Term Rewriting System R:
[X, Y, X1, X2]
af(g(X), Y) -> af(mark(X), f(g(X), Y))
af(X1, X2) -> f(X1, X2)
mark(f(X1, X2)) -> af(mark(X1), X2)
mark(g(X)) -> g(mark(X))

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

AF(g(X), Y) -> AF(mark(X), f(g(X), Y))
AF(g(X), Y) -> MARK(X)
MARK(f(X1, X2)) -> AF(mark(X1), X2)
MARK(f(X1, X2)) -> MARK(X1)
MARK(g(X)) -> MARK(X)

Furthermore, R contains one SCC.

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

Dependency Pairs:

MARK(g(X)) -> MARK(X)
MARK(f(X1, X2)) -> MARK(X1)
MARK(f(X1, X2)) -> AF(mark(X1), X2)
AF(g(X), Y) -> MARK(X)
AF(g(X), Y) -> AF(mark(X), f(g(X), Y))

Rules:

af(g(X), Y) -> af(mark(X), f(g(X), Y))
af(X1, X2) -> f(X1, X2)
mark(f(X1, X2)) -> af(mark(X1), X2)
mark(g(X)) -> g(mark(X))

Strategy:

innermost

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

AF(g(X), Y) -> AF(mark(X), f(g(X), Y))
two new Dependency Pairs are created:

AF(g(f(X1', X2')), Y) -> AF(af(mark(X1'), X2'), f(g(f(X1', X2')), Y))
AF(g(g(X'')), Y) -> AF(g(mark(X'')), f(g(g(X'')), Y))

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

AF(g(g(X'')), Y) -> AF(g(mark(X'')), f(g(g(X'')), Y))
AF(g(f(X1', X2')), Y) -> AF(af(mark(X1'), X2'), f(g(f(X1', X2')), Y))
MARK(f(X1, X2)) -> MARK(X1)
AF(g(X), Y) -> MARK(X)
MARK(f(X1, X2)) -> AF(mark(X1), X2)
MARK(g(X)) -> MARK(X)

Rules:

af(g(X), Y) -> af(mark(X), f(g(X), Y))
af(X1, X2) -> f(X1, X2)
mark(f(X1, X2)) -> af(mark(X1), X2)
mark(g(X)) -> g(mark(X))

Strategy:

innermost

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

MARK(f(X1, X2)) -> AF(mark(X1), X2)
two new Dependency Pairs are created:

MARK(f(f(X1'', X2''), X2)) -> AF(af(mark(X1''), X2''), X2)
MARK(f(g(X'), X2)) -> AF(g(mark(X')), X2)

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

MARK(f(g(X'), X2)) -> AF(g(mark(X')), X2)
AF(g(f(X1', X2')), Y) -> AF(af(mark(X1'), X2'), f(g(f(X1', X2')), Y))
MARK(f(f(X1'', X2''), X2)) -> AF(af(mark(X1''), X2''), X2)
MARK(g(X)) -> MARK(X)
MARK(f(X1, X2)) -> MARK(X1)
AF(g(X), Y) -> MARK(X)
AF(g(g(X'')), Y) -> AF(g(mark(X'')), f(g(g(X'')), Y))

Rules:

af(g(X), Y) -> af(mark(X), f(g(X), Y))
af(X1, X2) -> f(X1, X2)
mark(f(X1, X2)) -> af(mark(X1), X2)
mark(g(X)) -> g(mark(X))

Strategy:

innermost

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

AF(g(f(X1', X2')), Y) -> AF(af(mark(X1'), X2'), f(g(f(X1', X2')), Y))
three new Dependency Pairs are created:

AF(g(f(X1'', X2'')), Y) -> AF(f(mark(X1''), X2''), f(g(f(X1'', X2'')), Y))
AF(g(f(f(X1'', X2''), X2')), Y) -> AF(af(af(mark(X1''), X2''), X2'), f(g(f(f(X1'', X2''), X2')), Y))
AF(g(f(g(X'), X2')), Y) -> AF(af(g(mark(X')), X2'), f(g(f(g(X'), X2')), Y))

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

AF(g(f(g(X'), X2')), Y) -> AF(af(g(mark(X')), X2'), f(g(f(g(X'), X2')), Y))
AF(g(f(f(X1'', X2''), X2')), Y) -> AF(af(af(mark(X1''), X2''), X2'), f(g(f(f(X1'', X2''), X2')), Y))
AF(g(g(X'')), Y) -> AF(g(mark(X'')), f(g(g(X'')), Y))
MARK(f(f(X1'', X2''), X2)) -> AF(af(mark(X1''), X2''), X2)
MARK(g(X)) -> MARK(X)
MARK(f(X1, X2)) -> MARK(X1)
AF(g(X), Y) -> MARK(X)
MARK(f(g(X'), X2)) -> AF(g(mark(X')), X2)

Rules:

af(g(X), Y) -> af(mark(X), f(g(X), Y))
af(X1, X2) -> f(X1, X2)
mark(f(X1, X2)) -> af(mark(X1), X2)
mark(g(X)) -> g(mark(X))

Strategy:

innermost

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

MARK(f(f(X1'', X2''), X2)) -> AF(af(mark(X1''), X2''), X2)
three new Dependency Pairs are created:

MARK(f(f(X1''', X2'''), X2)) -> AF(f(mark(X1'''), X2'''), X2)
MARK(f(f(f(X1', X2'''), X2''), X2)) -> AF(af(af(mark(X1'), X2'''), X2''), X2)
MARK(f(f(g(X'), X2''), X2)) -> AF(af(g(mark(X')), X2''), X2)

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

MARK(f(f(g(X'), X2''), X2)) -> AF(af(g(mark(X')), X2''), X2)
MARK(f(f(f(X1', X2'''), X2''), X2)) -> AF(af(af(mark(X1'), X2'''), X2''), X2)
AF(g(f(f(X1'', X2''), X2')), Y) -> AF(af(af(mark(X1''), X2''), X2'), f(g(f(f(X1'', X2''), X2')), Y))
AF(g(g(X'')), Y) -> AF(g(mark(X'')), f(g(g(X'')), Y))
MARK(f(g(X'), X2)) -> AF(g(mark(X')), X2)
MARK(g(X)) -> MARK(X)
MARK(f(X1, X2)) -> MARK(X1)
AF(g(X), Y) -> MARK(X)
AF(g(f(g(X'), X2')), Y) -> AF(af(g(mark(X')), X2'), f(g(f(g(X'), X2')), Y))

Rules:

af(g(X), Y) -> af(mark(X), f(g(X), Y))
af(X1, X2) -> f(X1, X2)
mark(f(X1, X2)) -> af(mark(X1), X2)
mark(g(X)) -> g(mark(X))

Strategy:

innermost

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

AF(g(X), Y) -> MARK(X)
five new Dependency Pairs are created:

AF(g(f(X1'', X2'')), Y) -> MARK(f(X1'', X2''))
AF(g(g(X'')), Y) -> MARK(g(X''))
AF(g(f(g(X'''), X2'')), Y) -> MARK(f(g(X'''), X2''))
AF(g(f(f(f(X1''', X2'''''), X2''0'), X2'')), Y) -> MARK(f(f(f(X1''', X2'''''), X2''0'), X2''))
AF(g(f(f(g(X'''), X2''''), X2'')), Y) -> MARK(f(f(g(X'''), X2''''), X2''))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 6`
`                 ↳Narrowing Transformation`

Dependency Pairs:

AF(g(f(f(g(X'''), X2''''), X2'')), Y) -> MARK(f(f(g(X'''), X2''''), X2''))
MARK(f(f(f(X1', X2'''), X2''), X2)) -> AF(af(af(mark(X1'), X2'''), X2''), X2)
AF(g(f(f(f(X1''', X2'''''), X2''0'), X2'')), Y) -> MARK(f(f(f(X1''', X2'''''), X2''0'), X2''))
AF(g(f(g(X'''), X2'')), Y) -> MARK(f(g(X'''), X2''))
AF(g(g(X'')), Y) -> MARK(g(X''))
MARK(f(g(X'), X2)) -> AF(g(mark(X')), X2)
MARK(g(X)) -> MARK(X)
MARK(f(X1, X2)) -> MARK(X1)
AF(g(f(X1'', X2'')), Y) -> MARK(f(X1'', X2''))
AF(g(f(g(X'), X2')), Y) -> AF(af(g(mark(X')), X2'), f(g(f(g(X'), X2')), Y))
AF(g(f(f(X1'', X2''), X2')), Y) -> AF(af(af(mark(X1''), X2''), X2'), f(g(f(f(X1'', X2''), X2')), Y))
AF(g(g(X'')), Y) -> AF(g(mark(X'')), f(g(g(X'')), Y))
MARK(f(f(g(X'), X2''), X2)) -> AF(af(g(mark(X')), X2''), X2)

Rules:

af(g(X), Y) -> af(mark(X), f(g(X), Y))
af(X1, X2) -> f(X1, X2)
mark(f(X1, X2)) -> af(mark(X1), X2)
mark(g(X)) -> g(mark(X))

Strategy:

innermost

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

AF(g(g(X'')), Y) -> AF(g(mark(X'')), f(g(g(X'')), Y))
two new Dependency Pairs are created:

AF(g(g(f(X1', X2'))), Y) -> AF(g(af(mark(X1'), X2')), f(g(g(f(X1', X2'))), Y))
AF(g(g(g(X'))), Y) -> AF(g(g(mark(X'))), f(g(g(g(X'))), Y))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 7`
`                 ↳Narrowing Transformation`

Dependency Pairs:

MARK(f(f(g(X'), X2''), X2)) -> AF(af(g(mark(X')), X2''), X2)
AF(g(g(g(X'))), Y) -> AF(g(g(mark(X'))), f(g(g(g(X'))), Y))
AF(g(g(f(X1', X2'))), Y) -> AF(g(af(mark(X1'), X2')), f(g(g(f(X1', X2'))), Y))
AF(g(f(f(f(X1''', X2'''''), X2''0'), X2'')), Y) -> MARK(f(f(f(X1''', X2'''''), X2''0'), X2''))
AF(g(f(g(X'''), X2'')), Y) -> MARK(f(g(X'''), X2''))
AF(g(g(X'')), Y) -> MARK(g(X''))
MARK(f(f(f(X1', X2'''), X2''), X2)) -> AF(af(af(mark(X1'), X2'''), X2''), X2)
AF(g(f(X1'', X2'')), Y) -> MARK(f(X1'', X2''))
AF(g(f(g(X'), X2')), Y) -> AF(af(g(mark(X')), X2'), f(g(f(g(X'), X2')), Y))
AF(g(f(f(X1'', X2''), X2')), Y) -> AF(af(af(mark(X1''), X2''), X2'), f(g(f(f(X1'', X2''), X2')), Y))
MARK(f(g(X'), X2)) -> AF(g(mark(X')), X2)
MARK(g(X)) -> MARK(X)
MARK(f(X1, X2)) -> MARK(X1)
AF(g(f(f(g(X'''), X2''''), X2'')), Y) -> MARK(f(f(g(X'''), X2''''), X2''))

Rules:

af(g(X), Y) -> af(mark(X), f(g(X), Y))
af(X1, X2) -> f(X1, X2)
mark(f(X1, X2)) -> af(mark(X1), X2)
mark(g(X)) -> g(mark(X))

Strategy:

innermost

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

MARK(f(g(X'), X2)) -> AF(g(mark(X')), X2)
two new Dependency Pairs are created:

MARK(f(g(f(X1', X2'')), X2)) -> AF(g(af(mark(X1'), X2'')), X2)
MARK(f(g(g(X'')), X2)) -> AF(g(g(mark(X''))), X2)

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 8`
`                 ↳Forward Instantiation Transformation`

Dependency Pairs:

MARK(f(g(g(X'')), X2)) -> AF(g(g(mark(X''))), X2)
AF(g(g(g(X'))), Y) -> AF(g(g(mark(X'))), f(g(g(g(X'))), Y))
AF(g(g(f(X1', X2'))), Y) -> AF(g(af(mark(X1'), X2')), f(g(g(f(X1', X2'))), Y))
AF(g(f(f(g(X'''), X2''''), X2'')), Y) -> MARK(f(f(g(X'''), X2''''), X2''))
AF(g(f(f(f(X1''', X2'''''), X2''0'), X2'')), Y) -> MARK(f(f(f(X1''', X2'''''), X2''0'), X2''))
MARK(f(g(f(X1', X2'')), X2)) -> AF(g(af(mark(X1'), X2'')), X2)
AF(g(f(g(X'''), X2'')), Y) -> MARK(f(g(X'''), X2''))
AF(g(g(X'')), Y) -> MARK(g(X''))
MARK(f(f(f(X1', X2'''), X2''), X2)) -> AF(af(af(mark(X1'), X2'''), X2''), X2)
MARK(g(X)) -> MARK(X)
MARK(f(X1, X2)) -> MARK(X1)
AF(g(f(X1'', X2'')), Y) -> MARK(f(X1'', X2''))
AF(g(f(g(X'), X2')), Y) -> AF(af(g(mark(X')), X2'), f(g(f(g(X'), X2')), Y))
AF(g(f(f(X1'', X2''), X2')), Y) -> AF(af(af(mark(X1''), X2''), X2'), f(g(f(f(X1'', X2''), X2')), Y))
MARK(f(f(g(X'), X2''), X2)) -> AF(af(g(mark(X')), X2''), X2)

Rules:

af(g(X), Y) -> af(mark(X), f(g(X), Y))
af(X1, X2) -> f(X1, X2)
mark(f(X1, X2)) -> af(mark(X1), X2)
mark(g(X)) -> g(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(X1, X2)) -> MARK(X1)
six new Dependency Pairs are created:

MARK(f(f(X1'', X2''), X2)) -> MARK(f(X1'', X2''))
MARK(f(g(X''), X2)) -> MARK(g(X''))
MARK(f(f(f(f(X1''', X2'''''), X2''0'), X2''), X2)) -> MARK(f(f(f(X1''', X2'''''), X2''0'), X2''))
MARK(f(f(f(g(X'''), X2''''), X2''), X2)) -> MARK(f(f(g(X'''), X2''''), X2''))
MARK(f(f(g(f(X1''', X2'''')), X2''), X2)) -> MARK(f(g(f(X1''', X2'''')), X2''))
MARK(f(f(g(g(X'''')), X2''), X2)) -> MARK(f(g(g(X'''')), X2''))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 9`
`                 ↳Forward Instantiation Transformation`

Dependency Pairs:

AF(g(g(g(X'))), Y) -> AF(g(g(mark(X'))), f(g(g(g(X'))), Y))
AF(g(g(f(X1', X2'))), Y) -> AF(g(af(mark(X1'), X2')), f(g(g(f(X1', X2'))), Y))
AF(g(f(f(g(X'''), X2''''), X2'')), Y) -> MARK(f(f(g(X'''), X2''''), X2''))
MARK(f(f(g(g(X'''')), X2''), X2)) -> MARK(f(g(g(X'''')), X2''))
MARK(f(f(g(f(X1''', X2'''')), X2''), X2)) -> MARK(f(g(f(X1''', X2'''')), X2''))
MARK(f(f(f(g(X'''), X2''''), X2''), X2)) -> MARK(f(f(g(X'''), X2''''), X2''))
MARK(f(f(f(f(X1''', X2'''''), X2''0'), X2''), X2)) -> MARK(f(f(f(X1''', X2'''''), X2''0'), X2''))
MARK(f(g(X''), X2)) -> MARK(g(X''))
MARK(f(f(X1'', X2''), X2)) -> MARK(f(X1'', X2''))
AF(g(f(f(f(X1''', X2'''''), X2''0'), X2'')), Y) -> MARK(f(f(f(X1''', X2'''''), X2''0'), X2''))
MARK(f(g(f(X1', X2'')), X2)) -> AF(g(af(mark(X1'), X2'')), X2)
AF(g(f(g(X'''), X2'')), Y) -> MARK(f(g(X'''), X2''))
MARK(f(f(g(X'), X2''), X2)) -> AF(af(g(mark(X')), X2''), X2)
AF(g(f(X1'', X2'')), Y) -> MARK(f(X1'', X2''))
AF(g(f(g(X'), X2')), Y) -> AF(af(g(mark(X')), X2'), f(g(f(g(X'), X2')), Y))
AF(g(f(f(X1'', X2''), X2')), Y) -> AF(af(af(mark(X1''), X2''), X2'), f(g(f(f(X1'', X2''), X2')), Y))
MARK(f(f(f(X1', X2'''), X2''), X2)) -> AF(af(af(mark(X1'), X2'''), X2''), X2)
MARK(g(X)) -> MARK(X)
AF(g(g(X'')), Y) -> MARK(g(X''))
MARK(f(g(g(X'')), X2)) -> AF(g(g(mark(X''))), X2)

Rules:

af(g(X), Y) -> af(mark(X), f(g(X), Y))
af(X1, X2) -> f(X1, X2)
mark(f(X1, X2)) -> af(mark(X1), X2)
mark(g(X)) -> g(mark(X))

Strategy:

innermost

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

MARK(g(X)) -> MARK(X)
11 new Dependency Pairs are created:

MARK(g(g(X''))) -> MARK(g(X''))
MARK(g(f(f(f(X1''', X2'''''), X2''0'), X2''))) -> MARK(f(f(f(X1''', X2'''''), X2''0'), X2''))
MARK(g(f(f(g(X'''), X2''''), X2''))) -> MARK(f(f(g(X'''), X2''''), X2''))
MARK(g(f(g(f(X1''', X2'''')), X2''))) -> MARK(f(g(f(X1''', X2'''')), X2''))
MARK(g(f(g(g(X'''')), X2''))) -> MARK(f(g(g(X'''')), X2''))
MARK(g(f(f(X1'''', X2''''), X2''))) -> MARK(f(f(X1'''', X2''''), X2''))
MARK(g(f(g(X''''), X2''))) -> MARK(f(g(X''''), X2''))
MARK(g(f(f(f(f(X1''''', X2'''''''), X2''0'''), X2''''), X2''))) -> MARK(f(f(f(f(X1''''', X2'''''''), X2''0'''), X2''''), X2''))
MARK(g(f(f(f(g(X'''''), X2''''''), X2''''), X2''))) -> MARK(f(f(f(g(X'''''), X2''''''), X2''''), X2''))
MARK(g(f(f(g(f(X1''''', X2'''''')), X2''''), X2''))) -> MARK(f(f(g(f(X1''''', X2'''''')), X2''''), X2''))
MARK(g(f(f(g(g(X'''''')), X2''''), X2''))) -> MARK(f(f(g(g(X'''''')), X2''''), X2''))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 10`
`                 ↳Forward Instantiation Transformation`

Dependency Pairs:

