Termination of the given JBC could be proven:

0 JBC
↳1 JBCToGraph (⇒, 60 ms)
↳2 JBCTerminationGraph
↳3 TerminationGraphToSCCProof (⇒, 0 ms)
↳4 JBCTerminationSCC
↳5 SCCToIDPv1Proof (⇒, 70 ms)
↳6 IDP
↳7 IDPNonInfProof (⇒, 210 ms)
↳8 AND
    ↳9 IDP
        ↳10 IDependencyGraphProof (⇔, 0 ms)
        ↳11 TRUE
    ↳12 IDP
        ↳13 IDependencyGraphProof (⇔, 0 ms)
        ↳14 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) JBCToGraph (SOUND transformation)

Constructed TerminationGraph.

(2) Obligation:

Termination Graph based on JBC Program:
Nest.Nest.main([Ljava/lang/String;)V: Graph of 91 nodes with 0 SCCs.

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


(3) TerminationGraphToSCCProof (SOUND transformation)

Splitted TerminationGraph to 1 SCCs.

(4) Obligation:

SCC of termination graph based on JBC Program.
SCC contains nodes from the following methods: Nest.Nest.nest(I)I
SCC calls the following helper methods: Nest.Nest.nest(I)I
Performed SCC analyses: UsedFieldsAnalysis

(5) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 18 rules for P and 5 rules for R.


P rules:
127_0_nest_NE(EOS(STATIC_127), i22, i22) → 135_0_nest_NE(EOS(STATIC_135), i22, i22)
135_0_nest_NE(EOS(STATIC_135), i22, i22) → 139_0_nest_Load(EOS(STATIC_139), i22) | >(i22, 0)
139_0_nest_Load(EOS(STATIC_139), i22) → 146_0_nest_ConstantStackPush(EOS(STATIC_146), i22)
146_0_nest_ConstantStackPush(EOS(STATIC_146), i22) → 152_0_nest_IntArithmetic(EOS(STATIC_152), i22, 1)
152_0_nest_IntArithmetic(EOS(STATIC_152), i22, matching1) → 161_0_nest_InvokeMethod(EOS(STATIC_161), -(i22, 1)) | &&(>(i22, 0), =(matching1, 1))
161_0_nest_InvokeMethod(EOS(STATIC_161), i24) → 178_1_nest_InvokeMethod(178_0_nest_Load(EOS(STATIC_178), i24), i24)
178_0_nest_Load(EOS(STATIC_178), i24) → 187_0_nest_Load(EOS(STATIC_187), i24)
178_1_nest_InvokeMethod(147_0_nest_Return(EOS(STATIC_147), matching1, matching2), matching3) → 214_0_nest_Return(EOS(STATIC_214), 0, 0, 0) | &&(&&(=(matching1, 0), =(matching2, 0)), =(matching3, 0))
178_1_nest_InvokeMethod(278_0_nest_Return(EOS(STATIC_278), matching1), i41) → 295_0_nest_Return(EOS(STATIC_295), i41, 0) | =(matching1, 0)
187_0_nest_Load(EOS(STATIC_187), i24) → 122_0_nest_Load(EOS(STATIC_122), i24)
122_0_nest_Load(EOS(STATIC_122), i13) → 127_0_nest_NE(EOS(STATIC_127), i13, i13)
214_0_nest_Return(EOS(STATIC_214), matching1, matching2, matching3) → 220_0_nest_InvokeMethod(EOS(STATIC_220), 0) | &&(&&(=(matching1, 0), =(matching2, 0)), =(matching3, 0))
220_0_nest_InvokeMethod(EOS(STATIC_220), matching1) → 261_0_nest_InvokeMethod(EOS(STATIC_261), 0) | =(matching1, 0)
261_0_nest_InvokeMethod(EOS(STATIC_261), matching1) → 265_1_nest_InvokeMethod(265_0_nest_Load(EOS(STATIC_265), 0), 0) | =(matching1, 0)
265_0_nest_Load(EOS(STATIC_265), matching1) → 269_0_nest_Load(EOS(STATIC_269), 0) | =(matching1, 0)
269_0_nest_Load(EOS(STATIC_269), matching1) → 122_0_nest_Load(EOS(STATIC_122), 0) | =(matching1, 0)
295_0_nest_Return(EOS(STATIC_295), i41, matching1) → 252_0_nest_Return(EOS(STATIC_252), i41, 0) | =(matching1, 0)
252_0_nest_Return(EOS(STATIC_252), i34, matching1) → 261_0_nest_InvokeMethod(EOS(STATIC_261), 0) | =(matching1, 0)
R rules:
127_0_nest_NE(EOS(STATIC_127), matching1, matching2) → 136_0_nest_NE(EOS(STATIC_136), 0, 0) | &&(=(matching1, 0), =(matching2, 0))
136_0_nest_NE(EOS(STATIC_136), matching1, matching2) → 140_0_nest_ConstantStackPush(EOS(STATIC_140), 0) | &&(=(matching1, 0), =(matching2, 0))
140_0_nest_ConstantStackPush(EOS(STATIC_140), matching1) → 147_0_nest_Return(EOS(STATIC_147), 0, 0) | =(matching1, 0)
265_1_nest_InvokeMethod(147_0_nest_Return(EOS(STATIC_147), matching1, matching2), matching3) → 276_0_nest_Return(EOS(STATIC_276), 0, 0, 0) | &&(&&(=(matching1, 0), =(matching2, 0)), =(matching3, 0))
276_0_nest_Return(EOS(STATIC_276), matching1, matching2, matching3) → 278_0_nest_Return(EOS(STATIC_278), 0) | &&(&&(=(matching1, 0), =(matching2, 0)), =(matching3, 0))

