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

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

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

The following dependency pair can be strictly oriented:

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

The following rules can be oriented:

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

Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:
G(x1) -> G(x1)
s(x1) -> s(x1)
f(x1, x2, x3) -> f
g(x1) -> x1

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

Dependency Pair:

Rules:

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

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

Rules:

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

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