MARK(g(f(f(g(g(X'''''')), X2''''), X2''))) -> MARK(f(f(g(g(X'''''')), X2''''), X2''))
MARK(g(f(f(g(f(X1''''', X2'''''')), X2''''), X2''))) -> MARK(f(f(g(f(X1''''', X2'''''')), X2''''), X2''))
MARK(g(f(f(f(g(X'''''), X2''''''), X2''''), X2''))) -> MARK(f(f(f(g(X'''''), X2''''''), X2''''), X2''))
MARK(g(f(f(f(f(X1''''', X2'''''''), X2''0'''), X2''''), X2''))) -> MARK(f(f(f(f(X1''''', X2'''''''), X2''0'''), X2''''), X2''))
MARK(g(f(g(X''''), X2''))) -> MARK(f(g(X''''), X2''))
MARK(f(f(f(g(X'''), X2''''), X2''), X2)) -> MARK(f(f(g(X'''), X2''''), X2''))
MARK(f(f(f(f(X1''', X2'''''), X2''0'), X2''), X2)) -> MARK(f(f(f(X1''', X2'''''), X2''0'), X2''))
MARK(g(f(f(X1'''', X2''''), X2''))) -> MARK(f(f(X1'''', X2''''), X2''))
MARK(g(f(g(g(X'''')), X2''))) -> MARK(f(g(g(X'''')), X2''))
MARK(g(f(g(f(X1''', X2'''')), X2''))) -> MARK(f(g(f(X1''', X2'''')), X2''))
MARK(f(f(g(g(X'''')), X2''), X2)) -> MARK(f(g(g(X'''')), X2''))
MARK(g(f(f(g(X'''), X2''''), X2''))) -> MARK(f(f(g(X'''), X2''''), X2''))
MARK(f(g(X''), X2)) -> MARK(g(X''))
MARK(f(f(g(f(X1''', X2'''')), X2''), X2)) -> MARK(f(g(f(X1''', X2'''')), X2''))
AF(g(f(f(g(X'''), X2''''), X2'')), Y) -> MARK(f(f(g(X'''), X2''''), X2''))
AF(g(g(f(X1', X2'))), Y) -> AF(g(af(mark(X1'), X2')), f(g(g(f(X1', X2'))), Y))
MARK(f(g(g(X'')), X2)) -> AF(g(g(mark(X''))), X2)
MARK(f(f(X1'', X2''), X2)) -> MARK(f(X1'', X2''))
AF(g(f(f(f(X1''', X2'''''), X2''0'), X2'')), Y) -> MARK(f(f(f(X1''', X2'''''), X2''0'), X2''))
MARK(f(g(f(X1', X2'')), X2)) -> AF(g(af(mark(X1'), X2'')), X2)
AF(g(f(g(X'''), X2'')), Y) -> MARK(f(g(X'''), X2''))
MARK(f(f(g(X'), X2''), X2)) -> AF(af(g(mark(X')), X2''), X2)
AF(g(f(X1'', X2'')), Y) -> MARK(f(X1'', X2''))
AF(g(f(g(X'), X2')), Y) -> AF(af(g(mark(X')), X2'), f(g(f(g(X'), X2')), Y))
AF(g(f(f(X1'', X2''), X2')), Y) -> AF(af(af(mark(X1''), X2''), X2'), f(g(f(f(X1'', X2''), X2')), Y))
MARK(f(f(f(X1', X2'''), X2''), X2)) -> AF(af(af(mark(X1'), X2'''), X2''), X2)
MARK(g(f(f(f(X1''', X2'''''), X2''0'), X2''))) -> MARK(f(f(f(X1''', X2'''''), X2''0'), X2''))
MARK(g(g(X''))) -> MARK(g(X''))
AF(g(g(X'')), Y) -> MARK(g(X''))
AF(g(g(g(X'))), Y) -> AF(g(g(mark(X'))), f(g(g(g(X'))), Y))

Rules:

af(g(X), Y) -> af(mark(X), f(g(X), Y))
af(X1, X2) -> f(X1, X2)
mark(f(X1, X2)) -> af(mark(X1), X2)
mark(g(X)) -> g(mark(X))

Strategy:

innermost

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

AF(g(f(X1'', X2'')), Y) -> MARK(f(X1'', X2''))
10 new Dependency Pairs are created:

AF(g(f(f(f(X1'''', X2'''''), X2''0'), X2''')), Y) -> MARK(f(f(f(X1'''', X2'''''), X2''0'), X2'''))
AF(g(f(f(g(X'''), X2'''''), X2'''')), Y) -> MARK(f(f(g(X'''), X2'''''), X2''''))
AF(g(f(g(f(X1'''', X2''''')), X2'''')), Y) -> MARK(f(g(f(X1'''', X2''''')), X2''''))
AF(g(f(g(g(X'''')), X2''')), Y) -> MARK(f(g(g(X'''')), X2'''))
AF(g(f(f(X1'''', X2'''''), X2'''')), Y) -> MARK(f(f(X1'''', X2'''''), X2''''))
AF(g(f(g(X''''), X2''')), Y) -> MARK(f(g(X''''), X2'''))
AF(g(f(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''''), X2'''')), Y) -> MARK(f(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''''), X2''''))
AF(g(f(f(f(g(X'''''), X2''''''), X2'''0), X2'''')), Y) -> MARK(f(f(f(g(X'''''), X2''''''), X2'''0), X2''''))
AF(g(f(f(g(f(X1''''', X2'''''')), X2'''0), X2'''')), Y) -> MARK(f(f(g(f(X1''''', X2'''''')), X2'''0), X2''''))
AF(g(f(f(g(g(X'''''')), X2'''''), X2'''')), Y) -> MARK(f(f(g(g(X'''''')), X2'''''), X2''''))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 11`
`                 ↳Forward Instantiation Transformation`

Dependency Pairs:

AF(g(f(f(g(g(X'''''')), X2'''''), X2'''')), Y) -> MARK(f(f(g(g(X'''''')), X2'''''), X2''''))
AF(g(f(f(g(f(X1''''', X2'''''')), X2'''0), X2'''')), Y) -> MARK(f(f(g(f(X1''''', X2'''''')), X2'''0), X2''''))
AF(g(f(f(f(g(X'''''), X2''''''), X2'''0), X2'''')), Y) -> MARK(f(f(f(g(X'''''), X2''''''), X2'''0), X2''''))
AF(g(f(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''''), X2'''')), Y) -> MARK(f(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''''), X2''''))
AF(g(f(g(X''''), X2''')), Y) -> MARK(f(g(X''''), X2'''))
AF(g(f(f(X1'''', X2'''''), X2'''')), Y) -> MARK(f(f(X1'''', X2'''''), X2''''))
AF(g(f(g(g(X'''')), X2''')), Y) -> MARK(f(g(g(X'''')), X2'''))
AF(g(f(g(f(X1'''', X2''''')), X2'''')), Y) -> MARK(f(g(f(X1'''', X2''''')), X2''''))
AF(g(f(f(g(X'''), X2'''''), X2'''')), Y) -> MARK(f(f(g(X'''), X2'''''), X2''''))
AF(g(f(f(f(X1'''', X2'''''), X2''0'), X2''')), Y) -> MARK(f(f(f(X1'''', X2'''''), X2''0'), X2'''))
AF(g(g(g(X'))), Y) -> AF(g(g(mark(X'))), f(g(g(g(X'))), Y))
MARK(g(f(f(g(f(X1''''', X2'''''')), X2''''), X2''))) -> MARK(f(f(g(f(X1''''', X2'''''')), X2''''), X2''))
MARK(g(f(f(f(g(X'''''), X2''''''), X2''''), X2''))) -> MARK(f(f(f(g(X'''''), X2''''''), X2''''), X2''))
MARK(g(f(f(f(f(X1''''', X2'''''''), X2''0'''), X2''''), X2''))) -> MARK(f(f(f(f(X1''''', X2'''''''), X2''0'''), X2''''), X2''))
MARK(g(f(g(X''''), X2''))) -> MARK(f(g(X''''), X2''))
MARK(f(f(f(g(X'''), X2''''), X2''), X2)) -> MARK(f(f(g(X'''), X2''''), X2''))
MARK(f(f(f(f(X1''', X2'''''), X2''0'), X2''), X2)) -> MARK(f(f(f(X1''', X2'''''), X2''0'), X2''))
MARK(g(f(f(X1'''', X2''''), X2''))) -> MARK(f(f(X1'''', X2''''), X2''))
MARK(g(f(g(g(X'''')), X2''))) -> MARK(f(g(g(X'''')), X2''))
MARK(g(f(g(f(X1''', X2'''')), X2''))) -> MARK(f(g(f(X1''', X2'''')), X2''))
MARK(f(f(g(g(X'''')), X2''), X2)) -> MARK(f(g(g(X'''')), X2''))
MARK(g(f(f(g(X'''), X2''''), X2''))) -> MARK(f(f(g(X'''), X2''''), X2''))
MARK(f(g(X''), X2)) -> MARK(g(X''))
MARK(f(f(g(f(X1''', X2'''')), X2''), X2)) -> MARK(f(g(f(X1''', X2'''')), X2''))
AF(g(f(f(g(X'''), X2''''), X2'')), Y) -> MARK(f(f(g(X'''), X2''''), X2''))
AF(g(g(f(X1', X2'))), Y) -> AF(g(af(mark(X1'), X2')), f(g(g(f(X1', X2'))), Y))
MARK(f(g(g(X'')), X2)) -> AF(g(g(mark(X''))), X2)
MARK(f(f(X1'', X2''), X2)) -> MARK(f(X1'', X2''))
AF(g(f(f(f(X1''', X2'''''), X2''0'), X2'')), Y) -> MARK(f(f(f(X1''', X2'''''), X2''0'), X2''))
MARK(f(g(f(X1', X2'')), X2)) -> AF(g(af(mark(X1'), X2'')), X2)
AF(g(f(g(X'''), X2'')), Y) -> MARK(f(g(X'''), X2''))
MARK(f(f(f(X1', X2'''), X2''), X2)) -> AF(af(af(mark(X1'), X2'''), X2''), X2)
MARK(g(f(f(f(X1''', X2'''''), X2''0'), X2''))) -> MARK(f(f(f(X1''', X2'''''), X2''0'), X2''))
MARK(g(g(X''))) -> MARK(g(X''))
AF(g(g(X'')), Y) -> MARK(g(X''))
AF(g(f(g(X'), X2')), Y) -> AF(af(g(mark(X')), X2'), f(g(f(g(X'), X2')), Y))
AF(g(f(f(X1'', X2''), X2')), Y) -> AF(af(af(mark(X1''), X2''), X2'), f(g(f(f(X1'', X2''), X2')), Y))
MARK(f(f(g(X'), X2''), X2)) -> AF(af(g(mark(X')), X2''), X2)
MARK(g(f(f(g(g(X'''''')), X2''''), X2''))) -> MARK(f(f(g(g(X'''''')), X2''''), X2''))

Rules:

af(g(X), Y) -> af(mark(X), f(g(X), Y))
af(X1, X2) -> f(X1, X2)
mark(f(X1, X2)) -> af(mark(X1), X2)
mark(g(X)) -> g(mark(X))

Strategy:

innermost

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

AF(g(g(X'')), Y) -> MARK(g(X''))
11 new Dependency Pairs are created:

AF(g(g(g(X''''))), Y) -> MARK(g(g(X'''')))
AF(g(g(f(f(f(X1''''', X2'''''''), X2''0'''), X2''''))), Y) -> MARK(g(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''')))
AF(g(g(f(f(g(X'''''), X2''''''), X2''''))), Y) -> MARK(g(f(f(g(X'''''), X2''''''), X2'''')))
AF(g(g(f(g(f(X1''''', X2'''''')), X2''''))), Y) -> MARK(g(f(g(f(X1''''', X2'''''')), X2'''')))
AF(g(g(f(g(g(X'''''')), X2''''))), Y) -> MARK(g(f(g(g(X'''''')), X2'''')))
AF(g(g(f(f(X1'''''', X2''''''), X2''''))), Y) -> MARK(g(f(f(X1'''''', X2''''''), X2'''')))
AF(g(g(f(g(X''''''), X2''''))), Y) -> MARK(g(f(g(X''''''), X2'''')))
AF(g(g(f(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2''''''), X2''''))), Y) -> MARK(g(f(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2''''''), X2'''')))
AF(g(g(f(f(f(g(X'''''''), X2''''''''), X2''''''), X2''''))), Y) -> MARK(g(f(f(f(g(X'''''''), X2''''''''), X2''''''), X2'''')))
AF(g(g(f(f(g(f(X1''''''', X2'''''''')), X2''''''), X2''''))), Y) -> MARK(g(f(f(g(f(X1''''''', X2'''''''')), X2''''''), X2'''')))
AF(g(g(f(f(g(g(X'''''''')), X2''''''), X2''''))), Y) -> MARK(g(f(f(g(g(X'''''''')), X2''''''), X2'''')))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 12`
`                 ↳Forward Instantiation Transformation`

Dependency Pairs:

AF(g(f(f(g(f(X1''''', X2'''''')), X2'''0), X2'''')), Y) -> MARK(f(f(g(f(X1''''', X2'''''')), X2'''0), X2''''))
AF(g(f(f(f(g(X'''''), X2''''''), X2'''0), X2'''')), Y) -> MARK(f(f(f(g(X'''''), X2''''''), X2'''0), X2''''))
AF(g(f(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''''), X2'''')), Y) -> MARK(f(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''''), X2''''))
AF(g(f(g(X''''), X2''')), Y) -> MARK(f(g(X''''), X2'''))
AF(g(f(f(X1'''', X2'''''), X2'''')), Y) -> MARK(f(f(X1'''', X2'''''), X2''''))
AF(g(f(g(g(X'''')), X2''')), Y) -> MARK(f(g(g(X'''')), X2'''))
AF(g(f(g(f(X1'''', X2''''')), X2'''')), Y) -> MARK(f(g(f(X1'''', X2''''')), X2''''))
AF(g(f(f(g(X'''), X2'''''), X2'''')), Y) -> MARK(f(f(g(X'''), X2'''''), X2''''))
AF(g(f(f(f(X1'''', X2'''''), X2''0'), X2''')), Y) -> MARK(f(f(f(X1'''', X2'''''), X2''0'), X2'''))
AF(g(g(f(f(g(g(X'''''''')), X2''''''), X2''''))), Y) -> MARK(g(f(f(g(g(X'''''''')), X2''''''), X2'''')))
AF(g(g(f(f(g(f(X1''''''', X2'''''''')), X2''''''), X2''''))), Y) -> MARK(g(f(f(g(f(X1''''''', X2'''''''')), X2''''''), X2'''')))
AF(g(g(f(f(f(g(X'''''''), X2''''''''), X2''''''), X2''''))), Y) -> MARK(g(f(f(f(g(X'''''''), X2''''''''), X2''''''), X2'''')))
AF(g(g(f(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2''''''), X2''''))), Y) -> MARK(g(f(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2''''''), X2'''')))
AF(g(g(f(g(X''''''), X2''''))), Y) -> MARK(g(f(g(X''''''), X2'''')))
AF(g(g(f(f(X1'''''', X2''''''), X2''''))), Y) -> MARK(g(f(f(X1'''''', X2''''''), X2'''')))
AF(g(g(f(g(g(X'''''')), X2''''))), Y) -> MARK(g(f(g(g(X'''''')), X2'''')))
AF(g(g(f(g(f(X1''''', X2'''''')), X2''''))), Y) -> MARK(g(f(g(f(X1''''', X2'''''')), X2'''')))
AF(g(g(f(f(g(X'''''), X2''''''), X2''''))), Y) -> MARK(g(f(f(g(X'''''), X2''''''), X2'''')))
AF(g(g(f(f(f(X1''''', X2'''''''), X2''0'''), X2''''))), Y) -> MARK(g(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''')))
MARK(g(f(f(g(g(X'''''')), X2''''), X2''))) -> MARK(f(f(g(g(X'''''')), X2''''), X2''))
MARK(g(f(f(g(f(X1''''', X2'''''')), X2''''), X2''))) -> MARK(f(f(g(f(X1''''', X2'''''')), X2''''), X2''))
MARK(g(f(f(f(g(X'''''), X2''''''), X2''''), X2''))) -> MARK(f(f(f(g(X'''''), X2''''''), X2''''), X2''))
MARK(g(f(f(f(f(X1''''', X2'''''''), X2''0'''), X2''''), X2''))) -> MARK(f(f(f(f(X1''''', X2'''''''), X2''0'''), X2''''), X2''))
MARK(g(f(g(X''''), X2''))) -> MARK(f(g(X''''), X2''))
MARK(g(f(f(X1'''', X2''''), X2''))) -> MARK(f(f(X1'''', X2''''), X2''))
MARK(g(f(g(g(X'''')), X2''))) -> MARK(f(g(g(X'''')), X2''))
MARK(g(f(g(f(X1''', X2'''')), X2''))) -> MARK(f(g(f(X1''', X2'''')), X2''))
MARK(f(f(g(g(X'''')), X2''), X2)) -> MARK(f(g(g(X'''')), X2''))
MARK(g(f(f(g(X'''), X2''''), X2''))) -> MARK(f(f(g(X'''), X2''''), X2''))
MARK(f(g(X''), X2)) -> MARK(g(X''))
MARK(f(f(g(f(X1''', X2'''')), X2''), X2)) -> MARK(f(g(f(X1''', X2'''')), X2''))
MARK(f(f(f(g(X'''), X2''''), X2''), X2)) -> MARK(f(f(g(X'''), X2''''), X2''))
MARK(f(f(f(f(X1''', X2'''''), X2''0'), X2''), X2)) -> MARK(f(f(f(X1''', X2'''''), X2''0'), X2''))
MARK(g(f(f(f(X1''', X2'''''), X2''0'), X2''))) -> MARK(f(f(f(X1''', X2'''''), X2''0'), X2''))
MARK(g(g(X''))) -> MARK(g(X''))
AF(g(g(g(X''''))), Y) -> MARK(g(g(X'''')))
AF(g(g(g(X'))), Y) -> AF(g(g(mark(X'))), f(g(g(g(X'))), Y))
AF(g(g(f(X1', X2'))), Y) -> AF(g(af(mark(X1'), X2')), f(g(g(f(X1', X2'))), Y))
MARK(f(g(g(X'')), X2)) -> AF(g(g(mark(X''))), X2)
MARK(f(f(X1'', X2''), X2)) -> MARK(f(X1'', X2''))
AF(g(f(f(g(X'''), X2''''), X2'')), Y) -> MARK(f(f(g(X'''), X2''''), X2''))
MARK(f(f(f(X1', X2'''), X2''), X2)) -> AF(af(af(mark(X1'), X2'''), X2''), X2)
AF(g(f(f(f(X1''', X2'''''), X2''0'), X2'')), Y) -> MARK(f(f(f(X1''', X2'''''), X2''0'), X2''))
MARK(f(g(f(X1', X2'')), X2)) -> AF(g(af(mark(X1'), X2'')), X2)
AF(g(f(g(X'''), X2'')), Y) -> MARK(f(g(X'''), X2''))
AF(g(f(g(X'), X2')), Y) -> AF(af(g(mark(X')), X2'), f(g(f(g(X'), X2')), Y))
AF(g(f(f(X1'', X2''), X2')), Y) -> AF(af(af(mark(X1''), X2''), X2'), f(g(f(f(X1'', X2''), X2')), Y))
MARK(f(f(g(X'), X2''), X2)) -> AF(af(g(mark(X')), X2''), X2)
AF(g(f(f(g(g(X'''''')), X2'''''), X2'''')), Y) -> MARK(f(f(g(g(X'''''')), X2'''''), X2''''))

Rules:

af(g(X), Y) -> af(mark(X), f(g(X), Y))
af(X1, X2) -> f(X1, X2)
mark(f(X1, X2)) -> af(mark(X1), X2)
mark(g(X)) -> g(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(X1'', X2''), X2)) -> MARK(f(X1'', X2''))
10 new Dependency Pairs are created:

MARK(f(f(f(f(X1'''', X2'''''), X2''0'), X2'''), X2)) -> MARK(f(f(f(X1'''', X2'''''), X2''0'), X2'''))
MARK(f(f(f(g(X'''), X2'''''), X2''''), X2)) -> MARK(f(f(g(X'''), X2'''''), X2''''))
MARK(f(f(g(f(X1'''', X2''''')), X2''''), X2)) -> MARK(f(g(f(X1'''', X2''''')), X2''''))
MARK(f(f(g(g(X'''')), X2'''), X2)) -> MARK(f(g(g(X'''')), X2'''))
MARK(f(f(f(X1'''', X2'''''), X2''''), X2)) -> MARK(f(f(X1'''', X2'''''), X2''''))
MARK(f(f(g(X''''), X2'''), X2)) -> MARK(f(g(X''''), X2'''))
MARK(f(f(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''''), X2''''), X2)) -> MARK(f(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''''), X2''''))
MARK(f(f(f(f(g(X'''''), X2''''''), X2'''0), X2''''), X2)) -> MARK(f(f(f(g(X'''''), X2''''''), X2'''0), X2''''))
MARK(f(f(f(g(f(X1''''', X2'''''')), X2'''0), X2''''), X2)) -> MARK(f(f(g(f(X1''''', X2'''''')), X2'''0), X2''''))
MARK(f(f(f(g(g(X'''''')), X2'''''), X2''''), X2)) -> MARK(f(f(g(g(X'''''')), X2'''''), X2''''))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 13`
`                 ↳Forward Instantiation Transformation`

Dependency Pairs:

AF(g(f(f(g(g(X'''''')), X2'''''), X2'''')), Y) -> MARK(f(f(g(g(X'''''')), X2'''''), X2''''))
AF(g(f(f(f(g(X'''''), X2''''''), X2'''0), X2'''')), Y) -> MARK(f(f(f(g(X'''''), X2''''''), X2'''0), X2''''))
AF(g(f(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''''), X2'''')), Y) -> MARK(f(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''''), X2''''))
AF(g(f(g(X''''), X2''')), Y) -> MARK(f(g(X''''), X2'''))
AF(g(f(f(X1'''', X2'''''), X2'''')), Y) -> MARK(f(f(X1'''', X2'''''), X2''''))
AF(g(f(g(g(X'''')), X2''')), Y) -> MARK(f(g(g(X'''')), X2'''))
AF(g(f(g(f(X1'''', X2''''')), X2'''')), Y) -> MARK(f(g(f(X1'''', X2''''')), X2''''))
AF(g(f(f(g(X'''), X2'''''), X2'''')), Y) -> MARK(f(f(g(X'''), X2'''''), X2''''))
AF(g(f(f(f(X1'''', X2'''''), X2''0'), X2''')), Y) -> MARK(f(f(f(X1'''', X2'''''), X2''0'), X2'''))
AF(g(g(f(f(g(g(X'''''''')), X2''''''), X2''''))), Y) -> MARK(g(f(f(g(g(X'''''''')), X2''''''), X2'''')))
AF(g(g(f(f(g(f(X1''''''', X2'''''''')), X2''''''), X2''''))), Y) -> MARK(g(f(f(g(f(X1''''''', X2'''''''')), X2''''''), X2'''')))
AF(g(g(f(f(f(g(X'''''''), X2''''''''), X2''''''), X2''''))), Y) -> MARK(g(f(f(f(g(X'''''''), X2''''''''), X2''''''), X2'''')))
AF(g(g(f(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2''''''), X2''''))), Y) -> MARK(g(f(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2''''''), X2'''')))
AF(g(g(f(g(X''''''), X2''''))), Y) -> MARK(g(f(g(X''''''), X2'''')))
AF(g(g(f(f(X1'''''', X2''''''), X2''''))), Y) -> MARK(g(f(f(X1'''''', X2''''''), X2'''')))
MARK(g(f(g(g(X'''')), X2''))) -> MARK(f(g(g(X'''')), X2''))
AF(g(g(f(g(g(X'''''')), X2''''))), Y) -> MARK(g(f(g(g(X'''''')), X2'''')))
MARK(g(f(g(X''''), X2''))) -> MARK(f(g(X''''), X2''))
MARK(g(f(g(f(X1''', X2'''')), X2''))) -> MARK(f(g(f(X1''', X2'''')), X2''))
AF(g(g(f(g(f(X1''''', X2'''''')), X2''''))), Y) -> MARK(g(f(g(f(X1''''', X2'''''')), X2'''')))
MARK(g(f(f(g(g(X'''''')), X2''''), X2''))) -> MARK(f(f(g(g(X'''''')), X2''''), X2''))
MARK(g(f(f(g(f(X1''''', X2'''''')), X2''''), X2''))) -> MARK(f(f(g(f(X1''''', X2'''''')), X2''''), X2''))
MARK(g(f(f(g(X'''), X2''''), X2''))) -> MARK(f(f(g(X'''), X2''''), X2''))
AF(g(g(f(f(g(X'''''), X2''''''), X2''''))), Y) -> MARK(g(f(f(g(X'''''), X2''''''), X2'''')))
MARK(g(f(f(f(g(X'''''), X2''''''), X2''''), X2''))) -> MARK(f(f(f(g(X'''''), X2''''''), X2''''), X2''))
MARK(g(f(f(f(f(X1''''', X2'''''''), X2''0'''), X2''''), X2''))) -> MARK(f(f(f(f(X1''''', X2'''''''), X2''0'''), X2''''), X2''))
MARK(f(f(f(g(g(X'''''')), X2'''''), X2''''), X2)) -> MARK(f(f(g(g(X'''''')), X2'''''), X2''''))
MARK(f(f(f(g(f(X1''''', X2'''''')), X2'''0), X2''''), X2)) -> MARK(f(f(g(f(X1''''', X2'''''')), X2'''0), X2''''))
MARK(f(f(f(f(g(X'''''), X2''''''), X2'''0), X2''''), X2)) -> MARK(f(f(f(g(X'''''), X2''''''), X2'''0), X2''''))
MARK(f(f(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''''), X2''''), X2)) -> MARK(f(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''''), X2''''))
MARK(f(f(f(X1'''', X2'''''), X2''''), X2)) -> MARK(f(f(X1'''', X2'''''), X2''''))
MARK(f(f(g(X''''), X2'''), X2)) -> MARK(f(g(X''''), X2'''))
MARK(f(f(g(g(X'''')), X2'''), X2)) -> MARK(f(g(g(X'''')), X2'''))
MARK(f(f(g(f(X1'''', X2''''')), X2''''), X2)) -> MARK(f(g(f(X1'''', X2''''')), X2''''))
MARK(f(f(f(g(X'''), X2'''''), X2''''), X2)) -> MARK(f(f(g(X'''), X2'''''), X2''''))
MARK(f(f(f(f(X1'''', X2'''''), X2''0'), X2'''), X2)) -> MARK(f(f(f(X1'''', X2'''''), X2''0'), X2'''))
MARK(g(f(f(X1'''', X2''''), X2''))) -> MARK(f(f(X1'''', X2''''), X2''))
AF(g(g(f(f(f(X1''''', X2'''''''), X2''0'''), X2''''))), Y) -> MARK(g(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''')))
AF(g(g(g(X''''))), Y) -> MARK(g(g(X'''')))
AF(g(g(g(X'))), Y) -> AF(g(g(mark(X'))), f(g(g(g(X'))), Y))
AF(g(g(f(X1', X2'))), Y) -> AF(g(af(mark(X1'), X2')), f(g(g(f(X1', X2'))), Y))
MARK(f(g(g(X'')), X2)) -> AF(g(g(mark(X''))), X2)
MARK(f(f(g(g(X'''')), X2''), X2)) -> MARK(f(g(g(X'''')), X2''))
MARK(f(f(f(g(X'''), X2''''), X2''), X2)) -> MARK(f(f(g(X'''), X2''''), X2''))
MARK(f(f(f(f(X1''', X2'''''), X2''0'), X2''), X2)) -> MARK(f(f(f(X1''', X2'''''), X2''0'), X2''))
MARK(g(f(f(f(X1''', X2'''''), X2''0'), X2''))) -> MARK(f(f(f(X1''', X2'''''), X2''0'), X2''))
MARK(g(g(X''))) -> MARK(g(X''))
MARK(f(g(X''), X2)) -> MARK(g(X''))
MARK(f(f(g(f(X1''', X2'''')), X2''), X2)) -> MARK(f(g(f(X1''', X2'''')), X2''))
AF(g(f(f(g(X'''), X2''''), X2'')), Y) -> MARK(f(f(g(X'''), X2''''), X2''))
MARK(f(f(f(X1', X2'''), X2''), X2)) -> AF(af(af(mark(X1'), X2'''), X2''), X2)
AF(g(f(f(f(X1''', X2'''''), X2''0'), X2'')), Y) -> MARK(f(f(f(X1''', X2'''''), X2''0'), X2''))
MARK(f(g(f(X1', X2'')), X2)) -> AF(g(af(mark(X1'), X2'')), X2)
AF(g(f(g(X'''), X2'')), Y) -> MARK(f(g(X'''), X2''))
AF(g(f(g(X'), X2')), Y) -> AF(af(g(mark(X')), X2'), f(g(f(g(X'), X2')), Y))
AF(g(f(f(X1'', X2''), X2')), Y) -> AF(af(af(mark(X1''), X2''), X2'), f(g(f(f(X1'', X2''), X2')), Y))
MARK(f(f(g(X'), X2''), X2)) -> AF(af(g(mark(X')), X2''), X2)
AF(g(f(f(g(f(X1''''', X2'''''')), X2'''0), X2'''')), Y) -> MARK(f(f(g(f(X1''''', X2'''''')), X2'''0), X2''''))

Rules:

af(g(X), Y) -> af(mark(X), f(g(X), Y))
af(X1, X2) -> f(X1, X2)
mark(f(X1, X2)) -> af(mark(X1), X2)
mark(g(X)) -> g(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(g(X''), X2)) -> MARK(g(X''))
11 new Dependency Pairs are created:

MARK(f(g(g(X'''')), X2)) -> MARK(g(g(X'''')))
MARK(f(g(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''')), X2)) -> MARK(g(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''')))
MARK(f(g(f(f(g(X'''''), X2''''''), X2'''')), X2)) -> MARK(g(f(f(g(X'''''), X2''''''), X2'''')))
MARK(f(g(f(g(f(X1''''', X2'''''')), X2'''')), X2)) -> MARK(g(f(g(f(X1''''', X2'''''')), X2'''')))
MARK(f(g(f(g(g(X'''''')), X2'''')), X2)) -> MARK(g(f(g(g(X'''''')), X2'''')))
MARK(f(g(f(f(X1'''''', X2''''''), X2'''')), X2)) -> MARK(g(f(f(X1'''''', X2''''''), X2'''')))
MARK(f(g(f(g(X''''''), X2'''')), X2)) -> MARK(g(f(g(X''''''), X2'''')))
MARK(f(g(f(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2''''''), X2'''')), X2)) -> MARK(g(f(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2''''''), X2'''')))
MARK(f(g(f(f(f(g(X'''''''), X2''''''''), X2''''''), X2'''')), X2)) -> MARK(g(f(f(f(g(X'''''''), X2''''''''), X2''''''), X2'''')))
MARK(f(g(f(f(g(f(X1''''''', X2'''''''')), X2''''''), X2'''')), X2)) -> MARK(g(f(f(g(f(X1''''''', X2'''''''')), X2''''''), X2'''')))
MARK(f(g(f(f(g(g(X'''''''')), X2''''''), X2'''')), X2)) -> MARK(g(f(f(g(g(X'''''''')), X2''''''), X2'''')))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 14`
`                 ↳Forward Instantiation Transformation`

Dependency Pairs:

AF(g(f(f(g(f(X1''''', X2'''''')), X2'''0), X2'''')), Y) -> MARK(f(f(g(f(X1''''', X2'''''')), X2'''0), X2''''))
AF(g(f(f(f(g(X'''''), X2''''''), X2'''0), X2'''')), Y) -> MARK(f(f(f(g(X'''''), X2''''''), X2'''0), X2''''))
AF(g(f(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''''), X2'''')), Y) -> MARK(f(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''''), X2''''))
AF(g(f(g(X''''), X2''')), Y) -> MARK(f(g(X''''), X2'''))
AF(g(f(f(X1'''', X2'''''), X2'''')), Y) -> MARK(f(f(X1'''', X2'''''), X2''''))
AF(g(f(g(g(X'''')), X2''')), Y) -> MARK(f(g(g(X'''')), X2'''))
AF(g(f(g(f(X1'''', X2''''')), X2'''')), Y) -> MARK(f(g(f(X1'''', X2''''')), X2''''))
AF(g(f(f(g(X'''), X2'''''), X2'''')), Y) -> MARK(f(f(g(X'''), X2'''''), X2''''))
AF(g(f(f(f(X1'''', X2'''''), X2''0'), X2''')), Y) -> MARK(f(f(f(X1'''', X2'''''), X2''0'), X2'''))
AF(g(g(f(f(g(g(X'''''''')), X2''''''), X2''''))), Y) -> MARK(g(f(f(g(g(X'''''''')), X2''''''), X2'''')))
AF(g(g(f(f(g(f(X1''''''', X2'''''''')), X2''''''), X2''''))), Y) -> MARK(g(f(f(g(f(X1''''''', X2'''''''')), X2''''''), X2'''')))
AF(g(g(f(f(f(g(X'''''''), X2''''''''), X2''''''), X2''''))), Y) -> MARK(g(f(f(f(g(X'''''''), X2''''''''), X2''''''), X2'''')))
AF(g(g(f(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2''''''), X2''''))), Y) -> MARK(g(f(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2''''''), X2'''')))
AF(g(g(f(g(X''''''), X2''''))), Y) -> MARK(g(f(g(X''''''), X2'''')))
AF(g(g(f(f(X1'''''', X2''''''), X2''''))), Y) -> MARK(g(f(f(X1'''''', X2''''''), X2'''')))
AF(g(g(f(g(g(X'''''')), X2''''))), Y) -> MARK(g(f(g(g(X'''''')), X2'''')))
AF(g(g(f(g(f(X1''''', X2'''''')), X2''''))), Y) -> MARK(g(f(g(f(X1''''', X2'''''')), X2'''')))
AF(g(g(f(f(g(X'''''), X2''''''), X2''''))), Y) -> MARK(g(f(f(g(X'''''), X2''''''), X2'''')))
AF(g(g(f(f(f(X1''''', X2'''''''), X2''0'''), X2''''))), Y) -> MARK(g(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''')))
MARK(f(g(f(f(g(g(X'''''''')), X2''''''), X2'''')), X2)) -> MARK(g(f(f(g(g(X'''''''')), X2''''''), X2'''')))
MARK(f(g(f(f(g(f(X1''''''', X2'''''''')), X2''''''), X2'''')), X2)) -> MARK(g(f(f(g(f(X1''''''', X2'''''''')), X2''''''), X2'''')))
MARK(f(g(f(f(f(g(X'''''''), X2''''''''), X2''''''), X2'''')), X2)) -> MARK(g(f(f(f(g(X'''''''), X2''''''''), X2''''''), X2'''')))
MARK(f(g(f(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2''''''), X2'''')), X2)) -> MARK(g(f(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2''''''), X2'''')))
MARK(f(g(f(g(X''''''), X2'''')), X2)) -> MARK(g(f(g(X''''''), X2'''')))
MARK(g(f(f(g(g(X'''''')), X2''''), X2''))) -> MARK(f(f(g(g(X'''''')), X2''''), X2''))
MARK(g(f(f(g(f(X1''''', X2'''''')), X2''''), X2''))) -> MARK(f(f(g(f(X1''''', X2'''''')), X2''''), X2''))
MARK(g(f(f(f(g(X'''''), X2''''''), X2''''), X2''))) -> MARK(f(f(f(g(X'''''), X2''''''), X2''''), X2''))
MARK(f(f(f(g(g(X'''''')), X2'''''), X2''''), X2)) -> MARK(f(f(g(g(X'''''')), X2'''''), X2''''))
MARK(f(f(f(g(f(X1''''', X2'''''')), X2'''0), X2''''), X2)) -> MARK(f(f(g(f(X1''''', X2'''''')), X2'''0), X2''''))
MARK(f(f(f(f(g(X'''''), X2''''''), X2'''0), X2''''), X2)) -> MARK(f(f(f(g(X'''''), X2''''''), X2'''0), X2''''))
MARK(f(f(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''''), X2''''), X2)) -> MARK(f(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''''), X2''''))
MARK(f(f(f(X1'''', X2'''''), X2''''), X2)) -> MARK(f(f(X1'''', X2'''''), X2''''))
MARK(g(f(f(f(f(X1''''', X2'''''''), X2''0'''), X2''''), X2''))) -> MARK(f(f(f(f(X1''''', X2'''''''), X2''0'''), X2''''), X2''))
MARK(f(g(f(f(X1'''''', X2''''''), X2'''')), X2)) -> MARK(g(f(f(X1'''''', X2''''''), X2'''')))
MARK(g(f(g(X''''), X2''))) -> MARK(f(g(X''''), X2''))
MARK(g(f(g(g(X'''')), X2''))) -> MARK(f(g(g(X'''')), X2''))
MARK(f(g(f(g(g(X'''''')), X2'''')), X2)) -> MARK(g(f(g(g(X'''''')), X2'''')))
MARK(g(f(g(f(X1''', X2'''')), X2''))) -> MARK(f(g(f(X1''', X2'''')), X2''))
MARK(f(g(f(g(f(X1''''', X2'''''')), X2'''')), X2)) -> MARK(g(f(g(f(X1''''', X2'''''')), X2'''')))
MARK(f(f(g(X''''), X2'''), X2)) -> MARK(f(g(X''''), X2'''))
MARK(f(g(g(X'''')), X2)) -> MARK(g(g(X'''')))
MARK(f(f(g(g(X'''')), X2'''), X2)) -> MARK(f(g(g(X'''')), X2'''))
MARK(f(f(f(g(X'''), X2'''''), X2''''), X2)) -> MARK(f(f(g(X'''), X2'''''), X2''''))
MARK(f(f(f(f(X1'''', X2'''''), X2''0'), X2'''), X2)) -> MARK(f(f(f(X1'''', X2'''''), X2''0'), X2'''))
MARK(g(f(f(X1'''', X2''''), X2''))) -> MARK(f(f(X1'''', X2''''), X2''))
MARK(f(g(f(f(g(X'''''), X2''''''), X2'''')), X2)) -> MARK(g(f(f(g(X'''''), X2''''''), X2'''')))
MARK(f(f(g(f(X1'''', X2''''')), X2''''), X2)) -> MARK(f(g(f(X1'''', X2''''')), X2''''))
MARK(g(f(f(g(X'''), X2''''), X2''))) -> MARK(f(f(g(X'''), X2''''), X2''))
MARK(g(g(X''))) -> MARK(g(X''))
AF(g(g(g(X''''))), Y) -> MARK(g(g(X'''')))
AF(g(g(g(X'))), Y) -> AF(g(g(mark(X'))), f(g(g(g(X'))), Y))
AF(g(g(f(X1', X2'))), Y) -> AF(g(af(mark(X1'), X2')), f(g(g(f(X1', X2'))), Y))
MARK(f(g(g(X'')), X2)) -> AF(g(g(mark(X''))), X2)
MARK(f(f(g(g(X'''')), X2''), X2)) -> MARK(f(g(g(X'''')), X2''))
MARK(f(f(f(g(X'''), X2''''), X2''), X2)) -> MARK(f(f(g(X'''), X2''''), X2''))
MARK(f(f(f(f(X1''', X2'''''), X2''0'), X2''), X2)) -> MARK(f(f(f(X1''', X2'''''), X2''0'), X2''))
MARK(g(f(f(f(X1''', X2'''''), X2''0'), X2''))) -> MARK(f(f(f(X1''', X2'''''), X2''0'), X2''))
MARK(f(g(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''')), X2)) -> MARK(g(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''')))
MARK(f(f(g(f(X1''', X2'''')), X2''), X2)) -> MARK(f(g(f(X1''', X2'''')), X2''))
AF(g(f(f(g(X'''), X2''''), X2'')), Y) -> MARK(f(f(g(X'''), X2''''), X2''))
MARK(f(f(f(X1', X2'''), X2''), X2)) -> AF(af(af(mark(X1'), X2'''), X2''), X2)
AF(g(f(f(f(X1''', X2'''''), X2''0'), X2'')), Y) -> MARK(f(f(f(X1''', X2'''''), X2''0'), X2''))
MARK(f(g(f(X1', X2'')), X2)) -> AF(g(af(mark(X1'), X2'')), X2)
AF(g(f(g(X'''), X2'')), Y) -> MARK(f(g(X'''), X2''))
AF(g(f(g(X'), X2')), Y) -> AF(af(g(mark(X')), X2'), f(g(f(g(X'), X2')), Y))
AF(g(f(f(X1'', X2''), X2')), Y) -> AF(af(af(mark(X1''), X2''), X2'), f(g(f(f(X1'', X2''), X2')), Y))
MARK(f(f(g(X'), X2''), X2)) -> AF(af(g(mark(X')), X2''), X2)
AF(g(f(f(g(g(X'''''')), X2'''''), X2'''')), Y) -> MARK(f(f(g(g(X'''''')), X2'''''), X2''''))