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


P rules:
127_0_nest_NE(EOS(STATIC_127), x0, x0) → 178_1_nest_InvokeMethod(127_0_nest_NE(EOS(STATIC_127), -(x0, 1), -(x0, 1)), -(x0, 1)) | >(x0, 0)
178_1_nest_InvokeMethod(147_0_nest_Return(EOS(STATIC_147), 0, 0), 0) → 265_1_nest_InvokeMethod(127_0_nest_NE(EOS(STATIC_127), 0, 0), 0)
178_1_nest_InvokeMethod(278_0_nest_Return(EOS(STATIC_278), 0), x1) → 265_1_nest_InvokeMethod(127_0_nest_NE(EOS(STATIC_127), 0, 0), 0)
R rules:
127_0_nest_NE(EOS(STATIC_127), 0, 0) → 147_0_nest_Return(EOS(STATIC_147), 0, 0)
265_1_nest_InvokeMethod(147_0_nest_Return(EOS(STATIC_147), 0, 0), 0) → 278_0_nest_Return(EOS(STATIC_278), 0)

Filtered ground terms:



265_1_nest_InvokeMethod(x1, x2) → 265_1_nest_InvokeMethod(x1)
127_0_nest_NE(x1, x2, x3) → 127_0_nest_NE(x2, x3)
278_0_nest_Return(x1, x2) → 278_0_nest_Return
147_0_nest_Return(x1, x2, x3) → 147_0_nest_Return
Cond_127_0_nest_NE(x1, x2, x3, x4) → Cond_127_0_nest_NE(x1, x3, x4)

Filtered duplicate args:



127_0_nest_NE(x1, x2) → 127_0_nest_NE(x2)
Cond_127_0_nest_NE(x1, x2, x3) → Cond_127_0_nest_NE(x1, x3)

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


P rules:
127_0_nest_NE(x0) → 178_1_nest_InvokeMethod(127_0_nest_NE(-(x0, 1)), -(x0, 1)) | >(x0, 0)
178_1_nest_InvokeMethod(147_0_nest_Return, 0) → 265_1_nest_InvokeMethod(127_0_nest_NE(0))
178_1_nest_InvokeMethod(278_0_nest_Return, x1) → 265_1_nest_InvokeMethod(127_0_nest_NE(0))
R rules:
127_0_nest_NE(0) → 147_0_nest_Return
265_1_nest_InvokeMethod(147_0_nest_Return) → 278_0_nest_Return

Performed bisimulation on rules. Used the following equivalence classes: {[147_0_nest_Return, 278_0_nest_Return]=147_0_nest_Return}


Finished conversion. Obtained 5 rules for P and 2 rules for R. System has predefined symbols.


