Term Rewriting System R:
[x]
f(f(x)) -> f(x)
g(0) -> g(f(0))

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

G(0) -> G(f(0))
G(0) -> F(0)

Furthermore, R contains one SCC.

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

Dependency Pair:

G(0) -> G(f(0))

Rules:

f(f(x)) -> f(x)
g(0) -> g(f(0))

The following dependency pair can be strictly oriented:

G(0) -> G(f(0))

The following rules can be oriented:

f(f(x)) -> f(x)
g(0) -> g(f(0))

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
0 > f

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

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

Dependency Pair:

Rules:

f(f(x)) -> f(x)
g(0) -> g(f(0))

Using the Dependency Graph resulted in no new DP problems.

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