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