Termination of the given JBC could be proven:

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_22 (Sun Microsystems Inc.) Main-Class: TerminatorRec02

public class TerminatorRec02 {
	public static void main(String[] args) {
		fact(args.length);
	}

	public static int fact(int x) {
		if (x > 1) {
			int y = fact(x - 1);
			return y * x;
		}
		return 1;
	}
}

(1) JBCToGraph (SOUND transformation)

Constructed TerminationGraph.

(2) Obligation:

Termination Graph based on JBC Program:
TerminatorRec02.main([Ljava/lang/String;)V: Graph of 37 nodes with 0 SCCs.

TerminatorRec02.fact(I)I: Graph of 29 nodes with 0 SCCs.

(3) TerminationGraphToSCCProof (SOUND transformation)

Splitted TerminationGraph to 1 SCCs.

(4) Obligation:

SCC of termination graph based on JBC Program.
SCC contains nodes from the following methods: TerminatorRec02.fact(I)I
SCC calls the following helper methods: TerminatorRec02.fact(I)I
Performed SCC analyses: UsedFieldsAnalysis

(5) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 10 rules for P and 17 rules for R.

P rules:
68_0_fact_ConstantStackPush(EOS(STATIC_68), i1, i1) → 69_0_fact_LE(EOS(STATIC_69), i1, i1, 1)
69_0_fact_LE(EOS(STATIC_69), i8, i8, matching1) → 71_0_fact_LE(EOS(STATIC_71), i8, i8, 1) | =(matching1, 1)
71_0_fact_LE(EOS(STATIC_71), i8, i8, matching1) → 74_0_fact_Load(EOS(STATIC_74), i8) | &&(>(i8, 1), =(matching1, 1))
74_0_fact_Load(EOS(STATIC_74), i8) → 78_0_fact_ConstantStackPush(EOS(STATIC_78), i8, i8)
78_0_fact_ConstantStackPush(EOS(STATIC_78), i8, i8) → 84_0_fact_IntArithmetic(EOS(STATIC_84), i8, i8, 1)
84_0_fact_IntArithmetic(EOS(STATIC_84), i8, i8, matching1) → 96_0_fact_InvokeMethod(EOS(STATIC_96), i8, -(i8, 1)) | &&(>(i8, 0), =(matching1, 1))
96_0_fact_InvokeMethod(EOS(STATIC_96), i8, i12) → 100_1_fact_InvokeMethod(100_0_fact_Load(EOS(STATIC_100), i12), i8, i12)
100_0_fact_Load(EOS(STATIC_100), i12) → 105_0_fact_Load(EOS(STATIC_105), i12)
105_0_fact_Load(EOS(STATIC_105), i12) → 65_0_fact_Load(EOS(STATIC_65), i12)
65_0_fact_Load(EOS(STATIC_65), i1) → 68_0_fact_ConstantStackPush(EOS(STATIC_68), i1, i1)
R rules:
69_0_fact_LE(EOS(STATIC_69), i7, i7, matching1) → 70_0_fact_LE(EOS(STATIC_70), i7, i7, 1) | =(matching1, 1)
70_0_fact_LE(EOS(STATIC_70), i7, i7, matching1) → 72_0_fact_ConstantStackPush(EOS(STATIC_72)) | &&(<=(i7, 1), =(matching1, 1))
72_0_fact_ConstantStackPush(EOS(STATIC_72)) → 76_0_fact_Return(EOS(STATIC_76), 1)
100_1_fact_InvokeMethod(76_0_fact_Return(EOS(STATIC_76), matching1), i8, matching2) → 120_0_fact_Return(EOS(STATIC_120), i8, 1, 1) | &&(=(matching1, 1), =(matching2, 1))
100_1_fact_InvokeMethod(131_0_fact_Return(EOS(STATIC_131)), i8, i22) → 146_0_fact_Return(EOS(STATIC_146), i8, i22)
100_1_fact_InvokeMethod(167_0_fact_Return(EOS(STATIC_167)), i8, i30) → 186_0_fact_Return(EOS(STATIC_186), i8, i30)
120_0_fact_Return(EOS(STATIC_120), i8, matching1, matching2) → 122_0_fact_Store(EOS(STATIC_122), i8, 1) | &&(=(matching1, 1), =(matching2, 1))
122_0_fact_Store(EOS(STATIC_122), i8, matching1) → 124_0_fact_Load(EOS(STATIC_124), i8, 1) | =(matching1, 1)
124_0_fact_Load(EOS(STATIC_124), i8, matching1) → 126_0_fact_Load(EOS(STATIC_126), i8, 1) | =(matching1, 1)
126_0_fact_Load(EOS(STATIC_126), i8, matching1) → 129_0_fact_IntArithmetic(EOS(STATIC_129), 1, i8) | =(matching1, 1)
129_0_fact_IntArithmetic(EOS(STATIC_129), matching1, i8) → 131_0_fact_Return(EOS(STATIC_131)) | &&(>(i8, 1), =(matching1, 1))
146_0_fact_Return(EOS(STATIC_146), i8, i22) → 152_0_fact_Store(EOS(STATIC_152), i8)
152_0_fact_Store(EOS(STATIC_152), i8) → 157_0_fact_Load(EOS(STATIC_157), i8)
157_0_fact_Load(EOS(STATIC_157), i8) → 162_0_fact_Load(EOS(STATIC_162), i8)
162_0_fact_Load(EOS(STATIC_162), i8) → 164_0_fact_IntArithmetic(EOS(STATIC_164), i8)
164_0_fact_IntArithmetic(EOS(STATIC_164), i8) → 167_0_fact_Return(EOS(STATIC_167)) | >(i8, 1)
186_0_fact_Return(EOS(STATIC_186), i8, i30) → 146_0_fact_Return(EOS(STATIC_146), i8, i30)

