Termination proof of /tmp/tmpSi

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_16 (Sun Microsystems Inc.) Main-Class: PastaB5

/**
 * Example taken from "A Term Rewriting Approach to the Automated Termination
 * Analysis of Imperative Programs" (http://www.cs.unm.edu/~spf/papers/2009-02.pdf)
 * and converted to Java.
 */

public class PastaB5 {
    public static void main(String[] args) {
        Random.args = args;
        int x = Random.random();

        while (x > 0 && (x % 2) == 0) {
            x--;
        }
    }
}


public class Random {
  static String[] args;
  static int index = 0;

  public static int random() {
    String string = args[index];
    index++;
    return string.length();
  }
}

(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
PastaB5.main([Ljava/lang/String;)V: Graph of 92 nodes with 1 SCC.

(3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 11 rules for P and 5 rules for R.

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

Filtered ground terms:

118_0_main_LE(x1, x2, x3) → 118_0_main_LE(x2, x3)
Cond_118_0_main_LE(x1, x2, x3, x4) → Cond_118_0_main_LE(x1, x3, x4)
135_0_main_Return(x1) → 135_0_main_Return

Filtered duplicate args:

118_0_main_LE(x1, x2) → 118_0_main_LE(x2)
Cond_118_0_main_LE(x1, x2, x3) → Cond_118_0_main_LE(x1, x3)

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

Finished conversion. Obtained 1 rules for P and 1 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:
118_0_main_LE(0) → 135_0_main_Return

The integer pair graph contains the following rules and edges:
(0): 118_0_MAIN_LE(x0[0]) → COND_118_0_MAIN_LE(x0[0] > 0 && 0 = x0[0] % 2, x0[0])
(1): COND_118_0_MAIN_LE(TRUE, x0[1]) → 118_0_MAIN_LE(x0[1] + -1)

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

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

The set Q consists of the following terms:
118_0_main_LE(0)

(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 118_0_MAIN_LE(x0) → COND_118_0_MAIN_LE(&&(>(x0, 0), =(0, %(x0, 2))), x0) the following chains were created:

We consider the chain 118_0_MAIN_LE(x0[0]) → COND_118_0_MAIN_LE(&&(>(x0[0], 0), =(0, %(x0[0], 2))), x0[0]), COND_118_0_MAIN_LE(TRUE, x0[1]) → 118_0_MAIN_LE(+(x0[1], -1)) which results in the following constraint:
(1)    (&&(>(x0[0], 0), =(0, %(x0[0], 2)))=TRUE∧x0[0]=x0[1] ⇒ 118_0_MAIN_LE(x0[0])≥NonInfC∧118_0_MAIN_LE(x0[0])≥COND_118_0_MAIN_LE(&&(>(x0[0], 0), =(0, %(x0[0], 2))), x0[0])∧(U^Increasing(COND_118_0_MAIN_LE(&&(>(x0[0], 0), =(0, %(x0[0], 2))), x0[0])), ≥))

We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2)    (>(x0[0], 0)=TRUE∧>=(0, %(x0[0], 2))=TRUE∧<=(0, %(x0[0], 2))=TRUE ⇒ 118_0_MAIN_LE(x0[0])≥NonInfC∧118_0_MAIN_LE(x0[0])≥COND_118_0_MAIN_LE(&&(>(x0[0], 0), =(0, %(x0[0], 2))), x0[0])∧(U^Increasing(COND_118_0_MAIN_LE(&&(>(x0[0], 0), =(0, %(x0[0], 2))), x0[0])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x0[0] + [-1] ≥ 0∧[-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} ≥ 0 ⇒ (U^Increasing(COND_118_0_MAIN_LE(&&(>(x0[0], 0), =(0, %(x0[0], 2))), x0[0])), ≥)∧[(-1)Bound*bni_12] + [(2)bni_12]x0[0] ≥ 0∧[(-1)bso_13] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x0[0] + [-1] ≥ 0∧[-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} ≥ 0 ⇒ (U^Increasing(COND_118_0_MAIN_LE(&&(>(x0[0], 0), =(0, %(x0[0], 2))), x0[0])), ≥)∧[(-1)Bound*bni_12] + [(2)bni_12]x0[0] ≥ 0∧[(-1)bso_13] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x0[0] + [-1] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0 ⇒ (U^Increasing(COND_118_0_MAIN_LE(&&(>(x0[0], 0), =(0, %(x0[0], 2))), x0[0])), ≥)∧[(-1)Bound*bni_12] + [(2)bni_12]x0[0] ≥ 0∧[(-1)bso_13] ≥ 0)

We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6)    (x0[0] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0 ⇒ (U^Increasing(COND_118_0_MAIN_LE(&&(>(x0[0], 0), =(0, %(x0[0], 2))), x0[0])), ≥)∧[(-1)Bound*bni_12 + (2)bni_12] + [(2)bni_12]x0[0] ≥ 0∧[(-1)bso_13] ≥ 0)

