### (0) Obligation:

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

### (1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

### (2) Obligation:

FIGraph based on JBC Program:
Nest.Nest.main([Ljava/lang/String;)V: Graph of 69 nodes with 0 SCCs.

Nest.Nest.nest(I)I: Graph of 23 nodes with 0 SCCs.

### (3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 17 rules for P and 6 rules for R.

Combined rules. Obtained 3 rules for P and 2 rules for R.

Filtered ground terms:

242_1_nest_InvokeMethod(x1, x2) → 242_1_nest_InvokeMethod(x1)
89_0_nest_NE(x1, x2, x3) → 89_0_nest_NE(x2, x3)
320_0_nest_Return(x1, x2) → 320_0_nest_Return
135_0_nest_Return(x1, x2, x3) → 135_0_nest_Return
Cond_89_0_nest_NE(x1, x2, x3, x4) → Cond_89_0_nest_NE(x1, x3, x4)

Filtered duplicate args:

89_0_nest_NE(x1, x2) → 89_0_nest_NE(x2)
Cond_89_0_nest_NE(x1, x2, x3) → Cond_89_0_nest_NE(x1, x3)

Combined rules. Obtained 3 rules for P and 2 rules for R.

Finished conversion. Obtained 3 rules for P and 2 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

The ITRS R consists of the following rules:
89_0_nest_NE(0) → 135_0_nest_Return
242_1_nest_InvokeMethod(135_0_nest_Return) → 320_0_nest_Return

The integer pair graph contains the following rules and edges:
(0): 89_0_NEST_NE(x0[0]) → COND_89_0_NEST_NE(x0[0] > 0, x0[0])
(1): COND_89_0_NEST_NE(TRUE, x0[1]) → 200_1_NEST_INVOKEMETHOD(89_0_nest_NE(x0[1] - 1), x0[1] - 1)
(2): COND_89_0_NEST_NE(TRUE, x0[2]) → 89_0_NEST_NE(x0[2] - 1)
(3): 200_1_NEST_INVOKEMETHOD(135_0_nest_Return, 0) → 89_0_NEST_NE(0)
(4): 200_1_NEST_INVOKEMETHOD(320_0_nest_Return, x1[4]) → 89_0_NEST_NE(0)

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

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

(1) -> (3), if ((89_0_nest_NE(x0[1] - 1) →* 135_0_nest_Return)∧(x0[1] - 1* 0))

(1) -> (4), if ((89_0_nest_NE(x0[1] - 1) →* 320_0_nest_Return)∧(x0[1] - 1* x1[4]))

(2) -> (0), if ((x0[2] - 1* x0[0]))

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

(4) -> (0), if ((0* x0[0]))

The set Q consists of the following terms:
89_0_nest_NE(0)
242_1_nest_InvokeMethod(135_0_nest_Return)

### (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 89_0_NEST_NE(x0) → COND_89_0_NEST_NE(>(x0, 0), x0) the following chains were created:
• We consider the chain 89_0_NEST_NE(x0[0]) → COND_89_0_NEST_NE(>(x0[0], 0), x0[0]), COND_89_0_NEST_NE(TRUE, x0[1]) → 200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1)) which results in the following constraint:

(1)    (>(x0[0], 0)=TRUEx0[0]=x0[1]89_0_NEST_NE(x0[0])≥NonInfC∧89_0_NEST_NE(x0[0])≥COND_89_0_NEST_NE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥))

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

(2)    (>(x0[0], 0)=TRUE89_0_NEST_NE(x0[0])≥NonInfC∧89_0_NEST_NE(x0[0])≥COND_89_0_NEST_NE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥))

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

(3)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] ≥ 0∧[(-1)bso_17] ≥ 0)

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

(4)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] ≥ 0∧[(-1)bso_17] ≥ 0)

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

(5)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] ≥ 0∧[(-1)bso_17] ≥ 0)

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

(6)    (x0[0] ≥ 0 ⇒ (UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] ≥ 0∧[(-1)bso_17] ≥ 0)

• We consider the chain 89_0_NEST_NE(x0[0]) → COND_89_0_NEST_NE(>(x0[0], 0), x0[0]), COND_89_0_NEST_NE(TRUE, x0[2]) → 89_0_NEST_NE(-(x0[2], 1)) which results in the following constraint:

(7)    (>(x0[0], 0)=TRUEx0[0]=x0[2]89_0_NEST_NE(x0[0])≥NonInfC∧89_0_NEST_NE(x0[0])≥COND_89_0_NEST_NE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥))

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

(8)    (>(x0[0], 0)=TRUE89_0_NEST_NE(x0[0])≥NonInfC∧89_0_NEST_NE(x0[0])≥COND_89_0_NEST_NE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥))

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

(9)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] ≥ 0∧[(-1)bso_17] ≥ 0)

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

(10)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] ≥ 0∧[(-1)bso_17] ≥ 0)

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

(11)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] ≥ 0∧[(-1)bso_17] ≥ 0)

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

(12)    (x0[0] ≥ 0 ⇒ (UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] ≥ 0∧[(-1)bso_17] ≥ 0)

For Pair COND_89_0_NEST_NE(TRUE, x0) → 200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0, 1)), -(x0, 1)) the following chains were created:
• We consider the chain COND_89_0_NEST_NE(TRUE, x0[1]) → 200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1)) which results in the following constraint:

(13)    (COND_89_0_NEST_NE(TRUE, x0[1])≥NonInfC∧COND_89_0_NEST_NE(TRUE, x0[1])≥200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1))∧(UIncreasing(200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1))), ≥))

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

(14)    ((UIncreasing(200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1))), ≥)∧[(-1)bso_19] ≥ 0)

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

(15)    ((UIncreasing(200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1))), ≥)∧[(-1)bso_19] ≥ 0)

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

(16)    ((UIncreasing(200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1))), ≥)∧[(-1)bso_19] ≥ 0)

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

(17)    ((UIncreasing(200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1))), ≥)∧0 = 0∧[(-1)bso_19] ≥ 0)

For Pair COND_89_0_NEST_NE(TRUE, x0) → 89_0_NEST_NE(-(x0, 1)) the following chains were created:
• We consider the chain COND_89_0_NEST_NE(TRUE, x0[2]) → 89_0_NEST_NE(-(x0[2], 1)) which results in the following constraint:

(18)    (COND_89_0_NEST_NE(TRUE, x0[2])≥NonInfC∧COND_89_0_NEST_NE(TRUE, x0[2])≥89_0_NEST_NE(-(x0[2], 1))∧(UIncreasing(89_0_NEST_NE(-(x0[2], 1))), ≥))

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

(19)    ((UIncreasing(89_0_NEST_NE(-(x0[2], 1))), ≥)∧[(-1)bso_21] ≥ 0)

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

(20)    ((UIncreasing(89_0_NEST_NE(-(x0[2], 1))), ≥)∧[(-1)bso_21] ≥ 0)

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

(21)    ((UIncreasing(89_0_NEST_NE(-(x0[2], 1))), ≥)∧[(-1)bso_21] ≥ 0)

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

(22)    ((UIncreasing(89_0_NEST_NE(-(x0[2], 1))), ≥)∧0 = 0∧[(-1)bso_21] ≥ 0)

For Pair 200_1_NEST_INVOKEMETHOD(135_0_nest_Return, 0) → 89_0_NEST_NE(0) the following chains were created:
• We consider the chain 200_1_NEST_INVOKEMETHOD(135_0_nest_Return, 0) → 89_0_NEST_NE(0), 89_0_NEST_NE(x0[0]) → COND_89_0_NEST_NE(>(x0[0], 0), x0[0]) which results in the following constraint:

(23)    (0=x0[0]200_1_NEST_INVOKEMETHOD(135_0_nest_Return, 0)≥NonInfC∧200_1_NEST_INVOKEMETHOD(135_0_nest_Return, 0)≥89_0_NEST_NE(0)∧(UIncreasing(89_0_NEST_NE(0)), ≥))

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

(24)    (200_1_NEST_INVOKEMETHOD(135_0_nest_Return, 0)≥NonInfC∧200_1_NEST_INVOKEMETHOD(135_0_nest_Return, 0)≥89_0_NEST_NE(0)∧(UIncreasing(89_0_NEST_NE(0)), ≥))

