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_22 (Sun Microsystems Inc.) Main-Class: Fibonacci 
public class Fibonacci {
	public static void main(String[] args) {
		fib(args.length);
	}

	public static int fib(int x) {
		if (x == 0) {
			return 0;
		} else if (x == 1) {
			return 1;
		} else {
			return fib(x-1) + fib(x-2);
		}
	}
}


(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
Fibonacci.main([Ljava/lang/String;)V: Graph of 22 nodes with 0 SCCs.

Fibonacci.fib(I)I: Graph of 45 nodes with 0 SCCs.


(3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 27 rules for P and 17 rules for R.


Combined rules. Obtained 3 rules for P and 4 rules for R.


Filtered ground terms:


27_0_fib_NE(x1, x2, x3) → 27_0_fib_NE(x2, x3)
390_0_fib_Return(x1, x2) → 390_0_fib_Return(x2)
Cond_87_1_fib_InvokeMethod(x1, x2, x3, x4) → Cond_87_1_fib_InvokeMethod(x1, x3)
63_0_fib_Return(x1, x2, x3) → 63_0_fib_Return
Cond_27_0_fib_NE(x1, x2, x3, x4) → Cond_27_0_fib_NE(x1, x3, x4)
Cond_374_1_fib_InvokeMethod1(x1, x2, x3, x4) → Cond_374_1_fib_InvokeMethod1(x1, x3)
Cond_374_1_fib_InvokeMethod(x1, x2, x3, x4) → Cond_374_1_fib_InvokeMethod(x1, x3)
39_0_fib_Return(x1, x2, x3) → 39_0_fib_Return

Filtered duplicate args:


27_0_fib_NE(x1, x2) → 27_0_fib_NE(x2)
Cond_27_0_fib_NE(x1, x2, x3) → Cond_27_0_fib_NE(x1, x3)

Filtered unneeded arguments:


Cond_87_1_fib_InvokeMethod1(x1, x2, x3, x4) → Cond_87_1_fib_InvokeMethod1(x1, x2, x3)
Cond_374_1_fib_InvokeMethod2(x1, x2, x3, x4) → Cond_374_1_fib_InvokeMethod2(x1, x2, x3)

Combined rules. Obtained 3 rules for P and 4 rules for R.


Finished conversion. Obtained 3 rules for P and 4 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:
27_0_fib_NE(0) → 39_0_fib_Return
374_1_fib_InvokeMethod(39_0_fib_Return, x2, 0) → Cond_374_1_fib_InvokeMethod(x2 > 0, 39_0_fib_Return, x2, 0)
Cond_374_1_fib_InvokeMethod(TRUE, 39_0_fib_Return, x2, 0) → 390_0_fib_Return(x2 + 0)
374_1_fib_InvokeMethod(63_0_fib_Return, x2, 1) → Cond_374_1_fib_InvokeMethod1(x2 > 0 && 1 > 0, 63_0_fib_Return, x2, 1)
Cond_374_1_fib_InvokeMethod1(TRUE, 63_0_fib_Return, x2, 1) → 390_0_fib_Return(x2 + 1)
374_1_fib_InvokeMethod(390_0_fib_Return(x0), x1, x2) → Cond_374_1_fib_InvokeMethod2(x1 > 0 && x0 > 0, 390_0_fib_Return(x0), x1, x2)
Cond_374_1_fib_InvokeMethod2(TRUE, 390_0_fib_Return(x0), x1, x2) → 390_0_fib_Return(x1 + x0)

The integer pair graph contains the following rules and edges:
(0): 27_0_FIB_NE(x0[0]) → COND_27_0_FIB_NE(x0[0] > 0 && !(x0[0] = 1), x0[0])
(1): COND_27_0_FIB_NE(TRUE, x0[1]) → 87_1_FIB_INVOKEMETHOD(27_0_fib_NE(x0[1] - 1), x0[1], x0[1] - 1)
(2): COND_27_0_FIB_NE(TRUE, x0[2]) → 27_0_FIB_NE(x0[2] - 1)
(3): 87_1_FIB_INVOKEMETHOD(63_0_fib_Return, x2[3], 1) → COND_87_1_FIB_INVOKEMETHOD(x2[3] > 0, 63_0_fib_Return, x2[3], 1)
(4): COND_87_1_FIB_INVOKEMETHOD(TRUE, 63_0_fib_Return, x2[4], 1) → 27_0_FIB_NE(x2[4] - 2)
(5): 87_1_FIB_INVOKEMETHOD(390_0_fib_Return(x0[5]), x1[5], x2[5]) → COND_87_1_FIB_INVOKEMETHOD1(x1[5] > 0, 390_0_fib_Return(x0[5]), x1[5], x2[5])
(6): COND_87_1_FIB_INVOKEMETHOD1(TRUE, 390_0_fib_Return(x0[6]), x1[6], x2[6]) → 27_0_FIB_NE(x1[6] - 2)

(0) -> (1), if ((x0[0] > 0 && !(x0[0] = 1) →* TRUE)∧(x0[0] →* x0[1]))


(0) -> (2), if ((x0[0] > 0 && !(x0[0] = 1) →* TRUE)∧(x0[0] →* x0[2]))


(1) -> (3), if ((27_0_fib_NE(x0[1] - 1) →* 63_0_fib_Return)∧(x0[1] →* x2[3])∧(x0[1] - 1 →* 1))


