Termination of the given JBC could be proven:

0 JBC
↳1 JBC2FIG (⇒)
↳2 JBCTerminationGraph
↳3 FIGtoITRSProof (⇒)
↳4 IDP
↳5 IDPNonInfProof (⇒)
↳6 AND
    ↳7 IDP
        ↳8 IDPNonInfProof (⇒)
        ↳9 AND
            ↳10 IDP
                ↳11 IDependencyGraphProof (⇔)
                ↳12 TRUE
            ↳13 IDP
                ↳14 IDependencyGraphProof (⇔)
                ↳15 TRUE
    ↳16 IDP
        ↳17 IDependencyGraphProof (⇔)
        ↳18 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_22 (Sun Microsystems Inc.) Main-Class: Nest/Nest 
package Nest;

public class Nest{
    public static int nest(int x){
		if (x == 0) return 0;
		else return nest(nest(x-1));
    }

    
    public static void main(final String[] args) {
        final int x = args[0].length();
        final int y = nest(x);
    }
}


(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
Nest.Nest.main([Ljava/lang/String;)V: Graph of 69 nodes with 0 SCCs.

Nest.Nest.nest(I)I: Graph of 23 nodes with 0 SCCs.


(3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 17 rules for P and 6 rules for R.


Combined rules. Obtained 3 rules for P and 2 rules for R.


Filtered ground terms:


242_1_nest_InvokeMethod(x1, x2) → 242_1_nest_InvokeMethod(x1)
89_0_nest_NE(x1, x2, x3) → 89_0_nest_NE(x2, x3)
320_0_nest_Return(x1, x2) → 320_0_nest_Return
135_0_nest_Return(x1, x2, x3) → 135_0_nest_Return
Cond_89_0_nest_NE(x1, x2, x3, x4) → Cond_89_0_nest_NE(x1, x3, x4)

Filtered duplicate args:


89_0_nest_NE(x1, x2) → 89_0_nest_NE(x2)
Cond_89_0_nest_NE(x1, x2, x3) → Cond_89_0_nest_NE(x1, x3)

Combined rules. Obtained 3 rules for P and 2 rules for R.


Finished conversion. Obtained 3 rules for P and 2 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:
89_0_nest_NE(0) → 135_0_nest_Return
242_1_nest_InvokeMethod(135_0_nest_Return) → 320_0_nest_Return

The integer pair graph contains the following rules and edges:
(0): 89_0_NEST_NE(x0[0]) → COND_89_0_NEST_NE(x0[0] > 0, x0[0])
(1): COND_89_0_NEST_NE(TRUE, x0[1]) → 200_1_NEST_INVOKEMETHOD(89_0_nest_NE(x0[1] - 1), x0[1] - 1)
(2): COND_89_0_NEST_NE(TRUE, x0[2]) → 89_0_NEST_NE(x0[2] - 1)
(3): 200_1_NEST_INVOKEMETHOD(135_0_nest_Return, 0) → 89_0_NEST_NE(0)
(4): 200_1_NEST_INVOKEMETHOD(320_0_nest_Return, x1[4]) → 89_0_NEST_NE(0)

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


(0) -> (2), if ((x0[0] > 0 →* TRUE)∧(x0[0] →* x0[2]))


(1) -> (3), if ((89_0_nest_NE(x0[1] - 1) →* 135_0_nest_Return)∧(x0[1] - 1 →* 0))


(1) -> (4), if ((89_0_nest_NE(x0[1] - 1) →* 320_0_nest_Return)∧(x0[1] - 1 →* x1[4]))


(2) -> (0), if ((x0[2] - 1 →* x0[0]))


(3) -> (0), if ((0 →* x0[0]))


(4) -> (0), if ((0 →* x0[0]))