P rules:
127_0_NEST_NE(x0) → COND_127_0_NEST_NE(>(x0, 0), x0)
COND_127_0_NEST_NE(TRUE, x0) → 178_1_NEST_INVOKEMETHOD(127_0_nest_NE(-(x0, 1)), -(x0, 1))
COND_127_0_NEST_NE(TRUE, x0) → 127_0_NEST_NE(-(x0, 1))
178_1_NEST_INVOKEMETHOD(147_0_nest_Return, 0) → 127_0_NEST_NE(0)
178_1_NEST_INVOKEMETHOD(147_0_nest_Return, x1) → 127_0_NEST_NE(0)
R rules:
127_0_nest_NE(0) → 147_0_nest_Return
265_1_nest_InvokeMethod(147_0_nest_Return) → 147_0_nest_Return

(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:
127_0_nest_NE(0) → 147_0_nest_Return
265_1_nest_InvokeMethod(147_0_nest_Return) → 147_0_nest_Return

The integer pair graph contains the following rules and edges:
(0): 127_0_NEST_NE(x0[0]) → COND_127_0_NEST_NE(x0[0] > 0, x0[0])
(1): COND_127_0_NEST_NE(TRUE, x0[1]) → 178_1_NEST_INVOKEMETHOD(127_0_nest_NE(x0[1] - 1), x0[1] - 1)
(2): COND_127_0_NEST_NE(TRUE, x0[2]) → 127_0_NEST_NE(x0[2] - 1)
(3): 178_1_NEST_INVOKEMETHOD(147_0_nest_Return, 0) → 127_0_NEST_NE(0)
(4): 178_1_NEST_INVOKEMETHOD(147_0_nest_Return, x1[4]) → 127_0_NEST_NE(0)

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


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


(1) -> (3), if (127_0_nest_NE(x0[1] - 1) →* 147_0_nest_Return∧x0[1] - 1 →* 0)


(1) -> (4), if (127_0_nest_NE(x0[1] - 1) →* 147_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:
127_0_nest_NE(0)
265_1_nest_InvokeMethod(147_0_nest_Return)

(7) IDPNonInfProof (SOUND transformation)

Used the following options for this NonInfProof:
IDPGPoloSolver: Range: [(-1,2)] IsNat: false Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpCand1ShapeHeuristic@bb51061 Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 0 Max Right Steps: 0

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 127_0_NEST_NE(x0) → COND_127_0_NEST_NE(>(x0, 0), x0) the following chains were created:
  • We consider the chain 127_0_NEST_NE(x0[0]) → COND_127_0_NEST_NE(>(x0[0], 0), x0[0]), COND_127_0_NEST_NE(TRUE, x0[1]) → 178_1_NEST_INVOKEMETHOD(127_0_nest_NE(-(x0[1], 1)), -(x0[1], 1)) which results in the following constraint:

    (1)    (>(x0[0], 0)=TRUE∧x0[0]=x0[1] ⇒ 127_0_NEST_NE(x0[0])≥NonInfC∧127_0_NEST_NE(x0[0])≥COND_127_0_NEST_NE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_127_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 ⇒ 127_0_NEST_NE(x0[0])≥NonInfC∧127_0_NEST_NE(x0[0])≥COND_127_0_NEST_NE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_127_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_127_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(2)bni_15 + (-1)Bound*bni_15] + [bni_15]x0[0] ≥ 0∧[(-1)bso_16] ≥ 0)



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

    (4)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_127_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(2)bni_15 + (-1)Bound*bni_15] + [bni_15]x0[0] ≥ 0∧[(-1)bso_16] ≥ 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_127_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(2)bni_15 + (-1)Bound*bni_15] + [bni_15]x0[0] ≥ 0∧[(-1)bso_16] ≥ 0)



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

    (6)    (x0[0] ≥ 0 ⇒ (UIncreasing(COND_127_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(3)bni_15 + (-1)Bound*bni_15] + [bni_15]x0[0] ≥ 0∧[(-1)bso_16] ≥ 0)



  • We consider the chain 127_0_NEST_NE(x0[0]) → COND_127_0_NEST_NE(>(x0[0], 0), x0[0]), COND_127_0_NEST_NE(TRUE, x0[2]) → 127_0_NEST_NE(-(x0[2], 1)) which results in the following constraint:

    (7)    (>(x0[0], 0)=TRUE∧x0[0]=x0[2] ⇒ 127_0_NEST_NE(x0[0])≥NonInfC∧127_0_NEST_NE(x0[0])≥COND_127_0_NEST_NE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_127_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 ⇒ 127_0_NEST_NE(x0[0])≥NonInfC∧127_0_NEST_NE(x0[0])≥COND_127_0_NEST_NE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_127_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_127_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(2)bni_15 + (-1)Bound*bni_15] + [bni_15]x0[0] ≥ 0∧[(-1)bso_16] ≥ 0)



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

    (10)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_127_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(2)bni_15 + (-1)Bound*bni_15] + [bni_15]x0[0] ≥ 0∧[(-1)bso_16] ≥ 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_127_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(2)bni_15 + (-1)Bound*bni_15] + [bni_15]x0[0] ≥ 0∧[(-1)bso_16] ≥ 0)



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

    (12)    (x0[0] ≥ 0 ⇒ (UIncreasing(COND_127_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(3)bni_15 + (-1)Bound*bni_15] + [bni_15]x0[0] ≥ 0∧[(-1)bso_16] ≥ 0)







For Pair COND_127_0_NEST_NE(TRUE, x0) → 178_1_NEST_INVOKEMETHOD(127_0_nest_NE(-(x0, 1)), -(x0, 1)) the following chains were created:
  • We consider the chain COND_127_0_NEST_NE(TRUE, x0[1]) → 178_1_NEST_INVOKEMETHOD(127_0_nest_NE(-(x0[1], 1)), -(x0[1], 1)) which results in the following constraint:

    (13)    (COND_127_0_NEST_NE(TRUE, x0[1])≥NonInfC∧COND_127_0_NEST_NE(TRUE, x0[1])≥178_1_NEST_INVOKEMETHOD(127_0_nest_NE(-(x0[1], 1)), -(x0[1], 1))∧(UIncreasing(178_1_NEST_INVOKEMETHOD(127_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(178_1_NEST_INVOKEMETHOD(127_0_nest_NE(-(x0[1], 1)), -(x0[1], 1))), ≥)∧[bni_17] = 0∧[(-1)bso_18] ≥ 0)



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

    (15)    ((UIncreasing(178_1_NEST_INVOKEMETHOD(127_0_nest_NE(-(x0[1], 1)), -(x0[1], 1))), ≥)∧[bni_17] = 0∧[(-1)bso_18] ≥ 0)



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

    (16)    ((UIncreasing(178_1_NEST_INVOKEMETHOD(127_0_nest_NE(-(x0[1], 1)), -(x0[1], 1))), ≥)∧[bni_17] = 0∧[(-1)bso_18] ≥ 0)



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

    (17)    ((UIncreasing(178_1_NEST_INVOKEMETHOD(127_0_nest_NE(-(x0[1], 1)), -(x0[1], 1))), ≥)∧[bni_17] = 0∧0 = 0∧[(-1)bso_18] ≥ 0)







For Pair COND_127_0_NEST_NE(TRUE, x0) → 127_0_NEST_NE(-(x0, 1)) the following chains were created:
  • We consider the chain COND_127_0_NEST_NE(TRUE, x0[2]) → 127_0_NEST_NE(-(x0[2], 1)) which results in the following constraint:

    (18)    (COND_127_0_NEST_NE(TRUE, x0[2])≥NonInfC∧COND_127_0_NEST_NE(TRUE, x0[2])≥127_0_NEST_NE(-(x0[2], 1))∧(UIncreasing(127_0_NEST_NE(-(x0[2], 1))), ≥))



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

    (19)    ((UIncreasing(127_0_NEST_NE(-(x0[2], 1))), ≥)∧[bni_19] = 0∧[1 + (-1)bso_20] ≥ 0)



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

    (20)    ((UIncreasing(127_0_NEST_NE(-(x0[2], 1))), ≥)∧[bni_19] = 0∧[1 + (-1)bso_20] ≥ 0)



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

    (21)    ((UIncreasing(127_0_NEST_NE(-(x0[2], 1))), ≥)∧[bni_19] = 0∧[1 + (-1)bso_20] ≥ 0)



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

    (22)    ((UIncreasing(127_0_NEST_NE(-(x0[2], 1))), ≥)∧[bni_19] = 0∧0 = 0∧[1 + (-1)bso_20] ≥ 0)







