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)))
after(0, XS) → XS
after(s(N), cons(X, XS)) → after(N, activate(XS))
from(X) → n__from(X)
activate(n__from(X)) → from(X)
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)
after(x1, x2)  =  after(x1, x2)
0  =  0
activate(x1)  =  activate(x1)

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

after₂ > [from₁, activate₁] > s₁ > cons₂
0 > cons₂

Status:

from₁: [1]
cons₂: multiset
after₂: [1,2]
s₁: multiset
activate₁: [1]
0: 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)))
after(0, XS) → XS
after(s(N), cons(X, XS)) → after(N, activate(XS))
from(X) → n__from(X)
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)) = 2·x₁   
POL(from(x₁)) = 2 + x₁   
POL(n__from(x₁)) = 2 + 2·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