```* Step 1: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
activate(X) -> X
activate(n__first(X1,X2)) -> first(X1,X2)
activate(n__from(X)) -> from(X)
first(X1,X2) -> n__first(X1,X2)
first(0(),X) -> nil()
first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
- Signature:
{activate/1,first/2,from/1} / {0/0,cons/2,n__first/2,n__from/1,nil/0,s/1}
- Obligation:
runtime complexity wrt. defined symbols {activate,first,from} and constructors {0,cons,n__first,n__from,nil
,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(cons) = {2},
uargs(n__first) = {2}

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

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

activate(n__first(X1,X2)) =  X2 + 
>  X2 + 
= first(X1,X2)

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

Following rules are (at-least) weakly oriented:
first(X1,X2) =   X2 + 
>=  X2 + 
=  n__first(X1,X2)

first(0(),X) =   X + 
>= 
=  nil()

first(s(X),cons(Y,Z)) =   Z + 
>=  Z + 
=  cons(Y,n__first(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 2: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
first(X1,X2) -> n__first(X1,X2)
first(0(),X) -> nil()
first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
- Weak TRS:
activate(X) -> X
activate(n__first(X1,X2)) -> first(X1,X2)
activate(n__from(X)) -> from(X)
- Signature:
{activate/1,first/2,from/1} / {0/0,cons/2,n__first/2,n__from/1,nil/0,s/1}
- Obligation:
runtime complexity wrt. defined symbols {activate,first,from} and constructors {0,cons,n__first,n__from,nil
,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(cons) = {2},
uargs(n__first) = {2}

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

Following rules are strictly oriented:
first(s(X),cons(Y,Z)) =  X +  Z + 
>  X +  Z + 
= cons(Y,n__first(X,activate(Z)))

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

from(X) = 
> 
= n__from(X)

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

activate(n__first(X1,X2)) =   X1 +  X2 + 
>=  X1 +  X2 + 
=  first(X1,X2)

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

first(X1,X2) =   X1 +  X2 + 
>=  X1 +  X2 + 
=  n__first(X1,X2)

first(0(),X) =   X + 
>= 
=  nil()

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
first(X1,X2) -> n__first(X1,X2)
first(0(),X) -> nil()
- Weak TRS:
activate(X) -> X
activate(n__first(X1,X2)) -> first(X1,X2)
activate(n__from(X)) -> from(X)
first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
- Signature:
{activate/1,first/2,from/1} / {0/0,cons/2,n__first/2,n__from/1,nil/0,s/1}
- Obligation:
runtime complexity wrt. defined symbols {activate,first,from} and constructors {0,cons,n__first,n__from,nil
,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(cons) = {2},
uargs(n__first) = {2}

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

Following rules are strictly oriented:
first(X1,X2) =  X2 + 
>  X2 + 
= n__first(X1,X2)

first(0(),X) =  X + 
> 
= nil()

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

activate(n__first(X1,X2)) =   X2 + 
>=  X2 + 
=  first(X1,X2)

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

first(s(X),cons(Y,Z)) =   Z + 
>=  Z + 
=  cons(Y,n__first(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 4: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
activate(X) -> X
activate(n__first(X1,X2)) -> first(X1,X2)
activate(n__from(X)) -> from(X)
first(X1,X2) -> n__first(X1,X2)
first(0(),X) -> nil()
first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
- Signature:
{activate/1,first/2,from/1} / {0/0,cons/2,n__first/2,n__from/1,nil/0,s/1}
- Obligation:
runtime complexity wrt. defined symbols {activate,first,from} and constructors {0,cons,n__first,n__from,nil
,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

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