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

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC.

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

Dependency Pairs:

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

Rule:

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

Strategy:

innermost

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

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

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

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

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

Rule:

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

Strategy:

innermost

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

A(b(x)) -> A(x)
two new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

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

The following remains to be proven:
Dependency Pairs:

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

Rule:

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

Strategy:

innermost

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