Term Rewriting System R:
[x, y]
not(not(x)) -> x
not(or(x, y)) -> and(not(not(not(x))), not(not(not(y))))
not(and(x, y)) -> or(not(not(not(x))), not(not(not(y))))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

NOT(or(x, y)) -> NOT(not(not(x)))
NOT(or(x, y)) -> NOT(not(x))
NOT(or(x, y)) -> NOT(x)
NOT(or(x, y)) -> NOT(not(not(y)))
NOT(or(x, y)) -> NOT(not(y))
NOT(or(x, y)) -> NOT(y)
NOT(and(x, y)) -> NOT(not(not(x)))
NOT(and(x, y)) -> NOT(not(x))
NOT(and(x, y)) -> NOT(x)
NOT(and(x, y)) -> NOT(not(not(y)))
NOT(and(x, y)) -> NOT(not(y))
NOT(and(x, y)) -> NOT(y)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Argument Filtering and Ordering


Dependency Pairs:

NOT(and(x, y)) -> NOT(y)
NOT(and(x, y)) -> NOT(not(y))
NOT(and(x, y)) -> NOT(not(not(y)))
NOT(and(x, y)) -> NOT(x)
NOT(and(x, y)) -> NOT(not(x))
NOT(and(x, y)) -> NOT(not(not(x)))
NOT(or(x, y)) -> NOT(y)
NOT(or(x, y)) -> NOT(not(y))
NOT(or(x, y)) -> NOT(not(not(y)))
NOT(or(x, y)) -> NOT(x)
NOT(or(x, y)) -> NOT(not(x))
NOT(or(x, y)) -> NOT(not(not(x)))


Rules:


not(not(x)) -> x
not(or(x, y)) -> and(not(not(not(x))), not(not(not(y))))
not(and(x, y)) -> or(not(not(not(x))), not(not(not(y))))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

NOT(and(x, y)) -> NOT(y)
NOT(and(x, y)) -> NOT(not(y))
NOT(and(x, y)) -> NOT(not(not(y)))
NOT(and(x, y)) -> NOT(x)
NOT(and(x, y)) -> NOT(not(x))
NOT(and(x, y)) -> NOT(not(not(x)))
NOT(or(x, y)) -> NOT(y)
NOT(or(x, y)) -> NOT(not(y))
NOT(or(x, y)) -> NOT(not(not(y)))
NOT(or(x, y)) -> NOT(x)
NOT(or(x, y)) -> NOT(not(x))
NOT(or(x, y)) -> NOT(not(not(x)))


The following usable rules for innermost w.r.t. to the AFS can be oriented:

not(not(x)) -> x
not(or(x, y)) -> and(not(not(not(x))), not(not(not(y))))
not(and(x, y)) -> or(not(not(not(x))), not(not(not(y))))


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(and(x1, x2))=  1 + x1 + x2  
  POL(NOT(x1))=  1 + x1  
  POL(or(x1, x2))=  1 + x1 + x2  
  POL(not(x1))=  x1  

resulting in one new DP problem.
Used Argument Filtering System:
NOT(x1) -> NOT(x1)
or(x1, x2) -> or(x1, x2)
not(x1) -> not(x1)
and(x1, x2) -> and(x1, x2)


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
           →DP Problem 2
             ↳Dependency Graph


Dependency Pair:


Rules:


not(not(x)) -> x
not(or(x, y)) -> and(not(not(not(x))), not(not(not(y))))
not(and(x, y)) -> or(not(not(not(x))), not(not(not(y))))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:01 minutes