The set Q consists of the following terms:
89_0_nest_NE(0)
242_1_nest_InvokeMethod(135_0_nest_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 89_0_NEST_NE(x0) → COND_89_0_NEST_NE(>(x0, 0), x0) the following chains were created:
  • We consider the chain 89_0_NEST_NE(x0[0]) → COND_89_0_NEST_NE(>(x0[0], 0), x0[0]), COND_89_0_NEST_NE(TRUE, x0[1]) → 200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1)) which results in the following constraint:

    (1)    (>(x0[0], 0)=TRUE∧x0[0]=x0[1] ⇒ 89_0_NEST_NE(x0[0])≥NonInfC∧89_0_NEST_NE(x0[0])≥COND_89_0_NEST_NE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥))



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

    (2)    (>(x0[0], 0)=TRUE ⇒ 89_0_NEST_NE(x0[0])≥NonInfC∧89_0_NEST_NE(x0[0])≥COND_89_0_NEST_NE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥))



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

    (3)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] ≥ 0∧[(-1)bso_17] ≥ 0)



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

    (4)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] ≥ 0∧[(-1)bso_17] ≥ 0)



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

    (5)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] ≥ 0∧[(-1)bso_17] ≥ 0)



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

    (6)    (x0[0] ≥ 0 ⇒ (UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] ≥ 0∧[(-1)bso_17] ≥ 0)



  • We consider the chain 89_0_NEST_NE(x0[0]) → COND_89_0_NEST_NE(>(x0[0], 0), x0[0]), COND_89_0_NEST_NE(TRUE, x0[2]) → 89_0_NEST_NE(-(x0[2], 1)) which results in the following constraint:

    (7)    (>(x0[0], 0)=TRUE∧x0[0]=x0[2] ⇒ 89_0_NEST_NE(x0[0])≥NonInfC∧89_0_NEST_NE(x0[0])≥COND_89_0_NEST_NE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥))



    We simplified constraint (7) using rule (IV) which results in the following new constraint:

    (8)    (>(x0[0], 0)=TRUE ⇒ 89_0_NEST_NE(x0[0])≥NonInfC∧89_0_NEST_NE(x0[0])≥COND_89_0_NEST_NE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥))



    We simplified constraint (8) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (9)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] ≥ 0∧[(-1)bso_17] ≥ 0)



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

    (10)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] ≥ 0∧[(-1)bso_17] ≥ 0)



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

    (11)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] ≥ 0∧[(-1)bso_17] ≥ 0)



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

    (12)    (x0[0] ≥ 0 ⇒ (UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] ≥ 0∧[(-1)bso_17] ≥ 0)







For Pair COND_89_0_NEST_NE(TRUE, x0) → 200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0, 1)), -(x0, 1)) the following chains were created:
  • We consider the chain COND_89_0_NEST_NE(TRUE, x0[1]) → 200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1)) which results in the following constraint:

    (13)    (COND_89_0_NEST_NE(TRUE, x0[1])≥NonInfC∧COND_89_0_NEST_NE(TRUE, x0[1])≥200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1))∧(UIncreasing(200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1))), ≥))



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

    (14)    ((UIncreasing(200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1))), ≥)∧[(-1)bso_19] ≥ 0)



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

    (15)    ((UIncreasing(200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1))), ≥)∧[(-1)bso_19] ≥ 0)



    We simplified constraint (15) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (16)    ((UIncreasing(200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1))), ≥)∧[(-1)bso_19] ≥ 0)



    We simplified constraint (16) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (17)    ((UIncreasing(200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1))), ≥)∧0 = 0∧[(-1)bso_19] ≥ 0)







For Pair COND_89_0_NEST_NE(TRUE, x0) → 89_0_NEST_NE(-(x0, 1)) the following chains were created:
  • We consider the chain COND_89_0_NEST_NE(TRUE, x0[2]) → 89_0_NEST_NE(-(x0[2], 1)) which results in the following constraint:

    (18)    (COND_89_0_NEST_NE(TRUE, x0[2])≥NonInfC∧COND_89_0_NEST_NE(TRUE, x0[2])≥89_0_NEST_NE(-(x0[2], 1))∧(UIncreasing(89_0_NEST_NE(-(x0[2], 1))), ≥))



