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

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

A(b(a(b(x)))) -> A(b(a(a(b(x)))))
A(b(a(b(x)))) -> A(a(b(x)))

Furthermore, R contains one SCC.

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

Dependency Pairs:

A(b(a(b(x)))) -> A(a(b(x)))
A(b(a(b(x)))) -> A(b(a(a(b(x)))))

Rule:

a(b(a(b(x)))) -> b(a(b(a(a(b(x))))))

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

A(b(a(b(x)))) -> A(b(a(a(b(x)))))
one new Dependency Pair is created:

A(b(a(b(a(b(x'')))))) -> A(b(a(b(a(b(a(a(b(x'')))))))))

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

A(b(a(b(a(b(x'')))))) -> A(b(a(b(a(b(a(a(b(x'')))))))))
A(b(a(b(x)))) -> A(a(b(x)))

Rule:

a(b(a(b(x)))) -> b(a(b(a(a(b(x))))))

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

A(b(a(b(x)))) -> A(a(b(x)))
one new Dependency Pair is created:

A(b(a(b(a(b(x'')))))) -> A(b(a(b(a(a(b(x'')))))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 3`
`                 ↳Remaining Obligation(s)`

The following remains to be proven:
Dependency Pairs:

A(b(a(b(a(b(x'')))))) -> A(b(a(b(a(a(b(x'')))))))
A(b(a(b(a(b(x'')))))) -> A(b(a(b(a(b(a(a(b(x'')))))))))

Rule:

a(b(a(b(x)))) -> b(a(b(a(a(b(x))))))

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