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

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

AND(not(not(x)), y, not(z)) -> AND(y, band(x, z), x)

Furthermore, R contains one SCC. 


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


Dependency Pair:

AND(not(not(x)), y, not(z)) -> AND(y, band(x, z), x)


Rule:


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





The following dependency pair can be strictly oriented:

AND(not(not(x)), y, not(z)) -> AND(y, band(x, z), x)


The following rule can be oriented:

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


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(and(x1, x2, x3))=  1 + x1 + x2 + x3  
  POL(not(x1))=  1 + x1  
  POL(AND(x1, x2, x3))=  1 + x1 + x2 + x3  

resulting in one new DP problem.
Used Argument Filtering System:
AND(x1, x2, x3) -> AND(x1, x2, x3)
not(x1) -> not(x1)
band(x1, x2) -> x2
and(x1, x2, x3) -> and(x1, x2, x3)


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


Dependency Pair:


Rule:


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





Using the Dependency Graph resulted in no new DP problems.

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