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

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`
`         ↳Polynomial Ordering`

Dependency Pair:

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

Rules:

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

The following dependency pair can be strictly oriented:

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

Additionally, the following rules can be oriented:

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

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

resulting in one new DP problem.

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

Dependency Pair:

Rules:

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

Using the Dependency Graph resulted in no new DP problems.

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