Termination proof of /tmp/tmpEDrQzj/LogMult.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: LogMult

public class LogMult{

  public static int log(int x, int y) {

    int res = 1;

    if (x < 0 || y < 1) return 0;
    else {
      while (x > y) { 
        y = y*y;
        res = 2*res;
      }
    }
    return res;

  }





  public static void main(String[] args) {
    Random.args = args;

    int x = Random.random();
    log(x,2);
  }
}


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:
LogMult.main([Ljava/lang/String;)V: Graph of 109 nodes with 1 SCC.

(3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 14 rules for P and 5 rules for R.

Combined rules. Obtained 1 rules for P and 0 rules for R.

Filtered ground terms:

427_0_log_Load(x1, x2, x3, x4) → 427_0_log_Load(x2, x3, x4)

Filtered duplicate args:

427_0_log_Load(x1, x2, x3) → 427_0_log_Load(x2, x3)

Combined rules. Obtained 1 rules for P and 0 rules for R.

Finished conversion. Obtained 1 rules for P and 0 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

R is empty.

The integer pair graph contains the following rules and edges:
(0): 427_1_MAIN_INVOKEMETHOD(427_0_log_Load(x1[0], x0[0]), x0[0]) → COND_427_1_MAIN_INVOKEMETHOD(x1[0] > 1 && x1[0] < x0[0], 427_0_log_Load(x1[0], x0[0]), x0[0])
(1): COND_427_1_MAIN_INVOKEMETHOD(TRUE, 427_0_log_Load(x1[1], x0[1]), x0[1]) → 427_1_MAIN_INVOKEMETHOD(427_0_log_Load(x1[1] * x1[1], x0[1]), x0[1])

(0) -> (1), if ((x1[0] > 1 && x1[0] < x0[0] →^* TRUE)∧(427_0_log_Load(x1[0], x0[0]) →^* 427_0_log_Load(x1[1], x0[1]))∧(x0[0] →^* x0[1]))

(1) -> (0), if ((427_0_log_Load(x1[1] * x1[1], x0[1]) →^* 427_0_log_Load(x1[0], x0[0]))∧(x0[1] →^* x0[0]))

The set Q is empty.

(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 427_1_MAIN_INVOKEMETHOD(427_0_log_Load(x1, x0), x0) → COND_427_1_MAIN_INVOKEMETHOD(&&(>(x1, 1), <(x1, x0)), 427_0_log_Load(x1, x0), x0) the following chains were created:

We consider the chain 427_1_MAIN_INVOKEMETHOD(427_0_log_Load(x1[0], x0[0]), x0[0]) → COND_427_1_MAIN_INVOKEMETHOD(&&(>(x1[0], 1), <(x1[0], x0[0])), 427_0_log_Load(x1[0], x0[0]), x0[0]), COND_427_1_MAIN_INVOKEMETHOD(TRUE, 427_0_log_Load(x1[1], x0[1]), x0[1]) → 427_1_MAIN_INVOKEMETHOD(427_0_log_Load(*(x1[1], x1[1]), x0[1]), x0[1]) which results in the following constraint:
(1)    (&&(>(x1[0], 1), <(x1[0], x0[0]))=TRUE∧427_0_log_Load(x1[0], x0[0])=427_0_log_Load(x1[1], x0[1])∧x0[0]=x0[1] ⇒ 427_1_MAIN_INVOKEMETHOD(427_0_log_Load(x1[0], x0[0]), x0[0])≥NonInfC∧427_1_MAIN_INVOKEMETHOD(427_0_log_Load(x1[0], x0[0]), x0[0])≥COND_427_1_MAIN_INVOKEMETHOD(&&(>(x1[0], 1), <(x1[0], x0[0])), 427_0_log_Load(x1[0], x0[0]), x0[0])∧(U^Increasing(COND_427_1_MAIN_INVOKEMETHOD(&&(>(x1[0], 1), <(x1[0], x0[0])), 427_0_log_Load(x1[0], x0[0]), x0[0])), ≥))