We simplified constraint (24) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(25)    ((UIncreasing(89_0_NEST_NE(0)), ≥)∧[(-1)bso_23] ≥ 0)

We simplified constraint (25) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(26)    ((UIncreasing(89_0_NEST_NE(0)), ≥)∧[(-1)bso_23] ≥ 0)

We simplified constraint (26) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(27)    ((UIncreasing(89_0_NEST_NE(0)), ≥)∧[(-1)bso_23] ≥ 0)

For Pair 200_1_NEST_INVOKEMETHOD(320_0_nest_Return, x1) → 89_0_NEST_NE(0) the following chains were created:
• We consider the chain 200_1_NEST_INVOKEMETHOD(320_0_nest_Return, x1[4]) → 89_0_NEST_NE(0) which results in the following constraint:

(28)    (200_1_NEST_INVOKEMETHOD(320_0_nest_Return, x1[4])≥NonInfC∧200_1_NEST_INVOKEMETHOD(320_0_nest_Return, x1[4])≥89_0_NEST_NE(0)∧(UIncreasing(89_0_NEST_NE(0)), ≥))

We simplified constraint (28) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(29)    ((UIncreasing(89_0_NEST_NE(0)), ≥)∧[3 + (-1)bso_25] ≥ 0)

We simplified constraint (29) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(30)    ((UIncreasing(89_0_NEST_NE(0)), ≥)∧[3 + (-1)bso_25] ≥ 0)

We simplified constraint (30) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(31)    ((UIncreasing(89_0_NEST_NE(0)), ≥)∧[3 + (-1)bso_25] ≥ 0)

We simplified constraint (31) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(32)    ((UIncreasing(89_0_NEST_NE(0)), ≥)∧0 = 0∧[3 + (-1)bso_25] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 89_0_NEST_NE(x0) → COND_89_0_NEST_NE(>(x0, 0), x0)
• (x0[0] ≥ 0 ⇒ (UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] ≥ 0∧[(-1)bso_17] ≥ 0)
• (x0[0] ≥ 0 ⇒ (UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] ≥ 0∧[(-1)bso_17] ≥ 0)

• COND_89_0_NEST_NE(TRUE, x0) → 200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0, 1)), -(x0, 1))
• ((UIncreasing(200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1))), ≥)∧0 = 0∧[(-1)bso_19] ≥ 0)

• COND_89_0_NEST_NE(TRUE, x0) → 89_0_NEST_NE(-(x0, 1))
• ((UIncreasing(89_0_NEST_NE(-(x0[2], 1))), ≥)∧0 = 0∧[(-1)bso_21] ≥ 0)

• 200_1_NEST_INVOKEMETHOD(135_0_nest_Return, 0) → 89_0_NEST_NE(0)
• ((UIncreasing(89_0_NEST_NE(0)), ≥)∧[(-1)bso_23] ≥ 0)

• 200_1_NEST_INVOKEMETHOD(320_0_nest_Return, x1) → 89_0_NEST_NE(0)
• ((UIncreasing(89_0_NEST_NE(0)), ≥)∧0 = 0∧[3 + (-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(89_0_nest_NE(x1)) = [2]
POL(0) = 0
POL(135_0_nest_Return) = [2]
POL(242_1_nest_InvokeMethod(x1)) = [-1]
POL(320_0_nest_Return) = [-1]
POL(89_0_NEST_NE(x1)) = [-1]
POL(COND_89_0_NEST_NE(x1, x2)) = [-1]
POL(>(x1, x2)) = [-1]
POL(200_1_NEST_INVOKEMETHOD(x1, x2)) = [1] + [-1]x1
POL(-(x1, x2)) = x1 + [-1]x2
POL(1) = [1]

The following pairs are in P>:

200_1_NEST_INVOKEMETHOD(320_0_nest_Return, x1[4]) → 89_0_NEST_NE(0)

The following pairs are in Pbound:

89_0_NEST_NE(x0[0]) → COND_89_0_NEST_NE(>(x0[0], 0), x0[0])

The following pairs are in P:

89_0_NEST_NE(x0[0]) → COND_89_0_NEST_NE(>(x0[0], 0), x0[0])
COND_89_0_NEST_NE(TRUE, x0[1]) → 200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1))
COND_89_0_NEST_NE(TRUE, x0[2]) → 89_0_NEST_NE(-(x0[2], 1))
200_1_NEST_INVOKEMETHOD(135_0_nest_Return, 0) → 89_0_NEST_NE(0)

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

89_0_nest_NE(0)1135_0_nest_Return1

### (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

The ITRS R consists of the following rules:
89_0_nest_NE(0) → 135_0_nest_Return
242_1_nest_InvokeMethod(135_0_nest_Return) → 320_0_nest_Return

The integer pair graph contains the following rules and edges:
(0): 89_0_NEST_NE(x0[0]) → COND_89_0_NEST_NE(x0[0] > 0, x0[0])
(1): COND_89_0_NEST_NE(TRUE, x0[1]) → 200_1_NEST_INVOKEMETHOD(89_0_nest_NE(x0[1] - 1), x0[1] - 1)
(2): COND_89_0_NEST_NE(TRUE, x0[2]) → 89_0_NEST_NE(x0[2] - 1)
(3): 200_1_NEST_INVOKEMETHOD(135_0_nest_Return, 0) → 89_0_NEST_NE(0)

(2) -> (0), if ((x0[2] - 1* x0[0]))

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

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

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

(1) -> (3), if ((89_0_nest_NE(x0[1] - 1) →* 135_0_nest_Return)∧(x0[1] - 1* 0))

The set Q consists of the following terms:
89_0_nest_NE(0)
242_1_nest_InvokeMethod(135_0_nest_Return)

### (8) 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 89_0_NEST_NE(x0[0]) → COND_89_0_NEST_NE(>(x0[0], 0), x0[0]) the following chains were created:
• We consider the chain 89_0_NEST_NE(x0[0]) → COND_89_0_NEST_NE(>(x0[0], 0), x0[0]), COND_89_0_NEST_NE(TRUE, x0[1]) → 200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1)) which results in the following constraint:

(1)    (>(x0[0], 0)=TRUEx0[0]=x0[1]89_0_NEST_NE(x0[0])≥NonInfC∧89_0_NEST_NE(x0[0])≥COND_89_0_NEST_NE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥))

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

(2)    (>(x0[0], 0)=TRUE89_0_NEST_NE(x0[0])≥NonInfC∧89_0_NEST_NE(x0[0])≥COND_89_0_NEST_NE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥))

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

(3)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 0)

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

(4)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 0)

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

(5)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 0)

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

(6)    (x0[0] ≥ 0 ⇒ (UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 0)

• We consider the chain 89_0_NEST_NE(x0[0]) → COND_89_0_NEST_NE(>(x0[0], 0), x0[0]), COND_89_0_NEST_NE(TRUE, x0[2]) → 89_0_NEST_NE(-(x0[2], 1)) which results in the following constraint:

(7)    (>(x0[0], 0)=TRUEx0[0]=x0[2]89_0_NEST_NE(x0[0])≥NonInfC∧89_0_NEST_NE(x0[0])≥COND_89_0_NEST_NE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥))

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

(8)    (>(x0[0], 0)=TRUE89_0_NEST_NE(x0[0])≥NonInfC∧89_0_NEST_NE(x0[0])≥COND_89_0_NEST_NE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥))

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

(9)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 0)

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

(10)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 0)

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

(11)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 0)

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

(12)    (x0[0] ≥ 0 ⇒ (UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 0)

For Pair COND_89_0_NEST_NE(TRUE, x0[1]) → 200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1)) the following chains were created:
• We consider the chain COND_89_0_NEST_NE(TRUE, x0[1]) → 200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1)) which results in the following constraint:

(13)    (COND_89_0_NEST_NE(TRUE, x0[1])≥NonInfC∧COND_89_0_NEST_NE(TRUE, x0[1])≥200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1))∧(UIncreasing(200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1))), ≥))

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

(14)    ((UIncreasing(200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1))), ≥)∧[(-1)bso_14] ≥ 0)

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

(15)    ((UIncreasing(200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1))), ≥)∧[(-1)bso_14] ≥ 0)

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

(16)    ((UIncreasing(200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1))), ≥)∧[(-1)bso_14] ≥ 0)