    We simplified constraint (18) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (19)    ((UIncreasing(89_0_NEST_NE(-(x0[2], 1))), ≥)∧[(-1)bso_21] ≥ 0)



    We simplified constraint (19) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (20)    ((UIncreasing(89_0_NEST_NE(-(x0[2], 1))), ≥)∧[(-1)bso_21] ≥ 0)



    We simplified constraint (20) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (21)    ((UIncreasing(89_0_NEST_NE(-(x0[2], 1))), ≥)∧[(-1)bso_21] ≥ 0)



    We simplified constraint (21) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (22)    ((UIncreasing(89_0_NEST_NE(-(x0[2], 1))), ≥)∧0 = 0∧[(-1)bso_21] ≥ 0)







For Pair 200_1_NEST_INVOKEMETHOD(135_0_nest_Return, 0) → 89_0_NEST_NE(0) the following chains were created:
  • We consider the chain 200_1_NEST_INVOKEMETHOD(135_0_nest_Return, 0) → 89_0_NEST_NE(0), 89_0_NEST_NE(x0[0]) → COND_89_0_NEST_NE(>(x0[0], 0), x0[0]) which results in the following constraint:

    (23)    (0=x0[0] ⇒ 200_1_NEST_INVOKEMETHOD(135_0_nest_Return, 0)≥NonInfC∧200_1_NEST_INVOKEMETHOD(135_0_nest_Return, 0)≥89_0_NEST_NE(0)∧(UIncreasing(89_0_NEST_NE(0)), ≥))



    We simplified constraint (23) using rule (IV) which results in the following new constraint:

    (24)    (200_1_NEST_INVOKEMETHOD(135_0_nest_Return, 0)≥NonInfC∧200_1_NEST_INVOKEMETHOD(135_0_nest_Return, 0)≥89_0_NEST_NE(0)∧(UIncreasing(89_0_NEST_NE(0)), ≥))



    We simplified constraint (24) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (25)    ((UIncreasing(89_0_NEST_NE(0)), ≥)∧[(-1)bso_23] ≥ 0)



    We simplified constraint (25) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (26)    ((UIncreasing(89_0_NEST_NE(0)), ≥)∧[(-1)bso_23] ≥ 0)



    We simplified constraint (26) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (27)    ((UIncreasing(89_0_NEST_NE(0)), ≥)∧[(-1)bso_23] ≥ 0)







For Pair 200_1_NEST_INVOKEMETHOD(320_0_nest_Return, x1) → 89_0_NEST_NE(0) the following chains were created:
  • We consider the chain 200_1_NEST_INVOKEMETHOD(320_0_nest_Return, x1[4]) → 89_0_NEST_NE(0) which results in the following constraint:

    (28)    (200_1_NEST_INVOKEMETHOD(320_0_nest_Return, x1[4])≥NonInfC∧200_1_NEST_INVOKEMETHOD(320_0_nest_Return, x1[4])≥89_0_NEST_NE(0)∧(UIncreasing(89_0_NEST_NE(0)), ≥))



    We simplified constraint (28) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (29)    ((UIncreasing(89_0_NEST_NE(0)), ≥)∧[3 + (-1)bso_25] ≥ 0)



    We simplified constraint (29) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (30)    ((UIncreasing(89_0_NEST_NE(0)), ≥)∧[3 + (-1)bso_25] ≥ 0)



    We simplified constraint (30) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (31)    ((UIncreasing(89_0_NEST_NE(0)), ≥)∧[3 + (-1)bso_25] ≥ 0)



    We simplified constraint (31) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (32)    ((UIncreasing(89_0_NEST_NE(0)), ≥)∧0 = 0∧[3 + (-1)bso_25] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • 89_0_NEST_NE(x0) → COND_89_0_NEST_NE(>(x0, 0), x0)
    • (x0[0] ≥ 0 ⇒ (UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] ≥ 0∧[(-1)bso_17] ≥ 0)
    • (x0[0] ≥ 0 ⇒ (UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] ≥ 0∧[(-1)bso_17] ≥ 0)

