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

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

F(s(0), g(x)) -> F(x, g(x))
G(s(x)) -> G(x)

Furthermore, R contains two SCCs.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Argument Filtering and Ordering`
`       →DP Problem 2`
`         ↳Remaining`

Dependency Pair:

G(s(x)) -> G(x)

Rules:

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

The following dependency pair can be strictly oriented:

G(s(x)) -> G(x)

There are no usable rules w.r.t. to the AFS that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
trivial

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

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`           →DP Problem 3`
`             ↳Dependency Graph`
`       →DP Problem 2`
`         ↳Remaining`

Dependency Pair:

Rules:

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

Using the Dependency Graph resulted in no new DP problems.

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

The following remains to be proven:
Dependency Pair:

F(s(0), g(x)) -> F(x, g(x))

Rules:

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

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