Termination proof of /tmp/tmpGp8Qds/PlusSwap.jar

Termination of the given JBC could be proven:

0 JBC

↳1 JBC2FIG (⇒)

↳2 JBCTerminationGraph

↳3 FIGtoITRSProof (⇒)

↳4 IDP

↳5 IDPNonInfProof (⇒)

↳6 IDP

↳7 IDependencyGraphProof (⇔)

↳8 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: PlusSwap

public class PlusSwap{
  public static void main(String[] args) {
    Random.args = args;
    int x = Random.random();
    int y = Random.random();
    int z;
    int res = 0;

    while (y > 0) {

      z = x;
      x = y-1;
      y = z;
      res++;

    }

    res = res + 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:
PlusSwap.main([Ljava/lang/String;)V: Graph of 169 nodes with 1 SCC.

(3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 14 rules for P and 6 rules for R.

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

Filtered ground terms:

913_0_main_LE(x1, x2, x3, x4) → 913_0_main_LE(x2, x3, x4)
Cond_913_0_main_LE(x1, x2, x3, x4, x5) → Cond_913_0_main_LE(x1, x3, x4, x5)
930_0_main_Return(x1) → 930_0_main_Return

Filtered duplicate args:

913_0_main_LE(x1, x2, x3) → 913_0_main_LE(x1, x3)
Cond_913_0_main_LE(x1, x2, x3, x4) → Cond_913_0_main_LE(x1, x2, x4)

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:

Integer

The ITRS R consists of the following rules:
913_0_main_LE(x0, 0) → Cond_913_0_main_LE(x0 >= 0, x0, 0)
Cond_913_0_main_LE(TRUE, x0, 0) → 930_0_main_Return

The integer pair graph contains the following rules and edges:
(0): 913_0_MAIN_LE(x0[0], x1[0]) → COND_913_0_MAIN_LE(x1[0] > 0, x0[0], x1[0])
(1): COND_913_0_MAIN_LE(TRUE, x0[1], x1[1]) → 913_0_MAIN_LE(x1[1] - 1, x0[1])

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

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

The set Q consists of the following terms:
913_0_main_LE(x0, 0)
Cond_913_0_main_LE(TRUE, x0, 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 913_0_MAIN_LE(x0, x1) → COND_913_0_MAIN_LE(>(x1, 0), x0, x1) the following chains were created:

We consider the chain 913_0_MAIN_LE(x0[0], x1[0]) → COND_913_0_MAIN_LE(>(x1[0], 0), x0[0], x1[0]), COND_913_0_MAIN_LE(TRUE, x0[1], x1[1]) → 913_0_MAIN_LE(-(x1[1], 1), x0[1]) which results in the following constraint:
(1)    (>(x1[0], 0)=TRUE∧x0[0]=x0[1]∧x1[0]=x1[1] ⇒ 913_0_MAIN_LE(x0[0], x1[0])≥NonInfC∧913_0_MAIN_LE(x0[0], x1[0])≥COND_913_0_MAIN_LE(>(x1[0], 0), x0[0], x1[0])∧(U^Increasing(COND_913_0_MAIN_LE(>(x1[0], 0), x0[0], x1[0])), ≥))

We simplified constraint (1) using rule (IV) which results in the following new constraint:
(2)    (>(x1[0], 0)=TRUE ⇒ 913_0_MAIN_LE(x0[0], x1[0])≥NonInfC∧913_0_MAIN_LE(x0[0], x1[0])≥COND_913_0_MAIN_LE(>(x1[0], 0), x0[0], x1[0])∧(U^Increasing(COND_913_0_MAIN_LE(>(x1[0], 0), x0[0], x1[0])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x1[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_913_0_MAIN_LE(>(x1[0], 0), x0[0], x1[0])), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]x1[0] + [bni_20]x0[0] ≥ 0∧[(-1)bso_21] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x1[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_913_0_MAIN_LE(>(x1[0], 0), x0[0], x1[0])), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]x1[0] + [bni_20]x0[0] ≥ 0∧[(-1)bso_21] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x1[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_913_0_MAIN_LE(>(x1[0], 0), x0[0], x1[0])), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]x1[0] + [bni_20]x0[0] ≥ 0∧[(-1)bso_21] ≥ 0)

