Termination Proof using AProVE

Term Rewriting System R:
[X, XS, N, X1, X2]
from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
head(cons(X, XS)) -> X
2nd(cons(X, XS)) -> head(activate(XS))
take(0, XS) -> nil
take(s(N), cons(X, XS)) -> cons(X, n_{_take}(N, activate(XS)))
take(X1, X2) -> n_{_take}(X1, X2)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_take}(X1, X2)) -> take(activate(X1), activate(X2))
activate(X) -> X

Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

2ND(cons(X, XS)) -> HEAD(activate(XS))
2ND(cons(X, XS)) -> ACTIVATE(XS)
TAKE(s(N), cons(X, XS)) -> ACTIVATE(XS)
SEL(s(N), cons(X, XS)) -> SEL(N, activate(XS))
SEL(s(N), cons(X, XS)) -> ACTIVATE(XS)
ACTIVATE(n_{_from}(X)) -> FROM(activate(X))
ACTIVATE(n_{_from}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_s}(X)) -> S(activate(X))
ACTIVATE(n_{_s}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_take}(X1, X2)) -> TAKE(activate(X1), activate(X2))
ACTIVATE(n_{_take}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_take}(X1, X2)) -> ACTIVATE(X2)

Furthermore, R contains two SCCs.

     ↳DPs

       →DP Problem 1

         ↳Narrowing Transformation

       →DP Problem 2

         ↳Remaining

Dependency Pairs:

ACTIVATE(n_{_take}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_take}(X1, X2)) -> ACTIVATE(X1)
TAKE(s(N), cons(X, XS)) -> ACTIVATE(XS)
ACTIVATE(n_{_take}(X1, X2)) -> TAKE(activate(X1), activate(X2))
ACTIVATE(n_{_s}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_from}(X)) -> ACTIVATE(X)

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
head(cons(X, XS)) -> X
2nd(cons(X, XS)) -> head(activate(XS))
take(0, XS) -> nil
take(s(N), cons(X, XS)) -> cons(X, n_{_take}(N, activate(XS)))
take(X1, X2) -> n_{_take}(X1, X2)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_take}(X1, X2)) -> take(activate(X1), activate(X2))
activate(X) -> X

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

ACTIVATE(n_{_take}(X1, X2)) -> TAKE(activate(X1), activate(X2))

eight new Dependency Pairs are created:

ACTIVATE(n_{_take}(n_{_from}(X'), X2)) -> TAKE(from(activate(X')), activate(X2))
ACTIVATE(n_{_take}(n_{_s}(X'), X2)) -> TAKE(s(activate(X')), activate(X2))
ACTIVATE(n_{_take}(n_{_take}(X1'', X2''), X2)) -> TAKE(take(activate(X1''), activate(X2'')), activate(X2))
ACTIVATE(n_{_take}(X1', X2)) -> TAKE(X1', activate(X2))
ACTIVATE(n_{_take}(X1, n_{_from}(X'))) -> TAKE(activate(X1), from(activate(X')))
ACTIVATE(n_{_take}(X1, n_{_s}(X'))) -> TAKE(activate(X1), s(activate(X')))
ACTIVATE(n_{_take}(X1, n_{_take}(X1'', X2''))) -> TAKE(activate(X1), take(activate(X1''), activate(X2'')))
ACTIVATE(n_{_take}(X1, X2')) -> TAKE(activate(X1), X2')

