```* Step 1: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
activate(X) -> X
activate(n__from(X)) -> from(X)
after(0(),XS) -> XS
after(s(N),cons(X,XS)) -> after(N,activate(XS))
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
- Signature:
{activate/1,after/2,from/1} / {0/0,cons/2,n__from/1,s/1}
- Obligation:
runtime complexity wrt. defined symbols {activate,after,from} and constructors {0,cons,n__from,s}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(after) = {2}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = 
p(activate) =  x1 + 
p(after) =  x2 + 
p(cons) =  x2 + 
p(from) =  x1 + 
p(n__from) =  x1 + 
p(s) =  x1 + 

Following rules are strictly oriented:
after(0(),XS) =  XS + 
>  XS + 
= XS

from(X) =  X + 
>  X + 
= cons(X,n__from(s(X)))

from(X) =  X + 
>  X + 
= n__from(X)

Following rules are (at-least) weakly oriented:
activate(X) =   X + 
>=  X + 
=  X

activate(n__from(X)) =   X + 
>=  X + 
=  from(X)

after(s(N),cons(X,XS)) =   XS + 
>=  XS + 
=  after(N,activate(XS))

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: MI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
activate(X) -> X
activate(n__from(X)) -> from(X)
after(s(N),cons(X,XS)) -> after(N,activate(XS))
- Weak TRS:
after(0(),XS) -> XS
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
- Signature:
{activate/1,after/2,from/1} / {0/0,cons/2,n__from/1,s/1}
- Obligation:
runtime complexity wrt. defined symbols {activate,after,from} and constructors {0,cons,n__from,s}
+ Applied Processor:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 1, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity Nothing)):

The following argument positions are considered usable:
uargs(after) = {2}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = 
p(activate) =  x_1 + 
p(after) =  x_1 +  x_2 + 
p(cons) =  x_2 + 
p(from) = 
p(n__from) = 
p(s) =  x_1 + 

Following rules are strictly oriented:
activate(X) =  X + 
>  X + 
= X

activate(n__from(X)) = 
> 
= from(X)

after(s(N),cons(X,XS)) =  N +  XS + 
>  N +  XS + 
= after(N,activate(XS))

Following rules are (at-least) weakly oriented:
after(0(),XS) =   XS + 
>=  XS + 
=  XS

from(X) =  
>= 
=  cons(X,n__from(s(X)))

from(X) =  
>= 
=  n__from(X)

* Step 3: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
activate(X) -> X
activate(n__from(X)) -> from(X)
after(0(),XS) -> XS
after(s(N),cons(X,XS)) -> after(N,activate(XS))
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
- Signature:
{activate/1,after/2,from/1} / {0/0,cons/2,n__from/1,s/1}
- Obligation:
runtime complexity wrt. defined symbols {activate,after,from} and constructors {0,cons,n__from,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))
```