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

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polynomial Ordering`

Dependency Pairs:

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

Rules:

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

The following dependency pairs can be strictly oriented:

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

There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(G(x1)) =  x1 POL(s(x1)) =  1 + x1 POL(F(x1)) =  x1

resulting in one new DP problem.

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

Dependency Pair:

Rules:

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

Using the Dependency Graph resulted in no new DP problems.

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