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 AND
    ↳11 IDP
        ↳12 IDependencyGraphProof (⇔)
        ↳13 TRUE
    ↳14 IDP
        ↳15 IDependencyGraphProof (⇔)
        ↳16 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: example_2/Test 
package example_2;


public class Test {

	public static int divBy(int x){
		int r = 0;
		int y;
		while (x > 0) {
			y = 2;
			x = x/y;
			r = r + x;
		}
		return r;
	}

	public static void main(String[] args) {
		if (args.length > 0) {
		        int x = args[0].length();
			int r = divBy(x);
			// System.out.println("Result: " + r);
		}
		// else System.out.println("Error: Incorrect call");
	}
}


(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
Graph of 106 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:
Load460(i52, i44) → Cond_Load460(i52 > 0, i52, i44)
Cond_Load460(TRUE, i52, i44) → Load460(i52 / 2, i44 + i52 / 2)
The set Q consists of the following terms:
Load460(x0, x1)
Cond_Load460(TRUE, x0, x1)

(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:

Integer


The ITRS R consists of the following rules:
Load460(i52, i44) → Cond_Load460(i52 > 0, i52, i44)
Cond_Load460(TRUE, i52, i44) → Load460(i52 / 2, i44 + i52 / 2)

The integer pair graph contains the following rules and edges:
(0): LOAD460(i52[0], i44[0]) → COND_LOAD460(i52[0] > 0, i52[0], i44[0])
(1): COND_LOAD460(TRUE, i52[1], i44[1]) → LOAD460(i52[1] / 2, i44[1] + i52[1] / 2)

(0) -> (1), if ((i52[0] →* i52[1])∧(i44[0] →* i44[1])∧(i52[0] > 0 →* TRUE))


(1) -> (0), if ((i52[1] / 2 →* i52[0])∧(i44[1] + i52[1] / 2 →* i44[0]))



The set Q consists of the following terms:
Load460(x0, x1)
Cond_Load460(TRUE, x0, x1)

(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:

Integer


R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD460(i52[0], i44[0]) → COND_LOAD460(i52[0] > 0, i52[0], i44[0])
(1): COND_LOAD460(TRUE, i52[1], i44[1]) → LOAD460(i52[1] / 2, i44[1] + i52[1] / 2)

(0) -> (1), if ((i52[0] →* i52[1])∧(i44[0] →* i44[1])∧(i52[0] > 0 →* TRUE))


(1) -> (0), if ((i52[1] / 2 →* i52[0])∧(i44[1] + i52[1] / 2 →* i44[0]))



The set Q consists of the following terms:
Load460(x0, x1)
Cond_Load460(TRUE, x0, x1)

(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 LOAD460(i52, i44) → COND_LOAD460(>(i52, 0), i52, i44) the following chains were created:
  • We consider the chain LOAD460(i52[0], i44[0]) → COND_LOAD460(>(i52[0], 0), i52[0], i44[0]), COND_LOAD460(TRUE, i52[1], i44[1]) → LOAD460(/(i52[1], 2), +(i44[1], /(i52[1], 2))) which results in the following constraint:

    (1)    (i52[0]=i52[1]∧i44[0]=i44[1]∧>(i52[0], 0)=TRUE ⇒ LOAD460(i52[0], i44[0])≥NonInfC∧LOAD460(i52[0], i44[0])≥COND_LOAD460(>(i52[0], 0), i52[0], i44[0])∧(UIncreasing(COND_LOAD460(>(i52[0], 0), i52[0], i44[0])), ≥))



    We simplified constraint (1) using rule (IV) which results in the following new constraint:

    (2)    (>(i52[0], 0)=TRUE ⇒ LOAD460(i52[0], i44[0])≥NonInfC∧LOAD460(i52[0], i44[0])≥COND_LOAD460(>(i52[0], 0), i52[0], i44[0])∧(UIncreasing(COND_LOAD460(>(i52[0], 0), i52[0], i44[0])), ≥))



    We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (3)    (i52[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD460(>(i52[0], 0), i52[0], i44[0])), ≥)∧[(2)bni_11 + (-1)Bound*bni_11] + [bni_11]i52[0] ≥ 0∧[(-1)bso_12] ≥ 0)



