Term Rewriting System R:
[x]
not(true) -> false
not(false) -> true
evenodd(x, 0) -> not(evenodd(x, s(0)))
evenodd(0, s(0)) -> false
evenodd(s(x), s(0)) -> evenodd(x, 0)

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

EVENODD(x, 0) -> NOT(evenodd(x, s(0)))
EVENODD(x, 0) -> EVENODD(x, s(0))
EVENODD(s(x), s(0)) -> EVENODD(x, 0)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Forward Instantiation Transformation


Dependency Pairs:

EVENODD(s(x), s(0)) -> EVENODD(x, 0)
EVENODD(x, 0) -> EVENODD(x, s(0))


Rules:


not(true) -> false
not(false) -> true
evenodd(x, 0) -> not(evenodd(x, s(0)))
evenodd(0, s(0)) -> false
evenodd(s(x), s(0)) -> evenodd(x, 0)





On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

EVENODD(x, 0) -> EVENODD(x, s(0))
one new Dependency Pair is created:

EVENODD(s(x''), 0) -> EVENODD(s(x''), s(0))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 2
             ↳Forward Instantiation Transformation


Dependency Pairs:

EVENODD(s(x''), 0) -> EVENODD(s(x''), s(0))
EVENODD(s(x), s(0)) -> EVENODD(x, 0)


Rules:


not(true) -> false
not(false) -> true
evenodd(x, 0) -> not(evenodd(x, s(0)))
evenodd(0, s(0)) -> false
evenodd(s(x), s(0)) -> evenodd(x, 0)





On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

EVENODD(s(x), s(0)) -> EVENODD(x, 0)
one new Dependency Pair is created:

EVENODD(s(s(x'''')), s(0)) -> EVENODD(s(x''''), 0)

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 3
                 ↳Forward Instantiation Transformation


Dependency Pairs:

EVENODD(s(s(x'''')), s(0)) -> EVENODD(s(x''''), 0)
EVENODD(s(x''), 0) -> EVENODD(s(x''), s(0))


Rules:


not(true) -> false
not(false) -> true
evenodd(x, 0) -> not(evenodd(x, s(0)))
evenodd(0, s(0)) -> false
evenodd(s(x), s(0)) -> evenodd(x, 0)





On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

EVENODD(s(x''), 0) -> EVENODD(s(x''), s(0))
one new Dependency Pair is created:

EVENODD(s(s(x'''''')), 0) -> EVENODD(s(s(x'''''')), s(0))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 4
                 ↳Forward Instantiation Transformation


Dependency Pairs:

EVENODD(s(s(x'''''')), 0) -> EVENODD(s(s(x'''''')), s(0))
EVENODD(s(s(x'''')), s(0)) -> EVENODD(s(x''''), 0)


Rules:


not(true) -> false
not(false) -> true
evenodd(x, 0) -> not(evenodd(x, s(0)))
evenodd(0, s(0)) -> false
evenodd(s(x), s(0)) -> evenodd(x, 0)





On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

EVENODD(s(s(x'''')), s(0)) -> EVENODD(s(x''''), 0)
one new Dependency Pair is created:

EVENODD(s(s(s(x''''''''))), s(0)) -> EVENODD(s(s(x'''''''')), 0)

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 5
                 ↳Forward Instantiation Transformation


Dependency Pairs:

EVENODD(s(s(s(x''''''''))), s(0)) -> EVENODD(s(s(x'''''''')), 0)
EVENODD(s(s(x'''''')), 0) -> EVENODD(s(s(x'''''')), s(0))


Rules:


not(true) -> false
not(false) -> true
evenodd(x, 0) -> not(evenodd(x, s(0)))
evenodd(0, s(0)) -> false
evenodd(s(x), s(0)) -> evenodd(x, 0)





On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

EVENODD(s(s(x'''''')), 0) -> EVENODD(s(s(x'''''')), s(0))
one new Dependency Pair is created:

EVENODD(s(s(s(x''''''''''))), 0) -> EVENODD(s(s(s(x''''''''''))), s(0))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 6
                 ↳Forward Instantiation Transformation


Dependency Pairs:

EVENODD(s(s(s(x''''''''''))), 0) -> EVENODD(s(s(s(x''''''''''))), s(0))
EVENODD(s(s(s(x''''''''))), s(0)) -> EVENODD(s(s(x'''''''')), 0)


Rules:


not(true) -> false
not(false) -> true
evenodd(x, 0) -> not(evenodd(x, s(0)))
evenodd(0, s(0)) -> false
evenodd(s(x), s(0)) -> evenodd(x, 0)





On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

EVENODD(s(s(s(x''''''''))), s(0)) -> EVENODD(s(s(x'''''''')), 0)
one new Dependency Pair is created:

EVENODD(s(s(s(s(x'''''''''''')))), s(0)) -> EVENODD(s(s(s(x''''''''''''))), 0)

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 7
                 ↳Forward Instantiation Transformation


Dependency Pairs:

EVENODD(s(s(s(s(x'''''''''''')))), s(0)) -> EVENODD(s(s(s(x''''''''''''))), 0)
EVENODD(s(s(s(x''''''''''))), 0) -> EVENODD(s(s(s(x''''''''''))), s(0))


Rules:


not(true) -> false
not(false) -> true
evenodd(x, 0) -> not(evenodd(x, s(0)))
evenodd(0, s(0)) -> false
evenodd(s(x), s(0)) -> evenodd(x, 0)





On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

EVENODD(s(s(s(x''''''''''))), 0) -> EVENODD(s(s(s(x''''''''''))), s(0))
one new Dependency Pair is created:

EVENODD(s(s(s(s(x'''''''''''''')))), 0) -> EVENODD(s(s(s(s(x'''''''''''''')))), s(0))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 8
                 ↳Forward Instantiation Transformation


Dependency Pairs:

EVENODD(s(s(s(s(x'''''''''''''')))), 0) -> EVENODD(s(s(s(s(x'''''''''''''')))), s(0))
EVENODD(s(s(s(s(x'''''''''''')))), s(0)) -> EVENODD(s(s(s(x''''''''''''))), 0)


Rules:


not(true) -> false
not(false) -> true
evenodd(x, 0) -> not(evenodd(x, s(0)))
evenodd(0, s(0)) -> false
evenodd(s(x), s(0)) -> evenodd(x, 0)





On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

EVENODD(s(s(s(s(x'''''''''''')))), s(0)) -> EVENODD(s(s(s(x''''''''''''))), 0)
one new Dependency Pair is created:

EVENODD(s(s(s(s(s(x''''''''''''''''))))), s(0)) -> EVENODD(s(s(s(s(x'''''''''''''''')))), 0)

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 9
                 ↳Forward Instantiation Transformation


Dependency Pairs:

EVENODD(s(s(s(s(s(x''''''''''''''''))))), s(0)) -> EVENODD(s(s(s(s(x'''''''''''''''')))), 0)
EVENODD(s(s(s(s(x'''''''''''''')))), 0) -> EVENODD(s(s(s(s(x'''''''''''''')))), s(0))


Rules:


not(true) -> false
not(false) -> true
evenodd(x, 0) -> not(evenodd(x, s(0)))
evenodd(0, s(0)) -> false
evenodd(s(x), s(0)) -> evenodd(x, 0)





On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

EVENODD(s(s(s(s(x'''''''''''''')))), 0) -> EVENODD(s(s(s(s(x'''''''''''''')))), s(0))
one new Dependency Pair is created:

EVENODD(s(s(s(s(s(x''''''''''''''''''))))), 0) -> EVENODD(s(s(s(s(s(x''''''''''''''''''))))), s(0))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 10
                 ↳Forward Instantiation Transformation


Dependency Pairs:

EVENODD(s(s(s(s(s(x''''''''''''''''''))))), 0) -> EVENODD(s(s(s(s(s(x''''''''''''''''''))))), s(0))
EVENODD(s(s(s(s(s(x''''''''''''''''))))), s(0)) -> EVENODD(s(s(s(s(x'''''''''''''''')))), 0)


Rules:


not(true) -> false
not(false) -> true
evenodd(x, 0) -> not(evenodd(x, s(0)))
evenodd(0, s(0)) -> false
evenodd(s(x), s(0)) -> evenodd(x, 0)





On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

EVENODD(s(s(s(s(s(x''''''''''''''''))))), s(0)) -> EVENODD(s(s(s(s(x'''''''''''''''')))), 0)
one new Dependency Pair is created:

EVENODD(s(s(s(s(s(s(x'''''''''''''''''''')))))), s(0)) -> EVENODD(s(s(s(s(s(x''''''''''''''''''''))))), 0)

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 11
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

EVENODD(s(s(s(s(s(s(x'''''''''''''''''''')))))), s(0)) -> EVENODD(s(s(s(s(s(x''''''''''''''''''''))))), 0)
EVENODD(s(s(s(s(s(x''''''''''''''''''))))), 0) -> EVENODD(s(s(s(s(s(x''''''''''''''''''))))), s(0))


Rules:


not(true) -> false
not(false) -> true
evenodd(x, 0) -> not(evenodd(x, s(0)))
evenodd(0, s(0)) -> false
evenodd(s(x), s(0)) -> evenodd(x, 0)




Termination of R could not be shown.
Duration:
0:00 minutes