Term Rewriting System R:
[X, Y]
f(X) -> if(X, c, nf(ntrue))
f(X) -> nf(X)
if(true, X, Y) -> X
if(false, X, Y) -> activate(Y)
true -> ntrue
activate(nf(X)) -> f(activate(X))
activate(ntrue) -> true
activate(X) -> X

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

F(X) -> IF(X, c, nf(ntrue))
IF(false, X, Y) -> ACTIVATE(Y)
ACTIVATE(nf(X)) -> F(activate(X))
ACTIVATE(nf(X)) -> ACTIVATE(X)
ACTIVATE(ntrue) -> TRUE

Furthermore, R contains one SCC.

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

Dependency Pairs:

ACTIVATE(nf(X)) -> ACTIVATE(X)
ACTIVATE(nf(X)) -> F(activate(X))
IF(false, X, Y) -> ACTIVATE(Y)
F(X) -> IF(X, c, nf(ntrue))

Rules:

f(X) -> if(X, c, nf(ntrue))
f(X) -> nf(X)
if(true, X, Y) -> X
if(false, X, Y) -> activate(Y)
true -> ntrue
activate(nf(X)) -> f(activate(X))
activate(ntrue) -> true
activate(X) -> X

Strategy:

innermost

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

ACTIVATE(nf(X)) -> F(activate(X))
three new Dependency Pairs are created:

ACTIVATE(nf(nf(X''))) -> F(f(activate(X'')))
ACTIVATE(nf(ntrue)) -> F(true)
ACTIVATE(nf(X'')) -> F(X'')

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

ACTIVATE(nf(X'')) -> F(X'')
ACTIVATE(nf(ntrue)) -> F(true)
IF(false, X, Y) -> ACTIVATE(Y)
F(X) -> IF(X, c, nf(ntrue))
ACTIVATE(nf(nf(X''))) -> F(f(activate(X'')))
ACTIVATE(nf(X)) -> ACTIVATE(X)

Rules:

f(X) -> if(X, c, nf(ntrue))
f(X) -> nf(X)
if(true, X, Y) -> X
if(false, X, Y) -> activate(Y)
true -> ntrue
activate(nf(X)) -> f(activate(X))
activate(ntrue) -> true
activate(X) -> X

Strategy:

innermost

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

ACTIVATE(nf(ntrue)) -> F(true)
one new Dependency Pair is created:

ACTIVATE(nf(ntrue)) -> F(ntrue)

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

ACTIVATE(nf(ntrue)) -> F(ntrue)
ACTIVATE(nf(nf(X''))) -> F(f(activate(X'')))
ACTIVATE(nf(X)) -> ACTIVATE(X)
IF(false, X, Y) -> ACTIVATE(Y)
F(X) -> IF(X, c, nf(ntrue))
ACTIVATE(nf(X'')) -> F(X'')

Rules:

f(X) -> if(X, c, nf(ntrue))
f(X) -> nf(X)
if(true, X, Y) -> X
if(false, X, Y) -> activate(Y)
true -> ntrue
activate(nf(X)) -> f(activate(X))
activate(ntrue) -> true
activate(X) -> X

Strategy:

innermost

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

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

ACTIVATE(nf(nf(X'''))) -> F(if(activate(X'''), c, nf(ntrue)))
ACTIVATE(nf(nf(X'''))) -> F(nf(activate(X''')))
ACTIVATE(nf(nf(nf(X')))) -> F(f(f(activate(X'))))
ACTIVATE(nf(nf(ntrue))) -> F(f(true))
ACTIVATE(nf(nf(X'''))) -> F(f(X'''))

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

ACTIVATE(nf(nf(X'''))) -> F(f(X'''))
ACTIVATE(nf(nf(ntrue))) -> F(f(true))
ACTIVATE(nf(nf(nf(X')))) -> F(f(f(activate(X'))))
ACTIVATE(nf(nf(X'''))) -> F(nf(activate(X''')))
ACTIVATE(nf(nf(X'''))) -> F(if(activate(X'''), c, nf(ntrue)))
ACTIVATE(nf(X'')) -> F(X'')
ACTIVATE(nf(X)) -> ACTIVATE(X)
IF(false, X, Y) -> ACTIVATE(Y)
F(X) -> IF(X, c, nf(ntrue))
ACTIVATE(nf(ntrue)) -> F(ntrue)