    We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (4)    (i52[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD460(>(i52[0], 0), i52[0], i44[0])), ≥)∧[(2)bni_11 + (-1)Bound*bni_11] + [bni_11]i52[0] ≥ 0∧[(-1)bso_12] ≥ 0)



    We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (5)    (i52[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD460(>(i52[0], 0), i52[0], i44[0])), ≥)∧[(2)bni_11 + (-1)Bound*bni_11] + [bni_11]i52[0] ≥ 0∧[(-1)bso_12] ≥ 0)



    We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (6)    (i52[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD460(>(i52[0], 0), i52[0], i44[0])), ≥)∧0 = 0∧[(2)bni_11 + (-1)Bound*bni_11] + [bni_11]i52[0] ≥ 0∧0 = 0∧[(-1)bso_12] ≥ 0)



    We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (7)    (i52[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD460(>(i52[0], 0), i52[0], i44[0])), ≥)∧0 = 0∧[(3)bni_11 + (-1)Bound*bni_11] + [bni_11]i52[0] ≥ 0∧0 = 0∧[(-1)bso_12] ≥ 0)







For Pair COND_LOAD460(TRUE, i52, i44) → LOAD460(/(i52, 2), +(i44, /(i52, 2))) the following chains were created:
  • We consider the chain LOAD460(i52[0], i44[0]) → COND_LOAD460(>(i52[0], 0), i52[0], i44[0]), COND_LOAD460(TRUE, i52[1], i44[1]) → LOAD460(/(i52[1], 2), +(i44[1], /(i52[1], 2))), LOAD460(i52[0], i44[0]) → COND_LOAD460(>(i52[0], 0), i52[0], i44[0]) which results in the following constraint:

    (8)    (i52[0]=i52[1]∧i44[0]=i44[1]∧>(i52[0], 0)=TRUE∧/(i52[1], 2)=i52[0]1∧+(i44[1], /(i52[1], 2))=i44[0]1 ⇒ COND_LOAD460(TRUE, i52[1], i44[1])≥NonInfC∧COND_LOAD460(TRUE, i52[1], i44[1])≥LOAD460(/(i52[1], 2), +(i44[1], /(i52[1], 2)))∧(UIncreasing(LOAD460(/(i52[1], 2), +(i44[1], /(i52[1], 2)))), ≥))



    We simplified constraint (8) using rules (III), (IV) which results in the following new constraint:

    (9)    (>(i52[0], 0)=TRUE ⇒ COND_LOAD460(TRUE, i52[0], i44[0])≥NonInfC∧COND_LOAD460(TRUE, i52[0], i44[0])≥LOAD460(/(i52[0], 2), +(i44[0], /(i52[0], 2)))∧(UIncreasing(LOAD460(/(i52[1], 2), +(i44[1], /(i52[1], 2)))), ≥))



    We simplified constraint (9) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (10)    (i52[0] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD460(/(i52[1], 2), +(i44[1], /(i52[1], 2)))), ≥)∧[(2)bni_13 + (-1)Bound*bni_13] + [bni_13]i52[0] ≥ 0∧[1 + (-1)bso_20] + i52[0] + [-1]max{i52[0], [-1]i52[0]} ≥ 0)



    We simplified constraint (10) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (11)    (i52[0] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD460(/(i52[1], 2), +(i44[1], /(i52[1], 2)))), ≥)∧[(2)bni_13 + (-1)Bound*bni_13] + [bni_13]i52[0] ≥ 0∧[1 + (-1)bso_20] + i52[0] + [-1]max{i52[0], [-1]i52[0]} ≥ 0)



    We simplified constraint (11) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (12)    (i52[0] + [-1] ≥ 0∧[2]i52[0] ≥ 0 ⇒ (UIncreasing(LOAD460(/(i52[1], 2), +(i44[1], /(i52[1], 2)))), ≥)∧[(2)bni_13 + (-1)Bound*bni_13] + [bni_13]i52[0] ≥ 0∧[1 + (-1)bso_20] ≥ 0)