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

       →DP Problem 2

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pairs:
ACTIVATE(n_{_take}(X1, X2')) -> TAKE(activate(X1), X2')
ACTIVATE(n_{_take}(X1, n_{_take}(X1'', X2''))) -> TAKE(activate(X1), take(activate(X1''), activate(X2'')))
ACTIVATE(n_{_take}(X1, n_{_s}(X'))) -> TAKE(activate(X1), s(activate(X')))
ACTIVATE(n_{_take}(X1, n_{_from}(X'))) -> TAKE(activate(X1), from(activate(X')))
ACTIVATE(n_{_take}(X1', X2)) -> TAKE(X1', activate(X2))
ACTIVATE(n_{_take}(n_{_take}(X1'', X2''), X2)) -> TAKE(take(activate(X1''), activate(X2'')), activate(X2))
ACTIVATE(n_{_take}(n_{_s}(X'), X2)) -> TAKE(s(activate(X')), activate(X2))
TAKE(s(N), cons(X, XS)) -> ACTIVATE(XS)
ACTIVATE(n_{_take}(n_{_from}(X'), X2)) -> TAKE(from(activate(X')), activate(X2))
ACTIVATE(n_{_take}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_s}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_from}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_take}(X1, X2)) -> ACTIVATE(X2)

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
head(cons(X, XS)) -> X
2nd(cons(X, XS)) -> head(activate(XS))
take(0, XS) -> nil
take(s(N), cons(X, XS)) -> cons(X, n_{_take}(N, activate(XS)))
take(X1, X2) -> n_{_take}(X1, X2)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_take}(X1, X2)) -> take(activate(X1), activate(X2))
activate(X) -> X
Dependency Pair:
SEL(s(N), cons(X, XS)) -> SEL(N, activate(XS))

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
head(cons(X, XS)) -> X
2nd(cons(X, XS)) -> head(activate(XS))
take(0, XS) -> nil
take(s(N), cons(X, XS)) -> cons(X, n_{_take}(N, activate(XS)))
take(X1, X2) -> n_{_take}(X1, X2)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_take}(X1, X2)) -> take(activate(X1), activate(X2))
activate(X) -> X

     ↳DPs

       →DP Problem 1

         ↳Nar

       →DP Problem 2

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pairs:
ACTIVATE(n_{_take}(X1, X2')) -> TAKE(activate(X1), X2')
ACTIVATE(n_{_take}(X1, n_{_take}(X1'', X2''))) -> TAKE(activate(X1), take(activate(X1''), activate(X2'')))
ACTIVATE(n_{_take}(X1, n_{_s}(X'))) -> TAKE(activate(X1), s(activate(X')))
ACTIVATE(n_{_take}(X1, n_{_from}(X'))) -> TAKE(activate(X1), from(activate(X')))
ACTIVATE(n_{_take}(X1', X2)) -> TAKE(X1', activate(X2))
ACTIVATE(n_{_take}(n_{_take}(X1'', X2''), X2)) -> TAKE(take(activate(X1''), activate(X2'')), activate(X2))
ACTIVATE(n_{_take}(n_{_s}(X'), X2)) -> TAKE(s(activate(X')), activate(X2))
TAKE(s(N), cons(X, XS)) -> ACTIVATE(XS)
ACTIVATE(n_{_take}(n_{_from}(X'), X2)) -> TAKE(from(activate(X')), activate(X2))
ACTIVATE(n_{_take}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_s}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_from}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_take}(X1, X2)) -> ACTIVATE(X2)

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
head(cons(X, XS)) -> X
2nd(cons(X, XS)) -> head(activate(XS))
take(0, XS) -> nil
take(s(N), cons(X, XS)) -> cons(X, n_{_take}(N, activate(XS)))
take(X1, X2) -> n_{_take}(X1, X2)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_take}(X1, X2)) -> take(activate(X1), activate(X2))
activate(X) -> X
Dependency Pair:
SEL(s(N), cons(X, XS)) -> SEL(N, activate(XS))

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
head(cons(X, XS)) -> X
2nd(cons(X, XS)) -> head(activate(XS))
take(0, XS) -> nil
take(s(N), cons(X, XS)) -> cons(X, n_{_take}(N, activate(XS)))
take(X1, X2) -> n_{_take}(X1, X2)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_take}(X1, X2)) -> take(activate(X1), activate(X2))
activate(X) -> X

Termination of R could not be shown.
Duration:
0:01 minutes