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

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

F(s(s(x))) -> F(f(s(x)))
F(s(s(x))) -> F(s(x))

Furthermore, R contains one SCC.

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

Dependency Pairs:

F(s(s(x))) -> F(s(x))
F(s(s(x))) -> F(f(s(x)))

Rules:

f(0) -> s(0)
f(s(0)) -> s(0)
f(s(s(x))) -> f(f(s(x)))

The following dependency pair can be strictly oriented:

F(s(s(x))) -> F(s(x))

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

f(0) -> s(0)
f(s(0)) -> s(0)
f(s(s(x))) -> f(f(s(x)))

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
{f, s}

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

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`           →DP Problem 2`
`             ↳Narrowing Transformation`

Dependency Pair:

F(s(s(x))) -> F(f(s(x)))

Rules:

f(0) -> s(0)
f(s(0)) -> s(0)
f(s(s(x))) -> f(f(s(x)))

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

F(s(s(x))) -> F(f(s(x)))
two new Dependency Pairs are created:

F(s(s(0))) -> F(s(0))
F(s(s(s(x'')))) -> F(f(f(s(x''))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 3`
`                 ↳Remaining Obligation(s)`

The following remains to be proven:
Dependency Pair:

F(s(s(s(x'')))) -> F(f(f(s(x''))))

Rules:

f(0) -> s(0)
f(s(0)) -> s(0)
f(s(s(x))) -> f(f(s(x)))

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