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`
`         ↳Argument Filtering and Ordering`

Dependency Pair:

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

Rule:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(b(x1)) =  1 + x1 POL(A(x1)) =  x1

resulting in one new DP problem.
Used Argument Filtering System:
A(x1) -> A(x1)
b(x1) -> b(x1)

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`           →DP Problem 2`
`             ↳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