We simplified constraint (1) using rules (I), (II), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2)    (>(x1[0], 1)=TRUE∧<(x1[0], x0[0])=TRUE ⇒ 427_1_MAIN_INVOKEMETHOD(427_0_log_Load(x1[0], x0[0]), x0[0])≥NonInfC∧427_1_MAIN_INVOKEMETHOD(427_0_log_Load(x1[0], x0[0]), x0[0])≥COND_427_1_MAIN_INVOKEMETHOD(&&(>(x1[0], 1), <(x1[0], x0[0])), 427_0_log_Load(x1[0], x0[0]), x0[0])∧(U^Increasing(COND_427_1_MAIN_INVOKEMETHOD(&&(>(x1[0], 1), <(x1[0], x0[0])), 427_0_log_Load(x1[0], x0[0]), x0[0])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x1[0] + [-2] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(COND_427_1_MAIN_INVOKEMETHOD(&&(>(x1[0], 1), <(x1[0], x0[0])), 427_0_log_Load(x1[0], x0[0]), x0[0])), ≥)∧[(-1)Bound*bni_16] + [(2)bni_16]x0[0] + [(-1)bni_16]x1[0] ≥ 0∧[(-1)bso_17] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x1[0] + [-2] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(COND_427_1_MAIN_INVOKEMETHOD(&&(>(x1[0], 1), <(x1[0], x0[0])), 427_0_log_Load(x1[0], x0[0]), x0[0])), ≥)∧[(-1)Bound*bni_16] + [(2)bni_16]x0[0] + [(-1)bni_16]x1[0] ≥ 0∧[(-1)bso_17] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x1[0] + [-2] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(COND_427_1_MAIN_INVOKEMETHOD(&&(>(x1[0], 1), <(x1[0], x0[0])), 427_0_log_Load(x1[0], x0[0]), x0[0])), ≥)∧[(-1)Bound*bni_16] + [(2)bni_16]x0[0] + [(-1)bni_16]x1[0] ≥ 0∧[(-1)bso_17] ≥ 0)

We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6)    (x1[0] ≥ 0∧x0[0] + [-3] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(COND_427_1_MAIN_INVOKEMETHOD(&&(>(x1[0], 1), <(x1[0], x0[0])), 427_0_log_Load(x1[0], x0[0]), x0[0])), ≥)∧[(-1)Bound*bni_16 + (-2)bni_16] + [(2)bni_16]x0[0] + [(-1)bni_16]x1[0] ≥ 0∧[(-1)bso_17] ≥ 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_427_1_MAIN_INVOKEMETHOD(&&(>(x1[0], 1), <(x1[0], x0[0])), 427_0_log_Load(x1[0], x0[0]), x0[0])), ≥)∧[(-1)Bound*bni_16 + (4)bni_16] + [bni_16]x1[0] + [(2)bni_16]x0[0] ≥ 0∧[(-1)bso_17] ≥ 0)

For Pair COND_427_1_MAIN_INVOKEMETHOD(TRUE, 427_0_log_Load(x1, x0), x0) → 427_1_MAIN_INVOKEMETHOD(427_0_log_Load(*(x1, x1), x0), x0) the following chains were created:

We consider the chain 427_1_MAIN_INVOKEMETHOD(427_0_log_Load(x1[0], x0[0]), x0[0]) → COND_427_1_MAIN_INVOKEMETHOD(&&(>(x1[0], 1), <(x1[0], x0[0])), 427_0_log_Load(x1[0], x0[0]), x0[0]), COND_427_1_MAIN_INVOKEMETHOD(TRUE, 427_0_log_Load(x1[1], x0[1]), x0[1]) → 427_1_MAIN_INVOKEMETHOD(427_0_log_Load(*(x1[1], x1[1]), x0[1]), x0[1]), 427_1_MAIN_INVOKEMETHOD(427_0_log_Load(x1[0], x0[0]), x0[0]) → COND_427_1_MAIN_INVOKEMETHOD(&&(>(x1[0], 1), <(x1[0], x0[0])), 427_0_log_Load(x1[0], x0[0]), x0[0]) which results in the following constraint:
(8)    (&&(>(x1[0], 1), <(x1[0], x0[0]))=TRUE∧427_0_log_Load(x1[0], x0[0])=427_0_log_Load(x1[1], x0[1])∧x0[0]=x0[1]∧427_0_log_Load(*(x1[1], x1[1]), x0[1])=427_0_log_Load(x1[0]1, x0[0]1)∧x0[1]=x0[0]1 ⇒ COND_427_1_MAIN_INVOKEMETHOD(TRUE, 427_0_log_Load(x1[1], x0[1]), x0[1])≥NonInfC∧COND_427_1_MAIN_INVOKEMETHOD(TRUE, 427_0_log_Load(x1[1], x0[1]), x0[1])≥427_1_MAIN_INVOKEMETHOD(427_0_log_Load(*(x1[1], x1[1]), x0[1]), x0[1])∧(U^Increasing(427_1_MAIN_INVOKEMETHOD(427_0_log_Load(*(x1[1], x1[1]), x0[1]), x0[1])), ≥))

We simplified constraint (8) using rules (I), (II), (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(9)    (>(x1[0], 1)=TRUE∧<(x1[0], x0[0])=TRUE ⇒ COND_427_1_MAIN_INVOKEMETHOD(TRUE, 427_0_log_Load(x1[0], x0[0]), x0[0])≥NonInfC∧COND_427_1_MAIN_INVOKEMETHOD(TRUE, 427_0_log_Load(x1[0], x0[0]), x0[0])≥427_1_MAIN_INVOKEMETHOD(427_0_log_Load(*(x1[0], x1[0]), x0[0]), x0[0])∧(U^Increasing(427_1_MAIN_INVOKEMETHOD(427_0_log_Load(*(x1[1], x1[1]), x0[1]), x0[1])), ≥))

We simplified constraint (9) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(10)    (x1[0] + [-2] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(427_1_MAIN_INVOKEMETHOD(427_0_log_Load(*(x1[1], x1[1]), x0[1]), x0[1])), ≥)∧[(-1)Bound*bni_18] + [(2)bni_18]x0[0] + [(-1)bni_18]x1[0] ≥ 0∧[(-1)bso_19] + [-1]x1[0] + x1[0]² ≥ 0)

We simplified constraint (10) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(11)    (x1[0] + [-2] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(427_1_MAIN_INVOKEMETHOD(427_0_log_Load(*(x1[1], x1[1]), x0[1]), x0[1])), ≥)∧[(-1)Bound*bni_18] + [(2)bni_18]x0[0] + [(-1)bni_18]x1[0] ≥ 0∧[(-1)bso_19] + [-1]x1[0] + x1[0]² ≥ 0)

We simplified constraint (11) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(12)    (x1[0] + [-2] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(427_1_MAIN_INVOKEMETHOD(427_0_log_Load(*(x1[1], x1[1]), x0[1]), x0[1])), ≥)∧[(-1)Bound*bni_18] + [(2)bni_18]x0[0] + [(-1)bni_18]x1[0] ≥ 0∧[(-1)bso_19] + [-1]x1[0] + x1[0]² ≥ 0)

We simplified constraint (12) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(13)    (x1[0] ≥ 0∧x0[0] + [-3] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(427_1_MAIN_INVOKEMETHOD(427_0_log_Load(*(x1[1], x1[1]), x0[1]), x0[1])), ≥)∧[(-1)Bound*bni_18 + (-2)bni_18] + [(2)bni_18]x0[0] + [(-1)bni_18]x1[0] ≥ 0∧[2 + (-1)bso_19] + [3]x1[0] + x1[0]² ≥ 0)

