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

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC.

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

Dependency Pairs:

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

Rules:

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

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

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

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

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

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

Rules:

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

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

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

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

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

Rules:

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

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

A(a(f, x''), g) -> A(f, a(g, a(x'', g)))
one new Dependency Pair is created:

A(a(f, x'''), g) -> A(f, a(g, a(f, 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 4`
`                 ↳Narrowing Transformation`

Dependency Pairs:

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

Rules:

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

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

A(g, g) -> A(f, a(g, a(f, a(g, a(f, f)))))
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`
`                 ↳Narrowing Transformation`

Dependency Pairs:

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

Rules:

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

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

A(a(f, x''), g) -> A(g, a(x'', g))
one new Dependency Pair is created:

A(a(f, x'''), g) -> A(g, a(f, 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 6`
`                 ↳Narrowing Transformation`

Dependency Pairs:

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

Rules:

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

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

A(g, g) -> A(g, a(f, a(g, a(f, f))))
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 7`
`                 ↳Forward Instantiation Transformation`

Dependency Pairs:

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

Rules:

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

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

A(a(f, x'''), g) -> A(g, a(f, a(g, a(f, 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 8`
`                 ↳Narrowing Transformation`

Dependency Pairs:

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

Rules:

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

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

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

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 9`
`                 ↳Narrowing Transformation`

Dependency Pairs:

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

Rules:

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

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

A(a(f, a(f, x')), g) -> A(f, a(g, a(f, a(g, a(x', g)))))
one new Dependency Pair is created:

A(a(f, a(f, x'')), g) -> A(f, a(g, a(f, a(g, a(f, 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 10`
`                 ↳Narrowing Transformation`

Dependency Pairs:

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

Rules:

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

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

A(a(f, g), g) -> A(f, a(g, a(f, a(g, a(f, a(g, a(f, f)))))))
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 11`
`                 ↳Narrowing Transformation`

Dependency Pairs:

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

Rules:

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

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

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

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 12`
`                 ↳Narrowing Transformation`

Dependency Pairs:

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

Rules:

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

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

A(a(f, a(f, g)), g) -> A(f, a(g, a(f, a(g, a(f, a(g, a(f, a(g, a(f, f)))))))))
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 13`
`                 ↳Remaining Obligation(s)`

The following remains to be proven:
Dependency Pairs:

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

Rules:

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

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