Termination proof of /tmp/tmpR_8bNI/TerminatorRec02.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_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) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
TerminatorRec02.main([Ljava/lang/String;)V: Graph of 15 nodes with 0 SCCs.

TerminatorRec02.fact(I)I: Graph of 27 nodes with 0 SCCs.

(3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 10 rules for P and 16 rules for R.

Combined rules. Obtained 1 rules for P and 2 rules for R.

Filtered ground terms:

26_0_fact_ConstantStackPush(x1, x2, x3) → 26_0_fact_ConstantStackPush(x2, x3)
Cond_26_0_fact_ConstantStackPush(x1, x2, x3, x4) → Cond_26_0_fact_ConstantStackPush(x1, x3, x4)
75_0_fact_Return(x1) → 75_0_fact_Return
Cond_57_1_fact_InvokeMethod1(x1, x2, x3, x4) → Cond_57_1_fact_InvokeMethod1(x1, x3, x4)
Cond_57_1_fact_InvokeMethod(x1, x2, x3, x4) → Cond_57_1_fact_InvokeMethod(x1, x3)
37_0_fact_Return(x1, x2) → 37_0_fact_Return

Filtered duplicate args:

26_0_fact_ConstantStackPush(x1, x2) → 26_0_fact_ConstantStackPush(x2)
Cond_26_0_fact_ConstantStackPush(x1, x2, x3) → Cond_26_0_fact_ConstantStackPush(x1, x3)

Filtered unneeded arguments:

Cond_57_1_fact_InvokeMethod(x1, x2) → Cond_57_1_fact_InvokeMethod(x1)
Cond_57_1_fact_InvokeMethod1(x1, x2, x3) → Cond_57_1_fact_InvokeMethod1(x1)

Combined rules. Obtained 1 rules for P and 2 rules for R.

Finished conversion. Obtained 1 rules for P and 2 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

The ITRS R consists of the following rules:
57_1_fact_InvokeMethod(37_0_fact_Return, x1, 1) → Cond_57_1_fact_InvokeMethod(x1 > 1, 37_0_fact_Return, x1, 1)
Cond_57_1_fact_InvokeMethod(TRUE, 37_0_fact_Return, x1, 1) → 75_0_fact_Return
57_1_fact_InvokeMethod(75_0_fact_Return, x0, x1) → Cond_57_1_fact_InvokeMethod1(x0 > 1, 75_0_fact_Return, x0, x1)
Cond_57_1_fact_InvokeMethod1(TRUE, 75_0_fact_Return, x0, x1) → 75_0_fact_Return

The integer pair graph contains the following rules and edges:
(0): 26_0_FACT_CONSTANTSTACKPUSH(x0[0]) → COND_26_0_FACT_CONSTANTSTACKPUSH(x0[0] > 1, x0[0])
(1): COND_26_0_FACT_CONSTANTSTACKPUSH(TRUE, x0[1]) → 26_0_FACT_CONSTANTSTACKPUSH(x0[1] - 1)

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

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

The set Q consists of the following terms:
57_1_fact_InvokeMethod(37_0_fact_Return, x0, 1)
Cond_57_1_fact_InvokeMethod(TRUE, 37_0_fact_Return, x0, 1)
57_1_fact_InvokeMethod(75_0_fact_Return, x0, x1)
Cond_57_1_fact_InvokeMethod1(TRUE, 75_0_fact_Return, x0, x1)

(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 26_0_FACT_CONSTANTSTACKPUSH(x0) → COND_26_0_FACT_CONSTANTSTACKPUSH(>(x0, 1), x0) the following chains were created:

We consider the chain 26_0_FACT_CONSTANTSTACKPUSH(x0[0]) → COND_26_0_FACT_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0]), COND_26_0_FACT_CONSTANTSTACKPUSH(TRUE, x0[1]) → 26_0_FACT_CONSTANTSTACKPUSH(-(x0[1], 1)) which results in the following constraint:
(1)    (>(x0[0], 1)=TRUE∧x0[0]=x0[1] ⇒ 26_0_FACT_CONSTANTSTACKPUSH(x0[0])≥NonInfC∧26_0_FACT_CONSTANTSTACKPUSH(x0[0])≥COND_26_0_FACT_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])∧(U^Increasing(COND_26_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 ⇒ 26_0_FACT_CONSTANTSTACKPUSH(x0[0])≥NonInfC∧26_0_FACT_CONSTANTSTACKPUSH(x0[0])≥COND_26_0_FACT_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])∧(U^Increasing(COND_26_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_26_0_FACT_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥)∧[(-1)Bound*bni_16] + [(2)bni_16]x0[0] ≥ 0∧[(-1)bso_17] ≥ 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_26_0_FACT_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥)∧[(-1)Bound*bni_16] + [(2)bni_16]x0[0] ≥ 0∧[(-1)bso_17] ≥ 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_26_0_FACT_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥)∧[(-1)Bound*bni_16] + [(2)bni_16]x0[0] ≥ 0∧[(-1)bso_17] ≥ 0)

We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6)    (x0[0] ≥ 0 ⇒ (U^Increasing(COND_26_0_FACT_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥)∧[(-1)Bound*bni_16 + (4)bni_16] + [(2)bni_16]x0[0] ≥ 0∧[(-1)bso_17] ≥ 0)

For Pair COND_26_0_FACT_CONSTANTSTACKPUSH(TRUE, x0) → 26_0_FACT_CONSTANTSTACKPUSH(-(x0, 1)) the following chains were created:

We consider the chain COND_26_0_FACT_CONSTANTSTACKPUSH(TRUE, x0[1]) → 26_0_FACT_CONSTANTSTACKPUSH(-(x0[1], 1)) which results in the following constraint:
(7)    (COND_26_0_FACT_CONSTANTSTACKPUSH(TRUE, x0[1])≥NonInfC∧COND_26_0_FACT_CONSTANTSTACKPUSH(TRUE, x0[1])≥26_0_FACT_CONSTANTSTACKPUSH(-(x0[1], 1))∧(U^Increasing(26_0_FACT_CONSTANTSTACKPUSH(-(x0[1], 1))), ≥))

We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8)    ((U^Increasing(26_0_FACT_CONSTANTSTACKPUSH(-(x0[1], 1))), ≥)∧[2 + (-1)bso_19] ≥ 0)

We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(9)    ((U^Increasing(26_0_FACT_CONSTANTSTACKPUSH(-(x0[1], 1))), ≥)∧[2 + (-1)bso_19] ≥ 0)

We simplified constraint (9) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(10)    ((U^Increasing(26_0_FACT_CONSTANTSTACKPUSH(-(x0[1], 1))), ≥)∧[2 + (-1)bso_19] ≥ 0)

We simplified constraint (10) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(11)    ((U^Increasing(26_0_FACT_CONSTANTSTACKPUSH(-(x0[1], 1))), ≥)∧0 = 0∧[2 + (-1)bso_19] ≥ 0)

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

26_0_FACT_CONSTANTSTACKPUSH(x0) → COND_26_0_FACT_CONSTANTSTACKPUSH(>(x0, 1), x0)
- (x0[0] ≥ 0 ⇒ (U^Increasing(COND_26_0_FACT_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥)∧[(-1)Bound*bni_16 + (4)bni_16] + [(2)bni_16]x0[0] ≥ 0∧[(-1)bso_17] ≥ 0)
COND_26_0_FACT_CONSTANTSTACKPUSH(TRUE, x0) → 26_0_FACT_CONSTANTSTACKPUSH(-(x0, 1))
- ((U^Increasing(26_0_FACT_CONSTANTSTACKPUSH(-(x0[1], 1))), ≥)∧0 = 0∧[2 + (-1)bso_19] ≥ 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(57_1_fact_InvokeMethod(x₁, x₂, x₃)) = [-1] + [-1]x₂
POL(37_0_fact_Return) = [-1]
POL(1) = [1]
POL(Cond_57_1_fact_InvokeMethod(x₁, x₂, x₃, x₄)) = [-1] + [-1]x₃
POL(>(x₁, x₂)) = [-1]
POL(75_0_fact_Return) = [-1]
POL(Cond_57_1_fact_InvokeMethod1(x₁, x₂, x₃, x₄)) = [-1] + [-1]x₃
POL(26_0_FACT_CONSTANTSTACKPUSH(x₁)) = [2]x₁
POL(COND_26_0_FACT_CONSTANTSTACKPUSH(x₁, x₂)) = [2]x₂
POL(-(x₁, x₂)) = x₁ + [-1]x₂

