Termination proof of /tmp/tmpTVp-dk/FibSLR.jar

Termination of the given JBC could be proven:

0 JBC

↳1 JBC2FIG (⇒)

↳2 JBCTerminationGraph

↳3 FIGtoITRSProof (⇒)

↳4 IDP

↳5 IDPNonInfProof (⇒)

↳6 AND

    ↳7 IDP

        ↳8 IDependencyGraphProof (⇔)

        ↳9 TRUE

    ↳10 IDP

        ↳11 IDependencyGraphProof (⇔)

        ↳12 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_20 (Apple Inc.) Main-Class: FibSLR

public class FibSLR {

    public static int fib(int n){
	if (n  < 2) return 1;
	else return fib(n-1) + fib(n-2);	  
    }

    public static int doSum(List x){
	if (x==null) return 1;
	else return fib(x.head) + doSum(x.tail);	
    }

    public static void main(String [] args) {
	Random.args = args;
	List l = List.mk(Random.random()*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:
FibSLR.main([Ljava/lang/String;)V: Graph of 165 nodes with 0 SCCs.

List.mk(I)LList;: Graph of 117 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:

1260_0_mk_Inc(x1, x2, x3, x4) → 1260_0_mk_Inc(x2, x3, x4)
List(x1) → List
Cond_1316_1_mk_InvokeMethod(x1, x2, x3, x4, x5, x6) → Cond_1316_1_mk_InvokeMethod(x1, x2, x3, x4)
1316_0_random_LT(x1, x2, x3) → 1316_0_random_LT(x2, x3)
1316_1_mk_InvokeMethod(x1, x2, x3, x4, x5) → 1316_1_mk_InvokeMethod(x1, x2, x3)
Cond_1345_1_mk_InvokeMethod(x1, x2, x3, x4, x5, x6) → Cond_1345_1_mk_InvokeMethod(x1, x2, x3, x4)
1345_0_random_IntArithmetic(x1, x2, x3, x4) → 1345_0_random_IntArithmetic(x2, x3)
1345_1_mk_InvokeMethod(x1, x2, x3, x4, x5) → 1345_1_mk_InvokeMethod(x1, x2, x3)
Cond_1327_1_mk_InvokeMethod(x1, x2, x3, x4, x5, x6) → Cond_1327_1_mk_InvokeMethod(x1, x2, x3, x4)
1327_0_random_ArrayAccess(x1, x2, x3) → 1327_0_random_ArrayAccess(x2, x3)
1327_1_mk_InvokeMethod(x1, x2, x3, x4, x5) → 1327_1_mk_InvokeMethod(x1, x2, x3)
Cond_1315_1_mk_InvokeMethod(x1, x2, x3, x4, x5, x6) → Cond_1315_1_mk_InvokeMethod(x1, x2, x3, x4)
1315_0_random_LT(x1, x2, x3) → 1315_0_random_LT(x2, x3)
1315_1_mk_InvokeMethod(x1, x2, x3, x4, x5) → 1315_1_mk_InvokeMethod(x1, x2, x3)
Cond_1260_0_mk_Inc1(x1, x2, x3, x4, x5) → Cond_1260_0_mk_Inc1(x1, x3, x4, x5)
Cond_1260_0_mk_Inc(x1, x2, x3, x4, x5) → Cond_1260_0_mk_Inc(x1, x3, x4, x5)

Filtered duplicate args:

1260_0_mk_Inc(x1, x2, x3) → 1260_0_mk_Inc(x2, x3)
Cond_1260_0_mk_Inc1(x1, x2, x3, x4) → Cond_1260_0_mk_Inc1(x1, x3, x4)
Cond_1260_0_mk_Inc(x1, x2, x3, x4) → Cond_1260_0_mk_Inc(x1, x3, x4)

Filtered unneeded arguments:

1260_0_mk_Inc(x1, x2) → 1260_0_mk_Inc(x2)
Cond_1260_0_mk_Inc(x1, x2, x3) → Cond_1260_0_mk_Inc(x1, x3)
Cond_1260_0_mk_Inc1(x1, x2, x3) → Cond_1260_0_mk_Inc1(x1, x3)
1315_1_mk_InvokeMethod(x1, x2, x3) → 1315_1_mk_InvokeMethod(x1, x2)
Cond_1315_1_mk_InvokeMethod(x1, x2, x3, x4) → Cond_1315_1_mk_InvokeMethod(x1, x2, x3)
1327_1_mk_InvokeMethod(x1, x2, x3) → 1327_1_mk_InvokeMethod(x1, x2)
Cond_1327_1_mk_InvokeMethod(x1, x2, x3, x4) → Cond_1327_1_mk_InvokeMethod(x1, x2, x3)
1345_1_mk_InvokeMethod(x1, x2, x3) → 1345_1_mk_InvokeMethod(x1, x2)
Cond_1345_1_mk_InvokeMethod(x1, x2, x3, x4) → Cond_1345_1_mk_InvokeMethod(x1, x2, x3)
1316_1_mk_InvokeMethod(x1, x2, x3) → 1316_1_mk_InvokeMethod(x1, x2)
Cond_1316_1_mk_InvokeMethod(x1, x2, x3, x4) → Cond_1316_1_mk_InvokeMethod(x1, x2, x3)

Filtered all free variables:

1315_1_mk_InvokeMethod(x1, x2) → 1315_1_mk_InvokeMethod(x2)
1316_1_mk_InvokeMethod(x1, x2) → 1316_1_mk_InvokeMethod(x2)
Cond_1315_1_mk_InvokeMethod(x1, x2, x3) → Cond_1315_1_mk_InvokeMethod(x1, x3)
1327_1_mk_InvokeMethod(x1, x2) → 1327_1_mk_InvokeMethod(x2)
Cond_1327_1_mk_InvokeMethod(x1, x2, x3) → Cond_1327_1_mk_InvokeMethod(x1, x3)
1345_1_mk_InvokeMethod(x1, x2) → 1345_1_mk_InvokeMethod(x2)
Cond_1345_1_mk_InvokeMethod(x1, x2, x3) → Cond_1345_1_mk_InvokeMethod(x1, x3)
Cond_1316_1_mk_InvokeMethod(x1, x2, x3) → Cond_1316_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): 1345_1_MK_INVOKEMETHOD(x4[0]) → 1260_0_MK_INC(x4[0])
(1): 1260_0_MK_INC(x0[1]) → COND_1260_0_MK_INC(x0[1] > 0, x0[1])
(2): COND_1260_0_MK_INC(TRUE, x0[2]) → 1260_0_MK_INC(x0[2] + -1)
(3): 1260_0_MK_INC(x0[3]) → COND_1260_0_MK_INC1(x0[3] > 0, x0[3])
(4): COND_1260_0_MK_INC1(TRUE, x0[4]) → 1345_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 1345_1_MK_INVOKEMETHOD(x4) → 1260_0_MK_INC(x4) the following chains were created:

We consider the chain 1345_1_MK_INVOKEMETHOD(x4[0]) → 1260_0_MK_INC(x4[0]), 1260_0_MK_INC(x0[1]) → COND_1260_0_MK_INC(>(x0[1], 0), x0[1]) which results in the following constraint:
(1)    (x4[0]=x0[1] ⇒ 1345_1_MK_INVOKEMETHOD(x4[0])≥NonInfC∧1345_1_MK_INVOKEMETHOD(x4[0])≥1260_0_MK_INC(x4[0])∧(U^Increasing(1260_0_MK_INC(x4[0])), ≥))

We simplified constraint (1) using rule (IV) which results in the following new constraint:
(2)    (1345_1_MK_INVOKEMETHOD(x4[0])≥NonInfC∧1345_1_MK_INVOKEMETHOD(x4[0])≥1260_0_MK_INC(x4[0])∧(U^Increasing(1260_0_MK_INC(x4[0])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    ((U^Increasing(1260_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)    ((U^Increasing(1260_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)    ((U^Increasing(1260_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)    ((U^Increasing(1260_0_MK_INC(x4[0])), ≥)∧0 = 0∧[(-1)bso_13] ≥ 0)
We consider the chain 1345_1_MK_INVOKEMETHOD(x4[0]) → 1260_0_MK_INC(x4[0]), 1260_0_MK_INC(x0[3]) → COND_1260_0_MK_INC1(>(x0[3], 0), x0[3]) which results in the following constraint:
(7)    (x4[0]=x0[3] ⇒ 1345_1_MK_INVOKEMETHOD(x4[0])≥NonInfC∧1345_1_MK_INVOKEMETHOD(x4[0])≥1260_0_MK_INC(x4[0])∧(U^Increasing(1260_0_MK_INC(x4[0])), ≥))

