```* 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) = [0]
p(activate) = [1] x1 + [0]
p(after) = [1] x2 + [11]
p(cons) = [1] x2 + [0]
p(from) = [1] x1 + [1]
p(n__from) = [1] x1 + [0]
p(s) = [1] x1 + [0]

Following rules are strictly oriented:
after(0(),XS) = [1] XS + [11]
> [1] XS + [0]
= XS

from(X) = [1] X + [1]
> [1] X + [0]
= cons(X,n__from(s(X)))

from(X) = [1] X + [1]
> [1] X + [0]
= n__from(X)

Following rules are (at-least) weakly oriented:
activate(X) =  [1] X + [0]
>= [1] X + [0]
=  X

activate(n__from(X)) =  [1] X + [0]
>= [1] X + [1]
=  from(X)

after(s(N),cons(X,XS)) =  [1] XS + [11]
>= [1] XS + [11]
=  after(N,activate(XS))

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: WeightGap 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:
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) = [0]
p(activate) = [1] x1 + [13]
p(after) = [1] x2 + [0]
p(cons) = [1] x2 + [0]
p(from) = [1] x1 + [0]
p(n__from) = [1] x1 + [0]
p(s) = [1] x1 + [0]

Following rules are strictly oriented:
activate(X) = [1] X + [13]
> [1] X + [0]
= X

activate(n__from(X)) = [1] X + [13]
> [1] X + [0]
= from(X)

Following rules are (at-least) weakly oriented:
after(0(),XS) =  [1] XS + [0]
>= [1] XS + [0]
=  XS

after(s(N),cons(X,XS)) =  [1] XS + [0]
>= [1] XS + [13]
=  after(N,activate(XS))

from(X) =  [1] X + [0]
>= [1] X + [0]
=  cons(X,n__from(s(X)))

from(X) =  [1] X + [0]
>= [1] X + [0]
=  n__from(X)

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: MI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
after(s(N),cons(X,XS)) -> after(N,activate(XS))
- Weak TRS:
activate(X) -> X
activate(n__from(X)) -> from(X)
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) = [0]
p(activate) = [1] x_1 + [4]
p(after) = [2] x_1 + [4] x_2 + [2]
p(cons) = [1] x_2 + [4]
p(from) = [4]
p(n__from) = [0]
p(s) = [1] x_1 + [4]

Following rules are strictly oriented:
after(s(N),cons(X,XS)) = [2] N + [4] XS + [26]
> [2] N + [4] XS + [18]
= after(N,activate(XS))

Following rules are (at-least) weakly oriented:
activate(X) =  [1] X + [4]
>= [1] X + [0]
=  X

activate(n__from(X)) =  [4]
>= [4]
=  from(X)

after(0(),XS) =  [4] XS + [2]
>= [1] XS + [0]
=  XS

from(X) =  [4]
>= [4]
=  cons(X,n__from(s(X)))

from(X) =  [4]
>= [0]
=  n__from(X)

* Step 4: 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))
```