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

Strict DPs
activate#(X) -> c_1(X)
activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
activate#(n__from(X)) -> c_3(from#(X))
first#(X1,X2) -> c_4(X1,X2)
first#(0(),Z) -> c_5()
first#(s(X),cons(Y,Z)) -> c_6(Y,X,activate#(Z))
from#(X) -> c_7(X,X)
from#(X) -> c_8(X)
sel#(0(),cons(X,Z)) -> c_9(X)
sel#(s(X),cons(Y,Z)) -> c_10(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__first(X1,X2)) -> c_2(first#(X1,X2))
activate#(n__from(X)) -> c_3(from#(X))
first#(X1,X2) -> c_4(X1,X2)
first#(0(),Z) -> c_5()
first#(s(X),cons(Y,Z)) -> c_6(Y,X,activate#(Z))
from#(X) -> c_7(X,X)
from#(X) -> c_8(X)
sel#(0(),cons(X,Z)) -> c_9(X)
sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)))
- 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(),Z) -> 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)
sel(0(),cons(X,Z)) -> X
sel(s(X),cons(Y,Z)) -> sel(X,activate(Z))
- Signature:
{activate/1,first/2,from/1,sel/2,activate#/1,first#/2,from#/1,sel#/2} / {0/0,cons/2,n__first/2,n__from/1
,nil/0,s/1,c_1/1,c_2/1,c_3/1,c_4/2,c_5/0,c_6/3,c_7/2,c_8/1,c_9/1,c_10/1}
- Obligation:
runtime complexity wrt. defined symbols {activate#,first#,from#,sel#} and constructors {0,cons,n__first
,n__from,nil,s}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
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(),Z) -> 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)
activate#(X) -> c_1(X)
activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
activate#(n__from(X)) -> c_3(from#(X))
first#(X1,X2) -> c_4(X1,X2)
first#(0(),Z) -> c_5()
first#(s(X),cons(Y,Z)) -> c_6(Y,X,activate#(Z))
from#(X) -> c_7(X,X)
from#(X) -> c_8(X)
sel#(0(),cons(X,Z)) -> c_9(X)
sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)))
* Step 3: PredecessorEstimation WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
activate#(X) -> c_1(X)
activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
activate#(n__from(X)) -> c_3(from#(X))
first#(X1,X2) -> c_4(X1,X2)
first#(0(),Z) -> c_5()
first#(s(X),cons(Y,Z)) -> c_6(Y,X,activate#(Z))
from#(X) -> c_7(X,X)
from#(X) -> c_8(X)
sel#(0(),cons(X,Z)) -> c_9(X)
sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)))
- 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(),Z) -> 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,sel/2,activate#/1,first#/2,from#/1,sel#/2} / {0/0,cons/2,n__first/2,n__from/1
,nil/0,s/1,c_1/1,c_2/1,c_3/1,c_4/2,c_5/0,c_6/3,c_7/2,c_8/1,c_9/1,c_10/1}
- Obligation:
runtime complexity wrt. defined symbols {activate#,first#,from#,sel#} and constructors {0,cons,n__first
,n__from,nil,s}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{5}
by application of
Pre({5}) = {1,2,4,6,7,8,9}.
Here rules are labelled as follows:
1: activate#(X) -> c_1(X)
2: activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
3: activate#(n__from(X)) -> c_3(from#(X))
4: first#(X1,X2) -> c_4(X1,X2)
5: first#(0(),Z) -> c_5()
6: first#(s(X),cons(Y,Z)) -> c_6(Y,X,activate#(Z))
7: from#(X) -> c_7(X,X)
8: from#(X) -> c_8(X)
9: sel#(0(),cons(X,Z)) -> c_9(X)
10: sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)))
* Step 4: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
activate#(X) -> c_1(X)
activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
activate#(n__from(X)) -> c_3(from#(X))
first#(X1,X2) -> c_4(X1,X2)
first#(s(X),cons(Y,Z)) -> c_6(Y,X,activate#(Z))
from#(X) -> c_7(X,X)
from#(X) -> c_8(X)
sel#(0(),cons(X,Z)) -> c_9(X)
sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)))
- 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(),Z) -> 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 DPs:
first#(0(),Z) -> c_5()
- Signature:
{activate/1,first/2,from/1,sel/2,activate#/1,first#/2,from#/1,sel#/2} / {0/0,cons/2,n__first/2,n__from/1
,nil/0,s/1,c_1/1,c_2/1,c_3/1,c_4/2,c_5/0,c_6/3,c_7/2,c_8/1,c_9/1,c_10/1}
- Obligation:
runtime complexity wrt. defined symbols {activate#,first#,from#,sel#} and constructors {0,cons,n__first
,n__from,nil,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(activate) = {1},
uargs(cons) = {2},
uargs(first) = {2},
uargs(n__first) = {2},
uargs(sel#) = {2},
uargs(c_2) = {1},
uargs(c_3) = {1},
uargs(c_6) = {3},
uargs(c_10) = {1}