We simplified constraint (13) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(14)    (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(427_1_MAIN_INVOKEMETHOD(427_0_log_Load(*(x1[1], x1[1]), x0[1]), x0[1])), ≥)∧[(-1)Bound*bni_18 + (4)bni_18] + [bni_18]x1[0] + [(2)bni_18]x0[0] ≥ 0∧[2 + (-1)bso_19] + [3]x1[0] + x1[0]² ≥ 0)

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

427_1_MAIN_INVOKEMETHOD(427_0_log_Load(x1, x0), x0) → COND_427_1_MAIN_INVOKEMETHOD(&&(>(x1, 1), <(x1, x0)), 427_0_log_Load(x1, x0), x0)
- (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_427_1_MAIN_INVOKEMETHOD(&&(>(x1[0], 1), <(x1[0], x0[0])), 427_0_log_Load(x1[0], x0[0]), x0[0])), ≥)∧[(-1)Bound*bni_16 + (4)bni_16] + [bni_16]x1[0] + [(2)bni_16]x0[0] ≥ 0∧[(-1)bso_17] ≥ 0)
COND_427_1_MAIN_INVOKEMETHOD(TRUE, 427_0_log_Load(x1, x0), x0) → 427_1_MAIN_INVOKEMETHOD(427_0_log_Load(*(x1, x1), x0), x0)
- (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(427_1_MAIN_INVOKEMETHOD(427_0_log_Load(*(x1[1], x1[1]), x0[1]), x0[1])), ≥)∧[(-1)Bound*bni_18 + (4)bni_18] + [bni_18]x1[0] + [(2)bni_18]x0[0] ≥ 0∧[2 + (-1)bso_19] + [3]x1[0] + x1[0]² ≥ 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) = [2]
POL(427_1_MAIN_INVOKEMETHOD(x₁, x₂)) = [-1] + x₂ + [-1]x₁
POL(427_0_log_Load(x₁, x₂)) = [-1] + [-1]x₂ + x₁
POL(COND_427_1_MAIN_INVOKEMETHOD(x₁, x₂, x₃)) = [-1] + x₃ + [-1]x₂ + [-1]x₁
POL(&&(x₁, x₂)) = 0
POL(>(x₁, x₂)) = [-1]
POL(1) = [1]
POL(<(x₁, x₂)) = [-1]
POL(*(x₁, x₂)) = x₁·x₂

The following pairs are in P_>:

COND_427_1_MAIN_INVOKEMETHOD(TRUE, 427_0_log_Load(x1[1], x0[1]), x0[1]) → 427_1_MAIN_INVOKEMETHOD(427_0_log_Load(*(x1[1], x1[1]), x0[1]), x0[1])

The following pairs are in P_bound:

427_1_MAIN_INVOKEMETHOD(427_0_log_Load(x1[0], x0[0]), x0[0]) → COND_427_1_MAIN_INVOKEMETHOD(&&(>(x1[0], 1), <(x1[0], x0[0])), 427_0_log_Load(x1[0], x0[0]), x0[0])
COND_427_1_MAIN_INVOKEMETHOD(TRUE, 427_0_log_Load(x1[1], x0[1]), x0[1]) → 427_1_MAIN_INVOKEMETHOD(427_0_log_Load(*(x1[1], x1[1]), x0[1]), x0[1])

The following pairs are in P_≥:

427_1_MAIN_INVOKEMETHOD(427_0_log_Load(x1[0], x0[0]), x0[0]) → COND_427_1_MAIN_INVOKEMETHOD(&&(>(x1[0], 1), <(x1[0], x0[0])), 427_0_log_Load(x1[0], x0[0]), x0[0])

At least the following rules have been oriented under context sensitive arithmetic replacement:

&&(TRUE, TRUE)¹ ↔ TRUE¹
FALSE¹ → &&(TRUE, FALSE)¹
FALSE¹ → &&(FALSE, TRUE)¹
FALSE¹ → &&(FALSE, FALSE)¹

(6) Obligation: