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

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC. 


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


Dependency Pairs:

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


Rule:


f(g(X), Y) -> f(X, f(g(X), Y))





The following dependency pair can be strictly oriented:

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


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

f(g(X), Y) -> f(X, f(g(X), Y))


Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:
F(x1, x2) -> F(x1, x2)
g(x1) -> g(x1)
f(x1, x2) -> x2


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
           →DP Problem 2
             ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pair:

F(g(X), Y) -> F(g(X), Y)


Rule:


f(g(X), Y) -> f(X, f(g(X), Y))




Termination of R could not be shown.
Duration:
0:00 minutes