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

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC.

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

Dependency Pair:

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

Rule:

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

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

The transformation is resulting in one new DP problem:

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

Dependency Pair:

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

Rule:

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

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 2`
`             ↳FwdInst`
`             ...`
`               →DP Problem 3`
`                 ↳Polynomial Ordering`

Dependency Pair:

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

Rule:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(b(x1)) =  1 + x1 POL(A(x1)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 2`
`             ↳FwdInst`
`             ...`
`               →DP Problem 4`
`                 ↳Dependency Graph`

Dependency Pair:

Rule:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:00 minutes