```* Step 1: DependencyPairs WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
activate(X) -> X
activate(n__from(X)) -> from(X)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
sel(0(),cons(X,Y)) -> X
sel(s(X),cons(Y,Z)) -> sel(X,activate(Z))
- Signature:
{activate/1,from/1,sel/2} / {0/0,cons/2,n__from/1,s/1}
- Obligation:
runtime complexity wrt. defined symbols {activate,from,sel} and constructors {0,cons,n__from,s}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
We add the following weak dependency pairs:

Strict DPs
activate#(X) -> c_1(X)
activate#(n__from(X)) -> c_2(from#(X))
from#(X) -> c_3(X,X)
from#(X) -> c_4(X)
sel#(0(),cons(X,Y)) -> c_5(X)
sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z)))
Weak DPs

and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
activate#(X) -> c_1(X)
activate#(n__from(X)) -> c_2(from#(X))
from#(X) -> c_3(X,X)
from#(X) -> c_4(X)
sel#(0(),cons(X,Y)) -> c_5(X)
sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z)))
- Strict TRS:
activate(X) -> X
activate(n__from(X)) -> from(X)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
sel(0(),cons(X,Y)) -> X
sel(s(X),cons(Y,Z)) -> sel(X,activate(Z))
- Signature:
{activate/1,from/1,sel/2,activate#/1,from#/1,sel#/2} / {0/0,cons/2,n__from/1,s/1,c_1/1,c_2/1,c_3/2,c_4/1
,c_5/1,c_6/1}
- Obligation:
runtime complexity wrt. defined symbols {activate#,from#,sel#} and constructors {0,cons,n__from,s}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
activate(X) -> X
activate(n__from(X)) -> from(X)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
activate#(X) -> c_1(X)
activate#(n__from(X)) -> c_2(from#(X))
from#(X) -> c_3(X,X)
from#(X) -> c_4(X)
sel#(0(),cons(X,Y)) -> c_5(X)
sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z)))
* Step 3: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
activate#(X) -> c_1(X)
activate#(n__from(X)) -> c_2(from#(X))
from#(X) -> c_3(X,X)
from#(X) -> c_4(X)
sel#(0(),cons(X,Y)) -> c_5(X)
sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z)))
- Strict TRS:
activate(X) -> X
activate(n__from(X)) -> from(X)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
- Signature:
{activate/1,from/1,sel/2,activate#/1,from#/1,sel#/2} / {0/0,cons/2,n__from/1,s/1,c_1/1,c_2/1,c_3/2,c_4/1
,c_5/1,c_6/1}
- Obligation:
runtime complexity wrt. defined symbols {activate#,from#,sel#} and constructors {0,cons,n__from,s}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 0, 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(sel#) = {2},
uargs(c_2) = {1},
uargs(c_6) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = 
p(activate) =  x1 + 
p(cons) =  x2 + 
p(from) =  x1 + 
p(n__from) =  x1 + 
p(s) =  x1 + 
p(sel) = 
p(activate#) =  x1 + 
p(from#) =  x1 + 
p(sel#) =  x2 + 
p(c_1) =  x1 + 
p(c_2) =  x1 + 
p(c_3) =  x1 + 
p(c_4) =  x1 + 
p(c_5) = 
p(c_6) =  x1 + 

Following rules are strictly oriented:
sel#(0(),cons(X,Y)) =  Y + 
> 
= c_5(X)

sel#(s(X),cons(Y,Z)) =  Z + 
>  Z + 
= c_6(sel#(X,activate(Z)))

Following rules are (at-least) weakly oriented:
activate#(X) =   X + 
>=  X + 
=  c_1(X)

activate#(n__from(X)) =   X + 
>=  X + 
=  c_2(from#(X))

from#(X) =   X + 
>=  X + 
=  c_3(X,X)

from#(X) =   X + 
>=  X + 
=  c_4(X)

activate(X) =   X + 
>=  X + 
=  X

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

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

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

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 4: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
activate#(X) -> c_1(X)
activate#(n__from(X)) -> c_2(from#(X))
from#(X) -> c_3(X,X)
from#(X) -> c_4(X)
- Strict TRS:
activate(X) -> X
activate(n__from(X)) -> from(X)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
- Weak DPs:
sel#(0(),cons(X,Y)) -> c_5(X)
sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z)))
- Signature:
{activate/1,from/1,sel/2,activate#/1,from#/1,sel#/2} / {0/0,cons/2,n__from/1,s/1,c_1/1,c_2/1,c_3/2,c_4/1
,c_5/1,c_6/1}
- Obligation:
runtime complexity wrt. defined symbols {activate#,from#,sel#} and constructors {0,cons,n__from,s}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 0, 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(sel#) = {2},
uargs(c_2) = {1},
uargs(c_6) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = 
p(activate) =  x1 + 
p(cons) =  x2 + 
p(from) = 
p(n__from) = 
p(s) =  x1 + 
p(sel) = 
p(activate#) =  x1 + 
p(from#) = 
p(sel#) =  x1 +  x2 + 
p(c_1) = 
p(c_2) =  x1 + 
p(c_3) = 
p(c_4) = 
p(c_5) = 
p(c_6) =  x1 + 

Following rules are strictly oriented:
activate#(n__from(X)) = 
> 
= c_2(from#(X))

activate(X) =  X + 
>  X + 
= X

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

Following rules are (at-least) weakly oriented:
activate#(X) =   X + 
>= 
=  c_1(X)

from#(X) =  
>= 
=  c_3(X,X)

from#(X) =  
>= 
=  c_4(X)

sel#(0(),cons(X,Y)) =   Y + 
>= 
=  c_5(X)

sel#(s(X),cons(Y,Z)) =   X +  Z + 
>=  X +  Z + 
=  c_6(sel#(X,activate(Z)))

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

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

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 5: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
activate#(X) -> c_1(X)
from#(X) -> c_3(X,X)
from#(X) -> c_4(X)
- Strict TRS:
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
- Weak DPs:
activate#(n__from(X)) -> c_2(from#(X))
sel#(0(),cons(X,Y)) -> c_5(X)
sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z)))
- Weak TRS:
activate(X) -> X
activate(n__from(X)) -> from(X)
- Signature:
{activate/1,from/1,sel/2,activate#/1,from#/1,sel#/2} / {0/0,cons/2,n__from/1,s/1,c_1/1,c_2/1,c_3/2,c_4/1
,c_5/1,c_6/1}
- Obligation:
runtime complexity wrt. defined symbols {activate#,from#,sel#} and constructors {0,cons,n__from,s}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 0, 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(sel#) = {2},
uargs(c_2) = {1},
uargs(c_6) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = 
p(activate) =  x1 + 
p(cons) =  x2 + 
p(from) =  x1 + 
p(n__from) =  x1 + 
p(s) =  x1 + 
p(sel) = 
p(activate#) = 
p(from#) = 
p(sel#) =  x2 + 
p(c_1) = 
p(c_2) =  x1 + 
p(c_3) = 
p(c_4) = 
p(c_5) = 
p(c_6) =  x1 + 

Following rules are strictly oriented:
activate#(X) = 
> 
= c_1(X)

from#(X) = 
> 
= c_3(X,X)

from#(X) = 
> 
= c_4(X)

Following rules are (at-least) weakly oriented:
activate#(n__from(X)) =  
>= 
=  c_2(from#(X))

sel#(0(),cons(X,Y)) =   Y + 
>= 
=  c_5(X)

sel#(s(X),cons(Y,Z)) =   Z + 
>=  Z + 
=  c_6(sel#(X,activate(Z)))

activate(X) =   X + 
>=  X + 
=  X

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

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

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

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 6: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
- Weak DPs:
activate#(X) -> c_1(X)
activate#(n__from(X)) -> c_2(from#(X))
from#(X) -> c_3(X,X)
from#(X) -> c_4(X)
sel#(0(),cons(X,Y)) -> c_5(X)
sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z)))
- Weak TRS:
activate(X) -> X
activate(n__from(X)) -> from(X)
- Signature:
{activate/1,from/1,sel/2,activate#/1,from#/1,sel#/2} / {0/0,cons/2,n__from/1,s/1,c_1/1,c_2/1,c_3/2,c_4/1
,c_5/1,c_6/1}
- Obligation:
runtime complexity wrt. defined symbols {activate#,from#,sel#} and constructors {0,cons,n__from,s}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 0, 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(sel#) = {2},
uargs(c_2) = {1},
uargs(c_6) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = 
p(activate) =  x1 + 
p(cons) =  x2 + 
p(from) = 
p(n__from) = 
p(s) =  x1 + 
p(sel) =  x1 +  x2 + 
p(activate#) = 
p(from#) = 
p(sel#) =  x1 +  x2 + 
p(c_1) = 
p(c_2) =  x1 + 
p(c_3) = 
p(c_4) = 
p(c_5) = 
p(c_6) =  x1 + 

Following rules are strictly oriented:
from(X) = 
> 
= cons(X,n__from(s(X)))

from(X) = 
> 
= n__from(X)

Following rules are (at-least) weakly oriented:
activate#(X) =  
>= 
=  c_1(X)

activate#(n__from(X)) =  
>= 
=  c_2(from#(X))

from#(X) =  
>= 
=  c_3(X,X)

from#(X) =  
>= 
=  c_4(X)

sel#(0(),cons(X,Y)) =   Y + 
>= 
=  c_5(X)

sel#(s(X),cons(Y,Z)) =   X +  Z + 
>=  X +  Z + 
=  c_6(sel#(X,activate(Z)))

activate(X) =   X + 
>=  X + 
=  X

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

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 7: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
activate#(X) -> c_1(X)
activate#(n__from(X)) -> c_2(from#(X))
from#(X) -> c_3(X,X)
from#(X) -> c_4(X)
sel#(0(),cons(X,Y)) -> c_5(X)
sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z)))
- Weak TRS:
activate(X) -> X
activate(n__from(X)) -> from(X)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
- Signature:
{activate/1,from/1,sel/2,activate#/1,from#/1,sel#/2} / {0/0,cons/2,n__from/1,s/1,c_1/1,c_2/1,c_3/2,c_4/1
,c_5/1,c_6/1}
- Obligation:
runtime complexity wrt. defined symbols {activate#,from#,sel#} 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))
```