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

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

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

Furthermore, R contains two SCCs.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Instantiation Transformation`
`       →DP Problem 2`
`         ↳Remaining`

Dependency Pair:

FROM(X) -> FROM(s(X))

Rules:

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

Strategy:

innermost

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

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

FROM(s(X'')) -> FROM(s(s(X'')))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Inst`
`           →DP Problem 3`
`             ↳Instantiation Transformation`
`       →DP Problem 2`
`         ↳Remaining`

Dependency Pair:

FROM(s(X'')) -> FROM(s(s(X'')))

Rules:

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

Strategy:

innermost

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

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

FROM(s(s(X''''))) -> FROM(s(s(s(X''''))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Inst`
`           →DP Problem 3`
`             ↳Inst`
`             ...`
`               →DP Problem 4`
`                 ↳Instantiation Transformation`
`       →DP Problem 2`
`         ↳Remaining`

Dependency Pair:

FROM(s(s(X''''))) -> FROM(s(s(s(X''''))))

Rules:

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

Strategy:

innermost

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

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

FROM(s(s(s(X'''''')))) -> FROM(s(s(s(s(X'''''')))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Inst`
`           →DP Problem 3`
`             ↳Inst`
`             ...`
`               →DP Problem 5`
`                 ↳Instantiation Transformation`
`       →DP Problem 2`
`         ↳Remaining`

Dependency Pair:

FROM(s(s(s(X'''''')))) -> FROM(s(s(s(s(X'''''')))))

Rules:

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

Strategy:

innermost

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

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

FROM(s(s(s(s(X''''''''))))) -> FROM(s(s(s(s(s(X''''''''))))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Inst`
`           →DP Problem 3`
`             ↳Inst`
`             ...`
`               →DP Problem 6`
`                 ↳Instantiation Transformation`
`       →DP Problem 2`
`         ↳Remaining`

Dependency Pair:

FROM(s(s(s(s(X''''''''))))) -> FROM(s(s(s(s(s(X''''''''))))))

Rules:

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

Strategy:

innermost

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

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

FROM(s(s(s(s(s(X'''''''''')))))) -> FROM(s(s(s(s(s(s(X'''''''''')))))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Inst`
`       →DP Problem 2`
`         ↳Remaining Obligation(s)`

The following remains to be proven:
• Dependency Pair:

FROM(s(s(s(s(s(X'''''''''')))))) -> FROM(s(s(s(s(s(s(X'''''''''')))))))

Rules:

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

Strategy:

innermost

• Dependency Pair:

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

Rules:

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

Strategy:

innermost

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Inst`
`       →DP Problem 2`
`         ↳Remaining Obligation(s)`

The following remains to be proven:
• Dependency Pair:

FROM(s(s(s(s(s(X'''''''''')))))) -> FROM(s(s(s(s(s(s(X'''''''''')))))))

Rules:

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

Strategy:

innermost

• Dependency Pair:

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

Rules:

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

Strategy:

innermost

Innermost Termination of R could not be shown.
Duration:
0:00 minutes