Rules:

f(X) -> if(X, c, nf(ntrue))
f(X) -> nf(X)
if(true, X, Y) -> X
if(false, X, Y) -> activate(Y)
true -> ntrue
activate(nf(X)) -> f(activate(X))
activate(ntrue) -> true
activate(X) -> X

Strategy:

innermost

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

ACTIVATE(nf(nf(ntrue))) -> F(f(true))
one new Dependency Pair is created:

ACTIVATE(nf(nf(ntrue))) -> F(f(ntrue))

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

ACTIVATE(nf(nf(ntrue))) -> F(f(ntrue))
ACTIVATE(nf(nf(nf(X')))) -> F(f(f(activate(X'))))
ACTIVATE(nf(nf(X'''))) -> F(nf(activate(X''')))
ACTIVATE(nf(nf(X'''))) -> F(if(activate(X'''), c, nf(ntrue)))
ACTIVATE(nf(ntrue)) -> F(ntrue)
ACTIVATE(nf(X'')) -> F(X'')
ACTIVATE(nf(X)) -> ACTIVATE(X)
IF(false, X, Y) -> ACTIVATE(Y)
F(X) -> IF(X, c, nf(ntrue))
ACTIVATE(nf(nf(X'''))) -> F(f(X'''))

Rules:

f(X) -> if(X, c, nf(ntrue))
f(X) -> nf(X)
if(true, X, Y) -> X
if(false, X, Y) -> activate(Y)
true -> ntrue
activate(nf(X)) -> f(activate(X))
activate(ntrue) -> true
activate(X) -> X

Strategy:

innermost

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

IF(false, X, Y) -> ACTIVATE(Y)
one new Dependency Pair is created:

IF(false, c, nf(ntrue)) -> ACTIVATE(nf(ntrue))

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

ACTIVATE(nf(nf(X'''))) -> F(f(X'''))
ACTIVATE(nf(nf(nf(X')))) -> F(f(f(activate(X'))))
ACTIVATE(nf(nf(X'''))) -> F(nf(activate(X''')))
ACTIVATE(nf(nf(X'''))) -> F(if(activate(X'''), c, nf(ntrue)))
ACTIVATE(nf(ntrue)) -> F(ntrue)
ACTIVATE(nf(X'')) -> F(X'')
ACTIVATE(nf(X)) -> ACTIVATE(X)
IF(false, c, nf(ntrue)) -> ACTIVATE(nf(ntrue))
F(X) -> IF(X, c, nf(ntrue))
ACTIVATE(nf(nf(ntrue))) -> F(f(ntrue))

Rules:

f(X) -> if(X, c, nf(ntrue))
f(X) -> nf(X)
if(true, X, Y) -> X
if(false, X, Y) -> activate(Y)
true -> ntrue
activate(nf(X)) -> f(activate(X))
activate(ntrue) -> true
activate(X) -> X

Strategy:

innermost

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

F(X) -> IF(X, c, nf(ntrue))
one new Dependency Pair is created:

F(false) -> IF(false, c, nf(ntrue))

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

ACTIVATE(nf(nf(ntrue))) -> F(f(ntrue))
ACTIVATE(nf(nf(nf(X')))) -> F(f(f(activate(X'))))
ACTIVATE(nf(nf(X'''))) -> F(if(activate(X'''), c, nf(ntrue)))
ACTIVATE(nf(X'')) -> F(X'')
ACTIVATE(nf(X)) -> ACTIVATE(X)
IF(false, c, nf(ntrue)) -> ACTIVATE(nf(ntrue))
F(false) -> IF(false, c, nf(ntrue))
ACTIVATE(nf(nf(X'''))) -> F(f(X'''))

Rules:

f(X) -> if(X, c, nf(ntrue))
f(X) -> nf(X)
if(true, X, Y) -> X
if(false, X, Y) -> activate(Y)
true -> ntrue
activate(nf(X)) -> f(activate(X))
activate(ntrue) -> true
activate(X) -> X

Strategy:

innermost

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

ACTIVATE(nf(X)) -> ACTIVATE(X)
five new Dependency Pairs are created:

ACTIVATE(nf(nf(X''))) -> ACTIVATE(nf(X''))
ACTIVATE(nf(nf(X''''))) -> ACTIVATE(nf(X''''))
ACTIVATE(nf(nf(nf(X''''')))) -> ACTIVATE(nf(nf(X''''')))
ACTIVATE(nf(nf(nf(nf(X'''))))) -> ACTIVATE(nf(nf(nf(X'''))))
ACTIVATE(nf(nf(nf(ntrue)))) -> ACTIVATE(nf(nf(ntrue)))