For Pair 178_1_NEST_INVOKEMETHOD(147_0_nest_Return, 0) → 127_0_NEST_NE(0) the following chains were created:
  • We consider the chain 178_1_NEST_INVOKEMETHOD(147_0_nest_Return, 0) → 127_0_NEST_NE(0), 127_0_NEST_NE(x0[0]) → COND_127_0_NEST_NE(>(x0[0], 0), x0[0]) which results in the following constraint:

    (23)    (0=x0[0] ⇒ 178_1_NEST_INVOKEMETHOD(147_0_nest_Return, 0)≥NonInfC∧178_1_NEST_INVOKEMETHOD(147_0_nest_Return, 0)≥127_0_NEST_NE(0)∧(UIncreasing(127_0_NEST_NE(0)), ≥))



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

    (24)    (178_1_NEST_INVOKEMETHOD(147_0_nest_Return, 0)≥NonInfC∧178_1_NEST_INVOKEMETHOD(147_0_nest_Return, 0)≥127_0_NEST_NE(0)∧(UIncreasing(127_0_NEST_NE(0)), ≥))



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

    (25)    ((UIncreasing(127_0_NEST_NE(0)), ≥)∧[bni_21] = 0∧[1 + (-1)bso_22] ≥ 0)



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

    (26)    ((UIncreasing(127_0_NEST_NE(0)), ≥)∧[bni_21] = 0∧[1 + (-1)bso_22] ≥ 0)



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

    (27)    ((UIncreasing(127_0_NEST_NE(0)), ≥)∧[bni_21] = 0∧[1 + (-1)bso_22] ≥ 0)







For Pair 178_1_NEST_INVOKEMETHOD(147_0_nest_Return, x1) → 127_0_NEST_NE(0) the following chains were created:
  • We consider the chain 178_1_NEST_INVOKEMETHOD(147_0_nest_Return, x1[4]) → 127_0_NEST_NE(0) which results in the following constraint:

    (28)    (178_1_NEST_INVOKEMETHOD(147_0_nest_Return, x1[4])≥NonInfC∧178_1_NEST_INVOKEMETHOD(147_0_nest_Return, x1[4])≥127_0_NEST_NE(0)∧(UIncreasing(127_0_NEST_NE(0)), ≥))



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

    (29)    ((UIncreasing(127_0_NEST_NE(0)), ≥)∧[bni_23] = 0∧[1 + (-1)bso_24] ≥ 0)



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

    (30)    ((UIncreasing(127_0_NEST_NE(0)), ≥)∧[bni_23] = 0∧[1 + (-1)bso_24] ≥ 0)



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

    (31)    ((UIncreasing(127_0_NEST_NE(0)), ≥)∧[bni_23] = 0∧[1 + (-1)bso_24] ≥ 0)



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

    (32)    ((UIncreasing(127_0_NEST_NE(0)), ≥)∧[bni_23] = 0∧0 = 0∧[1 + (-1)bso_24] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • 127_0_NEST_NE(x0) → COND_127_0_NEST_NE(>(x0, 0), x0)
    • (x0[0] ≥ 0 ⇒ (UIncreasing(COND_127_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(3)bni_15 + (-1)Bound*bni_15] + [bni_15]x0[0] ≥ 0∧[(-1)bso_16] ≥ 0)
    • (x0[0] ≥ 0 ⇒ (UIncreasing(COND_127_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(3)bni_15 + (-1)Bound*bni_15] + [bni_15]x0[0] ≥ 0∧[(-1)bso_16] ≥ 0)

  • COND_127_0_NEST_NE(TRUE, x0) → 178_1_NEST_INVOKEMETHOD(127_0_nest_NE(-(x0, 1)), -(x0, 1))
    • ((UIncreasing(178_1_NEST_INVOKEMETHOD(127_0_nest_NE(-(x0[1], 1)), -(x0[1], 1))), ≥)∧[bni_17] = 0∧0 = 0∧[(-1)bso_18] ≥ 0)

  • COND_127_0_NEST_NE(TRUE, x0) → 127_0_NEST_NE(-(x0, 1))
    • ((UIncreasing(127_0_NEST_NE(-(x0[2], 1))), ≥)∧[bni_19] = 0∧0 = 0∧[1 + (-1)bso_20] ≥ 0)

