Termination proof of /tmp/tmpu9sXL8/LogRecursive.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_14 (Sun Microsystems Inc.) Main-Class: LogRecursive

public class LogRecursive {
  public static void main(String[] args) {
    Random.args = args;
    log(Random.random(), Random.random());
  }

  public static int log(int x, int y) {
    if (x >= y && y > 1) {
      return 1 + log(x/y, y);
    }
    return 0;
  }
}


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

  public static int random() {
    if (args.length <= index) {
      return 0;
    }
    String string = args[index];
    index++;
    if (string == null) {
      return 0;
    }
    return string.length();
  }
}

(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
LogRecursive.main([Ljava/lang/String;)V: Graph of 118 nodes with 0 SCCs.

LogRecursive.log(II)I: Graph of 34 nodes with 0 SCCs.

(3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 16 rules for P and 17 rules for R.

Combined rules. Obtained 1 rules for P and 3 rules for R.

Filtered ground terms:

529_1_log_InvokeMethod(x1, x2, x3, x4) → 529_1_log_InvokeMethod(x1, x3, x4)
464_0_log_Load(x1, x2, x3, x4) → 464_0_log_Load(x2, x3, x4)
Cond_464_0_log_Load(x1, x2, x3, x4, x5) → Cond_464_0_log_Load(x1, x3, x4, x5)
688_0_log_Return(x1) → 688_0_log_Return
541_0_log_Return(x1, x2) → 541_0_log_Return
475_0_log_Return(x1) → 475_0_log_Return

Filtered duplicate args:

464_0_log_Load(x1, x2, x3) → 464_0_log_Load(x2, x3)
Cond_464_0_log_Load(x1, x2, x3, x4) → Cond_464_0_log_Load(x1, x3, x4)

Filtered unneeded arguments:

529_1_log_InvokeMethod(x1, x2, x3) → 529_1_log_InvokeMethod(x1)

Combined rules. Obtained 1 rules for P and 3 rules for R.

Finished conversion. Obtained 1 rules for P and 3 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:
529_1_log_InvokeMethod(475_0_log_Return) → 541_0_log_Return
529_1_log_InvokeMethod(541_0_log_Return) → 688_0_log_Return
529_1_log_InvokeMethod(688_0_log_Return) → 688_0_log_Return

The integer pair graph contains the following rules and edges:
(0): 464_0_LOG_LOAD(x1[0], x0[0]) → COND_464_0_LOG_LOAD(x1[0] > 1 && x1[0] <= x0[0], x1[0], x0[0])
(1): COND_464_0_LOG_LOAD(TRUE, x1[1], x0[1]) → 464_0_LOG_LOAD(x1[1], x0[1] / x1[1])

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

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

The set Q consists of the following terms:
529_1_log_InvokeMethod(475_0_log_Return)
529_1_log_InvokeMethod(541_0_log_Return)
529_1_log_InvokeMethod(688_0_log_Return)

(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 464_0_LOG_LOAD(x1, x0) → COND_464_0_LOG_LOAD(&&(>(x1, 1), <=(x1, x0)), x1, x0) the following chains were created:

We consider the chain 464_0_LOG_LOAD(x1[0], x0[0]) → COND_464_0_LOG_LOAD(&&(>(x1[0], 1), <=(x1[0], x0[0])), x1[0], x0[0]), COND_464_0_LOG_LOAD(TRUE, x1[1], x0[1]) → 464_0_LOG_LOAD(x1[1], /(x0[1], x1[1])) which results in the following constraint:
(1)    (&&(>(x1[0], 1), <=(x1[0], x0[0]))=TRUE∧x1[0]=x1[1]∧x0[0]=x0[1] ⇒ 464_0_LOG_LOAD(x1[0], x0[0])≥NonInfC∧464_0_LOG_LOAD(x1[0], x0[0])≥COND_464_0_LOG_LOAD(&&(>(x1[0], 1), <=(x1[0], x0[0])), x1[0], x0[0])∧(U^Increasing(COND_464_0_LOG_LOAD(&&(>(x1[0], 1), <=(x1[0], x0[0])), x1[0], x0[0])), ≥))

