Termination proof of /tmp/tmpJ5mEeD/TerminatorRec02.jar

Termination of the given JBC could be proven:

0 JBC

↳1 JBCToGraph (⇒, 60 ms)

↳2 JBCTerminationGraph

↳3 TerminationGraphToSCCProof (⇒, 0 ms)

↳4 JBCTerminationSCC

↳5 SCCToIDPv1Proof (⇒, 100 ms)

↳6 IDP

↳7 IDPNonInfProof (⇒, 140 ms)

↳8 AND

    ↳9 IDP

        ↳10 IDependencyGraphProof (⇔, 0 ms)

        ↳11 TRUE

    ↳12 IDP

        ↳13 IDependencyGraphProof (⇔, 0 ms)

        ↳14 TRUE

(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) → 75_0_fact_Load(EOS(STATIC_75), i8) | &&(>(i8, 1), =(matching1, 1))
75_0_fact_Load(EOS(STATIC_75), i8) → 79_0_fact_ConstantStackPush(EOS(STATIC_79), i8, i8)
79_0_fact_ConstantStackPush(EOS(STATIC_79), i8, i8) → 84_0_fact_IntArithmetic(EOS(STATIC_84), i8, i8, 1)
84_0_fact_IntArithmetic(EOS(STATIC_84), i8, i8, matching1) → 95_0_fact_InvokeMethod(EOS(STATIC_95), i8, -(i8, 1)) | &&(>(i8, 0), =(matching1, 1))
95_0_fact_InvokeMethod(EOS(STATIC_95), i8, i12) → 98_1_fact_InvokeMethod(98_0_fact_Load(EOS(STATIC_98), i12), i8, i12)
98_0_fact_Load(EOS(STATIC_98), i12) → 103_0_fact_Load(EOS(STATIC_103), i12)
103_0_fact_Load(EOS(STATIC_103), i12) → 67_0_fact_Load(EOS(STATIC_67), i12)
67_0_fact_Load(EOS(STATIC_67), 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) → 73_0_fact_ConstantStackPush(EOS(STATIC_73)) | &&(<=(i7, 1), =(matching1, 1))
73_0_fact_ConstantStackPush(EOS(STATIC_73)) → 77_0_fact_Return(EOS(STATIC_77), 1)
98_1_fact_InvokeMethod(77_0_fact_Return(EOS(STATIC_77), matching1), i8, matching2) → 118_0_fact_Return(EOS(STATIC_118), i8, 1, 1) | &&(=(matching1, 1), =(matching2, 1))
98_1_fact_InvokeMethod(128_0_fact_Return(EOS(STATIC_128)), i8, i22) → 142_0_fact_Return(EOS(STATIC_142), i8, i22)
98_1_fact_InvokeMethod(162_0_fact_Return(EOS(STATIC_162)), i8, i30) → 181_0_fact_Return(EOS(STATIC_181), i8, i30)
118_0_fact_Return(EOS(STATIC_118), i8, matching1, matching2) → 120_0_fact_Store(EOS(STATIC_120), i8, 1) | &&(=(matching1, 1), =(matching2, 1))
120_0_fact_Store(EOS(STATIC_120), i8, matching1) → 122_0_fact_Load(EOS(STATIC_122), i8, 1) | =(matching1, 1)
122_0_fact_Load(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_IntArithmetic(EOS(STATIC_126), 1, i8) | =(matching1, 1)
126_0_fact_IntArithmetic(EOS(STATIC_126), matching1, i8) → 128_0_fact_Return(EOS(STATIC_128)) | &&(>(i8, 1), =(matching1, 1))
142_0_fact_Return(EOS(STATIC_142), i8, i22) → 147_0_fact_Store(EOS(STATIC_147), i8)
147_0_fact_Store(EOS(STATIC_147), i8) → 152_0_fact_Load(EOS(STATIC_152), i8)
152_0_fact_Load(EOS(STATIC_152), i8) → 157_0_fact_Load(EOS(STATIC_157), i8)
157_0_fact_Load(EOS(STATIC_157), i8) → 160_0_fact_IntArithmetic(EOS(STATIC_160), i8)
160_0_fact_IntArithmetic(EOS(STATIC_160), i8) → 162_0_fact_Return(EOS(STATIC_162)) | >(i8, 1)
181_0_fact_Return(EOS(STATIC_181), i8, i30) → 142_0_fact_Return(EOS(STATIC_142), 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) → 98_1_fact_InvokeMethod(68_0_fact_ConstantStackPush(EOS(STATIC_68), -(x0, 1), -(x0, 1)), x0, -(x0, 1)) | >(x0, 1)
R rules:
98_1_fact_InvokeMethod(77_0_fact_Return(EOS(STATIC_77), 1), x1, 1) → 128_0_fact_Return(EOS(STATIC_128)) | >(x1, 1)
98_1_fact_InvokeMethod(128_0_fact_Return(EOS(STATIC_128)), x0, x1) → 162_0_fact_Return(EOS(STATIC_162)) | >(x0, 1)
98_1_fact_InvokeMethod(162_0_fact_Return(EOS(STATIC_162)), x0, x1) → 162_0_fact_Return(EOS(STATIC_162)) | >(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)
162_0_fact_Return(x1) → 162_0_fact_Return
Cond_98_1_fact_InvokeMethod2(x1, x2, x3, x4) → Cond_98_1_fact_InvokeMethod2(x1, x3, x4)
Cond_98_1_fact_InvokeMethod1(x1, x2, x3, x4) → Cond_98_1_fact_InvokeMethod1(x1, x3, x4)
128_0_fact_Return(x1) → 128_0_fact_Return
Cond_98_1_fact_InvokeMethod(x1, x2, x3, x4) → Cond_98_1_fact_InvokeMethod(x1, x3)
77_0_fact_Return(x1, x2) → 77_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_98_1_fact_InvokeMethod(x1, x2) → Cond_98_1_fact_InvokeMethod(x1)
Cond_98_1_fact_InvokeMethod1(x1, x2, x3) → Cond_98_1_fact_InvokeMethod1(x1)
Cond_98_1_fact_InvokeMethod2(x1, x2, x3) → Cond_98_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) → 98_1_fact_InvokeMethod(68_0_fact_ConstantStackPush(-(x0, 1)), x0, -(x0, 1)) | >(x0, 1)
R rules:
98_1_fact_InvokeMethod(77_0_fact_Return, x1, 1) → 128_0_fact_Return | >(x1, 1)
98_1_fact_InvokeMethod(128_0_fact_Return, x0, x1) → 162_0_fact_Return | >(x0, 1)
98_1_fact_InvokeMethod(162_0_fact_Return, x0, x1) → 162_0_fact_Return | >(x0, 1)