  • 178_1_NEST_INVOKEMETHOD(147_0_nest_Return, 0) → 127_0_NEST_NE(0)
    • ((UIncreasing(127_0_NEST_NE(0)), ≥)∧[bni_21] = 0∧[1 + (-1)bso_22] ≥ 0)

  • 178_1_NEST_INVOKEMETHOD(147_0_nest_Return, x1) → 127_0_NEST_NE(0)
    • ((UIncreasing(127_0_NEST_NE(0)), ≥)∧[bni_23] = 0∧0 = 0∧[1 + (-1)bso_24] ≥ 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(127_0_nest_NE(x1)) = [1] + x1   
POL(0) = 0   
POL(147_0_nest_Return) = [1]   
POL(265_1_nest_InvokeMethod(x1)) = [-1]   
POL(127_0_NEST_NE(x1)) = [2] + x1   
POL(COND_127_0_NEST_NE(x1, x2)) = [2] + x2   
POL(>(x1, x2)) = [-1]   
POL(178_1_NEST_INVOKEMETHOD(x1, x2)) = [2] + x1   
POL(-(x1, x2)) = x1 + [-1]x2   
POL(1) = [1]   

The following pairs are in P>:

COND_127_0_NEST_NE(TRUE, x0[2]) → 127_0_NEST_NE(-(x0[2], 1))
178_1_NEST_INVOKEMETHOD(147_0_nest_Return, 0) → 127_0_NEST_NE(0)
178_1_NEST_INVOKEMETHOD(147_0_nest_Return, x1[4]) → 127_0_NEST_NE(0)

The following pairs are in Pbound:

127_0_NEST_NE(x0[0]) → COND_127_0_NEST_NE(>(x0[0], 0), x0[0])

The following pairs are in P≥:

127_0_NEST_NE(x0[0]) → COND_127_0_NEST_NE(>(x0[0], 0), x0[0])
COND_127_0_NEST_NE(TRUE, x0[1]) → 178_1_NEST_INVOKEMETHOD(127_0_nest_NE(-(x0[1], 1)), -(x0[1], 1))

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

127_0_nest_NE(0)1 ↔ 147_0_nest_Return1

(8) Complex Obligation (AND)

(9) 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:
127_0_nest_NE(0) → 147_0_nest_Return
265_1_nest_InvokeMethod(147_0_nest_Return) → 147_0_nest_Return

The integer pair graph contains the following rules and edges:
(0): 127_0_NEST_NE(x0[0]) → COND_127_0_NEST_NE(x0[0] > 0, x0[0])
(1): COND_127_0_NEST_NE(TRUE, x0[1]) → 178_1_NEST_INVOKEMETHOD(127_0_nest_NE(x0[1] - 1), x0[1] - 1)

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



The set Q consists of the following terms:
127_0_nest_NE(0)
265_1_nest_InvokeMethod(147_0_nest_Return)

(10) IDependencyGraphProof (EQUIVALENT transformation)

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

(11) TRUE

(12) 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:
127_0_nest_NE(0) → 147_0_nest_Return
265_1_nest_InvokeMethod(147_0_nest_Return) → 147_0_nest_Return

The integer pair graph contains the following rules and edges:
(1): COND_127_0_NEST_NE(TRUE, x0[1]) → 178_1_NEST_INVOKEMETHOD(127_0_nest_NE(x0[1] - 1), x0[1] - 1)
(2): COND_127_0_NEST_NE(TRUE, x0[2]) → 127_0_NEST_NE(x0[2] - 1)
(3): 178_1_NEST_INVOKEMETHOD(147_0_nest_Return, 0) → 127_0_NEST_NE(0)
(4): 178_1_NEST_INVOKEMETHOD(147_0_nest_Return, x1[4]) → 127_0_NEST_NE(0)

(1) -> (3), if (127_0_nest_NE(x0[1] - 1) →* 147_0_nest_Return∧x0[1] - 1 →* 0)


(1) -> (4), if (127_0_nest_NE(x0[1] - 1) →* 147_0_nest_Return∧x0[1] - 1 →* x1[4])



The set Q consists of the following terms:
127_0_nest_NE(0)
265_1_nest_InvokeMethod(147_0_nest_Return)

(13) IDependencyGraphProof (EQUIVALENT transformation)

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

(14) TRUE