Termination w.r.t. Q of the given QTRS could be proven:

(0) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

from(X) → cons(X, n__from(s(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)))
sel(0, cons(X, XS)) → X
sel(s(N), cons(X, XS)) → sel(N, activate(XS))
from(X) → n__from(X)
take(X1, X2) → n__take(X1, X2)
activate(n__from(X)) → from(X)
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Combined order from the following AFS and order.
from(x1)  =  from(x1)
cons(x1, x2)  =  cons(x1, x2)
n__from(x1)  =  x1
s(x1)  =  s(x1)
head(x1)  =  head(x1)
2nd(x1)  =  2nd(x1)
activate(x1)  =  activate(x1)
take(x1, x2)  =  take(x1, x2)
0  =  0
nil  =  nil
n__take(x1, x2)  =  n__take(x1, x2)
sel(x1, x2)  =  sel(x1, x2)

Recursive path order with status [RPO].
Quasi-Precedence:

sel₂ > [from₁, 2nd₁, activate₁, take₂] > [cons₂, head₁] > n_take2
sel₂ > [from₁, 2nd₁, activate₁, take₂] > s₁ > n_take2
sel₂ > [from₁, 2nd₁, activate₁, take₂] > nil

Status:

sel₂: [1,2]
from₁: [1]
cons₂: multiset
head₁: [1]
2nd₁: [1]
n_take2: multiset
s₁: multiset
activate₁: [1]
take₂: [2,1]
0: multiset
nil: multiset

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

from(X) → cons(X, n__from(s(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)))
sel(0, cons(X, XS)) → X
sel(s(N), cons(X, XS)) → sel(N, activate(XS))
from(X) → n__from(X)
take(X1, X2) → n__take(X1, X2)
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

(2) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

activate(n__from(X)) → from(X)

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(activate(x₁)) = 1 + x₁   
POL(from(x₁)) = x₁   
POL(n__from(x₁)) = x₁   

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

activate(n__from(X)) → from(X)

(4) Obligation:

Q restricted rewrite system:
R is empty.
Q is empty.

(5) RisEmptyProof (EQUIVALENT transformation)

The TRS R is empty. Hence, termination is trivially proven.

(6) TRUE

(7) RisEmptyProof (EQUIVALENT transformation)

The TRS R is empty. Hence, termination is trivially proven.

(8) TRUE

(9) RisEmptyProof (EQUIVALENT transformation)

The TRS R is empty. Hence, termination is trivially proven.

(10) TRUE