Term Rewriting System R:
[X, Y, Z]
f(X) -> cons(X, nf(ng(X)))
f(X) -> nf(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> ng(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(nf(X)) -> f(activate(X))
activate(ng(X)) -> g(activate(X))
activate(X) -> X

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

G(s(X)) -> G(X)
SEL(s(X), cons(Y, Z)) -> SEL(X, activate(Z))
SEL(s(X), cons(Y, Z)) -> ACTIVATE(Z)
ACTIVATE(nf(X)) -> F(activate(X))
ACTIVATE(nf(X)) -> ACTIVATE(X)
ACTIVATE(ng(X)) -> G(activate(X))
ACTIVATE(ng(X)) -> ACTIVATE(X)

Furthermore, R contains three SCCs.

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

Dependency Pair:

G(s(X)) -> G(X)

Rules:

f(X) -> cons(X, nf(ng(X)))
f(X) -> nf(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> ng(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(nf(X)) -> f(activate(X))
activate(ng(X)) -> g(activate(X))
activate(X) -> X

Strategy:

innermost

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

G(s(X)) -> G(X)
one new Dependency Pair is created:

G(s(s(X''))) -> G(s(X''))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 4`
`             ↳Forward Instantiation Transformation`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`

Dependency Pair:

G(s(s(X''))) -> G(s(X''))

Rules:

f(X) -> cons(X, nf(ng(X)))
f(X) -> nf(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> ng(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(nf(X)) -> f(activate(X))
activate(ng(X)) -> g(activate(X))
activate(X) -> X

Strategy:

innermost

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

G(s(s(X''))) -> G(s(X''))
one new Dependency Pair is created:

G(s(s(s(X'''')))) -> G(s(s(X'''')))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 4`
`             ↳FwdInst`
`             ...`
`               →DP Problem 5`
`                 ↳Polynomial Ordering`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`

Dependency Pair:

G(s(s(s(X'''')))) -> G(s(s(X'''')))

Rules:

f(X) -> cons(X, nf(ng(X)))
f(X) -> nf(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> ng(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(nf(X)) -> f(activate(X))
activate(ng(X)) -> g(activate(X))
activate(X) -> X

Strategy:

innermost

The following dependency pair can be strictly oriented:

G(s(s(s(X'''')))) -> G(s(s(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(G(x1)) =  1 + x1 POL(s(x1)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 4`
`             ↳FwdInst`
`             ...`
`               →DP Problem 6`
`                 ↳Dependency Graph`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`

Dependency Pair:

Rules:

f(X) -> cons(X, nf(ng(X)))
f(X) -> nf(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> ng(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(nf(X)) -> f(activate(X))
activate(ng(X)) -> g(activate(X))
activate(X) -> X

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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

Dependency Pairs:

ACTIVATE(ng(X)) -> ACTIVATE(X)
ACTIVATE(nf(X)) -> ACTIVATE(X)

Rules:

f(X) -> cons(X, nf(ng(X)))
f(X) -> nf(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> ng(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(nf(X)) -> f(activate(X))
activate(ng(X)) -> g(activate(X))
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)
two new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`           →DP Problem 7`
`             ↳Forward Instantiation Transformation`
`       →DP Problem 3`
`         ↳Nar`

Dependency Pairs:

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

Rules:

f(X) -> cons(X, nf(ng(X)))
f(X) -> nf(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> ng(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(nf(X)) -> f(activate(X))
activate(ng(X)) -> g(activate(X))
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(ng(X)) -> ACTIVATE(X)
three new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`           →DP Problem 7`
`             ↳FwdInst`
`             ...`
`               →DP Problem 8`
`                 ↳Forward Instantiation Transformation`
`       →DP Problem 3`
`         ↳Nar`

Dependency Pairs:

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

Rules:

f(X) -> cons(X, nf(ng(X)))
f(X) -> nf(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> ng(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(nf(X)) -> f(activate(X))
activate(ng(X)) -> g(activate(X))
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''))
two new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`           →DP Problem 7`
`             ↳FwdInst`
`             ...`
`               →DP Problem 9`
`                 ↳Forward Instantiation Transformation`
`       →DP Problem 3`
`         ↳Nar`

Dependency Pairs:

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

Rules:

f(X) -> cons(X, nf(ng(X)))
f(X) -> nf(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> ng(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(nf(X)) -> f(activate(X))
activate(ng(X)) -> g(activate(X))
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(ng(X''))) -> ACTIVATE(ng(X''))
three new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`           →DP Problem 7`
`             ↳FwdInst`
`             ...`
`               →DP Problem 10`
`                 ↳Forward Instantiation Transformation`
`       →DP Problem 3`
`         ↳Nar`

