Termination proof of /tmp/tmpDNk4AB/PastaC7.jar

Termination of the given JBC could be proven:

0 JBC

↳1 JBC2FIG (⇐)

↳2 FIGraph

↳3 FIGtoITRSProof (⇐)

↳4 ITRS

↳5 ITRStoIDPProof (⇔)

↳6 IDP

↳7 UsableRulesProof (⇔)

↳8 IDP

↳9 IDPNonInfProof (⇐)

↳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: PastaC7

/**
 * 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 PastaC7 {
    public static void main(String[] args) {
        Random.args = args;
        int i = Random.random();
        int j = Random.random();
        int k = Random.random();

        while (i <= 100 && j <= k) {
            int t = i;
            i = j;
            j = i + 1;
            k--;
        }
    }
}


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:
Graph of 259 nodes with 1 SCC.

(3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph to ITRS rules

(4) Obligation:

ITRS 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 TRS R consists of the following rules:
Load1106(i14, i23, i51) → Cond_Load1106(i23 <= i51 && i14 <= 100, i14, i23, i51)
Cond_Load1106(TRUE, i14, i23, i51) → Load1106(i23, i23 + 1, i51 + -1)
The set Q consists of the following terms:
Load1106(x0, x1, x2)
Cond_Load1106(TRUE, x0, x1, x2)

(5) ITRStoIDPProof (EQUIVALENT transformation)

Added dependency pairs

(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:

Boolean, Integer

The ITRS R consists of the following rules:
Load1106(i14, i23, i51) → Cond_Load1106(i23 <= i51 && i14 <= 100, i14, i23, i51)
Cond_Load1106(TRUE, i14, i23, i51) → Load1106(i23, i23 + 1, i51 + -1)

The integer pair graph contains the following rules and edges:
(0): LOAD1106(i14[0], i23[0], i51[0]) → COND_LOAD1106(i23[0] <= i51[0] && i14[0] <= 100, i14[0], i23[0], i51[0])
(1): COND_LOAD1106(TRUE, i14[1], i23[1], i51[1]) → LOAD1106(i23[1], i23[1] + 1, i51[1] + -1)

(0) -> (1), if ((i14[0] →^* i14[1])∧(i23[0] <= i51[0] && i14[0] <= 100 →^* TRUE)∧(i51[0] →^* i51[1])∧(i23[0] →^* i23[1]))

(1) -> (0), if ((i23[1] →^* i14[0])∧(i51[1] + -1 →^* i51[0])∧(i23[1] + 1 →^* i23[0]))

The set Q consists of the following terms:
Load1106(x0, x1, x2)
Cond_Load1106(TRUE, x0, x1, x2)

(7) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(8) 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

R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD1106(i14[0], i23[0], i51[0]) → COND_LOAD1106(i23[0] <= i51[0] && i14[0] <= 100, i14[0], i23[0], i51[0])
(1): COND_LOAD1106(TRUE, i14[1], i23[1], i51[1]) → LOAD1106(i23[1], i23[1] + 1, i51[1] + -1)

(0) -> (1), if ((i14[0] →^* i14[1])∧(i23[0] <= i51[0] && i14[0] <= 100 →^* TRUE)∧(i51[0] →^* i51[1])∧(i23[0] →^* i23[1]))

(1) -> (0), if ((i23[1] →^* i14[0])∧(i51[1] + -1 →^* i51[0])∧(i23[1] + 1 →^* i23[0]))

The set Q consists of the following terms:
Load1106(x0, x1, x2)
Cond_Load1106(TRUE, x0, x1, x2)

(9) 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 LOAD1106(i14, i23, i51) → COND_LOAD1106(&&(<=(i23, i51), <=(i14, 100)), i14, i23, i51) the following chains were created:

We consider the chain LOAD1106(i14[0], i23[0], i51[0]) → COND_LOAD1106(&&(<=(i23[0], i51[0]), <=(i14[0], 100)), i14[0], i23[0], i51[0]), COND_LOAD1106(TRUE, i14[1], i23[1], i51[1]) → LOAD1106(i23[1], +(i23[1], 1), +(i51[1], -1)) which results in the following constraint:
(1)    (i14[0]=i14[1]∧&&(<=(i23[0], i51[0]), <=(i14[0], 100))=TRUE∧i51[0]=i51[1]∧i23[0]=i23[1] ⇒ LOAD1106(i14[0], i23[0], i51[0])≥NonInfC∧LOAD1106(i14[0], i23[0], i51[0])≥COND_LOAD1106(&&(<=(i23[0], i51[0]), <=(i14[0], 100)), i14[0], i23[0], i51[0])∧(U^Increasing(COND_LOAD1106(&&(<=(i23[0], i51[0]), <=(i14[0], 100)), i14[0], i23[0], i51[0])), ≥))

We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2)    (<=(i23[0], i51[0])=TRUE∧<=(i14[0], 100)=TRUE ⇒ LOAD1106(i14[0], i23[0], i51[0])≥NonInfC∧LOAD1106(i14[0], i23[0], i51[0])≥COND_LOAD1106(&&(<=(i23[0], i51[0]), <=(i14[0], 100)), i14[0], i23[0], i51[0])∧(U^Increasing(COND_LOAD1106(&&(<=(i23[0], i51[0]), <=(i14[0], 100)), i14[0], i23[0], i51[0])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (i51[0] + [-1]i23[0] ≥ 0∧[100] + [-1]i14[0] ≥ 0 ⇒ (U^Increasing(COND_LOAD1106(&&(<=(i23[0], i51[0]), <=(i14[0], 100)), i14[0], i23[0], i51[0])), ≥)∧[bni_10 + (-1)Bound*bni_10] + [bni_10]i51[0] + [(-1)bni_10]i23[0] ≥ 0∧[2 + (-1)bso_11] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (i51[0] + [-1]i23[0] ≥ 0∧[100] + [-1]i14[0] ≥ 0 ⇒ (U^Increasing(COND_LOAD1106(&&(<=(i23[0], i51[0]), <=(i14[0], 100)), i14[0], i23[0], i51[0])), ≥)∧[bni_10 + (-1)Bound*bni_10] + [bni_10]i51[0] + [(-1)bni_10]i23[0] ≥ 0∧[2 + (-1)bso_11] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (i51[0] + [-1]i23[0] ≥ 0∧[100] + [-1]i14[0] ≥ 0 ⇒ (U^Increasing(COND_LOAD1106(&&(<=(i23[0], i51[0]), <=(i14[0], 100)), i14[0], i23[0], i51[0])), ≥)∧[bni_10 + (-1)Bound*bni_10] + [bni_10]i51[0] + [(-1)bni_10]i23[0] ≥ 0∧[2 + (-1)bso_11] ≥ 0)

We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6)    (i51[0] ≥ 0∧[100] + [-1]i14[0] ≥ 0 ⇒ (U^Increasing(COND_LOAD1106(&&(<=(i23[0], i51[0]), <=(i14[0], 100)), i14[0], i23[0], i51[0])), ≥)∧[bni_10 + (-1)Bound*bni_10] + [bni_10]i51[0] ≥ 0∧[2 + (-1)bso_11] ≥ 0)

We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(7)    (i51[0] ≥ 0∧[100] + [-1]i14[0] ≥ 0∧i23[0] ≥ 0 ⇒ (U^Increasing(COND_LOAD1106(&&(<=(i23[0], i51[0]), <=(i14[0], 100)), i14[0], i23[0], i51[0])), ≥)∧[bni_10 + (-1)Bound*bni_10] + [bni_10]i51[0] ≥ 0∧[2 + (-1)bso_11] ≥ 0)