We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(6)    (x1[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_913_0_MAIN_LE(>(x1[0], 0), x0[0], x1[0])), ≥)∧[bni_20] = 0∧[(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]x1[0] ≥ 0∧0 = 0∧[(-1)bso_21] ≥ 0)

We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7)    (x1[0] ≥ 0 ⇒ (U^Increasing(COND_913_0_MAIN_LE(>(x1[0], 0), x0[0], x1[0])), ≥)∧[bni_20] = 0∧[(-1)Bound*bni_20] + [bni_20]x1[0] ≥ 0∧0 = 0∧[(-1)bso_21] ≥ 0)

For Pair COND_913_0_MAIN_LE(TRUE, x0, x1) → 913_0_MAIN_LE(-(x1, 1), x0) the following chains were created:

We consider the chain 913_0_MAIN_LE(x0[0], x1[0]) → COND_913_0_MAIN_LE(>(x1[0], 0), x0[0], x1[0]), COND_913_0_MAIN_LE(TRUE, x0[1], x1[1]) → 913_0_MAIN_LE(-(x1[1], 1), x0[1]), 913_0_MAIN_LE(x0[0], x1[0]) → COND_913_0_MAIN_LE(>(x1[0], 0), x0[0], x1[0]), COND_913_0_MAIN_LE(TRUE, x0[1], x1[1]) → 913_0_MAIN_LE(-(x1[1], 1), x0[1]), 913_0_MAIN_LE(x0[0], x1[0]) → COND_913_0_MAIN_LE(>(x1[0], 0), x0[0], x1[0]), COND_913_0_MAIN_LE(TRUE, x0[1], x1[1]) → 913_0_MAIN_LE(-(x1[1], 1), x0[1]) which results in the following constraint:
(8)    (>(x1[0], 0)=TRUE∧x0[0]=x0[1]∧x1[0]=x1[1]∧-(x1[1], 1)=x0[0]1∧x0[1]=x1[0]1∧>(x1[0]1, 0)=TRUE∧x0[0]1=x0[1]1∧x1[0]1=x1[1]1∧-(x1[1]1, 1)=x0[0]2∧x0[1]1=x1[0]2∧>(x1[0]2, 0)=TRUE∧x0[0]2=x0[1]2∧x1[0]2=x1[1]2 ⇒ COND_913_0_MAIN_LE(TRUE, x0[1]1, x1[1]1)≥NonInfC∧COND_913_0_MAIN_LE(TRUE, x0[1]1, x1[1]1)≥913_0_MAIN_LE(-(x1[1]1, 1), x0[1]1)∧(U^Increasing(913_0_MAIN_LE(-(x1[1]1, 1), x0[1]1)), ≥))

We simplified constraint (8) using rules (III), (IV) which results in the following new constraint:
(9)    (>(x1[0], 0)=TRUE∧>(x1[0]1, 0)=TRUE∧>(-(x1[0], 1), 0)=TRUE ⇒ COND_913_0_MAIN_LE(TRUE, -(x1[0], 1), x1[0]1)≥NonInfC∧COND_913_0_MAIN_LE(TRUE, -(x1[0], 1), x1[0]1)≥913_0_MAIN_LE(-(x1[0]1, 1), -(x1[0], 1))∧(U^Increasing(913_0_MAIN_LE(-(x1[1]1, 1), x0[1]1)), ≥))

We simplified constraint (9) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(10)    (x1[0] + [-1] ≥ 0∧x1[0]1 + [-1] ≥ 0∧x1[0] + [-2] ≥ 0 ⇒ (U^Increasing(913_0_MAIN_LE(-(x1[1]1, 1), x0[1]1)), ≥)∧[(-2)bni_22 + (-1)Bound*bni_22] + [bni_22]x1[0]1 + [bni_22]x1[0] ≥ 0∧[1 + (-1)bso_23] ≥ 0)

We simplified constraint (10) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(11)    (x1[0] + [-1] ≥ 0∧x1[0]1 + [-1] ≥ 0∧x1[0] + [-2] ≥ 0 ⇒ (U^Increasing(913_0_MAIN_LE(-(x1[1]1, 1), x0[1]1)), ≥)∧[(-2)bni_22 + (-1)Bound*bni_22] + [bni_22]x1[0]1 + [bni_22]x1[0] ≥ 0∧[1 + (-1)bso_23] ≥ 0)

