Term Rewriting System R:
[x, y, z]
and(x, or(y, z)) -> or(and(x, y), and(x, z))
and(x, and(y, y)) -> and(x, y)
and(x, true) -> x
and(false, y) -> false
and(x, x) -> x
or(or(x, y), and(y, z)) -> or(x, y)
or(x, and(x, y)) -> x
or(true, y) -> true
or(x, false) -> x
or(x, x) -> x
or(x, or(y, y)) -> or(x, y)

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains two SCCs. 


   R
     ↳DPs
       →DP Problem 1
         ↳Argument Filtering and Ordering
       →DP Problem 2
         ↳AFS


Dependency Pair:

OR(x, or(y, y)) -> OR(x, y)


Rules:


and(x, or(y, z)) -> or(and(x, y), and(x, z))
and(x, and(y, y)) -> and(x, y)
and(x, true) -> x
and(false, y) -> false
and(x, x) -> x
or(or(x, y), and(y, z)) -> or(x, y)
or(x, and(x, y)) -> x
or(true, y) -> true
or(x, false) -> x
or(x, x) -> x
or(x, or(y, y)) -> or(x, y)





The following dependency pair can be strictly oriented:

OR(x, or(y, y)) -> OR(x, y)


There are no usable rules using the Ce-refinement that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(OR(x1, x2))=  x1 + x2  
  POL(or(x1, x2))=  1 + x1 + x2  

resulting in one new DP problem.
Used Argument Filtering System:
OR(x1, x2) -> OR(x1, x2)
or(x1, x2) -> or(x1, x2)


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
           →DP Problem 3
             ↳Dependency Graph
       →DP Problem 2
         ↳AFS


Dependency Pair:


Rules:


and(x, or(y, z)) -> or(and(x, y), and(x, z))
and(x, and(y, y)) -> and(x, y)
and(x, true) -> x
and(false, y) -> false
and(x, x) -> x
or(or(x, y), and(y, z)) -> or(x, y)
or(x, and(x, y)) -> x
or(true, y) -> true
or(x, false) -> x
or(x, x) -> x
or(x, or(y, y)) -> or(x, y)





Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳Argument Filtering and Ordering


Dependency Pairs:

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


Rules:


and(x, or(y, z)) -> or(and(x, y), and(x, z))
and(x, and(y, y)) -> and(x, y)
and(x, true) -> x
and(false, y) -> false
and(x, x) -> x
or(or(x, y), and(y, z)) -> or(x, y)
or(x, and(x, y)) -> x
or(true, y) -> true
or(x, false) -> x
or(x, x) -> x
or(x, or(y, y)) -> or(x, y)





The following dependency pair can be strictly oriented:

AND(x, and(y, y)) -> AND(x, y)


There are no usable rules using the Ce-refinement that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(and(x1, x2))=  1 + x1 + x2  
  POL(or(x1, x2))=  x1 + x2  
  POL(AND(x1, x2))=  x1 + x2  

resulting in one new DP problem.
Used Argument Filtering System:
AND(x1, x2) -> AND(x1, x2)
and(x1, x2) -> and(x1, x2)
or(x1, x2) -> or(x1, x2)


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳AFS
           →DP Problem 4
             ↳Argument Filtering and Ordering


Dependency Pairs:

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


Rules:


and(x, or(y, z)) -> or(and(x, y), and(x, z))
and(x, and(y, y)) -> and(x, y)
and(x, true) -> x
and(false, y) -> false
and(x, x) -> x
or(or(x, y), and(y, z)) -> or(x, y)
or(x, and(x, y)) -> x
or(true, y) -> true
or(x, false) -> x
or(x, x) -> x
or(x, or(y, y)) -> or(x, y)





The following dependency pairs can be strictly oriented:

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


There are no usable rules using the Ce-refinement 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.
Used Argument Filtering System:
AND(x1, x2) -> AND(x1, x2)
or(x1, x2) -> or(x1, x2)


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳AFS
           →DP Problem 4
             ↳AFS
             ...
               →DP Problem 5
                 ↳Dependency Graph


Dependency Pair:


Rules:


and(x, or(y, z)) -> or(and(x, y), and(x, z))
and(x, and(y, y)) -> and(x, y)
and(x, true) -> x
and(false, y) -> false
and(x, x) -> x
or(or(x, y), and(y, z)) -> or(x, y)
or(x, and(x, y)) -> x
or(true, y) -> true
or(x, false) -> x
or(x, x) -> x
or(x, or(y, y)) -> or(x, y)





Using the Dependency Graph resulted in no new DP problems.

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