Termination proof of /tmp/tmpzI30MD/LeUserDefRec.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: LeUserDefRec

public class LeUserDefRec {
	public static void main(String[] args) {
		int x = args[0].length();
		int y = args[1].length();
		le(x, y);
	}

	public static boolean le(int x, int y) {
		if (x > 0 && y > 0) {
			return le(x-1, y-1);
		} else {
			return (x == 0);
		}
	}
}

(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
LeUserDefRec.main([Ljava/lang/String;)V: Graph of 129 nodes with 0 SCCs.

LeUserDefRec.le(II)Z: Graph of 35 nodes with 0 SCCs.

(3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 15 rules for P and 19 rules for R.

Combined rules. Obtained 1 rules for P and 4 rules for R.

Filtered ground terms:

415_0_le_LE(x1, x2, x3, x4) → 415_0_le_LE(x2, x3, x4)
Cond_415_0_le_LE(x1, x2, x3, x4, x5) → Cond_415_0_le_LE(x1, x3, x4, x5)
689_0_le_Return(x1, x2) → 689_0_le_Return(x2)
643_0_le_Return(x1, x2) → 643_0_le_Return
616_0_le_Return(x1) → 616_0_le_Return

Filtered duplicate args:

415_0_le_LE(x1, x2, x3) → 415_0_le_LE(x2, x3)
Cond_415_0_le_LE(x1, x2, x3, x4) → Cond_415_0_le_LE(x1, x3, x4)

Filtered unneeded arguments:

681_1_le_InvokeMethod(x1, x2, x3, x4) → 681_1_le_InvokeMethod(x1, x3, x4)

Combined rules. Obtained 1 rules for P and 4 rules for R.

Finished conversion. Obtained 1 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:

Boolean, Integer

The ITRS R consists of the following rules:
415_0_le_LE(x1, 0) → 616_0_le_Return
681_1_le_InvokeMethod(616_0_le_Return, 0, x2) → 689_0_le_Return(x0)
681_1_le_InvokeMethod(643_0_le_Return, x2, 0) → 689_0_le_Return(x1)
681_1_le_InvokeMethod(689_0_le_Return(x0), x0, x2) → 689_0_le_Return(x1)

The integer pair graph contains the following rules and edges:
(0): 415_0_LE_LE(x1[0], x0[0]) → COND_415_0_LE_LE(x1[0] > 0 && x0[0] > 0, x1[0], x0[0])
(1): COND_415_0_LE_LE(TRUE, x1[1], x0[1]) → 415_0_LE_LE(x1[1] - 1, x0[1] - 1)

(0) -> (1), if ((x1[0] > 0 && x0[0] > 0 →^* TRUE)∧(x1[0] →^* x1[1])∧(x0[0] →^* x0[1]))

(1) -> (0), if ((x1[1] - 1 →^* x1[0])∧(x0[1] - 1 →^* x0[0]))

The set Q consists of the following terms:
415_0_le_LE(x0, 0)
681_1_le_InvokeMethod(616_0_le_Return, 0, x0)
681_1_le_InvokeMethod(643_0_le_Return, x0, 0)
681_1_le_InvokeMethod(689_0_le_Return(x0), 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 415_0_LE_LE(x1, x0) → COND_415_0_LE_LE(&&(>(x1, 0), >(x0, 0)), x1, x0) the following chains were created:

We consider the chain 415_0_LE_LE(x1[0], x0[0]) → COND_415_0_LE_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[0]), COND_415_0_LE_LE(TRUE, x1[1], x0[1]) → 415_0_LE_LE(-(x1[1], 1), -(x0[1], 1)) which results in the following constraint:
(1)    (&&(>(x1[0], 0), >(x0[0], 0))=TRUE∧x1[0]=x1[1]∧x0[0]=x0[1] ⇒ 415_0_LE_LE(x1[0], x0[0])≥NonInfC∧415_0_LE_LE(x1[0], x0[0])≥COND_415_0_LE_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])∧(U^Increasing(COND_415_0_LE_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])), ≥))

We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2)    (>(x1[0], 0)=TRUE∧>(x0[0], 0)=TRUE ⇒ 415_0_LE_LE(x1[0], x0[0])≥NonInfC∧415_0_LE_LE(x1[0], x0[0])≥COND_415_0_LE_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])∧(U^Increasing(COND_415_0_LE_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x1[0] + [-1] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_415_0_LE_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[bni_14 + (-1)Bound*bni_14] + [(2)bni_14]x0[0] + [(2)bni_14]x1[0] ≥ 0∧[(-1)bso_15] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x1[0] + [-1] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_415_0_LE_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[bni_14 + (-1)Bound*bni_14] + [(2)bni_14]x0[0] + [(2)bni_14]x1[0] ≥ 0∧[(-1)bso_15] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x1[0] + [-1] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_415_0_LE_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[bni_14 + (-1)Bound*bni_14] + [(2)bni_14]x0[0] + [(2)bni_14]x1[0] ≥ 0∧[(-1)bso_15] ≥ 0)

We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6)    (x1[0] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_415_0_LE_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(3)bni_14 + (-1)Bound*bni_14] + [(2)bni_14]x0[0] + [(2)bni_14]x1[0] ≥ 0∧[(-1)bso_15] ≥ 0)

We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7)    (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_415_0_LE_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(5)bni_14 + (-1)Bound*bni_14] + [(2)bni_14]x0[0] + [(2)bni_14]x1[0] ≥ 0∧[(-1)bso_15] ≥ 0)