    We simplified constraint (12) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (13)    (i52[0] + [-1] ≥ 0∧[2]i52[0] ≥ 0 ⇒ (UIncreasing(LOAD460(/(i52[1], 2), +(i44[1], /(i52[1], 2)))), ≥)∧0 = 0∧[(2)bni_13 + (-1)Bound*bni_13] + [bni_13]i52[0] ≥ 0∧0 = 0∧[1 + (-1)bso_20] ≥ 0)



    We simplified constraint (13) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (14)    (i52[0] ≥ 0∧[2] + [2]i52[0] ≥ 0 ⇒ (UIncreasing(LOAD460(/(i52[1], 2), +(i44[1], /(i52[1], 2)))), ≥)∧0 = 0∧[(3)bni_13 + (-1)Bound*bni_13] + [bni_13]i52[0] ≥ 0∧0 = 0∧[1 + (-1)bso_20] ≥ 0)



    We simplified constraint (14) using rule (IDP_POLY_GCD) which results in the following new constraint:

    (15)    (i52[0] ≥ 0∧[1] + i52[0] ≥ 0 ⇒ (UIncreasing(LOAD460(/(i52[1], 2), +(i44[1], /(i52[1], 2)))), ≥)∧0 = 0∧[(3)bni_13 + (-1)Bound*bni_13] + [bni_13]i52[0] ≥ 0∧0 = 0∧[1 + (-1)bso_20] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • LOAD460(i52, i44) → COND_LOAD460(>(i52, 0), i52, i44)
    • (i52[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD460(>(i52[0], 0), i52[0], i44[0])), ≥)∧0 = 0∧[(3)bni_11 + (-1)Bound*bni_11] + [bni_11]i52[0] ≥ 0∧0 = 0∧[(-1)bso_12] ≥ 0)

  • COND_LOAD460(TRUE, i52, i44) → LOAD460(/(i52, 2), +(i44, /(i52, 2)))
    • (i52[0] ≥ 0∧[1] + i52[0] ≥ 0 ⇒ (UIncreasing(LOAD460(/(i52[1], 2), +(i44[1], /(i52[1], 2)))), ≥)∧0 = 0∧[(3)bni_13 + (-1)Bound*bni_13] + [bni_13]i52[0] ≥ 0∧0 = 0∧[1 + (-1)bso_20] ≥ 0)




The constraints for P> respective Pbound 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 Pbound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:

POL(TRUE) = [3]   
POL(FALSE) = 0   
POL(LOAD460(x1, x2)) = [2] + x1   
POL(COND_LOAD460(x1, x2, x3)) = [2] + x2   
POL(>(x1, x2)) = [-1]   
POL(0) = 0   
POL(2) = [2]   
POL(+(x1, x2)) = x1 + x2   

Polynomial Interpretations with Context Sensitive Arithemetic Replacement
POL(TermCSAR-Mode @ Context)

POL(/(x1, 2)1 @ {LOAD460_2/0}) = max{x1, [-1]x1} + [-1]   
POL(/(x1, 2)1 @ {LOAD460_2/1, +_2/1}) = [-1]max{x1, [-1]x1} + [1]   

The following pairs are in P>:

COND_LOAD460(TRUE, i52[1], i44[1]) → LOAD460(/(i52[1], 2), +(i44[1], /(i52[1], 2)))

The following pairs are in Pbound:

LOAD460(i52[0], i44[0]) → COND_LOAD460(>(i52[0], 0), i52[0], i44[0])
COND_LOAD460(TRUE, i52[1], i44[1]) → LOAD460(/(i52[1], 2), +(i44[1], /(i52[1], 2)))

The following pairs are in P≥:

LOAD460(i52[0], i44[0]) → COND_LOAD460(>(i52[0], 0), i52[0], i44[0])

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

/1 →

(10) Complex Obligation (AND)

(11) 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:
(0): LOAD460(i52[0], i44[0]) → COND_LOAD460(i52[0] > 0, i52[0], i44[0])


The set Q consists of the following terms:
Load460(x0, x1)
Cond_Load460(TRUE, x0, x1)

(12) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(13) TRUE

(14) 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:
none


R is empty.

The integer pair graph is empty.

The set Q consists of the following terms:
Load460(x0, x1)
Cond_Load460(TRUE, x0, x1)

(15) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs.

(16) TRUE