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

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC. 


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


Dependency Pairs:

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


Rules:


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





The following dependency pair can be strictly oriented:

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


The following usable rules using the Ce-refinement can be oriented:

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


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

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


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


Dependency Pair:

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


Rules:


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





The following dependency pair can be strictly oriented:

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


The following usable rules using the Ce-refinement can be oriented:

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


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

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


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


Dependency Pair:


Rules:


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





Using the Dependency Graph resulted in no new DP problems.

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