The transformation is resulting in two new DP problems:

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

Dependency Pairs:

ACTIVATE(nf(X'')) -> F(X'')
IF(false, c, nf(ntrue)) -> ACTIVATE(nf(ntrue))
F(false) -> IF(false, c, nf(ntrue))

Rules:

f(X) -> if(X, c, nf(ntrue))
f(X) -> nf(X)
if(true, X, Y) -> X
if(false, X, Y) -> activate(Y)
true -> ntrue
activate(nf(X)) -> f(activate(X))
activate(ntrue) -> true
activate(X) -> X

Strategy:

innermost

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

ACTIVATE(nf(X'')) -> F(X'')
one new Dependency Pair is created:

ACTIVATE(nf(ntrue)) -> F(ntrue)

The transformation is resulting in no new DP problems.

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

Dependency Pairs:

ACTIVATE(nf(nf(nf(ntrue)))) -> ACTIVATE(nf(nf(ntrue)))
ACTIVATE(nf(nf(nf(nf(X'''))))) -> ACTIVATE(nf(nf(nf(X'''))))
ACTIVATE(nf(nf(nf(X''''')))) -> ACTIVATE(nf(nf(X''''')))
ACTIVATE(nf(nf(X''''))) -> ACTIVATE(nf(X''''))
ACTIVATE(nf(nf(X''))) -> ACTIVATE(nf(X''))

Rules:

f(X) -> if(X, c, nf(ntrue))
f(X) -> nf(X)
if(true, X, Y) -> X
if(false, X, Y) -> activate(Y)
true -> ntrue
activate(nf(X)) -> f(activate(X))
activate(ntrue) -> true
activate(X) -> X

Strategy:

innermost

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

ACTIVATE(nf(nf(X''))) -> ACTIVATE(nf(X''))
five new Dependency Pairs are created:

ACTIVATE(nf(nf(nf(X'''')))) -> ACTIVATE(nf(nf(X'''')))
ACTIVATE(nf(nf(nf(X'''''')))) -> ACTIVATE(nf(nf(X'''''')))
ACTIVATE(nf(nf(nf(nf(X'''''''))))) -> ACTIVATE(nf(nf(nf(X'''''''))))
ACTIVATE(nf(nf(nf(nf(nf(X''''')))))) -> ACTIVATE(nf(nf(nf(nf(X''''')))))
ACTIVATE(nf(nf(nf(nf(ntrue))))) -> ACTIVATE(nf(nf(nf(ntrue))))

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

ACTIVATE(nf(nf(nf(nf(ntrue))))) -> ACTIVATE(nf(nf(nf(ntrue))))
ACTIVATE(nf(nf(nf(nf(nf(X''''')))))) -> ACTIVATE(nf(nf(nf(nf(X''''')))))
ACTIVATE(nf(nf(nf(nf(X'''''''))))) -> ACTIVATE(nf(nf(nf(X'''''''))))
ACTIVATE(nf(nf(nf(X'''''')))) -> ACTIVATE(nf(nf(X'''''')))
ACTIVATE(nf(nf(nf(X'''')))) -> ACTIVATE(nf(nf(X'''')))
ACTIVATE(nf(nf(nf(nf(X'''))))) -> ACTIVATE(nf(nf(nf(X'''))))
ACTIVATE(nf(nf(nf(X''''')))) -> ACTIVATE(nf(nf(X''''')))
ACTIVATE(nf(nf(X''''))) -> ACTIVATE(nf(X''''))
ACTIVATE(nf(nf(nf(ntrue)))) -> ACTIVATE(nf(nf(ntrue)))

Rules:

f(X) -> if(X, c, nf(ntrue))
f(X) -> nf(X)
if(true, X, Y) -> X
if(false, X, Y) -> activate(Y)
true -> ntrue
activate(nf(X)) -> f(activate(X))
activate(ntrue) -> true
activate(X) -> X

Strategy:

innermost

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

ACTIVATE(nf(nf(X''''))) -> ACTIVATE(nf(X''''))
nine new Dependency Pairs are created:

ACTIVATE(nf(nf(nf(X'''''')))) -> ACTIVATE(nf(nf(X'''''')))
ACTIVATE(nf(nf(nf(nf(X'''''''))))) -> ACTIVATE(nf(nf(nf(X'''''''))))
ACTIVATE(nf(nf(nf(nf(nf(X'''''')))))) -> ACTIVATE(nf(nf(nf(nf(X'''''')))))
ACTIVATE(nf(nf(nf(nf(ntrue))))) -> ACTIVATE(nf(nf(nf(ntrue))))
ACTIVATE(nf(nf(nf(nf(X''''''))))) -> ACTIVATE(nf(nf(nf(X''''''))))
ACTIVATE(nf(nf(nf(nf(X''''''''))))) -> ACTIVATE(nf(nf(nf(X''''''''))))
ACTIVATE(nf(nf(nf(nf(nf(X''''''''')))))) -> ACTIVATE(nf(nf(nf(nf(X''''''''')))))
ACTIVATE(nf(nf(nf(nf(nf(nf(X'''''''))))))) -> ACTIVATE(nf(nf(nf(nf(nf(X'''''''))))))
ACTIVATE(nf(nf(nf(nf(nf(ntrue)))))) -> ACTIVATE(nf(nf(nf(nf(ntrue)))))

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

ACTIVATE(nf(nf(nf(nf(nf(ntrue)))))) -> ACTIVATE(nf(nf(nf(nf(ntrue)))))
ACTIVATE(nf(nf(nf(nf(nf(nf(X'''''''))))))) -> ACTIVATE(nf(nf(nf(nf(nf(X'''''''))))))
ACTIVATE(nf(nf(nf(nf(nf(X''''''''')))))) -> ACTIVATE(nf(nf(nf(nf(X''''''''')))))
ACTIVATE(nf(nf(nf(nf(X''''''''))))) -> ACTIVATE(nf(nf(nf(X''''''''))))
ACTIVATE(nf(nf(nf(nf(X''''''))))) -> ACTIVATE(nf(nf(nf(X''''''))))
ACTIVATE(nf(nf(nf(nf(ntrue))))) -> ACTIVATE(nf(nf(nf(ntrue))))
ACTIVATE(nf(nf(nf(nf(nf(X'''''')))))) -> ACTIVATE(nf(nf(nf(nf(X'''''')))))
ACTIVATE(nf(nf(nf(nf(X'''''''))))) -> ACTIVATE(nf(nf(nf(X'''''''))))
ACTIVATE(nf(nf(nf(X'''''')))) -> ACTIVATE(nf(nf(X'''''')))
ACTIVATE(nf(nf(nf(nf(nf(X''''')))))) -> ACTIVATE(nf(nf(nf(nf(X''''')))))
ACTIVATE(nf(nf(nf(nf(X'''''''))))) -> ACTIVATE(nf(nf(nf(X'''''''))))
ACTIVATE(nf(nf(nf(X'''''')))) -> ACTIVATE(nf(nf(X'''''')))
ACTIVATE(nf(nf(nf(X'''')))) -> ACTIVATE(nf(nf(X'''')))
ACTIVATE(nf(nf(nf(nf(X'''))))) -> ACTIVATE(nf(nf(nf(X'''))))
ACTIVATE(nf(nf(nf(X''''')))) -> ACTIVATE(nf(nf(X''''')))
ACTIVATE(nf(nf(nf(nf(ntrue))))) -> ACTIVATE(nf(nf(nf(ntrue))))

Rules:

f(X) -> if(X, c, nf(ntrue))
f(X) -> nf(X)
if(true, X, Y) -> X
if(false, X, Y) -> activate(Y)
true -> ntrue
activate(nf(X)) -> f(activate(X))
activate(ntrue) -> true
activate(X) -> X

Strategy:

innermost

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

ACTIVATE(nf(nf(nf(X''''')))) -> ACTIVATE(nf(nf(X''''')))
12 new Dependency Pairs are created:

ACTIVATE(nf(nf(nf(nf(X'''''''))))) -> ACTIVATE(nf(nf(nf(X'''''''))))
ACTIVATE(nf(nf(nf(nf(nf(X'''''')))))) -> ACTIVATE(nf(nf(nf(nf(X'''''')))))
ACTIVATE(nf(nf(nf(nf(X''''''''))))) -> ACTIVATE(nf(nf(nf(X''''''''))))
ACTIVATE(nf(nf(nf(nf(nf(X''''''''')))))) -> ACTIVATE(nf(nf(nf(nf(X''''''''')))))
ACTIVATE(nf(nf(nf(nf(nf(nf(X'''''''))))))) -> ACTIVATE(nf(nf(nf(nf(nf(X'''''''))))))
ACTIVATE(nf(nf(nf(nf(nf(ntrue)))))) -> ACTIVATE(nf(nf(nf(nf(ntrue)))))
ACTIVATE(nf(nf(nf(nf(nf(nf(X''''''''))))))) -> ACTIVATE(nf(nf(nf(nf(nf(X''''''''))))))
ACTIVATE(nf(nf(nf(nf(nf(X'''''''')))))) -> ACTIVATE(nf(nf(nf(nf(X'''''''')))))
ACTIVATE(nf(nf(nf(nf(nf(X'''''''''')))))) -> ACTIVATE(nf(nf(nf(nf(X'''''''''')))))
ACTIVATE(nf(nf(nf(nf(nf(nf(X'''''''''''))))))) -> ACTIVATE(nf(nf(nf(nf(nf(X'''''''''''))))))
ACTIVATE(nf(nf(nf(nf(nf(nf(nf(X''''''''')))))))) -> ACTIVATE(nf(nf(nf(nf(nf(nf(X''''''''')))))))
ACTIVATE(nf(nf(nf(nf(nf(nf(ntrue))))))) -> ACTIVATE(nf(nf(nf(nf(nf(ntrue))))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Rw`
`             ...`
`               →DP Problem 12`
`                 ↳Polynomial Ordering`

Dependency Pairs:

ACTIVATE(nf(nf(nf(nf(nf(nf(ntrue))))))) -> ACTIVATE(nf(nf(nf(nf(nf(ntrue))))))
ACTIVATE(nf(nf(nf(nf(nf(nf(nf(X''''''''')))))))) -> ACTIVATE(nf(nf(nf(nf(nf(nf(X''''''''')))))))
ACTIVATE(nf(nf(nf(nf(nf(nf(X'''''''''''))))))) -> ACTIVATE(nf(nf(nf(nf(nf(X'''''''''''))))))
ACTIVATE(nf(nf(nf(nf(nf(X'''''''''')))))) -> ACTIVATE(nf(nf(nf(nf(X'''''''''')))))
ACTIVATE(nf(nf(nf(nf(nf(X'''''''')))))) -> ACTIVATE(nf(nf(nf(nf(X'''''''')))))
ACTIVATE(nf(nf(nf(nf(nf(nf(X''''''''))))))) -> ACTIVATE(nf(nf(nf(nf(nf(X''''''''))))))
ACTIVATE(nf(nf(nf(nf(nf(ntrue)))))) -> ACTIVATE(nf(nf(nf(nf(ntrue)))))
ACTIVATE(nf(nf(nf(nf(nf(nf(X'''''''))))))) -> ACTIVATE(nf(nf(nf(nf(nf(X'''''''))))))
ACTIVATE(nf(nf(nf(nf(nf(X''''''''')))))) -> ACTIVATE(nf(nf(nf(nf(X''''''''')))))
ACTIVATE(nf(nf(nf(nf(X''''''''))))) -> ACTIVATE(nf(nf(nf(X''''''''))))
ACTIVATE(nf(nf(nf(nf(nf(X'''''')))))) -> ACTIVATE(nf(nf(nf(nf(X'''''')))))
ACTIVATE(nf(nf(nf(nf(X'''''''))))) -> ACTIVATE(nf(nf(nf(X'''''''))))
ACTIVATE(nf(nf(nf(nf(nf(nf(X'''''''))))))) -> ACTIVATE(nf(nf(nf(nf(nf(X'''''''))))))
ACTIVATE(nf(nf(nf(nf(nf(X''''''''')))))) -> ACTIVATE(nf(nf(nf(nf(X''''''''')))))
ACTIVATE(nf(nf(nf(nf(X''''''''))))) -> ACTIVATE(nf(nf(nf(X''''''''))))
ACTIVATE(nf(nf(nf(nf(X''''''))))) -> ACTIVATE(nf(nf(nf(X''''''))))
ACTIVATE(nf(nf(nf(nf(ntrue))))) -> ACTIVATE(nf(nf(nf(ntrue))))
ACTIVATE(nf(nf(nf(nf(nf(X'''''')))))) -> ACTIVATE(nf(nf(nf(nf(X'''''')))))
ACTIVATE(nf(nf(nf(nf(X'''''''))))) -> ACTIVATE(nf(nf(nf(X'''''''))))
ACTIVATE(nf(nf(nf(X'''''')))) -> ACTIVATE(nf(nf(X'''''')))
ACTIVATE(nf(nf(nf(nf(ntrue))))) -> ACTIVATE(nf(nf(nf(ntrue))))
ACTIVATE(nf(nf(nf(nf(nf(X''''')))))) -> ACTIVATE(nf(nf(nf(nf(X''''')))))
ACTIVATE(nf(nf(nf(nf(X'''''''))))) -> ACTIVATE(nf(nf(nf(X'''''''))))
ACTIVATE(nf(nf(nf(X'''''')))) -> ACTIVATE(nf(nf(X'''''')))
ACTIVATE(nf(nf(nf(X'''')))) -> ACTIVATE(nf(nf(X'''')))
ACTIVATE(nf(nf(nf(nf(X'''))))) -> ACTIVATE(nf(nf(nf(X'''))))
ACTIVATE(nf(nf(nf(nf(nf(ntrue)))))) -> ACTIVATE(nf(nf(nf(nf(ntrue)))))

Rules:

f(X) -> if(X, c, nf(ntrue))
f(X) -> nf(X)
if(true, X, Y) -> X
if(false, X, Y) -> activate(Y)
true -> ntrue
activate(nf(X)) -> f(activate(X))
activate(ntrue) -> true
activate(X) -> X

Strategy:

innermost

The following dependency pairs can be strictly oriented:

ACTIVATE(nf(nf(nf(nf(nf(nf(ntrue))))))) -> ACTIVATE(nf(nf(nf(nf(nf(ntrue))))))
ACTIVATE(nf(nf(nf(nf(nf(nf(nf(X''''''''')))))))) -> ACTIVATE(nf(nf(nf(nf(nf(nf(X''''''''')))))))
ACTIVATE(nf(nf(nf(nf(nf(nf(X'''''''''''))))))) -> ACTIVATE(nf(nf(nf(nf(nf(X'''''''''''))))))
ACTIVATE(nf(nf(nf(nf(nf(X'''''''''')))))) -> ACTIVATE(nf(nf(nf(nf(X'''''''''')))))
ACTIVATE(nf(nf(nf(nf(nf(X'''''''')))))) -> ACTIVATE(nf(nf(nf(nf(X'''''''')))))
ACTIVATE(nf(nf(nf(nf(nf(nf(X''''''''))))))) -> ACTIVATE(nf(nf(nf(nf(nf(X''''''''))))))
ACTIVATE(nf(nf(nf(nf(nf(ntrue)))))) -> ACTIVATE(nf(nf(nf(nf(ntrue)))))
ACTIVATE(nf(nf(nf(nf(nf(nf(X'''''''))))))) -> ACTIVATE(nf(nf(nf(nf(nf(X'''''''))))))
ACTIVATE(nf(nf(nf(nf(nf(X''''''''')))))) -> ACTIVATE(nf(nf(nf(nf(X''''''''')))))
ACTIVATE(nf(nf(nf(nf(X''''''''))))) -> ACTIVATE(nf(nf(nf(X''''''''))))
ACTIVATE(nf(nf(nf(nf(nf(X'''''')))))) -> ACTIVATE(nf(nf(nf(nf(X'''''')))))
ACTIVATE(nf(nf(nf(nf(X'''''''))))) -> ACTIVATE(nf(nf(nf(X'''''''))))
ACTIVATE(nf(nf(nf(nf(X''''''))))) -> ACTIVATE(nf(nf(nf(X''''''))))
ACTIVATE(nf(nf(nf(nf(ntrue))))) -> ACTIVATE(nf(nf(nf(ntrue))))
ACTIVATE(nf(nf(nf(X'''''')))) -> ACTIVATE(nf(nf(X'''''')))
ACTIVATE(nf(nf(nf(nf(nf(X''''')))))) -> ACTIVATE(nf(nf(nf(nf(X''''')))))
ACTIVATE(nf(nf(nf(X'''')))) -> ACTIVATE(nf(nf(X'''')))
ACTIVATE(nf(nf(nf(nf(X'''))))) -> ACTIVATE(nf(nf(nf(X'''))))

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

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(n__f(x1)) =  1 + x1 POL(n__true) =  0 POL(ACTIVATE(x1)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Rw`
`             ...`
`               →DP Problem 13`
`                 ↳Dependency Graph`

Dependency Pair:

Rules:

f(X) -> if(X, c, nf(ntrue))
f(X) -> nf(X)
if(true, X, Y) -> X
if(false, X, Y) -> activate(Y)
true -> ntrue
activate(nf(X)) -> f(activate(X))
activate(ntrue) -> true
activate(X) -> X

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:15 minutes