod(931_0_average_Return, 0, x0)

### (10) 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.

### (11) Obligation:

IDP problem:
The following function symbols are pre-defined:
 != ~ Neq: (Integer, Integer) -> Boolean * ~ Mul: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean -1 ~ UnaryMinus: (Integer) -> Integer | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer = ~ Eq: (Integer, Integer) -> Boolean ~ Bwxor: (Integer, Integer) -> Integer || ~ Lor: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean < ~ Lt: (Integer, Integer) -> Boolean - ~ Sub: (Integer, Integer) -> Integer <= ~ Le: (Integer, Integer) -> Boolean > ~ Gt: (Integer, Integer) -> Boolean ~ ~ Bwnot: (Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer & ~ Bwand: (Integer, Integer) -> Integer + ~ Add: (Integer, Integer) -> Integer && ~ Land: (Boolean, Boolean) -> Boolean

The following domains are used:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_391_0_AVERAGE_LE(TRUE, x1[1], 0) → 391_0_AVERAGE_LE(x1[1] - 2, 1)
(0): 391_0_AVERAGE_LE(x1[0], 0) → COND_391_0_AVERAGE_LE(x1[0] > 2, x1[0], 0)

(1) -> (0), if ((x1[1] - 2* x1[0])∧(1* 0))

(0) -> (1), if ((x1[0] > 2* TRUE)∧(x1[0]* x1[1]))

The set Q consists of the following terms:
727_1_average_InvokeMethod(699_0_average_Return(1, x0), x0)
727_1_average_InvokeMethod(784_0_average_Return(1, x0), x0)
727_1_average_InvokeMethod(857_0_average_Return(1, x0), x0)
727_1_average_InvokeMethod(901_0_average_Return(1, x0), x0)
682_1_average_InvokeMethod(631_0_average_Return, 0, x0)
682_1_average_InvokeMethod(699_0_average_Return(x0, x1), x0, x1)
682_1_average_InvokeMethod(784_0_average_Return(x0, x1), x0, x1)
682_1_average_InvokeMethod(857_0_average_Return(x0, x1), x0, x1)
682_1_average_InvokeMethod(901_0_average_Return(x0, x1), x0, x1)
682_1_average_InvokeMethod(931_0_average_Return, 0, x0)

### (12) 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_391_0_AVERAGE_LE(TRUE, x1[1], 0) → 391_0_AVERAGE_LE(-(x1[1], 2), 1) the following chains were created:
• We consider the chain COND_391_0_AVERAGE_LE(TRUE, x1[1], 0) → 391_0_AVERAGE_LE(-(x1[1], 2), 1) which results in the following constraint:

(1)    (COND_391_0_AVERAGE_LE(TRUE, x1[1], 0)≥NonInfC∧COND_391_0_AVERAGE_LE(TRUE, x1[1], 0)≥391_0_AVERAGE_LE(-(x1[1], 2), 1)∧(UIncreasing(391_0_AVERAGE_LE(-(x1[1], 2), 1)), ≥))

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

(2)    ((UIncreasing(391_0_AVERAGE_LE(-(x1[1], 2), 1)), ≥)∧[2 + (-1)bso_7] ≥ 0)

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

(3)    ((UIncreasing(391_0_AVERAGE_LE(-(x1[1], 2), 1)), ≥)∧[2 + (-1)bso_7] ≥ 0)

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

(4)    ((UIncreasing(391_0_AVERAGE_LE(-(x1[1], 2), 1)), ≥)∧[2 + (-1)bso_7] ≥ 0)

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

(5)    ((UIncreasing(391_0_AVERAGE_LE(-(x1[1], 2), 1)), ≥)∧0 = 0∧[2 + (-1)bso_7] ≥ 0)

For Pair 391_0_AVERAGE_LE(x1[0], 0) → COND_391_0_AVERAGE_LE(>(x1[0], 2), x1[0], 0) the following chains were created:
• We consider the chain 391_0_AVERAGE_LE(x1[0], 0) → COND_391_0_AVERAGE_LE(>(x1[0], 2), x1[0], 0), COND_391_0_AVERAGE_LE(TRUE, x1[1], 0) → 391_0_AVERAGE_LE(-(x1[1], 2), 1) which results in the following constraint:

(6)    (>(x1[0], 2)=TRUEx1[0]=x1[1]391_0_AVERAGE_LE(x1[0], 0)≥NonInfC∧391_0_AVERAGE_LE(x1[0], 0)≥COND_391_0_AVERAGE_LE(>(x1[0], 2), x1[0], 0)∧(UIncreasing(COND_391_0_AVERAGE_LE(>(x1[0], 2), x1[0], 0)), ≥))

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

(7)    (>(x1[0], 2)=TRUE391_0_AVERAGE_LE(x1[0], 0)≥NonInfC∧391_0_AVERAGE_LE(x1[0], 0)≥COND_391_0_AVERAGE_LE(>(x1[0], 2), x1[0], 0)∧(UIncreasing(COND_391_0_AVERAGE_LE(>(x1[0], 2), x1[0], 0)), ≥))

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

(8)    (x1[0] + [-3] ≥ 0 ⇒ (UIncreasing(COND_391_0_AVERAGE_LE(>(x1[0], 2), x1[0], 0)), ≥)∧[(-1)bni_8 + (-1)Bound*bni_8] + [bni_8]x1[0] ≥ 0∧[(-1)bso_9] ≥ 0)

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

(9)    (x1[0] + [-3] ≥ 0 ⇒ (UIncreasing(COND_391_0_AVERAGE_LE(>(x1[0], 2), x1[0], 0)), ≥)∧[(-1)bni_8 + (-1)Bound*bni_8] + [bni_8]x1[0] ≥ 0∧[(-1)bso_9] ≥ 0)

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

(10)    (x1[0] + [-3] ≥ 0 ⇒ (UIncreasing(COND_391_0_AVERAGE_LE(>(x1[0], 2), x1[0], 0)), ≥)∧[(-1)bni_8 + (-1)Bound*bni_8] + [bni_8]x1[0] ≥ 0∧[(-1)bso_9] ≥ 0)

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

(11)    (x1[0] ≥ 0 ⇒ (UIncreasing(COND_391_0_AVERAGE_LE(>(x1[0], 2), x1[0], 0)), ≥)∧[(2)bni_8 + (-1)Bound*bni_8] + [bni_8]x1[0] ≥ 0∧[(-1)bso_9] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• COND_391_0_AVERAGE_LE(TRUE, x1[1], 0) → 391_0_AVERAGE_LE(-(x1[1], 2), 1)
• ((UIncreasing(391_0_AVERAGE_LE(-(x1[1], 2), 1)), ≥)∧0 = 0∧[2 + (-1)bso_7] ≥ 0)

• 391_0_AVERAGE_LE(x1[0], 0) → COND_391_0_AVERAGE_LE(>(x1[0], 2), x1[0], 0)
• (x1[0] ≥ 0 ⇒ (UIncreasing(COND_391_0_AVERAGE_LE(>(x1[0], 2), x1[0], 0)), ≥)∧[(2)bni_8 + (-1)Bound*bni_8] + [bni_8]x1[0] ≥ 0∧[(-1)bso_9] ≥ 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_391_0_AVERAGE_LE(x1, x2, x3)) = [-1] + x2
POL(0) = 0
POL(391_0_AVERAGE_LE(x1, x2)) = [-1] + x1
POL(-(x1, x2)) = x1 + [-1]x2
POL(2) = [2]
POL(1) = [1]
POL(>(x1, x2)) = [-1]

The following pairs are in P>:

COND_391_0_AVERAGE_LE(TRUE, x1[1], 0) → 391_0_AVERAGE_LE(-(x1[1], 2), 1)

The following pairs are in Pbound:

391_0_AVERAGE_LE(x1[0], 0) → COND_391_0_AVERAGE_LE(>(x1[0], 2), x1[0], 0)

The following pairs are in P:

391_0_AVERAGE_LE(x1[0], 0) → COND_391_0_AVERAGE_LE(>(x1[0], 2), x1[0], 0)

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:
(0): 391_0_AVERAGE_LE(x1[0], 0) → COND_391_0_AVERAGE_LE(x1[0] > 2, x1[0], 0)