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

(17)    ((UIncreasing(200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1))), ≥)∧0 = 0∧[(-1)bso_14] ≥ 0)

For Pair COND_89_0_NEST_NE(TRUE, x0[2]) → 89_0_NEST_NE(-(x0[2], 1)) the following chains were created:
• We consider the chain COND_89_0_NEST_NE(TRUE, x0[2]) → 89_0_NEST_NE(-(x0[2], 1)) which results in the following constraint:

(18)    (COND_89_0_NEST_NE(TRUE, x0[2])≥NonInfC∧COND_89_0_NEST_NE(TRUE, x0[2])≥89_0_NEST_NE(-(x0[2], 1))∧(UIncreasing(89_0_NEST_NE(-(x0[2], 1))), ≥))

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

(19)    ((UIncreasing(89_0_NEST_NE(-(x0[2], 1))), ≥)∧[1 + (-1)bso_16] ≥ 0)

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

(20)    ((UIncreasing(89_0_NEST_NE(-(x0[2], 1))), ≥)∧[1 + (-1)bso_16] ≥ 0)

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

(21)    ((UIncreasing(89_0_NEST_NE(-(x0[2], 1))), ≥)∧[1 + (-1)bso_16] ≥ 0)

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

(22)    ((UIncreasing(89_0_NEST_NE(-(x0[2], 1))), ≥)∧0 = 0∧[1 + (-1)bso_16] ≥ 0)

For Pair 200_1_NEST_INVOKEMETHOD(135_0_nest_Return, 0) → 89_0_NEST_NE(0) the following chains were created:
• We consider the chain 200_1_NEST_INVOKEMETHOD(135_0_nest_Return, 0) → 89_0_NEST_NE(0), 89_0_NEST_NE(x0[0]) → COND_89_0_NEST_NE(>(x0[0], 0), x0[0]) which results in the following constraint:

(23)    (0=x0[0]200_1_NEST_INVOKEMETHOD(135_0_nest_Return, 0)≥NonInfC∧200_1_NEST_INVOKEMETHOD(135_0_nest_Return, 0)≥89_0_NEST_NE(0)∧(UIncreasing(89_0_NEST_NE(0)), ≥))

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

(24)    (200_1_NEST_INVOKEMETHOD(135_0_nest_Return, 0)≥NonInfC∧200_1_NEST_INVOKEMETHOD(135_0_nest_Return, 0)≥89_0_NEST_NE(0)∧(UIncreasing(89_0_NEST_NE(0)), ≥))

We simplified constraint (24) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(25)    ((UIncreasing(89_0_NEST_NE(0)), ≥)∧[1 + (-1)bso_18] ≥ 0)

We simplified constraint (25) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(26)    ((UIncreasing(89_0_NEST_NE(0)), ≥)∧[1 + (-1)bso_18] ≥ 0)

We simplified constraint (26) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(27)    ((UIncreasing(89_0_NEST_NE(0)), ≥)∧[1 + (-1)bso_18] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 89_0_NEST_NE(x0[0]) → COND_89_0_NEST_NE(>(x0[0], 0), x0[0])
• (x0[0] ≥ 0 ⇒ (UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 0)
• (x0[0] ≥ 0 ⇒ (UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 0)

• COND_89_0_NEST_NE(TRUE, x0[1]) → 200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1))
• ((UIncreasing(200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1))), ≥)∧0 = 0∧[(-1)bso_14] ≥ 0)

• COND_89_0_NEST_NE(TRUE, x0[2]) → 89_0_NEST_NE(-(x0[2], 1))
• ((UIncreasing(89_0_NEST_NE(-(x0[2], 1))), ≥)∧0 = 0∧[1 + (-1)bso_16] ≥ 0)

• 200_1_NEST_INVOKEMETHOD(135_0_nest_Return, 0) → 89_0_NEST_NE(0)
• ((UIncreasing(89_0_NEST_NE(0)), ≥)∧[1 + (-1)bso_18] ≥ 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(89_0_nest_NE(x1)) = [-1] + [-1]x1
POL(0) = 0
POL(135_0_nest_Return) = [-1]
POL(242_1_nest_InvokeMethod(x1)) = [-1]
POL(320_0_nest_Return) = [-1]
POL(89_0_NEST_NE(x1)) = [-1] + x1
POL(COND_89_0_NEST_NE(x1, x2)) = [-1] + x2
POL(>(x1, x2)) = [1]
POL(200_1_NEST_INVOKEMETHOD(x1, x2)) = [-1] + [-1]x1
POL(-(x1, x2)) = x1 + [-1]x2
POL(1) = [1]