Dependency Pairs:

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

Rules:

f(X) -> cons(X, nf(ng(X)))
f(X) -> nf(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> ng(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(nf(X)) -> f(activate(X))
activate(ng(X)) -> g(activate(X))
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(ng(ng(X''))) -> ACTIVATE(ng(X''))
three new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`           →DP Problem 7`
`             ↳FwdInst`
`             ...`
`               →DP Problem 11`
`                 ↳Forward Instantiation Transformation`
`       →DP Problem 3`
`         ↳Nar`

Dependency Pairs:

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

Rules:

f(X) -> cons(X, nf(ng(X)))
f(X) -> nf(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> ng(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(nf(X)) -> f(activate(X))
activate(ng(X)) -> g(activate(X))
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(ng(nf(nf(X'''')))) -> ACTIVATE(nf(nf(X'''')))
two new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`           →DP Problem 7`
`             ↳FwdInst`
`             ...`
`               →DP Problem 12`
`                 ↳Forward Instantiation Transformation`
`       →DP Problem 3`
`         ↳Nar`

Dependency Pairs:

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

Rules:

f(X) -> cons(X, nf(ng(X)))
f(X) -> nf(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> ng(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(nf(X)) -> f(activate(X))
activate(ng(X)) -> g(activate(X))
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(ng(nf(ng(X'''')))) -> ACTIVATE(nf(ng(X'''')))
three new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`           →DP Problem 7`
`             ↳FwdInst`
`             ...`
`               →DP Problem 13`
`                 ↳Polynomial Ordering`
`       →DP Problem 3`
`         ↳Nar`

Dependency Pairs:

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

Rules:

f(X) -> cons(X, nf(ng(X)))
f(X) -> nf(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> ng(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(nf(X)) -> f(activate(X))
activate(ng(X)) -> g(activate(X))
activate(X) -> X

Strategy:

innermost

The following dependency pairs can be strictly oriented:

ACTIVATE(ng(nf(ng(nf(ng(X'''''''')))))) -> ACTIVATE(nf(ng(nf(ng(X'''''''')))))
ACTIVATE(ng(nf(ng(nf(nf(X'''''''')))))) -> ACTIVATE(nf(ng(nf(nf(X'''''''')))))
ACTIVATE(ng(ng(nf(ng(X''''''))))) -> ACTIVATE(ng(nf(ng(X''''''))))
ACTIVATE(ng(nf(nf(ng(X''''''))))) -> ACTIVATE(nf(nf(ng(X''''''))))
ACTIVATE(ng(nf(nf(nf(X''''''))))) -> ACTIVATE(nf(nf(nf(X''''''))))
ACTIVATE(ng(ng(nf(nf(X''''''))))) -> ACTIVATE(ng(nf(nf(X''''''))))
ACTIVATE(ng(ng(ng(X'''')))) -> ACTIVATE(ng(ng(X'''')))
ACTIVATE(ng(nf(ng(ng(X''''''))))) -> ACTIVATE(nf(ng(ng(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)) =  x1 POL(n__g(x1)) =  1 + x1 POL(ACTIVATE(x1)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`           →DP Problem 7`
`             ↳FwdInst`
`             ...`
`               →DP Problem 14`
`                 ↳Dependency Graph`
`       →DP Problem 3`
`         ↳Nar`

Dependency Pairs:

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

Rules:

f(X) -> cons(X, nf(ng(X)))
f(X) -> nf(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> ng(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(nf(X)) -> f(activate(X))
activate(ng(X)) -> g(activate(X))
activate(X) -> X

Strategy:

innermost

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

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`           →DP Problem 7`
`             ↳FwdInst`
`             ...`
`               →DP Problem 15`
`                 ↳Polynomial Ordering`
`       →DP Problem 3`
`         ↳Nar`

Dependency Pair:

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

Rules:

f(X) -> cons(X, nf(ng(X)))
f(X) -> nf(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> ng(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(nf(X)) -> f(activate(X))
activate(ng(X)) -> g(activate(X))
activate(X) -> X

Strategy:

innermost

The following dependency pair can be strictly oriented:

ACTIVATE(nf(nf(nf(X'''')))) -> ACTIVATE(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(ACTIVATE(x1)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`           →DP Problem 7`
`             ↳FwdInst`
`             ...`
`               →DP Problem 16`
`                 ↳Dependency Graph`
`       →DP Problem 3`
`         ↳Nar`

Dependency Pair:

Rules:

f(X) -> cons(X, nf(ng(X)))
f(X) -> nf(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> ng(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(nf(X)) -> f(activate(X))
activate(ng(X)) -> g(activate(X))
activate(X) -> X

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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

Dependency Pair:

SEL(s(X), cons(Y, Z)) -> SEL(X, activate(Z))

Rules:

f(X) -> cons(X, nf(ng(X)))
f(X) -> nf(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> ng(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(nf(X)) -> f(activate(X))
activate(ng(X)) -> g(activate(X))
activate(X) -> X

Strategy:

innermost

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

SEL(s(X), cons(Y, Z)) -> SEL(X, activate(Z))
three new Dependency Pairs are created:

SEL(s(X), cons(Y, nf(X''))) -> SEL(X, f(activate(X'')))
SEL(s(X), cons(Y, ng(X''))) -> SEL(X, g(activate(X'')))
SEL(s(X), cons(Y, Z')) -> SEL(X, Z')

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`           →DP Problem 17`
`             ↳Narrowing Transformation`

Dependency Pairs:

SEL(s(X), cons(Y, Z')) -> SEL(X, Z')
SEL(s(X), cons(Y, ng(X''))) -> SEL(X, g(activate(X'')))
SEL(s(X), cons(Y, nf(X''))) -> SEL(X, f(activate(X'')))

Rules:

f(X) -> cons(X, nf(ng(X)))
f(X) -> nf(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> ng(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(nf(X)) -> f(activate(X))
activate(ng(X)) -> g(activate(X))
activate(X) -> X

Strategy:

innermost

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

SEL(s(X), cons(Y, nf(X''))) -> SEL(X, f(activate(X'')))
five new Dependency Pairs are created:

SEL(s(X), cons(Y, nf(X'''))) -> SEL(X, cons(activate(X'''), nf(ng(activate(X''')))))
SEL(s(X), cons(Y, nf(X'''))) -> SEL(X, nf(activate(X''')))
SEL(s(X), cons(Y, nf(nf(X''')))) -> SEL(X, f(f(activate(X'''))))
SEL(s(X), cons(Y, nf(ng(X''')))) -> SEL(X, f(g(activate(X'''))))
SEL(s(X), cons(Y, nf(X'''))) -> SEL(X, f(X'''))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`           →DP Problem 17`
`             ↳Nar`
`             ...`
`               →DP Problem 18`
`                 ↳Narrowing Transformation`

Dependency Pairs:

SEL(s(X), cons(Y, nf(X'''))) -> SEL(X, f(X'''))
SEL(s(X), cons(Y, nf(nf(X''')))) -> SEL(X, f(f(activate(X'''))))
SEL(s(X), cons(Y, nf(ng(X''')))) -> SEL(X, f(g(activate(X'''))))
SEL(s(X), cons(Y, nf(X'''))) -> SEL(X, cons(activate(X'''), nf(ng(activate(X''')))))
SEL(s(X), cons(Y, ng(X''))) -> SEL(X, g(activate(X'')))
SEL(s(X), cons(Y, Z')) -> SEL(X, Z')

Rules:

f(X) -> cons(X, nf(ng(X)))
f(X) -> nf(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> ng(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(nf(X)) -> f(activate(X))
activate(ng(X)) -> g(activate(X))
activate(X) -> X

Strategy:

innermost

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

SEL(s(X), cons(Y, ng(X''))) -> SEL(X, g(activate(X'')))
four new Dependency Pairs are created:

SEL(s(X), cons(Y, ng(X'''))) -> SEL(X, ng(activate(X''')))
SEL(s(X), cons(Y, ng(nf(X''')))) -> SEL(X, g(f(activate(X'''))))
SEL(s(X), cons(Y, ng(ng(X''')))) -> SEL(X, g(g(activate(X'''))))
SEL(s(X), cons(Y, ng(X'''))) -> SEL(X, g(X'''))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`           →DP Problem 17`
`             ↳Nar`
`             ...`
`               →DP Problem 19`
`                 ↳Forward Instantiation Transformation`

Dependency Pairs:

SEL(s(X), cons(Y, ng(X'''))) -> SEL(X, g(X'''))
SEL(s(X), cons(Y, ng(ng(X''')))) -> SEL(X, g(g(activate(X'''))))
SEL(s(X), cons(Y, ng(nf(X''')))) -> SEL(X, g(f(activate(X'''))))
SEL(s(X), cons(Y, nf(nf(X''')))) -> SEL(X, f(f(activate(X'''))))
SEL(s(X), cons(Y, nf(ng(X''')))) -> SEL(X, f(g(activate(X'''))))
SEL(s(X), cons(Y, nf(X'''))) -> SEL(X, cons(activate(X'''), nf(ng(activate(X''')))))
SEL(s(X), cons(Y, Z')) -> SEL(X, Z')
SEL(s(X), cons(Y, nf(X'''))) -> SEL(X, f(X'''))

Rules:

f(X) -> cons(X, nf(ng(X)))
f(X) -> nf(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> ng(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(nf(X)) -> f(activate(X))
activate(ng(X)) -> g(activate(X))
activate(X) -> X

Strategy:

innermost

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

SEL(s(X), cons(Y, Z')) -> SEL(X, Z')
seven new Dependency Pairs are created:

SEL(s(s(X'')), cons(Y, cons(Y'', Z'''))) -> SEL(s(X''), cons(Y'', Z'''))
SEL(s(s(X'')), cons(Y, cons(Y'', nf(X''''')))) -> SEL(s(X''), cons(Y'', nf(X''''')))
SEL(s(s(X'')), cons(Y, cons(Y'', nf(nf(X'''''))))) -> SEL(s(X''), cons(Y'', nf(nf(X'''''))))
SEL(s(s(X'')), cons(Y, cons(Y'', nf(ng(X'''''))))) -> SEL(s(X''), cons(Y'', nf(ng(X'''''))))
SEL(s(s(X'')), cons(Y, cons(Y'', ng(nf(X'''''))))) -> SEL(s(X''), cons(Y'', ng(nf(X'''''))))
SEL(s(s(X'')), cons(Y, cons(Y'', ng(ng(X'''''))))) -> SEL(s(X''), cons(Y'', ng(ng(X'''''))))
SEL(s(s(X'')), cons(Y, cons(Y'', ng(X''''')))) -> SEL(s(X''), cons(Y'', ng(X''''')))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`           →DP Problem 17`
`             ↳Nar`
`             ...`
`               →DP Problem 20`
`                 ↳Forward Instantiation Transformation`

Dependency Pairs:

SEL(s(s(X'')), cons(Y, cons(Y'', ng(X''''')))) -> SEL(s(X''), cons(Y'', ng(X''''')))
SEL(s(s(X'')), cons(Y, cons(Y'', ng(ng(X'''''))))) -> SEL(s(X''), cons(Y'', ng(ng(X'''''))))
SEL(s(s(X'')), cons(Y, cons(Y'', ng(nf(X'''''))))) -> SEL(s(X''), cons(Y'', ng(nf(X'''''))))
SEL(s(s(X'')), cons(Y, cons(Y'', nf(ng(X'''''))))) -> SEL(s(X''), cons(Y'', nf(ng(X'''''))))
SEL(s(s(X'')), cons(Y, cons(Y'', nf(nf(X'''''))))) -> SEL(s(X''), cons(Y'', nf(nf(X'''''))))
SEL(s(s(X'')), cons(Y, cons(Y'', nf(X''''')))) -> SEL(s(X''), cons(Y'', nf(X''''')))
SEL(s(s(X'')), cons(Y, cons(Y'', Z'''))) -> SEL(s(X''), cons(Y'', Z'''))
SEL(s(X), cons(Y, ng(ng(X''')))) -> SEL(X, g(g(activate(X'''))))
SEL(s(X), cons(Y, ng(nf(X''')))) -> SEL(X, g(f(activate(X'''))))
SEL(s(X), cons(Y, nf(X'''))) -> SEL(X, f(X'''))
SEL(s(X), cons(Y, nf(nf(X''')))) -> SEL(X, f(f(activate(X'''))))
SEL(s(X), cons(Y, nf(ng(X''')))) -> SEL(X, f(g(activate(X'''))))
SEL(s(X), cons(Y, nf(X'''))) -> SEL(X, cons(activate(X'''), nf(ng(activate(X''')))))
SEL(s(X), cons(Y, ng(X'''))) -> SEL(X, g(X'''))

Rules:

f(X) -> cons(X, nf(ng(X)))
f(X) -> nf(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> ng(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(nf(X)) -> f(activate(X))
activate(ng(X)) -> g(activate(X))
activate(X) -> X

Strategy:

innermost

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

SEL(s(X), cons(Y, ng(X'''))) -> SEL(X, g(X'''))
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`           →DP Problem 17`
`             ↳Nar`
`             ...`
`               →DP Problem 21`
`                 ↳Polynomial Ordering`

Dependency Pairs:

SEL(s(s(X'')), cons(Y, cons(Y'', ng(ng(X'''''))))) -> SEL(s(X''), cons(Y'', ng(ng(X'''''))))
SEL(s(s(X'')), cons(Y, cons(Y'', ng(nf(X'''''))))) -> SEL(s(X''), cons(Y'', ng(nf(X'''''))))
SEL(s(s(X'')), cons(Y, cons(Y'', nf(ng(X'''''))))) -> SEL(s(X''), cons(Y'', nf(ng(X'''''))))
SEL(s(s(X'')), cons(Y, cons(Y'', nf(nf(X'''''))))) -> SEL(s(X''), cons(Y'', nf(nf(X'''''))))
SEL(s(s(X'')), cons(Y, cons(Y'', nf(X''''')))) -> SEL(s(X''), cons(Y'', nf(X''''')))
SEL(s(s(X'')), cons(Y, cons(Y'', Z'''))) -> SEL(s(X''), cons(Y'', Z'''))
SEL(s(X), cons(Y, ng(ng(X''')))) -> SEL(X, g(g(activate(X'''))))
SEL(s(X), cons(Y, nf(X'''))) -> SEL(X, f(X'''))
SEL(s(X), cons(Y, nf(nf(X''')))) -> SEL(X, f(f(activate(X'''))))
SEL(s(X), cons(Y, nf(ng(X''')))) -> SEL(X, f(g(activate(X'''))))
SEL(s(X), cons(Y, nf(X'''))) -> SEL(X, cons(activate(X'''), nf(ng(activate(X''')))))
SEL(s(X), cons(Y, ng(nf(X''')))) -> SEL(X, g(f(activate(X'''))))
SEL(s(s(X'')), cons(Y, cons(Y'', ng(X''''')))) -> SEL(s(X''), cons(Y'', ng(X''''')))

Rules:

f(X) -> cons(X, nf(ng(X)))
f(X) -> nf(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> ng(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(nf(X)) -> f(activate(X))
activate(ng(X)) -> g(activate(X))
activate(X) -> X

Strategy:

innermost

The following dependency pairs can be strictly oriented:

SEL(s(s(X'')), cons(Y, cons(Y'', ng(ng(X'''''))))) -> SEL(s(X''), cons(Y'', ng(ng(X'''''))))
SEL(s(s(X'')), cons(Y, cons(Y'', ng(nf(X'''''))))) -> SEL(s(X''), cons(Y'', ng(nf(X'''''))))
SEL(s(s(X'')), cons(Y, cons(Y'', nf(ng(X'''''))))) -> SEL(s(X''), cons(Y'', nf(ng(X'''''))))
SEL(s(s(X'')), cons(Y, cons(Y'', nf(nf(X'''''))))) -> SEL(s(X''), cons(Y'', nf(nf(X'''''))))
SEL(s(s(X'')), cons(Y, cons(Y'', nf(X''''')))) -> SEL(s(X''), cons(Y'', nf(X''''')))
SEL(s(s(X'')), cons(Y, cons(Y'', Z'''))) -> SEL(s(X''), cons(Y'', Z'''))
SEL(s(X), cons(Y, ng(ng(X''')))) -> SEL(X, g(g(activate(X'''))))
SEL(s(X), cons(Y, nf(X'''))) -> SEL(X, f(X'''))
SEL(s(X), cons(Y, nf(nf(X''')))) -> SEL(X, f(f(activate(X'''))))
SEL(s(X), cons(Y, nf(ng(X''')))) -> SEL(X, f(g(activate(X'''))))
SEL(s(X), cons(Y, nf(X'''))) -> SEL(X, cons(activate(X'''), nf(ng(activate(X''')))))
SEL(s(X), cons(Y, ng(nf(X''')))) -> SEL(X, g(f(activate(X'''))))
SEL(s(s(X'')), cons(Y, cons(Y'', ng(X''''')))) -> SEL(s(X''), cons(Y'', ng(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(activate(x1)) =  0 POL(n__f(x1)) =  0 POL(0) =  0 POL(g(x1)) =  0 POL(cons(x1, x2)) =  0 POL(SEL(x1, x2)) =  x1 POL(n__g(x1)) =  0 POL(s(x1)) =  1 + x1 POL(f(x1)) =  0

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`           →DP Problem 17`
`             ↳Nar`
`             ...`
`               →DP Problem 22`
`                 ↳Dependency Graph`

Dependency Pair:

Rules:

f(X) -> cons(X, nf(ng(X)))
f(X) -> nf(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> ng(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(nf(X)) -> f(activate(X))
activate(ng(X)) -> g(activate(X))
activate(X) -> X

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:09 minutes