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

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Polynomial Ordering


Dependency Pairs:

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


Rules:


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


Strategy:

innermost




The following dependency pairs can be strictly oriented:

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


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(and(x1, x2))=  x1 + x2  
  POL(NOT(x1))=  x1  
  POL(or(x1, x2))=  1 + x1 + x2  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
           →DP Problem 2
             ↳Polynomial Ordering


Dependency Pairs:

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


Rules:


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


Strategy:

innermost




The following dependency pairs can be strictly oriented:

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


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(and(x1, x2))=  1 + x1 + x2  
  POL(NOT(x1))=  x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
           →DP Problem 2
             ↳Polo
             ...
               →DP Problem 3
                 ↳Dependency Graph


Dependency Pair:


Rules:


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


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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