### (0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_20 (Apple Inc.) Main-Class: FactSLR
`public class FactSLR {    public static int factorial(int n){	if (n  < 1) return 1;	else return n*factorial(n-1);	      }    public static int doSum(List x){	if (x==null) return 0;	else return factorial(x.head) + doSum(x.tail);	    }    public static void main(String [] args) {	Random.args = args;	List l = List.mk(3*Random.random());	//System.out.println(doSum(l));    }}public class List {    public int head;    public List tail;    public List(int head, List tail) {	this.head = head;	this.tail = tail;    }    public List getTail() {	return tail;    }    public static List mk(int len) {	List result = null;	while (len-- > 0)	    result = new List(Random.random(), result);	return result;    }}public class Random {  static String[] args;  static int index = 0;  public static int random() {      if (index >= args.length)	  return 0;      String string = args[index];      index++;      return string.length();  }}`

### (1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

### (2) Obligation:

FIGraph based on JBC Program:
FactSLR.main([Ljava/lang/String;)V: Graph of 93 nodes with 0 SCCs.

List.mk(I)LList;: Graph of 118 nodes with 1 SCC.

### (3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 61 rules for P and 49 rules for R.

Combined rules. Obtained 6 rules for P and 0 rules for R.

Filtered ground terms:

1303_0_mk_Inc(x1, x2, x3, x4) → 1303_0_mk_Inc(x2, x3, x4)
List(x1) → List
Cond_1329_1_mk_InvokeMethod(x1, x2, x3, x4, x5, x6) → Cond_1329_1_mk_InvokeMethod(x1, x2, x3, x4)
1329_0_random_LT(x1, x2, x3) → 1329_0_random_LT(x2, x3)
1329_1_mk_InvokeMethod(x1, x2, x3, x4, x5) → 1329_1_mk_InvokeMethod(x1, x2, x3)
Cond_1358_1_mk_InvokeMethod(x1, x2, x3, x4, x5, x6) → Cond_1358_1_mk_InvokeMethod(x1, x2, x3, x4)
1358_0_random_IntArithmetic(x1, x2, x3, x4) → 1358_0_random_IntArithmetic(x2, x3)
1358_1_mk_InvokeMethod(x1, x2, x3, x4, x5) → 1358_1_mk_InvokeMethod(x1, x2, x3)
Cond_1340_1_mk_InvokeMethod(x1, x2, x3, x4, x5, x6) → Cond_1340_1_mk_InvokeMethod(x1, x2, x3, x4)
1340_0_random_ArrayAccess(x1, x2, x3) → 1340_0_random_ArrayAccess(x2, x3)
1340_1_mk_InvokeMethod(x1, x2, x3, x4, x5) → 1340_1_mk_InvokeMethod(x1, x2, x3)
Cond_1328_1_mk_InvokeMethod(x1, x2, x3, x4, x5, x6) → Cond_1328_1_mk_InvokeMethod(x1, x2, x3, x4)
1328_0_random_LT(x1, x2, x3) → 1328_0_random_LT(x2, x3)
1328_1_mk_InvokeMethod(x1, x2, x3, x4, x5) → 1328_1_mk_InvokeMethod(x1, x2, x3)
Cond_1303_0_mk_Inc1(x1, x2, x3, x4, x5) → Cond_1303_0_mk_Inc1(x1, x3, x4, x5)
Cond_1303_0_mk_Inc(x1, x2, x3, x4, x5) → Cond_1303_0_mk_Inc(x1, x3, x4, x5)

Filtered duplicate args:

1303_0_mk_Inc(x1, x2, x3) → 1303_0_mk_Inc(x2, x3)
Cond_1303_0_mk_Inc1(x1, x2, x3, x4) → Cond_1303_0_mk_Inc1(x1, x3, x4)
Cond_1303_0_mk_Inc(x1, x2, x3, x4) → Cond_1303_0_mk_Inc(x1, x3, x4)

Filtered unneeded arguments:

1303_0_mk_Inc(x1, x2) → 1303_0_mk_Inc(x2)
Cond_1303_0_mk_Inc(x1, x2, x3) → Cond_1303_0_mk_Inc(x1, x3)
Cond_1303_0_mk_Inc1(x1, x2, x3) → Cond_1303_0_mk_Inc1(x1, x3)
1328_1_mk_InvokeMethod(x1, x2, x3) → 1328_1_mk_InvokeMethod(x1, x2)
Cond_1328_1_mk_InvokeMethod(x1, x2, x3, x4) → Cond_1328_1_mk_InvokeMethod(x1, x2, x3)
1340_1_mk_InvokeMethod(x1, x2, x3) → 1340_1_mk_InvokeMethod(x1, x2)
Cond_1340_1_mk_InvokeMethod(x1, x2, x3, x4) → Cond_1340_1_mk_InvokeMethod(x1, x2, x3)
1358_1_mk_InvokeMethod(x1, x2, x3) → 1358_1_mk_InvokeMethod(x1, x2)
Cond_1358_1_mk_InvokeMethod(x1, x2, x3, x4) → Cond_1358_1_mk_InvokeMethod(x1, x2, x3)
1329_1_mk_InvokeMethod(x1, x2, x3) → 1329_1_mk_InvokeMethod(x1, x2)
Cond_1329_1_mk_InvokeMethod(x1, x2, x3, x4) → Cond_1329_1_mk_InvokeMethod(x1, x2, x3)

Filtered all free variables:

1328_1_mk_InvokeMethod(x1, x2) → 1328_1_mk_InvokeMethod(x2)
1329_1_mk_InvokeMethod(x1, x2) → 1329_1_mk_InvokeMethod(x2)
Cond_1328_1_mk_InvokeMethod(x1, x2, x3) → Cond_1328_1_mk_InvokeMethod(x1, x3)
1340_1_mk_InvokeMethod(x1, x2) → 1340_1_mk_InvokeMethod(x2)
Cond_1340_1_mk_InvokeMethod(x1, x2, x3) → Cond_1340_1_mk_InvokeMethod(x1, x3)
1358_1_mk_InvokeMethod(x1, x2) → 1358_1_mk_InvokeMethod(x2)
Cond_1358_1_mk_InvokeMethod(x1, x2, x3) → Cond_1358_1_mk_InvokeMethod(x1, x3)
Cond_1329_1_mk_InvokeMethod(x1, x2, x3) → Cond_1329_1_mk_InvokeMethod(x1, x3)

Combined rules. Obtained 3 rules for P and 0 rules for R.

Finished conversion. Obtained 3 rules for P and 0 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

R is empty.

The integer pair graph contains the following rules and edges:
(0): 1358_1_MK_INVOKEMETHOD(x4[0]) → 1303_0_MK_INC(x4[0])
(1): 1303_0_MK_INC(x0[1]) → COND_1303_0_MK_INC(x0[1] > 0, x0[1])
(2): COND_1303_0_MK_INC(TRUE, x0[2]) → 1303_0_MK_INC(x0[2] + -1)
(3): 1303_0_MK_INC(x0[3]) → COND_1303_0_MK_INC1(x0[3] > 0, x0[3])
(4): COND_1303_0_MK_INC1(TRUE, x0[4]) → 1358_1_MK_INVOKEMETHOD(x0[4] + -1)

(0) -> (1), if ((x4[0]* x0[1]))

(0) -> (3), if ((x4[0]* x0[3]))

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

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

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

(3) -> (4), if ((x0[3] > 0* TRUE)∧(x0[3]* x0[4]))

(4) -> (0), if ((x0[4] + -1* x4[0]))

The set Q is empty.

### (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 1358_1_MK_INVOKEMETHOD(x4) → 1303_0_MK_INC(x4) the following chains were created:
• We consider the chain 1358_1_MK_INVOKEMETHOD(x4[0]) → 1303_0_MK_INC(x4[0]), 1303_0_MK_INC(x0[1]) → COND_1303_0_MK_INC(>(x0[1], 0), x0[1]) which results in the following constraint:

(1)    (x4[0]=x0[1]1358_1_MK_INVOKEMETHOD(x4[0])≥NonInfC∧1358_1_MK_INVOKEMETHOD(x4[0])≥1303_0_MK_INC(x4[0])∧(UIncreasing(1303_0_MK_INC(x4[0])), ≥))

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

(2)    (1358_1_MK_INVOKEMETHOD(x4[0])≥NonInfC∧1358_1_MK_INVOKEMETHOD(x4[0])≥1303_0_MK_INC(x4[0])∧(UIncreasing(1303_0_MK_INC(x4[0])), ≥))

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

(3)    ((UIncreasing(1303_0_MK_INC(x4[0])), ≥)∧[(-1)bso_13] ≥ 0)