Rules:

af(g(X), Y) -> af(mark(X), f(g(X), Y))
af(X1, X2) -> f(X1, X2)
mark(f(X1, X2)) -> af(mark(X1), X2)
mark(g(X)) -> g(mark(X))

Strategy:

innermost

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

AF(g(f(g(X'''), X2'')), Y) -> MARK(f(g(X'''), X2''))
13 new Dependency Pairs are created:

AF(g(f(g(f(X1''', X2''''')), X2'''')), Y) -> MARK(f(g(f(X1''', X2''''')), X2''''))
AF(g(f(g(g(X''''')), X2''')), Y) -> MARK(f(g(g(X''''')), X2'''))
AF(g(f(g(g(X'''''')), X2''')), Y) -> MARK(f(g(g(X'''''')), X2'''))
AF(g(f(g(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2'''''')), X2''')), Y) -> MARK(f(g(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2'''''')), X2'''))
AF(g(f(g(f(f(g(X'''''''), X2''''''''), X2'''''')), X2''')), Y) -> MARK(f(g(f(f(g(X'''''''), X2''''''''), X2'''''')), X2'''))
AF(g(f(g(f(g(f(X1''''''', X2'''''''')), X2'''''')), X2''')), Y) -> MARK(f(g(f(g(f(X1''''''', X2'''''''')), X2'''''')), X2'''))
AF(g(f(g(f(g(g(X'''''''')), X2'''''')), X2''')), Y) -> MARK(f(g(f(g(g(X'''''''')), X2'''''')), X2'''))
AF(g(f(g(f(f(X1'''''''', X2''''''''), X2'''''')), X2''')), Y) -> MARK(f(g(f(f(X1'''''''', X2''''''''), X2'''''')), X2'''))
AF(g(f(g(f(g(X''''''''), X2'''''')), X2''')), Y) -> MARK(f(g(f(g(X''''''''), X2'''''')), X2'''))
AF(g(f(g(f(f(f(f(X1''''''''', X2'''''''''''), X2''0'''''''), X2''''''''), X2'''''')), X2''')), Y) -> MARK(f(g(f(f(f(f(X1''''''''', X2'''''''''''), X2''0'''''''), X2''''''''), X2'''''')), X2'''))
AF(g(f(g(f(f(f(g(X'''''''''), X2''''''''''), X2''''''''), X2'''''')), X2''')), Y) -> MARK(f(g(f(f(f(g(X'''''''''), X2''''''''''), X2''''''''), X2'''''')), X2'''))
AF(g(f(g(f(f(g(f(X1''''''''', X2'''''''''')), X2''''''''), X2'''''')), X2''')), Y) -> MARK(f(g(f(f(g(f(X1''''''''', X2'''''''''')), X2''''''''), X2'''''')), X2'''))
AF(g(f(g(f(f(g(g(X'''''''''')), X2''''''''), X2'''''')), X2''')), Y) -> MARK(f(g(f(f(g(g(X'''''''''')), X2''''''''), X2'''''')), X2'''))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 15`
`                 ↳Forward Instantiation Transformation`

Dependency Pairs:

AF(g(f(g(f(f(g(g(X'''''''''')), X2''''''''), X2'''''')), X2''')), Y) -> MARK(f(g(f(f(g(g(X'''''''''')), X2''''''''), X2'''''')), X2'''))
AF(g(f(g(f(f(g(f(X1''''''''', X2'''''''''')), X2''''''''), X2'''''')), X2''')), Y) -> MARK(f(g(f(f(g(f(X1''''''''', X2'''''''''')), X2''''''''), X2'''''')), X2'''))
AF(g(f(g(f(f(f(g(X'''''''''), X2''''''''''), X2''''''''), X2'''''')), X2''')), Y) -> MARK(f(g(f(f(f(g(X'''''''''), X2''''''''''), X2''''''''), X2'''''')), X2'''))
AF(g(f(g(f(f(f(f(X1''''''''', X2'''''''''''), X2''0'''''''), X2''''''''), X2'''''')), X2''')), Y) -> MARK(f(g(f(f(f(f(X1''''''''', X2'''''''''''), X2''0'''''''), X2''''''''), X2'''''')), X2'''))
AF(g(f(g(f(g(X''''''''), X2'''''')), X2''')), Y) -> MARK(f(g(f(g(X''''''''), X2'''''')), X2'''))
AF(g(f(g(f(f(X1'''''''', X2''''''''), X2'''''')), X2''')), Y) -> MARK(f(g(f(f(X1'''''''', X2''''''''), X2'''''')), X2'''))
AF(g(f(g(f(g(g(X'''''''')), X2'''''')), X2''')), Y) -> MARK(f(g(f(g(g(X'''''''')), X2'''''')), X2'''))
AF(g(f(g(f(g(f(X1''''''', X2'''''''')), X2'''''')), X2''')), Y) -> MARK(f(g(f(g(f(X1''''''', X2'''''''')), X2'''''')), X2'''))
AF(g(f(g(f(f(g(X'''''''), X2''''''''), X2'''''')), X2''')), Y) -> MARK(f(g(f(f(g(X'''''''), X2''''''''), X2'''''')), X2'''))
AF(g(f(g(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2'''''')), X2''')), Y) -> MARK(f(g(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2'''''')), X2'''))
AF(g(f(g(g(X'''''')), X2''')), Y) -> MARK(f(g(g(X'''''')), X2'''))
AF(g(f(g(g(X''''')), X2''')), Y) -> MARK(f(g(g(X''''')), X2'''))
AF(g(f(g(f(X1''', X2''''')), X2'''')), Y) -> MARK(f(g(f(X1''', X2''''')), X2''''))
AF(g(f(f(g(g(X'''''')), X2'''''), X2'''')), Y) -> MARK(f(f(g(g(X'''''')), X2'''''), X2''''))
AF(g(f(f(f(g(X'''''), X2''''''), X2'''0), X2'''')), Y) -> MARK(f(f(f(g(X'''''), X2''''''), X2'''0), X2''''))
AF(g(f(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''''), X2'''')), Y) -> MARK(f(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''''), X2''''))
AF(g(f(g(X''''), X2''')), Y) -> MARK(f(g(X''''), X2'''))
AF(g(f(f(X1'''', X2'''''), X2'''')), Y) -> MARK(f(f(X1'''', X2'''''), X2''''))
AF(g(f(g(g(X'''')), X2''')), Y) -> MARK(f(g(g(X'''')), X2'''))
AF(g(f(g(f(X1'''', X2''''')), X2'''')), Y) -> MARK(f(g(f(X1'''', X2''''')), X2''''))
AF(g(f(f(g(X'''), X2'''''), X2'''')), Y) -> MARK(f(f(g(X'''), X2'''''), X2''''))
AF(g(f(f(f(X1'''', X2'''''), X2''0'), X2''')), Y) -> MARK(f(f(f(X1'''', X2'''''), X2''0'), X2'''))
AF(g(g(f(f(g(g(X'''''''')), X2''''''), X2''''))), Y) -> MARK(g(f(f(g(g(X'''''''')), X2''''''), X2'''')))
AF(g(g(f(f(g(f(X1''''''', X2'''''''')), X2''''''), X2''''))), Y) -> MARK(g(f(f(g(f(X1''''''', X2'''''''')), X2''''''), X2'''')))
AF(g(g(f(f(f(g(X'''''''), X2''''''''), X2''''''), X2''''))), Y) -> MARK(g(f(f(f(g(X'''''''), X2''''''''), X2''''''), X2'''')))
AF(g(g(f(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2''''''), X2''''))), Y) -> MARK(g(f(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2''''''), X2'''')))
AF(g(g(f(g(X''''''), X2''''))), Y) -> MARK(g(f(g(X''''''), X2'''')))
AF(g(g(f(f(X1'''''', X2''''''), X2''''))), Y) -> MARK(g(f(f(X1'''''', X2''''''), X2'''')))
AF(g(g(f(g(g(X'''''')), X2''''))), Y) -> MARK(g(f(g(g(X'''''')), X2'''')))
AF(g(g(f(g(f(X1''''', X2'''''')), X2''''))), Y) -> MARK(g(f(g(f(X1''''', X2'''''')), X2'''')))
AF(g(g(f(f(g(X'''''), X2''''''), X2''''))), Y) -> MARK(g(f(f(g(X'''''), X2''''''), X2'''')))
MARK(f(f(f(f(g(X'''''), X2''''''), X2'''0), X2''''), X2)) -> MARK(f(f(f(g(X'''''), X2''''''), X2'''0), X2''''))
MARK(f(f(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''''), X2''''), X2)) -> MARK(f(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''''), X2''''))
MARK(f(g(f(f(g(g(X'''''''')), X2''''''), X2'''')), X2)) -> MARK(g(f(f(g(g(X'''''''')), X2''''''), X2'''')))
MARK(f(g(f(f(g(f(X1''''''', X2'''''''')), X2''''''), X2'''')), X2)) -> MARK(g(f(f(g(f(X1''''''', X2'''''''')), X2''''''), X2'''')))
MARK(f(g(f(f(f(g(X'''''''), X2''''''''), X2''''''), X2'''')), X2)) -> MARK(g(f(f(f(g(X'''''''), X2''''''''), X2''''''), X2'''')))
MARK(f(g(f(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2''''''), X2'''')), X2)) -> MARK(g(f(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2''''''), X2'''')))
MARK(f(g(f(g(X''''''), X2'''')), X2)) -> MARK(g(f(g(X''''''), X2'''')))
MARK(f(f(f(g(g(X'''''')), X2'''''), X2''''), X2)) -> MARK(f(f(g(g(X'''''')), X2'''''), X2''''))
MARK(f(f(f(g(f(X1''''', X2'''''')), X2'''0), X2''''), X2)) -> MARK(f(f(g(f(X1''''', X2'''''')), X2'''0), X2''''))
MARK(g(f(f(f(g(X'''''), X2''''''), X2''''), X2''))) -> MARK(f(f(f(g(X'''''), X2''''''), X2''''), X2''))
MARK(f(g(f(f(X1'''''', X2''''''), X2'''')), X2)) -> MARK(g(f(f(X1'''''', X2''''''), X2'''')))
MARK(g(f(g(X''''), X2''))) -> MARK(f(g(X''''), X2''))
MARK(g(f(g(g(X'''')), X2''))) -> MARK(f(g(g(X'''')), X2''))
MARK(f(g(f(g(g(X'''''')), X2'''')), X2)) -> MARK(g(f(g(g(X'''''')), X2'''')))
MARK(g(f(g(f(X1''', X2'''')), X2''))) -> MARK(f(g(f(X1''', X2'''')), X2''))
MARK(f(g(f(g(f(X1''''', X2'''''')), X2'''')), X2)) -> MARK(g(f(g(f(X1''''', X2'''''')), X2'''')))
MARK(g(f(f(g(g(X'''''')), X2''''), X2''))) -> MARK(f(f(g(g(X'''''')), X2''''), X2''))
MARK(g(f(f(g(f(X1''''', X2'''''')), X2''''), X2''))) -> MARK(f(f(g(f(X1''''', X2'''''')), X2''''), X2''))
MARK(g(f(f(g(X'''), X2''''), X2''))) -> MARK(f(f(g(X'''), X2''''), X2''))
MARK(f(g(f(f(g(X'''''), X2''''''), X2'''')), X2)) -> MARK(g(f(f(g(X'''''), X2''''''), X2'''')))
MARK(f(f(g(X''''), X2'''), X2)) -> MARK(f(g(X''''), X2'''))
MARK(f(g(g(X'''')), X2)) -> MARK(g(g(X'''')))
MARK(f(f(g(g(X'''')), X2'''), X2)) -> MARK(f(g(g(X'''')), X2'''))
MARK(f(f(f(X1'''', X2'''''), X2''''), X2)) -> MARK(f(f(X1'''', X2'''''), X2''''))
MARK(g(f(f(f(f(X1''''', X2'''''''), X2''0'''), X2''''), X2''))) -> MARK(f(f(f(f(X1''''', X2'''''''), X2''0'''), X2''''), X2''))
MARK(f(g(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''')), X2)) -> MARK(g(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''')))
MARK(f(f(g(f(X1'''', X2''''')), X2''''), X2)) -> MARK(f(g(f(X1'''', X2''''')), X2''''))
MARK(f(f(f(g(X'''), X2'''''), X2''''), X2)) -> MARK(f(f(g(X'''), X2'''''), X2''''))
MARK(f(f(f(f(X1'''', X2'''''), X2''0'), X2'''), X2)) -> MARK(f(f(f(X1'''', X2'''''), X2''0'), X2'''))
MARK(g(f(f(X1'''', X2''''), X2''))) -> MARK(f(f(X1'''', X2''''), X2''))
AF(g(g(f(f(f(X1''''', X2'''''''), X2''0'''), X2''''))), Y) -> MARK(g(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''')))
MARK(f(g(g(X'')), X2)) -> AF(g(g(mark(X''))), X2)
MARK(f(f(g(g(X'''')), X2''), X2)) -> MARK(f(g(g(X'''')), X2''))
MARK(f(f(f(g(X'''), X2''''), X2''), X2)) -> MARK(f(f(g(X'''), X2''''), X2''))
MARK(f(f(f(f(X1''', X2'''''), X2''0'), X2''), X2)) -> MARK(f(f(f(X1''', X2'''''), X2''0'), X2''))
MARK(g(f(f(f(X1''', X2'''''), X2''0'), X2''))) -> MARK(f(f(f(X1''', X2'''''), X2''0'), X2''))
MARK(g(g(X''))) -> MARK(g(X''))
AF(g(g(g(X''''))), Y) -> MARK(g(g(X'''')))
AF(g(g(g(X'))), Y) -> AF(g(g(mark(X'))), f(g(g(g(X'))), Y))
AF(g(g(f(X1', X2'))), Y) -> AF(g(af(mark(X1'), X2')), f(g(g(f(X1', X2'))), Y))
MARK(f(g(f(X1', X2'')), X2)) -> AF(g(af(mark(X1'), X2'')), X2)
MARK(f(f(g(f(X1''', X2'''')), X2''), X2)) -> MARK(f(g(f(X1''', X2'''')), X2''))
AF(g(f(f(g(X'''), X2''''), X2'')), Y) -> MARK(f(f(g(X'''), X2''''), X2''))
MARK(f(f(f(X1', X2'''), X2''), X2)) -> AF(af(af(mark(X1'), X2'''), X2''), X2)
AF(g(f(f(f(X1''', X2'''''), X2''0'), X2'')), Y) -> MARK(f(f(f(X1''', X2'''''), X2''0'), X2''))
AF(g(f(g(X'), X2')), Y) -> AF(af(g(mark(X')), X2'), f(g(f(g(X'), X2')), Y))
AF(g(f(f(X1'', X2''), X2')), Y) -> AF(af(af(mark(X1''), X2''), X2'), f(g(f(f(X1'', X2''), X2')), Y))
MARK(f(f(g(X'), X2''), X2)) -> AF(af(g(mark(X')), X2''), X2)
AF(g(f(f(g(f(X1''''', X2'''''')), X2'''0), X2'''')), Y) -> MARK(f(f(g(f(X1''''', X2'''''')), X2'''0), X2''''))

Rules:

af(g(X), Y) -> af(mark(X), f(g(X), Y))
af(X1, X2) -> f(X1, X2)
mark(f(X1, X2)) -> af(mark(X1), X2)
mark(g(X)) -> g(mark(X))

Strategy:

innermost

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

MARK(g(g(X''))) -> MARK(g(X''))
11 new Dependency Pairs are created:

MARK(g(g(g(X'''')))) -> MARK(g(g(X'''')))
MARK(g(g(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''')))) -> MARK(g(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''')))
MARK(g(g(f(f(g(X'''''), X2''''''), X2'''')))) -> MARK(g(f(f(g(X'''''), X2''''''), X2'''')))
MARK(g(g(f(g(f(X1''''', X2'''''')), X2'''')))) -> MARK(g(f(g(f(X1''''', X2'''''')), X2'''')))
MARK(g(g(f(g(g(X'''''')), X2'''')))) -> MARK(g(f(g(g(X'''''')), X2'''')))
MARK(g(g(f(f(X1'''''', X2''''''), X2'''')))) -> MARK(g(f(f(X1'''''', X2''''''), X2'''')))
MARK(g(g(f(g(X''''''), X2'''')))) -> MARK(g(f(g(X''''''), X2'''')))
MARK(g(g(f(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2''''''), X2'''')))) -> MARK(g(f(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2''''''), X2'''')))
MARK(g(g(f(f(f(g(X'''''''), X2''''''''), X2''''''), X2'''')))) -> MARK(g(f(f(f(g(X'''''''), X2''''''''), X2''''''), X2'''')))
MARK(g(g(f(f(g(f(X1''''''', X2'''''''')), X2''''''), X2'''')))) -> MARK(g(f(f(g(f(X1''''''', X2'''''''')), X2''''''), X2'''')))
MARK(g(g(f(f(g(g(X'''''''')), X2''''''), X2'''')))) -> MARK(g(f(f(g(g(X'''''''')), X2''''''), X2'''')))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 16`
`                 ↳Forward Instantiation Transformation`

Dependency Pairs:

AF(g(f(g(f(f(g(f(X1''''''''', X2'''''''''')), X2''''''''), X2'''''')), X2''')), Y) -> MARK(f(g(f(f(g(f(X1''''''''', X2'''''''''')), X2''''''''), X2'''''')), X2'''))
AF(g(f(g(f(f(f(g(X'''''''''), X2''''''''''), X2''''''''), X2'''''')), X2''')), Y) -> MARK(f(g(f(f(f(g(X'''''''''), X2''''''''''), X2''''''''), X2'''''')), X2'''))
AF(g(f(g(f(f(f(f(X1''''''''', X2'''''''''''), X2''0'''''''), X2''''''''), X2'''''')), X2''')), Y) -> MARK(f(g(f(f(f(f(X1''''''''', X2'''''''''''), X2''0'''''''), X2''''''''), X2'''''')), X2'''))
AF(g(f(g(f(g(X''''''''), X2'''''')), X2''')), Y) -> MARK(f(g(f(g(X''''''''), X2'''''')), X2'''))
AF(g(f(g(f(f(X1'''''''', X2''''''''), X2'''''')), X2''')), Y) -> MARK(f(g(f(f(X1'''''''', X2''''''''), X2'''''')), X2'''))
AF(g(f(g(f(g(g(X'''''''')), X2'''''')), X2''')), Y) -> MARK(f(g(f(g(g(X'''''''')), X2'''''')), X2'''))
AF(g(f(g(f(g(f(X1''''''', X2'''''''')), X2'''''')), X2''')), Y) -> MARK(f(g(f(g(f(X1''''''', X2'''''''')), X2'''''')), X2'''))
AF(g(f(g(f(f(g(X'''''''), X2''''''''), X2'''''')), X2''')), Y) -> MARK(f(g(f(f(g(X'''''''), X2''''''''), X2'''''')), X2'''))
AF(g(f(g(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2'''''')), X2''')), Y) -> MARK(f(g(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2'''''')), X2'''))
AF(g(f(g(g(X'''''')), X2''')), Y) -> MARK(f(g(g(X'''''')), X2'''))
AF(g(f(g(g(X''''')), X2''')), Y) -> MARK(f(g(g(X''''')), X2'''))
AF(g(f(g(f(X1''', X2''''')), X2'''')), Y) -> MARK(f(g(f(X1''', X2''''')), X2''''))
AF(g(f(f(g(g(X'''''')), X2'''''), X2'''')), Y) -> MARK(f(f(g(g(X'''''')), X2'''''), X2''''))
AF(g(f(f(g(f(X1''''', X2'''''')), X2'''0), X2'''')), Y) -> MARK(f(f(g(f(X1''''', X2'''''')), X2'''0), X2''''))
AF(g(f(f(f(g(X'''''), X2''''''), X2'''0), X2'''')), Y) -> MARK(f(f(f(g(X'''''), X2''''''), X2'''0), X2''''))
AF(g(f(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''''), X2'''')), Y) -> MARK(f(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''''), X2''''))
AF(g(f(g(X''''), X2''')), Y) -> MARK(f(g(X''''), X2'''))
AF(g(f(f(X1'''', X2'''''), X2'''')), Y) -> MARK(f(f(X1'''', X2'''''), X2''''))
AF(g(f(g(g(X'''')), X2''')), Y) -> MARK(f(g(g(X'''')), X2'''))
AF(g(f(g(f(X1'''', X2''''')), X2'''')), Y) -> MARK(f(g(f(X1'''', X2''''')), X2''''))
AF(g(f(f(g(X'''), X2'''''), X2'''')), Y) -> MARK(f(f(g(X'''), X2'''''), X2''''))
AF(g(f(f(f(X1'''', X2'''''), X2''0'), X2''')), Y) -> MARK(f(f(f(X1'''', X2'''''), X2''0'), X2'''))
AF(g(g(f(f(g(g(X'''''''')), X2''''''), X2''''))), Y) -> MARK(g(f(f(g(g(X'''''''')), X2''''''), X2'''')))
AF(g(g(f(f(g(f(X1''''''', X2'''''''')), X2''''''), X2''''))), Y) -> MARK(g(f(f(g(f(X1''''''', X2'''''''')), X2''''''), X2'''')))
AF(g(g(f(f(f(g(X'''''''), X2''''''''), X2''''''), X2''''))), Y) -> MARK(g(f(f(f(g(X'''''''), X2''''''''), X2''''''), X2'''')))
AF(g(g(f(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2''''''), X2''''))), Y) -> MARK(g(f(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2''''''), X2'''')))
AF(g(g(f(g(X''''''), X2''''))), Y) -> MARK(g(f(g(X''''''), X2'''')))
AF(g(g(f(f(X1'''''', X2''''''), X2''''))), Y) -> MARK(g(f(f(X1'''''', X2''''''), X2'''')))
AF(g(g(f(g(g(X'''''')), X2''''))), Y) -> MARK(g(f(g(g(X'''''')), X2'''')))
AF(g(g(f(g(f(X1''''', X2'''''')), X2''''))), Y) -> MARK(g(f(g(f(X1''''', X2'''''')), X2'''')))
AF(g(g(f(f(g(X'''''), X2''''''), X2''''))), Y) -> MARK(g(f(f(g(X'''''), X2''''''), X2'''')))
MARK(g(g(f(f(g(g(X'''''''')), X2''''''), X2'''')))) -> MARK(g(f(f(g(g(X'''''''')), X2''''''), X2'''')))
MARK(g(g(f(f(g(f(X1''''''', X2'''''''')), X2''''''), X2'''')))) -> MARK(g(f(f(g(f(X1''''''', X2'''''''')), X2''''''), X2'''')))
MARK(g(g(f(f(f(g(X'''''''), X2''''''''), X2''''''), X2'''')))) -> MARK(g(f(f(f(g(X'''''''), X2''''''''), X2''''''), X2'''')))
MARK(g(g(f(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2''''''), X2'''')))) -> MARK(g(f(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2''''''), X2'''')))
MARK(g(g(f(g(X''''''), X2'''')))) -> MARK(g(f(g(X''''''), X2'''')))
MARK(g(g(f(f(X1'''''', X2''''''), X2'''')))) -> MARK(g(f(f(X1'''''', X2''''''), X2'''')))
MARK(g(g(f(g(g(X'''''')), X2'''')))) -> MARK(g(f(g(g(X'''''')), X2'''')))
MARK(g(g(f(g(f(X1''''', X2'''''')), X2'''')))) -> MARK(g(f(g(f(X1''''', X2'''''')), X2'''')))
MARK(f(g(f(f(g(g(X'''''''')), X2''''''), X2'''')), X2)) -> MARK(g(f(f(g(g(X'''''''')), X2''''''), X2'''')))
MARK(f(g(f(f(g(f(X1''''''', X2'''''''')), X2''''''), X2'''')), X2)) -> MARK(g(f(f(g(f(X1''''''', X2'''''''')), X2''''''), X2'''')))
MARK(f(g(f(f(f(g(X'''''''), X2''''''''), X2''''''), X2'''')), X2)) -> MARK(g(f(f(f(g(X'''''''), X2''''''''), X2''''''), X2'''')))
MARK(f(g(f(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2''''''), X2'''')), X2)) -> MARK(g(f(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2''''''), X2'''')))
MARK(f(g(f(g(X''''''), X2'''')), X2)) -> MARK(g(f(g(X''''''), X2'''')))
MARK(g(f(f(g(g(X'''''')), X2''''), X2''))) -> MARK(f(f(g(g(X'''''')), X2''''), X2''))
MARK(f(f(f(g(g(X'''''')), X2'''''), X2''''), X2)) -> MARK(f(f(g(g(X'''''')), X2'''''), X2''''))
MARK(f(f(f(g(f(X1''''', X2'''''')), X2'''0), X2''''), X2)) -> MARK(f(f(g(f(X1''''', X2'''''')), X2'''0), X2''''))
MARK(f(f(f(f(g(X'''''), X2''''''), X2'''0), X2''''), X2)) -> MARK(f(f(f(g(X'''''), X2''''''), X2'''0), X2''''))
MARK(f(f(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''''), X2''''), X2)) -> MARK(f(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''''), X2''''))
MARK(f(f(f(X1'''', X2'''''), X2''''), X2)) -> MARK(f(f(X1'''', X2'''''), X2''''))
MARK(g(f(f(f(g(X'''''), X2''''''), X2''''), X2''))) -> MARK(f(f(f(g(X'''''), X2''''''), X2''''), X2''))
MARK(f(g(f(f(X1'''''', X2''''''), X2'''')), X2)) -> MARK(g(f(f(X1'''''', X2''''''), X2'''')))
MARK(g(f(g(X''''), X2''))) -> MARK(f(g(X''''), X2''))
MARK(g(f(g(g(X'''')), X2''))) -> MARK(f(g(g(X'''')), X2''))
MARK(f(g(f(g(g(X'''''')), X2'''')), X2)) -> MARK(g(f(g(g(X'''''')), X2'''')))
MARK(g(f(g(f(X1''', X2'''')), X2''))) -> MARK(f(g(f(X1''', X2'''')), X2''))
MARK(f(g(f(g(f(X1''''', X2'''''')), X2'''')), X2)) -> MARK(g(f(g(f(X1''''', X2'''''')), X2'''')))
MARK(f(f(g(X''''), X2'''), X2)) -> MARK(f(g(X''''), X2'''))
MARK(g(f(f(g(f(X1''''', X2'''''')), X2''''), X2''))) -> MARK(f(f(g(f(X1''''', X2'''''')), X2''''), X2''))
MARK(g(g(f(f(g(X'''''), X2''''''), X2'''')))) -> MARK(g(f(f(g(X'''''), X2''''''), X2'''')))
MARK(f(g(g(X'''')), X2)) -> MARK(g(g(X'''')))
MARK(f(f(g(g(X'''')), X2'''), X2)) -> MARK(f(g(g(X'''')), X2'''))
MARK(g(f(f(g(X'''), X2''''), X2''))) -> MARK(f(f(g(X'''), X2''''), X2''))
MARK(f(g(f(f(g(X'''''), X2''''''), X2'''')), X2)) -> MARK(g(f(f(g(X'''''), X2''''''), X2'''')))
MARK(f(f(g(f(X1'''', X2''''')), X2''''), X2)) -> MARK(f(g(f(X1'''', X2''''')), X2''''))
MARK(f(f(f(g(X'''), X2'''''), X2''''), X2)) -> MARK(f(f(g(X'''), X2'''''), X2''''))
MARK(f(f(f(f(X1'''', X2'''''), X2''0'), X2'''), X2)) -> MARK(f(f(f(X1'''', X2'''''), X2''0'), X2'''))
MARK(g(f(f(f(f(X1''''', X2'''''''), X2''0'''), X2''''), X2''))) -> MARK(f(f(f(f(X1''''', X2'''''''), X2''0'''), X2''''), X2''))
AF(g(g(f(f(f(X1''''', X2'''''''), X2''0'''), X2''''))), Y) -> MARK(g(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''')))
MARK(f(g(g(X'')), X2)) -> AF(g(g(mark(X''))), X2)
MARK(f(f(g(g(X'''')), X2''), X2)) -> MARK(f(g(g(X'''')), X2''))
MARK(g(f(f(X1'''', X2''''), X2''))) -> MARK(f(f(X1'''', X2''''), X2''))
MARK(f(g(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''')), X2)) -> MARK(g(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''')))
MARK(f(f(g(f(X1''', X2'''')), X2''), X2)) -> MARK(f(g(f(X1''', X2'''')), X2''))
MARK(f(f(f(g(X'''), X2''''), X2''), X2)) -> MARK(f(f(g(X'''), X2''''), X2''))
MARK(f(f(f(f(X1''', X2'''''), X2''0'), X2''), X2)) -> MARK(f(f(f(X1''', X2'''''), X2''0'), X2''))
MARK(g(f(f(f(X1''', X2'''''), X2''0'), X2''))) -> MARK(f(f(f(X1''', X2'''''), X2''0'), X2''))
MARK(g(g(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''')))) -> MARK(g(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''')))
MARK(g(g(g(X'''')))) -> MARK(g(g(X'''')))
AF(g(g(g(X''''))), Y) -> MARK(g(g(X'''')))
AF(g(g(g(X'))), Y) -> AF(g(g(mark(X'))), f(g(g(g(X'))), Y))
AF(g(g(f(X1', X2'))), Y) -> AF(g(af(mark(X1'), X2')), f(g(g(f(X1', X2'))), Y))
MARK(f(f(g(X'), X2''), X2)) -> AF(af(g(mark(X')), X2''), X2)
AF(g(f(f(g(X'''), X2''''), X2'')), Y) -> MARK(f(f(g(X'''), X2''''), X2''))
MARK(f(f(f(X1', X2'''), X2''), X2)) -> AF(af(af(mark(X1'), X2'''), X2''), X2)
AF(g(f(f(f(X1''', X2'''''), X2''0'), X2'')), Y) -> MARK(f(f(f(X1''', X2'''''), X2''0'), X2''))
AF(g(f(g(X'), X2')), Y) -> AF(af(g(mark(X')), X2'), f(g(f(g(X'), X2')), Y))
AF(g(f(f(X1'', X2''), X2')), Y) -> AF(af(af(mark(X1''), X2''), X2'), f(g(f(f(X1'', X2''), X2')), Y))
MARK(f(g(f(X1', X2'')), X2)) -> AF(g(af(mark(X1'), X2'')), X2)
AF(g(f(g(f(f(g(g(X'''''''''')), X2''''''''), X2'''''')), X2''')), Y) -> MARK(f(g(f(f(g(g(X'''''''''')), X2''''''''), X2'''''')), X2'''))

Rules:

af(g(X), Y) -> af(mark(X), f(g(X), Y))
af(X1, X2) -> f(X1, X2)
mark(f(X1, X2)) -> af(mark(X1), X2)
mark(g(X)) -> g(mark(X))

Strategy:

innermost

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

MARK(g(f(f(X1'''', X2''''), X2''))) -> MARK(f(f(X1'''', X2''''), X2''))
16 new Dependency Pairs are created:

MARK(g(f(f(f(X1''', X2'''''''), X2''''''), X2'''))) -> MARK(f(f(f(X1''', X2'''''''), X2''''''), X2'''))
MARK(g(f(f(g(X'''), X2'''''), X2''0))) -> MARK(f(f(g(X'''), X2'''''), X2''0))
MARK(g(f(f(f(f(X1'''''', X2'''''''), X2''0'''), X2'''''), X2''0))) -> MARK(f(f(f(f(X1'''''', X2'''''''), X2''0'''), X2'''''), X2''0))
MARK(g(f(f(f(g(X'''''), X2'''''''), X2''''''), X2''0))) -> MARK(f(f(f(g(X'''''), X2'''''''), X2''''''), X2''0))
MARK(g(f(f(g(f(X1'''''', X2''''''')), X2''''''), X2''0))) -> MARK(f(f(g(f(X1'''''', X2''''''')), X2''''''), X2''0))
MARK(g(f(f(g(g(X'''''')), X2'''''), X2''0))) -> MARK(f(f(g(g(X'''''')), X2'''''), X2''0))
MARK(g(f(f(f(f(X1'''''', X2'''''''), X2''0'''), X2'''''), X2'''))) -> MARK(f(f(f(f(X1'''''', X2'''''''), X2''0'''), X2'''''), X2'''))
MARK(g(f(f(f(g(X'''''), X2'''''''), X2'''''), X2'''))) -> MARK(f(f(f(g(X'''''), X2'''''''), X2'''''), X2'''))
MARK(g(f(f(g(f(X1'''''', X2''''''')), X2'''''), X2'''))) -> MARK(f(f(g(f(X1'''''', X2''''''')), X2'''''), X2'''))
MARK(g(f(f(g(g(X'''''')), X2''''''), X2'''))) -> MARK(f(f(g(g(X'''''')), X2''''''), X2'''))
MARK(g(f(f(f(X1'''''', X2'''''''), X2'''''), X2'''))) -> MARK(f(f(f(X1'''''', X2'''''''), X2'''''), X2'''))
MARK(g(f(f(g(X''''''), X2''''''), X2'''))) -> MARK(f(f(g(X''''''), X2''''''), X2'''))
MARK(g(f(f(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2'''''''), X2'''''), X2'''))) -> MARK(f(f(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2'''''''), X2'''''), X2'''))
MARK(g(f(f(f(f(g(X'''''''), X2''''''''), X2'''0''), X2''''''), X2'''))) -> MARK(f(f(f(f(g(X'''''''), X2''''''''), X2'''0''), X2''''''), X2'''))
MARK(g(f(f(f(g(f(X1''''''', X2'''''''')), X2'''0''), X2''''''), X2'''))) -> MARK(f(f(f(g(f(X1''''''', X2'''''''')), X2'''0''), X2''''''), X2'''))
MARK(g(f(f(f(g(g(X'''''''')), X2'''''''), X2'''''), X2'''))) -> MARK(f(f(f(g(g(X'''''''')), X2'''''''), X2'''''), X2'''))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 17`
`                 ↳Forward Instantiation Transformation`

Dependency Pairs:

AF(g(f(g(f(f(g(g(X'''''''''')), X2''''''''), X2'''''')), X2''')), Y) -> MARK(f(g(f(f(g(g(X'''''''''')), X2''''''''), X2'''''')), X2'''))
AF(g(f(g(f(f(f(g(X'''''''''), X2''''''''''), X2''''''''), X2'''''')), X2''')), Y) -> MARK(f(g(f(f(f(g(X'''''''''), X2''''''''''), X2''''''''), X2'''''')), X2'''))
AF(g(f(g(f(f(f(f(X1''''''''', X2'''''''''''), X2''0'''''''), X2''''''''), X2'''''')), X2''')), Y) -> MARK(f(g(f(f(f(f(X1''''''''', X2'''''''''''), X2''0'''''''), X2''''''''), X2'''''')), X2'''))
AF(g(f(g(f(g(X''''''''), X2'''''')), X2''')), Y) -> MARK(f(g(f(g(X''''''''), X2'''''')), X2'''))
AF(g(f(g(f(f(X1'''''''', X2''''''''), X2'''''')), X2''')), Y) -> MARK(f(g(f(f(X1'''''''', X2''''''''), X2'''''')), X2'''))
AF(g(f(g(f(g(g(X'''''''')), X2'''''')), X2''')), Y) -> MARK(f(g(f(g(g(X'''''''')), X2'''''')), X2'''))
AF(g(f(g(f(g(f(X1''''''', X2'''''''')), X2'''''')), X2''')), Y) -> MARK(f(g(f(g(f(X1''''''', X2'''''''')), X2'''''')), X2'''))
AF(g(f(g(f(f(g(X'''''''), X2''''''''), X2'''''')), X2''')), Y) -> MARK(f(g(f(f(g(X'''''''), X2''''''''), X2'''''')), X2'''))
AF(g(f(g(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2'''''')), X2''')), Y) -> MARK(f(g(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2'''''')), X2'''))
AF(g(f(g(g(X'''''')), X2''')), Y) -> MARK(f(g(g(X'''''')), X2'''))
AF(g(f(g(g(X''''')), X2''')), Y) -> MARK(f(g(g(X''''')), X2'''))
AF(g(f(g(f(X1''', X2''''')), X2'''')), Y) -> MARK(f(g(f(X1''', X2''''')), X2''''))
AF(g(f(f(g(g(X'''''')), X2'''''), X2'''')), Y) -> MARK(f(f(g(g(X'''''')), X2'''''), X2''''))
AF(g(f(f(g(f(X1''''', X2'''''')), X2'''0), X2'''')), Y) -> MARK(f(f(g(f(X1''''', X2'''''')), X2'''0), X2''''))
AF(g(f(f(f(g(X'''''), X2''''''), X2'''0), X2'''')), Y) -> MARK(f(f(f(g(X'''''), X2''''''), X2'''0), X2''''))
AF(g(f(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''''), X2'''')), Y) -> MARK(f(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''''), X2''''))
AF(g(f(g(X''''), X2''')), Y) -> MARK(f(g(X''''), X2'''))
AF(g(f(f(X1'''', X2'''''), X2'''')), Y) -> MARK(f(f(X1'''', X2'''''), X2''''))
AF(g(f(g(g(X'''')), X2''')), Y) -> MARK(f(g(g(X'''')), X2'''))
AF(g(f(g(f(X1'''', X2''''')), X2'''')), Y) -> MARK(f(g(f(X1'''', X2''''')), X2''''))
AF(g(f(f(g(X'''), X2'''''), X2'''')), Y) -> MARK(f(f(g(X'''), X2'''''), X2''''))
AF(g(f(f(f(X1'''', X2'''''), X2''0'), X2''')), Y) -> MARK(f(f(f(X1'''', X2'''''), X2''0'), X2'''))
AF(g(g(f(f(g(g(X'''''''')), X2''''''), X2''''))), Y) -> MARK(g(f(f(g(g(X'''''''')), X2''''''), X2'''')))
AF(g(g(f(f(g(f(X1''''''', X2'''''''')), X2''''''), X2''''))), Y) -> MARK(g(f(f(g(f(X1''''''', X2'''''''')), X2''''''), X2'''')))
AF(g(g(f(f(f(g(X'''''''), X2''''''''), X2''''''), X2''''))), Y) -> MARK(g(f(f(f(g(X'''''''), X2''''''''), X2''''''), X2'''')))
AF(g(g(f(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2''''''), X2''''))), Y) -> MARK(g(f(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2''''''), X2'''')))
AF(g(g(f(g(X''''''), X2''''))), Y) -> MARK(g(f(g(X''''''), X2'''')))
AF(g(g(f(f(X1'''''', X2''''''), X2''''))), Y) -> MARK(g(f(f(X1'''''', X2''''''), X2'''')))
AF(g(g(f(g(g(X'''''')), X2''''))), Y) -> MARK(g(f(g(g(X'''''')), X2'''')))
AF(g(g(f(g(f(X1''''', X2'''''')), X2''''))), Y) -> MARK(g(f(g(f(X1''''', X2'''''')), X2'''')))
AF(g(g(f(f(g(X'''''), X2''''''), X2''''))), Y) -> MARK(g(f(f(g(X'''''), X2''''''), X2'''')))
MARK(g(g(f(f(g(g(X'''''''')), X2''''''), X2'''')))) -> MARK(g(f(f(g(g(X'''''''')), X2''''''), X2'''')))
MARK(g(g(f(f(g(f(X1''''''', X2'''''''')), X2''''''), X2'''')))) -> MARK(g(f(f(g(f(X1''''''', X2'''''''')), X2''''''), X2'''')))
MARK(g(g(f(f(f(g(X'''''''), X2''''''''), X2''''''), X2'''')))) -> MARK(g(f(f(f(g(X'''''''), X2''''''''), X2''''''), X2'''')))
MARK(g(g(f(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2''''''), X2'''')))) -> MARK(g(f(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2''''''), X2'''')))
MARK(g(g(f(g(X''''''), X2'''')))) -> MARK(g(f(g(X''''''), X2'''')))
MARK(g(g(f(f(X1'''''', X2''''''), X2'''')))) -> MARK(g(f(f(X1'''''', X2''''''), X2'''')))
MARK(g(g(f(g(g(X'''''')), X2'''')))) -> MARK(g(f(g(g(X'''''')), X2'''')))
MARK(g(g(f(g(f(X1''''', X2'''''')), X2'''')))) -> MARK(g(f(g(f(X1''''', X2'''''')), X2'''')))
MARK(f(g(f(f(g(g(X'''''''')), X2''''''), X2'''')), X2)) -> MARK(g(f(f(g(g(X'''''''')), X2''''''), X2'''')))
MARK(f(g(f(f(g(f(X1''''''', X2'''''''')), X2''''''), X2'''')), X2)) -> MARK(g(f(f(g(f(X1''''''', X2'''''''')), X2''''''), X2'''')))
MARK(f(g(f(f(f(g(X'''''''), X2''''''''), X2''''''), X2'''')), X2)) -> MARK(g(f(f(f(g(X'''''''), X2''''''''), X2''''''), X2'''')))
MARK(f(g(f(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2''''''), X2'''')), X2)) -> MARK(g(f(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2''''''), X2'''')))
MARK(f(g(f(g(X''''''), X2'''')), X2)) -> MARK(g(f(g(X''''''), X2'''')))
MARK(g(f(f(f(g(g(X'''''''')), X2'''''''), X2'''''), X2'''))) -> MARK(f(f(f(g(g(X'''''''')), X2'''''''), X2'''''), X2'''))
MARK(g(f(f(f(g(f(X1''''''', X2'''''''')), X2'''0''), X2''''''), X2'''))) -> MARK(f(f(f(g(f(X1''''''', X2'''''''')), X2'''0''), X2''''''), X2'''))
MARK(g(f(f(f(f(g(X'''''''), X2''''''''), X2'''0''), X2''''''), X2'''))) -> MARK(f(f(f(f(g(X'''''''), X2''''''''), X2'''0''), X2''''''), X2'''))
MARK(g(f(f(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2'''''''), X2'''''), X2'''))) -> MARK(f(f(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2'''''''), X2'''''), X2'''))
MARK(g(f(f(g(X''''''), X2''''''), X2'''))) -> MARK(f(f(g(X''''''), X2''''''), X2'''))
MARK(g(f(f(f(X1'''''', X2'''''''), X2'''''), X2'''))) -> MARK(f(f(f(X1'''''', X2'''''''), X2'''''), X2'''))
MARK(g(f(f(g(g(X'''''')), X2''''''), X2'''))) -> MARK(f(f(g(g(X'''''')), X2''''''), X2'''))
MARK(g(f(f(g(f(X1'''''', X2''''''')), X2'''''), X2'''))) -> MARK(f(f(g(f(X1'''''', X2''''''')), X2'''''), X2'''))
MARK(g(f(f(f(g(X'''''), X2'''''''), X2'''''), X2'''))) -> MARK(f(f(f(g(X'''''), X2'''''''), X2'''''), X2'''))
MARK(g(f(f(f(f(X1'''''', X2'''''''), X2''0'''), X2'''''), X2'''))) -> MARK(f(f(f(f(X1'''''', X2'''''''), X2''0'''), X2'''''), X2'''))
MARK(g(f(f(g(g(X'''''')), X2'''''), X2''0))) -> MARK(f(f(g(g(X'''''')), X2'''''), X2''0))
MARK(g(f(f(g(f(X1'''''', X2''''''')), X2''''''), X2''0))) -> MARK(f(f(g(f(X1'''''', X2''''''')), X2''''''), X2''0))
MARK(g(f(f(f(g(X'''''), X2'''''''), X2''''''), X2''0))) -> MARK(f(f(f(g(X'''''), X2'''''''), X2''''''), X2''0))
MARK(g(f(f(f(f(X1'''''', X2'''''''), X2''0'''), X2'''''), X2''0))) -> MARK(f(f(f(f(X1'''''', X2'''''''), X2''0'''), X2'''''), X2''0))
MARK(g(f(f(g(X'''), X2'''''), X2''0))) -> MARK(f(f(g(X'''), X2'''''), X2''0))
MARK(f(f(f(g(g(X'''''')), X2'''''), X2''''), X2)) -> MARK(f(f(g(g(X'''''')), X2'''''), X2''''))
MARK(f(f(f(g(f(X1''''', X2'''''')), X2'''0), X2''''), X2)) -> MARK(f(f(g(f(X1''''', X2'''''')), X2'''0), X2''''))
MARK(f(f(f(f(g(X'''''), X2''''''), X2'''0), X2''''), X2)) -> MARK(f(f(f(g(X'''''), X2''''''), X2'''0), X2''''))
MARK(f(f(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''''), X2''''), X2)) -> MARK(f(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''''), X2''''))
MARK(g(f(f(f(X1''', X2'''''''), X2''''''), X2'''))) -> MARK(f(f(f(X1''', X2'''''''), X2''''''), X2'''))
MARK(g(f(f(g(g(X'''''')), X2''''), X2''))) -> MARK(f(f(g(g(X'''''')), X2''''), X2''))
MARK(f(g(f(f(X1'''''', X2''''''), X2'''')), X2)) -> MARK(g(f(f(X1'''''', X2''''''), X2'''')))
MARK(g(f(g(X''''), X2''))) -> MARK(f(g(X''''), X2''))
MARK(g(f(g(g(X'''')), X2''))) -> MARK(f(g(g(X'''')), X2''))
MARK(f(g(f(g(g(X'''''')), X2'''')), X2)) -> MARK(g(f(g(g(X'''''')), X2'''')))
MARK(g(f(g(f(X1''', X2'''')), X2''))) -> MARK(f(g(f(X1''', X2'''')), X2''))
MARK(f(g(f(g(f(X1''''', X2'''''')), X2'''')), X2)) -> MARK(g(f(g(f(X1''''', X2'''''')), X2'''')))
MARK(f(f(g(X''''), X2'''), X2)) -> MARK(f(g(X''''), X2'''))
MARK(g(f(f(g(f(X1''''', X2'''''')), X2''''), X2''))) -> MARK(f(f(g(f(X1''''', X2'''''')), X2''''), X2''))
MARK(g(g(f(f(g(X'''''), X2''''''), X2'''')))) -> MARK(g(f(f(g(X'''''), X2''''''), X2'''')))
MARK(f(g(g(X'''')), X2)) -> MARK(g(g(X'''')))
MARK(f(f(g(g(X'''')), X2'''), X2)) -> MARK(f(g(g(X'''')), X2'''))
MARK(g(f(f(g(X'''), X2''''), X2''))) -> MARK(f(f(g(X'''), X2''''), X2''))
MARK(f(g(f(f(g(X'''''), X2''''''), X2'''')), X2)) -> MARK(g(f(f(g(X'''''), X2''''''), X2'''')))
MARK(f(f(g(f(X1'''', X2''''')), X2''''), X2)) -> MARK(f(g(f(X1'''', X2''''')), X2''''))
MARK(f(f(f(X1'''', X2'''''), X2''''), X2)) -> MARK(f(f(X1'''', X2'''''), X2''''))
MARK(g(f(f(f(g(X'''''), X2''''''), X2''''), X2''))) -> MARK(f(f(f(g(X'''''), X2''''''), X2''''), X2''))
AF(g(g(f(f(f(X1''''', X2'''''''), X2''0'''), X2''''))), Y) -> MARK(g(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''')))
MARK(f(g(g(X'')), X2)) -> AF(g(g(mark(X''))), X2)
MARK(f(f(g(g(X'''')), X2''), X2)) -> MARK(f(g(g(X'''')), X2''))
MARK(f(f(f(g(X'''), X2'''''), X2''''), X2)) -> MARK(f(f(g(X'''), X2'''''), X2''''))
MARK(f(f(f(f(X1'''', X2'''''), X2''0'), X2'''), X2)) -> MARK(f(f(f(X1'''', X2'''''), X2''0'), X2'''))
MARK(g(f(f(f(f(X1''''', X2'''''''), X2''0'''), X2''''), X2''))) -> MARK(f(f(f(f(X1''''', X2'''''''), X2''0'''), X2''''), X2''))
MARK(f(g(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''')), X2)) -> MARK(g(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''')))
MARK(f(f(g(f(X1''', X2'''')), X2''), X2)) -> MARK(f(g(f(X1''', X2'''')), X2''))
MARK(f(f(f(g(X'''), X2''''), X2''), X2)) -> MARK(f(f(g(X'''), X2''''), X2''))
MARK(f(f(f(f(X1''', X2'''''), X2''0'), X2''), X2)) -> MARK(f(f(f(X1''', X2'''''), X2''0'), X2''))
MARK(g(f(f(f(X1''', X2'''''), X2''0'), X2''))) -> MARK(f(f(f(X1''', X2'''''), X2''0'), X2''))
MARK(g(g(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''')))) -> MARK(g(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''')))
MARK(g(g(g(X'''')))) -> MARK(g(g(X'''')))
AF(g(g(g(X''''))), Y) -> MARK(g(g(X'''')))
AF(g(g(g(X'))), Y) -> AF(g(g(mark(X'))), f(g(g(g(X'))), Y))
AF(g(g(f(X1', X2'))), Y) -> AF(g(af(mark(X1'), X2')), f(g(g(f(X1', X2'))), Y))
MARK(f(f(g(X'), X2''), X2)) -> AF(af(g(mark(X')), X2''), X2)
AF(g(f(f(g(X'''), X2''''), X2'')), Y) -> MARK(f(f(g(X'''), X2''''), X2''))
MARK(f(f(f(X1', X2'''), X2''), X2)) -> AF(af(af(mark(X1'), X2'''), X2''), X2)
AF(g(f(f(f(X1''', X2'''''), X2''0'), X2'')), Y) -> MARK(f(f(f(X1''', X2'''''), X2''0'), X2''))
AF(g(f(g(X'), X2')), Y) -> AF(af(g(mark(X')), X2'), f(g(f(g(X'), X2')), Y))
AF(g(f(f(X1'', X2''), X2')), Y) -> AF(af(af(mark(X1''), X2''), X2'), f(g(f(f(X1'', X2''), X2')), Y))
MARK(f(g(f(X1', X2'')), X2)) -> AF(g(af(mark(X1'), X2'')), X2)
AF(g(f(g(f(f(g(f(X1''''''''', X2'''''''''')), X2''''''''), X2'''''')), X2''')), Y) -> MARK(f(g(f(f(g(f(X1''''''''', X2'''''''''')), X2''''''''), X2'''''')), X2'''))

Rules:

af(g(X), Y) -> af(mark(X), f(g(X), Y))
af(X1, X2) -> f(X1, X2)
mark(f(X1, X2)) -> af(mark(X1), X2)
mark(g(X)) -> g(mark(X))

Strategy:

innermost

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

MARK(g(f(g(X''''), X2''))) -> MARK(f(g(X''''), X2''))
13 new Dependency Pairs are created:

MARK(g(f(g(f(X1''', X2''''')), X2''''))) -> MARK(f(g(f(X1''', X2''''')), X2''''))
MARK(g(f(g(g(X''''')), X2'''))) -> MARK(f(g(g(X''''')), X2'''))
MARK(g(f(g(g(X'''''')), X2'''))) -> MARK(f(g(g(X'''''')), X2'''))
MARK(g(f(g(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2'''''')), X2'''))) -> MARK(f(g(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2'''''')), X2'''))
MARK(g(f(g(f(f(g(X'''''''), X2''''''''), X2'''''')), X2'''))) -> MARK(f(g(f(f(g(X'''''''), X2''''''''), X2'''''')), X2'''))
MARK(g(f(g(f(g(f(X1''''''', X2'''''''')), X2'''''')), X2'''))) -> MARK(f(g(f(g(f(X1''''''', X2'''''''')), X2'''''')), X2'''))
MARK(g(f(g(f(g(g(X'''''''')), X2'''''')), X2'''))) -> MARK(f(g(f(g(g(X'''''''')), X2'''''')), X2'''))
MARK(g(f(g(f(f(X1'''''''', X2''''''''), X2'''''')), X2'''))) -> MARK(f(g(f(f(X1'''''''', X2''''''''), X2'''''')), X2'''))
MARK(g(f(g(f(g(X''''''''), X2'''''')), X2'''))) -> MARK(f(g(f(g(X''''''''), X2'''''')), X2'''))
MARK(g(f(g(f(f(f(f(X1''''''''', X2'''''''''''), X2''0'''''''), X2''''''''), X2'''''')), X2'''))) -> MARK(f(g(f(f(f(f(X1''''''''', X2'''''''''''), X2''0'''''''), X2''''''''), X2'''''')), X2'''))
MARK(g(f(g(f(f(f(g(X'''''''''), X2''''''''''), X2''''''''), X2'''''')), X2'''))) -> MARK(f(g(f(f(f(g(X'''''''''), X2''''''''''), X2''''''''), X2'''''')), X2'''))
MARK(g(f(g(f(f(g(f(X1''''''''', X2'''''''''')), X2''''''''), X2'''''')), X2'''))) -> MARK(f(g(f(f(g(f(X1''''''''', X2'''''''''')), X2''''''''), X2'''''')), X2'''))
MARK(g(f(g(f(f(g(g(X'''''''''')), X2''''''''), X2'''''')), X2'''))) -> MARK(f(g(f(f(g(g(X'''''''''')), X2''''''''), X2'''''')), X2'''))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 18`
`                 ↳Polynomial Ordering`

Dependency Pairs:

AF(g(f(g(f(f(g(f(X1''''''''', X2'''''''''')), X2''''''''), X2'''''')), X2''')), Y) -> MARK(f(g(f(f(g(f(X1''''''''', X2'''''''''')), X2''''''''), X2'''''')), X2'''))
AF(g(f(g(f(f(f(g(X'''''''''), X2''''''''''), X2''''''''), X2'''''')), X2''')), Y) -> MARK(f(g(f(f(f(g(X'''''''''), X2''''''''''), X2''''''''), X2'''''')), X2'''))
AF(g(f(g(f(f(f(f(X1''''''''', X2'''''''''''), X2''0'''''''), X2''''''''), X2'''''')), X2''')), Y) -> MARK(f(g(f(f(f(f(X1''''''''', X2'''''''''''), X2''0'''''''), X2''''''''), X2'''''')), X2'''))
AF(g(f(g(f(g(X''''''''), X2'''''')), X2''')), Y) -> MARK(f(g(f(g(X''''''''), X2'''''')), X2'''))
AF(g(f(g(f(f(X1'''''''', X2''''''''), X2'''''')), X2''')), Y) -> MARK(f(g(f(f(X1'''''''', X2''''''''), X2'''''')), X2'''))
AF(g(f(g(f(g(g(X'''''''')), X2'''''')), X2''')), Y) -> MARK(f(g(f(g(g(X'''''''')), X2'''''')), X2'''))
AF(g(f(g(f(g(f(X1''''''', X2'''''''')), X2'''''')), X2''')), Y) -> MARK(f(g(f(g(f(X1''''''', X2'''''''')), X2'''''')), X2'''))
AF(g(f(g(f(f(g(X'''''''), X2''''''''), X2'''''')), X2''')), Y) -> MARK(f(g(f(f(g(X'''''''), X2''''''''), X2'''''')), X2'''))
AF(g(f(g(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2'''''')), X2''')), Y) -> MARK(f(g(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2'''''')), X2'''))
AF(g(f(g(g(X'''''')), X2''')), Y) -> MARK(f(g(g(X'''''')), X2'''))
AF(g(f(g(g(X''''')), X2''')), Y) -> MARK(f(g(g(X''''')), X2'''))
AF(g(f(g(f(X1''', X2''''')), X2'''')), Y) -> MARK(f(g(f(X1''', X2''''')), X2''''))
AF(g(f(f(g(g(X'''''')), X2'''''), X2'''')), Y) -> MARK(f(f(g(g(X'''''')), X2'''''), X2''''))
AF(g(f(f(g(f(X1''''', X2'''''')), X2'''0), X2'''')), Y) -> MARK(f(f(g(f(X1''''', X2'''''')), X2'''0), X2''''))
AF(g(f(f(f(g(X'''''), X2''''''), X2'''0), X2'''')), Y) -> MARK(f(f(f(g(X'''''), X2''''''), X2'''0), X2''''))
AF(g(f(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''''), X2'''')), Y) -> MARK(f(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''''), X2''''))
AF(g(f(g(X''''), X2''')), Y) -> MARK(f(g(X''''), X2'''))
AF(g(f(f(X1'''', X2'''''), X2'''')), Y) -> MARK(f(f(X1'''', X2'''''), X2''''))
AF(g(f(g(g(X'''')), X2''')), Y) -> MARK(f(g(g(X'''')), X2'''))
AF(g(f(g(f(X1'''', X2''''')), X2'''')), Y) -> MARK(f(g(f(X1'''', X2''''')), X2''''))
AF(g(f(f(g(X'''), X2'''''), X2'''')), Y) -> MARK(f(f(g(X'''), X2'''''), X2''''))
AF(g(f(f(f(X1'''', X2'''''), X2''0'), X2''')), Y) -> MARK(f(f(f(X1'''', X2'''''), X2''0'), X2'''))
AF(g(g(f(f(g(g(X'''''''')), X2''''''), X2''''))), Y) -> MARK(g(f(f(g(g(X'''''''')), X2''''''), X2'''')))
AF(g(g(f(f(g(f(X1''''''', X2'''''''')), X2''''''), X2''''))), Y) -> MARK(g(f(f(g(f(X1''''''', X2'''''''')), X2''''''), X2'''')))
AF(g(g(f(f(f(g(X'''''''), X2''''''''), X2''''''), X2''''))), Y) -> MARK(g(f(f(f(g(X'''''''), X2''''''''), X2''''''), X2'''')))
AF(g(g(f(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2''''''), X2''''))), Y) -> MARK(g(f(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2''''''), X2'''')))
AF(g(g(f(g(X''''''), X2''''))), Y) -> MARK(g(f(g(X''''''), X2'''')))
AF(g(g(f(f(X1'''''', X2''''''), X2''''))), Y) -> MARK(g(f(f(X1'''''', X2''''''), X2'''')))
AF(g(g(f(g(g(X'''''')), X2''''))), Y) -> MARK(g(f(g(g(X'''''')), X2'''')))
AF(g(g(f(g(f(X1''''', X2'''''')), X2''''))), Y) -> MARK(g(f(g(f(X1''''', X2'''''')), X2'''')))
AF(g(g(f(f(g(X'''''), X2''''''), X2''''))), Y) -> MARK(g(f(f(g(X'''''), X2''''''), X2'''')))
MARK(g(g(f(f(g(g(X'''''''')), X2''''''), X2'''')))) -> MARK(g(f(f(g(g(X'''''''')), X2''''''), X2'''')))
MARK(g(g(f(f(g(f(X1''''''', X2'''''''')), X2''''''), X2'''')))) -> MARK(g(f(f(g(f(X1''''''', X2'''''''')), X2''''''), X2'''')))
MARK(g(g(f(f(f(g(X'''''''), X2''''''''), X2''''''), X2'''')))) -> MARK(g(f(f(f(g(X'''''''), X2''''''''), X2''''''), X2'''')))
MARK(g(g(f(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2''''''), X2'''')))) -> MARK(g(f(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2''''''), X2'''')))
MARK(g(g(f(g(X''''''), X2'''')))) -> MARK(g(f(g(X''''''), X2'''')))
MARK(g(g(f(f(X1'''''', X2''''''), X2'''')))) -> MARK(g(f(f(X1'''''', X2''''''), X2'''')))
MARK(g(g(f(g(g(X'''''')), X2'''')))) -> MARK(g(f(g(g(X'''''')), X2'''')))
MARK(g(g(f(g(f(X1''''', X2'''''')), X2'''')))) -> MARK(g(f(g(f(X1''''', X2'''''')), X2'''')))
MARK(g(f(g(f(f(g(g(X'''''''''')), X2''''''''), X2'''''')), X2'''))) -> MARK(f(g(f(f(g(g(X'''''''''')), X2''''''''), X2'''''')), X2'''))
MARK(g(f(g(f(f(g(f(X1''''''''', X2'''''''''')), X2''''''''), X2'''''')), X2'''))) -> MARK(f(g(f(f(g(f(X1''''''''', X2'''''''''')), X2''''''''), X2'''''')), X2'''))
MARK(g(f(g(f(f(f(g(X'''''''''), X2''''''''''), X2''''''''), X2'''''')), X2'''))) -> MARK(f(g(f(f(f(g(X'''''''''), X2''''''''''), X2''''''''), X2'''''')), X2'''))
MARK(g(f(g(f(f(f(f(X1''''''''', X2'''''''''''), X2''0'''''''), X2''''''''), X2'''''')), X2'''))) -> MARK(f(g(f(f(f(f(X1''''''''', X2'''''''''''), X2''0'''''''), X2''''''''), X2'''''')), X2'''))
MARK(g(f(g(f(g(X''''''''), X2'''''')), X2'''))) -> MARK(f(g(f(g(X''''''''), X2'''''')), X2'''))
MARK(g(f(g(f(f(X1'''''''', X2''''''''), X2'''''')), X2'''))) -> MARK(f(g(f(f(X1'''''''', X2''''''''), X2'''''')), X2'''))
MARK(g(f(g(f(g(g(X'''''''')), X2'''''')), X2'''))) -> MARK(f(g(f(g(g(X'''''''')), X2'''''')), X2'''))
MARK(g(f(g(f(g(f(X1''''''', X2'''''''')), X2'''''')), X2'''))) -> MARK(f(g(f(g(f(X1''''''', X2'''''''')), X2'''''')), X2'''))
MARK(g(f(g(f(f(g(X'''''''), X2''''''''), X2'''''')), X2'''))) -> MARK(f(g(f(f(g(X'''''''), X2''''''''), X2'''''')), X2'''))
MARK(g(f(g(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2'''''')), X2'''))) -> MARK(f(g(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2'''''')), X2'''))
MARK(f(g(f(f(g(g(X'''''''')), X2''''''), X2'''')), X2)) -> MARK(g(f(f(g(g(X'''''''')), X2''''''), X2'''')))
MARK(f(g(f(f(g(f(X1''''''', X2'''''''')), X2''''''), X2'''')), X2)) -> MARK(g(f(f(g(f(X1''''''', X2'''''''')), X2''''''), X2'''')))
MARK(f(g(f(f(f(g(X'''''''), X2''''''''), X2''''''), X2'''')), X2)) -> MARK(g(f(f(f(g(X'''''''), X2''''''''), X2''''''), X2'''')))
MARK(f(g(f(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2''''''), X2'''')), X2)) -> MARK(g(f(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2''''''), X2'''')))
MARK(g(f(g(f(X1''', X2''''')), X2''''))) -> MARK(f(g(f(X1''', X2''''')), X2''''))
MARK(f(g(f(g(X''''''), X2'''')), X2)) -> MARK(g(f(g(X''''''), X2'''')))
MARK(g(f(f(f(g(g(X'''''''')), X2'''''''), X2'''''), X2'''))) -> MARK(f(f(f(g(g(X'''''''')), X2'''''''), X2'''''), X2'''))
MARK(g(f(f(f(g(f(X1''''''', X2'''''''')), X2'''0''), X2''''''), X2'''))) -> MARK(f(f(f(g(f(X1''''''', X2'''''''')), X2'''0''), X2''''''), X2'''))
MARK(g(f(f(f(f(g(X'''''''), X2''''''''), X2'''0''), X2''''''), X2'''))) -> MARK(f(f(f(f(g(X'''''''), X2''''''''), X2'''0''), X2''''''), X2'''))
MARK(g(f(f(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2'''''''), X2'''''), X2'''))) -> MARK(f(f(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2'''''''), X2'''''), X2'''))
MARK(g(f(f(g(X''''''), X2''''''), X2'''))) -> MARK(f(f(g(X''''''), X2''''''), X2'''))
MARK(g(f(f(f(X1'''''', X2'''''''), X2'''''), X2'''))) -> MARK(f(f(f(X1'''''', X2'''''''), X2'''''), X2'''))
MARK(g(f(f(g(g(X'''''')), X2''''''), X2'''))) -> MARK(f(f(g(g(X'''''')), X2''''''), X2'''))
MARK(g(f(f(g(f(X1'''''', X2''''''')), X2'''''), X2'''))) -> MARK(f(f(g(f(X1'''''', X2''''''')), X2'''''), X2'''))
MARK(g(f(f(f(g(X'''''), X2'''''''), X2'''''), X2'''))) -> MARK(f(f(f(g(X'''''), X2'''''''), X2'''''), X2'''))
MARK(g(f(f(f(f(X1'''''', X2'''''''), X2''0'''), X2'''''), X2'''))) -> MARK(f(f(f(f(X1'''''', X2'''''''), X2''0'''), X2'''''), X2'''))
MARK(g(f(f(g(g(X'''''')), X2'''''), X2''0))) -> MARK(f(f(g(g(X'''''')), X2'''''), X2''0))
MARK(g(f(f(g(f(X1'''''', X2''''''')), X2''''''), X2''0))) -> MARK(f(f(g(f(X1'''''', X2''''''')), X2''''''), X2''0))
MARK(g(f(f(f(g(X'''''), X2'''''''), X2''''''), X2''0))) -> MARK(f(f(f(g(X'''''), X2'''''''), X2''''''), X2''0))
MARK(g(f(f(f(f(X1'''''', X2'''''''), X2''0'''), X2'''''), X2''0))) -> MARK(f(f(f(f(X1'''''', X2'''''''), X2''0'''), X2'''''), X2''0))
MARK(g(f(f(g(X'''), X2'''''), X2''0))) -> MARK(f(f(g(X'''), X2'''''), X2''0))
MARK(f(f(f(g(g(X'''''')), X2'''''), X2''''), X2)) -> MARK(f(f(g(g(X'''''')), X2'''''), X2''''))
MARK(f(f(f(g(f(X1''''', X2'''''')), X2'''0), X2''''), X2)) -> MARK(f(f(g(f(X1''''', X2'''''')), X2'''0), X2''''))
MARK(f(f(f(f(g(X'''''), X2''''''), X2'''0), X2''''), X2)) -> MARK(f(f(f(g(X'''''), X2''''''), X2'''0), X2''''))
MARK(f(f(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''''), X2''''), X2)) -> MARK(f(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''''), X2''''))
MARK(g(f(f(f(X1''', X2'''''''), X2''''''), X2'''))) -> MARK(f(f(f(X1''', X2'''''''), X2''''''), X2'''))
MARK(g(f(f(g(g(X'''''')), X2''''), X2''))) -> MARK(f(f(g(g(X'''''')), X2''''), X2''))
MARK(f(g(f(f(X1'''''', X2''''''), X2'''')), X2)) -> MARK(g(f(f(X1'''''', X2''''''), X2'''')))
MARK(g(f(g(g(X'''''')), X2'''))) -> MARK(f(g(g(X'''''')), X2'''))
MARK(g(f(g(g(X''''')), X2'''))) -> MARK(f(g(g(X''''')), X2'''))
MARK(g(f(g(g(X'''')), X2''))) -> MARK(f(g(g(X'''')), X2''))
MARK(f(g(f(g(g(X'''''')), X2'''')), X2)) -> MARK(g(f(g(g(X'''''')), X2'''')))
MARK(g(f(g(f(X1''', X2'''')), X2''))) -> MARK(f(g(f(X1''', X2'''')), X2''))
MARK(f(g(f(g(f(X1''''', X2'''''')), X2'''')), X2)) -> MARK(g(f(g(f(X1''''', X2'''''')), X2'''')))
MARK(f(f(g(X''''), X2'''), X2)) -> MARK(f(g(X''''), X2'''))
MARK(g(f(f(g(f(X1''''', X2'''''')), X2''''), X2''))) -> MARK(f(f(g(f(X1''''', X2'''''')), X2''''), X2''))
MARK(g(g(f(f(g(X'''''), X2''''''), X2'''')))) -> MARK(g(f(f(g(X'''''), X2''''''), X2'''')))
MARK(f(g(g(X'''')), X2)) -> MARK(g(g(X'''')))
MARK(f(f(g(g(X'''')), X2'''), X2)) -> MARK(f(g(g(X'''')), X2'''))
MARK(g(f(f(g(X'''), X2''''), X2''))) -> MARK(f(f(g(X'''), X2''''), X2''))
MARK(f(g(f(f(g(X'''''), X2''''''), X2'''')), X2)) -> MARK(g(f(f(g(X'''''), X2''''''), X2'''')))
MARK(f(f(g(f(X1'''', X2''''')), X2''''), X2)) -> MARK(f(g(f(X1'''', X2''''')), X2''''))
MARK(f(f(f(X1'''', X2'''''), X2''''), X2)) -> MARK(f(f(X1'''', X2'''''), X2''''))
MARK(g(f(f(f(g(X'''''), X2''''''), X2''''), X2''))) -> MARK(f(f(f(g(X'''''), X2''''''), X2''''), X2''))
AF(g(g(f(f(f(X1''''', X2'''''''), X2''0'''), X2''''))), Y) -> MARK(g(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''')))
MARK(f(g(g(X'')), X2)) -> AF(g(g(mark(X''))), X2)
MARK(f(f(g(g(X'''')), X2''), X2)) -> MARK(f(g(g(X'''')), X2''))
MARK(f(f(f(g(X'''), X2'''''), X2''''), X2)) -> MARK(f(f(g(X'''), X2'''''), X2''''))
MARK(f(f(f(f(X1'''', X2'''''), X2''0'), X2'''), X2)) -> MARK(f(f(f(X1'''', X2'''''), X2''0'), X2'''))
MARK(g(f(f(f(f(X1''''', X2'''''''), X2''0'''), X2''''), X2''))) -> MARK(f(f(f(f(X1''''', X2'''''''), X2''0'''), X2''''), X2''))
MARK(f(g(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''')), X2)) -> MARK(g(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''')))
MARK(f(f(g(f(X1''', X2'''')), X2''), X2)) -> MARK(f(g(f(X1''', X2'''')), X2''))
MARK(f(f(f(g(X'''), X2''''), X2''), X2)) -> MARK(f(f(g(X'''), X2''''), X2''))
MARK(f(f(f(f(X1''', X2'''''), X2''0'), X2''), X2)) -> MARK(f(f(f(X1''', X2'''''), X2''0'), X2''))
MARK(g(f(f(f(X1''', X2'''''), X2''0'), X2''))) -> MARK(f(f(f(X1''', X2'''''), X2''0'), X2''))
MARK(g(g(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''')))) -> MARK(g(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''')))
MARK(g(g(g(X'''')))) -> MARK(g(g(X'''')))
AF(g(g(g(X''''))), Y) -> MARK(g(g(X'''')))
AF(g(g(g(X'))), Y) -> AF(g(g(mark(X'))), f(g(g(g(X'))), Y))
AF(g(g(f(X1', X2'))), Y) -> AF(g(af(mark(X1'), X2')), f(g(g(f(X1', X2'))), Y))
MARK(f(f(g(X'), X2''), X2)) -> AF(af(g(mark(X')), X2''), X2)
AF(g(f(f(g(X'''), X2''''), X2'')), Y) -> MARK(f(f(g(X'''), X2''''), X2''))
MARK(f(f(f(X1', X2'''), X2''), X2)) -> AF(af(af(mark(X1'), X2'''), X2''), X2)
AF(g(f(f(f(X1''', X2'''''), X2''0'), X2'')), Y) -> MARK(f(f(f(X1''', X2'''''), X2''0'), X2''))
AF(g(f(g(X'), X2')), Y) -> AF(af(g(mark(X')), X2'), f(g(f(g(X'), X2')), Y))
AF(g(f(f(X1'', X2''), X2')), Y) -> AF(af(af(mark(X1''), X2''), X2'), f(g(f(f(X1'', X2''), X2')), Y))
MARK(f(g(f(X1', X2'')), X2)) -> AF(g(af(mark(X1'), X2'')), X2)
AF(g(f(g(f(f(g(g(X'''''''''')), X2''''''''), X2'''''')), X2''')), Y) -> MARK(f(g(f(f(g(g(X'''''''''')), X2''''''''), X2'''''')), X2'''))

Rules:

af(g(X), Y) -> af(mark(X), f(g(X), Y))
af(X1, X2) -> f(X1, X2)
mark(f(X1, X2)) -> af(mark(X1), X2)
mark(g(X)) -> g(mark(X))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