Combined rules. Obtained 1 conditional rules for P and 3 conditional rules for R.

P rules:
68_0_fact_ConstantStackPush(EOS(STATIC_68), x0, x0) → 100_1_fact_InvokeMethod(68_0_fact_ConstantStackPush(EOS(STATIC_68), -(x0, 1), -(x0, 1)), x0, -(x0, 1)) | >(x0, 1)
R rules:
100_1_fact_InvokeMethod(76_0_fact_Return(EOS(STATIC_76), 1), x1, 1) → 131_0_fact_Return(EOS(STATIC_131)) | >(x1, 1)
100_1_fact_InvokeMethod(131_0_fact_Return(EOS(STATIC_131)), x0, x1) → 167_0_fact_Return(EOS(STATIC_167)) | >(x0, 1)
100_1_fact_InvokeMethod(167_0_fact_Return(EOS(STATIC_167)), x0, x1) → 167_0_fact_Return(EOS(STATIC_167)) | >(x0, 1)

Filtered ground terms:

68_0_fact_ConstantStackPush(x1, x2, x3) → 68_0_fact_ConstantStackPush(x2, x3)
Cond_68_0_fact_ConstantStackPush(x1, x2, x3, x4) → Cond_68_0_fact_ConstantStackPush(x1, x3, x4)
167_0_fact_Return(x1) → 167_0_fact_Return
Cond_100_1_fact_InvokeMethod2(x1, x2, x3, x4) → Cond_100_1_fact_InvokeMethod2(x1, x3, x4)
Cond_100_1_fact_InvokeMethod1(x1, x2, x3, x4) → Cond_100_1_fact_InvokeMethod1(x1, x3, x4)
131_0_fact_Return(x1) → 131_0_fact_Return
Cond_100_1_fact_InvokeMethod(x1, x2, x3, x4) → Cond_100_1_fact_InvokeMethod(x1, x3)
76_0_fact_Return(x1, x2) → 76_0_fact_Return

Filtered duplicate args:

68_0_fact_ConstantStackPush(x1, x2) → 68_0_fact_ConstantStackPush(x2)
Cond_68_0_fact_ConstantStackPush(x1, x2, x3) → Cond_68_0_fact_ConstantStackPush(x1, x3)

Filtered unneeded arguments:

Cond_100_1_fact_InvokeMethod(x1, x2) → Cond_100_1_fact_InvokeMethod(x1)
Cond_100_1_fact_InvokeMethod1(x1, x2, x3) → Cond_100_1_fact_InvokeMethod1(x1)
Cond_100_1_fact_InvokeMethod2(x1, x2, x3) → Cond_100_1_fact_InvokeMethod2(x1)

Combined rules. Obtained 1 conditional rules for P and 3 conditional rules for R.

