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

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

F(g(a)) -> F(s(g(b)))
F(g(a)) -> G(b)
G(x) -> F(g(x))
G(x) -> G(x)

Furthermore, R contains one SCC.

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

Dependency Pairs:

G(x) -> G(x)
G(x) -> F(g(x))
F(g(a)) -> G(b)

Rules:

f(g(a)) -> f(s(g(b)))
f(f(x)) -> b
g(x) -> f(g(x))

The following dependency pairs can be strictly oriented:

G(x) -> F(g(x))
F(g(a)) -> G(b)

The following usable rules using the Ce-refinement can be oriented:

g(x) -> f(g(x))
f(g(a)) -> f(s(g(b)))
f(f(x)) -> b

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
a > G > F > b
a > G > g > b
a > s > b
f > b

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

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

The following remains to be proven:
Dependency Pair:

G(x) -> G(x)

Rules:

f(g(a)) -> f(s(g(b)))
f(f(x)) -> b
g(x) -> f(g(x))

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