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

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

MARK(f(X)) -> AF(mark(X))
MARK(f(X)) -> MARK(X)
MARK(h(X)) -> MARK(X)

Furthermore, R contains one SCC.

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

Dependency Pairs:

MARK(h(X)) -> MARK(X)
MARK(f(X)) -> MARK(X)

Rules:

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

Strategy:

innermost

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

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

MARK(f(f(X''))) -> MARK(f(X''))
MARK(f(h(X''))) -> MARK(h(X''))

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

MARK(f(h(X''))) -> MARK(h(X''))
MARK(f(f(X''))) -> MARK(f(X''))
MARK(h(X)) -> MARK(X)

Rules:

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

Strategy:

innermost

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

MARK(h(X)) -> MARK(X)
three new Dependency Pairs are created:

MARK(h(h(X''))) -> MARK(h(X''))
MARK(h(f(f(X'''')))) -> MARK(f(f(X'''')))
MARK(h(f(h(X'''')))) -> MARK(f(h(X'''')))

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

MARK(h(f(h(X'''')))) -> MARK(f(h(X'''')))
MARK(f(f(X''))) -> MARK(f(X''))
MARK(h(f(f(X'''')))) -> MARK(f(f(X'''')))
MARK(h(h(X''))) -> MARK(h(X''))
MARK(f(h(X''))) -> MARK(h(X''))

Rules:

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

Strategy:

innermost

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

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

MARK(f(f(f(X'''')))) -> MARK(f(f(X'''')))
MARK(f(f(h(X'''')))) -> MARK(f(h(X'''')))

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

MARK(f(f(h(X'''')))) -> MARK(f(h(X'''')))
MARK(f(f(f(X'''')))) -> MARK(f(f(X'''')))
MARK(h(f(f(X'''')))) -> MARK(f(f(X'''')))
MARK(h(h(X''))) -> MARK(h(X''))
MARK(f(h(X''))) -> MARK(h(X''))
MARK(h(f(h(X'''')))) -> MARK(f(h(X'''')))

Rules:

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

Strategy:

innermost

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

MARK(f(h(X''))) -> MARK(h(X''))
three new Dependency Pairs are created:

MARK(f(h(h(X'''')))) -> MARK(h(h(X'''')))
MARK(f(h(f(f(X''''''))))) -> MARK(h(f(f(X''''''))))
MARK(f(h(f(h(X''''''))))) -> MARK(h(f(h(X''''''))))

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

MARK(f(h(f(h(X''''''))))) -> MARK(h(f(h(X''''''))))
MARK(f(h(f(f(X''''''))))) -> MARK(h(f(f(X''''''))))
MARK(h(f(h(X'''')))) -> MARK(f(h(X'''')))
MARK(f(f(f(X'''')))) -> MARK(f(f(X'''')))
MARK(h(f(f(X'''')))) -> MARK(f(f(X'''')))
MARK(h(h(X''))) -> MARK(h(X''))
MARK(f(h(h(X'''')))) -> MARK(h(h(X'''')))
MARK(f(f(h(X'''')))) -> MARK(f(h(X'''')))

Rules:

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

Strategy:

innermost

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

MARK(h(h(X''))) -> MARK(h(X''))
three new Dependency Pairs are created:

MARK(h(h(h(X'''')))) -> MARK(h(h(X'''')))
MARK(h(h(f(f(X''''''))))) -> MARK(h(f(f(X''''''))))
MARK(h(h(f(h(X''''''))))) -> MARK(h(f(h(X''''''))))

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

MARK(h(h(f(h(X''''''))))) -> MARK(h(f(h(X''''''))))
MARK(f(h(f(f(X''''''))))) -> MARK(h(f(f(X''''''))))
MARK(f(f(h(X'''')))) -> MARK(f(h(X'''')))
MARK(f(f(f(X'''')))) -> MARK(f(f(X'''')))
MARK(h(f(f(X'''')))) -> MARK(f(f(X'''')))
MARK(h(h(f(f(X''''''))))) -> MARK(h(f(f(X''''''))))
MARK(h(h(h(X'''')))) -> MARK(h(h(X'''')))
MARK(f(h(h(X'''')))) -> MARK(h(h(X'''')))
MARK(h(f(h(X'''')))) -> MARK(f(h(X'''')))
MARK(f(h(f(h(X''''''))))) -> MARK(h(f(h(X''''''))))

Rules:

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

Strategy:

innermost

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

MARK(h(f(f(X'''')))) -> MARK(f(f(X'''')))
two new Dependency Pairs are created:

MARK(h(f(f(f(X''''''))))) -> MARK(f(f(f(X''''''))))
MARK(h(f(f(h(X''''''))))) -> MARK(f(f(h(X''''''))))

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

MARK(f(h(f(h(X''''''))))) -> MARK(h(f(h(X''''''))))
MARK(h(f(f(h(X''''''))))) -> MARK(f(f(h(X''''''))))
MARK(f(h(f(f(X''''''))))) -> MARK(h(f(f(X''''''))))
MARK(f(f(h(X'''')))) -> MARK(f(h(X'''')))
MARK(f(f(f(X'''')))) -> MARK(f(f(X'''')))
MARK(h(f(f(f(X''''''))))) -> MARK(f(f(f(X''''''))))
MARK(h(h(f(f(X''''''))))) -> MARK(h(f(f(X''''''))))
MARK(h(h(h(X'''')))) -> MARK(h(h(X'''')))
MARK(f(h(h(X'''')))) -> MARK(h(h(X'''')))
MARK(h(f(h(X'''')))) -> MARK(f(h(X'''')))
MARK(h(h(f(h(X''''''))))) -> MARK(h(f(h(X''''''))))

Rules:

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

Strategy:

innermost

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

MARK(h(f(h(X'''')))) -> MARK(f(h(X'''')))
three new Dependency Pairs are created:

MARK(h(f(h(h(X''''''))))) -> MARK(f(h(h(X''''''))))
MARK(h(f(h(f(f(X'''''''')))))) -> MARK(f(h(f(f(X'''''''')))))
MARK(h(f(h(f(h(X'''''''')))))) -> MARK(f(h(f(h(X'''''''')))))

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

MARK(h(f(h(f(h(X'''''''')))))) -> MARK(f(h(f(h(X'''''''')))))
MARK(h(f(h(f(f(X'''''''')))))) -> MARK(f(h(f(f(X'''''''')))))
MARK(h(h(f(h(X''''''))))) -> MARK(h(f(h(X''''''))))
MARK(h(f(f(h(X''''''))))) -> MARK(f(f(h(X''''''))))
MARK(f(h(f(f(X''''''))))) -> MARK(h(f(f(X''''''))))
MARK(f(f(h(X'''')))) -> MARK(f(h(X'''')))
MARK(f(f(f(X'''')))) -> MARK(f(f(X'''')))
MARK(h(f(f(f(X''''''))))) -> MARK(f(f(f(X''''''))))
MARK(h(h(f(f(X''''''))))) -> MARK(h(f(f(X''''''))))
MARK(h(h(h(X'''')))) -> MARK(h(h(X'''')))
MARK(f(h(h(X'''')))) -> MARK(h(h(X'''')))
MARK(h(f(h(h(X''''''))))) -> MARK(f(h(h(X''''''))))
MARK(f(h(f(h(X''''''))))) -> MARK(h(f(h(X''''''))))

Rules:

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

Strategy:

innermost

The following dependency pairs can be strictly oriented:

MARK(h(f(h(f(h(X'''''''')))))) -> MARK(f(h(f(h(X'''''''')))))
MARK(h(f(h(f(f(X'''''''')))))) -> MARK(f(h(f(f(X'''''''')))))
MARK(h(h(f(h(X''''''))))) -> MARK(h(f(h(X''''''))))
MARK(h(f(f(h(X''''''))))) -> MARK(f(f(h(X''''''))))
MARK(h(f(f(f(X''''''))))) -> MARK(f(f(f(X''''''))))
MARK(h(h(f(f(X''''''))))) -> MARK(h(f(f(X''''''))))
MARK(h(h(h(X'''')))) -> MARK(h(h(X'''')))
MARK(h(f(h(h(X''''''))))) -> MARK(f(h(h(X''''''))))

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(MARK(x1)) =  1 + x1 POL(h(x1)) =  1 + x1 POL(f(x1)) =  x1

resulting in one new DP problem.

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

Dependency Pairs:

MARK(f(h(f(f(X''''''))))) -> MARK(h(f(f(X''''''))))
MARK(f(f(h(X'''')))) -> MARK(f(h(X'''')))
MARK(f(f(f(X'''')))) -> MARK(f(f(X'''')))
MARK(f(h(h(X'''')))) -> MARK(h(h(X'''')))
MARK(f(h(f(h(X''''''))))) -> MARK(h(f(h(X''''''))))

Rules:

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

Strategy:

innermost

Using the Dependency Graph the DP problem was split into 1 DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 2`
`             ↳FwdInst`
`             ...`
`               →DP Problem 10`
`                 ↳Polynomial Ordering`

Dependency Pair:

MARK(f(f(f(X'''')))) -> MARK(f(f(X'''')))

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

MARK(f(f(f(X'''')))) -> MARK(f(f(X'''')))

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(MARK(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 11`
`                 ↳Dependency Graph`

Dependency Pair:

Rules:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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