Term Rewriting System R:
[x, y]
*(x, *(minus(y), y)) -> *(minus(*(y, y)), x)

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

*'(x, *(minus(y), y)) -> *'(minus(*(y, y)), x)
*'(x, *(minus(y), y)) -> *'(y, y)

Furthermore, R contains one SCC. 


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


Dependency Pairs:

*'(x, *(minus(y), y)) -> *'(y, y)
*'(x, *(minus(y), y)) -> *'(minus(*(y, y)), x)


Rule:


*(x, *(minus(y), y)) -> *(minus(*(y, y)), x)





The following dependency pair can be strictly oriented:

*'(x, *(minus(y), y)) -> *'(y, y)


The following usable rule w.r.t. to the AFS can be oriented:

*(x, *(minus(y), y)) -> *(minus(*(y, y)), x)


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(*'(x1, x2))=  1 + x1 + x2  
  POL(*(x1, x2))=  x1 + x2  
  POL(minus(x1))=  1 + x1  

resulting in one new DP problem.
Used Argument Filtering System:
*'(x1, x2) -> *'(x1, x2)
*(x1, x2) -> *(x1, x2)
minus(x1) -> minus(x1)


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


Dependency Pair:

*'(x, *(minus(y), y)) -> *'(minus(*(y, y)), x)


Rule:


*(x, *(minus(y), y)) -> *(minus(*(y, y)), x)





The following dependency pair can be strictly oriented:

*'(x, *(minus(y), y)) -> *'(minus(*(y, y)), x)


There are no usable rules w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(*'(x1, x2))=  x1 + x2  
  POL(*(x1, x2))=  1 + x1 + x2  
  POL(minus)=  0  

resulting in one new DP problem.
Used Argument Filtering System:
*'(x1, x2) -> *'(x1, x2)
*(x1, x2) -> *(x1, x2)
minus(x1) -> minus


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


Dependency Pair:


Rule:


*(x, *(minus(y), y)) -> *(minus(*(y, y)), x)





Using the Dependency Graph resulted in no new DP problems.

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