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

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

F(a, f(f(a, a), x)) -> F(f(a, a), f(a, f(a, x)))
F(a, f(f(a, a), x)) -> F(a, f(a, x))
F(a, f(f(a, a), x)) -> F(a, x)

Furthermore, R contains one SCC.

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

Dependency Pairs:

F(a, f(f(a, a), x)) -> F(a, x)
F(a, f(f(a, a), x)) -> F(a, f(a, x))
F(a, f(f(a, a), x)) -> F(f(a, a), f(a, f(a, x)))

Rule:

f(a, f(f(a, a), x)) -> f(f(a, a), f(a, f(a, x)))

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

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

F(a, f(f(a, a), f(f(a, a), x''))) -> F(f(a, a), f(a, f(f(a, a), f(a, f(a, x'')))))

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

F(a, f(f(a, a), f(f(a, a), x''))) -> F(f(a, a), f(a, f(f(a, a), f(a, f(a, x'')))))
F(a, f(f(a, a), x)) -> F(a, f(a, x))
F(a, f(f(a, a), x)) -> F(a, x)

Rule:

f(a, f(f(a, a), x)) -> f(f(a, a), f(a, f(a, x)))

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

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

F(a, f(f(a, a), f(f(a, a), x''))) -> F(a, f(f(a, a), f(a, f(a, 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:

F(a, f(f(a, a), f(f(a, a), x''))) -> F(a, f(f(a, a), f(a, f(a, x''))))
F(a, f(f(a, a), x)) -> F(a, x)
F(a, f(f(a, a), f(f(a, a), x''))) -> F(f(a, a), f(a, f(f(a, a), f(a, f(a, x'')))))

Rule:

f(a, f(f(a, a), x)) -> f(f(a, a), f(a, f(a, x)))

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

F(a, f(f(a, a), f(f(a, a), x''))) -> F(f(a, a), f(a, f(f(a, a), f(a, f(a, x'')))))
two new Dependency Pairs are created:

F(a, f(f(a, a), f(f(a, a), x'''))) -> F(f(a, a), f(f(a, a), f(a, f(a, f(a, f(a, x'''))))))
F(a, f(f(a, a), f(f(a, a), f(f(a, a), x')))) -> F(f(a, a), f(a, f(f(a, a), f(a, f(f(a, a), f(a, f(a, x')))))))

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

F(a, f(f(a, a), f(f(a, a), f(f(a, a), x')))) -> F(f(a, a), f(a, f(f(a, a), f(a, f(f(a, a), f(a, f(a, x')))))))
F(a, f(f(a, a), f(f(a, a), x'''))) -> F(f(a, a), f(f(a, a), f(a, f(a, f(a, f(a, x'''))))))
F(a, f(f(a, a), x)) -> F(a, x)
F(a, f(f(a, a), f(f(a, a), x''))) -> F(a, f(f(a, a), f(a, f(a, x''))))

Rule:

f(a, f(f(a, a), x)) -> f(f(a, a), f(a, f(a, x)))

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

F(a, f(f(a, a), f(f(a, a), x'''))) -> F(f(a, a), f(f(a, a), f(a, f(a, f(a, f(a, x'''))))))
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 5`
`                 ↳Forward Instantiation Transformation`

Dependency Pairs:

F(a, f(f(a, a), f(f(a, a), x''))) -> F(a, f(f(a, a), f(a, f(a, x''))))
F(a, f(f(a, a), x)) -> F(a, x)
F(a, f(f(a, a), f(f(a, a), f(f(a, a), x')))) -> F(f(a, a), f(a, f(f(a, a), f(a, f(f(a, a), f(a, f(a, x')))))))

Rule:

f(a, f(f(a, a), x)) -> f(f(a, a), f(a, f(a, x)))

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

F(a, f(f(a, a), f(f(a, a), f(f(a, a), x')))) -> F(f(a, a), f(a, f(f(a, a), f(a, f(f(a, a), f(a, f(a, x')))))))
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 6`
`                 ↳Remaining Obligation(s)`

The following remains to be proven:
Dependency Pairs:

F(a, f(f(a, a), x)) -> F(a, x)
F(a, f(f(a, a), f(f(a, a), x''))) -> F(a, f(f(a, a), f(a, f(a, x''))))

Rule:

f(a, f(f(a, a), x)) -> f(f(a, a), f(a, f(a, x)))

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