We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2)    (>(x1[0], 1)=TRUE∧<=(x1[0], x0[0])=TRUE ⇒ 464_0_LOG_LOAD(x1[0], x0[0])≥NonInfC∧464_0_LOG_LOAD(x1[0], x0[0])≥COND_464_0_LOG_LOAD(&&(>(x1[0], 1), <=(x1[0], x0[0])), x1[0], x0[0])∧(U^Increasing(COND_464_0_LOG_LOAD(&&(>(x1[0], 1), <=(x1[0], x0[0])), x1[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]x1[0] ≥ 0 ⇒ (U^Increasing(COND_464_0_LOG_LOAD(&&(>(x1[0], 1), <=(x1[0], x0[0])), x1[0], x0[0])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] + [(-1)bni_18]x1[0] ≥ 0∧[(-1)bso_19] ≥ 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]x1[0] ≥ 0 ⇒ (U^Increasing(COND_464_0_LOG_LOAD(&&(>(x1[0], 1), <=(x1[0], x0[0])), x1[0], x0[0])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] + [(-1)bni_18]x1[0] ≥ 0∧[(-1)bso_19] ≥ 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]x1[0] ≥ 0 ⇒ (U^Increasing(COND_464_0_LOG_LOAD(&&(>(x1[0], 1), <=(x1[0], x0[0])), x1[0], x0[0])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] + [(-1)bni_18]x1[0] ≥ 0∧[(-1)bso_19] ≥ 0)

We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6)    (x1[0] ≥ 0∧x0[0] + [-2] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(COND_464_0_LOG_LOAD(&&(>(x1[0], 1), <=(x1[0], x0[0])), x1[0], x0[0])), ≥)∧[(-3)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] + [(-1)bni_18]x1[0] ≥ 0∧[(-1)bso_19] ≥ 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_464_0_LOG_LOAD(&&(>(x1[0], 1), <=(x1[0], x0[0])), x1[0], x0[0])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[(-1)bso_19] ≥ 0)

For Pair COND_464_0_LOG_LOAD(TRUE, x1, x0) → 464_0_LOG_LOAD(x1, /(x0, x1)) the following chains were created:

We consider the chain 464_0_LOG_LOAD(x1[0], x0[0]) → COND_464_0_LOG_LOAD(&&(>(x1[0], 1), <=(x1[0], x0[0])), x1[0], x0[0]), COND_464_0_LOG_LOAD(TRUE, x1[1], x0[1]) → 464_0_LOG_LOAD(x1[1], /(x0[1], x1[1])), 464_0_LOG_LOAD(x1[0], x0[0]) → COND_464_0_LOG_LOAD(&&(>(x1[0], 1), <=(x1[0], x0[0])), x1[0], x0[0]) which results in the following constraint:
(8)    (&&(>(x1[0], 1), <=(x1[0], x0[0]))=TRUE∧x1[0]=x1[1]∧x0[0]=x0[1]∧x1[1]=x1[0]1∧/(x0[1], x1[1])=x0[0]1 ⇒ COND_464_0_LOG_LOAD(TRUE, x1[1], x0[1])≥NonInfC∧COND_464_0_LOG_LOAD(TRUE, x1[1], x0[1])≥464_0_LOG_LOAD(x1[1], /(x0[1], x1[1]))∧(U^Increasing(464_0_LOG_LOAD(x1[1], /(x0[1], x1[1]))), ≥))

We simplified constraint (8) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(9)    (>(x1[0], 1)=TRUE∧<=(x1[0], x0[0])=TRUE ⇒ COND_464_0_LOG_LOAD(TRUE, x1[0], x0[0])≥NonInfC∧COND_464_0_LOG_LOAD(TRUE, x1[0], x0[0])≥464_0_LOG_LOAD(x1[0], /(x0[0], x1[0]))∧(U^Increasing(464_0_LOG_LOAD(x1[1], /(x0[1], x1[1]))), ≥))

We simplified constraint (9) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(10)    (x1[0] + [-2] ≥ 0∧x0[0] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(464_0_LOG_LOAD(x1[1], /(x0[1], x1[1]))), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]x0[0] + [(-1)bni_20]x1[0] ≥ 0∧[(-1)bso_24] + x0[0] + [-1]max{x0[0], [-1]x0[0]} + min{max{x1[0], [-1]x1[0]} + [-1], max{x0[0], [-1]x0[0]}} ≥ 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]x1[0] ≥ 0 ⇒ (U^Increasing(464_0_LOG_LOAD(x1[1], /(x0[1], x1[1]))), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]x0[0] + [(-1)bni_20]x1[0] ≥ 0∧[(-1)bso_24] + x0[0] + [-1]max{x0[0], [-1]x0[0]} + min{max{x1[0], [-1]x1[0]} + [-1], max{x0[0], [-1]x0[0]}} ≥ 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]x1[0] ≥ 0∧[2]x0[0] ≥ 0∧[2]x1[0] ≥ 0 ⇒ (U^Increasing(464_0_LOG_LOAD(x1[1], /(x0[1], x1[1]))), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]x0[0] + [(-1)bni_20]x1[0] ≥ 0∧[-1 + (-1)bso_24] + x1[0] ≥ 0)

