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)

Innermost 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)


Strategy:

innermost




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)


Strategy:

innermost




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)


Strategy:

innermost




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)


Strategy:

innermost




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
                 ↳Polynomial Ordering


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)


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(0)=  0  
  POL(EVENODD(x1, x2))=  x1  
  POL(s(x1))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 6
                 ↳Dependency Graph


Dependency Pair:

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)


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:00 minutes