Term Rewriting System R:
[X, XS, N]
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
after(0, XS) -> XS
after(s(N), cons(X, XS)) -> after(N, activate(XS))
activate(nfrom(X)) -> from(X)
activate(X) -> X

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

AFTER(s(N), cons(X, XS)) -> AFTER(N, activate(XS))
AFTER(s(N), cons(X, XS)) -> ACTIVATE(XS)
ACTIVATE(nfrom(X)) -> FROM(X)

Furthermore, R contains one SCC.

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

Dependency Pair:

AFTER(s(N), cons(X, XS)) -> AFTER(N, activate(XS))

Rules:

from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
after(0, XS) -> XS
after(s(N), cons(X, XS)) -> after(N, activate(XS))
activate(nfrom(X)) -> from(X)
activate(X) -> X

Strategy:

innermost

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

AFTER(s(N), cons(X, XS)) -> AFTER(N, activate(XS))
two new Dependency Pairs are created:

AFTER(s(N), cons(X, nfrom(X''))) -> AFTER(N, from(X''))
AFTER(s(N), cons(X, XS')) -> AFTER(N, XS')

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

AFTER(s(N), cons(X, XS')) -> AFTER(N, XS')
AFTER(s(N), cons(X, nfrom(X''))) -> AFTER(N, from(X''))

Rules:

from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
after(0, XS) -> XS
after(s(N), cons(X, XS)) -> after(N, activate(XS))
activate(nfrom(X)) -> from(X)
activate(X) -> X

Strategy:

innermost

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

AFTER(s(N), cons(X, nfrom(X''))) -> AFTER(N, from(X''))
two new Dependency Pairs are created:

AFTER(s(N), cons(X, nfrom(X'''))) -> AFTER(N, cons(X''', nfrom(s(X'''))))
AFTER(s(N), cons(X, nfrom(X'''))) -> AFTER(N, nfrom(X'''))

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

AFTER(s(N), cons(X, nfrom(X'''))) -> AFTER(N, cons(X''', nfrom(s(X'''))))
AFTER(s(N), cons(X, XS')) -> AFTER(N, XS')

Rules:

from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
after(0, XS) -> XS
after(s(N), cons(X, XS)) -> after(N, activate(XS))
activate(nfrom(X)) -> from(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

AFTER(s(N), cons(X, XS')) -> AFTER(N, XS')
two new Dependency Pairs are created:

AFTER(s(s(N'')), cons(X, cons(X'', XS'''))) -> AFTER(s(N''), cons(X'', XS'''))
AFTER(s(s(N'')), cons(X, cons(X'', nfrom(X''''')))) -> AFTER(s(N''), cons(X'', nfrom(X''''')))

The transformation is resulting in two new DP problems:

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

Dependency Pair:

AFTER(s(N), cons(X, nfrom(X'''))) -> AFTER(N, cons(X''', nfrom(s(X'''))))

Rules:

from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
after(0, XS) -> XS
after(s(N), cons(X, XS)) -> after(N, activate(XS))
activate(nfrom(X)) -> from(X)
activate(X) -> X

Strategy:

innermost

The following dependency pair can be strictly oriented:

AFTER(s(N), cons(X, nfrom(X'''))) -> AFTER(N, cons(X''', nfrom(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(n__from(x1)) =  0 POL(AFTER(x1, x2)) =  x1 POL(cons(x1, x2)) =  0 POL(s(x1)) =  1 + x1

resulting in one new DP problem.

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

Dependency Pair:

Rules:

from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
after(0, XS) -> XS
after(s(N), cons(X, XS)) -> after(N, activate(XS))
activate(nfrom(X)) -> from(X)
activate(X) -> X

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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

Dependency Pair:

AFTER(s(s(N'')), cons(X, cons(X'', XS'''))) -> AFTER(s(N''), cons(X'', XS'''))

Rules:

from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
after(0, XS) -> XS
after(s(N), cons(X, XS)) -> after(N, activate(XS))
activate(nfrom(X)) -> from(X)
activate(X) -> X

Strategy:

innermost

The following dependency pair can be strictly oriented:

AFTER(s(s(N'')), cons(X, cons(X'', XS'''))) -> AFTER(s(N''), cons(X'', XS'''))

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(AFTER(x1, x2)) =  x1 POL(cons(x1, x2)) =  0 POL(s(x1)) =  1 + x1

resulting in one new DP problem.

Innermost Termination of R successfully shown.
Duration:
0:00 minutes