AF(g(f(g(f(f(g(f(X1''''''''', X2'''''''''')), X2''''''''), X2'''''')), X2''')), Y) -> MARK(f(g(f(f(g(f(X1''''''''', X2'''''''''')), X2''''''''), X2'''''')), X2'''))
AF(g(f(g(f(f(f(g(X'''''''''), X2''''''''''), X2''''''''), X2'''''')), X2''')), Y) -> MARK(f(g(f(f(f(g(X'''''''''), X2''''''''''), X2''''''''), X2'''''')), X2'''))
AF(g(f(g(f(f(f(f(X1''''''''', X2'''''''''''), X2''0'''''''), X2''''''''), X2'''''')), X2''')), Y) -> MARK(f(g(f(f(f(f(X1''''''''', X2'''''''''''), X2''0'''''''), X2''''''''), X2'''''')), X2'''))
AF(g(f(g(f(g(X''''''''), X2'''''')), X2''')), Y) -> MARK(f(g(f(g(X''''''''), X2'''''')), X2'''))
AF(g(f(g(f(f(X1'''''''', X2''''''''), X2'''''')), X2''')), Y) -> MARK(f(g(f(f(X1'''''''', X2''''''''), X2'''''')), X2'''))
AF(g(f(g(f(g(g(X'''''''')), X2'''''')), X2''')), Y) -> MARK(f(g(f(g(g(X'''''''')), X2'''''')), X2'''))
AF(g(f(g(f(g(f(X1''''''', X2'''''''')), X2'''''')), X2''')), Y) -> MARK(f(g(f(g(f(X1''''''', X2'''''''')), X2'''''')), X2'''))
AF(g(f(g(f(f(g(X'''''''), X2''''''''), X2'''''')), X2''')), Y) -> MARK(f(g(f(f(g(X'''''''), X2''''''''), X2'''''')), X2'''))
AF(g(f(g(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2'''''')), X2''')), Y) -> MARK(f(g(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2'''''')), X2'''))
AF(g(f(g(g(X'''''')), X2''')), Y) -> MARK(f(g(g(X'''''')), X2'''))
AF(g(f(g(g(X''''')), X2''')), Y) -> MARK(f(g(g(X''''')), X2'''))
AF(g(f(g(f(X1''', X2''''')), X2'''')), Y) -> MARK(f(g(f(X1''', X2''''')), X2''''))
AF(g(f(f(g(g(X'''''')), X2'''''), X2'''')), Y) -> MARK(f(f(g(g(X'''''')), X2'''''), X2''''))
AF(g(f(f(g(f(X1''''', X2'''''')), X2'''0), X2'''')), Y) -> MARK(f(f(g(f(X1''''', X2'''''')), X2'''0), X2''''))
AF(g(f(f(f(g(X'''''), X2''''''), X2'''0), X2'''')), Y) -> MARK(f(f(f(g(X'''''), X2''''''), X2'''0), X2''''))
AF(g(f(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''''), X2'''')), Y) -> MARK(f(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''''), X2''''))
AF(g(f(g(X''''), X2''')), Y) -> MARK(f(g(X''''), X2'''))
AF(g(f(f(X1'''', X2'''''), X2'''')), Y) -> MARK(f(f(X1'''', X2'''''), X2''''))
AF(g(f(g(g(X'''')), X2''')), Y) -> MARK(f(g(g(X'''')), X2'''))
AF(g(f(g(f(X1'''', X2''''')), X2'''')), Y) -> MARK(f(g(f(X1'''', X2''''')), X2''''))
AF(g(f(f(g(X'''), X2'''''), X2'''')), Y) -> MARK(f(f(g(X'''), X2'''''), X2''''))
AF(g(f(f(f(X1'''', X2'''''), X2''0'), X2''')), Y) -> MARK(f(f(f(X1'''', X2'''''), X2''0'), X2'''))
AF(g(g(f(f(g(g(X'''''''')), X2''''''), X2''''))), Y) -> MARK(g(f(f(g(g(X'''''''')), X2''''''), X2'''')))
AF(g(g(f(f(g(f(X1''''''', X2'''''''')), X2''''''), X2''''))), Y) -> MARK(g(f(f(g(f(X1''''''', X2'''''''')), X2''''''), X2'''')))
AF(g(g(f(f(f(g(X'''''''), X2''''''''), X2''''''), X2''''))), Y) -> MARK(g(f(f(f(g(X'''''''), X2''''''''), X2''''''), X2'''')))
AF(g(g(f(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2''''''), X2''''))), Y) -> MARK(g(f(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2''''''), X2'''')))
AF(g(g(f(g(X''''''), X2''''))), Y) -> MARK(g(f(g(X''''''), X2'''')))
AF(g(g(f(f(X1'''''', X2''''''), X2''''))), Y) -> MARK(g(f(f(X1'''''', X2''''''), X2'''')))
AF(g(g(f(g(g(X'''''')), X2''''))), Y) -> MARK(g(f(g(g(X'''''')), X2'''')))
AF(g(g(f(g(f(X1''''', X2'''''')), X2''''))), Y) -> MARK(g(f(g(f(X1''''', X2'''''')), X2'''')))
AF(g(g(f(f(g(X'''''), X2''''''), X2''''))), Y) -> MARK(g(f(f(g(X'''''), X2''''''), X2'''')))
MARK(g(g(f(f(g(g(X'''''''')), X2''''''), X2'''')))) -> MARK(g(f(f(g(g(X'''''''')), X2''''''), X2'''')))
MARK(g(g(f(f(g(f(X1''''''', X2'''''''')), X2''''''), X2'''')))) -> MARK(g(f(f(g(f(X1''''''', X2'''''''')), X2''''''), X2'''')))
MARK(g(g(f(f(f(g(X'''''''), X2''''''''), X2''''''), X2'''')))) -> MARK(g(f(f(f(g(X'''''''), X2''''''''), X2''''''), X2'''')))
MARK(g(g(f(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2''''''), X2'''')))) -> MARK(g(f(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2''''''), X2'''')))
MARK(g(g(f(g(X''''''), X2'''')))) -> MARK(g(f(g(X''''''), X2'''')))
MARK(g(g(f(f(X1'''''', X2''''''), X2'''')))) -> MARK(g(f(f(X1'''''', X2''''''), X2'''')))
MARK(g(g(f(g(g(X'''''')), X2'''')))) -> MARK(g(f(g(g(X'''''')), X2'''')))
MARK(g(g(f(g(f(X1''''', X2'''''')), X2'''')))) -> MARK(g(f(g(f(X1''''', X2'''''')), X2'''')))
MARK(g(f(g(f(f(g(g(X'''''''''')), X2''''''''), X2'''''')), X2'''))) -> MARK(f(g(f(f(g(g(X'''''''''')), X2''''''''), X2'''''')), X2'''))
MARK(g(f(g(f(f(g(f(X1''''''''', X2'''''''''')), X2''''''''), X2'''''')), X2'''))) -> MARK(f(g(f(f(g(f(X1''''''''', X2'''''''''')), X2''''''''), X2'''''')), X2'''))
MARK(g(f(g(f(f(f(g(X'''''''''), X2''''''''''), X2''''''''), X2'''''')), X2'''))) -> MARK(f(g(f(f(f(g(X'''''''''), X2''''''''''), X2''''''''), X2'''''')), X2'''))
MARK(g(f(g(f(f(f(f(X1''''''''', X2'''''''''''), X2''0'''''''), X2''''''''), X2'''''')), X2'''))) -> MARK(f(g(f(f(f(f(X1''''''''', X2'''''''''''), X2''0'''''''), X2''''''''), X2'''''')), X2'''))
MARK(g(f(g(f(g(X''''''''), X2'''''')), X2'''))) -> MARK(f(g(f(g(X''''''''), X2'''''')), X2'''))
MARK(g(f(g(f(f(X1'''''''', X2''''''''), X2'''''')), X2'''))) -> MARK(f(g(f(f(X1'''''''', X2''''''''), X2'''''')), X2'''))
MARK(g(f(g(f(g(g(X'''''''')), X2'''''')), X2'''))) -> MARK(f(g(f(g(g(X'''''''')), X2'''''')), X2'''))
MARK(g(f(g(f(g(f(X1''''''', X2'''''''')), X2'''''')), X2'''))) -> MARK(f(g(f(g(f(X1''''''', X2'''''''')), X2'''''')), X2'''))
MARK(g(f(g(f(f(g(X'''''''), X2''''''''), X2'''''')), X2'''))) -> MARK(f(g(f(f(g(X'''''''), X2''''''''), X2'''''')), X2'''))
MARK(g(f(g(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2'''''')), X2'''))) -> MARK(f(g(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2'''''')), X2'''))
MARK(g(f(g(f(X1''', X2''''')), X2''''))) -> MARK(f(g(f(X1''', X2''''')), X2''''))
MARK(g(f(f(f(g(g(X'''''''')), X2'''''''), X2'''''), X2'''))) -> MARK(f(f(f(g(g(X'''''''')), X2'''''''), X2'''''), X2'''))
MARK(g(f(f(f(g(f(X1''''''', X2'''''''')), X2'''0''), X2''''''), X2'''))) -> MARK(f(f(f(g(f(X1''''''', X2'''''''')), X2'''0''), X2''''''), X2'''))
MARK(g(f(f(f(f(g(X'''''''), X2''''''''), X2'''0''), X2''''''), X2'''))) -> MARK(f(f(f(f(g(X'''''''), X2''''''''), X2'''0''), X2''''''), X2'''))
MARK(g(f(f(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2'''''''), X2'''''), X2'''))) -> MARK(f(f(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2'''''''), X2'''''), X2'''))
MARK(g(f(f(g(X''''''), X2''''''), X2'''))) -> MARK(f(f(g(X''''''), X2''''''), X2'''))
MARK(g(f(f(f(X1'''''', X2'''''''), X2'''''), X2'''))) -> MARK(f(f(f(X1'''''', X2'''''''), X2'''''), X2'''))
MARK(g(f(f(g(g(X'''''')), X2''''''), X2'''))) -> MARK(f(f(g(g(X'''''')), X2''''''), X2'''))
MARK(g(f(f(g(f(X1'''''', X2''''''')), X2'''''), X2'''))) -> MARK(f(f(g(f(X1'''''', X2''''''')), X2'''''), X2'''))
MARK(g(f(f(f(g(X'''''), X2'''''''), X2'''''), X2'''))) -> MARK(f(f(f(g(X'''''), X2'''''''), X2'''''), X2'''))
MARK(g(f(f(f(f(X1'''''', X2'''''''), X2''0'''), X2'''''), X2'''))) -> MARK(f(f(f(f(X1'''''', X2'''''''), X2''0'''), X2'''''), X2'''))
MARK(g(f(f(g(g(X'''''')), X2'''''), X2''0))) -> MARK(f(f(g(g(X'''''')), X2'''''), X2''0))
MARK(g(f(f(g(f(X1'''''', X2''''''')), X2''''''), X2''0))) -> MARK(f(f(g(f(X1'''''', X2''''''')), X2''''''), X2''0))
MARK(g(f(f(f(g(X'''''), X2'''''''), X2''''''), X2''0))) -> MARK(f(f(f(g(X'''''), X2'''''''), X2''''''), X2''0))
MARK(g(f(f(f(f(X1'''''', X2'''''''), X2''0'''), X2'''''), X2''0))) -> MARK(f(f(f(f(X1'''''', X2'''''''), X2''0'''), X2'''''), X2''0))
MARK(g(f(f(g(X'''), X2'''''), X2''0))) -> MARK(f(f(g(X'''), X2'''''), X2''0))
MARK(g(f(f(f(X1''', X2'''''''), X2''''''), X2'''))) -> MARK(f(f(f(X1''', X2'''''''), X2''''''), X2'''))
MARK(g(f(f(g(g(X'''''')), X2''''), X2''))) -> MARK(f(f(g(g(X'''''')), X2''''), X2''))
MARK(g(f(g(g(X'''''')), X2'''))) -> MARK(f(g(g(X'''''')), X2'''))
MARK(g(f(g(g(X''''')), X2'''))) -> MARK(f(g(g(X''''')), X2'''))
MARK(g(f(g(g(X'''')), X2''))) -> MARK(f(g(g(X'''')), X2''))
MARK(g(f(g(f(X1''', X2'''')), X2''))) -> MARK(f(g(f(X1''', X2'''')), X2''))
MARK(g(f(f(g(f(X1''''', X2'''''')), X2''''), X2''))) -> MARK(f(f(g(f(X1''''', X2'''''')), X2''''), X2''))
MARK(g(g(f(f(g(X'''''), X2''''''), X2'''')))) -> MARK(g(f(f(g(X'''''), X2''''''), X2'''')))
MARK(g(f(f(g(X'''), X2''''), X2''))) -> MARK(f(f(g(X'''), X2''''), X2''))
MARK(g(f(f(f(g(X'''''), X2''''''), X2''''), X2''))) -> MARK(f(f(f(g(X'''''), X2''''''), X2''''), X2''))
AF(g(g(f(f(f(X1''''', X2'''''''), X2''0'''), X2''''))), Y) -> MARK(g(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''')))
MARK(g(f(f(f(f(X1''''', X2'''''''), X2''0'''), X2''''), X2''))) -> MARK(f(f(f(f(X1''''', X2'''''''), X2''0'''), X2''''), X2''))
MARK(g(f(f(f(X1''', X2'''''), X2''0'), X2''))) -> MARK(f(f(f(X1''', X2'''''), X2''0'), X2''))
MARK(g(g(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''')))) -> MARK(g(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''')))
MARK(g(g(g(X'''')))) -> MARK(g(g(X'''')))
AF(g(g(g(X''''))), Y) -> MARK(g(g(X'''')))
AF(g(g(g(X'))), Y) -> AF(g(g(mark(X'))), f(g(g(g(X'))), Y))
AF(g(g(f(X1', X2'))), Y) -> AF(g(af(mark(X1'), X2')), f(g(g(f(X1', X2'))), Y))
AF(g(f(f(g(X'''), X2''''), X2'')), Y) -> MARK(f(f(g(X'''), X2''''), X2''))
AF(g(f(f(f(X1''', X2'''''), X2''0'), X2'')), Y) -> MARK(f(f(f(X1''', X2'''''), X2''0'), X2''))
AF(g(f(g(X'), X2')), Y) -> AF(af(g(mark(X')), X2'), f(g(f(g(X'), X2')), Y))
AF(g(f(f(X1'', X2''), X2')), Y) -> AF(af(af(mark(X1''), X2''), X2'), f(g(f(f(X1'', X2''), X2')), Y))
AF(g(f(g(f(f(g(g(X'''''''''')), X2''''''''), X2'''''')), X2''')), Y) -> MARK(f(g(f(f(g(g(X'''''''''')), X2''''''''), X2'''''')), X2'''))

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

mark(f(X1, X2)) -> af(mark(X1), X2)
mark(g(X)) -> g(mark(X))
af(g(X), Y) -> af(mark(X), f(g(X), Y))
af(X1, X2) -> f(X1, X2)

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(MARK(x1)) =  x1 POL(g(x1)) =  1 + x1 POL(A__F(x1, x2)) =  x1 POL(mark(x1)) =  x1 POL(f(x1, x2)) =  x1 POL(a__f(x1, x2)) =  x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 19`
`                 ↳Dependency Graph`

Dependency Pairs:

MARK(f(g(f(f(g(g(X'''''''')), X2''''''), X2'''')), X2)) -> MARK(g(f(f(g(g(X'''''''')), X2''''''), X2'''')))
MARK(f(g(f(f(g(f(X1''''''', X2'''''''')), X2''''''), X2'''')), X2)) -> MARK(g(f(f(g(f(X1''''''', X2'''''''')), X2''''''), X2'''')))
MARK(f(g(f(f(f(g(X'''''''), X2''''''''), X2''''''), X2'''')), X2)) -> MARK(g(f(f(f(g(X'''''''), X2''''''''), X2''''''), X2'''')))
MARK(f(g(f(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2''''''), X2'''')), X2)) -> MARK(g(f(f(f(f(X1''''''', X2'''''''''), X2''0'''''), X2''''''), X2'''')))
MARK(f(g(f(g(X''''''), X2'''')), X2)) -> MARK(g(f(g(X''''''), X2'''')))
MARK(f(f(f(g(g(X'''''')), X2'''''), X2''''), X2)) -> MARK(f(f(g(g(X'''''')), X2'''''), X2''''))
MARK(f(f(f(g(f(X1''''', X2'''''')), X2'''0), X2''''), X2)) -> MARK(f(f(g(f(X1''''', X2'''''')), X2'''0), X2''''))
MARK(f(f(f(f(g(X'''''), X2''''''), X2'''0), X2''''), X2)) -> MARK(f(f(f(g(X'''''), X2''''''), X2'''0), X2''''))
MARK(f(f(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''''), X2''''), X2)) -> MARK(f(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''''), X2''''))
MARK(f(g(f(f(X1'''''', X2''''''), X2'''')), X2)) -> MARK(g(f(f(X1'''''', X2''''''), X2'''')))
MARK(f(g(f(g(g(X'''''')), X2'''')), X2)) -> MARK(g(f(g(g(X'''''')), X2'''')))
MARK(f(g(f(g(f(X1''''', X2'''''')), X2'''')), X2)) -> MARK(g(f(g(f(X1''''', X2'''''')), X2'''')))
MARK(f(f(g(X''''), X2'''), X2)) -> MARK(f(g(X''''), X2'''))
MARK(f(g(g(X'''')), X2)) -> MARK(g(g(X'''')))
MARK(f(f(g(g(X'''')), X2'''), X2)) -> MARK(f(g(g(X'''')), X2'''))
MARK(f(g(f(f(g(X'''''), X2''''''), X2'''')), X2)) -> MARK(g(f(f(g(X'''''), X2''''''), X2'''')))
MARK(f(f(g(f(X1'''', X2''''')), X2''''), X2)) -> MARK(f(g(f(X1'''', X2''''')), X2''''))
MARK(f(f(f(X1'''', X2'''''), X2''''), X2)) -> MARK(f(f(X1'''', X2'''''), X2''''))
MARK(f(g(g(X'')), X2)) -> AF(g(g(mark(X''))), X2)
MARK(f(f(g(g(X'''')), X2''), X2)) -> MARK(f(g(g(X'''')), X2''))
MARK(f(f(f(g(X'''), X2'''''), X2''''), X2)) -> MARK(f(f(g(X'''), X2'''''), X2''''))
MARK(f(f(f(f(X1'''', X2'''''), X2''0'), X2'''), X2)) -> MARK(f(f(f(X1'''', X2'''''), X2''0'), X2'''))
MARK(f(g(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''')), X2)) -> MARK(g(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''')))
MARK(f(f(g(f(X1''', X2'''')), X2''), X2)) -> MARK(f(g(f(X1''', X2'''')), X2''))
MARK(f(f(f(g(X'''), X2''''), X2''), X2)) -> MARK(f(f(g(X'''), X2''''), X2''))
MARK(f(f(f(f(X1''', X2'''''), X2''0'), X2''), X2)) -> MARK(f(f(f(X1''', X2'''''), X2''0'), X2''))
MARK(f(f(g(X'), X2''), X2)) -> AF(af(g(mark(X')), X2''), X2)
MARK(f(f(f(X1', X2'''), X2''), X2)) -> AF(af(af(mark(X1'), X2'''), X2''), X2)
MARK(f(g(f(X1', X2'')), X2)) -> AF(g(af(mark(X1'), X2'')), X2)

Rules:

af(g(X), Y) -> af(mark(X), f(g(X), Y))
af(X1, X2) -> f(X1, X2)
mark(f(X1, X2)) -> af(mark(X1), X2)
mark(g(X)) -> g(mark(X))

Strategy:

innermost

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

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 20`
`                 ↳Polynomial Ordering`

Dependency Pairs:

MARK(f(f(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''''), X2''''), X2)) -> MARK(f(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''''), X2''''))
MARK(f(f(f(f(X1'''', X2'''''), X2''0'), X2'''), X2)) -> MARK(f(f(f(X1'''', X2'''''), X2''0'), X2'''))
MARK(f(f(f(f(X1''', X2'''''), X2''0'), X2''), X2)) -> MARK(f(f(f(X1''', X2'''''), X2''0'), X2''))
MARK(f(f(f(X1'''', X2'''''), X2''''), X2)) -> MARK(f(f(X1'''', X2'''''), X2''''))
MARK(f(f(f(f(g(X'''''), X2''''''), X2'''0), X2''''), X2)) -> MARK(f(f(f(g(X'''''), X2''''''), X2'''0), X2''''))

Rules:

af(g(X), Y) -> af(mark(X), f(g(X), Y))
af(X1, X2) -> f(X1, X2)
mark(f(X1, X2)) -> af(mark(X1), X2)
mark(g(X)) -> g(mark(X))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

MARK(f(f(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''''), X2''''), X2)) -> MARK(f(f(f(f(X1''''', X2'''''''), X2''0'''), X2'''''), X2''''))
MARK(f(f(f(f(X1'''', X2'''''), X2''0'), X2'''), X2)) -> MARK(f(f(f(X1'''', X2'''''), X2''0'), X2'''))
MARK(f(f(f(f(X1''', X2'''''), X2''0'), X2''), X2)) -> MARK(f(f(f(X1''', X2'''''), X2''0'), X2''))
MARK(f(f(f(X1'''', X2'''''), X2''''), X2)) -> MARK(f(f(X1'''', X2'''''), X2''''))
MARK(f(f(f(f(g(X'''''), X2''''''), X2'''0), X2''''), X2)) -> MARK(f(f(f(g(X'''''), X2''''''), X2'''0), X2''''))

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(g(x1)) =  0 POL(f(x1, x2)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 21`
`                 ↳Dependency Graph`

Dependency Pair:

Rules:

af(g(X), Y) -> af(mark(X), f(g(X), Y))
af(X1, X2) -> f(X1, X2)
mark(f(X1, X2)) -> af(mark(X1), X2)
mark(g(X)) -> g(mark(X))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:45 minutes