We simplified constraint (11) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(12)    (x1[0] + [-1] ≥ 0∧x1[0]1 + [-1] ≥ 0∧x1[0] + [-2] ≥ 0 ⇒ (U^Increasing(913_0_MAIN_LE(-(x1[1]1, 1), x0[1]1)), ≥)∧[(-2)bni_22 + (-1)Bound*bni_22] + [bni_22]x1[0]1 + [bni_22]x1[0] ≥ 0∧[1 + (-1)bso_23] ≥ 0)

We simplified constraint (12) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(13)    (x1[0] ≥ 0∧x1[0]1 + [-1] ≥ 0∧[-1] + x1[0] ≥ 0 ⇒ (U^Increasing(913_0_MAIN_LE(-(x1[1]1, 1), x0[1]1)), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]x1[0]1 + [bni_22]x1[0] ≥ 0∧[1 + (-1)bso_23] ≥ 0)

We simplified constraint (13) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(14)    ([1] + x1[0] ≥ 0∧x1[0]1 + [-1] ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(913_0_MAIN_LE(-(x1[1]1, 1), x0[1]1)), ≥)∧[(-1)Bound*bni_22] + [bni_22]x1[0]1 + [bni_22]x1[0] ≥ 0∧[1 + (-1)bso_23] ≥ 0)

We simplified constraint (14) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(15)    ([1] + x1[0] ≥ 0∧x1[0]1 ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(913_0_MAIN_LE(-(x1[1]1, 1), x0[1]1)), ≥)∧[bni_22 + (-1)Bound*bni_22] + [bni_22]x1[0]1 + [bni_22]x1[0] ≥ 0∧[1 + (-1)bso_23] ≥ 0)

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

913_0_MAIN_LE(x0, x1) → COND_913_0_MAIN_LE(>(x1, 0), x0, x1)
- (x1[0] ≥ 0 ⇒ (U^Increasing(COND_913_0_MAIN_LE(>(x1[0], 0), x0[0], x1[0])), ≥)∧[bni_20] = 0∧[(-1)Bound*bni_20] + [bni_20]x1[0] ≥ 0∧0 = 0∧[(-1)bso_21] ≥ 0)
COND_913_0_MAIN_LE(TRUE, x0, x1) → 913_0_MAIN_LE(-(x1, 1), x0)
- ([1] + x1[0] ≥ 0∧x1[0]1 ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(913_0_MAIN_LE(-(x1[1]1, 1), x0[1]1)), ≥)∧[bni_22 + (-1)Bound*bni_22] + [bni_22]x1[0]1 + [bni_22]x1[0] ≥ 0∧[1 + (-1)bso_23] ≥ 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(913_0_main_LE(x₁, x₂)) = [-1] + [-1]x₂ + [-1]x₁
POL(0) = 0
POL(Cond_913_0_main_LE(x₁, x₂, x₃)) = [-1] + [-1]x₃ + [-1]x₂ + [-1]x₁
POL(>=(x₁, x₂)) = [-1]
POL(930_0_main_Return) = [-1]
POL(913_0_MAIN_LE(x₁, x₂)) = [-1] + x₂ + x₁
POL(COND_913_0_MAIN_LE(x₁, x₂, x₃)) = [-1] + x₃ + x₂
POL(>(x₁, x₂)) = [-1]
POL(-(x₁, x₂)) = x₁ + [-1]x₂
POL(1) = [1]

The following pairs are in P_>:

COND_913_0_MAIN_LE(TRUE, x0[1], x1[1]) → 913_0_MAIN_LE(-(x1[1], 1), x0[1])

The following pairs are in P_bound:

COND_913_0_MAIN_LE(TRUE, x0[1], x1[1]) → 913_0_MAIN_LE(-(x1[1], 1), x0[1])

The following pairs are in P_≥:

913_0_MAIN_LE(x0[0], x1[0]) → COND_913_0_MAIN_LE(>(x1[0], 0), x0[0], x1[0])

There are no usable rules.

(6) Obligation: