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

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Argument Filtering and Ordering`

Dependency Pair:

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

Rule:

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

The following dependency pair can be strictly oriented:

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

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

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

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

Dependency Pair:

Rule:

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

Using the Dependency Graph resulted in no new DP problems.

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