We simplified constraint (6) using rule (IDP_POLY_GCD) which results in the following new constraint:
(7)    (x0[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(COND_118_0_MAIN_LE(&&(>(x0[0], 0), =(0, %(x0[0], 2))), x0[0])), ≥)∧[(-1)Bound*bni_12 + (2)bni_12] + [(2)bni_12]x0[0] ≥ 0∧[(-1)bso_13] ≥ 0)

For Pair COND_118_0_MAIN_LE(TRUE, x0) → 118_0_MAIN_LE(+(x0, -1)) the following chains were created:

We consider the chain COND_118_0_MAIN_LE(TRUE, x0[1]) → 118_0_MAIN_LE(+(x0[1], -1)) which results in the following constraint:
(8)    (COND_118_0_MAIN_LE(TRUE, x0[1])≥NonInfC∧COND_118_0_MAIN_LE(TRUE, x0[1])≥118_0_MAIN_LE(+(x0[1], -1))∧(U^Increasing(118_0_MAIN_LE(+(x0[1], -1))), ≥))

We simplified constraint (8) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(9)    ((U^Increasing(118_0_MAIN_LE(+(x0[1], -1))), ≥)∧[2 + (-1)bso_15] ≥ 0)

We simplified constraint (9) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(10)    ((U^Increasing(118_0_MAIN_LE(+(x0[1], -1))), ≥)∧[2 + (-1)bso_15] ≥ 0)

We simplified constraint (10) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(11)    ((U^Increasing(118_0_MAIN_LE(+(x0[1], -1))), ≥)∧[2 + (-1)bso_15] ≥ 0)

We simplified constraint (11) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(12)    ((U^Increasing(118_0_MAIN_LE(+(x0[1], -1))), ≥)∧0 = 0∧[2 + (-1)bso_15] ≥ 0)

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

118_0_MAIN_LE(x0) → COND_118_0_MAIN_LE(&&(>(x0, 0), =(0, %(x0, 2))), x0)
- (x0[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(COND_118_0_MAIN_LE(&&(>(x0[0], 0), =(0, %(x0[0], 2))), x0[0])), ≥)∧[(-1)Bound*bni_12 + (2)bni_12] + [(2)bni_12]x0[0] ≥ 0∧[(-1)bso_13] ≥ 0)
COND_118_0_MAIN_LE(TRUE, x0) → 118_0_MAIN_LE(+(x0, -1))
- ((U^Increasing(118_0_MAIN_LE(+(x0[1], -1))), ≥)∧0 = 0∧[2 + (-1)bso_15] ≥ 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(118_0_main_LE(x₁)) = [-1]
POL(0) = 0
POL(135_0_main_Return) = [-1]
POL(118_0_MAIN_LE(x₁)) = [2]x₁
POL(COND_118_0_MAIN_LE(x₁, x₂)) = [2]x₂
POL(&&(x₁, x₂)) = [-1]
POL(>(x₁, x₂)) = [-1]
POL(=(x₁, x₂)) = [-1]
POL(2) = [2]
POL(+(x₁, x₂)) = x₁ + x₂
POL(-1) = [-1]

Polynomial Interpretations with Context Sensitive Arithemetic Replacement
POL(Term^CSAR-Mode @ Context)

POL(%(x₁, 2)^-1 @ {}) = min{x₂, [-1]x₂}
POL(%(x₁, 2)¹ @ {}) = max{x₂, [-1]x₂}

The following pairs are in P_>:

COND_118_0_MAIN_LE(TRUE, x0[1]) → 118_0_MAIN_LE(+(x0[1], -1))

The following pairs are in P_bound:

118_0_MAIN_LE(x0[0]) → COND_118_0_MAIN_LE(&&(>(x0[0], 0), =(0, %(x0[0], 2))), x0[0])

The following pairs are in P_≥:

118_0_MAIN_LE(x0[0]) → COND_118_0_MAIN_LE(&&(>(x0[0], 0), =(0, %(x0[0], 2))), 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:
118_0_main_LE(0) → 135_0_main_Return

The integer pair graph contains the following rules and edges:
(0): 118_0_MAIN_LE(x0[0]) → COND_118_0_MAIN_LE(x0[0] > 0 && 0 = x0[0] % 2, x0[0])

The set Q consists of the following terms:
118_0_main_LE(0)

(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:
118_0_main_LE(0) → 135_0_main_Return

The integer pair graph contains the following rules and edges:
(1): COND_118_0_MAIN_LE(TRUE, x0[1]) → 118_0_MAIN_LE(x0[1] + -1)

The set Q consists of the following terms:
118_0_main_LE(0)

(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