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

Innermost 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)))

Strategy:

innermost

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

The following usable rules for innermost w.r.t. to the 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.
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`
`             ↳Dependency Graph`

Dependency Pair:

Rules:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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