Term Rewriting System R:
[X, Y, X1, X2]
f(g(X), Y) -> f(X, nf(ng(X), activate(Y)))
f(X1, X2) -> nf(X1, X2)
g(X) -> ng(X)
activate(nf(X1, X2)) -> f(activate(X1), X2)
activate(ng(X)) -> g(activate(X))
activate(X) -> X

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

F(g(X), Y) -> F(X, nf(ng(X), activate(Y)))
F(g(X), Y) -> ACTIVATE(Y)
ACTIVATE(nf(X1, X2)) -> F(activate(X1), X2)
ACTIVATE(nf(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(ng(X)) -> G(activate(X))
ACTIVATE(ng(X)) -> ACTIVATE(X)

Furthermore, R contains one SCC.

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

Dependency Pairs:

ACTIVATE(ng(X)) -> ACTIVATE(X)
ACTIVATE(nf(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(nf(X1, X2)) -> F(activate(X1), X2)
F(g(X), Y) -> ACTIVATE(Y)
F(g(X), Y) -> F(X, nf(ng(X), activate(Y)))

Rules:

f(g(X), Y) -> f(X, nf(ng(X), activate(Y)))
f(X1, X2) -> nf(X1, X2)
g(X) -> ng(X)
activate(nf(X1, X2)) -> f(activate(X1), X2)
activate(ng(X)) -> g(activate(X))
activate(X) -> X

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

ACTIVATE(nf(X1, X2)) -> F(activate(X1), X2)
three new Dependency Pairs are created:

ACTIVATE(nf(nf(X1'', X2''), X2)) -> F(f(activate(X1''), X2''), X2)
ACTIVATE(nf(ng(X'), X2)) -> F(g(activate(X')), X2)
ACTIVATE(nf(X1', X2)) -> F(X1', X2)

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

ACTIVATE(nf(X1', X2)) -> F(X1', X2)
ACTIVATE(nf(ng(X'), X2)) -> F(g(activate(X')), X2)
F(g(X), Y) -> ACTIVATE(Y)
F(g(X), Y) -> F(X, nf(ng(X), activate(Y)))
ACTIVATE(nf(nf(X1'', X2''), X2)) -> F(f(activate(X1''), X2''), X2)
ACTIVATE(nf(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(ng(X)) -> ACTIVATE(X)

Rules:

f(g(X), Y) -> f(X, nf(ng(X), activate(Y)))
f(X1, X2) -> nf(X1, X2)
g(X) -> ng(X)
activate(nf(X1, X2)) -> f(activate(X1), X2)
activate(ng(X)) -> g(activate(X))
activate(X) -> X

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

ACTIVATE(nf(nf(X1'', X2''), X2)) -> F(f(activate(X1''), X2''), X2)
four new Dependency Pairs are created:

ACTIVATE(nf(nf(X1''', X2'''), X2)) -> F(nf(activate(X1'''), X2'''), X2)
ACTIVATE(nf(nf(nf(X1', X2'''), X2''), X2)) -> F(f(f(activate(X1'), X2'''), X2''), X2)
ACTIVATE(nf(nf(ng(X'), X2''), X2)) -> F(f(g(activate(X')), X2''), X2)
ACTIVATE(nf(nf(X1''', X2''), X2)) -> F(f(X1''', X2''), 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:

ACTIVATE(nf(nf(X1''', X2''), X2)) -> F(f(X1''', X2''), X2)
ACTIVATE(nf(nf(ng(X'), X2''), X2)) -> F(f(g(activate(X')), X2''), X2)
ACTIVATE(nf(nf(nf(X1', X2'''), X2''), X2)) -> F(f(f(activate(X1'), X2'''), X2''), X2)
ACTIVATE(nf(ng(X'), X2)) -> F(g(activate(X')), X2)
ACTIVATE(ng(X)) -> ACTIVATE(X)
ACTIVATE(nf(X1, X2)) -> ACTIVATE(X1)
F(g(X), Y) -> ACTIVATE(Y)
F(g(X), Y) -> F(X, nf(ng(X), activate(Y)))
ACTIVATE(nf(X1', X2)) -> F(X1', X2)

Rules:

f(g(X), Y) -> f(X, nf(ng(X), activate(Y)))
f(X1, X2) -> nf(X1, X2)
g(X) -> ng(X)
activate(nf(X1, X2)) -> f(activate(X1), X2)
activate(ng(X)) -> g(activate(X))
activate(X) -> X

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

ACTIVATE(nf(nf(nf(X1', X2'''), X2''), X2)) -> F(f(f(activate(X1'), X2'''), X2''), X2)
five new Dependency Pairs are created:

ACTIVATE(nf(nf(nf(X1'', X2'''''), X2''''), X2)) -> F(nf(f(activate(X1''), X2'''''), X2''''), X2)
ACTIVATE(nf(nf(nf(X1'', X2''''), X2''), X2)) -> F(f(nf(activate(X1''), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1'', X2'0), X2'''), X2''), X2)) -> F(f(f(f(activate(X1''), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X'), X2'''), X2''), X2)) -> F(f(f(g(activate(X')), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(X1'', X2'''), X2''), X2)) -> F(f(f(X1'', X2'''), X2''), X2)

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:

ACTIVATE(nf(nf(nf(X1'', X2'''), X2''), X2)) -> F(f(f(X1'', X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X'), X2'''), X2''), X2)) -> F(f(f(g(activate(X')), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1'', X2'0), X2'''), X2''), X2)) -> F(f(f(f(activate(X1''), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(X1'', X2''''), X2''), X2)) -> F(f(nf(activate(X1''), X2''''), X2''), X2)
ACTIVATE(nf(nf(ng(X'), X2''), X2)) -> F(f(g(activate(X')), X2''), X2)
ACTIVATE(nf(X1', X2)) -> F(X1', X2)
ACTIVATE(nf(ng(X'), X2)) -> F(g(activate(X')), X2)
ACTIVATE(ng(X)) -> ACTIVATE(X)
ACTIVATE(nf(X1, X2)) -> ACTIVATE(X1)
F(g(X), Y) -> ACTIVATE(Y)
F(g(X), Y) -> F(X, nf(ng(X), activate(Y)))
ACTIVATE(nf(nf(X1''', X2''), X2)) -> F(f(X1''', X2''), X2)

Rules:

f(g(X), Y) -> f(X, nf(ng(X), activate(Y)))
f(X1, X2) -> nf(X1, X2)
g(X) -> ng(X)
activate(nf(X1, X2)) -> f(activate(X1), X2)
activate(ng(X)) -> g(activate(X))
activate(X) -> X

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

ACTIVATE(nf(nf(ng(X'), X2''), X2)) -> F(f(g(activate(X')), X2''), X2)
six new Dependency Pairs are created:

ACTIVATE(nf(nf(ng(X''), X2'''), X2)) -> F(f(activate(X''), nf(ng(activate(X'')), activate(X2'''))), X2)
ACTIVATE(nf(nf(ng(X''), X2'''), X2)) -> F(nf(g(activate(X'')), X2'''), X2)
ACTIVATE(nf(nf(ng(X''), X2''), X2)) -> F(f(ng(activate(X'')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(X1', X2''')), X2''), X2)) -> F(f(g(f(activate(X1'), X2''')), X2''), X2)
ACTIVATE(nf(nf(ng(ng(X'')), X2''), X2)) -> F(f(g(g(activate(X''))), X2''), X2)
ACTIVATE(nf(nf(ng(X''), X2''), X2)) -> 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 5`
`                 ↳Narrowing Transformation`

Dependency Pairs:

ACTIVATE(nf(nf(ng(X''), X2''), X2)) -> F(f(g(X''), X2''), X2)
ACTIVATE(nf(nf(ng(ng(X'')), X2''), X2)) -> F(f(g(g(activate(X''))), X2''), X2)
ACTIVATE(nf(nf(ng(nf(X1', X2''')), X2''), X2)) -> F(f(g(f(activate(X1'), X2''')), X2''), X2)
ACTIVATE(nf(nf(ng(X''), X2''), X2)) -> F(f(ng(activate(X'')), X2''), X2)
ACTIVATE(nf(nf(ng(X''), X2'''), X2)) -> F(f(activate(X''), nf(ng(activate(X'')), activate(X2'''))), X2)
ACTIVATE(nf(nf(nf(ng(X'), X2'''), X2''), X2)) -> F(f(f(g(activate(X')), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1'', X2'0), X2'''), X2''), X2)) -> F(f(f(f(activate(X1''), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(X1'', X2''''), X2''), X2)) -> F(f(nf(activate(X1''), X2''''), X2''), X2)
ACTIVATE(nf(nf(X1''', X2''), X2)) -> F(f(X1''', X2''), X2)
ACTIVATE(nf(X1', X2)) -> F(X1', X2)
ACTIVATE(nf(ng(X'), X2)) -> F(g(activate(X')), X2)
ACTIVATE(ng(X)) -> ACTIVATE(X)
ACTIVATE(nf(X1, X2)) -> ACTIVATE(X1)
F(g(X), Y) -> ACTIVATE(Y)
F(g(X), Y) -> F(X, nf(ng(X), activate(Y)))
ACTIVATE(nf(nf(nf(X1'', X2'''), X2''), X2)) -> F(f(f(X1'', X2'''), X2''), X2)

Rules:

f(g(X), Y) -> f(X, nf(ng(X), activate(Y)))
f(X1, X2) -> nf(X1, X2)
g(X) -> ng(X)
activate(nf(X1, X2)) -> f(activate(X1), X2)
activate(ng(X)) -> g(activate(X))
activate(X) -> X

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

ACTIVATE(nf(nf(X1''', X2''), X2)) -> F(f(X1''', X2''), X2)
two new Dependency Pairs are created:

ACTIVATE(nf(nf(g(X'), X2'''), X2)) -> F(f(X', nf(ng(X'), activate(X2'''))), X2)
ACTIVATE(nf(nf(X1'''', X2'''), X2)) -> F(nf(X1'''', 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:

ACTIVATE(nf(nf(g(X'), X2'''), X2)) -> F(f(X', nf(ng(X'), activate(X2'''))), X2)
ACTIVATE(nf(nf(ng(ng(X'')), X2''), X2)) -> F(f(g(g(activate(X''))), X2''), X2)
ACTIVATE(nf(nf(ng(nf(X1', X2''')), X2''), X2)) -> F(f(g(f(activate(X1'), X2''')), X2''), X2)
ACTIVATE(nf(nf(ng(X''), X2''), X2)) -> F(f(ng(activate(X'')), X2''), X2)
ACTIVATE(nf(nf(ng(X''), X2'''), X2)) -> F(f(activate(X''), nf(ng(activate(X'')), activate(X2'''))), X2)
ACTIVATE(nf(nf(nf(X1'', X2'''), X2''), X2)) -> F(f(f(X1'', X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X'), X2'''), X2''), X2)) -> F(f(f(g(activate(X')), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1'', X2'0), X2'''), X2''), X2)) -> F(f(f(f(activate(X1''), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(X1'', X2''''), X2''), X2)) -> F(f(nf(activate(X1''), X2''''), X2''), X2)
ACTIVATE(nf(X1', X2)) -> F(X1', X2)
ACTIVATE(nf(ng(X'), X2)) -> F(g(activate(X')), X2)
ACTIVATE(ng(X)) -> ACTIVATE(X)
ACTIVATE(nf(X1, X2)) -> ACTIVATE(X1)
F(g(X), Y) -> ACTIVATE(Y)
F(g(X), Y) -> F(X, nf(ng(X), activate(Y)))
ACTIVATE(nf(nf(ng(X''), X2''), X2)) -> F(f(g(X''), X2''), X2)

Rules:

f(g(X), Y) -> f(X, nf(ng(X), activate(Y)))
f(X1, X2) -> nf(X1, X2)
g(X) -> ng(X)
activate(nf(X1, X2)) -> f(activate(X1), X2)
activate(ng(X)) -> g(activate(X))
activate(X) -> X

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

ACTIVATE(nf(nf(nf(X1'', X2''''), X2''), X2)) -> F(f(nf(activate(X1''), X2''''), X2''), X2)
four new Dependency Pairs are created:

ACTIVATE(nf(nf(nf(X1''', X2'''''), X2'''), X2)) -> F(nf(nf(activate(X1'''), X2'''''), X2'''), X2)
ACTIVATE(nf(nf(nf(nf(X1', X2'''), X2''''), X2''), X2)) -> F(f(nf(f(activate(X1'), X2'''), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X'), X2''''), X2''), X2)) -> F(f(nf(g(activate(X')), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(X1''', X2''''), X2''), X2)) -> F(f(nf(X1''', X2''''), X2''), X2)

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:

ACTIVATE(nf(nf(nf(X1''', X2''''), X2''), X2)) -> F(f(nf(X1''', X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X'), X2''''), X2''), X2)) -> F(f(nf(g(activate(X')), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1', X2'''), X2''''), X2''), X2)) -> F(f(nf(f(activate(X1'), X2'''), X2''''), X2''), X2)
ACTIVATE(nf(nf(ng(X''), X2''), X2)) -> F(f(g(X''), X2''), X2)
ACTIVATE(nf(nf(ng(ng(X'')), X2''), X2)) -> F(f(g(g(activate(X''))), X2''), X2)
ACTIVATE(nf(nf(ng(nf(X1', X2''')), X2''), X2)) -> F(f(g(f(activate(X1'), X2''')), X2''), X2)
ACTIVATE(nf(nf(ng(X''), X2''), X2)) -> F(f(ng(activate(X'')), X2''), X2)
ACTIVATE(nf(nf(ng(X''), X2'''), X2)) -> F(f(activate(X''), nf(ng(activate(X'')), activate(X2'''))), X2)
ACTIVATE(nf(nf(nf(X1'', X2'''), X2''), X2)) -> F(f(f(X1'', X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X'), X2'''), X2''), X2)) -> F(f(f(g(activate(X')), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1'', X2'0), X2'''), X2''), X2)) -> F(f(f(f(activate(X1''), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(X1', X2)) -> F(X1', X2)
ACTIVATE(nf(ng(X'), X2)) -> F(g(activate(X')), X2)
ACTIVATE(ng(X)) -> ACTIVATE(X)
ACTIVATE(nf(X1, X2)) -> ACTIVATE(X1)
F(g(X), Y) -> ACTIVATE(Y)
F(g(X), Y) -> F(X, nf(ng(X), activate(Y)))
ACTIVATE(nf(nf(g(X'), X2'''), X2)) -> F(f(X', nf(ng(X'), activate(X2'''))), X2)

Rules:

f(g(X), Y) -> f(X, nf(ng(X), activate(Y)))
f(X1, X2) -> nf(X1, X2)
g(X) -> ng(X)
activate(nf(X1, X2)) -> f(activate(X1), X2)
activate(ng(X)) -> g(activate(X))
activate(X) -> X

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

ACTIVATE(nf(nf(nf(nf(X1'', X2'0), X2'''), X2''), X2)) -> F(f(f(f(activate(X1''), X2'0), X2'''), X2''), X2)
six new Dependency Pairs are created:

ACTIVATE(nf(nf(nf(nf(X1''', X2'0'), X2'''''), X2''''), X2)) -> F(nf(f(f(activate(X1'''), X2'0'), X2'''''), X2''''), X2)
ACTIVATE(nf(nf(nf(nf(X1''', X2'0'), X2''''), X2''), X2)) -> F(f(nf(f(activate(X1'''), X2'0'), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1''', X2'0'), X2'''), X2''), X2)) -> F(f(f(nf(activate(X1'''), X2'0'), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(X1', X2'1), X2'0), X2'''), X2''), X2)) -> F(f(f(f(f(activate(X1'), X2'1), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X'), X2'0), X2'''), X2''), X2)) -> F(f(f(f(g(activate(X')), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1''', X2'0), X2'''), X2''), X2)) -> F(f(f(f(X1''', X2'0), X2'''), X2''), X2)