We simplified constraint (12) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(13)    (x1[0] ≥ 0∧x0[0] + [-2] + [-1]x1[0] ≥ 0∧[2]x0[0] ≥ 0∧[4] + [2]x1[0] ≥ 0 ⇒ (U^Increasing(464_0_LOG_LOAD(x1[1], /(x0[1], x1[1]))), ≥)∧[(-3)bni_20 + (-1)Bound*bni_20] + [bni_20]x0[0] + [(-1)bni_20]x1[0] ≥ 0∧[1 + (-1)bso_24] + x1[0] ≥ 0)

We simplified constraint (13) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(14)    (x1[0] ≥ 0∧x0[0] ≥ 0∧[4] + [2]x1[0] + [2]x0[0] ≥ 0∧[4] + [2]x1[0] ≥ 0 ⇒ (U^Increasing(464_0_LOG_LOAD(x1[1], /(x0[1], x1[1]))), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]x0[0] ≥ 0∧[1 + (-1)bso_24] + x1[0] ≥ 0)

We simplified constraint (14) using rule (IDP_POLY_GCD) which results in the following new constraint:
(15)    (x1[0] ≥ 0∧x0[0] ≥ 0∧[2] + x1[0] + x0[0] ≥ 0∧[2] + x1[0] ≥ 0 ⇒ (U^Increasing(464_0_LOG_LOAD(x1[1], /(x0[1], x1[1]))), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]x0[0] ≥ 0∧[1 + (-1)bso_24] + x1[0] ≥ 0)

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

464_0_LOG_LOAD(x1, x0) → COND_464_0_LOG_LOAD(&&(>(x1, 1), <=(x1, x0)), x1, x0)
- (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_464_0_LOG_LOAD(&&(>(x1[0], 1), <=(x1[0], x0[0])), x1[0], x0[0])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[(-1)bso_19] ≥ 0)
COND_464_0_LOG_LOAD(TRUE, x1, x0) → 464_0_LOG_LOAD(x1, /(x0, x1))
- (x1[0] ≥ 0∧x0[0] ≥ 0∧[2] + x1[0] + x0[0] ≥ 0∧[2] + x1[0] ≥ 0 ⇒ (U^Increasing(464_0_LOG_LOAD(x1[1], /(x0[1], x1[1]))), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]x0[0] ≥ 0∧[1 + (-1)bso_24] + x1[0] ≥ 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) = [2]
POL(FALSE) = [3]
POL(529_1_log_InvokeMethod(x₁)) = [-1] + [-1]x₁
POL(475_0_log_Return) = [-1]
POL(541_0_log_Return) = [-1]
POL(688_0_log_Return) = [-1]
POL(464_0_LOG_LOAD(x₁, x₂)) = [-1] + x₂ + [-1]x₁
POL(COND_464_0_LOG_LOAD(x₁, x₂, x₃)) = [-1] + x₃ + [-1]x₂
POL(&&(x₁, x₂)) = 0
POL(>(x₁, x₂)) = [-1]
POL(1) = [1]
POL(<=(x₁, x₂)) = [-1]

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

POL(/(x₁, x1[0])¹ @ {464_0_LOG_LOAD_2/1}) = max{x₁, [-1]x₁} + [-1]min{max{x₂, [-1]x₂} + [-1], max{x₁, [-1]x₁}}

The following pairs are in P_>:

COND_464_0_LOG_LOAD(TRUE, x1[1], x0[1]) → 464_0_LOG_LOAD(x1[1], /(x0[1], x1[1]))

The following pairs are in P_bound:

464_0_LOG_LOAD(x1[0], x0[0]) → COND_464_0_LOG_LOAD(&&(>(x1[0], 1), <=(x1[0], x0[0])), x1[0], x0[0])
COND_464_0_LOG_LOAD(TRUE, x1[1], x0[1]) → 464_0_LOG_LOAD(x1[1], /(x0[1], x1[1]))

The following pairs are in P_≥:

464_0_LOG_LOAD(x1[0], x0[0]) → COND_464_0_LOG_LOAD(&&(>(x1[0], 1), <=(x1[0], x0[0])), x1[0], x0[0])

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

FALSE¹ → &&(FALSE, FALSE)¹
/¹ →

(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:
529_1_log_InvokeMethod(475_0_log_Return) → 541_0_log_Return
529_1_log_InvokeMethod(541_0_log_Return) → 688_0_log_Return
529_1_log_InvokeMethod(688_0_log_Return) → 688_0_log_Return

The integer pair graph contains the following rules and edges:
(0): 464_0_LOG_LOAD(x1[0], x0[0]) → COND_464_0_LOG_LOAD(x1[0] > 1 && x1[0] <= x0[0], x1[0], x0[0])

The set Q consists of the following terms:
529_1_log_InvokeMethod(475_0_log_Return)
529_1_log_InvokeMethod(541_0_log_Return)
529_1_log_InvokeMethod(688_0_log_Return)

(7) 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) Obligation:

(7) IDependencyGraphProof (EQUIVALENT transformation)

(8) TRUE