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

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC.

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

Dependency Pairs:

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

Rules:

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

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

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

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

Rules:

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

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

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

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

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

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

Rules:

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

The following dependency pairs can be strictly oriented:

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

Additionally, the following usable rules w.r.t. to the implicit AFS can be oriented:

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

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

resulting in one new DP problem.

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

Dependency Pair:

Rules:

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

Using the Dependency Graph resulted in no new DP problems.

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