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

ACTIVATE(nf(nf(nf(nf(X1''', X2'0), X2'''), X2''), X2)) -> F(f(f(f(X1''', X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X'), X2'0), X2'''), X2''), X2)) -> F(f(f(f(g(activate(X')), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(X1', X2'1), X2'0), X2'''), X2''), X2)) -> F(f(f(f(f(activate(X1'), X2'1), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1''', X2'0'), X2'''), X2''), X2)) -> F(f(f(nf(activate(X1'''), X2'0'), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1''', X2'0'), X2''''), X2''), X2)) -> F(f(nf(f(activate(X1'''), X2'0'), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X'), X2''''), X2''), X2)) -> F(f(nf(g(activate(X')), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1', X2'''), X2''''), X2''), X2)) -> F(f(nf(f(activate(X1'), X2'''), X2''''), X2''), X2)
ACTIVATE(nf(nf(g(X'), X2'''), X2)) -> F(f(X', nf(ng(X'), activate(X2'''))), X2)
ACTIVATE(nf(nf(ng(X''), X2''), X2)) -> F(f(g(X''), X2''), X2)
ACTIVATE(nf(nf(ng(ng(X'')), X2''), X2)) -> F(f(g(g(activate(X''))), X2''), X2)
ACTIVATE(nf(nf(ng(nf(X1', X2''')), X2''), X2)) -> F(f(g(f(activate(X1'), X2''')), X2''), X2)
ACTIVATE(nf(nf(ng(X''), X2''), X2)) -> F(f(ng(activate(X'')), X2''), X2)
ACTIVATE(nf(nf(ng(X''), X2'''), X2)) -> F(f(activate(X''), nf(ng(activate(X'')), activate(X2'''))), X2)
ACTIVATE(nf(nf(nf(X1'', X2'''), X2''), X2)) -> F(f(f(X1'', X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X'), X2'''), X2''), X2)) -> F(f(f(g(activate(X')), X2'''), X2''), X2)
ACTIVATE(nf(X1', X2)) -> F(X1', X2)
ACTIVATE(nf(ng(X'), X2)) -> F(g(activate(X')), X2)
ACTIVATE(ng(X)) -> ACTIVATE(X)
ACTIVATE(nf(X1, X2)) -> ACTIVATE(X1)
F(g(X), Y) -> ACTIVATE(Y)
F(g(X), Y) -> F(X, nf(ng(X), activate(Y)))
ACTIVATE(nf(nf(nf(X1''', X2''''), X2''), X2)) -> F(f(nf(X1''', X2''''), X2''), X2)

Rules:

f(g(X), Y) -> f(X, nf(ng(X), activate(Y)))
f(X1, X2) -> nf(X1, X2)
g(X) -> ng(X)
activate(nf(X1, X2)) -> f(activate(X1), X2)
activate(ng(X)) -> g(activate(X))
activate(X) -> X

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

ACTIVATE(nf(nf(nf(ng(X'), X2'''), X2''), X2)) -> F(f(f(g(activate(X')), X2'''), X2''), X2)
seven new Dependency Pairs are created:

ACTIVATE(nf(nf(nf(ng(X''), X2'''''), X2''''), X2)) -> F(nf(f(g(activate(X'')), X2'''''), X2''''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2''''), X2''), X2)) -> F(f(f(activate(X''), nf(ng(activate(X'')), activate(X2''''))), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2''''), X2''), X2)) -> F(f(nf(g(activate(X'')), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2'''), X2''), X2)) -> F(f(f(ng(activate(X'')), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(nf(X1', X2'0)), X2'''), X2''), X2)) -> F(f(f(g(f(activate(X1'), X2'0)), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(ng(X'')), X2'''), X2''), X2)) -> F(f(f(g(g(activate(X''))), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2'''), X2''), X2)) -> F(f(f(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 9`
`                 ↳Narrowing Transformation`

Dependency Pairs:

ACTIVATE(nf(nf(nf(ng(X''), X2'''), X2''), X2)) -> F(f(f(g(X''), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(ng(X'')), X2'''), X2''), X2)) -> F(f(f(g(g(activate(X''))), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(nf(X1', X2'0)), X2'''), X2''), X2)) -> F(f(f(g(f(activate(X1'), X2'0)), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2'''), X2''), X2)) -> F(f(f(ng(activate(X'')), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2''''), X2''), X2)) -> F(f(nf(g(activate(X'')), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2''''), X2''), X2)) -> F(f(f(activate(X''), nf(ng(activate(X'')), activate(X2''''))), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X'), X2'0), X2'''), X2''), X2)) -> F(f(f(f(g(activate(X')), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(X1', X2'1), X2'0), X2'''), X2''), X2)) -> F(f(f(f(f(activate(X1'), X2'1), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1''', X2'0'), X2'''), X2''), X2)) -> F(f(f(nf(activate(X1'''), X2'0'), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1''', X2'0'), X2''''), X2''), X2)) -> F(f(nf(f(activate(X1'''), X2'0'), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(X1''', X2''''), X2''), X2)) -> F(f(nf(X1''', X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X'), X2''''), X2''), X2)) -> F(f(nf(g(activate(X')), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1', X2'''), X2''''), X2''), X2)) -> F(f(nf(f(activate(X1'), X2'''), X2''''), X2''), X2)
ACTIVATE(nf(nf(g(X'), X2'''), X2)) -> F(f(X', nf(ng(X'), activate(X2'''))), X2)
ACTIVATE(nf(nf(ng(X''), X2''), X2)) -> F(f(g(X''), X2''), X2)
ACTIVATE(nf(nf(ng(ng(X'')), X2''), X2)) -> F(f(g(g(activate(X''))), X2''), X2)
ACTIVATE(nf(nf(ng(nf(X1', X2''')), X2''), X2)) -> F(f(g(f(activate(X1'), X2''')), X2''), X2)
ACTIVATE(nf(nf(ng(X''), X2''), X2)) -> F(f(ng(activate(X'')), X2''), X2)
ACTIVATE(nf(nf(ng(X''), X2'''), X2)) -> F(f(activate(X''), nf(ng(activate(X'')), activate(X2'''))), X2)
ACTIVATE(nf(nf(nf(X1'', X2'''), X2''), X2)) -> F(f(f(X1'', X2'''), X2''), X2)
ACTIVATE(nf(X1', X2)) -> F(X1', X2)
ACTIVATE(nf(ng(X'), X2)) -> F(g(activate(X')), X2)
ACTIVATE(ng(X)) -> ACTIVATE(X)
ACTIVATE(nf(X1, X2)) -> ACTIVATE(X1)
F(g(X), Y) -> ACTIVATE(Y)
F(g(X), Y) -> F(X, nf(ng(X), activate(Y)))
ACTIVATE(nf(nf(nf(nf(X1''', X2'0), X2'''), X2''), X2)) -> F(f(f(f(X1''', X2'0), X2'''), X2''), X2)

Rules:

f(g(X), Y) -> f(X, nf(ng(X), activate(Y)))
f(X1, X2) -> nf(X1, X2)
g(X) -> ng(X)
activate(nf(X1, X2)) -> f(activate(X1), X2)
activate(ng(X)) -> g(activate(X))
activate(X) -> X

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

ACTIVATE(nf(nf(nf(X1'', X2'''), X2''), X2)) -> F(f(f(X1'', X2'''), X2''), X2)
three new Dependency Pairs are created:

ACTIVATE(nf(nf(nf(X1''', X2'''''), X2''''), X2)) -> F(nf(f(X1''', X2'''''), X2''''), X2)
ACTIVATE(nf(nf(nf(g(X'), X2''''), X2''), X2)) -> F(f(f(X', nf(ng(X'), activate(X2''''))), X2''), X2)
ACTIVATE(nf(nf(nf(X1''', X2''''), X2''), X2)) -> F(f(nf(X1''', X2''''), X2''), X2)

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

ACTIVATE(nf(nf(nf(X1''', X2''''), X2''), X2)) -> F(f(nf(X1''', X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(g(X'), X2''''), X2''), X2)) -> F(f(f(X', nf(ng(X'), activate(X2''''))), X2''), X2)
ACTIVATE(nf(nf(nf(ng(ng(X'')), X2'''), X2''), X2)) -> F(f(f(g(g(activate(X''))), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(nf(X1', X2'0)), X2'''), X2''), X2)) -> F(f(f(g(f(activate(X1'), X2'0)), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2'''), X2''), X2)) -> F(f(f(ng(activate(X'')), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2''''), X2''), X2)) -> F(f(nf(g(activate(X'')), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2''''), X2''), X2)) -> F(f(f(activate(X''), nf(ng(activate(X'')), activate(X2''''))), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1''', X2'0), X2'''), X2''), X2)) -> F(f(f(f(X1''', X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X'), X2'0), X2'''), X2''), X2)) -> F(f(f(f(g(activate(X')), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(X1', X2'1), X2'0), X2'''), X2''), X2)) -> F(f(f(f(f(activate(X1'), X2'1), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1''', X2'0'), X2'''), X2''), X2)) -> F(f(f(nf(activate(X1'''), X2'0'), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1''', X2'0'), X2''''), X2''), X2)) -> F(f(nf(f(activate(X1'''), X2'0'), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(X1''', X2''''), X2''), X2)) -> F(f(nf(X1''', X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X'), X2''''), X2''), X2)) -> F(f(nf(g(activate(X')), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1', X2'''), X2''''), X2''), X2)) -> F(f(nf(f(activate(X1'), X2'''), X2''''), X2''), X2)
ACTIVATE(nf(nf(g(X'), X2'''), X2)) -> F(f(X', nf(ng(X'), activate(X2'''))), X2)
ACTIVATE(nf(nf(ng(X''), X2''), X2)) -> F(f(g(X''), X2''), X2)
ACTIVATE(nf(nf(ng(ng(X'')), X2''), X2)) -> F(f(g(g(activate(X''))), X2''), X2)
ACTIVATE(nf(nf(ng(nf(X1', X2''')), X2''), X2)) -> F(f(g(f(activate(X1'), X2''')), X2''), X2)
ACTIVATE(nf(nf(ng(X''), X2''), X2)) -> F(f(ng(activate(X'')), X2''), X2)
ACTIVATE(nf(nf(ng(X''), X2'''), X2)) -> F(f(activate(X''), nf(ng(activate(X'')), activate(X2'''))), X2)
ACTIVATE(nf(X1', X2)) -> F(X1', X2)
ACTIVATE(nf(ng(X'), X2)) -> F(g(activate(X')), X2)
ACTIVATE(ng(X)) -> ACTIVATE(X)
ACTIVATE(nf(X1, X2)) -> ACTIVATE(X1)
F(g(X), Y) -> ACTIVATE(Y)
F(g(X), Y) -> F(X, nf(ng(X), activate(Y)))
ACTIVATE(nf(nf(nf(ng(X''), X2'''), X2''), X2)) -> F(f(f(g(X''), X2'''), X2''), X2)

Rules:

f(g(X), Y) -> f(X, nf(ng(X), activate(Y)))
f(X1, X2) -> nf(X1, X2)
g(X) -> ng(X)
activate(nf(X1, X2)) -> f(activate(X1), X2)
activate(ng(X)) -> g(activate(X))
activate(X) -> X

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

ACTIVATE(nf(nf(ng(X''), X2''), X2)) -> F(f(ng(activate(X'')), X2''), X2)
four new Dependency Pairs are created:

ACTIVATE(nf(nf(ng(X'''), X2'''), X2)) -> F(nf(ng(activate(X''')), X2'''), X2)
ACTIVATE(nf(nf(ng(nf(X1', X2''')), X2''), X2)) -> F(f(ng(f(activate(X1'), X2''')), X2''), X2)
ACTIVATE(nf(nf(ng(ng(X')), X2''), X2)) -> F(f(ng(g(activate(X'))), X2''), X2)
ACTIVATE(nf(nf(ng(X'''), X2''), X2)) -> F(f(ng(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`
`                 ↳Narrowing Transformation`

Dependency Pairs:

ACTIVATE(nf(nf(ng(X'''), X2''), X2)) -> F(f(ng(X'''), X2''), X2)
ACTIVATE(nf(nf(ng(ng(X')), X2''), X2)) -> F(f(ng(g(activate(X'))), X2''), X2)
ACTIVATE(nf(nf(ng(nf(X1', X2''')), X2''), X2)) -> F(f(ng(f(activate(X1'), X2''')), X2''), X2)
ACTIVATE(nf(nf(nf(g(X'), X2''''), X2''), X2)) -> F(f(f(X', nf(ng(X'), activate(X2''''))), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2'''), X2''), X2)) -> F(f(f(g(X''), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(ng(X'')), X2'''), X2''), X2)) -> F(f(f(g(g(activate(X''))), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(nf(X1', X2'0)), X2'''), X2''), X2)) -> F(f(f(g(f(activate(X1'), X2'0)), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2'''), X2''), X2)) -> F(f(f(ng(activate(X'')), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2''''), X2''), X2)) -> F(f(nf(g(activate(X'')), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2''''), X2''), X2)) -> F(f(f(activate(X''), nf(ng(activate(X'')), activate(X2''''))), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1''', X2'0), X2'''), X2''), X2)) -> F(f(f(f(X1''', X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X'), X2'0), X2'''), X2''), X2)) -> F(f(f(f(g(activate(X')), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(X1', X2'1), X2'0), X2'''), X2''), X2)) -> F(f(f(f(f(activate(X1'), X2'1), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1''', X2'0'), X2'''), X2''), X2)) -> F(f(f(nf(activate(X1'''), X2'0'), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1''', X2'0'), X2''''), X2''), X2)) -> F(f(nf(f(activate(X1'''), X2'0'), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(X1''', X2''''), X2''), X2)) -> F(f(nf(X1''', X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X'), X2''''), X2''), X2)) -> F(f(nf(g(activate(X')), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1', X2'''), X2''''), X2''), X2)) -> F(f(nf(f(activate(X1'), X2'''), X2''''), X2''), X2)
ACTIVATE(nf(nf(g(X'), X2'''), X2)) -> F(f(X', nf(ng(X'), activate(X2'''))), X2)
ACTIVATE(nf(nf(ng(X''), X2''), X2)) -> F(f(g(X''), X2''), X2)
ACTIVATE(nf(nf(ng(ng(X'')), X2''), X2)) -> F(f(g(g(activate(X''))), X2''), X2)
ACTIVATE(nf(nf(ng(nf(X1', X2''')), X2''), X2)) -> F(f(g(f(activate(X1'), X2''')), X2''), X2)
ACTIVATE(nf(nf(ng(X''), X2'''), X2)) -> F(f(activate(X''), nf(ng(activate(X'')), activate(X2'''))), X2)
ACTIVATE(nf(X1', X2)) -> F(X1', X2)
ACTIVATE(nf(ng(X'), X2)) -> F(g(activate(X')), X2)
ACTIVATE(ng(X)) -> ACTIVATE(X)
ACTIVATE(nf(X1, X2)) -> ACTIVATE(X1)
F(g(X), Y) -> ACTIVATE(Y)
F(g(X), Y) -> F(X, nf(ng(X), activate(Y)))
ACTIVATE(nf(nf(nf(X1''', X2''''), X2''), X2)) -> F(f(nf(X1''', X2''''), X2''), X2)

Rules:

f(g(X), Y) -> f(X, nf(ng(X), activate(Y)))
f(X1, X2) -> nf(X1, X2)
g(X) -> ng(X)
activate(nf(X1, X2)) -> f(activate(X1), X2)
activate(ng(X)) -> g(activate(X))
activate(X) -> X

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

ACTIVATE(nf(nf(ng(nf(X1', X2''')), X2''), X2)) -> F(f(g(f(activate(X1'), X2''')), X2''), X2)
seven new Dependency Pairs are created:

ACTIVATE(nf(nf(ng(nf(X1'', X2''''')), X2''''), X2)) -> F(f(f(activate(X1''), X2'''''), nf(ng(f(activate(X1''), X2''''')), activate(X2''''))), X2)
ACTIVATE(nf(nf(ng(nf(X1'', X2''''')), X2''''), X2)) -> F(nf(g(f(activate(X1''), X2''''')), X2''''), X2)
ACTIVATE(nf(nf(ng(nf(X1'', X2'''')), X2''), X2)) -> F(f(ng(f(activate(X1''), X2'''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(X1'', X2'''')), X2''), X2)) -> F(f(g(nf(activate(X1''), X2'''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(nf(X1'', X2'0), X2''')), X2''), X2)) -> F(f(g(f(f(activate(X1''), X2'0), X2''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(ng(X'), X2''')), X2''), X2)) -> F(f(g(f(g(activate(X')), X2''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(X1'', X2''')), X2''), X2)) -> F(f(g(f(X1'', X2''')), X2''), X2)

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

ACTIVATE(nf(nf(ng(nf(X1'', X2''')), X2''), X2)) -> F(f(g(f(X1'', X2''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(ng(X'), X2''')), X2''), X2)) -> F(f(g(f(g(activate(X')), X2''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(nf(X1'', X2'0), X2''')), X2''), X2)) -> F(f(g(f(f(activate(X1''), X2'0), X2''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(X1'', X2'''')), X2''), X2)) -> F(f(g(nf(activate(X1''), X2'''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(X1'', X2'''')), X2''), X2)) -> F(f(ng(f(activate(X1''), X2'''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(X1'', X2''''')), X2''''), X2)) -> F(f(f(activate(X1''), X2'''''), nf(ng(f(activate(X1''), X2''''')), activate(X2''''))), X2)
ACTIVATE(nf(nf(ng(ng(X')), X2''), X2)) -> F(f(ng(g(activate(X'))), X2''), X2)
ACTIVATE(nf(nf(ng(nf(X1', X2''')), X2''), X2)) -> F(f(ng(f(activate(X1'), X2''')), X2''), X2)
ACTIVATE(nf(nf(nf(X1''', X2''''), X2''), X2)) -> F(f(nf(X1''', X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(g(X'), X2''''), X2''), X2)) -> F(f(f(X', nf(ng(X'), activate(X2''''))), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2'''), X2''), X2)) -> F(f(f(g(X''), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(ng(X'')), X2'''), X2''), X2)) -> F(f(f(g(g(activate(X''))), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(nf(X1', X2'0)), X2'''), X2''), X2)) -> F(f(f(g(f(activate(X1'), X2'0)), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2'''), X2''), X2)) -> F(f(f(ng(activate(X'')), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2''''), X2''), X2)) -> F(f(nf(g(activate(X'')), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2''''), X2''), X2)) -> F(f(f(activate(X''), nf(ng(activate(X'')), activate(X2''''))), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1''', X2'0), X2'''), X2''), X2)) -> F(f(f(f(X1''', X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X'), X2'0), X2'''), X2''), X2)) -> F(f(f(f(g(activate(X')), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(X1', X2'1), X2'0), X2'''), X2''), X2)) -> F(f(f(f(f(activate(X1'), X2'1), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1''', X2'0'), X2'''), X2''), X2)) -> F(f(f(nf(activate(X1'''), X2'0'), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1''', X2'0'), X2''''), X2''), X2)) -> F(f(nf(f(activate(X1'''), X2'0'), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(X1''', X2''''), X2''), X2)) -> F(f(nf(X1''', X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X'), X2''''), X2''), X2)) -> F(f(nf(g(activate(X')), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1', X2'''), X2''''), X2''), X2)) -> F(f(nf(f(activate(X1'), X2'''), X2''''), X2''), X2)
ACTIVATE(nf(nf(g(X'), X2'''), X2)) -> F(f(X', nf(ng(X'), activate(X2'''))), X2)
ACTIVATE(nf(nf(ng(X''), X2''), X2)) -> F(f(g(X''), X2''), X2)
ACTIVATE(nf(nf(ng(ng(X'')), X2''), X2)) -> F(f(g(g(activate(X''))), X2''), X2)
ACTIVATE(nf(nf(ng(X''), X2'''), X2)) -> F(f(activate(X''), nf(ng(activate(X'')), activate(X2'''))), X2)
ACTIVATE(nf(X1', X2)) -> F(X1', X2)
ACTIVATE(nf(ng(X'), X2)) -> F(g(activate(X')), X2)
ACTIVATE(ng(X)) -> ACTIVATE(X)
ACTIVATE(nf(X1, X2)) -> ACTIVATE(X1)
F(g(X), Y) -> ACTIVATE(Y)
F(g(X), Y) -> F(X, nf(ng(X), activate(Y)))
ACTIVATE(nf(nf(ng(X'''), X2''), X2)) -> F(f(ng(X'''), X2''), X2)

Rules:

f(g(X), Y) -> f(X, nf(ng(X), activate(Y)))
f(X1, X2) -> nf(X1, X2)
g(X) -> ng(X)
activate(nf(X1, X2)) -> f(activate(X1), X2)
activate(ng(X)) -> g(activate(X))
activate(X) -> X

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

ACTIVATE(nf(nf(ng(ng(X'')), X2''), X2)) -> F(f(g(g(activate(X''))), X2''), X2)
seven new Dependency Pairs are created:

ACTIVATE(nf(nf(ng(ng(X''')), X2'''), X2)) -> F(f(g(activate(X''')), nf(ng(g(activate(X'''))), activate(X2'''))), X2)
ACTIVATE(nf(nf(ng(ng(X''')), X2'''), X2)) -> F(nf(g(g(activate(X'''))), X2'''), X2)
ACTIVATE(nf(nf(ng(ng(X''')), X2''), X2)) -> F(f(ng(g(activate(X'''))), X2''), X2)
ACTIVATE(nf(nf(ng(ng(X''')), X2''), X2)) -> F(f(g(ng(activate(X'''))), X2''), X2)
ACTIVATE(nf(nf(ng(ng(nf(X1', X2'''))), X2''), X2)) -> F(f(g(g(f(activate(X1'), X2'''))), X2''), X2)
ACTIVATE(nf(nf(ng(ng(ng(X'))), X2''), X2)) -> F(f(g(g(g(activate(X')))), X2''), X2)
ACTIVATE(nf(nf(ng(ng(X''')), X2''), X2)) -> 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`
`                 ↳Narrowing Transformation`

Dependency Pairs:

ACTIVATE(nf(nf(ng(ng(X''')), X2''), X2)) -> F(f(g(g(X''')), X2''), X2)
ACTIVATE(nf(nf(ng(ng(ng(X'))), X2''), X2)) -> F(f(g(g(g(activate(X')))), X2''), X2)
ACTIVATE(nf(nf(ng(ng(nf(X1', X2'''))), X2''), X2)) -> F(f(g(g(f(activate(X1'), X2'''))), X2''), X2)
ACTIVATE(nf(nf(ng(ng(X''')), X2''), X2)) -> F(f(g(ng(activate(X'''))), X2''), X2)
ACTIVATE(nf(nf(ng(ng(X''')), X2''), X2)) -> F(f(ng(g(activate(X'''))), X2''), X2)
ACTIVATE(nf(nf(ng(ng(X''')), X2'''), X2)) -> F(f(g(activate(X''')), nf(ng(g(activate(X'''))), activate(X2'''))), X2)
ACTIVATE(nf(nf(ng(nf(ng(X'), X2''')), X2''), X2)) -> F(f(g(f(g(activate(X')), X2''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(nf(X1'', X2'0), X2''')), X2''), X2)) -> F(f(g(f(f(activate(X1''), X2'0), X2''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(X1'', X2'''')), X2''), X2)) -> F(f(g(nf(activate(X1''), X2'''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(X1'', X2'''')), X2''), X2)) -> F(f(ng(f(activate(X1''), X2'''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(X1'', X2''''')), X2''''), X2)) -> F(f(f(activate(X1''), X2'''''), nf(ng(f(activate(X1''), X2''''')), activate(X2''''))), X2)
ACTIVATE(nf(nf(ng(X'''), X2''), X2)) -> F(f(ng(X'''), X2''), X2)
ACTIVATE(nf(nf(ng(ng(X')), X2''), X2)) -> F(f(ng(g(activate(X'))), X2''), X2)
ACTIVATE(nf(nf(ng(nf(X1', X2''')), X2''), X2)) -> F(f(ng(f(activate(X1'), X2''')), X2''), X2)
ACTIVATE(nf(nf(nf(X1''', X2''''), X2''), X2)) -> F(f(nf(X1''', X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(g(X'), X2''''), X2''), X2)) -> F(f(f(X', nf(ng(X'), activate(X2''''))), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2'''), X2''), X2)) -> F(f(f(g(X''), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(ng(X'')), X2'''), X2''), X2)) -> F(f(f(g(g(activate(X''))), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(nf(X1', X2'0)), X2'''), X2''), X2)) -> F(f(f(g(f(activate(X1'), X2'0)), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2'''), X2''), X2)) -> F(f(f(ng(activate(X'')), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2''''), X2''), X2)) -> F(f(nf(g(activate(X'')), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2''''), X2''), X2)) -> F(f(f(activate(X''), nf(ng(activate(X'')), activate(X2''''))), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1''', X2'0), X2'''), X2''), X2)) -> F(f(f(f(X1''', X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X'), X2'0), X2'''), X2''), X2)) -> F(f(f(f(g(activate(X')), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(X1', X2'1), X2'0), X2'''), X2''), X2)) -> F(f(f(f(f(activate(X1'), X2'1), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1''', X2'0'), X2'''), X2''), X2)) -> F(f(f(nf(activate(X1'''), X2'0'), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1''', X2'0'), X2''''), X2''), X2)) -> F(f(nf(f(activate(X1'''), X2'0'), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(X1''', X2''''), X2''), X2)) -> F(f(nf(X1''', X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X'), X2''''), X2''), X2)) -> F(f(nf(g(activate(X')), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1', X2'''), X2''''), X2''), X2)) -> F(f(nf(f(activate(X1'), X2'''), X2''''), X2''), X2)
ACTIVATE(nf(nf(g(X'), X2'''), X2)) -> F(f(X', nf(ng(X'), activate(X2'''))), X2)
ACTIVATE(nf(nf(ng(X''), X2''), X2)) -> F(f(g(X''), X2''), X2)
ACTIVATE(nf(nf(ng(X''), X2'''), X2)) -> F(f(activate(X''), nf(ng(activate(X'')), activate(X2'''))), X2)
ACTIVATE(nf(X1', X2)) -> F(X1', X2)
ACTIVATE(nf(ng(X'), X2)) -> F(g(activate(X')), X2)
ACTIVATE(ng(X)) -> ACTIVATE(X)
ACTIVATE(nf(X1, X2)) -> ACTIVATE(X1)
F(g(X), Y) -> ACTIVATE(Y)
F(g(X), Y) -> F(X, nf(ng(X), activate(Y)))
ACTIVATE(nf(nf(ng(nf(X1'', X2''')), X2''), X2)) -> F(f(g(f(X1'', X2''')), X2''), X2)

Rules:

f(g(X), Y) -> f(X, nf(ng(X), activate(Y)))
f(X1, X2) -> nf(X1, X2)
g(X) -> ng(X)
activate(nf(X1, X2)) -> f(activate(X1), X2)
activate(ng(X)) -> g(activate(X))
activate(X) -> X

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

ACTIVATE(nf(nf(ng(X''), X2''), X2)) -> F(f(g(X''), X2''), X2)
three new Dependency Pairs are created:

ACTIVATE(nf(nf(ng(X'''), X2'''), X2)) -> F(f(X''', nf(ng(X'''), activate(X2'''))), X2)
ACTIVATE(nf(nf(ng(X'''), X2'''), X2)) -> F(nf(g(X'''), X2'''), X2)
ACTIVATE(nf(nf(ng(X'''), X2''), X2)) -> F(f(ng(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`
`                 ↳Narrowing Transformation`

Dependency Pairs:

ACTIVATE(nf(nf(ng(X'''), X2''), X2)) -> F(f(ng(X'''), X2''), X2)
ACTIVATE(nf(nf(ng(X'''), X2'''), X2)) -> F(f(X''', nf(ng(X'''), activate(X2'''))), X2)
ACTIVATE(nf(nf(ng(ng(ng(X'))), X2''), X2)) -> F(f(g(g(g(activate(X')))), X2''), X2)
ACTIVATE(nf(nf(ng(ng(nf(X1', X2'''))), X2''), X2)) -> F(f(g(g(f(activate(X1'), X2'''))), X2''), X2)
ACTIVATE(nf(nf(ng(ng(X''')), X2''), X2)) -> F(f(g(ng(activate(X'''))), X2''), X2)
ACTIVATE(nf(nf(ng(ng(X''')), X2''), X2)) -> F(f(ng(g(activate(X'''))), X2''), X2)
ACTIVATE(nf(nf(ng(ng(X''')), X2'''), X2)) -> F(f(g(activate(X''')), nf(ng(g(activate(X'''))), activate(X2'''))), X2)
ACTIVATE(nf(nf(ng(nf(X1'', X2''')), X2''), X2)) -> F(f(g(f(X1'', X2''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(ng(X'), X2''')), X2''), X2)) -> F(f(g(f(g(activate(X')), X2''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(nf(X1'', X2'0), X2''')), X2''), X2)) -> F(f(g(f(f(activate(X1''), X2'0), X2''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(X1'', X2'''')), X2''), X2)) -> F(f(g(nf(activate(X1''), X2'''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(X1'', X2'''')), X2''), X2)) -> F(f(ng(f(activate(X1''), X2'''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(X1'', X2''''')), X2''''), X2)) -> F(f(f(activate(X1''), X2'''''), nf(ng(f(activate(X1''), X2''''')), activate(X2''''))), X2)
ACTIVATE(nf(nf(ng(X'''), X2''), X2)) -> F(f(ng(X'''), X2''), X2)
ACTIVATE(nf(nf(ng(ng(X')), X2''), X2)) -> F(f(ng(g(activate(X'))), X2''), X2)
ACTIVATE(nf(nf(ng(nf(X1', X2''')), X2''), X2)) -> F(f(ng(f(activate(X1'), X2''')), X2''), X2)
ACTIVATE(nf(nf(nf(X1''', X2''''), X2''), X2)) -> F(f(nf(X1''', X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(g(X'), X2''''), X2''), X2)) -> F(f(f(X', nf(ng(X'), activate(X2''''))), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2'''), X2''), X2)) -> F(f(f(g(X''), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(ng(X'')), X2'''), X2''), X2)) -> F(f(f(g(g(activate(X''))), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(nf(X1', X2'0)), X2'''), X2''), X2)) -> F(f(f(g(f(activate(X1'), X2'0)), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2'''), X2''), X2)) -> F(f(f(ng(activate(X'')), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2''''), X2''), X2)) -> F(f(nf(g(activate(X'')), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2''''), X2''), X2)) -> F(f(f(activate(X''), nf(ng(activate(X'')), activate(X2''''))), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1''', X2'0), X2'''), X2''), X2)) -> F(f(f(f(X1''', X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X'), X2'0), X2'''), X2''), X2)) -> F(f(f(f(g(activate(X')), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(X1', X2'1), X2'0), X2'''), X2''), X2)) -> F(f(f(f(f(activate(X1'), X2'1), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1''', X2'0'), X2'''), X2''), X2)) -> F(f(f(nf(activate(X1'''), X2'0'), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1''', X2'0'), X2''''), X2''), X2)) -> F(f(nf(f(activate(X1'''), X2'0'), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(X1''', X2''''), X2''), X2)) -> F(f(nf(X1''', X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X'), X2''''), X2''), X2)) -> F(f(nf(g(activate(X')), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1', X2'''), X2''''), X2''), X2)) -> F(f(nf(f(activate(X1'), X2'''), X2''''), X2''), X2)
ACTIVATE(nf(nf(g(X'), X2'''), X2)) -> F(f(X', nf(ng(X'), activate(X2'''))), X2)
ACTIVATE(nf(nf(ng(X''), X2'''), X2)) -> F(f(activate(X''), nf(ng(activate(X'')), activate(X2'''))), X2)
ACTIVATE(nf(X1', X2)) -> F(X1', X2)
ACTIVATE(nf(ng(X'), X2)) -> F(g(activate(X')), X2)
ACTIVATE(ng(X)) -> ACTIVATE(X)
ACTIVATE(nf(X1, X2)) -> ACTIVATE(X1)
F(g(X), Y) -> ACTIVATE(Y)
F(g(X), Y) -> F(X, nf(ng(X), activate(Y)))
ACTIVATE(nf(nf(ng(ng(X''')), X2''), X2)) -> F(f(g(g(X''')), X2''), X2)

Rules:

f(g(X), Y) -> f(X, nf(ng(X), activate(Y)))
f(X1, X2) -> nf(X1, X2)
g(X) -> ng(X)
activate(nf(X1, X2)) -> f(activate(X1), X2)
activate(ng(X)) -> g(activate(X))
activate(X) -> X

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

ACTIVATE(nf(nf(nf(nf(X1', X2'''), X2''''), X2''), X2)) -> F(f(nf(f(activate(X1'), X2'''), X2''''), X2''), X2)
five new Dependency Pairs are created:

ACTIVATE(nf(nf(nf(nf(X1'', X2'''''), X2''''''), X2''0), X2)) -> F(nf(nf(f(activate(X1''), X2'''''), X2''''''), X2''0), X2)
ACTIVATE(nf(nf(nf(nf(X1'', X2'''''), X2''''), X2''), X2)) -> F(f(nf(nf(activate(X1''), X2'''''), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(X1'', X2'0), X2'''), X2''''), X2''), X2)) -> F(f(nf(f(f(activate(X1''), X2'0), X2'''), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X'), X2'''), X2''''), X2''), X2)) -> F(f(nf(f(g(activate(X')), X2'''), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1'', X2'''), X2''''), X2''), X2)) -> F(f(nf(f(X1'', X2'''), 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`
`                 ↳Narrowing Transformation`

Dependency Pairs:

ACTIVATE(nf(nf(nf(nf(X1'', X2'''), X2''''), X2''), X2)) -> F(f(nf(f(X1'', X2'''), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X'), X2'''), X2''''), X2''), X2)) -> F(f(nf(f(g(activate(X')), X2'''), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(X1'', X2'0), X2'''), X2''''), X2''), X2)) -> F(f(nf(f(f(activate(X1''), X2'0), X2'''), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1'', X2'''''), X2''''), X2''), X2)) -> F(f(nf(nf(activate(X1''), X2'''''), X2''''), X2''), X2)
ACTIVATE(nf(nf(ng(X'''), X2'''), X2)) -> F(f(X''', nf(ng(X'''), activate(X2'''))), X2)
ACTIVATE(nf(nf(ng(ng(X''')), X2''), X2)) -> F(f(g(g(X''')), X2''), X2)
ACTIVATE(nf(nf(ng(ng(ng(X'))), X2''), X2)) -> F(f(g(g(g(activate(X')))), X2''), X2)
ACTIVATE(nf(nf(ng(ng(nf(X1', X2'''))), X2''), X2)) -> F(f(g(g(f(activate(X1'), X2'''))), X2''), X2)
ACTIVATE(nf(nf(ng(ng(X''')), X2''), X2)) -> F(f(g(ng(activate(X'''))), X2''), X2)
ACTIVATE(nf(nf(ng(ng(X''')), X2''), X2)) -> F(f(ng(g(activate(X'''))), X2''), X2)
ACTIVATE(nf(nf(ng(ng(X''')), X2'''), X2)) -> F(f(g(activate(X''')), nf(ng(g(activate(X'''))), activate(X2'''))), X2)
ACTIVATE(nf(nf(ng(nf(X1'', X2''')), X2''), X2)) -> F(f(g(f(X1'', X2''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(ng(X'), X2''')), X2''), X2)) -> F(f(g(f(g(activate(X')), X2''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(nf(X1'', X2'0), X2''')), X2''), X2)) -> F(f(g(f(f(activate(X1''), X2'0), X2''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(X1'', X2'''')), X2''), X2)) -> F(f(g(nf(activate(X1''), X2'''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(X1'', X2'''')), X2''), X2)) -> F(f(ng(f(activate(X1''), X2'''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(X1'', X2''''')), X2''''), X2)) -> F(f(f(activate(X1''), X2'''''), nf(ng(f(activate(X1''), X2''''')), activate(X2''''))), X2)
ACTIVATE(nf(nf(ng(X'''), X2''), X2)) -> F(f(ng(X'''), X2''), X2)
ACTIVATE(nf(nf(ng(ng(X')), X2''), X2)) -> F(f(ng(g(activate(X'))), X2''), X2)
ACTIVATE(nf(nf(ng(nf(X1', X2''')), X2''), X2)) -> F(f(ng(f(activate(X1'), X2''')), X2''), X2)
ACTIVATE(nf(nf(nf(X1''', X2''''), X2''), X2)) -> F(f(nf(X1''', X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(g(X'), X2''''), X2''), X2)) -> F(f(f(X', nf(ng(X'), activate(X2''''))), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2'''), X2''), X2)) -> F(f(f(g(X''), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(ng(X'')), X2'''), X2''), X2)) -> F(f(f(g(g(activate(X''))), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(nf(X1', X2'0)), X2'''), X2''), X2)) -> F(f(f(g(f(activate(X1'), X2'0)), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2'''), X2''), X2)) -> F(f(f(ng(activate(X'')), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2''''), X2''), X2)) -> F(f(nf(g(activate(X'')), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2''''), X2''), X2)) -> F(f(f(activate(X''), nf(ng(activate(X'')), activate(X2''''))), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1''', X2'0), X2'''), X2''), X2)) -> F(f(f(f(X1''', X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X'), X2'0), X2'''), X2''), X2)) -> F(f(f(f(g(activate(X')), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(X1', X2'1), X2'0), X2'''), X2''), X2)) -> F(f(f(f(f(activate(X1'), X2'1), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1''', X2'0'), X2'''), X2''), X2)) -> F(f(f(nf(activate(X1'''), X2'0'), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1''', X2'0'), X2''''), X2''), X2)) -> F(f(nf(f(activate(X1'''), X2'0'), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(X1''', X2''''), X2''), X2)) -> F(f(nf(X1''', X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X'), X2''''), X2''), X2)) -> F(f(nf(g(activate(X')), X2''''), X2''), X2)
ACTIVATE(nf(nf(g(X'), X2'''), X2)) -> F(f(X', nf(ng(X'), activate(X2'''))), X2)
ACTIVATE(nf(nf(ng(X''), X2'''), X2)) -> F(f(activate(X''), nf(ng(activate(X'')), activate(X2'''))), X2)
ACTIVATE(nf(X1', X2)) -> F(X1', X2)
ACTIVATE(nf(ng(X'), X2)) -> F(g(activate(X')), X2)
ACTIVATE(ng(X)) -> ACTIVATE(X)
ACTIVATE(nf(X1, X2)) -> ACTIVATE(X1)
F(g(X), Y) -> ACTIVATE(Y)
F(g(X), Y) -> F(X, nf(ng(X), activate(Y)))
ACTIVATE(nf(nf(ng(X'''), X2''), X2)) -> F(f(ng(X'''), X2''), X2)

Rules:

f(g(X), Y) -> f(X, nf(ng(X), activate(Y)))
f(X1, X2) -> nf(X1, X2)
g(X) -> ng(X)
activate(nf(X1, X2)) -> f(activate(X1), X2)
activate(ng(X)) -> g(activate(X))
activate(X) -> X

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

ACTIVATE(nf(nf(nf(ng(X'), X2''''), X2''), X2)) -> F(f(nf(g(activate(X')), X2''''), X2''), X2)
five new Dependency Pairs are created:

ACTIVATE(nf(nf(nf(ng(X''), X2'''''), X2'''), X2)) -> F(nf(nf(g(activate(X'')), X2'''''), X2'''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2''''), X2''), X2)) -> F(f(nf(ng(activate(X'')), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(nf(X1', X2''')), X2''''), X2''), X2)) -> F(f(nf(g(f(activate(X1'), X2''')), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(ng(X'')), X2''''), X2''), X2)) -> F(f(nf(g(g(activate(X''))), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2''''), X2''), X2)) -> F(f(nf(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 16`
`                 ↳Narrowing Transformation`

Dependency Pairs:

ACTIVATE(nf(nf(nf(ng(X''), X2''''), X2''), X2)) -> F(f(nf(g(X''), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(ng(X'')), X2''''), X2''), X2)) -> F(f(nf(g(g(activate(X''))), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(nf(X1', X2''')), X2''''), X2''), X2)) -> F(f(nf(g(f(activate(X1'), X2''')), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2''''), X2''), X2)) -> F(f(nf(ng(activate(X'')), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X'), X2'''), X2''''), X2''), X2)) -> F(f(nf(f(g(activate(X')), X2'''), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(X1'', X2'0), X2'''), X2''''), X2''), X2)) -> F(f(nf(f(f(activate(X1''), X2'0), X2'''), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1'', X2'''''), X2''''), X2''), X2)) -> F(f(nf(nf(activate(X1''), X2'''''), X2''''), X2''), X2)
ACTIVATE(nf(nf(ng(X'''), X2''), X2)) -> F(f(ng(X'''), X2''), X2)
ACTIVATE(nf(nf(ng(X'''), X2'''), X2)) -> F(f(X''', nf(ng(X'''), activate(X2'''))), X2)
ACTIVATE(nf(nf(ng(ng(X''')), X2''), X2)) -> F(f(g(g(X''')), X2''), X2)
ACTIVATE(nf(nf(ng(ng(ng(X'))), X2''), X2)) -> F(f(g(g(g(activate(X')))), X2''), X2)
ACTIVATE(nf(nf(ng(ng(nf(X1', X2'''))), X2''), X2)) -> F(f(g(g(f(activate(X1'), X2'''))), X2''), X2)
ACTIVATE(nf(nf(ng(ng(X''')), X2''), X2)) -> F(f(g(ng(activate(X'''))), X2''), X2)
ACTIVATE(nf(nf(ng(ng(X''')), X2''), X2)) -> F(f(ng(g(activate(X'''))), X2''), X2)
ACTIVATE(nf(nf(ng(ng(X''')), X2'''), X2)) -> F(f(g(activate(X''')), nf(ng(g(activate(X'''))), activate(X2'''))), X2)
ACTIVATE(nf(nf(ng(nf(X1'', X2''')), X2''), X2)) -> F(f(g(f(X1'', X2''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(ng(X'), X2''')), X2''), X2)) -> F(f(g(f(g(activate(X')), X2''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(nf(X1'', X2'0), X2''')), X2''), X2)) -> F(f(g(f(f(activate(X1''), X2'0), X2''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(X1'', X2'''')), X2''), X2)) -> F(f(g(nf(activate(X1''), X2'''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(X1'', X2'''')), X2''), X2)) -> F(f(ng(f(activate(X1''), X2'''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(X1'', X2''''')), X2''''), X2)) -> F(f(f(activate(X1''), X2'''''), nf(ng(f(activate(X1''), X2''''')), activate(X2''''))), X2)
ACTIVATE(nf(nf(ng(X'''), X2''), X2)) -> F(f(ng(X'''), X2''), X2)
ACTIVATE(nf(nf(ng(ng(X')), X2''), X2)) -> F(f(ng(g(activate(X'))), X2''), X2)
ACTIVATE(nf(nf(ng(nf(X1', X2''')), X2''), X2)) -> F(f(ng(f(activate(X1'), X2''')), X2''), X2)
ACTIVATE(nf(nf(nf(X1''', X2''''), X2''), X2)) -> F(f(nf(X1''', X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(g(X'), X2''''), X2''), X2)) -> F(f(f(X', nf(ng(X'), activate(X2''''))), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2'''), X2''), X2)) -> F(f(f(g(X''), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(ng(X'')), X2'''), X2''), X2)) -> F(f(f(g(g(activate(X''))), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(nf(X1', X2'0)), X2'''), X2''), X2)) -> F(f(f(g(f(activate(X1'), X2'0)), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2'''), X2''), X2)) -> F(f(f(ng(activate(X'')), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2''''), X2''), X2)) -> F(f(nf(g(activate(X'')), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2''''), X2''), X2)) -> F(f(f(activate(X''), nf(ng(activate(X'')), activate(X2''''))), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1''', X2'0), X2'''), X2''), X2)) -> F(f(f(f(X1''', X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X'), X2'0), X2'''), X2''), X2)) -> F(f(f(f(g(activate(X')), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(X1', X2'1), X2'0), X2'''), X2''), X2)) -> F(f(f(f(f(activate(X1'), X2'1), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1''', X2'0'), X2'''), X2''), X2)) -> F(f(f(nf(activate(X1'''), X2'0'), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1''', X2'0'), X2''''), X2''), X2)) -> F(f(nf(f(activate(X1'''), X2'0'), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(X1''', X2''''), X2''), X2)) -> F(f(nf(X1''', X2''''), X2''), X2)
ACTIVATE(nf(nf(g(X'), X2'''), X2)) -> F(f(X', nf(ng(X'), activate(X2'''))), X2)
ACTIVATE(nf(nf(ng(X''), X2'''), X2)) -> F(f(activate(X''), nf(ng(activate(X'')), activate(X2'''))), X2)
ACTIVATE(nf(X1', X2)) -> F(X1', X2)
ACTIVATE(nf(ng(X'), X2)) -> F(g(activate(X')), X2)
ACTIVATE(ng(X)) -> ACTIVATE(X)
ACTIVATE(nf(X1, X2)) -> ACTIVATE(X1)
F(g(X), Y) -> ACTIVATE(Y)
F(g(X), Y) -> F(X, nf(ng(X), activate(Y)))
ACTIVATE(nf(nf(nf(nf(X1'', X2'''), X2''''), X2''), X2)) -> F(f(nf(f(X1'', X2'''), X2''''), X2''), X2)

Rules:

f(g(X), Y) -> f(X, nf(ng(X), activate(Y)))
f(X1, X2) -> nf(X1, X2)
g(X) -> ng(X)
activate(nf(X1, X2)) -> f(activate(X1), X2)
activate(ng(X)) -> g(activate(X))
activate(X) -> X

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

ACTIVATE(nf(nf(nf(X1''', X2''''), X2''), X2)) -> F(f(nf(X1''', X2''''), X2''), X2)
one new Dependency Pair is created:

ACTIVATE(nf(nf(nf(X1'''', X2'''''), X2'''), X2)) -> F(nf(nf(X1'''', 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`
`                 ↳Narrowing Transformation`

Dependency Pairs:

ACTIVATE(nf(nf(nf(ng(ng(X'')), X2''''), X2''), X2)) -> F(f(nf(g(g(activate(X''))), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(nf(X1', X2''')), X2''''), X2''), X2)) -> F(f(nf(g(f(activate(X1'), X2''')), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2''''), X2''), X2)) -> F(f(nf(ng(activate(X'')), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1'', X2'''), X2''''), X2''), X2)) -> F(f(nf(f(X1'', X2'''), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X'), X2'''), X2''''), X2''), X2)) -> F(f(nf(f(g(activate(X')), X2'''), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(X1'', X2'0), X2'''), X2''''), X2''), X2)) -> F(f(nf(f(f(activate(X1''), X2'0), X2'''), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1'', X2'''''), X2''''), X2''), X2)) -> F(f(nf(nf(activate(X1''), X2'''''), X2''''), X2''), X2)
ACTIVATE(nf(nf(ng(X'''), X2''), X2)) -> F(f(ng(X'''), X2''), X2)
ACTIVATE(nf(nf(ng(X'''), X2'''), X2)) -> F(f(X''', nf(ng(X'''), activate(X2'''))), X2)
ACTIVATE(nf(nf(ng(ng(X''')), X2''), X2)) -> F(f(g(g(X''')), X2''), X2)
ACTIVATE(nf(nf(ng(ng(ng(X'))), X2''), X2)) -> F(f(g(g(g(activate(X')))), X2''), X2)
ACTIVATE(nf(nf(ng(ng(nf(X1', X2'''))), X2''), X2)) -> F(f(g(g(f(activate(X1'), X2'''))), X2''), X2)
ACTIVATE(nf(nf(ng(ng(X''')), X2''), X2)) -> F(f(g(ng(activate(X'''))), X2''), X2)
ACTIVATE(nf(nf(ng(ng(X''')), X2''), X2)) -> F(f(ng(g(activate(X'''))), X2''), X2)
ACTIVATE(nf(nf(ng(ng(X''')), X2'''), X2)) -> F(f(g(activate(X''')), nf(ng(g(activate(X'''))), activate(X2'''))), X2)
ACTIVATE(nf(nf(ng(nf(X1'', X2''')), X2''), X2)) -> F(f(g(f(X1'', X2''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(ng(X'), X2''')), X2''), X2)) -> F(f(g(f(g(activate(X')), X2''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(nf(X1'', X2'0), X2''')), X2''), X2)) -> F(f(g(f(f(activate(X1''), X2'0), X2''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(X1'', X2'''')), X2''), X2)) -> F(f(g(nf(activate(X1''), X2'''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(X1'', X2'''')), X2''), X2)) -> F(f(ng(f(activate(X1''), X2'''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(X1'', X2''''')), X2''''), X2)) -> F(f(f(activate(X1''), X2'''''), nf(ng(f(activate(X1''), X2''''')), activate(X2''''))), X2)
ACTIVATE(nf(nf(ng(X'''), X2''), X2)) -> F(f(ng(X'''), X2''), X2)
ACTIVATE(nf(nf(ng(ng(X')), X2''), X2)) -> F(f(ng(g(activate(X'))), X2''), X2)
ACTIVATE(nf(nf(ng(nf(X1', X2''')), X2''), X2)) -> F(f(ng(f(activate(X1'), X2''')), X2''), X2)
ACTIVATE(nf(nf(nf(X1''', X2''''), X2''), X2)) -> F(f(nf(X1''', X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(g(X'), X2''''), X2''), X2)) -> F(f(f(X', nf(ng(X'), activate(X2''''))), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2'''), X2''), X2)) -> F(f(f(g(X''), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(ng(X'')), X2'''), X2''), X2)) -> F(f(f(g(g(activate(X''))), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(nf(X1', X2'0)), X2'''), X2''), X2)) -> F(f(f(g(f(activate(X1'), X2'0)), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2'''), X2''), X2)) -> F(f(f(ng(activate(X'')), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2''''), X2''), X2)) -> F(f(nf(g(activate(X'')), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2''''), X2''), X2)) -> F(f(f(activate(X''), nf(ng(activate(X'')), activate(X2''''))), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1''', X2'0), X2'''), X2''), X2)) -> F(f(f(f(X1''', X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X'), X2'0), X2'''), X2''), X2)) -> F(f(f(f(g(activate(X')), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(X1', X2'1), X2'0), X2'''), X2''), X2)) -> F(f(f(f(f(activate(X1'), X2'1), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1''', X2'0'), X2'''), X2''), X2)) -> F(f(f(nf(activate(X1'''), X2'0'), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1''', X2'0'), X2''''), X2''), X2)) -> F(f(nf(f(activate(X1'''), X2'0'), X2''''), X2''), X2)
ACTIVATE(nf(nf(g(X'), X2'''), X2)) -> F(f(X', nf(ng(X'), activate(X2'''))), X2)
ACTIVATE(nf(nf(ng(X''), X2'''), X2)) -> F(f(activate(X''), nf(ng(activate(X'')), activate(X2'''))), X2)
ACTIVATE(nf(X1', X2)) -> F(X1', X2)
ACTIVATE(nf(ng(X'), X2)) -> F(g(activate(X')), X2)
ACTIVATE(ng(X)) -> ACTIVATE(X)
ACTIVATE(nf(X1, X2)) -> ACTIVATE(X1)
F(g(X), Y) -> ACTIVATE(Y)
F(g(X), Y) -> F(X, nf(ng(X), activate(Y)))
ACTIVATE(nf(nf(nf(ng(X''), X2''''), X2''), X2)) -> F(f(nf(g(X''), X2''''), X2''), X2)

Rules:

f(g(X), Y) -> f(X, nf(ng(X), activate(Y)))
f(X1, X2) -> nf(X1, X2)
g(X) -> ng(X)
activate(nf(X1, X2)) -> f(activate(X1), X2)
activate(ng(X)) -> g(activate(X))
activate(X) -> X

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

ACTIVATE(nf(nf(nf(nf(X1''', X2'0'), X2''''), X2''), X2)) -> F(f(nf(f(activate(X1'''), X2'0'), X2''''), X2''), X2)
five new Dependency Pairs are created:

ACTIVATE(nf(nf(nf(nf(X1'''', X2'0''), X2'''''), X2'''), X2)) -> F(nf(nf(f(activate(X1''''), X2'0''), X2'''''), X2'''), X2)
ACTIVATE(nf(nf(nf(nf(X1'''', X2'0''), X2''''), X2''), X2)) -> F(f(nf(nf(activate(X1''''), X2'0''), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(X1', X2'''), X2'0'), X2''''), X2''), X2)) -> F(f(nf(f(f(activate(X1'), X2'''), X2'0'), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X'), X2'0'), X2''''), X2''), X2)) -> F(f(nf(f(g(activate(X')), X2'0'), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1'''', X2'0'), X2''''), X2''), X2)) -> F(f(nf(f(X1'''', X2'0'), 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`
`                 ↳Narrowing Transformation`

Dependency Pairs:

ACTIVATE(nf(nf(nf(nf(X1'''', X2'0'), X2''''), X2''), X2)) -> F(f(nf(f(X1'''', X2'0'), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X'), X2'0'), X2''''), X2''), X2)) -> F(f(nf(f(g(activate(X')), X2'0'), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(X1', X2'''), X2'0'), X2''''), X2''), X2)) -> F(f(nf(f(f(activate(X1'), X2'''), X2'0'), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1'''', X2'0''), X2''''), X2''), X2)) -> F(f(nf(nf(activate(X1''''), X2'0''), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2''''), X2''), X2)) -> F(f(nf(g(X''), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(nf(X1', X2''')), X2''''), X2''), X2)) -> F(f(nf(g(f(activate(X1'), X2''')), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2''''), X2''), X2)) -> F(f(nf(ng(activate(X'')), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1'', X2'''), X2''''), X2''), X2)) -> F(f(nf(f(X1'', X2'''), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X'), X2'''), X2''''), X2''), X2)) -> F(f(nf(f(g(activate(X')), X2'''), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(X1'', X2'0), X2'''), X2''''), X2''), X2)) -> F(f(nf(f(f(activate(X1''), X2'0), X2'''), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1'', X2'''''), X2''''), X2''), X2)) -> F(f(nf(nf(activate(X1''), X2'''''), X2''''), X2''), X2)
ACTIVATE(nf(nf(ng(X'''), X2''), X2)) -> F(f(ng(X'''), X2''), X2)
ACTIVATE(nf(nf(ng(X'''), X2'''), X2)) -> F(f(X''', nf(ng(X'''), activate(X2'''))), X2)
ACTIVATE(nf(nf(ng(ng(X''')), X2''), X2)) -> F(f(g(g(X''')), X2''), X2)
ACTIVATE(nf(nf(ng(ng(ng(X'))), X2''), X2)) -> F(f(g(g(g(activate(X')))), X2''), X2)
ACTIVATE(nf(nf(ng(ng(nf(X1', X2'''))), X2''), X2)) -> F(f(g(g(f(activate(X1'), X2'''))), X2''), X2)
ACTIVATE(nf(nf(ng(ng(X''')), X2''), X2)) -> F(f(g(ng(activate(X'''))), X2''), X2)
ACTIVATE(nf(nf(ng(ng(X''')), X2''), X2)) -> F(f(ng(g(activate(X'''))), X2''), X2)
ACTIVATE(nf(nf(ng(ng(X''')), X2'''), X2)) -> F(f(g(activate(X''')), nf(ng(g(activate(X'''))), activate(X2'''))), X2)
ACTIVATE(nf(nf(ng(nf(X1'', X2''')), X2''), X2)) -> F(f(g(f(X1'', X2''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(ng(X'), X2''')), X2''), X2)) -> F(f(g(f(g(activate(X')), X2''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(nf(X1'', X2'0), X2''')), X2''), X2)) -> F(f(g(f(f(activate(X1''), X2'0), X2''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(X1'', X2'''')), X2''), X2)) -> F(f(g(nf(activate(X1''), X2'''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(X1'', X2'''')), X2''), X2)) -> F(f(ng(f(activate(X1''), X2'''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(X1'', X2''''')), X2''''), X2)) -> F(f(f(activate(X1''), X2'''''), nf(ng(f(activate(X1''), X2''''')), activate(X2''''))), X2)
ACTIVATE(nf(nf(ng(X'''), X2''), X2)) -> F(f(ng(X'''), X2''), X2)
ACTIVATE(nf(nf(ng(ng(X')), X2''), X2)) -> F(f(ng(g(activate(X'))), X2''), X2)
ACTIVATE(nf(nf(ng(nf(X1', X2''')), X2''), X2)) -> F(f(ng(f(activate(X1'), X2''')), X2''), X2)
ACTIVATE(nf(nf(nf(X1''', X2''''), X2''), X2)) -> F(f(nf(X1''', X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(g(X'), X2''''), X2''), X2)) -> F(f(f(X', nf(ng(X'), activate(X2''''))), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2'''), X2''), X2)) -> F(f(f(g(X''), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(ng(X'')), X2'''), X2''), X2)) -> F(f(f(g(g(activate(X''))), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(nf(X1', X2'0)), X2'''), X2''), X2)) -> F(f(f(g(f(activate(X1'), X2'0)), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2'''), X2''), X2)) -> F(f(f(ng(activate(X'')), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2''''), X2''), X2)) -> F(f(nf(g(activate(X'')), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2''''), X2''), X2)) -> F(f(f(activate(X''), nf(ng(activate(X'')), activate(X2''''))), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1''', X2'0), X2'''), X2''), X2)) -> F(f(f(f(X1''', X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X'), X2'0), X2'''), X2''), X2)) -> F(f(f(f(g(activate(X')), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(X1', X2'1), X2'0), X2'''), X2''), X2)) -> F(f(f(f(f(activate(X1'), X2'1), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1''', X2'0'), X2'''), X2''), X2)) -> F(f(f(nf(activate(X1'''), X2'0'), X2'''), X2''), X2)
ACTIVATE(nf(nf(g(X'), X2'''), X2)) -> F(f(X', nf(ng(X'), activate(X2'''))), X2)
ACTIVATE(nf(nf(ng(X''), X2'''), X2)) -> F(f(activate(X''), nf(ng(activate(X'')), activate(X2'''))), X2)
ACTIVATE(nf(X1', X2)) -> F(X1', X2)
ACTIVATE(nf(ng(X'), X2)) -> F(g(activate(X')), X2)
ACTIVATE(ng(X)) -> ACTIVATE(X)
ACTIVATE(nf(X1, X2)) -> ACTIVATE(X1)
F(g(X), Y) -> ACTIVATE(Y)
F(g(X), Y) -> F(X, nf(ng(X), activate(Y)))
ACTIVATE(nf(nf(nf(ng(ng(X'')), X2''''), X2''), X2)) -> F(f(nf(g(g(activate(X''))), X2''''), X2''), X2)

Rules:

f(g(X), Y) -> f(X, nf(ng(X), activate(Y)))
f(X1, X2) -> nf(X1, X2)
g(X) -> ng(X)
activate(nf(X1, X2)) -> f(activate(X1), X2)
activate(ng(X)) -> g(activate(X))
activate(X) -> X

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

ACTIVATE(nf(nf(nf(nf(X1''', X2'0'), X2'''), X2''), X2)) -> F(f(f(nf(activate(X1'''), X2'0'), X2'''), X2''), X2)
five new Dependency Pairs are created:

ACTIVATE(nf(nf(nf(nf(X1'''', X2'0''), X2'''''), X2''''), X2)) -> F(nf(f(nf(activate(X1''''), X2'0''), X2'''''), X2''''), X2)
ACTIVATE(nf(nf(nf(nf(X1'''', X2'0''), X2''''), X2''), X2)) -> F(f(nf(nf(activate(X1''''), X2'0''), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(X1', X2'0), X2'0'), X2'''), X2''), X2)) -> F(f(f(nf(f(activate(X1'), X2'0), X2'0'), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X'), X2'0'), X2'''), X2''), X2)) -> F(f(f(nf(g(activate(X')), X2'0'), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1'''', X2'0'), X2'''), X2''), X2)) -> F(f(f(nf(X1'''', X2'0'), X2'''), X2''), X2)

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 19`
`                 ↳Narrowing Transformation`

Dependency Pairs:

ACTIVATE(nf(nf(nf(nf(X1'''', X2'0'), X2'''), X2''), X2)) -> F(f(f(nf(X1'''', X2'0'), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X'), X2'0'), X2'''), X2''), X2)) -> F(f(f(nf(g(activate(X')), X2'0'), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(X1', X2'0), X2'0'), X2'''), X2''), X2)) -> F(f(f(nf(f(activate(X1'), X2'0), X2'0'), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1'''', X2'0''), X2''''), X2''), X2)) -> F(f(nf(nf(activate(X1''''), X2'0''), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X'), X2'0'), X2''''), X2''), X2)) -> F(f(nf(f(g(activate(X')), X2'0'), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(X1', X2'''), X2'0'), X2''''), X2''), X2)) -> F(f(nf(f(f(activate(X1'), X2'''), X2'0'), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1'''', X2'0''), X2''''), X2''), X2)) -> F(f(nf(nf(activate(X1''''), X2'0''), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2''''), X2''), X2)) -> F(f(nf(g(X''), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(ng(X'')), X2''''), X2''), X2)) -> F(f(nf(g(g(activate(X''))), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(nf(X1', X2''')), X2''''), X2''), X2)) -> F(f(nf(g(f(activate(X1'), X2''')), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2''''), X2''), X2)) -> F(f(nf(ng(activate(X'')), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1'', X2'''), X2''''), X2''), X2)) -> F(f(nf(f(X1'', X2'''), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X'), X2'''), X2''''), X2''), X2)) -> F(f(nf(f(g(activate(X')), X2'''), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(X1'', X2'0), X2'''), X2''''), X2''), X2)) -> F(f(nf(f(f(activate(X1''), X2'0), X2'''), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1'', X2'''''), X2''''), X2''), X2)) -> F(f(nf(nf(activate(X1''), X2'''''), X2''''), X2''), X2)
ACTIVATE(nf(nf(ng(X'''), X2''), X2)) -> F(f(ng(X'''), X2''), X2)
ACTIVATE(nf(nf(ng(X'''), X2'''), X2)) -> F(f(X''', nf(ng(X'''), activate(X2'''))), X2)
ACTIVATE(nf(nf(ng(ng(X''')), X2''), X2)) -> F(f(g(g(X''')), X2''), X2)
ACTIVATE(nf(nf(ng(ng(ng(X'))), X2''), X2)) -> F(f(g(g(g(activate(X')))), X2''), X2)
ACTIVATE(nf(nf(ng(ng(nf(X1', X2'''))), X2''), X2)) -> F(f(g(g(f(activate(X1'), X2'''))), X2''), X2)
ACTIVATE(nf(nf(ng(ng(X''')), X2''), X2)) -> F(f(g(ng(activate(X'''))), X2''), X2)
ACTIVATE(nf(nf(ng(ng(X''')), X2''), X2)) -> F(f(ng(g(activate(X'''))), X2''), X2)
ACTIVATE(nf(nf(ng(ng(X''')), X2'''), X2)) -> F(f(g(activate(X''')), nf(ng(g(activate(X'''))), activate(X2'''))), X2)
ACTIVATE(nf(nf(ng(nf(X1'', X2''')), X2''), X2)) -> F(f(g(f(X1'', X2''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(ng(X'), X2''')), X2''), X2)) -> F(f(g(f(g(activate(X')), X2''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(nf(X1'', X2'0), X2''')), X2''), X2)) -> F(f(g(f(f(activate(X1''), X2'0), X2''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(X1'', X2'''')), X2''), X2)) -> F(f(g(nf(activate(X1''), X2'''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(X1'', X2'''')), X2''), X2)) -> F(f(ng(f(activate(X1''), X2'''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(X1'', X2''''')), X2''''), X2)) -> F(f(f(activate(X1''), X2'''''), nf(ng(f(activate(X1''), X2''''')), activate(X2''''))), X2)
ACTIVATE(nf(nf(ng(X'''), X2''), X2)) -> F(f(ng(X'''), X2''), X2)
ACTIVATE(nf(nf(ng(ng(X')), X2''), X2)) -> F(f(ng(g(activate(X'))), X2''), X2)
ACTIVATE(nf(nf(ng(nf(X1', X2''')), X2''), X2)) -> F(f(ng(f(activate(X1'), X2''')), X2''), X2)
ACTIVATE(nf(nf(nf(X1''', X2''''), X2''), X2)) -> F(f(nf(X1''', X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(g(X'), X2''''), X2''), X2)) -> F(f(f(X', nf(ng(X'), activate(X2''''))), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2'''), X2''), X2)) -> F(f(f(g(X''), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(ng(X'')), X2'''), X2''), X2)) -> F(f(f(g(g(activate(X''))), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(nf(X1', X2'0)), X2'''), X2''), X2)) -> F(f(f(g(f(activate(X1'), X2'0)), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2'''), X2''), X2)) -> F(f(f(ng(activate(X'')), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2''''), X2''), X2)) -> F(f(nf(g(activate(X'')), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2''''), X2''), X2)) -> F(f(f(activate(X''), nf(ng(activate(X'')), activate(X2''''))), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1''', X2'0), X2'''), X2''), X2)) -> F(f(f(f(X1''', X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X'), X2'0), X2'''), X2''), X2)) -> F(f(f(f(g(activate(X')), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(X1', X2'1), X2'0), X2'''), X2''), X2)) -> F(f(f(f(f(activate(X1'), X2'1), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(g(X'), X2'''), X2)) -> F(f(X', nf(ng(X'), activate(X2'''))), X2)
ACTIVATE(nf(nf(ng(X''), X2'''), X2)) -> F(f(activate(X''), nf(ng(activate(X'')), activate(X2'''))), X2)
ACTIVATE(nf(X1', X2)) -> F(X1', X2)
ACTIVATE(nf(ng(X'), X2)) -> F(g(activate(X')), X2)
ACTIVATE(ng(X)) -> ACTIVATE(X)
ACTIVATE(nf(X1, X2)) -> ACTIVATE(X1)
F(g(X), Y) -> ACTIVATE(Y)
F(g(X), Y) -> F(X, nf(ng(X), activate(Y)))
ACTIVATE(nf(nf(nf(nf(X1'''', X2'0'), X2''''), X2''), X2)) -> F(f(nf(f(X1'''', X2'0'), X2''''), X2''), X2)

Rules:

f(g(X), Y) -> f(X, nf(ng(X), activate(Y)))
f(X1, X2) -> nf(X1, X2)
g(X) -> ng(X)
activate(nf(X1, X2)) -> f(activate(X1), X2)
activate(ng(X)) -> g(activate(X))
activate(X) -> X

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

ACTIVATE(nf(nf(nf(nf(nf(X1', X2'1), X2'0), X2'''), X2''), X2)) -> F(f(f(f(f(activate(X1'), X2'1), X2'0), X2'''), X2''), X2)
seven new Dependency Pairs are created:

ACTIVATE(nf(nf(nf(nf(nf(X1'', X2'1'), X2'0'), X2'''''), X2''''), X2)) -> F(nf(f(f(f(activate(X1''), X2'1'), X2'0'), X2'''''), X2''''), X2)
ACTIVATE(nf(nf(nf(nf(nf(X1'', X2'1'), X2'0'), X2''''), X2''), X2)) -> F(f(nf(f(f(activate(X1''), X2'1'), X2'0'), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(X1'', X2'1'), X2'0'), X2'''), X2''), X2)) -> F(f(f(nf(f(activate(X1''), X2'1'), X2'0'), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(X1'', X2'1'), X2'0), X2'''), X2''), X2)) -> F(f(f(f(nf(activate(X1''), X2'1'), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(nf(X1'', X2'2), X2'1), X2'0), X2'''), X2''), X2)) -> F(f(f(f(f(f(activate(X1''), X2'2), X2'1), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(ng(X'), X2'1), X2'0), X2'''), X2''), X2)) -> F(f(f(f(f(g(activate(X')), X2'1), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(X1'', X2'1), X2'0), X2'''), X2''), X2)) -> F(f(f(f(f(X1'', X2'1), X2'0), X2'''), X2''), X2)

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 20`
`                 ↳Narrowing Transformation`

Dependency Pairs:

ACTIVATE(nf(nf(nf(nf(nf(X1'', X2'1), X2'0), X2'''), X2''), X2)) -> F(f(f(f(f(X1'', X2'1), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(ng(X'), X2'1), X2'0), X2'''), X2''), X2)) -> F(f(f(f(f(g(activate(X')), X2'1), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(nf(X1'', X2'2), X2'1), X2'0), X2'''), X2''), X2)) -> F(f(f(f(f(f(activate(X1''), X2'2), X2'1), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(X1'', X2'1'), X2'0), X2'''), X2''), X2)) -> F(f(f(f(nf(activate(X1''), X2'1'), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(X1'', X2'1'), X2'0'), X2'''), X2''), X2)) -> F(f(f(nf(f(activate(X1''), X2'1'), X2'0'), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(X1'', X2'1'), X2'0'), X2''''), X2''), X2)) -> F(f(nf(f(f(activate(X1''), X2'1'), X2'0'), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X'), X2'0'), X2'''), X2''), X2)) -> F(f(f(nf(g(activate(X')), X2'0'), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(X1', X2'0), X2'0'), X2'''), X2''), X2)) -> F(f(f(nf(f(activate(X1'), X2'0), X2'0'), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1'''', X2'0''), X2''''), X2''), X2)) -> F(f(nf(nf(activate(X1''''), X2'0''), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1'''', X2'0'), X2''''), X2''), X2)) -> F(f(nf(f(X1'''', X2'0'), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X'), X2'0'), X2''''), X2''), X2)) -> F(f(nf(f(g(activate(X')), X2'0'), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(X1', X2'''), X2'0'), X2''''), X2''), X2)) -> F(f(nf(f(f(activate(X1'), X2'''), X2'0'), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1'''', X2'0''), X2''''), X2''), X2)) -> F(f(nf(nf(activate(X1''''), X2'0''), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2''''), X2''), X2)) -> F(f(nf(g(X''), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(ng(X'')), X2''''), X2''), X2)) -> F(f(nf(g(g(activate(X''))), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(nf(X1', X2''')), X2''''), X2''), X2)) -> F(f(nf(g(f(activate(X1'), X2''')), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2''''), X2''), X2)) -> F(f(nf(ng(activate(X'')), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1'', X2'''), X2''''), X2''), X2)) -> F(f(nf(f(X1'', X2'''), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X'), X2'''), X2''''), X2''), X2)) -> F(f(nf(f(g(activate(X')), X2'''), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(X1'', X2'0), X2'''), X2''''), X2''), X2)) -> F(f(nf(f(f(activate(X1''), X2'0), X2'''), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1'', X2'''''), X2''''), X2''), X2)) -> F(f(nf(nf(activate(X1''), X2'''''), X2''''), X2''), X2)
ACTIVATE(nf(nf(ng(X'''), X2''), X2)) -> F(f(ng(X'''), X2''), X2)
ACTIVATE(nf(nf(ng(X'''), X2'''), X2)) -> F(f(X''', nf(ng(X'''), activate(X2'''))), X2)
ACTIVATE(nf(nf(ng(ng(X''')), X2''), X2)) -> F(f(g(g(X''')), X2''), X2)
ACTIVATE(nf(nf(ng(ng(ng(X'))), X2''), X2)) -> F(f(g(g(g(activate(X')))), X2''), X2)
ACTIVATE(nf(nf(ng(ng(nf(X1', X2'''))), X2''), X2)) -> F(f(g(g(f(activate(X1'), X2'''))), X2''), X2)
ACTIVATE(nf(nf(ng(ng(X''')), X2''), X2)) -> F(f(g(ng(activate(X'''))), X2''), X2)
ACTIVATE(nf(nf(ng(ng(X''')), X2''), X2)) -> F(f(ng(g(activate(X'''))), X2''), X2)
ACTIVATE(nf(nf(ng(ng(X''')), X2'''), X2)) -> F(f(g(activate(X''')), nf(ng(g(activate(X'''))), activate(X2'''))), X2)
ACTIVATE(nf(nf(ng(nf(X1'', X2''')), X2''), X2)) -> F(f(g(f(X1'', X2''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(ng(X'), X2''')), X2''), X2)) -> F(f(g(f(g(activate(X')), X2''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(nf(X1'', X2'0), X2''')), X2''), X2)) -> F(f(g(f(f(activate(X1''), X2'0), X2''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(X1'', X2'''')), X2''), X2)) -> F(f(g(nf(activate(X1''), X2'''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(X1'', X2'''')), X2''), X2)) -> F(f(ng(f(activate(X1''), X2'''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(X1'', X2''''')), X2''''), X2)) -> F(f(f(activate(X1''), X2'''''), nf(ng(f(activate(X1''), X2''''')), activate(X2''''))), X2)
ACTIVATE(nf(nf(ng(X'''), X2''), X2)) -> F(f(ng(X'''), X2''), X2)
ACTIVATE(nf(nf(ng(ng(X')), X2''), X2)) -> F(f(ng(g(activate(X'))), X2''), X2)
ACTIVATE(nf(nf(ng(nf(X1', X2''')), X2''), X2)) -> F(f(ng(f(activate(X1'), X2''')), X2''), X2)
ACTIVATE(nf(nf(nf(X1''', X2''''), X2''), X2)) -> F(f(nf(X1''', X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(g(X'), X2''''), X2''), X2)) -> F(f(f(X', nf(ng(X'), activate(X2''''))), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2'''), X2''), X2)) -> F(f(f(g(X''), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(ng(X'')), X2'''), X2''), X2)) -> F(f(f(g(g(activate(X''))), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(nf(X1', X2'0)), X2'''), X2''), X2)) -> F(f(f(g(f(activate(X1'), X2'0)), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2'''), X2''), X2)) -> F(f(f(ng(activate(X'')), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2''''), X2''), X2)) -> F(f(nf(g(activate(X'')), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2''''), X2''), X2)) -> F(f(f(activate(X''), nf(ng(activate(X'')), activate(X2''''))), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1''', X2'0), X2'''), X2''), X2)) -> F(f(f(f(X1''', X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X'), X2'0), X2'''), X2''), X2)) -> F(f(f(f(g(activate(X')), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(g(X'), X2'''), X2)) -> F(f(X', nf(ng(X'), activate(X2'''))), X2)
ACTIVATE(nf(nf(ng(X''), X2'''), X2)) -> F(f(activate(X''), nf(ng(activate(X'')), activate(X2'''))), X2)
ACTIVATE(nf(X1', X2)) -> F(X1', X2)
ACTIVATE(nf(ng(X'), X2)) -> F(g(activate(X')), X2)
ACTIVATE(ng(X)) -> ACTIVATE(X)
ACTIVATE(nf(X1, X2)) -> ACTIVATE(X1)
F(g(X), Y) -> ACTIVATE(Y)
F(g(X), Y) -> F(X, nf(ng(X), activate(Y)))
ACTIVATE(nf(nf(nf(nf(X1'''', X2'0'), X2'''), X2''), X2)) -> F(f(f(nf(X1'''', X2'0'), X2'''), X2''), X2)

Rules:

f(g(X), Y) -> f(X, nf(ng(X), activate(Y)))
f(X1, X2) -> nf(X1, X2)
g(X) -> ng(X)
activate(nf(X1, X2)) -> f(activate(X1), X2)
activate(ng(X)) -> g(activate(X))
activate(X) -> X

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

ACTIVATE(nf(nf(nf(nf(ng(X'), X2'0), X2'''), X2''), X2)) -> F(f(f(f(g(activate(X')), X2'0), X2'''), X2''), X2)
eight new Dependency Pairs are created:

ACTIVATE(nf(nf(nf(nf(ng(X''), X2'0'), X2'''''), X2''''), X2)) -> F(nf(f(f(g(activate(X'')), X2'0'), X2'''''), X2''''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X''), X2'0'), X2''''), X2''), X2)) -> F(f(nf(f(g(activate(X'')), X2'0'), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X''), X2'0'), X2'''), X2''), X2)) -> F(f(f(f(activate(X''), nf(ng(activate(X'')), activate(X2'0'))), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X''), X2'0'), X2'''), X2''), X2)) -> F(f(f(nf(g(activate(X'')), X2'0'), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X''), X2'0), X2'''), X2''), X2)) -> F(f(f(f(ng(activate(X'')), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(nf(X1', X2'1)), X2'0), X2'''), X2''), X2)) -> F(f(f(f(g(f(activate(X1'), X2'1)), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(ng(X'')), X2'0), X2'''), X2''), X2)) -> F(f(f(f(g(g(activate(X''))), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X''), X2'0), X2'''), X2''), X2)) -> F(f(f(f(g(X''), X2'0), X2'''), X2''), X2)

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 21`
`                 ↳Narrowing Transformation`

Dependency Pairs:

ACTIVATE(nf(nf(nf(nf(ng(X''), X2'0), X2'''), X2''), X2)) -> F(f(f(f(g(X''), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(ng(X'')), X2'0), X2'''), X2''), X2)) -> F(f(f(f(g(g(activate(X''))), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(nf(X1', X2'1)), X2'0), X2'''), X2''), X2)) -> F(f(f(f(g(f(activate(X1'), X2'1)), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X''), X2'0), X2'''), X2''), X2)) -> F(f(f(f(ng(activate(X'')), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X''), X2'0'), X2'''), X2''), X2)) -> F(f(f(nf(g(activate(X'')), X2'0'), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X''), X2'0'), X2'''), X2''), X2)) -> F(f(f(f(activate(X''), nf(ng(activate(X'')), activate(X2'0'))), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X''), X2'0'), X2''''), X2''), X2)) -> F(f(nf(f(g(activate(X'')), X2'0'), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(ng(X'), X2'1), X2'0), X2'''), X2''), X2)) -> F(f(f(f(f(g(activate(X')), X2'1), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(nf(X1'', X2'2), X2'1), X2'0), X2'''), X2''), X2)) -> F(f(f(f(f(f(activate(X1''), X2'2), X2'1), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(X1'', X2'1'), X2'0), X2'''), X2''), X2)) -> F(f(f(f(nf(activate(X1''), X2'1'), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(X1'', X2'1'), X2'0'), X2'''), X2''), X2)) -> F(f(f(nf(f(activate(X1''), X2'1'), X2'0'), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(X1'', X2'1'), X2'0'), X2''''), X2''), X2)) -> F(f(nf(f(f(activate(X1''), X2'1'), X2'0'), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1'''', X2'0'), X2'''), X2''), X2)) -> F(f(f(nf(X1'''', X2'0'), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X'), X2'0'), X2'''), X2''), X2)) -> F(f(f(nf(g(activate(X')), X2'0'), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(X1', X2'0), X2'0'), X2'''), X2''), X2)) -> F(f(f(nf(f(activate(X1'), X2'0), X2'0'), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1'''', X2'0''), X2''''), X2''), X2)) -> F(f(nf(nf(activate(X1''''), X2'0''), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1'''', X2'0'), X2''''), X2''), X2)) -> F(f(nf(f(X1'''', X2'0'), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X'), X2'0'), X2''''), X2''), X2)) -> F(f(nf(f(g(activate(X')), X2'0'), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(X1', X2'''), X2'0'), X2''''), X2''), X2)) -> F(f(nf(f(f(activate(X1'), X2'''), X2'0'), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1'''', X2'0''), X2''''), X2''), X2)) -> F(f(nf(nf(activate(X1''''), X2'0''), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2''''), X2''), X2)) -> F(f(nf(g(X''), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(ng(X'')), X2''''), X2''), X2)) -> F(f(nf(g(g(activate(X''))), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(nf(X1', X2''')), X2''''), X2''), X2)) -> F(f(nf(g(f(activate(X1'), X2''')), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2''''), X2''), X2)) -> F(f(nf(ng(activate(X'')), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1'', X2'''), X2''''), X2''), X2)) -> F(f(nf(f(X1'', X2'''), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X'), X2'''), X2''''), X2''), X2)) -> F(f(nf(f(g(activate(X')), X2'''), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(X1'', X2'0), X2'''), X2''''), X2''), X2)) -> F(f(nf(f(f(activate(X1''), X2'0), X2'''), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1'', X2'''''), X2''''), X2''), X2)) -> F(f(nf(nf(activate(X1''), X2'''''), X2''''), X2''), X2)
ACTIVATE(nf(nf(ng(X'''), X2''), X2)) -> F(f(ng(X'''), X2''), X2)
ACTIVATE(nf(nf(ng(X'''), X2'''), X2)) -> F(f(X''', nf(ng(X'''), activate(X2'''))), X2)
ACTIVATE(nf(nf(ng(ng(X''')), X2''), X2)) -> F(f(g(g(X''')), X2''), X2)
ACTIVATE(nf(nf(ng(ng(ng(X'))), X2''), X2)) -> F(f(g(g(g(activate(X')))), X2''), X2)
ACTIVATE(nf(nf(ng(ng(nf(X1', X2'''))), X2''), X2)) -> F(f(g(g(f(activate(X1'), X2'''))), X2''), X2)
ACTIVATE(nf(nf(ng(ng(X''')), X2''), X2)) -> F(f(g(ng(activate(X'''))), X2''), X2)
ACTIVATE(nf(nf(ng(ng(X''')), X2''), X2)) -> F(f(ng(g(activate(X'''))), X2''), X2)
ACTIVATE(nf(nf(ng(ng(X''')), X2'''), X2)) -> F(f(g(activate(X''')), nf(ng(g(activate(X'''))), activate(X2'''))), X2)
ACTIVATE(nf(nf(ng(nf(X1'', X2''')), X2''), X2)) -> F(f(g(f(X1'', X2''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(ng(X'), X2''')), X2''), X2)) -> F(f(g(f(g(activate(X')), X2''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(nf(X1'', X2'0), X2''')), X2''), X2)) -> F(f(g(f(f(activate(X1''), X2'0), X2''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(X1'', X2'''')), X2''), X2)) -> F(f(g(nf(activate(X1''), X2'''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(X1'', X2'''')), X2''), X2)) -> F(f(ng(f(activate(X1''), X2'''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(X1'', X2''''')), X2''''), X2)) -> F(f(f(activate(X1''), X2'''''), nf(ng(f(activate(X1''), X2''''')), activate(X2''''))), X2)
ACTIVATE(nf(nf(ng(X'''), X2''), X2)) -> F(f(ng(X'''), X2''), X2)
ACTIVATE(nf(nf(ng(ng(X')), X2''), X2)) -> F(f(ng(g(activate(X'))), X2''), X2)
ACTIVATE(nf(nf(ng(nf(X1', X2''')), X2''), X2)) -> F(f(ng(f(activate(X1'), X2''')), X2''), X2)
ACTIVATE(nf(nf(nf(X1''', X2''''), X2''), X2)) -> F(f(nf(X1''', X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(g(X'), X2''''), X2''), X2)) -> F(f(f(X', nf(ng(X'), activate(X2''''))), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2'''), X2''), X2)) -> F(f(f(g(X''), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(ng(X'')), X2'''), X2''), X2)) -> F(f(f(g(g(activate(X''))), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(nf(X1', X2'0)), X2'''), X2''), X2)) -> F(f(f(g(f(activate(X1'), X2'0)), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2'''), X2''), X2)) -> F(f(f(ng(activate(X'')), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2''''), X2''), X2)) -> F(f(nf(g(activate(X'')), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2''''), X2''), X2)) -> F(f(f(activate(X''), nf(ng(activate(X'')), activate(X2''''))), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1''', X2'0), X2'''), X2''), X2)) -> F(f(f(f(X1''', X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(g(X'), X2'''), X2)) -> F(f(X', nf(ng(X'), activate(X2'''))), X2)
ACTIVATE(nf(nf(ng(X''), X2'''), X2)) -> F(f(activate(X''), nf(ng(activate(X'')), activate(X2'''))), X2)
ACTIVATE(nf(X1', X2)) -> F(X1', X2)
ACTIVATE(nf(ng(X'), X2)) -> F(g(activate(X')), X2)
ACTIVATE(ng(X)) -> ACTIVATE(X)
ACTIVATE(nf(X1, X2)) -> ACTIVATE(X1)
F(g(X), Y) -> ACTIVATE(Y)
F(g(X), Y) -> F(X, nf(ng(X), activate(Y)))
ACTIVATE(nf(nf(nf(nf(nf(X1'', X2'1), X2'0), X2'''), X2''), X2)) -> F(f(f(f(f(X1'', X2'1), X2'0), X2'''), X2''), X2)

Rules:

f(g(X), Y) -> f(X, nf(ng(X), activate(Y)))
f(X1, X2) -> nf(X1, X2)
g(X) -> ng(X)
activate(nf(X1, X2)) -> f(activate(X1), X2)
activate(ng(X)) -> g(activate(X))
activate(X) -> X

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

ACTIVATE(nf(nf(nf(nf(X1''', X2'0), X2'''), X2''), X2)) -> F(f(f(f(X1''', X2'0), X2'''), X2''), X2)
four new Dependency Pairs are created:

ACTIVATE(nf(nf(nf(nf(X1'''', X2'0'), X2'''''), X2''''), X2)) -> F(nf(f(f(X1'''', X2'0'), X2'''''), X2''''), X2)
ACTIVATE(nf(nf(nf(nf(X1'''', X2'0'), X2''''), X2''), X2)) -> F(f(nf(f(X1'''', X2'0'), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(g(X'), X2'0'), X2'''), X2''), X2)) -> F(f(f(f(X', nf(ng(X'), activate(X2'0'))), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1'''', X2'0'), X2'''), X2''), X2)) -> F(f(f(nf(X1'''', X2'0'), X2'''), X2''), X2)

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 22`
`                 ↳Narrowing Transformation`

Dependency Pairs:

ACTIVATE(nf(nf(nf(nf(X1'''', X2'0'), X2'''), X2''), X2)) -> F(f(f(nf(X1'''', X2'0'), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(g(X'), X2'0'), X2'''), X2''), X2)) -> F(f(f(f(X', nf(ng(X'), activate(X2'0'))), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1'''', X2'0'), X2''''), X2''), X2)) -> F(f(nf(f(X1'''', X2'0'), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(ng(X'')), X2'0), X2'''), X2''), X2)) -> F(f(f(f(g(g(activate(X''))), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(nf(X1', X2'1)), X2'0), X2'''), X2''), X2)) -> F(f(f(f(g(f(activate(X1'), X2'1)), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X''), X2'0), X2'''), X2''), X2)) -> F(f(f(f(ng(activate(X'')), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X''), X2'0'), X2'''), X2''), X2)) -> F(f(f(nf(g(activate(X'')), X2'0'), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X''), X2'0'), X2'''), X2''), X2)) -> F(f(f(f(activate(X''), nf(ng(activate(X'')), activate(X2'0'))), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X''), X2'0'), X2''''), X2''), X2)) -> F(f(nf(f(g(activate(X'')), X2'0'), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(X1'', X2'1), X2'0), X2'''), X2''), X2)) -> F(f(f(f(f(X1'', X2'1), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(ng(X'), X2'1), X2'0), X2'''), X2''), X2)) -> F(f(f(f(f(g(activate(X')), X2'1), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(nf(X1'', X2'2), X2'1), X2'0), X2'''), X2''), X2)) -> F(f(f(f(f(f(activate(X1''), X2'2), X2'1), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(X1'', X2'1'), X2'0), X2'''), X2''), X2)) -> F(f(f(f(nf(activate(X1''), X2'1'), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(X1'', X2'1'), X2'0'), X2'''), X2''), X2)) -> F(f(f(nf(f(activate(X1''), X2'1'), X2'0'), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(X1'', X2'1'), X2'0'), X2''''), X2''), X2)) -> F(f(nf(f(f(activate(X1''), X2'1'), X2'0'), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1'''', X2'0'), X2'''), X2''), X2)) -> F(f(f(nf(X1'''', X2'0'), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X'), X2'0'), X2'''), X2''), X2)) -> F(f(f(nf(g(activate(X')), X2'0'), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(X1', X2'0), X2'0'), X2'''), X2''), X2)) -> F(f(f(nf(f(activate(X1'), X2'0), X2'0'), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1'''', X2'0''), X2''''), X2''), X2)) -> F(f(nf(nf(activate(X1''''), X2'0''), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1'''', X2'0'), X2''''), X2''), X2)) -> F(f(nf(f(X1'''', X2'0'), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X'), X2'0'), X2''''), X2''), X2)) -> F(f(nf(f(g(activate(X')), X2'0'), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(X1', X2'''), X2'0'), X2''''), X2''), X2)) -> F(f(nf(f(f(activate(X1'), X2'''), X2'0'), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1'''', X2'0''), X2''''), X2''), X2)) -> F(f(nf(nf(activate(X1''''), X2'0''), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2''''), X2''), X2)) -> F(f(nf(g(X''), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(ng(X'')), X2''''), X2''), X2)) -> F(f(nf(g(g(activate(X''))), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(nf(X1', X2''')), X2''''), X2''), X2)) -> F(f(nf(g(f(activate(X1'), X2''')), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2''''), X2''), X2)) -> F(f(nf(ng(activate(X'')), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1'', X2'''), X2''''), X2''), X2)) -> F(f(nf(f(X1'', X2'''), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X'), X2'''), X2''''), X2''), X2)) -> F(f(nf(f(g(activate(X')), X2'''), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(X1'', X2'0), X2'''), X2''''), X2''), X2)) -> F(f(nf(f(f(activate(X1''), X2'0), X2'''), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1'', X2'''''), X2''''), X2''), X2)) -> F(f(nf(nf(activate(X1''), X2'''''), X2''''), X2''), X2)
ACTIVATE(nf(nf(ng(X'''), X2''), X2)) -> F(f(ng(X'''), X2''), X2)
ACTIVATE(nf(nf(ng(X'''), X2'''), X2)) -> F(f(X''', nf(ng(X'''), activate(X2'''))), X2)
ACTIVATE(nf(nf(ng(ng(X''')), X2''), X2)) -> F(f(g(g(X''')), X2''), X2)
ACTIVATE(nf(nf(ng(ng(ng(X'))), X2''), X2)) -> F(f(g(g(g(activate(X')))), X2''), X2)
ACTIVATE(nf(nf(ng(ng(nf(X1', X2'''))), X2''), X2)) -> F(f(g(g(f(activate(X1'), X2'''))), X2''), X2)
ACTIVATE(nf(nf(ng(ng(X''')), X2''), X2)) -> F(f(g(ng(activate(X'''))), X2''), X2)
ACTIVATE(nf(nf(ng(ng(X''')), X2''), X2)) -> F(f(ng(g(activate(X'''))), X2''), X2)
ACTIVATE(nf(nf(ng(ng(X''')), X2'''), X2)) -> F(f(g(activate(X''')), nf(ng(g(activate(X'''))), activate(X2'''))), X2)
ACTIVATE(nf(nf(ng(nf(X1'', X2''')), X2''), X2)) -> F(f(g(f(X1'', X2''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(ng(X'), X2''')), X2''), X2)) -> F(f(g(f(g(activate(X')), X2''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(nf(X1'', X2'0), X2''')), X2''), X2)) -> F(f(g(f(f(activate(X1''), X2'0), X2''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(X1'', X2'''')), X2''), X2)) -> F(f(g(nf(activate(X1''), X2'''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(X1'', X2'''')), X2''), X2)) -> F(f(ng(f(activate(X1''), X2'''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(X1'', X2''''')), X2''''), X2)) -> F(f(f(activate(X1''), X2'''''), nf(ng(f(activate(X1''), X2''''')), activate(X2''''))), X2)
ACTIVATE(nf(nf(ng(X'''), X2''), X2)) -> F(f(ng(X'''), X2''), X2)
ACTIVATE(nf(nf(ng(ng(X')), X2''), X2)) -> F(f(ng(g(activate(X'))), X2''), X2)
ACTIVATE(nf(nf(ng(nf(X1', X2''')), X2''), X2)) -> F(f(ng(f(activate(X1'), X2''')), X2''), X2)
ACTIVATE(nf(nf(nf(X1''', X2''''), X2''), X2)) -> F(f(nf(X1''', X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(g(X'), X2''''), X2''), X2)) -> F(f(f(X', nf(ng(X'), activate(X2''''))), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2'''), X2''), X2)) -> F(f(f(g(X''), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(ng(X'')), X2'''), X2''), X2)) -> F(f(f(g(g(activate(X''))), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(nf(X1', X2'0)), X2'''), X2''), X2)) -> F(f(f(g(f(activate(X1'), X2'0)), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2'''), X2''), X2)) -> F(f(f(ng(activate(X'')), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2''''), X2''), X2)) -> F(f(nf(g(activate(X'')), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2''''), X2''), X2)) -> F(f(f(activate(X''), nf(ng(activate(X'')), activate(X2''''))), X2''), X2)
ACTIVATE(nf(nf(g(X'), X2'''), X2)) -> F(f(X', nf(ng(X'), activate(X2'''))), X2)
ACTIVATE(nf(nf(ng(X''), X2'''), X2)) -> F(f(activate(X''), nf(ng(activate(X'')), activate(X2'''))), X2)
ACTIVATE(nf(X1', X2)) -> F(X1', X2)
ACTIVATE(nf(ng(X'), X2)) -> F(g(activate(X')), X2)
ACTIVATE(ng(X)) -> ACTIVATE(X)
ACTIVATE(nf(X1, X2)) -> ACTIVATE(X1)
F(g(X), Y) -> ACTIVATE(Y)
F(g(X), Y) -> F(X, nf(ng(X), activate(Y)))
ACTIVATE(nf(nf(nf(nf(ng(X''), X2'0), X2'''), X2''), X2)) -> F(f(f(f(g(X''), X2'0), X2'''), X2''), X2)

Rules:

f(g(X), Y) -> f(X, nf(ng(X), activate(Y)))
f(X1, X2) -> nf(X1, X2)
g(X) -> ng(X)
activate(nf(X1, X2)) -> f(activate(X1), X2)
activate(ng(X)) -> g(activate(X))
activate(X) -> X

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

ACTIVATE(nf(nf(nf(ng(X''), X2''''), X2''), X2)) -> F(f(nf(g(activate(X'')), X2''''), X2''), X2)
five new Dependency Pairs are created:

ACTIVATE(nf(nf(nf(ng(X'''), X2'''''), X2'''), X2)) -> F(nf(nf(g(activate(X''')), X2'''''), X2'''), X2)
ACTIVATE(nf(nf(nf(ng(X'''), X2''''), X2''), X2)) -> F(f(nf(ng(activate(X''')), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(nf(X1', X2''')), X2''''), X2''), X2)) -> F(f(nf(g(f(activate(X1'), X2''')), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(ng(X')), X2''''), X2''), X2)) -> F(f(nf(g(g(activate(X'))), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X'''), X2''''), X2''), X2)) -> F(f(nf(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 23`
`                 ↳Narrowing Transformation`

Dependency Pairs:

ACTIVATE(nf(nf(nf(ng(X'''), X2''''), X2''), X2)) -> F(f(nf(g(X'''), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(ng(X')), X2''''), X2''), X2)) -> F(f(nf(g(g(activate(X'))), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(nf(X1', X2''')), X2''''), X2''), X2)) -> F(f(nf(g(f(activate(X1'), X2''')), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X'''), X2''''), X2''), X2)) -> F(f(nf(ng(activate(X''')), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(g(X'), X2'0'), X2'''), X2''), X2)) -> F(f(f(f(X', nf(ng(X'), activate(X2'0'))), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1'''', X2'0'), X2''''), X2''), X2)) -> F(f(nf(f(X1'''', X2'0'), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X''), X2'0), X2'''), X2''), X2)) -> F(f(f(f(g(X''), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(ng(X'')), X2'0), X2'''), X2''), X2)) -> F(f(f(f(g(g(activate(X''))), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(nf(X1', X2'1)), X2'0), X2'''), X2''), X2)) -> F(f(f(f(g(f(activate(X1'), X2'1)), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X''), X2'0), X2'''), X2''), X2)) -> F(f(f(f(ng(activate(X'')), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X''), X2'0'), X2'''), X2''), X2)) -> F(f(f(nf(g(activate(X'')), X2'0'), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X''), X2'0'), X2'''), X2''), X2)) -> F(f(f(f(activate(X''), nf(ng(activate(X'')), activate(X2'0'))), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X''), X2'0'), X2''''), X2''), X2)) -> F(f(nf(f(g(activate(X'')), X2'0'), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(X1'', X2'1), X2'0), X2'''), X2''), X2)) -> F(f(f(f(f(X1'', X2'1), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(ng(X'), X2'1), X2'0), X2'''), X2''), X2)) -> F(f(f(f(f(g(activate(X')), X2'1), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(nf(X1'', X2'2), X2'1), X2'0), X2'''), X2''), X2)) -> F(f(f(f(f(f(activate(X1''), X2'2), X2'1), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(X1'', X2'1'), X2'0), X2'''), X2''), X2)) -> F(f(f(f(nf(activate(X1''), X2'1'), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(X1'', X2'1'), X2'0'), X2'''), X2''), X2)) -> F(f(f(nf(f(activate(X1''), X2'1'), X2'0'), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(X1'', X2'1'), X2'0'), X2''''), X2''), X2)) -> F(f(nf(f(f(activate(X1''), X2'1'), X2'0'), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1'''', X2'0'), X2'''), X2''), X2)) -> F(f(f(nf(X1'''', X2'0'), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X'), X2'0'), X2'''), X2''), X2)) -> F(f(f(nf(g(activate(X')), X2'0'), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(X1', X2'0), X2'0'), X2'''), X2''), X2)) -> F(f(f(nf(f(activate(X1'), X2'0), X2'0'), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1'''', X2'0''), X2''''), X2''), X2)) -> F(f(nf(nf(activate(X1''''), X2'0''), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1'''', X2'0'), X2''''), X2''), X2)) -> F(f(nf(f(X1'''', X2'0'), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X'), X2'0'), X2''''), X2''), X2)) -> F(f(nf(f(g(activate(X')), X2'0'), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(X1', X2'''), X2'0'), X2''''), X2''), X2)) -> F(f(nf(f(f(activate(X1'), X2'''), X2'0'), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1'''', X2'0''), X2''''), X2''), X2)) -> F(f(nf(nf(activate(X1''''), X2'0''), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2''''), X2''), X2)) -> F(f(nf(g(X''), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(ng(X'')), X2''''), X2''), X2)) -> F(f(nf(g(g(activate(X''))), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(nf(X1', X2''')), X2''''), X2''), X2)) -> F(f(nf(g(f(activate(X1'), X2''')), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2''''), X2''), X2)) -> F(f(nf(ng(activate(X'')), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1'', X2'''), X2''''), X2''), X2)) -> F(f(nf(f(X1'', X2'''), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X'), X2'''), X2''''), X2''), X2)) -> F(f(nf(f(g(activate(X')), X2'''), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(X1'', X2'0), X2'''), X2''''), X2''), X2)) -> F(f(nf(f(f(activate(X1''), X2'0), X2'''), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1'', X2'''''), X2''''), X2''), X2)) -> F(f(nf(nf(activate(X1''), X2'''''), X2''''), X2''), X2)
ACTIVATE(nf(nf(ng(X'''), X2''), X2)) -> F(f(ng(X'''), X2''), X2)
ACTIVATE(nf(nf(ng(X'''), X2'''), X2)) -> F(f(X''', nf(ng(X'''), activate(X2'''))), X2)
ACTIVATE(nf(nf(ng(ng(X''')), X2''), X2)) -> F(f(g(g(X''')), X2''), X2)
ACTIVATE(nf(nf(ng(ng(ng(X'))), X2''), X2)) -> F(f(g(g(g(activate(X')))), X2''), X2)
ACTIVATE(nf(nf(ng(ng(nf(X1', X2'''))), X2''), X2)) -> F(f(g(g(f(activate(X1'), X2'''))), X2''), X2)
ACTIVATE(nf(nf(ng(ng(X''')), X2''), X2)) -> F(f(g(ng(activate(X'''))), X2''), X2)
ACTIVATE(nf(nf(ng(ng(X''')), X2''), X2)) -> F(f(ng(g(activate(X'''))), X2''), X2)
ACTIVATE(nf(nf(ng(ng(X''')), X2'''), X2)) -> F(f(g(activate(X''')), nf(ng(g(activate(X'''))), activate(X2'''))), X2)
ACTIVATE(nf(nf(ng(nf(X1'', X2''')), X2''), X2)) -> F(f(g(f(X1'', X2''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(ng(X'), X2''')), X2''), X2)) -> F(f(g(f(g(activate(X')), X2''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(nf(X1'', X2'0), X2''')), X2''), X2)) -> F(f(g(f(f(activate(X1''), X2'0), X2''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(X1'', X2'''')), X2''), X2)) -> F(f(g(nf(activate(X1''), X2'''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(X1'', X2'''')), X2''), X2)) -> F(f(ng(f(activate(X1''), X2'''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(X1'', X2''''')), X2''''), X2)) -> F(f(f(activate(X1''), X2'''''), nf(ng(f(activate(X1''), X2''''')), activate(X2''''))), X2)
ACTIVATE(nf(nf(ng(X'''), X2''), X2)) -> F(f(ng(X'''), X2''), X2)
ACTIVATE(nf(nf(ng(ng(X')), X2''), X2)) -> F(f(ng(g(activate(X'))), X2''), X2)
ACTIVATE(nf(nf(ng(nf(X1', X2''')), X2''), X2)) -> F(f(ng(f(activate(X1'), X2''')), X2''), X2)
ACTIVATE(nf(nf(nf(X1''', X2''''), X2''), X2)) -> F(f(nf(X1''', X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(g(X'), X2''''), X2''), X2)) -> F(f(f(X', nf(ng(X'), activate(X2''''))), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2'''), X2''), X2)) -> F(f(f(g(X''), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(ng(X'')), X2'''), X2''), X2)) -> F(f(f(g(g(activate(X''))), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(nf(X1', X2'0)), X2'''), X2''), X2)) -> F(f(f(g(f(activate(X1'), X2'0)), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2'''), X2''), X2)) -> F(f(f(ng(activate(X'')), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2''''), X2''), X2)) -> F(f(f(activate(X''), nf(ng(activate(X'')), activate(X2''''))), X2''), X2)
ACTIVATE(nf(nf(g(X'), X2'''), X2)) -> F(f(X', nf(ng(X'), activate(X2'''))), X2)
ACTIVATE(nf(nf(ng(X''), X2'''), X2)) -> F(f(activate(X''), nf(ng(activate(X'')), activate(X2'''))), X2)
ACTIVATE(nf(X1', X2)) -> F(X1', X2)
ACTIVATE(nf(ng(X'), X2)) -> F(g(activate(X')), X2)
ACTIVATE(ng(X)) -> ACTIVATE(X)
ACTIVATE(nf(X1, X2)) -> ACTIVATE(X1)
F(g(X), Y) -> ACTIVATE(Y)
F(g(X), Y) -> F(X, nf(ng(X), activate(Y)))
ACTIVATE(nf(nf(nf(nf(X1'''', X2'0'), X2'''), X2''), X2)) -> F(f(f(nf(X1'''', X2'0'), X2'''), X2''), X2)

Rules:

f(g(X), Y) -> f(X, nf(ng(X), activate(Y)))
f(X1, X2) -> nf(X1, X2)
g(X) -> ng(X)
activate(nf(X1, X2)) -> f(activate(X1), X2)
activate(ng(X)) -> g(activate(X))
activate(X) -> X

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

ACTIVATE(nf(nf(nf(ng(X''), X2'''), X2''), X2)) -> F(f(f(ng(activate(X'')), X2'''), X2''), X2)
five new Dependency Pairs are created:

ACTIVATE(nf(nf(nf(ng(X'''), X2'''''), X2''''), X2)) -> F(nf(f(ng(activate(X''')), X2'''''), X2''''), X2)
ACTIVATE(nf(nf(nf(ng(X'''), X2''''), X2''), X2)) -> F(f(nf(ng(activate(X''')), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(nf(X1', X2'0)), X2'''), X2''), X2)) -> F(f(f(ng(f(activate(X1'), X2'0)), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(ng(X')), X2'''), X2''), X2)) -> F(f(f(ng(g(activate(X'))), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X'''), X2'''), X2''), X2)) -> F(f(f(ng(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 24`
`                 ↳Narrowing Transformation`

Dependency Pairs:

ACTIVATE(nf(nf(nf(ng(X'''), X2'''), X2''), X2)) -> F(f(f(ng(X'''), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(ng(X')), X2'''), X2''), X2)) -> F(f(f(ng(g(activate(X'))), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(nf(X1', X2'0)), X2'''), X2''), X2)) -> F(f(f(ng(f(activate(X1'), X2'0)), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X'''), X2''''), X2''), X2)) -> F(f(nf(ng(activate(X''')), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(ng(X')), X2''''), X2''), X2)) -> F(f(nf(g(g(activate(X'))), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(nf(X1', X2''')), X2''''), X2''), X2)) -> F(f(nf(g(f(activate(X1'), X2''')), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X'''), X2''''), X2''), X2)) -> F(f(nf(ng(activate(X''')), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1'''', X2'0'), X2'''), X2''), X2)) -> F(f(f(nf(X1'''', X2'0'), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(g(X'), X2'0'), X2'''), X2''), X2)) -> F(f(f(f(X', nf(ng(X'), activate(X2'0'))), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1'''', X2'0'), X2''''), X2''), X2)) -> F(f(nf(f(X1'''', X2'0'), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X''), X2'0), X2'''), X2''), X2)) -> F(f(f(f(g(X''), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(ng(X'')), X2'0), X2'''), X2''), X2)) -> F(f(f(f(g(g(activate(X''))), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(nf(X1', X2'1)), X2'0), X2'''), X2''), X2)) -> F(f(f(f(g(f(activate(X1'), X2'1)), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X''), X2'0), X2'''), X2''), X2)) -> F(f(f(f(ng(activate(X'')), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X''), X2'0'), X2'''), X2''), X2)) -> F(f(f(nf(g(activate(X'')), X2'0'), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X''), X2'0'), X2'''), X2''), X2)) -> F(f(f(f(activate(X''), nf(ng(activate(X'')), activate(X2'0'))), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X''), X2'0'), X2''''), X2''), X2)) -> F(f(nf(f(g(activate(X'')), X2'0'), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(X1'', X2'1), X2'0), X2'''), X2''), X2)) -> F(f(f(f(f(X1'', X2'1), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(ng(X'), X2'1), X2'0), X2'''), X2''), X2)) -> F(f(f(f(f(g(activate(X')), X2'1), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(nf(X1'', X2'2), X2'1), X2'0), X2'''), X2''), X2)) -> F(f(f(f(f(f(activate(X1''), X2'2), X2'1), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(X1'', X2'1'), X2'0), X2'''), X2''), X2)) -> F(f(f(f(nf(activate(X1''), X2'1'), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(X1'', X2'1'), X2'0'), X2'''), X2''), X2)) -> F(f(f(nf(f(activate(X1''), X2'1'), X2'0'), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(X1'', X2'1'), X2'0'), X2''''), X2''), X2)) -> F(f(nf(f(f(activate(X1''), X2'1'), X2'0'), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1'''', X2'0'), X2'''), X2''), X2)) -> F(f(f(nf(X1'''', X2'0'), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X'), X2'0'), X2'''), X2''), X2)) -> F(f(f(nf(g(activate(X')), X2'0'), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(X1', X2'0), X2'0'), X2'''), X2''), X2)) -> F(f(f(nf(f(activate(X1'), X2'0), X2'0'), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1'''', X2'0''), X2''''), X2''), X2)) -> F(f(nf(nf(activate(X1''''), X2'0''), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1'''', X2'0'), X2''''), X2''), X2)) -> F(f(nf(f(X1'''', X2'0'), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X'), X2'0'), X2''''), X2''), X2)) -> F(f(nf(f(g(activate(X')), X2'0'), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(X1', X2'''), X2'0'), X2''''), X2''), X2)) -> F(f(nf(f(f(activate(X1'), X2'''), X2'0'), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1'''', X2'0''), X2''''), X2''), X2)) -> F(f(nf(nf(activate(X1''''), X2'0''), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2''''), X2''), X2)) -> F(f(nf(g(X''), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(ng(X'')), X2''''), X2''), X2)) -> F(f(nf(g(g(activate(X''))), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(nf(X1', X2''')), X2''''), X2''), X2)) -> F(f(nf(g(f(activate(X1'), X2''')), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2''''), X2''), X2)) -> F(f(nf(ng(activate(X'')), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1'', X2'''), X2''''), X2''), X2)) -> F(f(nf(f(X1'', X2'''), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X'), X2'''), X2''''), X2''), X2)) -> F(f(nf(f(g(activate(X')), X2'''), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(X1'', X2'0), X2'''), X2''''), X2''), X2)) -> F(f(nf(f(f(activate(X1''), X2'0), X2'''), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1'', X2'''''), X2''''), X2''), X2)) -> F(f(nf(nf(activate(X1''), X2'''''), X2''''), X2''), X2)
ACTIVATE(nf(nf(ng(X'''), X2''), X2)) -> F(f(ng(X'''), X2''), X2)
ACTIVATE(nf(nf(ng(X'''), X2'''), X2)) -> F(f(X''', nf(ng(X'''), activate(X2'''))), X2)
ACTIVATE(nf(nf(ng(ng(X''')), X2''), X2)) -> F(f(g(g(X''')), X2''), X2)
ACTIVATE(nf(nf(ng(ng(ng(X'))), X2''), X2)) -> F(f(g(g(g(activate(X')))), X2''), X2)
ACTIVATE(nf(nf(ng(ng(nf(X1', X2'''))), X2''), X2)) -> F(f(g(g(f(activate(X1'), X2'''))), X2''), X2)
ACTIVATE(nf(nf(ng(ng(X''')), X2''), X2)) -> F(f(g(ng(activate(X'''))), X2''), X2)
ACTIVATE(nf(nf(ng(ng(X''')), X2''), X2)) -> F(f(ng(g(activate(X'''))), X2''), X2)
ACTIVATE(nf(nf(ng(ng(X''')), X2'''), X2)) -> F(f(g(activate(X''')), nf(ng(g(activate(X'''))), activate(X2'''))), X2)
ACTIVATE(nf(nf(ng(nf(X1'', X2''')), X2''), X2)) -> F(f(g(f(X1'', X2''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(ng(X'), X2''')), X2''), X2)) -> F(f(g(f(g(activate(X')), X2''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(nf(X1'', X2'0), X2''')), X2''), X2)) -> F(f(g(f(f(activate(X1''), X2'0), X2''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(X1'', X2'''')), X2''), X2)) -> F(f(g(nf(activate(X1''), X2'''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(X1'', X2'''')), X2''), X2)) -> F(f(ng(f(activate(X1''), X2'''')), X2''), X2)
ACTIVATE(nf(nf(ng(nf(X1'', X2''''')), X2''''), X2)) -> F(f(f(activate(X1''), X2'''''), nf(ng(f(activate(X1''), X2''''')), activate(X2''''))), X2)
ACTIVATE(nf(nf(ng(X'''), X2''), X2)) -> F(f(ng(X'''), X2''), X2)
ACTIVATE(nf(nf(ng(ng(X')), X2''), X2)) -> F(f(ng(g(activate(X'))), X2''), X2)
ACTIVATE(nf(nf(ng(nf(X1', X2''')), X2''), X2)) -> F(f(ng(f(activate(X1'), X2''')), X2''), X2)
ACTIVATE(nf(nf(nf(X1''', X2''''), X2''), X2)) -> F(f(nf(X1''', X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(g(X'), X2''''), X2''), X2)) -> F(f(f(X', nf(ng(X'), activate(X2''''))), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2'''), X2''), X2)) -> F(f(f(g(X''), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(ng(X'')), X2'''), X2''), X2)) -> F(f(f(g(g(activate(X''))), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(nf(X1', X2'0)), X2'''), X2''), X2)) -> F(f(f(g(f(activate(X1'), X2'0)), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X''), X2''''), X2''), X2)) -> F(f(f(activate(X''), nf(ng(activate(X'')), activate(X2''''))), X2''), X2)
ACTIVATE(nf(nf(g(X'), X2'''), X2)) -> F(f(X', nf(ng(X'), activate(X2'''))), X2)
ACTIVATE(nf(nf(ng(X''), X2'''), X2)) -> F(f(activate(X''), nf(ng(activate(X'')), activate(X2'''))), X2)
ACTIVATE(nf(X1', X2)) -> F(X1', X2)
ACTIVATE(nf(ng(X'), X2)) -> F(g(activate(X')), X2)
ACTIVATE(ng(X)) -> ACTIVATE(X)
ACTIVATE(nf(X1, X2)) -> ACTIVATE(X1)
F(g(X), Y) -> ACTIVATE(Y)
F(g(X), Y) -> F(X, nf(ng(X), activate(Y)))
ACTIVATE(nf(nf(nf(ng(X'''), X2''''), X2''), X2)) -> F(f(nf(g(X'''), X2''''), X2''), X2)

Rules:

f(g(X), Y) -> f(X, nf(ng(X), activate(Y)))
f(X1, X2) -> nf(X1, X2)
g(X) -> ng(X)
activate(nf(X1, X2)) -> f(activate(X1), X2)
activate(ng(X)) -> g(activate(X))
activate(X) -> X

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

ACTIVATE(nf(nf(nf(ng(nf(X1', X2'0)), X2'''), X2''), X2)) -> F(f(f(g(f(activate(X1'), X2'0)), X2'''), X2''), X2)
eight new Dependency Pairs are created:

ACTIVATE(nf(nf(nf(ng(nf(X1'', X2'0')), X2'''''), X2''''), X2)) -> F(nf(f(g(f(activate(X1''), X2'0')), X2'''''), X2''''), X2)
ACTIVATE(nf(nf(nf(ng(nf(X1'', X2'0')), X2''''), X2''), X2)) -> F(f(f(f(activate(X1''), X2'0'), nf(ng(f(activate(X1''), X2'0')), activate(X2''''))), X2''), X2)
ACTIVATE(nf(nf(nf(ng(nf(X1'', X2'0')), X2''''), X2''), X2)) -> F(f(nf(g(f(activate(X1''), X2'0')), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(nf(X1'', X2'0')), X2'''), X2''), X2)) -> F(f(f(ng(f(activate(X1''), X2'0')), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(nf(X1'', X2'0')), X2'''), X2''), X2)) -> F(f(f(g(nf(activate(X1''), X2'0')), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(nf(nf(X1'', X2'1), X2'0)), X2'''), X2''), X2)) -> F(f(f(g(f(f(activate(X1''), X2'1), X2'0)), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(nf(ng(X'), X2'0)), X2'''), X2''), X2)) -> F(f(f(g(f(g(activate(X')), X2'0)), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(nf(X1'', X2'0)), X2'''), X2''), X2)) -> F(f(f(g(f(X1'', X2'0)), X2'''), X2''), X2)

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 25`
`                 ↳Narrowing Transformation`

Dependency Pairs:

ACTIVATE(nf(nf(nf(ng(nf(X1'', X2'0)), X2'''), X2''), X2)) -> F(f(f(g(f(X1'', X2'0)), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(nf(ng(X'), X2'0)), X2'''), X2''), X2)) -> F(f(f(g(f(g(activate(X')), X2'0)), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(nf(nf(X1'', X2'1), X2'0)), X2'''), X2''), X2)) -> F(f(f(g(f(f(activate(X1''), X2'1), X2'0)), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(nf(X1'', X2'0')), X2'''), X2''), X2)) -> F(f(f(g(nf(activate(X1''), X2'0')), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(nf(X1'', X2'0')), X2'''), X2''), X2)) -> F(f(f(ng(f(activate(X1''), X2'0')), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(nf(X1'', X2'0')), X2''''), X2''), X2)) -> F(f(nf(g(f(activate(X1''), X2'0')), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(nf(X1'', X2'0')), X2''''), X2''), X2)) -> F(f(f(f(activate(X1''), X2'0'), nf(ng(f(activate(X1''), X2'0')), activate(X2''''))), X2''), X2)
ACTIVATE(nf(nf(nf(ng(ng(X')), X2'''), X2''), X2)) -> F(f(f(ng(g(activate(X'))), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(nf(X1', X2'0)), X2'''), X2''), X2)) -> F(f(f(ng(f(activate(X1'), X2'0)), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X'''), X2''''), X2''), X2)) -> F(f(nf(ng(activate(X''')), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X'''), X2''''), X2''), X2)) -> F(f(nf(g(X'''), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(ng(X')), X2''''), X2''), X2)) -> F(f(nf(g(g(activate(X'))), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(nf(X1', X2''')), X2''''), X2''), X2)) -> F(f(nf(g(f(activate(X1'), X2''')), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(ng(X'''), X2''''), X2''), X2)) -> F(f(nf(ng(activate(X''')), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1'''', X2'0'), X2'''), X2''), X2)) -> F(f(f(nf(X1'''', X2'0'), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(g(X'), X2'0'), X2'''), X2''), X2)) -> F(f(f(f(X', nf(ng(X'), activate(X2'0'))), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(X1'''', X2'0'), X2''''), X2''), X2)) -> F(f(nf(f(X1'''', X2'0'), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X''), X2'0), X2'''), X2''), X2)) -> F(f(f(f(g(X''), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(ng(X'')), X2'0), X2'''), X2''), X2)) -> F(f(f(f(g(g(activate(X''))), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(nf(X1', X2'1)), X2'0), X2'''), X2''), X2)) -> F(f(f(f(g(f(activate(X1'), X2'1)), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X''), X2'0), X2'''), X2''), X2)) -> F(f(f(f(ng(activate(X'')), X2'0), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X''), X2'0'), X2'''), X2''), X2)) -> F(f(f(nf(g(activate(X'')), X2'0'), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X''), X2'0'), X2'''), X2''), X2)) -> F(f(f(f(activate(X''), nf(ng(activate(X'')), activate(X2'0'))), X2'''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(ng(X''), X2'0'), X2''''), X2''), X2)) -> F(f(nf(f(g(activate(X'')), X2'0'), X2''''), X2''), X2)
ACTIVATE(nf(nf(nf(nf(nf(X1'', X2'1), X2'0), X2'''), X2''), X2)) -> F(f(f(f(f(X1'', X2'1), X2'0), X2'''), X2''), X2)
ACTIVATE(n