Term Rewriting System R:
[X]
a -> g(c)
g(a) -> b
f(g(X), b) -> f(a, X)

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

A -> G(c)
F(g(X), b) -> F(a, X)
F(g(X), b) -> A

Furthermore, R contains one SCC.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Narrowing Transformation`

Dependency Pair:

F(g(X), b) -> F(a, X)

Rules:

a -> g(c)
g(a) -> b
f(g(X), b) -> f(a, X)

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

F(g(X), b) -> F(a, X)
one new Dependency Pair is created:

F(g(X), b) -> F(g(c), X)

The transformation is resulting in one new DP problem:

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

Dependency Pair:

F(g(X), b) -> F(g(c), X)

Rules:

a -> g(c)
g(a) -> b
f(g(X), b) -> f(a, X)

The following dependency pair can be strictly oriented:

F(g(X), b) -> F(g(c), X)

Additionally, the following usable rule w.r.t. to the implicit AFS can be oriented:

g(a) -> b

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(c) =  0 POL(g(x1)) =  1 + x1 POL(b) =  1 POL(a) =  0 POL(F(x1, x2)) =  1 + x1 + x2

resulting in one new DP problem.

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

Dependency Pair:

Rules:

a -> g(c)
g(a) -> b
f(g(X), b) -> f(a, X)

Using the Dependency Graph resulted in no new DP problems.

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