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

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

F(a, f(f(a, x), a)) -> F(f(a, f(a, x)), a)
F(a, f(f(a, x), a)) -> F(a, f(a, x))

Furthermore, R contains one SCC.

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

Dependency Pair:

F(a, f(f(a, x), a)) -> F(a, f(a, x))

Rule:

f(a, f(f(a, x), a)) -> f(f(a, f(a, x)), a)

The following dependency pair can be strictly oriented:

F(a, f(f(a, x), a)) -> F(a, f(a, x))

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

f(a, f(f(a, x), a)) -> f(f(a, f(a, x)), a)

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(a) =  0 POL(F(x1, x2)) =  1 + x1 + x2 POL(f(x1, x2)) =  1 + x1 + x2

resulting in one new DP problem.
Used Argument Filtering System:
F(x1, x2) -> F(x1, x2)
f(x1, x2) -> f(x1, x2)

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

Dependency Pair:

Rule:

f(a, f(f(a, x), a)) -> f(f(a, f(a, x)), a)

Using the Dependency Graph resulted in no new DP problems.

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