The following pairs are in P>:

COND_89_0_NEST_NE(TRUE, x0[2]) → 89_0_NEST_NE(-(x0[2], 1))
200_1_NEST_INVOKEMETHOD(135_0_nest_Return, 0) → 89_0_NEST_NE(0)

The following pairs are in Pbound:

89_0_NEST_NE(x0[0]) → COND_89_0_NEST_NE(>(x0[0], 0), x0[0])

The following pairs are in P:

89_0_NEST_NE(x0[0]) → COND_89_0_NEST_NE(>(x0[0], 0), x0[0])
COND_89_0_NEST_NE(TRUE, x0[1]) → 200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1))

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

89_0_nest_NE(0)1135_0_nest_Return1

### (10) Obligation:

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

The following domains are used:

Integer

The ITRS R consists of the following rules:
89_0_nest_NE(0) → 135_0_nest_Return
242_1_nest_InvokeMethod(135_0_nest_Return) → 320_0_nest_Return

The integer pair graph contains the following rules and edges:
(0): 89_0_NEST_NE(x0[0]) → COND_89_0_NEST_NE(x0[0] > 0, x0[0])
(1): COND_89_0_NEST_NE(TRUE, x0[1]) → 200_1_NEST_INVOKEMETHOD(89_0_nest_NE(x0[1] - 1), x0[1] - 1)

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

The set Q consists of the following terms:
89_0_nest_NE(0)
242_1_nest_InvokeMethod(135_0_nest_Return)

### (11) IDependencyGraphProof (EQUIVALENT transformation)

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

### (13) 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:
89_0_nest_NE(0) → 135_0_nest_Return
242_1_nest_InvokeMethod(135_0_nest_Return) → 320_0_nest_Return

The integer pair graph contains the following rules and edges:
(1): COND_89_0_NEST_NE(TRUE, x0[1]) → 200_1_NEST_INVOKEMETHOD(89_0_nest_NE(x0[1] - 1), x0[1] - 1)
(2): COND_89_0_NEST_NE(TRUE, x0[2]) → 89_0_NEST_NE(x0[2] - 1)
(3): 200_1_NEST_INVOKEMETHOD(135_0_nest_Return, 0) → 89_0_NEST_NE(0)

(1) -> (3), if ((89_0_nest_NE(x0[1] - 1) →* 135_0_nest_Return)∧(x0[1] - 1* 0))

The set Q consists of the following terms:
89_0_nest_NE(0)
242_1_nest_InvokeMethod(135_0_nest_Return)

### (14) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.

### (16) 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:
89_0_nest_NE(0) → 135_0_nest_Return
242_1_nest_InvokeMethod(135_0_nest_Return) → 320_0_nest_Return

The integer pair graph contains the following rules and edges:
(1): COND_89_0_NEST_NE(TRUE, x0[1]) → 200_1_NEST_INVOKEMETHOD(89_0_nest_NE(x0[1] - 1), x0[1] - 1)
(2): COND_89_0_NEST_NE(TRUE, x0[2]) → 89_0_NEST_NE(x0[2] - 1)
(3): 200_1_NEST_INVOKEMETHOD(135_0_nest_Return, 0) → 89_0_NEST_NE(0)
(4): 200_1_NEST_INVOKEMETHOD(320_0_nest_Return, x1[4]) → 89_0_NEST_NE(0)

(1) -> (3), if ((89_0_nest_NE(x0[1] - 1) →* 135_0_nest_Return)∧(x0[1] - 1* 0))

(1) -> (4), if ((89_0_nest_NE(x0[1] - 1) →* 320_0_nest_Return)∧(x0[1] - 1* x1[4]))

The set Q consists of the following terms:
89_0_nest_NE(0)
242_1_nest_InvokeMethod(135_0_nest_Return)

### (17) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 4 less nodes.