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

(4)    ((UIncreasing(1303_0_MK_INC(x4[0])), ≥)∧[(-1)bso_13] ≥ 0)

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

(5)    ((UIncreasing(1303_0_MK_INC(x4[0])), ≥)∧[(-1)bso_13] ≥ 0)

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

(6)    ((UIncreasing(1303_0_MK_INC(x4[0])), ≥)∧0 = 0∧[(-1)bso_13] ≥ 0)

• We consider the chain 1358_1_MK_INVOKEMETHOD(x4[0]) → 1303_0_MK_INC(x4[0]), 1303_0_MK_INC(x0[3]) → COND_1303_0_MK_INC1(>(x0[3], 0), x0[3]) which results in the following constraint:

(7)    (x4[0]=x0[3]1358_1_MK_INVOKEMETHOD(x4[0])≥NonInfC∧1358_1_MK_INVOKEMETHOD(x4[0])≥1303_0_MK_INC(x4[0])∧(UIncreasing(1303_0_MK_INC(x4[0])), ≥))

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

(8)    (1358_1_MK_INVOKEMETHOD(x4[0])≥NonInfC∧1358_1_MK_INVOKEMETHOD(x4[0])≥1303_0_MK_INC(x4[0])∧(UIncreasing(1303_0_MK_INC(x4[0])), ≥))

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

(9)    ((UIncreasing(1303_0_MK_INC(x4[0])), ≥)∧[(-1)bso_13] ≥ 0)

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

(10)    ((UIncreasing(1303_0_MK_INC(x4[0])), ≥)∧[(-1)bso_13] ≥ 0)

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

(11)    ((UIncreasing(1303_0_MK_INC(x4[0])), ≥)∧[(-1)bso_13] ≥ 0)

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

(12)    ((UIncreasing(1303_0_MK_INC(x4[0])), ≥)∧0 = 0∧[(-1)bso_13] ≥ 0)

For Pair 1303_0_MK_INC(x0) → COND_1303_0_MK_INC(>(x0, 0), x0) the following chains were created:
• We consider the chain 1303_0_MK_INC(x0[1]) → COND_1303_0_MK_INC(>(x0[1], 0), x0[1]), COND_1303_0_MK_INC(TRUE, x0[2]) → 1303_0_MK_INC(+(x0[2], -1)) which results in the following constraint:

(13)    (>(x0[1], 0)=TRUEx0[1]=x0[2]1303_0_MK_INC(x0[1])≥NonInfC∧1303_0_MK_INC(x0[1])≥COND_1303_0_MK_INC(>(x0[1], 0), x0[1])∧(UIncreasing(COND_1303_0_MK_INC(>(x0[1], 0), x0[1])), ≥))

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

(14)    (>(x0[1], 0)=TRUE1303_0_MK_INC(x0[1])≥NonInfC∧1303_0_MK_INC(x0[1])≥COND_1303_0_MK_INC(>(x0[1], 0), x0[1])∧(UIncreasing(COND_1303_0_MK_INC(>(x0[1], 0), x0[1])), ≥))

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

(15)    (x0[1] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1303_0_MK_INC(>(x0[1], 0), x0[1])), ≥)∧[(-1)Bound*bni_14] + [(2)bni_14]x0[1] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

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

(16)    (x0[1] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1303_0_MK_INC(>(x0[1], 0), x0[1])), ≥)∧[(-1)Bound*bni_14] + [(2)bni_14]x0[1] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

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

(17)    (x0[1] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1303_0_MK_INC(>(x0[1], 0), x0[1])), ≥)∧[(-1)Bound*bni_14] + [(2)bni_14]x0[1] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

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

(18)    (x0[1] ≥ 0 ⇒ (UIncreasing(COND_1303_0_MK_INC(>(x0[1], 0), x0[1])), ≥)∧[(-1)Bound*bni_14 + (2)bni_14] + [(2)bni_14]x0[1] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

For Pair COND_1303_0_MK_INC(TRUE, x0) → 1303_0_MK_INC(+(x0, -1)) the following chains were created:
• We consider the chain COND_1303_0_MK_INC(TRUE, x0[2]) → 1303_0_MK_INC(+(x0[2], -1)) which results in the following constraint:

(19)    (COND_1303_0_MK_INC(TRUE, x0[2])≥NonInfC∧COND_1303_0_MK_INC(TRUE, x0[2])≥1303_0_MK_INC(+(x0[2], -1))∧(UIncreasing(1303_0_MK_INC(+(x0[2], -1))), ≥))

