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

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

H(f(f(x))) -> H(f(g(f(x))))
H(f(f(x))) -> F(g(f(x)))
F(g(f(x))) -> F(f(x))

Furthermore, R contains two SCCs.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Narrowing Transformation`
`       →DP Problem 2`
`         ↳Nar`

Dependency Pair:

F(g(f(x))) -> F(f(x))

Rules:

h(f(f(x))) -> h(f(g(f(x))))
f(g(f(x))) -> f(f(x))

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

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

F(g(f(g(f(x''))))) -> F(f(f(x'')))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 3`
`             ↳Narrowing Transformation`
`       →DP Problem 2`
`         ↳Nar`

Dependency Pair:

F(g(f(g(f(x''))))) -> F(f(f(x'')))

Rules:

h(f(f(x))) -> h(f(g(f(x))))
f(g(f(x))) -> f(f(x))

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

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

F(g(f(g(f(g(f(x'))))))) -> F(f(f(f(x'))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 3`
`             ↳Nar`
`             ...`
`               →DP Problem 4`
`                 ↳Forward Instantiation Transformation`
`       →DP Problem 2`
`         ↳Nar`

Dependency Pair:

F(g(f(g(f(g(f(x'))))))) -> F(f(f(f(x'))))

Rules:

h(f(f(x))) -> h(f(g(f(x))))
f(g(f(x))) -> f(f(x))

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

F(g(f(g(f(g(f(x'))))))) -> F(f(f(f(x'))))
no new Dependency Pairs are created.
The transformation is resulting in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`       →DP Problem 2`
`         ↳Narrowing Transformation`

Dependency Pair:

H(f(f(x))) -> H(f(g(f(x))))

Rules:

h(f(f(x))) -> h(f(g(f(x))))
f(g(f(x))) -> f(f(x))

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

H(f(f(x))) -> H(f(g(f(x))))
two new Dependency Pairs are created:

H(f(f(x''))) -> H(f(f(x'')))
H(f(f(g(f(x''))))) -> H(f(g(f(f(x'')))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`       →DP Problem 2`
`         ↳Nar`
`           →DP Problem 5`
`             ↳Narrowing Transformation`

Dependency Pairs:

H(f(f(g(f(x''))))) -> H(f(g(f(f(x'')))))
H(f(f(x''))) -> H(f(f(x'')))

Rules:

h(f(f(x))) -> h(f(g(f(x))))
f(g(f(x))) -> f(f(x))

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

H(f(f(g(f(x''))))) -> H(f(g(f(f(x'')))))
two new Dependency Pairs are created:

H(f(f(g(f(x'''))))) -> H(f(f(f(x'''))))
H(f(f(g(f(g(f(x'))))))) -> H(f(g(f(f(f(x'))))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`       →DP Problem 2`
`         ↳Nar`
`           →DP Problem 5`
`             ↳Nar`
`             ...`
`               →DP Problem 6`
`                 ↳Polynomial Ordering`

Dependency Pairs:

H(f(f(g(f(g(f(x'))))))) -> H(f(g(f(f(f(x'))))))
H(f(f(g(f(x'''))))) -> H(f(f(f(x'''))))
H(f(f(x''))) -> H(f(f(x'')))

Rules:

h(f(f(x))) -> h(f(g(f(x))))
f(g(f(x))) -> f(f(x))

The following dependency pairs can be strictly oriented:

H(f(f(g(f(g(f(x'))))))) -> H(f(g(f(f(f(x'))))))
H(f(f(g(f(x'''))))) -> H(f(f(f(x'''))))

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

f(g(f(x))) -> f(f(x))

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(g(x1)) =  1 + x1 POL(H(x1)) =  1 + x1 POL(f(x1)) =  x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`       →DP Problem 2`
`         ↳Nar`
`           →DP Problem 5`
`             ↳Nar`
`             ...`
`               →DP Problem 7`
`                 ↳Remaining Obligation(s)`

The following remains to be proven:
Dependency Pair:

H(f(f(x''))) -> H(f(f(x'')))

Rules:

h(f(f(x))) -> h(f(g(f(x))))
f(g(f(x))) -> f(f(x))

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