(8)    (i51[0] ≥ 0∧[100] + [-1]i14[0] ≥ 0∧i23[0] ≥ 0 ⇒ (U^Increasing(COND_LOAD1106(&&(<=(i23[0], i51[0]), <=(i14[0], 100)), i14[0], i23[0], i51[0])), ≥)∧[bni_10 + (-1)Bound*bni_10] + [bni_10]i51[0] ≥ 0∧[2 + (-1)bso_11] ≥ 0)

We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(9)    (i51[0] ≥ 0∧[100] + [-1]i14[0] ≥ 0∧i23[0] ≥ 0∧i14[0] ≥ 0 ⇒ (U^Increasing(COND_LOAD1106(&&(<=(i23[0], i51[0]), <=(i14[0], 100)), i14[0], i23[0], i51[0])), ≥)∧[bni_10 + (-1)Bound*bni_10] + [bni_10]i51[0] ≥ 0∧[2 + (-1)bso_11] ≥ 0)

(10)    (i51[0] ≥ 0∧[100] + i14[0] ≥ 0∧i23[0] ≥ 0∧i14[0] ≥ 0 ⇒ (U^Increasing(COND_LOAD1106(&&(<=(i23[0], i51[0]), <=(i14[0], 100)), i14[0], i23[0], i51[0])), ≥)∧[bni_10 + (-1)Bound*bni_10] + [bni_10]i51[0] ≥ 0∧[2 + (-1)bso_11] ≥ 0)

We simplified constraint (8) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(11)    (i51[0] ≥ 0∧[100] + i14[0] ≥ 0∧i23[0] ≥ 0∧i14[0] ≥ 0 ⇒ (U^Increasing(COND_LOAD1106(&&(<=(i23[0], i51[0]), <=(i14[0], 100)), i14[0], i23[0], i51[0])), ≥)∧[bni_10 + (-1)Bound*bni_10] + [bni_10]i51[0] ≥ 0∧[2 + (-1)bso_11] ≥ 0)

(12)    (i51[0] ≥ 0∧[100] + [-1]i14[0] ≥ 0∧i23[0] ≥ 0∧i14[0] ≥ 0 ⇒ (U^Increasing(COND_LOAD1106(&&(<=(i23[0], i51[0]), <=(i14[0], 100)), i14[0], i23[0], i51[0])), ≥)∧[bni_10 + (-1)Bound*bni_10] + [bni_10]i51[0] ≥ 0∧[2 + (-1)bso_11] ≥ 0)

For Pair COND_LOAD1106(TRUE, i14, i23, i51) → LOAD1106(i23, +(i23, 1), +(i51, -1)) the following chains were created:

We consider the chain COND_LOAD1106(TRUE, i14[1], i23[1], i51[1]) → LOAD1106(i23[1], +(i23[1], 1), +(i51[1], -1)) which results in the following constraint:
(13)    (COND_LOAD1106(TRUE, i14[1], i23[1], i51[1])≥NonInfC∧COND_LOAD1106(TRUE, i14[1], i23[1], i51[1])≥LOAD1106(i23[1], +(i23[1], 1), +(i51[1], -1))∧(U^Increasing(LOAD1106(i23[1], +(i23[1], 1), +(i51[1], -1))), ≥))

We simplified constraint (13) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(14)    ((U^Increasing(LOAD1106(i23[1], +(i23[1], 1), +(i51[1], -1))), ≥)∧[(-1)bso_13] ≥ 0)

We simplified constraint (14) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(15)    ((U^Increasing(LOAD1106(i23[1], +(i23[1], 1), +(i51[1], -1))), ≥)∧[(-1)bso_13] ≥ 0)

We simplified constraint (15) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(16)    ((U^Increasing(LOAD1106(i23[1], +(i23[1], 1), +(i51[1], -1))), ≥)∧[(-1)bso_13] ≥ 0)

We simplified constraint (16) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(17)    ((U^Increasing(LOAD1106(i23[1], +(i23[1], 1), +(i51[1], -1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_13] ≥ 0)

