Term Rewriting System R:
[x]
f(g(x)) -> f(a(g(g(f(x))), g(f(x))))

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

F(g(x)) -> F(a(g(g(f(x))), g(f(x))))
F(g(x)) -> F(x)

Furthermore, R contains one SCC.

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

Dependency Pair:

F(g(x)) -> F(x)

Rule:

f(g(x)) -> f(a(g(g(f(x))), g(f(x))))

The following dependency pair can be strictly oriented:

F(g(x)) -> F(x)

There are no usable rules w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(g(x1)) =  1 + x1 POL(F(x1)) =  x1

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

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

Dependency Pair:

Rule:

f(g(x)) -> f(a(g(g(f(x))), g(f(x))))

Using the Dependency Graph resulted in no new DP problems.

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