Term Rewriting System R:
[X]
f(f(X)) -> f(g(f(g(f(X)))))
f(g(f(X))) -> f(g(X))

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(f(X)) -> F(g(f(g(f(X)))))
F(f(X)) -> F(g(f(X)))
F(g(f(X))) -> F(g(X))

Furthermore, R contains one SCC. 


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


Dependency Pair:

F(g(f(X))) -> F(g(X))


Rules:


f(f(X)) -> f(g(f(g(f(X)))))
f(g(f(X))) -> f(g(X))





The following dependency pair can be strictly oriented:

F(g(f(X))) -> F(g(X))


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

resulting in one new DP problem.
Used Argument Filtering System:
F(x1) -> F(x1)
g(x1) -> g(x1)
f(x1) -> f(x1)


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


Dependency Pair:


Rules:


f(f(X)) -> f(g(f(g(f(X)))))
f(g(f(X))) -> f(g(X))





Using the Dependency Graph resulted in no new DP problems.

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