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`
`         ↳Forward Instantiation Transformation`

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))

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

G(x) -> F(g(x))
one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 2`
`             ↳Polynomial Ordering`

Dependency Pairs:

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

Rules:

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

The following dependency pair can be strictly oriented:

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

There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(g(x1)) =  0 POL(G(x1)) =  x1 POL(b) =  0 POL(s(x1)) =  0 POL(a) =  1 POL(f(x1)) =  0 POL(F(x1)) =  0

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 2`
`             ↳Polo`
`             ...`
`               →DP Problem 3`
`                 ↳Dependency Graph`

Dependency Pairs:

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

Rules:

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

Using the Dependency Graph the DP problem was split into 1 DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 2`
`             ↳Polo`
`             ...`
`               →DP Problem 4`
`                 ↳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