Term Rewriting System R:
[X, Y, X1, X2]
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
length(nnil) -> 0
length(ncons(X, Y)) -> s(length1(activate(Y)))
length1(X) -> length(activate(X))
nil -> nnil
cons(X1, X2) -> ncons(X1, X2)
activate(nfrom(X)) -> from(X)
activate(nnil) -> nil
activate(ncons(X1, X2)) -> cons(X1, X2)
activate(X) -> X

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

FROM(X) -> CONS(X, nfrom(s(X)))
LENGTH(ncons(X, Y)) -> LENGTH1(activate(Y))
LENGTH(ncons(X, Y)) -> ACTIVATE(Y)
LENGTH1(X) -> LENGTH(activate(X))
LENGTH1(X) -> ACTIVATE(X)
ACTIVATE(nfrom(X)) -> FROM(X)
ACTIVATE(nnil) -> NIL
ACTIVATE(ncons(X1, X2)) -> CONS(X1, X2)

Furthermore, R contains one SCC.

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

Dependency Pairs:

LENGTH1(X) -> LENGTH(activate(X))
LENGTH(ncons(X, Y)) -> LENGTH1(activate(Y))

Rules:

from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
length(nnil) -> 0
length(ncons(X, Y)) -> s(length1(activate(Y)))
length1(X) -> length(activate(X))
nil -> nnil
cons(X1, X2) -> ncons(X1, X2)
activate(nfrom(X)) -> from(X)
activate(nnil) -> nil
activate(ncons(X1, X2)) -> cons(X1, X2)
activate(X) -> X

Strategy:

innermost

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

LENGTH(ncons(X, Y)) -> LENGTH1(activate(Y))
four new Dependency Pairs are created:

LENGTH(ncons(X, nfrom(X''))) -> LENGTH1(from(X''))
LENGTH(ncons(X, nnil)) -> LENGTH1(nil)
LENGTH(ncons(X, ncons(X1', X2'))) -> LENGTH1(cons(X1', X2'))
LENGTH(ncons(X, Y')) -> LENGTH1(Y')

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

LENGTH(ncons(X, Y')) -> LENGTH1(Y')
LENGTH(ncons(X, ncons(X1', X2'))) -> LENGTH1(cons(X1', X2'))
LENGTH(ncons(X, nnil)) -> LENGTH1(nil)
LENGTH(ncons(X, nfrom(X''))) -> LENGTH1(from(X''))
LENGTH1(X) -> LENGTH(activate(X))

Rules:

from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
length(nnil) -> 0
length(ncons(X, Y)) -> s(length1(activate(Y)))
length1(X) -> length(activate(X))
nil -> nnil
cons(X1, X2) -> ncons(X1, X2)
activate(nfrom(X)) -> from(X)
activate(nnil) -> nil
activate(ncons(X1, X2)) -> cons(X1, X2)
activate(X) -> X

Strategy:

innermost

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

LENGTH(ncons(X, nnil)) -> LENGTH1(nil)
one new Dependency Pair is created:

LENGTH(ncons(X, nnil)) -> LENGTH1(nnil)

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

LENGTH(ncons(X, nnil)) -> LENGTH1(nnil)
LENGTH(ncons(X, ncons(X1', X2'))) -> LENGTH1(cons(X1', X2'))
LENGTH(ncons(X, nfrom(X''))) -> LENGTH1(from(X''))
LENGTH1(X) -> LENGTH(activate(X))
LENGTH(ncons(X, Y')) -> LENGTH1(Y')

Rules:

from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
length(nnil) -> 0
length(ncons(X, Y)) -> s(length1(activate(Y)))
length1(X) -> length(activate(X))
nil -> nnil
cons(X1, X2) -> ncons(X1, X2)
activate(nfrom(X)) -> from(X)
activate(nnil) -> nil
activate(ncons(X1, X2)) -> cons(X1, X2)
activate(X) -> X

Strategy:

innermost

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

LENGTH(ncons(X, ncons(X1', X2'))) -> LENGTH1(cons(X1', X2'))
one new Dependency Pair is created:

LENGTH(ncons(X, ncons(X1', X2'))) -> LENGTH1(ncons(X1', X2'))

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

LENGTH(ncons(X, ncons(X1', X2'))) -> LENGTH1(ncons(X1', X2'))
LENGTH(ncons(X, Y')) -> LENGTH1(Y')
LENGTH(ncons(X, nfrom(X''))) -> LENGTH1(from(X''))
LENGTH1(X) -> LENGTH(activate(X))
LENGTH(ncons(X, nnil)) -> LENGTH1(nnil)