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) =  x1 + 
p(n__first) =  x1 +  x2 + 
p(n__from) =  x1 + 
p(nil) = 
p(s) =  x1 + 
p(sel) = 
p(activate#) = 
p(first#) = 
p(from#) = 
p(sel#) =  x2 + 
p(c_1) = 
p(c_2) =  x1 + 
p(c_3) =  x1 + 
p(c_4) = 
p(c_5) = 
p(c_6) =  x3 + 
p(c_7) = 
p(c_8) = 
p(c_9) = 
p(c_10) =  x1 + 

Following rules are strictly oriented:
sel#(0(),cons(X,Z)) =  Z + 
> 
= c_9(X)

sel#(s(X),cons(Y,Z)) =  Z + 
>  Z + 
= c_10(sel#(X,activate(Z)))

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

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

activate#(n__first(X1,X2)) =  
>= 
=  c_2(first#(X1,X2))

activate#(n__from(X)) =  
>= 
=  c_3(from#(X))

first#(X1,X2) =  
>= 
=  c_4(X1,X2)

first#(0(),Z) =  
>= 
=  c_5()

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

from#(X) =  
>= 
=  c_7(X,X)

from#(X) =  
>= 
=  c_8(X)

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

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

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

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

first(s(X),cons(Y,Z)) =   X +  Z + 
>=  X +  Z + 
=  cons(Y,n__first(X,activate(Z)))

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 5: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
activate#(X) -> c_1(X)
activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
activate#(n__from(X)) -> c_3(from#(X))
first#(X1,X2) -> c_4(X1,X2)
first#(s(X),cons(Y,Z)) -> c_6(Y,X,activate#(Z))
from#(X) -> c_7(X,X)
from#(X) -> c_8(X)
- Strict TRS:
activate(X) -> X
activate(n__from(X)) -> from(X)
first(X1,X2) -> n__first(X1,X2)
first(0(),Z) -> 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 DPs:
first#(0(),Z) -> c_5()
sel#(0(),cons(X,Z)) -> c_9(X)
sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)))
- Weak TRS:
activate(n__first(X1,X2)) -> first(X1,X2)
- Signature:
{activate/1,first/2,from/1,sel/2,activate#/1,first#/2,from#/1,sel#/2} / {0/0,cons/2,n__first/2,n__from/1
,nil/0,s/1,c_1/1,c_2/1,c_3/1,c_4/2,c_5/0,c_6/3,c_7/2,c_8/1,c_9/1,c_10/1}
- Obligation:
runtime complexity wrt. defined symbols {activate#,first#,from#,sel#} and constructors {0,cons,n__first
,n__from,nil,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(activate) = {1},
uargs(cons) = {2},
uargs(first) = {2},
uargs(n__first) = {2},
uargs(sel#) = {2},
uargs(c_2) = {1},
uargs(c_3) = {1},
uargs(c_6) = {3},
uargs(c_10) = {1}

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) =  x1 + 
p(n__first) =  x1 +  x2 + 
p(n__from) =  x1 + 
p(nil) = 
p(s) =  x1 + 
p(sel) = 
p(activate#) = 
p(first#) = 
p(from#) = 
p(sel#) =  x2 + 
p(c_1) = 
p(c_2) =  x1 + 
p(c_3) =  x1 + 
p(c_4) = 
p(c_5) = 
p(c_6) =  x3 + 
p(c_7) = 
p(c_8) = 
p(c_9) = 
p(c_10) =  x1 + 

Following rules are strictly oriented:
from#(X) = 
> 
= c_7(X,X)

from#(X) = 
> 
= c_8(X)

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

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

activate#(n__first(X1,X2)) =  
>= 
=  c_2(first#(X1,X2))

activate#(n__from(X)) =  
>= 
=  c_3(from#(X))

first#(X1,X2) =  
>= 
=  c_4(X1,X2)

first#(0(),Z) =  
>= 
=  c_5()

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

sel#(0(),cons(X,Z)) =   Z + 
>= 
=  c_9(X)

sel#(s(X),cons(Y,Z)) =   Z + 
>=  Z + 
=  c_10(sel#(X,activate(Z)))

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

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

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

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

first(s(X),cons(Y,Z)) =   X +  Z + 
>=  X +  Z + 
=  cons(Y,n__first(X,activate(Z)))