Performed bisimulation on rules. Used the following equivalence classes: {[77_0_fact_Return, 128_0_fact_Return, 162_0_fact_Return]=77_0_fact_Return, [Cond_98_1_fact_InvokeMethod1_4, Cond_98_1_fact_InvokeMethod2_4]=Cond_98_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:
98_1_fact_InvokeMethod(77_0_fact_Return, x1, 1) → Cond_98_1_fact_InvokeMethod(>(x1, 1), 77_0_fact_Return, x1, 1)
Cond_98_1_fact_InvokeMethod(TRUE, 77_0_fact_Return, x1, 1) → 77_0_fact_Return
98_1_fact_InvokeMethod(77_0_fact_Return, x0, x1) → Cond_98_1_fact_InvokeMethod1(>(x0, 1), 77_0_fact_Return, x0, x1)
Cond_98_1_fact_InvokeMethod1(TRUE, 77_0_fact_Return, x0, x1) → 77_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:
98_1_fact_InvokeMethod(77_0_fact_Return, x1, 1) → Cond_98_1_fact_InvokeMethod(x1 > 1, 77_0_fact_Return, x1, 1)
Cond_98_1_fact_InvokeMethod(TRUE, 77_0_fact_Return, x1, 1) → 77_0_fact_Return
98_1_fact_InvokeMethod(77_0_fact_Return, x0, x1) → Cond_98_1_fact_InvokeMethod1(x0 > 1, 77_0_fact_Return, x0, x1)
Cond_98_1_fact_InvokeMethod1(TRUE, 77_0_fact_Return, x0, x1) → 77_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_98_1_fact_InvokeMethod(TRUE, 77_0_fact_Return, x0, 1)
98_1_fact_InvokeMethod(77_0_fact_Return, x0, x1)
Cond_98_1_fact_InvokeMethod1(TRUE, 77_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@140de648 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(98_1_fact_InvokeMethod(x₁, x₂, x₃)) = [-1] + [-1]x₂
POL(77_0_fact_Return) = [-1]
POL(1) = [1]
POL(Cond_98_1_fact_InvokeMethod(x₁, x₂, x₃, x₄)) = [-1] + [-1]x₃
POL(>(x₁, x₂)) = [-1]
POL(Cond_98_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:
98_1_fact_InvokeMethod(77_0_fact_Return, x1, 1) → Cond_98_1_fact_InvokeMethod(x1 > 1, 77_0_fact_Return, x1, 1)
Cond_98_1_fact_InvokeMethod(TRUE, 77_0_fact_Return, x1, 1) → 77_0_fact_Return
98_1_fact_InvokeMethod(77_0_fact_Return, x0, x1) → Cond_98_1_fact_InvokeMethod1(x0 > 1, 77_0_fact_Return, x0, x1)
Cond_98_1_fact_InvokeMethod1(TRUE, 77_0_fact_Return, x0, x1) → 77_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_98_1_fact_InvokeMethod(TRUE, 77_0_fact_Return, x0, 1)
98_1_fact_InvokeMethod(77_0_fact_Return, x0, x1)
Cond_98_1_fact_InvokeMethod1(TRUE, 77_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:
98_1_fact_InvokeMethod(77_0_fact_Return, x1, 1) → Cond_98_1_fact_InvokeMethod(x1 > 1, 77_0_fact_Return, x1, 1)
Cond_98_1_fact_InvokeMethod(TRUE, 77_0_fact_Return, x1, 1) → 77_0_fact_Return
98_1_fact_InvokeMethod(77_0_fact_Return, x0, x1) → Cond_98_1_fact_InvokeMethod1(x0 > 1, 77_0_fact_Return, x0, x1)
Cond_98_1_fact_InvokeMethod1(TRUE, 77_0_fact_Return, x0, x1) → 77_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_98_1_fact_InvokeMethod(TRUE, 77_0_fact_Return, x0, 1)
98_1_fact_InvokeMethod(77_0_fact_Return, x0, x1)
Cond_98_1_fact_InvokeMethod1(TRUE, 77_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.

(0) Obligation:

(1) JBCToGraph (SOUND transformation)

(2) Obligation:

(3) TerminationGraphToSCCProof (SOUND transformation)

(4) Obligation:

(5) SCCToIDPv1Proof (SOUND transformation)

(6) Obligation:

(7) IDPNonInfProof (SOUND transformation)

(8) Complex Obligation (AND)

(9) Obligation:

(10) IDependencyGraphProof (EQUIVALENT transformation)

(11) TRUE

(12) Obligation:

(13) IDependencyGraphProof (EQUIVALENT transformation)

(14) TRUE