Rules:

from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
length(nnil) -> 0
length(ncons(X, Y)) -> s(length1(activate(Y)))
length1(X) -> length(activate(X))
nil -> nnil
cons(X1, X2) -> ncons(X1, X2)
activate(nfrom(X)) -> from(X)
activate(nnil) -> nil
activate(ncons(X1, X2)) -> cons(X1, X2)
activate(X) -> X

Strategy:

innermost

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

LENGTH1(X) -> LENGTH(activate(X))
four new Dependency Pairs are created:

LENGTH1(nfrom(X'')) -> LENGTH(from(X''))
LENGTH1(nnil) -> LENGTH(nil)
LENGTH1(ncons(X1', X2')) -> LENGTH(cons(X1', X2'))
LENGTH1(X'') -> LENGTH(X'')

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

LENGTH1(X'') -> LENGTH(X'')
LENGTH(ncons(X, nnil)) -> LENGTH1(nnil)
LENGTH1(nnil) -> LENGTH(nil)
LENGTH(ncons(X, Y')) -> LENGTH1(Y')
LENGTH1(nfrom(X'')) -> LENGTH(from(X''))
LENGTH(ncons(X, nfrom(X''))) -> LENGTH1(from(X''))
LENGTH1(ncons(X1', X2')) -> LENGTH(cons(X1', X2'))
LENGTH(ncons(X, ncons(X1', X2'))) -> LENGTH1(ncons(X1', X2'))

Rules:

from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
length(nnil) -> 0
length(ncons(X, Y)) -> s(length1(activate(Y)))
length1(X) -> length(activate(X))
nil -> nnil
cons(X1, X2) -> ncons(X1, X2)
activate(nfrom(X)) -> from(X)
activate(nnil) -> nil
activate(ncons(X1, X2)) -> cons(X1, X2)
activate(X) -> X

Strategy:

innermost

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

LENGTH1(nnil) -> LENGTH(nil)
one new Dependency Pair is created:

LENGTH1(nnil) -> LENGTH(nnil)

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

LENGTH(ncons(X, ncons(X1', X2'))) -> LENGTH1(ncons(X1', X2'))
LENGTH(ncons(X, nnil)) -> LENGTH1(nnil)
LENGTH1(ncons(X1', X2')) -> LENGTH(cons(X1', X2'))
LENGTH(ncons(X, Y')) -> LENGTH1(Y')
LENGTH1(nfrom(X'')) -> LENGTH(from(X''))
LENGTH(ncons(X, nfrom(X''))) -> LENGTH1(from(X''))
LENGTH1(X'') -> LENGTH(X'')

Rules:

from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
length(nnil) -> 0
length(ncons(X, Y)) -> s(length1(activate(Y)))
length1(X) -> length(activate(X))
nil -> nnil
cons(X1, X2) -> ncons(X1, X2)
activate(nfrom(X)) -> from(X)
activate(nnil) -> nil
activate(ncons(X1, X2)) -> cons(X1, X2)
activate(X) -> X

Strategy:

innermost

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

LENGTH1(ncons(X1', X2')) -> LENGTH(cons(X1', X2'))
one new Dependency Pair is created:

LENGTH1(ncons(X1', X2')) -> LENGTH(ncons(X1', X2'))

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:

LENGTH(ncons(X, nnil)) -> LENGTH1(nnil)
LENGTH1(ncons(X1', X2')) -> LENGTH(ncons(X1', X2'))
LENGTH(ncons(X, Y')) -> LENGTH1(Y')
LENGTH1(nfrom(X'')) -> LENGTH(from(X''))
LENGTH(ncons(X, nfrom(X''))) -> LENGTH1(from(X''))
LENGTH1(X'') -> LENGTH(X'')
LENGTH(ncons(X, ncons(X1', X2'))) -> LENGTH1(ncons(X1', X2'))

Rules:

from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
length(nnil) -> 0
length(ncons(X, Y)) -> s(length1(activate(Y)))
length1(X) -> length(activate(X))
nil -> nnil
cons(X1, X2) -> ncons(X1, X2)
activate(nfrom(X)) -> from(X)
activate(nnil) -> nil
activate(ncons(X1, X2)) -> cons(X1, X2)
activate(X) -> X

Strategy:

innermost

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

LENGTH1(X'') -> LENGTH(X'')
four new Dependency Pairs are created:

LENGTH1(ncons(X'0, nfrom(X''''))) -> LENGTH(ncons(X'0, nfrom(X'''')))
LENGTH1(ncons(X''', Y''')) -> LENGTH(ncons(X''', Y'''))
LENGTH1(ncons(X''', nnil)) -> LENGTH(ncons(X''', nnil))
LENGTH1(ncons(X''', ncons(X1''', X2'''))) -> LENGTH(ncons(X''', ncons(X1''', X2''')))

