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

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC. 


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


Dependency Pair:

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


Rules:


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





The following dependency pair can be strictly oriented:

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


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


Dependency Pair:


Rules:


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





Using the Dependency Graph resulted in no new DP problems.

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