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

(20)    ((UIncreasing(1303_0_MK_INC(+(x0[2], -1))), ≥)∧[1 + (-1)bso_17] ≥ 0)

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

(21)    ((UIncreasing(1303_0_MK_INC(+(x0[2], -1))), ≥)∧[1 + (-1)bso_17] ≥ 0)

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

(22)    ((UIncreasing(1303_0_MK_INC(+(x0[2], -1))), ≥)∧[1 + (-1)bso_17] ≥ 0)

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

(23)    ((UIncreasing(1303_0_MK_INC(+(x0[2], -1))), ≥)∧0 = 0∧[1 + (-1)bso_17] ≥ 0)

For Pair 1303_0_MK_INC(x0) → COND_1303_0_MK_INC1(>(x0, 0), x0) the following chains were created:
• We consider the chain 1303_0_MK_INC(x0[3]) → COND_1303_0_MK_INC1(>(x0[3], 0), x0[3]), COND_1303_0_MK_INC1(TRUE, x0[4]) → 1358_1_MK_INVOKEMETHOD(+(x0[4], -1)) which results in the following constraint:

(24)    (>(x0[3], 0)=TRUEx0[3]=x0[4]1303_0_MK_INC(x0[3])≥NonInfC∧1303_0_MK_INC(x0[3])≥COND_1303_0_MK_INC1(>(x0[3], 0), x0[3])∧(UIncreasing(COND_1303_0_MK_INC1(>(x0[3], 0), x0[3])), ≥))

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

(25)    (>(x0[3], 0)=TRUE1303_0_MK_INC(x0[3])≥NonInfC∧1303_0_MK_INC(x0[3])≥COND_1303_0_MK_INC1(>(x0[3], 0), x0[3])∧(UIncreasing(COND_1303_0_MK_INC1(>(x0[3], 0), x0[3])), ≥))

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

(26)    (x0[3] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1303_0_MK_INC1(>(x0[3], 0), x0[3])), ≥)∧[(-1)Bound*bni_18] + [(2)bni_18]x0[3] ≥ 0∧[(-1)bso_19] ≥ 0)

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

(27)    (x0[3] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1303_0_MK_INC1(>(x0[3], 0), x0[3])), ≥)∧[(-1)Bound*bni_18] + [(2)bni_18]x0[3] ≥ 0∧[(-1)bso_19] ≥ 0)

We simplified constraint (27) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(28)    (x0[3] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1303_0_MK_INC1(>(x0[3], 0), x0[3])), ≥)∧[(-1)Bound*bni_18] + [(2)bni_18]x0[3] ≥ 0∧[(-1)bso_19] ≥ 0)

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

(29)    (x0[3] ≥ 0 ⇒ (UIncreasing(COND_1303_0_MK_INC1(>(x0[3], 0), x0[3])), ≥)∧[(-1)Bound*bni_18 + (2)bni_18] + [(2)bni_18]x0[3] ≥ 0∧[(-1)bso_19] ≥ 0)

For Pair COND_1303_0_MK_INC1(TRUE, x0) → 1358_1_MK_INVOKEMETHOD(+(x0, -1)) the following chains were created:
• We consider the chain COND_1303_0_MK_INC1(TRUE, x0[4]) → 1358_1_MK_INVOKEMETHOD(+(x0[4], -1)) which results in the following constraint:

(30)    (COND_1303_0_MK_INC1(TRUE, x0[4])≥NonInfC∧COND_1303_0_MK_INC1(TRUE, x0[4])≥1358_1_MK_INVOKEMETHOD(+(x0[4], -1))∧(UIncreasing(1358_1_MK_INVOKEMETHOD(+(x0[4], -1))), ≥))

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

(31)    ((UIncreasing(1358_1_MK_INVOKEMETHOD(+(x0[4], -1))), ≥)∧[2 + (-1)bso_21] ≥ 0)

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

(32)    ((UIncreasing(1358_1_MK_INVOKEMETHOD(+(x0[4], -1))), ≥)∧[2 + (-1)bso_21] ≥ 0)

We simplified constraint (32) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(33)    ((UIncreasing(1358_1_MK_INVOKEMETHOD(+(x0[4], -1))), ≥)∧[2 + (-1)bso_21] ≥ 0)