P rules:
68_0_fact_ConstantStackPush(x0) → 100_1_fact_InvokeMethod(68_0_fact_ConstantStackPush(-(x0, 1)), x0, -(x0, 1)) | >(x0, 1)
R rules:
100_1_fact_InvokeMethod(76_0_fact_Return, x1, 1) → 131_0_fact_Return | >(x1, 1)
100_1_fact_InvokeMethod(131_0_fact_Return, x0, x1) → 167_0_fact_Return | >(x0, 1)
100_1_fact_InvokeMethod(167_0_fact_Return, x0, x1) → 167_0_fact_Return | >(x0, 1)

Performed bisimulation on rules. Used the following equivalence classes: {[76_0_fact_Return, 131_0_fact_Return, 167_0_fact_Return]=76_0_fact_Return, [Cond_100_1_fact_InvokeMethod1_4, Cond_100_1_fact_InvokeMethod2_4]=Cond_100_1_fact_InvokeMethod1_4}

Finished conversion. Obtained 2 rules for P and 4 rules for R. System has predefined symbols.

P rules:
68_0_FACT_CONSTANTSTACKPUSH(x0) → COND_68_0_FACT_CONSTANTSTACKPUSH(>(x0, 1), x0)
COND_68_0_FACT_CONSTANTSTACKPUSH(TRUE, x0) → 68_0_FACT_CONSTANTSTACKPUSH(-(x0, 1))
R rules:
100_1_fact_InvokeMethod(76_0_fact_Return, x1, 1) → Cond_100_1_fact_InvokeMethod(>(x1, 1), 76_0_fact_Return, x1, 1)
Cond_100_1_fact_InvokeMethod(TRUE, 76_0_fact_Return, x1, 1) → 76_0_fact_Return
100_1_fact_InvokeMethod(76_0_fact_Return, x0, x1) → Cond_100_1_fact_InvokeMethod1(>(x0, 1), 76_0_fact_Return, x0, x1)
Cond_100_1_fact_InvokeMethod1(TRUE, 76_0_fact_Return, x0, x1) → 76_0_fact_Return

(6) Obligation:

IDP problem:
The following function symbols are pre-defined:

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

The following domains are used:

Integer

The ITRS R consists of the following rules:
100_1_fact_InvokeMethod(76_0_fact_Return, x1, 1) → Cond_100_1_fact_InvokeMethod(x1 > 1, 76_0_fact_Return, x1, 1)
Cond_100_1_fact_InvokeMethod(TRUE, 76_0_fact_Return, x1, 1) → 76_0_fact_Return
100_1_fact_InvokeMethod(76_0_fact_Return, x0, x1) → Cond_100_1_fact_InvokeMethod1(x0 > 1, 76_0_fact_Return, x0, x1)
Cond_100_1_fact_InvokeMethod1(TRUE, 76_0_fact_Return, x0, x1) → 76_0_fact_Return

The integer pair graph contains the following rules and edges:
(0): 68_0_FACT_CONSTANTSTACKPUSH(x0[0]) → COND_68_0_FACT_CONSTANTSTACKPUSH(x0[0] > 1, x0[0])
(1): COND_68_0_FACT_CONSTANTSTACKPUSH(TRUE, x0[1]) → 68_0_FACT_CONSTANTSTACKPUSH(x0[1] - 1)

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

(1) -> (0), if (x0[1] - 1 →^* x0[0])

The set Q consists of the following terms:
Cond_100_1_fact_InvokeMethod(TRUE, 76_0_fact_Return, x0, 1)
100_1_fact_InvokeMethod(76_0_fact_Return, x0, x1)
Cond_100_1_fact_InvokeMethod1(TRUE, 76_0_fact_Return, x0, x1)

(7) IDPNonInfProof (SOUND transformation)

Used the following options for this NonInfProof:
IDPGPoloSolver: Range: [(-1,2)] IsNat: false Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpCand1ShapeHeuristic@5e34d46a Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 0 Max Right Steps: 0

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 68_0_FACT_CONSTANTSTACKPUSH(x0) → COND_68_0_FACT_CONSTANTSTACKPUSH(>(x0, 1), x0) the following chains were created:

• We consider the chain 68_0_FACT_CONSTANTSTACKPUSH(x0[0]) → COND_68_0_FACT_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0]), COND_68_0_FACT_CONSTANTSTACKPUSH(TRUE, x0[1]) → 68_0_FACT_CONSTANTSTACKPUSH(-(x0[1], 1)) which results in the following constraint:
  

