(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.

(6) Complex Obligation (AND)

(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.

(9) TRUE

(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.

(12) TRUE