  • COND_89_0_NEST_NE(TRUE, x0) → 200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0, 1)), -(x0, 1))
    • ((UIncreasing(200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1))), ≥)∧0 = 0∧[(-1)bso_19] ≥ 0)

  • COND_89_0_NEST_NE(TRUE, x0) → 89_0_NEST_NE(-(x0, 1))
    • ((UIncreasing(89_0_NEST_NE(-(x0[2], 1))), ≥)∧0 = 0∧[(-1)bso_21] ≥ 0)

  • 200_1_NEST_INVOKEMETHOD(135_0_nest_Return, 0) → 89_0_NEST_NE(0)
    • ((UIncreasing(89_0_NEST_NE(0)), ≥)∧[(-1)bso_23] ≥ 0)

  • 200_1_NEST_INVOKEMETHOD(320_0_nest_Return, x1) → 89_0_NEST_NE(0)
    • ((UIncreasing(89_0_NEST_NE(0)), ≥)∧0 = 0∧[3 + (-1)bso_25] ≥ 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) = 0   
POL(FALSE) = 0   
POL(89_0_nest_NE(x1)) = [2]   
POL(0) = 0   
POL(135_0_nest_Return) = [2]   
POL(242_1_nest_InvokeMethod(x1)) = [-1]   
POL(320_0_nest_Return) = [-1]   
POL(89_0_NEST_NE(x1)) = [-1]   
POL(COND_89_0_NEST_NE(x1, x2)) = [-1]   
POL(>(x1, x2)) = [-1]   
POL(200_1_NEST_INVOKEMETHOD(x1, x2)) = [1] + [-1]x1   
POL(-(x1, x2)) = x1 + [-1]x2   
POL(1) = [1]   

The following pairs are in P>:

200_1_NEST_INVOKEMETHOD(320_0_nest_Return, x1[4]) → 89_0_NEST_NE(0)

The following pairs are in Pbound:

89_0_NEST_NE(x0[0]) → COND_89_0_NEST_NE(>(x0[0], 0), x0[0])

The following pairs are in P≥:

89_0_NEST_NE(x0[0]) → COND_89_0_NEST_NE(>(x0[0], 0), x0[0])
COND_89_0_NEST_NE(TRUE, x0[1]) → 200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1))
COND_89_0_NEST_NE(TRUE, x0[2]) → 89_0_NEST_NE(-(x0[2], 1))
200_1_NEST_INVOKEMETHOD(135_0_nest_Return, 0) → 89_0_NEST_NE(0)

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

89_0_nest_NE(0)1 ↔ 135_0_nest_Return1

(6) Complex Obligation (AND)

(7) 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:
89_0_nest_NE(0) → 135_0_nest_Return
242_1_nest_InvokeMethod(135_0_nest_Return) → 320_0_nest_Return

The integer pair graph contains the following rules and edges:
(0): 89_0_NEST_NE(x0[0]) → COND_89_0_NEST_NE(x0[0] > 0, x0[0])
(1): COND_89_0_NEST_NE(TRUE, x0[1]) → 200_1_NEST_INVOKEMETHOD(89_0_nest_NE(x0[1] - 1), x0[1] - 1)
(2): COND_89_0_NEST_NE(TRUE, x0[2]) → 89_0_NEST_NE(x0[2] - 1)
(3): 200_1_NEST_INVOKEMETHOD(135_0_nest_Return, 0) → 89_0_NEST_NE(0)

(2) -> (0), if ((x0[2] - 1 →* x0[0]))


(3) -> (0), if ((0 →* x0[0]))


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


(0) -> (2), if ((x0[0] > 0 →* TRUE)∧(x0[0] →* x0[2]))


(1) -> (3), if ((89_0_nest_NE(x0[1] - 1) →* 135_0_nest_Return)∧(x0[1] - 1 →* 0))