(1) -> (5), if ((27_0_fib_NE(x0[1] - 1) →* 390_0_fib_Return(x0[5]))∧(x0[1] →* x1[5])∧(x0[1] - 1 →* x2[5]))


(2) -> (0), if ((x0[2] - 1 →* x0[0]))


(3) -> (4), if ((x2[3] > 0 →* TRUE)∧(x2[3] →* x2[4]))


(4) -> (0), if ((x2[4] - 2 →* x0[0]))


(5) -> (6), if ((x1[5] > 0 →* TRUE)∧(390_0_fib_Return(x0[5]) →* 390_0_fib_Return(x0[6]))∧(x1[5] →* x1[6])∧(x2[5] →* x2[6]))


(6) -> (0), if ((x1[6] - 2 →* x0[0]))



The set Q consists of the following terms:
27_0_fib_NE(0)
374_1_fib_InvokeMethod(39_0_fib_Return, x0, 0)
Cond_374_1_fib_InvokeMethod(TRUE, 39_0_fib_Return, x0, 0)
374_1_fib_InvokeMethod(63_0_fib_Return, x0, 1)
Cond_374_1_fib_InvokeMethod1(TRUE, 63_0_fib_Return, x0, 1)
374_1_fib_InvokeMethod(390_0_fib_Return(x0), x1, x2)
Cond_374_1_fib_InvokeMethod2(TRUE, 390_0_fib_Return(x0), x1, x2)

(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 27_0_FIB_NE(x0) → COND_27_0_FIB_NE(&&(>(x0, 0), !(=(x0, 1))), x0) the following chains were created:
  • We consider the chain 27_0_FIB_NE(x0[0]) → COND_27_0_FIB_NE(&&(>(x0[0], 0), !(=(x0[0], 1))), x0[0]), COND_27_0_FIB_NE(TRUE, x0[1]) → 87_1_FIB_INVOKEMETHOD(27_0_fib_NE(-(x0[1], 1)), x0[1], -(x0[1], 1)) which results in the following constraint:

    (1)    (&&(>(x0[0], 0), !(=(x0[0], 1)))=TRUE∧x0[0]=x0[1] ⇒ 27_0_FIB_NE(x0[0])≥NonInfC∧27_0_FIB_NE(x0[0])≥COND_27_0_FIB_NE(&&(>(x0[0], 0), !(=(x0[0], 1))), x0[0])∧(UIncreasing(COND_27_0_FIB_NE(&&(>(x0[0], 0), !(=(x0[0], 1))), x0[0])), ≥))



    We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraints:

    (2)    (>(x0[0], 0)=TRUE∧<(x0[0], 1)=TRUE ⇒ 27_0_FIB_NE(x0[0])≥NonInfC∧27_0_FIB_NE(x0[0])≥COND_27_0_FIB_NE(&&(>(x0[0], 0), !(=(x0[0], 1))), x0[0])∧(UIncreasing(COND_27_0_FIB_NE(&&(>(x0[0], 0), !(=(x0[0], 1))), x0[0])), ≥))


    (3)    (>(x0[0], 0)=TRUE∧>(x0[0], 1)=TRUE ⇒ 27_0_FIB_NE(x0[0])≥NonInfC∧27_0_FIB_NE(x0[0])≥COND_27_0_FIB_NE(&&(>(x0[0], 0), !(=(x0[0], 1))), x0[0])∧(UIncreasing(COND_27_0_FIB_NE(&&(>(x0[0], 0), !(=(x0[0], 1))), x0[0])), ≥))



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

    (4)    (x0[0] + [-1] ≥ 0∧[-1]x0[0] ≥ 0 ⇒ (UIncreasing(COND_27_0_FIB_NE(&&(>(x0[0], 0), !(=(x0[0], 1))), x0[0])), ≥)∧[(2)bni_29 + (-1)Bound*bni_29] + [(2)bni_29]x0[0] ≥ 0∧[(-1)bso_30] ≥ 0)



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

    (5)    (x0[0] + [-1] ≥ 0∧x0[0] + [-2] ≥ 0 ⇒ (UIncreasing(COND_27_0_FIB_NE(&&(>(x0[0], 0), !(=(x0[0], 1))), x0[0])), ≥)∧[(2)bni_29 + (-1)Bound*bni_29] + [(2)bni_29]x0[0] ≥ 0∧[(-1)bso_30] ≥ 0)



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

    (6)    (x0[0] + [-1] ≥ 0∧[-1]x0[0] ≥ 0 ⇒ (UIncreasing(COND_27_0_FIB_NE(&&(>(x0[0], 0), !(=(x0[0], 1))), x0[0])), ≥)∧[(2)bni_29 + (-1)Bound*bni_29] + [(2)bni_29]x0[0] ≥ 0∧[(-1)bso_30] ≥ 0)



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

    (7)    (x0[0] + [-1] ≥ 0∧x0[0] + [-2] ≥ 0 ⇒ (UIncreasing(COND_27_0_FIB_NE(&&(>(x0[0], 0), !(=(x0[0], 1))), x0[0])), ≥)∧[(2)bni_29 + (-1)Bound*bni_29] + [(2)bni_29]x0[0] ≥ 0∧[(-1)bso_30] ≥ 0)



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

    (8)    (x0[0] + [-1] ≥ 0∧[-1]x0[0] ≥ 0 ⇒ (UIncreasing(COND_27_0_FIB_NE(&&(>(x0[0], 0), !(=(x0[0], 1))), x0[0])), ≥)∧[(2)bni_29 + (-1)Bound*bni_29] + [(2)bni_29]x0[0] ≥ 0∧[(-1)bso_30] ≥ 0)