For Pair COND_415_0_LE_LE(TRUE, x1, x0) → 415_0_LE_LE(-(x1, 1), -(x0, 1)) the following chains were created:

We consider the chain COND_415_0_LE_LE(TRUE, x1[1], x0[1]) → 415_0_LE_LE(-(x1[1], 1), -(x0[1], 1)) which results in the following constraint:
(8)    (COND_415_0_LE_LE(TRUE, x1[1], x0[1])≥NonInfC∧COND_415_0_LE_LE(TRUE, x1[1], x0[1])≥415_0_LE_LE(-(x1[1], 1), -(x0[1], 1))∧(U^Increasing(415_0_LE_LE(-(x1[1], 1), -(x0[1], 1))), ≥))

We simplified constraint (8) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(9)    ((U^Increasing(415_0_LE_LE(-(x1[1], 1), -(x0[1], 1))), ≥)∧[4 + (-1)bso_17] ≥ 0)

We simplified constraint (9) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(10)    ((U^Increasing(415_0_LE_LE(-(x1[1], 1), -(x0[1], 1))), ≥)∧[4 + (-1)bso_17] ≥ 0)

We simplified constraint (10) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(11)    ((U^Increasing(415_0_LE_LE(-(x1[1], 1), -(x0[1], 1))), ≥)∧[4 + (-1)bso_17] ≥ 0)

We simplified constraint (11) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(12)    ((U^Increasing(415_0_LE_LE(-(x1[1], 1), -(x0[1], 1))), ≥)∧0 = 0∧0 = 0∧[4 + (-1)bso_17] ≥ 0)

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

415_0_LE_LE(x1, x0) → COND_415_0_LE_LE(&&(>(x1, 0), >(x0, 0)), x1, x0)
- (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_415_0_LE_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(5)bni_14 + (-1)Bound*bni_14] + [(2)bni_14]x0[0] + [(2)bni_14]x1[0] ≥ 0∧[(-1)bso_15] ≥ 0)
COND_415_0_LE_LE(TRUE, x1, x0) → 415_0_LE_LE(-(x1, 1), -(x0, 1))
- ((U^Increasing(415_0_LE_LE(-(x1[1], 1), -(x0[1], 1))), ≥)∧0 = 0∧0 = 0∧[4 + (-1)bso_17] ≥ 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(415_0_le_LE(x₁, x₂)) = [-1]
POL(0) = 0
POL(616_0_le_Return) = [-1]
POL(681_1_le_InvokeMethod(x₁, x₂, x₃)) = [-1]
POL(689_0_le_Return(x₁)) = [-1]
POL(643_0_le_Return) = [-1]
POL(415_0_LE_LE(x₁, x₂)) = [1] + [2]x₂ + [2]x₁
POL(COND_415_0_LE_LE(x₁, x₂, x₃)) = [1] + [2]x₃ + [2]x₂
POL(&&(x₁, x₂)) = [-1]
POL(>(x₁, x₂)) = [-1]
POL(-(x₁, x₂)) = x₁ + [-1]x₂
POL(1) = [1]

The following pairs are in P_>:

COND_415_0_LE_LE(TRUE, x1[1], x0[1]) → 415_0_LE_LE(-(x1[1], 1), -(x0[1], 1))

The following pairs are in P_bound:

415_0_LE_LE(x1[0], x0[0]) → COND_415_0_LE_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])

The following pairs are in P_≥:

415_0_LE_LE(x1[0], x0[0]) → COND_415_0_LE_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], 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:

Boolean, Integer

The ITRS R consists of the following rules:
415_0_le_LE(x1, 0) → 616_0_le_Return
681_1_le_InvokeMethod(616_0_le_Return, 0, x2) → 689_0_le_Return(x0)
681_1_le_InvokeMethod(643_0_le_Return, x2, 0) → 689_0_le_Return(x1)
681_1_le_InvokeMethod(689_0_le_Return(x0), x0, x2) → 689_0_le_Return(x1)

The integer pair graph contains the following rules and edges:
(0): 415_0_LE_LE(x1[0], x0[0]) → COND_415_0_LE_LE(x1[0] > 0 && x0[0] > 0, x1[0], x0[0])

The set Q consists of the following terms:
415_0_le_LE(x0, 0)
681_1_le_InvokeMethod(616_0_le_Return, 0, x0)
681_1_le_InvokeMethod(643_0_le_Return, x0, 0)
681_1_le_InvokeMethod(689_0_le_Return(x0), 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:
415_0_le_LE(x1, 0) → 616_0_le_Return
681_1_le_InvokeMethod(616_0_le_Return, 0, x2) → 689_0_le_Return(x0)
681_1_le_InvokeMethod(643_0_le_Return, x2, 0) → 689_0_le_Return(x1)
681_1_le_InvokeMethod(689_0_le_Return(x0), x0, x2) → 689_0_le_Return(x1)

The integer pair graph contains the following rules and edges:
(1): COND_415_0_LE_LE(TRUE, x1[1], x0[1]) → 415_0_LE_LE(x1[1] - 1, x0[1] - 1)

The set Q consists of the following terms:
415_0_le_LE(x0, 0)
681_1_le_InvokeMethod(616_0_le_Return, 0, x0)
681_1_le_InvokeMethod(643_0_le_Return, x0, 0)
681_1_le_InvokeMethod(689_0_le_Return(x0), 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