We simplified constraint (7) using rule (IV) which results in the following new constraint:
(8)    (1345_1_MK_INVOKEMETHOD(x4[0])≥NonInfC∧1345_1_MK_INVOKEMETHOD(x4[0])≥1260_0_MK_INC(x4[0])∧(U^Increasing(1260_0_MK_INC(x4[0])), ≥))

We simplified constraint (8) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(9)    ((U^Increasing(1260_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)    ((U^Increasing(1260_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)    ((U^Increasing(1260_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)    ((U^Increasing(1260_0_MK_INC(x4[0])), ≥)∧0 = 0∧[(-1)bso_13] ≥ 0)

For Pair 1260_0_MK_INC(x0) → COND_1260_0_MK_INC(>(x0, 0), x0) the following chains were created:

We consider the chain 1260_0_MK_INC(x0[1]) → COND_1260_0_MK_INC(>(x0[1], 0), x0[1]), COND_1260_0_MK_INC(TRUE, x0[2]) → 1260_0_MK_INC(+(x0[2], -1)) which results in the following constraint:
(13)    (>(x0[1], 0)=TRUE∧x0[1]=x0[2] ⇒ 1260_0_MK_INC(x0[1])≥NonInfC∧1260_0_MK_INC(x0[1])≥COND_1260_0_MK_INC(>(x0[1], 0), x0[1])∧(U^Increasing(COND_1260_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)=TRUE ⇒ 1260_0_MK_INC(x0[1])≥NonInfC∧1260_0_MK_INC(x0[1])≥COND_1260_0_MK_INC(>(x0[1], 0), x0[1])∧(U^Increasing(COND_1260_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 ⇒ (U^Increasing(COND_1260_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 ⇒ (U^Increasing(COND_1260_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 ⇒ (U^Increasing(COND_1260_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 ⇒ (U^Increasing(COND_1260_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_1260_0_MK_INC(TRUE, x0) → 1260_0_MK_INC(+(x0, -1)) the following chains were created:

We consider the chain COND_1260_0_MK_INC(TRUE, x0[2]) → 1260_0_MK_INC(+(x0[2], -1)) which results in the following constraint:
(19)    (COND_1260_0_MK_INC(TRUE, x0[2])≥NonInfC∧COND_1260_0_MK_INC(TRUE, x0[2])≥1260_0_MK_INC(+(x0[2], -1))∧(U^Increasing(1260_0_MK_INC(+(x0[2], -1))), ≥))

We simplified constraint (19) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(20)    ((U^Increasing(1260_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)    ((U^Increasing(1260_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)    ((U^Increasing(1260_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)    ((U^Increasing(1260_0_MK_INC(+(x0[2], -1))), ≥)∧0 = 0∧[1 + (-1)bso_17] ≥ 0)

For Pair 1260_0_MK_INC(x0) → COND_1260_0_MK_INC1(>(x0, 0), x0) the following chains were created:

We consider the chain 1260_0_MK_INC(x0[3]) → COND_1260_0_MK_INC1(>(x0[3], 0), x0[3]), COND_1260_0_MK_INC1(TRUE, x0[4]) → 1345_1_MK_INVOKEMETHOD(+(x0[4], -1)) which results in the following constraint:
(24)    (>(x0[3], 0)=TRUE∧x0[3]=x0[4] ⇒ 1260_0_MK_INC(x0[3])≥NonInfC∧1260_0_MK_INC(x0[3])≥COND_1260_0_MK_INC1(>(x0[3], 0), x0[3])∧(U^Increasing(COND_1260_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)=TRUE ⇒ 1260_0_MK_INC(x0[3])≥NonInfC∧1260_0_MK_INC(x0[3])≥COND_1260_0_MK_INC1(>(x0[3], 0), x0[3])∧(U^Increasing(COND_1260_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 ⇒ (U^Increasing(COND_1260_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 ⇒ (U^Increasing(COND_1260_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 ⇒ (U^Increasing(COND_1260_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 ⇒ (U^Increasing(COND_1260_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_1260_0_MK_INC1(TRUE, x0) → 1345_1_MK_INVOKEMETHOD(+(x0, -1)) the following chains were created:

We consider the chain COND_1260_0_MK_INC1(TRUE, x0[4]) → 1345_1_MK_INVOKEMETHOD(+(x0[4], -1)) which results in the following constraint:
(30)    (COND_1260_0_MK_INC1(TRUE, x0[4])≥NonInfC∧COND_1260_0_MK_INC1(TRUE, x0[4])≥1345_1_MK_INVOKEMETHOD(+(x0[4], -1))∧(U^Increasing(1345_1_MK_INVOKEMETHOD(+(x0[4], -1))), ≥))

We simplified constraint (30) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(31)    ((U^Increasing(1345_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)    ((U^Increasing(1345_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)    ((U^Increasing(1345_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)    ((U^Increasing(1345_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.

1345_1_MK_INVOKEMETHOD(x4) → 1260_0_MK_INC(x4)
- ((U^Increasing(1260_0_MK_INC(x4[0])), ≥)∧0 = 0∧[(-1)bso_13] ≥ 0)
- ((U^Increasing(1260_0_MK_INC(x4[0])), ≥)∧0 = 0∧[(-1)bso_13] ≥ 0)
1260_0_MK_INC(x0) → COND_1260_0_MK_INC(>(x0, 0), x0)
- (x0[1] ≥ 0 ⇒ (U^Increasing(COND_1260_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_1260_0_MK_INC(TRUE, x0) → 1260_0_MK_INC(+(x0, -1))
- ((U^Increasing(1260_0_MK_INC(+(x0[2], -1))), ≥)∧0 = 0∧[1 + (-1)bso_17] ≥ 0)
1260_0_MK_INC(x0) → COND_1260_0_MK_INC1(>(x0, 0), x0)
- (x0[3] ≥ 0 ⇒ (U^Increasing(COND_1260_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_1260_0_MK_INC1(TRUE, x0) → 1345_1_MK_INVOKEMETHOD(+(x0, -1))
- ((U^Increasing(1345_1_MK_INVOKEMETHOD(+(x0[4], -1))), ≥)∧0 = 0∧[2 + (-1)bso_21] ≥ 0)

The constraints for P_> respective P_bound 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 P_bound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:

POL(TRUE) = 0
POL(FALSE) = 0
POL(1345_1_MK_INVOKEMETHOD(x₁)) = [2]x₁
POL(1260_0_MK_INC(x₁)) = [2]x₁
POL(COND_1260_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_1260_0_MK_INC1(x₁, x₂)) = [2]x₂

The following pairs are in P_>:

1260_0_MK_INC(x0[1]) → COND_1260_0_MK_INC(>(x0[1], 0), x0[1])
COND_1260_0_MK_INC(TRUE, x0[2]) → 1260_0_MK_INC(+(x0[2], -1))
COND_1260_0_MK_INC1(TRUE, x0[4]) → 1345_1_MK_INVOKEMETHOD(+(x0[4], -1))

The following pairs are in P_bound:

1260_0_MK_INC(x0[1]) → COND_1260_0_MK_INC(>(x0[1], 0), x0[1])
1260_0_MK_INC(x0[3]) → COND_1260_0_MK_INC1(>(x0[3], 0), x0[3])

The following pairs are in P_≥:

1345_1_MK_INVOKEMETHOD(x4[0]) → 1260_0_MK_INC(x4[0])
1260_0_MK_INC(x0[3]) → COND_1260_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): 1345_1_MK_INVOKEMETHOD(x4[0]) → 1260_0_MK_INC(x4[0])
(3): 1260_0_MK_INC(x0[3]) → COND_1260_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): 1345_1_MK_INVOKEMETHOD(x4[0]) → 1260_0_MK_INC(x4[0])
(2): COND_1260_0_MK_INC(TRUE, x0[2]) → 1260_0_MK_INC(x0[2] + -1)
(4): COND_1260_0_MK_INC1(TRUE, x0[4]) → 1345_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.

(0) Obligation:

(1) JBC2FIG (SOUND transformation)

(2) Obligation:

(3) FIGtoITRSProof (SOUND transformation)

(4) Obligation:

(5) IDPNonInfProof (SOUND transformation)

(6) Complex Obligation (AND)

(7) Obligation:

(8) IDependencyGraphProof (EQUIVALENT transformation)

(9) TRUE

(10) Obligation:

(11) IDependencyGraphProof (EQUIVALENT transformation)

(12) TRUE