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

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Narrowing Transformation`

Dependency Pair:

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

Rules:

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

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

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

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Remaining Obligation(s)`

The following remains to be proven:
Dependency Pair:

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

Rules:

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

Termination of R could not be shown.
Duration:
0:00 minutes