  (1)    (>(x0[0], 1)=TRUE∧x0[0]=x0[1] ⇒ 68_0_FACT_CONSTANTSTACKPUSH(x0[0])≥NonInfC∧68_0_FACT_CONSTANTSTACKPUSH(x0[0])≥COND_68_0_FACT_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])∧(U^Increasing(COND_68_0_FACT_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥))

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

  (2)    (>(x0[0], 1)=TRUE ⇒ 68_0_FACT_CONSTANTSTACKPUSH(x0[0])≥NonInfC∧68_0_FACT_CONSTANTSTACKPUSH(x0[0])≥COND_68_0_FACT_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])∧(U^Increasing(COND_68_0_FACT_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥))

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

  (3)    (x0[0] + [-2] ≥ 0 ⇒ (U^Increasing(COND_68_0_FACT_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥)∧[(-1)Bound*bni_15] + [(2)bni_15]x0[0] ≥ 0∧[(-1)bso_16] ≥ 0)

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

  (4)    (x0[0] + [-2] ≥ 0 ⇒ (U^Increasing(COND_68_0_FACT_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥)∧[(-1)Bound*bni_15] + [(2)bni_15]x0[0] ≥ 0∧[(-1)bso_16] ≥ 0)

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

  (5)    (x0[0] + [-2] ≥ 0 ⇒ (U^Increasing(COND_68_0_FACT_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥)∧[(-1)Bound*bni_15] + [(2)bni_15]x0[0] ≥ 0∧[(-1)bso_16] ≥ 0)

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

  (6)    (x0[0] ≥ 0 ⇒ (U^Increasing(COND_68_0_FACT_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥)∧[(-1)Bound*bni_15 + (4)bni_15] + [(2)bni_15]x0[0] ≥ 0∧[(-1)bso_16] ≥ 0)

  
  
  

For Pair COND_68_0_FACT_CONSTANTSTACKPUSH(TRUE, x0) → 68_0_FACT_CONSTANTSTACKPUSH(-(x0, 1)) the following chains were created:

• We consider the chain COND_68_0_FACT_CONSTANTSTACKPUSH(TRUE, x0[1]) → 68_0_FACT_CONSTANTSTACKPUSH(-(x0[1], 1)) which results in the following constraint:
  

  (7)    (COND_68_0_FACT_CONSTANTSTACKPUSH(TRUE, x0[1])≥NonInfC∧COND_68_0_FACT_CONSTANTSTACKPUSH(TRUE, x0[1])≥68_0_FACT_CONSTANTSTACKPUSH(-(x0[1], 1))∧(U^Increasing(68_0_FACT_CONSTANTSTACKPUSH(-(x0[1], 1))), ≥))

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

  (8)    ((U^Increasing(68_0_FACT_CONSTANTSTACKPUSH(-(x0[1], 1))), ≥)∧[bni_17] = 0∧[2 + (-1)bso_18] ≥ 0)

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

  (9)    ((U^Increasing(68_0_FACT_CONSTANTSTACKPUSH(-(x0[1], 1))), ≥)∧[bni_17] = 0∧[2 + (-1)bso_18] ≥ 0)

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

  (10)    ((U^Increasing(68_0_FACT_CONSTANTSTACKPUSH(-(x0[1], 1))), ≥)∧[bni_17] = 0∧[2 + (-1)bso_18] ≥ 0)

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

  (11)    ((U^Increasing(68_0_FACT_CONSTANTSTACKPUSH(-(x0[1], 1))), ≥)∧[bni_17] = 0∧0 = 0∧[2 + (-1)bso_18] ≥ 0)

  
  
  

To summarize, we get the following constraints P_≥ for the following pairs.

• 68_0_FACT_CONSTANTSTACKPUSH(x0) → COND_68_0_FACT_CONSTANTSTACKPUSH(>(x0, 1), x0)
  
  • (x0[0] ≥ 0 ⇒ (U^Increasing(COND_68_0_FACT_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥)∧[(-1)Bound*bni_15 + (4)bni_15] + [(2)bni_15]x0[0] ≥ 0∧[(-1)bso_16] ≥ 0)
    
  
  
• COND_68_0_FACT_CONSTANTSTACKPUSH(TRUE, x0) → 68_0_FACT_CONSTANTSTACKPUSH(-(x0, 1))
  
