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
     ↳Removing Redundant Rules



Removing the following rules from R which fullfill a polynomial ordering:

evenodd(0, s(0)) -> false

where the Polynomial interpretation:
  POL(0)=  0  
  POL(evenodd(x1, x2))=  1 + x1 + x2  
  POL(false)=  0  
  POL(true)=  0  
  POL(s(x1))=  x1  
  POL(not(x1))=  x1  
was used.

Not all Rules of R can be deleted, so we still have to regard a part of R. 


   R
     ↳RRRPolo
       →TRS2
         ↳Overlay and local confluence Check



The TRS is overlay and locally confluent (all critical pairs are trivially joinable).Hence, we can switch to innermost. 


   R
     ↳RRRPolo
       →TRS2
         ↳OC
           →TRS3
             ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC. 


   R
     ↳RRRPolo
       →TRS2
         ↳OC
           →TRS3
             ↳DPs
             ...
               →DP Problem 1
                 ↳Usable Rules (Innermost)


Dependency Pairs:

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


Rules:


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


Strategy:

innermost




As we are in the innermost case, we can delete all 4 non-usable-rules.


   R
     ↳RRRPolo
       →TRS2
         ↳OC
           →TRS3
             ↳DPs
             ...
               →DP Problem 2
                 ↳Size-Change Principle


Dependency Pairs:

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


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. EVENODD(x, 0) -> EVENODD(x, s(0))
  2. EVENODD(s(x), s(0)) -> EVENODD(x, 0)
and get the following Size-Change Graph(s):
{1} , {1}
1=1
{2} , {2}
1>1
2>2

which lead(s) to this/these maximal multigraph(s):
{2} , {1}
1>1
{1} , {2}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
s(x1) -> s(x1)

We obtain no new DP problems.

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