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

    (9)    (x0[0] + [-1] ≥ 0∧x0[0] + [-2] ≥ 0 ⇒ (UIncreasing(COND_27_0_FIB_NE(&&(>(x0[0], 0), !(=(x0[0], 1))), x0[0])), ≥)∧[(2)bni_29 + (-1)Bound*bni_29] + [(2)bni_29]x0[0] ≥ 0∧[(-1)bso_30] ≥ 0)



    We solved constraint (8) using rule (IDP_SMT_SPLIT).We simplified constraint (9) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (10)    (x0[0] ≥ 0∧[-1] + x0[0] ≥ 0 ⇒ (UIncreasing(COND_27_0_FIB_NE(&&(>(x0[0], 0), !(=(x0[0], 1))), x0[0])), ≥)∧[(4)bni_29 + (-1)Bound*bni_29] + [(2)bni_29]x0[0] ≥ 0∧[(-1)bso_30] ≥ 0)



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

    (11)    ([1] + x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_27_0_FIB_NE(&&(>(x0[0], 0), !(=(x0[0], 1))), x0[0])), ≥)∧[(6)bni_29 + (-1)Bound*bni_29] + [(2)bni_29]x0[0] ≥ 0∧[(-1)bso_30] ≥ 0)



  • We consider the chain 27_0_FIB_NE(x0[0]) → COND_27_0_FIB_NE(&&(>(x0[0], 0), !(=(x0[0], 1))), x0[0]), COND_27_0_FIB_NE(TRUE, x0[2]) → 27_0_FIB_NE(-(x0[2], 1)) which results in the following constraint:

    (12)    (&&(>(x0[0], 0), !(=(x0[0], 1)))=TRUE∧x0[0]=x0[2] ⇒ 27_0_FIB_NE(x0[0])≥NonInfC∧27_0_FIB_NE(x0[0])≥COND_27_0_FIB_NE(&&(>(x0[0], 0), !(=(x0[0], 1))), x0[0])∧(UIncreasing(COND_27_0_FIB_NE(&&(>(x0[0], 0), !(=(x0[0], 1))), x0[0])), ≥))



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

    (13)    (>(x0[0], 0)=TRUE∧<(x0[0], 1)=TRUE ⇒ 27_0_FIB_NE(x0[0])≥NonInfC∧27_0_FIB_NE(x0[0])≥COND_27_0_FIB_NE(&&(>(x0[0], 0), !(=(x0[0], 1))), x0[0])∧(UIncreasing(COND_27_0_FIB_NE(&&(>(x0[0], 0), !(=(x0[0], 1))), x0[0])), ≥))


    (14)    (>(x0[0], 0)=TRUE∧>(x0[0], 1)=TRUE ⇒ 27_0_FIB_NE(x0[0])≥NonInfC∧27_0_FIB_NE(x0[0])≥COND_27_0_FIB_NE(&&(>(x0[0], 0), !(=(x0[0], 1))), x0[0])∧(UIncreasing(COND_27_0_FIB_NE(&&(>(x0[0], 0), !(=(x0[0], 1))), x0[0])), ≥))



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

    (15)    (x0[0] + [-1] ≥ 0∧[-1]x0[0] ≥ 0 ⇒ (UIncreasing(COND_27_0_FIB_NE(&&(>(x0[0], 0), !(=(x0[0], 1))), x0[0])), ≥)∧[(2)bni_29 + (-1)Bound*bni_29] + [(2)bni_29]x0[0] ≥ 0∧[(-1)bso_30] ≥ 0)



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

    (16)    (x0[0] + [-1] ≥ 0∧x0[0] + [-2] ≥ 0 ⇒ (UIncreasing(COND_27_0_FIB_NE(&&(>(x0[0], 0), !(=(x0[0], 1))), x0[0])), ≥)∧[(2)bni_29 + (-1)Bound*bni_29] + [(2)bni_29]x0[0] ≥ 0∧[(-1)bso_30] ≥ 0)



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

    (17)    (x0[0] + [-1] ≥ 0∧[-1]x0[0] ≥ 0 ⇒ (UIncreasing(COND_27_0_FIB_NE(&&(>(x0[0], 0), !(=(x0[0], 1))), x0[0])), ≥)∧[(2)bni_29 + (-1)Bound*bni_29] + [(2)bni_29]x0[0] ≥ 0∧[(-1)bso_30] ≥ 0)



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

    (18)    (x0[0] + [-1] ≥ 0∧x0[0] + [-2] ≥ 0 ⇒ (UIncreasing(COND_27_0_FIB_NE(&&(>(x0[0], 0), !(=(x0[0], 1))), x0[0])), ≥)∧[(2)bni_29 + (-1)Bound*bni_29] + [(2)bni_29]x0[0] ≥ 0∧[(-1)bso_30] ≥ 0)



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

    (19)    (x0[0] + [-1] ≥ 0∧[-1]x0[0] ≥ 0 ⇒ (UIncreasing(COND_27_0_FIB_NE(&&(>(x0[0], 0), !(=(x0[0], 1))), x0[0])), ≥)∧[(2)bni_29 + (-1)Bound*bni_29] + [(2)bni_29]x0[0] ≥ 0∧[(-1)bso_30] ≥ 0)



