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`
`         ↳Argument Filtering and Ordering`

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)))

The following dependency pairs can be strictly oriented:

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

The following usable rules using the Ce-refinement can be oriented:

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

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
{A, B}
{a, b}

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

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`           →DP Problem 2`
`             ↳Argument Filtering and Ordering`

Dependency Pairs:

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)))

The following dependency pairs can be strictly oriented:

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

The following usable rules using the Ce-refinement can be oriented:

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

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
{A, B}

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

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`           →DP Problem 2`
`             ↳AFS`
`             ...`
`               →DP Problem 3`
`                 ↳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