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

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polynomial Ordering`

Dependency Pair:

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

Rule:

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

The following dependency pair can be strictly oriented:

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

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

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

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

Dependency Pair:

Rule:

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

Using the Dependency Graph resulted in no new DP problems.

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