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

    (20)    (x0[0] + [-1] ≥ 0∧x0[0] + [-2] ≥ 0 ⇒ (UIncreasing(COND_27_0_FIB_NE(&&(>(x0[0], 0), !(=(x0[0], 1))), x0[0])), ≥)∧[(2)bni_29 + (-1)Bound*bni_29] + [(2)bni_29]x0[0] ≥ 0∧[(-1)bso_30] ≥ 0)



    We solved constraint (19) using rule (IDP_SMT_SPLIT).We simplified constraint (20) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (21)    (x0[0] ≥ 0∧[-1] + x0[0] ≥ 0 ⇒ (UIncreasing(COND_27_0_FIB_NE(&&(>(x0[0], 0), !(=(x0[0], 1))), x0[0])), ≥)∧[(4)bni_29 + (-1)Bound*bni_29] + [(2)bni_29]x0[0] ≥ 0∧[(-1)bso_30] ≥ 0)



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

    (22)    ([1] + x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_27_0_FIB_NE(&&(>(x0[0], 0), !(=(x0[0], 1))), x0[0])), ≥)∧[(6)bni_29 + (-1)Bound*bni_29] + [(2)bni_29]x0[0] ≥ 0∧[(-1)bso_30] ≥ 0)







For Pair COND_27_0_FIB_NE(TRUE, x0) → 87_1_FIB_INVOKEMETHOD(27_0_fib_NE(-(x0, 1)), x0, -(x0, 1)) the following chains were created:
  • We consider the chain COND_27_0_FIB_NE(TRUE, x0[1]) → 87_1_FIB_INVOKEMETHOD(27_0_fib_NE(-(x0[1], 1)), x0[1], -(x0[1], 1)) which results in the following constraint:

    (23)    (COND_27_0_FIB_NE(TRUE, x0[1])≥NonInfC∧COND_27_0_FIB_NE(TRUE, x0[1])≥87_1_FIB_INVOKEMETHOD(27_0_fib_NE(-(x0[1], 1)), x0[1], -(x0[1], 1))∧(UIncreasing(87_1_FIB_INVOKEMETHOD(27_0_fib_NE(-(x0[1], 1)), x0[1], -(x0[1], 1))), ≥))



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

    (24)    ((UIncreasing(87_1_FIB_INVOKEMETHOD(27_0_fib_NE(-(x0[1], 1)), x0[1], -(x0[1], 1))), ≥)∧[2 + (-1)bso_32] ≥ 0)



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

    (25)    ((UIncreasing(87_1_FIB_INVOKEMETHOD(27_0_fib_NE(-(x0[1], 1)), x0[1], -(x0[1], 1))), ≥)∧[2 + (-1)bso_32] ≥ 0)



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

    (26)    ((UIncreasing(87_1_FIB_INVOKEMETHOD(27_0_fib_NE(-(x0[1], 1)), x0[1], -(x0[1], 1))), ≥)∧[2 + (-1)bso_32] ≥ 0)



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

    (27)    ((UIncreasing(87_1_FIB_INVOKEMETHOD(27_0_fib_NE(-(x0[1], 1)), x0[1], -(x0[1], 1))), ≥)∧0 = 0∧[2 + (-1)bso_32] ≥ 0)







For Pair COND_27_0_FIB_NE(TRUE, x0) → 27_0_FIB_NE(-(x0, 1)) the following chains were created:
  • We consider the chain COND_27_0_FIB_NE(TRUE, x0[2]) → 27_0_FIB_NE(-(x0[2], 1)) which results in the following constraint:

    (28)    (COND_27_0_FIB_NE(TRUE, x0[2])≥NonInfC∧COND_27_0_FIB_NE(TRUE, x0[2])≥27_0_FIB_NE(-(x0[2], 1))∧(UIncreasing(27_0_FIB_NE(-(x0[2], 1))), ≥))



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

    (29)    ((UIncreasing(27_0_FIB_NE(-(x0[2], 1))), ≥)∧[2 + (-1)bso_34] ≥ 0)



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

    (30)    ((UIncreasing(27_0_FIB_NE(-(x0[2], 1))), ≥)∧[2 + (-1)bso_34] ≥ 0)



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

    (31)    ((UIncreasing(27_0_FIB_NE(-(x0[2], 1))), ≥)∧[2 + (-1)bso_34] ≥ 0)



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

    (32)    ((UIncreasing(27_0_FIB_NE(-(x0[2], 1))), ≥)∧0 = 0∧[2 + (-1)bso_34] ≥ 0)







For Pair 87_1_FIB_INVOKEMETHOD(63_0_fib_Return, x2, 1) → COND_87_1_FIB_INVOKEMETHOD(>(x2, 0), 63_0_fib_Return, x2, 1) the following chains were created:
  • We consider the chain 87_1_FIB_INVOKEMETHOD(63_0_fib_Return, x2[3], 1) → COND_87_1_FIB_INVOKEMETHOD(>(x2[3], 0), 63_0_fib_Return, x2[3], 1), COND_87_1_FIB_INVOKEMETHOD(TRUE, 63_0_fib_Return, x2[4], 1) → 27_0_FIB_NE(-(x2[4], 2)) which results in the following constraint:

    (33)    (>(x2[3], 0)=TRUE∧x2[3]=x2[4] ⇒ 87_1_FIB_INVOKEMETHOD(63_0_fib_Return, x2[3], 1)≥NonInfC∧87_1_FIB_INVOKEMETHOD(63_0_fib_Return, x2[3], 1)≥COND_87_1_FIB_INVOKEMETHOD(>(x2[3], 0), 63_0_fib_Return, x2[3], 1)∧(UIncreasing(COND_87_1_FIB_INVOKEMETHOD(>(x2[3], 0), 63_0_fib_Return, x2[3], 1)), ≥))