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

LENGTH1(ncons(X''', ncons(X1''', X2'''))) -> LENGTH(ncons(X''', ncons(X1''', X2''')))
LENGTH1(ncons(X''', nnil)) -> LENGTH(ncons(X''', nnil))
LENGTH(ncons(X, ncons(X1', X2'))) -> LENGTH1(ncons(X1', X2'))
LENGTH1(ncons(X''', Y''')) -> LENGTH(ncons(X''', Y'''))
LENGTH1(ncons(X'0, nfrom(X''''))) -> LENGTH(ncons(X'0, nfrom(X'''')))
LENGTH(ncons(X, Y')) -> LENGTH1(Y')
LENGTH1(nfrom(X'')) -> LENGTH(from(X''))
LENGTH(ncons(X, nfrom(X''))) -> LENGTH1(from(X''))
LENGTH1(ncons(X1', X2')) -> LENGTH(ncons(X1', X2'))

Rules:

from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
length(nnil) -> 0
length(ncons(X, Y)) -> s(length1(activate(Y)))
length1(X) -> length(activate(X))
nil -> nnil
cons(X1, X2) -> ncons(X1, X2)
activate(nfrom(X)) -> from(X)
activate(nnil) -> nil
activate(ncons(X1, X2)) -> cons(X1, X2)
activate(X) -> X

Strategy:

innermost

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

LENGTH(ncons(X, Y')) -> LENGTH1(Y')
six new Dependency Pairs are created:

LENGTH(ncons(X, nfrom(X''''))) -> LENGTH1(nfrom(X''''))
LENGTH(ncons(X, ncons(X1''', X2'''))) -> LENGTH1(ncons(X1''', X2'''))
LENGTH(ncons(X, ncons(X'0'', nfrom(X'''''')))) -> LENGTH1(ncons(X'0'', nfrom(X'''''')))
LENGTH(ncons(X, ncons(X''''', Y'''''))) -> LENGTH1(ncons(X''''', Y'''''))
LENGTH(ncons(X, ncons(X''''', nnil))) -> LENGTH1(ncons(X''''', nnil))
LENGTH(ncons(X, ncons(X''''', ncons(X1''''', X2''''')))) -> LENGTH1(ncons(X''''', ncons(X1''''', X2''''')))

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

LENGTH(ncons(X, ncons(X''''', ncons(X1''''', X2''''')))) -> LENGTH1(ncons(X''''', ncons(X1''''', X2''''')))
LENGTH(ncons(X, ncons(X''''', nnil))) -> LENGTH1(ncons(X''''', nnil))
LENGTH(ncons(X, ncons(X''''', Y'''''))) -> LENGTH1(ncons(X''''', Y'''''))
LENGTH(ncons(X, ncons(X'0'', nfrom(X'''''')))) -> LENGTH1(ncons(X'0'', nfrom(X'''''')))
LENGTH1(ncons(X''', Y''')) -> LENGTH(ncons(X''', Y'''))
LENGTH1(ncons(X'0, nfrom(X''''))) -> LENGTH(ncons(X'0, nfrom(X'''')))
LENGTH(ncons(X, ncons(X1''', X2'''))) -> LENGTH1(ncons(X1''', X2'''))
LENGTH(ncons(X, nfrom(X''''))) -> LENGTH1(nfrom(X''''))
LENGTH1(nfrom(X'')) -> LENGTH(from(X''))
LENGTH(ncons(X, nfrom(X''))) -> LENGTH1(from(X''))
LENGTH1(ncons(X1', X2')) -> LENGTH(ncons(X1', X2'))
LENGTH(ncons(X, ncons(X1', X2'))) -> LENGTH1(ncons(X1', X2'))
LENGTH1(ncons(X''', ncons(X1''', X2'''))) -> LENGTH(ncons(X''', ncons(X1''', X2''')))

Rules:

from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
length(nnil) -> 0
length(ncons(X, Y)) -> s(length1(activate(Y)))
length1(X) -> length(activate(X))
nil -> nnil
cons(X1, X2) -> ncons(X1, X2)
activate(nfrom(X)) -> from(X)
activate(nnil) -> nil
activate(ncons(X1, X2)) -> cons(X1, X2)
activate(X) -> X

Strategy:

innermost

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

LENGTH1(ncons(X1', X2')) -> LENGTH(ncons(X1', X2'))
eight new Dependency Pairs are created:

LENGTH1(ncons(X1'', nfrom(X''''))) -> LENGTH(ncons(X1'', nfrom(X'''')))
LENGTH1(ncons(X1'0, ncons(X1''', X2'''))) -> LENGTH(ncons(X1'0, ncons(X1''', X2''')))
LENGTH1(ncons(X1'', nfrom(X''''''))) -> LENGTH(ncons(X1'', nfrom(X'''''')))
LENGTH1(ncons(X1'', ncons(X1''''', X2'''''))) -> LENGTH(ncons(X1'', ncons(X1''''', X2''''')))
LENGTH1(ncons(X1'', ncons(X'0'''', nfrom(X'''''''')))) -> LENGTH(ncons(X1'', ncons(X'0'''', nfrom(X''''''''))))
LENGTH1(ncons(X1'', ncons(X''''''', Y'''''''))) -> LENGTH(ncons(X1'', ncons(X''''''', Y''''''')))
LENGTH1(ncons(X1'', ncons(X''''''', nnil))) -> LENGTH(ncons(X1'', ncons(X''''''', nnil)))
LENGTH1(ncons(X1'', ncons(X''''''', ncons(X1''''''', X2''''''')))) -> LENGTH(ncons(X1'', ncons(X''''''', ncons(X1''''''', X2'''''''))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Rw`
`             ...`
`               →DP Problem 10`
`                 ↳Remaining Obligation(s)`

The following remains to be proven:
Dependency Pairs:

LENGTH1(ncons(X1'', ncons(X''''''', ncons(X1''''''', X2''''''')))) -> LENGTH(ncons(X1'', ncons(X''''''', ncons(X1''''''', X2'''''''))))
LENGTH1(ncons(X1'', ncons(X''''''', nnil))) -> LENGTH(ncons(X1'', ncons(X''''''', nnil)))
LENGTH1(ncons(X1'', ncons(X''''''', Y'''''''))) -> LENGTH(ncons(X1'', ncons(X''''''', Y''''''')))
LENGTH1(ncons(X1'', ncons(X'0'''', nfrom(X'''''''')))) -> LENGTH(ncons(X1'', ncons(X'0'''', nfrom(X''''''''))))
LENGTH(ncons(X, ncons(X''''', nnil))) -> LENGTH1(ncons(X''''', nnil))
LENGTH1(ncons(X1'', ncons(X1''''', X2'''''))) -> LENGTH(ncons(X1'', ncons(X1''''', X2''''')))
LENGTH(ncons(X, ncons(X''''', Y'''''))) -> LENGTH1(ncons(X''''', Y'''''))
LENGTH1(ncons(X1'', nfrom(X''''''))) -> LENGTH(ncons(X1'', nfrom(X'''''')))
LENGTH(ncons(X, ncons(X'0'', nfrom(X'''''')))) -> LENGTH1(ncons(X'0'', nfrom(X'''''')))
LENGTH1(ncons(X1'0, ncons(X1''', X2'''))) -> LENGTH(ncons(X1'0, ncons(X1''', X2''')))
LENGTH1(ncons(X1'', nfrom(X''''))) -> LENGTH(ncons(X1'', nfrom(X'''')))
LENGTH(ncons(X, ncons(X1''', X2'''))) -> LENGTH1(ncons(X1''', X2'''))
LENGTH1(ncons(X''', ncons(X1''', X2'''))) -> LENGTH(ncons(X''', ncons(X1''', X2''')))
LENGTH(ncons(X, nfrom(X''''))) -> LENGTH1(nfrom(X''''))
LENGTH1(ncons(X'0, nfrom(X''''))) -> LENGTH(ncons(X'0, nfrom(X'''')))
LENGTH(ncons(X, ncons(X1', X2'))) -> LENGTH1(ncons(X1', X2'))
LENGTH1(nfrom(X'')) -> LENGTH(from(X''))
LENGTH(ncons(X, nfrom(X''))) -> LENGTH1(from(X''))
LENGTH1(ncons(X''', Y''')) -> LENGTH(ncons(X''', Y'''))
LENGTH(ncons(X, ncons(X''''', ncons(X1''''', X2''''')))) -> LENGTH1(ncons(X''''', ncons(X1''''', X2''''')))

Rules:

from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
length(nnil) -> 0
length(ncons(X, Y)) -> s(length1(activate(Y)))
length1(X) -> length(activate(X))
nil -> nnil
cons(X1, X2) -> ncons(X1, X2)
activate(nfrom(X)) -> from(X)
activate(nnil) -> nil
activate(ncons(X1, X2)) -> cons(X1, X2)
activate(X) -> X

Strategy:

innermost

Innermost Termination of R could not be shown.
Duration:
0:02 minutes