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

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains three SCCs. 


   R
     ↳DPs
       →DP Problem 1
         ↳Polynomial Ordering
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo


Dependency Pairs:

AND(or(y, z), x) -> AND(x, z)
AND(or(y, z), x) -> AND(x, y)
AND(x, or(y, z)) -> AND(x, z)
AND(x, or(y, z)) -> AND(x, y)


Rules:


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


Strategy:

innermost




The following dependency pairs can be strictly oriented:

AND(or(y, z), x) -> AND(x, z)
AND(or(y, z), x) -> AND(x, y)
AND(x, or(y, z)) -> AND(x, z)
AND(x, or(y, z)) -> AND(x, y)


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(or(x1, x2))=  1 + x1 + x2  
  POL(AND(x1, x2))=  x1 + x2  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
           →DP Problem 4
             ↳Dependency Graph
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo


Dependency Pair:


Rules:


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


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polynomial Ordering
       →DP Problem 3
         ↳Polo


Dependency Pairs:

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


Rules:


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


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(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
         ↳Polo
           →DP Problem 5
             ↳Dependency Graph
       →DP Problem 3
         ↳Polo


Dependency Pair:


Rules:


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


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polynomial Ordering


Dependency Pairs:

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


Rules:


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


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
         ↳Polo
           →DP Problem 6
             ↳Dependency Graph


Dependency Pair:


Rules:


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


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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