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

    (34)    (>(x2[3], 0)=TRUE ⇒ 87_1_FIB_INVOKEMETHOD(63_0_fib_Return, x2[3], 1)≥NonInfC∧87_1_FIB_INVOKEMETHOD(63_0_fib_Return, x2[3], 1)≥COND_87_1_FIB_INVOKEMETHOD(>(x2[3], 0), 63_0_fib_Return, x2[3], 1)∧(UIncreasing(COND_87_1_FIB_INVOKEMETHOD(>(x2[3], 0), 63_0_fib_Return, x2[3], 1)), ≥))



    We simplified constraint (34) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (35)    (x2[3] + [-1] ≥ 0 ⇒ (UIncreasing(COND_87_1_FIB_INVOKEMETHOD(>(x2[3], 0), 63_0_fib_Return, x2[3], 1)), ≥)∧[(-1)Bound*bni_35] + [(2)bni_35]x2[3] ≥ 0∧[1 + (-1)bso_36] ≥ 0)



    We simplified constraint (35) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (36)    (x2[3] + [-1] ≥ 0 ⇒ (UIncreasing(COND_87_1_FIB_INVOKEMETHOD(>(x2[3], 0), 63_0_fib_Return, x2[3], 1)), ≥)∧[(-1)Bound*bni_35] + [(2)bni_35]x2[3] ≥ 0∧[1 + (-1)bso_36] ≥ 0)



    We simplified constraint (36) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (37)    (x2[3] + [-1] ≥ 0 ⇒ (UIncreasing(COND_87_1_FIB_INVOKEMETHOD(>(x2[3], 0), 63_0_fib_Return, x2[3], 1)), ≥)∧[(-1)Bound*bni_35] + [(2)bni_35]x2[3] ≥ 0∧[1 + (-1)bso_36] ≥ 0)



    We simplified constraint (37) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (38)    (x2[3] ≥ 0 ⇒ (UIncreasing(COND_87_1_FIB_INVOKEMETHOD(>(x2[3], 0), 63_0_fib_Return, x2[3], 1)), ≥)∧[(-1)Bound*bni_35 + (2)bni_35] + [(2)bni_35]x2[3] ≥ 0∧[1 + (-1)bso_36] ≥ 0)







For Pair COND_87_1_FIB_INVOKEMETHOD(TRUE, 63_0_fib_Return, x2, 1) → 27_0_FIB_NE(-(x2, 2)) the following chains were created:
  • We consider the chain COND_87_1_FIB_INVOKEMETHOD(TRUE, 63_0_fib_Return, x2[4], 1) → 27_0_FIB_NE(-(x2[4], 2)) which results in the following constraint:

    (39)    (COND_87_1_FIB_INVOKEMETHOD(TRUE, 63_0_fib_Return, x2[4], 1)≥NonInfC∧COND_87_1_FIB_INVOKEMETHOD(TRUE, 63_0_fib_Return, x2[4], 1)≥27_0_FIB_NE(-(x2[4], 2))∧(UIncreasing(27_0_FIB_NE(-(x2[4], 2))), ≥))



    We simplified constraint (39) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (40)    ((UIncreasing(27_0_FIB_NE(-(x2[4], 2))), ≥)∧[1 + (-1)bso_38] ≥ 0)



    We simplified constraint (40) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (41)    ((UIncreasing(27_0_FIB_NE(-(x2[4], 2))), ≥)∧[1 + (-1)bso_38] ≥ 0)



    We simplified constraint (41) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (42)    ((UIncreasing(27_0_FIB_NE(-(x2[4], 2))), ≥)∧[1 + (-1)bso_38] ≥ 0)



    We simplified constraint (42) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (43)    ((UIncreasing(27_0_FIB_NE(-(x2[4], 2))), ≥)∧0 = 0∧[1 + (-1)bso_38] ≥ 0)







For Pair 87_1_FIB_INVOKEMETHOD(390_0_fib_Return(x0), x1, x2) → COND_87_1_FIB_INVOKEMETHOD1(>(x1, 0), 390_0_fib_Return(x0), x1, x2) the following chains were created:
  • We consider the chain 87_1_FIB_INVOKEMETHOD(390_0_fib_Return(x0[5]), x1[5], x2[5]) → COND_87_1_FIB_INVOKEMETHOD1(>(x1[5], 0), 390_0_fib_Return(x0[5]), x1[5], x2[5]), COND_87_1_FIB_INVOKEMETHOD1(TRUE, 390_0_fib_Return(x0[6]), x1[6], x2[6]) → 27_0_FIB_NE(-(x1[6], 2)) which results in the following constraint:

    (44)    (>(x1[5], 0)=TRUE∧390_0_fib_Return(x0[5])=390_0_fib_Return(x0[6])∧x1[5]=x1[6]∧x2[5]=x2[6] ⇒ 87_1_FIB_INVOKEMETHOD(390_0_fib_Return(x0[5]), x1[5], x2[5])≥NonInfC∧87_1_FIB_INVOKEMETHOD(390_0_fib_Return(x0[5]), x1[5], x2[5])≥COND_87_1_FIB_INVOKEMETHOD1(>(x1[5], 0), 390_0_fib_Return(x0[5]), x1[5], x2[5])∧(UIncreasing(COND_87_1_FIB_INVOKEMETHOD1(>(x1[5], 0), 390_0_fib_Return(x0[5]), x1[5], x2[5])), ≥))



