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

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

F(g(X)) -> F(f(X))
F(g(X)) -> F(X)

Furthermore, R contains one SCC.

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

Dependency Pairs:

F(g(X)) -> F(X)
F(g(X)) -> F(f(X))

Rules:

f(g(X)) -> g(f(f(X)))
f(h(X)) -> h(g(X))

Strategy:

innermost

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

F(g(X)) -> F(f(X))
two new Dependency Pairs are created:

F(g(g(X''))) -> F(g(f(f(X''))))
F(g(h(X''))) -> F(h(g(X'')))

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

F(g(g(X''))) -> F(g(f(f(X''))))
F(g(X)) -> F(X)

Rules:

f(g(X)) -> g(f(f(X)))
f(h(X)) -> h(g(X))

Strategy:

innermost

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

F(g(X)) -> F(X)
two new Dependency Pairs are created:

F(g(g(X''))) -> F(g(X''))
F(g(g(g(X'''')))) -> F(g(g(X'''')))

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 Pairs:

F(g(g(g(X'''')))) -> F(g(g(X'''')))
F(g(g(X''))) -> F(g(X''))
F(g(g(X''))) -> F(g(f(f(X''))))

Rules:

f(g(X)) -> g(f(f(X)))
f(h(X)) -> h(g(X))

Strategy:

innermost

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

F(g(g(X''))) -> F(g(f(f(X''))))
two new Dependency Pairs are created:

F(g(g(g(X')))) -> F(g(f(g(f(f(X'))))))
F(g(g(h(X')))) -> F(g(f(h(g(X')))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳FwdInst`
`             ...`
`               →DP Problem 4`
`                 ↳Rewriting Transformation`

Dependency Pairs:

F(g(g(h(X')))) -> F(g(f(h(g(X')))))
F(g(g(g(X')))) -> F(g(f(g(f(f(X'))))))
F(g(g(X''))) -> F(g(X''))
F(g(g(g(X'''')))) -> F(g(g(X'''')))

Rules:

f(g(X)) -> g(f(f(X)))
f(h(X)) -> h(g(X))

Strategy:

innermost

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

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

F(g(g(g(X')))) -> F(g(g(f(f(f(f(X')))))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳FwdInst`
`             ...`
`               →DP Problem 5`
`                 ↳Rewriting Transformation`

Dependency Pairs:

F(g(g(g(X')))) -> F(g(g(f(f(f(f(X')))))))
F(g(g(g(X'''')))) -> F(g(g(X'''')))
F(g(g(X''))) -> F(g(X''))
F(g(g(h(X')))) -> F(g(f(h(g(X')))))

Rules:

f(g(X)) -> g(f(f(X)))
f(h(X)) -> h(g(X))

Strategy:

innermost

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

F(g(g(h(X')))) -> F(g(f(h(g(X')))))
one new Dependency Pair is created:

F(g(g(h(X')))) -> F(g(h(g(g(X')))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳FwdInst`
`             ...`
`               →DP Problem 6`
`                 ↳Forward Instantiation Transformation`

Dependency Pairs:

F(g(g(g(X'''')))) -> F(g(g(X'''')))
F(g(g(X''))) -> F(g(X''))
F(g(g(g(X')))) -> F(g(g(f(f(f(f(X')))))))

Rules:

f(g(X)) -> g(f(f(X)))
f(h(X)) -> h(g(X))

Strategy:

innermost

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''))
three new Dependency Pairs are created:

F(g(g(g(X'''')))) -> F(g(g(X'''')))
F(g(g(g(g(X''''''))))) -> F(g(g(g(X''''''))))
F(g(g(g(g(X''''))))) -> F(g(g(g(X''''))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳FwdInst`
`             ...`
`               →DP Problem 7`
`                 ↳Forward Instantiation Transformation`

Dependency Pairs:

F(g(g(g(g(X''''))))) -> F(g(g(g(X''''))))
F(g(g(g(g(X''''''))))) -> F(g(g(g(X''''''))))
F(g(g(g(X'''')))) -> F(g(g(X'''')))
F(g(g(g(X')))) -> F(g(g(f(f(f(f(X')))))))
F(g(g(g(X'''')))) -> F(g(g(X'''')))

Rules:

f(g(X)) -> g(f(f(X)))
f(h(X)) -> h(g(X))

Strategy:

innermost

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

F(g(g(g(X'''')))) -> F(g(g(X'''')))
four new Dependency Pairs are created:

F(g(g(g(g(X''''''))))) -> F(g(g(g(X''''''))))
F(g(g(g(g(X'''))))) -> F(g(g(g(X'''))))
F(g(g(g(g(g(X'''''''')))))) -> F(g(g(g(g(X'''''''')))))
F(g(g(g(g(g(X'''''')))))) -> F(g(g(g(g(X'''''')))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳FwdInst`
`             ...`
`               →DP Problem 8`
`                 ↳Polynomial Ordering`

Dependency Pairs:

F(g(g(g(g(g(X'''''')))))) -> F(g(g(g(g(X'''''')))))
F(g(g(g(g(g(X'''''''')))))) -> F(g(g(g(g(X'''''''')))))
F(g(g(g(g(X'''))))) -> F(g(g(g(X'''))))
F(g(g(g(g(X''''''))))) -> F(g(g(g(X''''''))))
F(g(g(g(g(X''''''))))) -> F(g(g(g(X''''''))))
F(g(g(g(X'''')))) -> F(g(g(X'''')))
F(g(g(g(X')))) -> F(g(g(f(f(f(f(X')))))))
F(g(g(g(g(X''''))))) -> F(g(g(g(X''''))))

Rules:

f(g(X)) -> g(f(f(X)))
f(h(X)) -> h(g(X))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

F(g(g(g(g(g(X'''''')))))) -> F(g(g(g(g(X'''''')))))
F(g(g(g(g(g(X'''''''')))))) -> F(g(g(g(g(X'''''''')))))
F(g(g(g(g(X'''))))) -> F(g(g(g(X'''))))
F(g(g(g(g(X''''''))))) -> F(g(g(g(X''''''))))
F(g(g(g(X'''')))) -> F(g(g(X'''')))
F(g(g(g(X')))) -> F(g(g(f(f(f(f(X')))))))
F(g(g(g(g(X''''))))) -> F(g(g(g(X''''))))

Additionally, the following usable rules for innermost w.r.t. to the implicit AFS can be oriented:

f(g(X)) -> g(f(f(X)))
f(h(X)) -> h(g(X))

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(g(x1)) =  1 + x1 POL(h(x1)) =  0 POL(f(x1)) =  x1 POL(F(x1)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳FwdInst`
`             ...`
`               →DP Problem 9`
`                 ↳Dependency Graph`

Dependency Pair:

Rules:

f(g(X)) -> g(f(f(X)))
f(h(X)) -> h(g(X))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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