The set Q consists of the following terms:
89_0_nest_NE(0)
242_1_nest_InvokeMethod(135_0_nest_Return)

(8) 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 89_0_NEST_NE(x0[0]) → COND_89_0_NEST_NE(>(x0[0], 0), x0[0]) the following chains were created:
  • We consider the chain 89_0_NEST_NE(x0[0]) → COND_89_0_NEST_NE(>(x0[0], 0), x0[0]), COND_89_0_NEST_NE(TRUE, x0[1]) → 200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1)) which results in the following constraint:

    (1)    (>(x0[0], 0)=TRUE∧x0[0]=x0[1] ⇒ 89_0_NEST_NE(x0[0])≥NonInfC∧89_0_NEST_NE(x0[0])≥COND_89_0_NEST_NE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥))



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

    (2)    (>(x0[0], 0)=TRUE ⇒ 89_0_NEST_NE(x0[0])≥NonInfC∧89_0_NEST_NE(x0[0])≥COND_89_0_NEST_NE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥))



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

    (3)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 0)



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

    (4)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 0)



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

    (5)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 0)



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

    (6)    (x0[0] ≥ 0 ⇒ (UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 0)



  • We consider the chain 89_0_NEST_NE(x0[0]) → COND_89_0_NEST_NE(>(x0[0], 0), x0[0]), COND_89_0_NEST_NE(TRUE, x0[2]) → 89_0_NEST_NE(-(x0[2], 1)) which results in the following constraint:

    (7)    (>(x0[0], 0)=TRUE∧x0[0]=x0[2] ⇒ 89_0_NEST_NE(x0[0])≥NonInfC∧89_0_NEST_NE(x0[0])≥COND_89_0_NEST_NE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥))



    We simplified constraint (7) using rule (IV) which results in the following new constraint:

    (8)    (>(x0[0], 0)=TRUE ⇒ 89_0_NEST_NE(x0[0])≥NonInfC∧89_0_NEST_NE(x0[0])≥COND_89_0_NEST_NE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥))



    We simplified constraint (8) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (9)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 0)



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

    (10)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 0)



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

    (11)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 0)



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

    (12)    (x0[0] ≥ 0 ⇒ (UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 0)







For Pair COND_89_0_NEST_NE(TRUE, x0[1]) → 200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1)) the following chains were created:
  • We consider the chain COND_89_0_NEST_NE(TRUE, x0[1]) → 200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1)) which results in the following constraint:

    (13)    (COND_89_0_NEST_NE(TRUE, x0[1])≥NonInfC∧COND_89_0_NEST_NE(TRUE, x0[1])≥200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1))∧(UIncreasing(200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1))), ≥))



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

    (14)    ((UIncreasing(200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1))), ≥)∧[(-1)bso_14] ≥ 0)



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

    (15)    ((UIncreasing(200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1))), ≥)∧[(-1)bso_14] ≥ 0)



    We simplified constraint (15) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (16)    ((UIncreasing(200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1))), ≥)∧[(-1)bso_14] ≥ 0)



    We simplified constraint (16) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (17)    ((UIncreasing(200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1))), ≥)∧0 = 0∧[(-1)bso_14] ≥ 0)







For Pair COND_89_0_NEST_NE(TRUE, x0[2]) → 89_0_NEST_NE(-(x0[2], 1)) the following chains were created:
  • We consider the chain COND_89_0_NEST_NE(TRUE, x0[2]) → 89_0_NEST_NE(-(x0[2], 1)) which results in the following constraint:

    (18)    (COND_89_0_NEST_NE(TRUE, x0[2])≥NonInfC∧COND_89_0_NEST_NE(TRUE, x0[2])≥89_0_NEST_NE(-(x0[2], 1))∧(UIncreasing(89_0_NEST_NE(-(x0[2], 1))), ≥))



