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

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC.

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

Dependency Pair:

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

Rule:

f(x) -> f(g(x))

Strategy:

innermost

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

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

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Inst`
`           →DP Problem 2`
`             ↳Instantiation Transformation`

Dependency Pair:

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

Rule:

f(x) -> f(g(x))

Strategy:

innermost

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

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

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Inst`
`           →DP Problem 2`
`             ↳Inst`
`             ...`
`               →DP Problem 3`
`                 ↳Instantiation Transformation`

Dependency Pair:

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

Rule:

f(x) -> f(g(x))

Strategy:

innermost

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

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

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Inst`
`           →DP Problem 2`
`             ↳Inst`
`             ...`
`               →DP Problem 4`
`                 ↳Instantiation Transformation`

Dependency Pair:

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

Rule:

f(x) -> f(g(x))

Strategy:

innermost

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

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

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Inst`
`           →DP Problem 2`
`             ↳Inst`
`             ...`
`               →DP Problem 5`
`                 ↳Instantiation Transformation`

Dependency Pair:

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

Rule:

f(x) -> f(g(x))

Strategy:

innermost

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

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

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Inst`
`           →DP Problem 2`
`             ↳Inst`
`             ...`
`               →DP Problem 6`
`                 ↳Remaining Obligation(s)`

The following remains to be proven:
Dependency Pair:

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

Rule:

f(x) -> f(g(x))

Strategy:

innermost

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