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

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC.

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

Dependency Pair:

F(g(x)) -> F(x)

Rule:

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

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

F(g(x)) -> F(x)
one new Dependency Pair is created:

F(g(g(x''))) -> F(g(x''))

The transformation is resulting in one new DP problem:

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

Dependency Pair:

F(g(g(x''))) -> F(g(x''))

Rule:

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

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

F(g(g(x''))) -> F(g(x''))
one new Dependency Pair is created:

F(g(g(g(x'''')))) -> F(g(g(x'''')))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 2`
`             ↳FwdInst`
`             ...`
`               →DP Problem 3`
`                 ↳Polynomial Ordering`

Dependency Pair:

F(g(g(g(x'''')))) -> F(g(g(x'''')))

Rule:

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

The following dependency pair can be strictly oriented:

F(g(g(g(x'''')))) -> F(g(g(x'''')))

There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(g(x1)) =  1 + x1 POL(F(x1)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 2`
`             ↳FwdInst`
`             ...`
`               →DP Problem 4`
`                 ↳Dependency Graph`

Dependency Pair:

Rule:

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

Using the Dependency Graph resulted in no new DP problems.

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