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

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

F(f(x)) -> G(f(x))
G(g(x)) -> F(x)

Furthermore, R contains one SCC.

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

Dependency Pairs:

G(g(x)) -> F(x)
F(f(x)) -> G(f(x))

Rules:

f(f(x)) -> g(f(x))
g(g(x)) -> f(x)

The following dependency pairs can be strictly oriented:

G(g(x)) -> F(x)
F(f(x)) -> G(f(x))

The following usable rules w.r.t. to the AFS can be oriented:

g(g(x)) -> f(x)
f(f(x)) -> g(f(x))

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
{F, f, g} > G

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

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

Dependency Pair:

Rules:

f(f(x)) -> g(f(x))
g(g(x)) -> f(x)

Using the Dependency Graph resulted in no new DP problems.

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