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

LOAD1106(i14, i23, i51) → COND_LOAD1106(&&(<=(i23, i51), <=(i14, 100)), i14, i23, i51)
- (i51[0] ≥ 0∧[100] + [-1]i14[0] ≥ 0∧i23[0] ≥ 0∧i14[0] ≥ 0 ⇒ (U^Increasing(COND_LOAD1106(&&(<=(i23[0], i51[0]), <=(i14[0], 100)), i14[0], i23[0], i51[0])), ≥)∧[bni_10 + (-1)Bound*bni_10] + [bni_10]i51[0] ≥ 0∧[2 + (-1)bso_11] ≥ 0)
- (i51[0] ≥ 0∧[100] + i14[0] ≥ 0∧i23[0] ≥ 0∧i14[0] ≥ 0 ⇒ (U^Increasing(COND_LOAD1106(&&(<=(i23[0], i51[0]), <=(i14[0], 100)), i14[0], i23[0], i51[0])), ≥)∧[bni_10 + (-1)Bound*bni_10] + [bni_10]i51[0] ≥ 0∧[2 + (-1)bso_11] ≥ 0)
- (i51[0] ≥ 0∧[100] + i14[0] ≥ 0∧i23[0] ≥ 0∧i14[0] ≥ 0 ⇒ (U^Increasing(COND_LOAD1106(&&(<=(i23[0], i51[0]), <=(i14[0], 100)), i14[0], i23[0], i51[0])), ≥)∧[bni_10 + (-1)Bound*bni_10] + [bni_10]i51[0] ≥ 0∧[2 + (-1)bso_11] ≥ 0)
- (i51[0] ≥ 0∧[100] + [-1]i14[0] ≥ 0∧i23[0] ≥ 0∧i14[0] ≥ 0 ⇒ (U^Increasing(COND_LOAD1106(&&(<=(i23[0], i51[0]), <=(i14[0], 100)), i14[0], i23[0], i51[0])), ≥)∧[bni_10 + (-1)Bound*bni_10] + [bni_10]i51[0] ≥ 0∧[2 + (-1)bso_11] ≥ 0)
COND_LOAD1106(TRUE, i14, i23, i51) → LOAD1106(i23, +(i23, 1), +(i51, -1))
- ((U^Increasing(LOAD1106(i23[1], +(i23[1], 1), +(i51[1], -1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_13] ≥ 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(LOAD1106(x₁, x₂, x₃)) = [1] + x₃ + [-1]x₂
POL(COND_LOAD1106(x₁, x₂, x₃, x₄)) = [-1] + x₄ + [-1]x₃
POL(&&(x₁, x₂)) = [-1]
POL(<=(x₁, x₂)) = [-1]
POL(100) = [100]
POL(+(x₁, x₂)) = x₁ + x₂
POL(1) = [1]
POL(-1) = [-1]

The following pairs are in P_>:

LOAD1106(i14[0], i23[0], i51[0]) → COND_LOAD1106(&&(<=(i23[0], i51[0]), <=(i14[0], 100)), i14[0], i23[0], i51[0])

The following pairs are in P_bound:

LOAD1106(i14[0], i23[0], i51[0]) → COND_LOAD1106(&&(<=(i23[0], i51[0]), <=(i14[0], 100)), i14[0], i23[0], i51[0])

The following pairs are in P_≥:

COND_LOAD1106(TRUE, i14[1], i23[1], i51[1]) → LOAD1106(i23[1], +(i23[1], 1), +(i51[1], -1))

There are no usable rules.

(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

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_LOAD1106(TRUE, i14[1], i23[1], i51[1]) → LOAD1106(i23[1], i23[1] + 1, i51[1] + -1)

The set Q consists of the following terms:
Load1106(x0, x1, x2)
Cond_Load1106(TRUE, x0, x1, x2)

(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) ITRStoIDPProof (EQUIVALENT transformation)

(6) Obligation:

(7) UsableRulesProof (EQUIVALENT transformation)

(8) Obligation:

(9) IDPNonInfProof (SOUND transformation)

(10) Obligation:

(11) IDependencyGraphProof (EQUIVALENT transformation)

(12) TRUE