    We simplified constraint (18) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (19)    ((UIncreasing(89_0_NEST_NE(-(x0[2], 1))), ≥)∧[1 + (-1)bso_16] ≥ 0)



    We simplified constraint (19) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (20)    ((UIncreasing(89_0_NEST_NE(-(x0[2], 1))), ≥)∧[1 + (-1)bso_16] ≥ 0)



    We simplified constraint (20) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (21)    ((UIncreasing(89_0_NEST_NE(-(x0[2], 1))), ≥)∧[1 + (-1)bso_16] ≥ 0)



    We simplified constraint (21) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (22)    ((UIncreasing(89_0_NEST_NE(-(x0[2], 1))), ≥)∧0 = 0∧[1 + (-1)bso_16] ≥ 0)







For Pair 200_1_NEST_INVOKEMETHOD(135_0_nest_Return, 0) → 89_0_NEST_NE(0) the following chains were created:
  • We consider the chain 200_1_NEST_INVOKEMETHOD(135_0_nest_Return, 0) → 89_0_NEST_NE(0), 89_0_NEST_NE(x0[0]) → COND_89_0_NEST_NE(>(x0[0], 0), x0[0]) which results in the following constraint:

    (23)    (0=x0[0] ⇒ 200_1_NEST_INVOKEMETHOD(135_0_nest_Return, 0)≥NonInfC∧200_1_NEST_INVOKEMETHOD(135_0_nest_Return, 0)≥89_0_NEST_NE(0)∧(UIncreasing(89_0_NEST_NE(0)), ≥))



    We simplified constraint (23) using rule (IV) which results in the following new constraint:

    (24)    (200_1_NEST_INVOKEMETHOD(135_0_nest_Return, 0)≥NonInfC∧200_1_NEST_INVOKEMETHOD(135_0_nest_Return, 0)≥89_0_NEST_NE(0)∧(UIncreasing(89_0_NEST_NE(0)), ≥))



    We simplified constraint (24) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (25)    ((UIncreasing(89_0_NEST_NE(0)), ≥)∧[1 + (-1)bso_18] ≥ 0)



    We simplified constraint (25) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (26)    ((UIncreasing(89_0_NEST_NE(0)), ≥)∧[1 + (-1)bso_18] ≥ 0)



    We simplified constraint (26) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (27)    ((UIncreasing(89_0_NEST_NE(0)), ≥)∧[1 + (-1)bso_18] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • 89_0_NEST_NE(x0[0]) → COND_89_0_NEST_NE(>(x0[0], 0), x0[0])
    • (x0[0] ≥ 0 ⇒ (UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 0)
    • (x0[0] ≥ 0 ⇒ (UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 0)

  • COND_89_0_NEST_NE(TRUE, x0[1]) → 200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1))
    • ((UIncreasing(200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1))), ≥)∧0 = 0∧[(-1)bso_14] ≥ 0)

  • COND_89_0_NEST_NE(TRUE, x0[2]) → 89_0_NEST_NE(-(x0[2], 1))
    • ((UIncreasing(89_0_NEST_NE(-(x0[2], 1))), ≥)∧0 = 0∧[1 + (-1)bso_16] ≥ 0)

  • 200_1_NEST_INVOKEMETHOD(135_0_nest_Return, 0) → 89_0_NEST_NE(0)
    • ((UIncreasing(89_0_NEST_NE(0)), ≥)∧[1 + (-1)bso_18] ≥ 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) = 0   
POL(FALSE) = 0   
POL(89_0_nest_NE(x1)) = [-1] + [-1]x1   
POL(0) = 0   
POL(135_0_nest_Return) = [-1]   
POL(242_1_nest_InvokeMethod(x1)) = [-1]   
POL(320_0_nest_Return) = [-1]   
POL(89_0_NEST_NE(x1)) = [-1] + x1   
POL(COND_89_0_NEST_NE(x1, x2)) = [-1] + x2   
POL(>(x1, x2)) = [1]   
POL(200_1_NEST_INVOKEMETHOD(x1, x2)) = [-1] + [-1]x1   
POL(-(x1, x2)) = x1 + [-1]x2   
POL(1) = [1]   