    We simplified constraint (44) using rules (I), (II), (IV) which results in the following new constraint:

    (45)    (>(x1[5], 0)=TRUE ⇒ 87_1_FIB_INVOKEMETHOD(390_0_fib_Return(x0[5]), x1[5], x2[5])≥NonInfC∧87_1_FIB_INVOKEMETHOD(390_0_fib_Return(x0[5]), x1[5], x2[5])≥COND_87_1_FIB_INVOKEMETHOD1(>(x1[5], 0), 390_0_fib_Return(x0[5]), x1[5], x2[5])∧(UIncreasing(COND_87_1_FIB_INVOKEMETHOD1(>(x1[5], 0), 390_0_fib_Return(x0[5]), x1[5], x2[5])), ≥))



    We simplified constraint (45) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (46)    (x1[5] + [-1] ≥ 0 ⇒ (UIncreasing(COND_87_1_FIB_INVOKEMETHOD1(>(x1[5], 0), 390_0_fib_Return(x0[5]), x1[5], x2[5])), ≥)∧[(-1)Bound*bni_39] + [(2)bni_39]x1[5] ≥ 0∧[1 + (-1)bso_40] ≥ 0)



    We simplified constraint (46) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (47)    (x1[5] + [-1] ≥ 0 ⇒ (UIncreasing(COND_87_1_FIB_INVOKEMETHOD1(>(x1[5], 0), 390_0_fib_Return(x0[5]), x1[5], x2[5])), ≥)∧[(-1)Bound*bni_39] + [(2)bni_39]x1[5] ≥ 0∧[1 + (-1)bso_40] ≥ 0)



    We simplified constraint (47) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (48)    (x1[5] + [-1] ≥ 0 ⇒ (UIncreasing(COND_87_1_FIB_INVOKEMETHOD1(>(x1[5], 0), 390_0_fib_Return(x0[5]), x1[5], x2[5])), ≥)∧[(-1)Bound*bni_39] + [(2)bni_39]x1[5] ≥ 0∧[1 + (-1)bso_40] ≥ 0)



    We simplified constraint (48) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (49)    (x1[5] + [-1] ≥ 0 ⇒ (UIncreasing(COND_87_1_FIB_INVOKEMETHOD1(>(x1[5], 0), 390_0_fib_Return(x0[5]), x1[5], x2[5])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_39] + [(2)bni_39]x1[5] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_40] ≥ 0)



    We simplified constraint (49) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (50)    (x1[5] ≥ 0 ⇒ (UIncreasing(COND_87_1_FIB_INVOKEMETHOD1(>(x1[5], 0), 390_0_fib_Return(x0[5]), x1[5], x2[5])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_39 + (2)bni_39] + [(2)bni_39]x1[5] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_40] ≥ 0)







For Pair COND_87_1_FIB_INVOKEMETHOD1(TRUE, 390_0_fib_Return(x0), x1, x2) → 27_0_FIB_NE(-(x1, 2)) the following chains were created:
  • We consider the chain COND_87_1_FIB_INVOKEMETHOD1(TRUE, 390_0_fib_Return(x0[6]), x1[6], x2[6]) → 27_0_FIB_NE(-(x1[6], 2)) which results in the following constraint:

    (51)    (COND_87_1_FIB_INVOKEMETHOD1(TRUE, 390_0_fib_Return(x0[6]), x1[6], x2[6])≥NonInfC∧COND_87_1_FIB_INVOKEMETHOD1(TRUE, 390_0_fib_Return(x0[6]), x1[6], x2[6])≥27_0_FIB_NE(-(x1[6], 2))∧(UIncreasing(27_0_FIB_NE(-(x1[6], 2))), ≥))



    We simplified constraint (51) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (52)    ((UIncreasing(27_0_FIB_NE(-(x1[6], 2))), ≥)∧[1 + (-1)bso_42] ≥ 0)



    We simplified constraint (52) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (53)    ((UIncreasing(27_0_FIB_NE(-(x1[6], 2))), ≥)∧[1 + (-1)bso_42] ≥ 0)



    We simplified constraint (53) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (54)    ((UIncreasing(27_0_FIB_NE(-(x1[6], 2))), ≥)∧[1 + (-1)bso_42] ≥ 0)



    We simplified constraint (54) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (55)    ((UIncreasing(27_0_FIB_NE(-(x1[6], 2))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_42] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • 27_0_FIB_NE(x0) → COND_27_0_FIB_NE(&&(>(x0, 0), !(=(x0, 1))), x0)
    • ([1] + x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_27_0_FIB_NE(&&(>(x0[0], 0), !(=(x0[0], 1))), x0[0])), ≥)∧[(6)bni_29 + (-1)Bound*bni_29] + [(2)bni_29]x0[0] ≥ 0∧[(-1)bso_30] ≥ 0)
    • ([1] + x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_27_0_FIB_NE(&&(>(x0[0], 0), !(=(x0[0], 1))), x0[0])), ≥)∧[(6)bni_29 + (-1)Bound*bni_29] + [(2)bni_29]x0[0] ≥ 0∧[(-1)bso_30] ≥ 0)

  • COND_27_0_FIB_NE(TRUE, x0) → 87_1_FIB_INVOKEMETHOD(27_0_fib_NE(-(x0, 1)), x0, -(x0, 1))
    • ((UIncreasing(87_1_FIB_INVOKEMETHOD(27_0_fib_NE(-(x0[1], 1)), x0[1], -(x0[1], 1))), ≥)∧0 = 0∧[2 + (-1)bso_32] ≥ 0)

  • COND_27_0_FIB_NE(TRUE, x0) → 27_0_FIB_NE(-(x0, 1))
    • ((UIncreasing(27_0_FIB_NE(-(x0[2], 1))), ≥)∧0 = 0∧[2 + (-1)bso_34] ≥ 0)

  • 87_1_FIB_INVOKEMETHOD(63_0_fib_Return, x2, 1) → COND_87_1_FIB_INVOKEMETHOD(>(x2, 0), 63_0_fib_Return, x2, 1)
    • (x2[3] ≥ 0 ⇒ (UIncreasing(COND_87_1_FIB_INVOKEMETHOD(>(x2[3], 0), 63_0_fib_Return, x2[3], 1)), ≥)∧[(-1)Bound*bni_35 + (2)bni_35] + [(2)bni_35]x2[3] ≥ 0∧[1 + (-1)bso_36] ≥ 0)

  • COND_87_1_FIB_INVOKEMETHOD(TRUE, 63_0_fib_Return, x2, 1) → 27_0_FIB_NE(-(x2, 2))
    • ((UIncreasing(27_0_FIB_NE(-(x2[4], 2))), ≥)∧0 = 0∧[1 + (-1)bso_38] ≥ 0)

  • 87_1_FIB_INVOKEMETHOD(390_0_fib_Return(x0), x1, x2) → COND_87_1_FIB_INVOKEMETHOD1(>(x1, 0), 390_0_fib_Return(x0), x1, x2)
    • (x1[5] ≥ 0 ⇒ (UIncreasing(COND_87_1_FIB_INVOKEMETHOD1(>(x1[5], 0), 390_0_fib_Return(x0[5]), x1[5], x2[5])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_39 + (2)bni_39] + [(2)bni_39]x1[5] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_40] ≥ 0)

  • COND_87_1_FIB_INVOKEMETHOD1(TRUE, 390_0_fib_Return(x0), x1, x2) → 27_0_FIB_NE(-(x1, 2))
    • ((UIncreasing(27_0_FIB_NE(-(x1[6], 2))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_42] ≥ 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(27_0_fib_NE(x1)) = [-1] + x1   
POL(0) = 0   
POL(39_0_fib_Return) = [2]   
POL(374_1_fib_InvokeMethod(x1, x2, x3)) = [-1] + [-1]x2 + [-1]x1   
POL(Cond_374_1_fib_InvokeMethod(x1, x2, x3, x4)) = [-1] + [-1]x3   
POL(>(x1, x2)) = [-1]   
POL(390_0_fib_Return(x1)) = x1   
POL(+(x1, x2)) = x1 + x2   
POL(63_0_fib_Return) = [-1]   
POL(1) = [1]   
POL(Cond_374_1_fib_InvokeMethod1(x1, x2, x3, x4)) = [-1] + [-1]x3   
POL(&&(x1, x2)) = [-1]   
POL(Cond_374_1_fib_InvokeMethod2(x1, x2, x3, x4)) = [-1] + [-1]x3 + [-1]x2   
POL(27_0_FIB_NE(x1)) = [2] + [2]x1   
POL(COND_27_0_FIB_NE(x1, x2)) = [2] + [2]x2   
POL(!(x1)) = [-1]   
POL(=(x1, x2)) = [-1]   
POL(87_1_FIB_INVOKEMETHOD(x1, x2, x3)) = [2]x2   
POL(-(x1, x2)) = x1 + [-1]x2   
POL(COND_87_1_FIB_INVOKEMETHOD(x1, x2, x3, x4)) = [-1] + [2]x3   
POL(2) = [2]   
POL(COND_87_1_FIB_INVOKEMETHOD1(x1, x2, x3, x4)) = [-1] + [2]x3   

The following pairs are in P>:

COND_27_0_FIB_NE(TRUE, x0[1]) → 87_1_FIB_INVOKEMETHOD(27_0_fib_NE(-(x0[1], 1)), x0[1], -(x0[1], 1))
COND_27_0_FIB_NE(TRUE, x0[2]) → 27_0_FIB_NE(-(x0[2], 1))
87_1_FIB_INVOKEMETHOD(63_0_fib_Return, x2[3], 1) → COND_87_1_FIB_INVOKEMETHOD(>(x2[3], 0), 63_0_fib_Return, x2[3], 1)
COND_87_1_FIB_INVOKEMETHOD(TRUE, 63_0_fib_Return, x2[4], 1) → 27_0_FIB_NE(-(x2[4], 2))
87_1_FIB_INVOKEMETHOD(390_0_fib_Return(x0[5]), x1[5], x2[5]) → COND_87_1_FIB_INVOKEMETHOD1(>(x1[5], 0), 390_0_fib_Return(x0[5]), x1[5], x2[5])
COND_87_1_FIB_INVOKEMETHOD1(TRUE, 390_0_fib_Return(x0[6]), x1[6], x2[6]) → 27_0_FIB_NE(-(x1[6], 2))

The following pairs are in Pbound:

27_0_FIB_NE(x0[0]) → COND_27_0_FIB_NE(&&(>(x0[0], 0), !(=(x0[0], 1))), x0[0])
87_1_FIB_INVOKEMETHOD(63_0_fib_Return, x2[3], 1) → COND_87_1_FIB_INVOKEMETHOD(>(x2[3], 0), 63_0_fib_Return, x2[3], 1)
87_1_FIB_INVOKEMETHOD(390_0_fib_Return(x0[5]), x1[5], x2[5]) → COND_87_1_FIB_INVOKEMETHOD1(>(x1[5], 0), 390_0_fib_Return(x0[5]), x1[5], x2[5])

The following pairs are in P≥:

27_0_FIB_NE(x0[0]) → COND_27_0_FIB_NE(&&(>(x0[0], 0), !(=(x0[0], 1))), x0[0])

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, Boolean


The ITRS R consists of the following rules:
27_0_fib_NE(0) → 39_0_fib_Return
374_1_fib_InvokeMethod(39_0_fib_Return, x2, 0) → Cond_374_1_fib_InvokeMethod(x2 > 0, 39_0_fib_Return, x2, 0)
Cond_374_1_fib_InvokeMethod(TRUE, 39_0_fib_Return, x2, 0) → 390_0_fib_Return(x2 + 0)
374_1_fib_InvokeMethod(63_0_fib_Return, x2, 1) → Cond_374_1_fib_InvokeMethod1(x2 > 0 && 1 > 0, 63_0_fib_Return, x2, 1)
Cond_374_1_fib_InvokeMethod1(TRUE, 63_0_fib_Return, x2, 1) → 390_0_fib_Return(x2 + 1)
374_1_fib_InvokeMethod(390_0_fib_Return(x0), x1, x2) → Cond_374_1_fib_InvokeMethod2(x1 > 0 && x0 > 0, 390_0_fib_Return(x0), x1, x2)
Cond_374_1_fib_InvokeMethod2(TRUE, 390_0_fib_Return(x0), x1, x2) → 390_0_fib_Return(x1 + x0)

The integer pair graph contains the following rules and edges:
(0): 27_0_FIB_NE(x0[0]) → COND_27_0_FIB_NE(x0[0] > 0 && !(x0[0] = 1), x0[0])


The set Q consists of the following terms:
27_0_fib_NE(0)
374_1_fib_InvokeMethod(39_0_fib_Return, x0, 0)
Cond_374_1_fib_InvokeMethod(TRUE, 39_0_fib_Return, x0, 0)
374_1_fib_InvokeMethod(63_0_fib_Return, x0, 1)
Cond_374_1_fib_InvokeMethod1(TRUE, 63_0_fib_Return, x0, 1)
374_1_fib_InvokeMethod(390_0_fib_Return(x0), x1, x2)
Cond_374_1_fib_InvokeMethod2(TRUE, 390_0_fib_Return(x0), x1, x2)

(8) IDependencyGraphProof (EQUIVALENT transformation)

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

(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, Boolean


The ITRS R consists of the following rules:
27_0_fib_NE(0) → 39_0_fib_Return
374_1_fib_InvokeMethod(39_0_fib_Return, x2, 0) → Cond_374_1_fib_InvokeMethod(x2 > 0, 39_0_fib_Return, x2, 0)
Cond_374_1_fib_InvokeMethod(TRUE, 39_0_fib_Return, x2, 0) → 390_0_fib_Return(x2 + 0)
374_1_fib_InvokeMethod(63_0_fib_Return, x2, 1) → Cond_374_1_fib_InvokeMethod1(x2 > 0 && 1 > 0, 63_0_fib_Return, x2, 1)
Cond_374_1_fib_InvokeMethod1(TRUE, 63_0_fib_Return, x2, 1) → 390_0_fib_Return(x2 + 1)
374_1_fib_InvokeMethod(390_0_fib_Return(x0), x1, x2) → Cond_374_1_fib_InvokeMethod2(x1 > 0 && x0 > 0, 390_0_fib_Return(x0), x1, x2)
Cond_374_1_fib_InvokeMethod2(TRUE, 390_0_fib_Return(x0), x1, x2) → 390_0_fib_Return(x1 + x0)

The integer pair graph contains the following rules and edges:
(1): COND_27_0_FIB_NE(TRUE, x0[1]) → 87_1_FIB_INVOKEMETHOD(27_0_fib_NE(x0[1] - 1), x0[1], x0[1] - 1)
(2): COND_27_0_FIB_NE(TRUE, x0[2]) → 27_0_FIB_NE(x0[2] - 1)
(4): COND_87_1_FIB_INVOKEMETHOD(TRUE, 63_0_fib_Return, x2[4], 1) → 27_0_FIB_NE(x2[4] - 2)
(6): COND_87_1_FIB_INVOKEMETHOD1(TRUE, 390_0_fib_Return(x0[6]), x1[6], x2[6]) → 27_0_FIB_NE(x1[6] - 2)


The set Q consists of the following terms:
27_0_fib_NE(0)
374_1_fib_InvokeMethod(39_0_fib_Return, x0, 0)
Cond_374_1_fib_InvokeMethod(TRUE, 39_0_fib_Return, x0, 0)
374_1_fib_InvokeMethod(63_0_fib_Return, x0, 1)
Cond_374_1_fib_InvokeMethod1(TRUE, 63_0_fib_Return, x0, 1)
374_1_fib_InvokeMethod(390_0_fib_Return(x0), x1, x2)
Cond_374_1_fib_InvokeMethod2(TRUE, 390_0_fib_Return(x0), x1, x2)

(11) IDependencyGraphProof (EQUIVALENT transformation)

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

(12) TRUE