The following pairs are in P_>:

COND_26_0_FACT_CONSTANTSTACKPUSH(TRUE, x0[1]) → 26_0_FACT_CONSTANTSTACKPUSH(-(x0[1], 1))

The following pairs are in P_bound:

26_0_FACT_CONSTANTSTACKPUSH(x0[0]) → COND_26_0_FACT_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])

The following pairs are in P_≥:

26_0_FACT_CONSTANTSTACKPUSH(x0[0]) → COND_26_0_FACT_CONSTANTSTACKPUSH(>(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

The ITRS R consists of the following rules:
57_1_fact_InvokeMethod(37_0_fact_Return, x1, 1) → Cond_57_1_fact_InvokeMethod(x1 > 1, 37_0_fact_Return, x1, 1)
Cond_57_1_fact_InvokeMethod(TRUE, 37_0_fact_Return, x1, 1) → 75_0_fact_Return
57_1_fact_InvokeMethod(75_0_fact_Return, x0, x1) → Cond_57_1_fact_InvokeMethod1(x0 > 1, 75_0_fact_Return, x0, x1)
Cond_57_1_fact_InvokeMethod1(TRUE, 75_0_fact_Return, x0, x1) → 75_0_fact_Return

The integer pair graph contains the following rules and edges:
(0): 26_0_FACT_CONSTANTSTACKPUSH(x0[0]) → COND_26_0_FACT_CONSTANTSTACKPUSH(x0[0] > 1, x0[0])

The set Q consists of the following terms:
57_1_fact_InvokeMethod(37_0_fact_Return, x0, 1)
Cond_57_1_fact_InvokeMethod(TRUE, 37_0_fact_Return, x0, 1)
57_1_fact_InvokeMethod(75_0_fact_Return, x0, x1)
Cond_57_1_fact_InvokeMethod1(TRUE, 75_0_fact_Return, x0, x1)

(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

The ITRS R consists of the following rules:
57_1_fact_InvokeMethod(37_0_fact_Return, x1, 1) → Cond_57_1_fact_InvokeMethod(x1 > 1, 37_0_fact_Return, x1, 1)
Cond_57_1_fact_InvokeMethod(TRUE, 37_0_fact_Return, x1, 1) → 75_0_fact_Return
57_1_fact_InvokeMethod(75_0_fact_Return, x0, x1) → Cond_57_1_fact_InvokeMethod1(x0 > 1, 75_0_fact_Return, x0, x1)
Cond_57_1_fact_InvokeMethod1(TRUE, 75_0_fact_Return, x0, x1) → 75_0_fact_Return

The integer pair graph contains the following rules and edges:
(1): COND_26_0_FACT_CONSTANTSTACKPUSH(TRUE, x0[1]) → 26_0_FACT_CONSTANTSTACKPUSH(x0[1] - 1)

The set Q consists of the following terms:
57_1_fact_InvokeMethod(37_0_fact_Return, x0, 1)
Cond_57_1_fact_InvokeMethod(TRUE, 37_0_fact_Return, x0, 1)
57_1_fact_InvokeMethod(75_0_fact_Return, x0, x1)
Cond_57_1_fact_InvokeMethod1(TRUE, 75_0_fact_Return, x0, x1)

(11) IDependencyGraphProof (EQUIVALENT transformation)

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

(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