The set Q consists of the following terms:
727_1_average_InvokeMethod(699_0_average_Return(1, x0), x0)
727_1_average_InvokeMethod(784_0_average_Return(1, x0), x0)
727_1_average_InvokeMethod(857_0_average_Return(1, x0), x0)
727_1_average_InvokeMethod(901_0_average_Return(1, x0), x0)
682_1_average_InvokeMethod(631_0_average_Return, 0, x0)
682_1_average_InvokeMethod(699_0_average_Return(x0, x1), x0, x1)
682_1_average_InvokeMethod(784_0_average_Return(x0, x1), x0, x1)
682_1_average_InvokeMethod(857_0_average_Return(x0, x1), x0, x1)
682_1_average_InvokeMethod(901_0_average_Return(x0, x1), x0, x1)
682_1_average_InvokeMethod(931_0_average_Return, 0, x0)

### (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:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_391_0_AVERAGE_LE(TRUE, x1[1], 0) → 391_0_AVERAGE_LE(x1[1] - 2, 1)

The set Q consists of the following terms:
727_1_average_InvokeMethod(699_0_average_Return(1, x0), x0)
727_1_average_InvokeMethod(784_0_average_Return(1, x0), x0)
727_1_average_InvokeMethod(857_0_average_Return(1, x0), x0)
727_1_average_InvokeMethod(901_0_average_Return(1, x0), x0)
682_1_average_InvokeMethod(631_0_average_Return, 0, x0)
682_1_average_InvokeMethod(699_0_average_Return(x0, x1), x0, x1)
682_1_average_InvokeMethod(784_0_average_Return(x0, x1), x0, x1)
682_1_average_InvokeMethod(857_0_average_Return(x0, x1), x0, x1)
682_1_average_InvokeMethod(901_0_average_Return(x0, x1), x0, x1)
682_1_average_InvokeMethod(931_0_average_Return, 0, x0)

### (18) IDependencyGraphProof (EQUIVALENT transformation)

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

### (20) 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:
727_1_average_InvokeMethod(699_0_average_Return(1, x1), x1) → 931_0_average_Return
727_1_average_InvokeMethod(784_0_average_Return(1, x1), x1) → 931_0_average_Return
727_1_average_InvokeMethod(857_0_average_Return(1, x1), x1) → 931_0_average_Return
727_1_average_InvokeMethod(901_0_average_Return(1, x1), x1) → 931_0_average_Return
682_1_average_InvokeMethod(631_0_average_Return, 0, x3) → 901_0_average_Return(x0, x1)
682_1_average_InvokeMethod(699_0_average_Return(x0, x1), x0, x1) → 901_0_average_Return(x2, x3)
682_1_average_InvokeMethod(784_0_average_Return(x0, x1), x0, x1) → 901_0_average_Return(x2, x3)
682_1_average_InvokeMethod(857_0_average_Return(x0, x1), x0, x1) → 901_0_average_Return(x2, x3)
682_1_average_InvokeMethod(901_0_average_Return(x0, x1), x0, x1) → 901_0_average_Return(x2, x3)
682_1_average_InvokeMethod(931_0_average_Return, 0, x3) → 901_0_average_Return(x0, x1)

The integer pair graph contains the following rules and edges:
(1): COND_391_0_AVERAGE_LE(TRUE, x1[1], 0) → 391_0_AVERAGE_LE(x1[1] - 2, 1)
(3): COND_391_0_AVERAGE_LE1(TRUE, x1[3], x0[3]) → 391_0_AVERAGE_LE(x1[3] + 1, x0[3] - 1)

The set Q consists of the following terms:
727_1_average_InvokeMethod(699_0_average_Return(1, x0), x0)
727_1_average_InvokeMethod(784_0_average_Return(1, x0), x0)
727_1_average_InvokeMethod(857_0_average_Return(1, x0), x0)
727_1_average_InvokeMethod(901_0_average_Return(1, x0), x0)
682_1_average_InvokeMethod(631_0_average_Return, 0, x0)
682_1_average_InvokeMethod(699_0_average_Return(x0, x1), x0, x1)
682_1_average_InvokeMethod(784_0_average_Return(x0, x1), x0, x1)
682_1_average_InvokeMethod(857_0_average_Return(x0, x1), x0, x1)
682_1_average_InvokeMethod(901_0_average_Return(x0, x1), x0, x1)
682_1_average_InvokeMethod(931_0_average_Return, 0, x0)

### (21) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.