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

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

G(g(x)) -> G(h(g(x)))
G(g(x)) -> H(g(x))
H(h(x)) -> H(f(h(x), x))

Furthermore, R contains one SCC.

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

Dependency Pair:

G(g(x)) -> G(h(g(x)))

Rules:

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

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

G(g(x)) -> G(h(g(x)))
two new Dependency Pairs are created:

G(g(h(g(x'')))) -> G(h(g(x'')))
G(g(g(x''))) -> G(h(g(h(g(x'')))))

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

G(g(g(x''))) -> G(h(g(h(g(x'')))))
G(g(h(g(x'')))) -> G(h(g(x'')))

Rules:

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

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

G(g(h(g(x'')))) -> G(h(g(x'')))
two new Dependency Pairs are created:

G(g(h(g(h(g(x')))))) -> G(h(g(x')))
G(g(h(g(g(x'))))) -> G(h(g(h(g(x')))))

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

G(g(h(g(g(x'))))) -> G(h(g(h(g(x')))))
G(g(h(g(h(g(x')))))) -> G(h(g(x')))
G(g(g(x''))) -> G(h(g(h(g(x'')))))

Rules:

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

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

G(g(g(x''))) -> G(h(g(h(g(x'')))))
three new Dependency Pairs are created:

G(g(g(x'''))) -> G(h(g(x''')))
G(g(g(h(g(x'))))) -> G(h(g(h(g(x')))))
G(g(g(g(x')))) -> G(h(g(h(g(h(g(x')))))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 4`
`                 ↳Polynomial Ordering`

Dependency Pairs:

G(g(g(g(x')))) -> G(h(g(h(g(h(g(x')))))))
G(g(g(h(g(x'))))) -> G(h(g(h(g(x')))))
G(g(g(x'''))) -> G(h(g(x''')))
G(g(h(g(h(g(x')))))) -> G(h(g(x')))
G(g(h(g(g(x'))))) -> G(h(g(h(g(x')))))

Rules:

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

The following dependency pairs can be strictly oriented:

G(g(g(g(x')))) -> G(h(g(h(g(h(g(x')))))))
G(g(g(h(g(x'))))) -> G(h(g(h(g(x')))))
G(g(g(x'''))) -> G(h(g(x''')))
G(g(h(g(h(g(x')))))) -> G(h(g(x')))
G(g(h(g(g(x'))))) -> G(h(g(h(g(x')))))

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

h(h(x)) -> h(f(h(x), x))

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(g(x1)) =  1 POL(G(x1)) =  x1 POL(h(x1)) =  0 POL(f(x1, x2)) =  0

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 5`
`                 ↳Dependency Graph`

Dependency Pair:

Rules:

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

Using the Dependency Graph resulted in no new DP problems.

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