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

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC.

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

Dependency Pairs:

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

Rules:

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

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

A(f, a(g, a(f, x))) -> A(f, a(g, a(g, a(f, x))))
two new Dependency Pairs are created:

A(f, a(g, a(f, a(g, x'')))) -> A(f, a(g, a(g, a(f, a(f, a(g, x''))))))
A(f, a(g, a(f, a(g, a(f, x''))))) -> A(f, a(g, a(g, a(f, a(g, a(g, a(f, x'')))))))

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

A(f, a(g, a(f, a(g, a(f, x''))))) -> A(f, a(g, a(g, a(f, a(g, a(g, a(f, x'')))))))
A(f, a(g, a(f, a(g, x'')))) -> A(f, a(g, a(g, a(f, a(f, a(g, x''))))))
A(g, a(f, a(g, x))) -> A(g, a(f, a(f, a(g, x))))
A(f, a(g, a(f, x))) -> A(g, a(g, a(f, x)))
A(g, a(f, a(g, x))) -> A(f, a(f, a(g, x)))

Rules:

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

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

A(f, a(g, a(f, x))) -> A(g, a(g, a(f, x)))
two new Dependency Pairs are created:

A(f, a(g, a(f, a(g, x'')))) -> A(g, a(g, a(f, a(f, a(g, x'')))))
A(f, a(g, a(f, a(g, a(f, x''))))) -> A(g, a(g, a(f, a(g, a(g, a(f, 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:

A(f, a(g, a(f, a(g, a(f, x''))))) -> A(g, a(g, a(f, a(g, a(g, a(f, x''))))))
A(g, a(f, a(g, x))) -> A(f, a(f, a(g, x)))
A(g, a(f, a(g, x))) -> A(g, a(f, a(f, a(g, x))))
A(f, a(g, a(f, a(g, x'')))) -> A(g, a(g, a(f, a(f, a(g, x'')))))
A(f, a(g, a(f, a(g, x'')))) -> A(f, a(g, a(g, a(f, a(f, a(g, x''))))))
A(f, a(g, a(f, a(g, a(f, x''))))) -> A(f, a(g, a(g, a(f, a(g, a(g, a(f, x'')))))))

Rules:

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

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

A(g, a(f, a(g, x))) -> A(g, a(f, a(f, a(g, x))))
two new Dependency Pairs are created:

A(g, a(f, a(g, a(f, x'')))) -> A(g, a(f, a(f, a(g, a(g, a(f, x''))))))
A(g, a(f, a(g, a(f, a(g, x''))))) -> A(g, a(f, a(f, a(g, a(f, a(f, a(g, x'')))))))

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

A(g, a(f, a(g, a(f, a(g, x''))))) -> A(g, a(f, a(f, a(g, a(f, a(f, a(g, x'')))))))
A(g, a(f, a(g, a(f, x'')))) -> A(g, a(f, a(f, a(g, a(g, a(f, x''))))))
A(f, a(g, a(f, a(g, x'')))) -> A(g, a(g, a(f, a(f, a(g, x'')))))
A(f, a(g, a(f, a(g, a(f, x''))))) -> A(f, a(g, a(g, a(f, a(g, a(g, a(f, x'')))))))
A(f, a(g, a(f, a(g, x'')))) -> A(f, a(g, a(g, a(f, a(f, a(g, x''))))))
A(g, a(f, a(g, x))) -> A(f, a(f, a(g, x)))
A(f, a(g, a(f, a(g, a(f, x''))))) -> A(g, a(g, a(f, a(g, a(g, a(f, x''))))))

Rules:

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

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

A(g, a(f, a(g, x))) -> A(f, a(f, a(g, x)))
two new Dependency Pairs are created:

A(g, a(f, a(g, a(f, x'')))) -> A(f, a(f, a(g, a(g, a(f, x'')))))
A(g, a(f, a(g, a(f, a(g, x''))))) -> A(f, a(f, a(g, a(f, a(f, a(g, x''))))))

The transformation is resulting in one new DP problem:

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

The following remains to be proven:
Dependency Pairs:

A(f, a(g, a(f, a(g, a(f, x''))))) -> A(g, a(g, a(f, a(g, a(g, a(f, x''))))))
A(g, a(f, a(g, a(f, a(g, x''))))) -> A(f, a(f, a(g, a(f, a(f, a(g, x''))))))
A(f, a(g, a(f, a(g, x'')))) -> A(g, a(g, a(f, a(f, a(g, x'')))))
A(f, a(g, a(f, a(g, a(f, x''))))) -> A(f, a(g, a(g, a(f, a(g, a(g, a(f, x'')))))))
A(f, a(g, a(f, a(g, x'')))) -> A(f, a(g, a(g, a(f, a(f, a(g, x''))))))
A(g, a(f, a(g, a(f, x'')))) -> A(f, a(f, a(g, a(g, a(f, x'')))))
A(g, a(f, a(g, a(f, x'')))) -> A(g, a(f, a(f, a(g, a(g, a(f, x''))))))
A(g, a(f, a(g, a(f, a(g, x''))))) -> A(g, a(f, a(f, a(g, a(f, a(f, a(g, x'')))))))

Rules:

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

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