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

Termination of R to be shown.

`   R`
`     ↳Overlay and local confluence Check`

The TRS is overlay and locally confluent (all critical pairs are trivially joinable).Hence, we can switch to innermost.

`   R`
`     ↳OC`
`       →TRS2`
`         ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC.

`   R`
`     ↳OC`
`       →TRS2`
`         ↳DPs`
`           →DP Problem 1`
`             ↳Usable Rules (Innermost)`

Dependency Pair:

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

Rule:

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

Strategy:

innermost

As we are in the innermost case, we can delete all 1 non-usable-rules.

`   R`
`     ↳OC`
`       →TRS2`
`         ↳DPs`
`           →DP Problem 1`
`             ↳UsableRules`
`             ...`
`               →DP Problem 2`
`                 ↳Instantiation Transformation`

Dependency Pair:

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

Rule:

none

Strategy:

innermost

On this DP problem, an Instantiation SCC transformation can be performed.
As a result of transforming the rule

F(f(a, x), a) -> F(f(x, f(a, a)), a)
one new Dependency Pair is created:

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

The transformation is resulting in no new DP problems.

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