  • ((U^Increasing(68_0_FACT_CONSTANTSTACKPUSH(-(x0[1], 1))), ≥)∧[bni_17] = 0∧0 = 0∧[2 + (-1)bso_18] ≥ 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(100_1_fact_InvokeMethod(x₁, x₂, x₃)) = [-1] + [-1]x₂   
POL(76_0_fact_Return) = [-1]   
POL(1) = [1]   
POL(Cond_100_1_fact_InvokeMethod(x₁, x₂, x₃, x₄)) = [-1] + [-1]x₃   
POL(>(x₁, x₂)) = [-1]   
POL(Cond_100_1_fact_InvokeMethod1(x₁, x₂, x₃, x₄)) = [-1] + [-1]x₃   
POL(68_0_FACT_CONSTANTSTACKPUSH(x₁)) = [2]x₁   
POL(COND_68_0_FACT_CONSTANTSTACKPUSH(x₁, x₂)) = [2]x₂   
POL(-(x₁, x₂)) = x₁ + [-1]x₂   

The following pairs are in P_>:

COND_68_0_FACT_CONSTANTSTACKPUSH(TRUE, x0[1]) → 68_0_FACT_CONSTANTSTACKPUSH(-(x0[1], 1))

The following pairs are in P_bound:

68_0_FACT_CONSTANTSTACKPUSH(x0[0]) → COND_68_0_FACT_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])

The following pairs are in P_≥:

68_0_FACT_CONSTANTSTACKPUSH(x0[0]) → COND_68_0_FACT_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])

There are no usable rules.

(8) Complex Obligation (AND)

(9) Obligation:

IDP problem:
The following function symbols are pre-defined:

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

The following domains are used:

Integer

The ITRS R consists of the following rules:
100_1_fact_InvokeMethod(76_0_fact_Return, x1, 1) → Cond_100_1_fact_InvokeMethod(x1 > 1, 76_0_fact_Return, x1, 1)
Cond_100_1_fact_InvokeMethod(TRUE, 76_0_fact_Return, x1, 1) → 76_0_fact_Return
100_1_fact_InvokeMethod(76_0_fact_Return, x0, x1) → Cond_100_1_fact_InvokeMethod1(x0 > 1, 76_0_fact_Return, x0, x1)
Cond_100_1_fact_InvokeMethod1(TRUE, 76_0_fact_Return, x0, x1) → 76_0_fact_Return

The integer pair graph contains the following rules and edges:
(0): 68_0_FACT_CONSTANTSTACKPUSH(x0[0]) → COND_68_0_FACT_CONSTANTSTACKPUSH(x0[0] > 1, x0[0])


The set Q consists of the following terms:
Cond_100_1_fact_InvokeMethod(TRUE, 76_0_fact_Return, x0, 1)
100_1_fact_InvokeMethod(76_0_fact_Return, x0, x1)
Cond_100_1_fact_InvokeMethod1(TRUE, 76_0_fact_Return, x0, x1)

(10) IDependencyGraphProof (EQUIVALENT transformation)

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

(11) TRUE

(12) Obligation:

IDP problem:
The following function symbols are pre-defined:

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

The following domains are used:

Integer

The ITRS R consists of the following rules:
100_1_fact_InvokeMethod(76_0_fact_Return, x1, 1) → Cond_100_1_fact_InvokeMethod(x1 > 1, 76_0_fact_Return, x1, 1)
Cond_100_1_fact_InvokeMethod(TRUE, 76_0_fact_Return, x1, 1) → 76_0_fact_Return
100_1_fact_InvokeMethod(76_0_fact_Return, x0, x1) → Cond_100_1_fact_InvokeMethod1(x0 > 1, 76_0_fact_Return, x0, x1)
Cond_100_1_fact_InvokeMethod1(TRUE, 76_0_fact_Return, x0, x1) → 76_0_fact_Return

The integer pair graph contains the following rules and edges:
(1): COND_68_0_FACT_CONSTANTSTACKPUSH(TRUE, x0[1]) → 68_0_FACT_CONSTANTSTACKPUSH(x0[1] - 1)


The set Q consists of the following terms:
Cond_100_1_fact_InvokeMethod(TRUE, 76_0_fact_Return, x0, 1)
100_1_fact_InvokeMethod(76_0_fact_Return, x0, x1)
Cond_100_1_fact_InvokeMethod1(TRUE, 76_0_fact_Return, x0, x1)

(13) IDependencyGraphProof (EQUIVALENT transformation)

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

(14) TRUE