The following pairs are in P>:

COND_89_0_NEST_NE(TRUE, x0[2]) → 89_0_NEST_NE(-(x0[2], 1))
200_1_NEST_INVOKEMETHOD(135_0_nest_Return, 0) → 89_0_NEST_NE(0)

The following pairs are in Pbound:

89_0_NEST_NE(x0[0]) → COND_89_0_NEST_NE(>(x0[0], 0), x0[0])

The following pairs are in P≥:

89_0_NEST_NE(x0[0]) → COND_89_0_NEST_NE(>(x0[0], 0), x0[0])
COND_89_0_NEST_NE(TRUE, x0[1]) → 200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1))

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

89_0_nest_NE(0)1 ↔ 135_0_nest_Return1

(9) Complex Obligation (AND)

(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


The ITRS R consists of the following rules:
89_0_nest_NE(0) → 135_0_nest_Return
242_1_nest_InvokeMethod(135_0_nest_Return) → 320_0_nest_Return

The integer pair graph contains the following rules and edges:
(0): 89_0_NEST_NE(x0[0]) → COND_89_0_NEST_NE(x0[0] > 0, x0[0])
(1): COND_89_0_NEST_NE(TRUE, x0[1]) → 200_1_NEST_INVOKEMETHOD(89_0_nest_NE(x0[1] - 1), x0[1] - 1)

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



The set Q consists of the following terms:
89_0_nest_NE(0)
242_1_nest_InvokeMethod(135_0_nest_Return)

(11) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.

(12) TRUE

(13) 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:
89_0_nest_NE(0) → 135_0_nest_Return
242_1_nest_InvokeMethod(135_0_nest_Return) → 320_0_nest_Return

The integer pair graph contains the following rules and edges:
(1): COND_89_0_NEST_NE(TRUE, x0[1]) → 200_1_NEST_INVOKEMETHOD(89_0_nest_NE(x0[1] - 1), x0[1] - 1)
(2): COND_89_0_NEST_NE(TRUE, x0[2]) → 89_0_NEST_NE(x0[2] - 1)
(3): 200_1_NEST_INVOKEMETHOD(135_0_nest_Return, 0) → 89_0_NEST_NE(0)

(1) -> (3), if ((89_0_nest_NE(x0[1] - 1) →* 135_0_nest_Return)∧(x0[1] - 1 →* 0))



The set Q consists of the following terms:
89_0_nest_NE(0)
242_1_nest_InvokeMethod(135_0_nest_Return)

(14) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.

(15) TRUE

(16) 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:
89_0_nest_NE(0) → 135_0_nest_Return
242_1_nest_InvokeMethod(135_0_nest_Return) → 320_0_nest_Return

The integer pair graph contains the following rules and edges:
(1): COND_89_0_NEST_NE(TRUE, x0[1]) → 200_1_NEST_INVOKEMETHOD(89_0_nest_NE(x0[1] - 1), x0[1] - 1)
(2): COND_89_0_NEST_NE(TRUE, x0[2]) → 89_0_NEST_NE(x0[2] - 1)
(3): 200_1_NEST_INVOKEMETHOD(135_0_nest_Return, 0) → 89_0_NEST_NE(0)
(4): 200_1_NEST_INVOKEMETHOD(320_0_nest_Return, x1[4]) → 89_0_NEST_NE(0)

(1) -> (3), if ((89_0_nest_NE(x0[1] - 1) →* 135_0_nest_Return)∧(x0[1] - 1 →* 0))


(1) -> (4), if ((89_0_nest_NE(x0[1] - 1) →* 320_0_nest_Return)∧(x0[1] - 1 →* x1[4]))



The set Q consists of the following terms:
89_0_nest_NE(0)
242_1_nest_InvokeMethod(135_0_nest_Return)

(17) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 4 less nodes.

(18) TRUE