We simplified constraint (33) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(34)    ((UIncreasing(1358_1_MK_INVOKEMETHOD(+(x0[4], -1))), ≥)∧0 = 0∧[2 + (-1)bso_21] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 1358_1_MK_INVOKEMETHOD(x4) → 1303_0_MK_INC(x4)
• ((UIncreasing(1303_0_MK_INC(x4[0])), ≥)∧0 = 0∧[(-1)bso_13] ≥ 0)
• ((UIncreasing(1303_0_MK_INC(x4[0])), ≥)∧0 = 0∧[(-1)bso_13] ≥ 0)

• 1303_0_MK_INC(x0) → COND_1303_0_MK_INC(>(x0, 0), x0)
• (x0[1] ≥ 0 ⇒ (UIncreasing(COND_1303_0_MK_INC(>(x0[1], 0), x0[1])), ≥)∧[(-1)Bound*bni_14 + (2)bni_14] + [(2)bni_14]x0[1] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

• COND_1303_0_MK_INC(TRUE, x0) → 1303_0_MK_INC(+(x0, -1))
• ((UIncreasing(1303_0_MK_INC(+(x0[2], -1))), ≥)∧0 = 0∧[1 + (-1)bso_17] ≥ 0)

• 1303_0_MK_INC(x0) → COND_1303_0_MK_INC1(>(x0, 0), x0)
• (x0[3] ≥ 0 ⇒ (UIncreasing(COND_1303_0_MK_INC1(>(x0[3], 0), x0[3])), ≥)∧[(-1)Bound*bni_18 + (2)bni_18] + [(2)bni_18]x0[3] ≥ 0∧[(-1)bso_19] ≥ 0)

• COND_1303_0_MK_INC1(TRUE, x0) → 1358_1_MK_INVOKEMETHOD(+(x0, -1))
• ((UIncreasing(1358_1_MK_INVOKEMETHOD(+(x0[4], -1))), ≥)∧0 = 0∧[2 + (-1)bso_21] ≥ 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(1358_1_MK_INVOKEMETHOD(x1)) = [2]x1
POL(1303_0_MK_INC(x1)) = [2]x1
POL(COND_1303_0_MK_INC(x1, x2)) = [-1] + [2]x2
POL(>(x1, x2)) = [-1]
POL(0) = 0
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]
POL(COND_1303_0_MK_INC1(x1, x2)) = [2]x2

The following pairs are in P>:

1303_0_MK_INC(x0[1]) → COND_1303_0_MK_INC(>(x0[1], 0), x0[1])
COND_1303_0_MK_INC(TRUE, x0[2]) → 1303_0_MK_INC(+(x0[2], -1))
COND_1303_0_MK_INC1(TRUE, x0[4]) → 1358_1_MK_INVOKEMETHOD(+(x0[4], -1))

The following pairs are in Pbound:

1303_0_MK_INC(x0[1]) → COND_1303_0_MK_INC(>(x0[1], 0), x0[1])
1303_0_MK_INC(x0[3]) → COND_1303_0_MK_INC1(>(x0[3], 0), x0[3])

The following pairs are in P:

1358_1_MK_INVOKEMETHOD(x4[0]) → 1303_0_MK_INC(x4[0])
1303_0_MK_INC(x0[3]) → COND_1303_0_MK_INC1(>(x0[3], 0), x0[3])

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

R is empty.

The integer pair graph contains the following rules and edges:
(0): 1358_1_MK_INVOKEMETHOD(x4[0]) → 1303_0_MK_INC(x4[0])
(3): 1303_0_MK_INC(x0[3]) → COND_1303_0_MK_INC1(x0[3] > 0, x0[3])

(0) -> (3), if ((x4[0]* x0[3]))

The set Q is empty.

### (8) IDependencyGraphProof (EQUIVALENT transformation)

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

### (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

R is empty.

The integer pair graph contains the following rules and edges:
(0): 1358_1_MK_INVOKEMETHOD(x4[0]) → 1303_0_MK_INC(x4[0])
(2): COND_1303_0_MK_INC(TRUE, x0[2]) → 1303_0_MK_INC(x0[2] + -1)
(4): COND_1303_0_MK_INC1(TRUE, x0[4]) → 1358_1_MK_INVOKEMETHOD(x0[4] + -1)

(4) -> (0), if ((x0[4] + -1* x4[0]))

The set Q is empty.

### (11) IDependencyGraphProof (EQUIVALENT transformation)

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