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

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC.

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

Dependency Pairs:

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

Rule:

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

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

F(f(x, a), a) -> F(f(a, f(a, a)), a)
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Forward Instantiation Transformation`

Dependency Pair:

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

Rule:

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

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

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

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳FwdInst`
`             ...`
`               →DP Problem 3`
`                 ↳Narrowing Transformation`

Dependency Pair:

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

Rule:

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

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

F(f(a, a), a) -> F(f(f(a, f(a, a)), a), a)
no new Dependency Pairs are created.
The transformation is resulting in no new DP problems.

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