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 DPs:
activate#(X) -> c_1(X)
activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
activate#(n__from(X)) -> c_3(from#(X))
first#(X1,X2) -> c_4(X1,X2)
first#(s(X),cons(Y,Z)) -> c_6(Y,X,activate#(Z))
- Strict TRS:
activate(X) -> X
activate(n__from(X)) -> from(X)
first(X1,X2) -> n__first(X1,X2)
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 DPs:
first#(0(),Z) -> c_5()
from#(X) -> c_7(X,X)
from#(X) -> c_8(X)
sel#(0(),cons(X,Z)) -> c_9(X)
sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)))
- Weak TRS:
activate(n__first(X1,X2)) -> first(X1,X2)
first(0(),Z) -> nil()
- Signature:
{activate/1,first/2,from/1,sel/2,activate#/1,first#/2,from#/1,sel#/2} / {0/0,cons/2,n__first/2,n__from/1
,nil/0,s/1,c_1/1,c_2/1,c_3/1,c_4/2,c_5/0,c_6/3,c_7/2,c_8/1,c_9/1,c_10/1}
- Obligation:
runtime complexity wrt. defined symbols {activate#,first#,from#,sel#} and constructors {0,cons,n__first
,n__from,nil,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(activate) = {1},
uargs(cons) = {2},
uargs(first) = {2},
uargs(n__first) = {2},
uargs(sel#) = {2},
uargs(c_2) = {1},
uargs(c_3) = {1},
uargs(c_6) = {3},
uargs(c_10) = {1}

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 + 
p(sel) = 
p(activate#) =  x1 + 
p(first#) =  x1 +  x2 + 
p(from#) = 
p(sel#) =  x2 + 
p(c_1) =  x1 + 
p(c_2) =  x1 + 
p(c_3) =  x1 + 
p(c_4) =  x1 +  x2 + 
p(c_5) = 
p(c_6) =  x2 +  x3 + 
p(c_7) = 
p(c_8) = 
p(c_9) = 
p(c_10) =  x1 + 

Following rules are strictly oriented:
first#(s(X),cons(Y,Z)) =  X +  Z + 
>  X +  Z + 
= c_6(Y,X,activate#(Z))

first(s(X),cons(Y,Z)) =  X +  Z + 
>  X +  Z + 
= cons(Y,n__first(X,activate(Z)))

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

activate#(n__first(X1,X2)) =   X1 +  X2 + 
>=  X1 +  X2 + 
=  c_2(first#(X1,X2))

activate#(n__from(X)) =  
>= 
=  c_3(from#(X))

first#(X1,X2) =   X1 +  X2 + 
>=  X1 +  X2 + 
=  c_4(X1,X2)

first#(0(),Z) =   Z + 
>= 
=  c_5()

from#(X) =  
>= 
=  c_7(X,X)

from#(X) =  
>= 
=  c_8(X)

sel#(0(),cons(X,Z)) =   Z + 
>= 
=  c_9(X)

sel#(s(X),cons(Y,Z)) =   Z + 
>=  Z + 
=  c_10(sel#(X,activate(Z)))

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(),Z) =   Z + 
>= 
=  nil()

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 7: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
activate#(X) -> c_1(X)
activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
activate#(n__from(X)) -> c_3(from#(X))
first#(X1,X2) -> c_4(X1,X2)
- Strict TRS:
activate(X) -> X
activate(n__from(X)) -> from(X)
first(X1,X2) -> n__first(X1,X2)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
- Weak DPs:
first#(0(),Z) -> c_5()
first#(s(X),cons(Y,Z)) -> c_6(Y,X,activate#(Z))
from#(X) -> c_7(X,X)
from#(X) -> c_8(X)
sel#(0(),cons(X,Z)) -> c_9(X)
sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)))
- Weak TRS:
activate(n__first(X1,X2)) -> first(X1,X2)
first(0(),Z) -> nil()
first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
- Signature:
{activate/1,first/2,from/1,sel/2,activate#/1,first#/2,from#/1,sel#/2} / {0/0,cons/2,n__first/2,n__from/1
,nil/0,s/1,c_1/1,c_2/1,c_3/1,c_4/2,c_5/0,c_6/3,c_7/2,c_8/1,c_9/1,c_10/1}
- Obligation:
runtime complexity wrt. defined symbols {activate#,first#,from#,sel#} and constructors {0,cons,n__first
,n__from,nil,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(activate) = {1},
uargs(cons) = {2},
uargs(first) = {2},
uargs(n__first) = {2},
uargs(sel#) = {2},
uargs(c_2) = {1},
uargs(c_3) = {1},
uargs(c_6) = {3},
uargs(c_10) = {1}

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) =  x1 + 
p(n__first) =  x2 + 
p(n__from) =  x1 + 
p(nil) = 
p(s) =  x1 + 
p(sel) =  x1 + 
p(activate#) = 
p(first#) = 
p(from#) = 
p(sel#) =  x1 +  x2 + 
p(c_1) = 
p(c_2) =  x1 + 
p(c_3) =  x1 + 
p(c_4) = 
p(c_5) = 
p(c_6) =  x3 + 
p(c_7) = 
p(c_8) = 
p(c_9) = 
p(c_10) =  x1 + 

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

activate#(n__from(X)) = 
> 
= c_3(from#(X))

first#(X1,X2) = 
> 
= c_4(X1,X2)

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

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

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

Following rules are (at-least) weakly oriented:
activate#(n__first(X1,X2)) =  
>= 
=  c_2(first#(X1,X2))

first#(0(),Z) =  
>= 
=  c_5()

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

from#(X) =  
>= 
=  c_7(X,X)

from#(X) =  
>= 
=  c_8(X)

sel#(0(),cons(X,Z)) =   Z + 
>= 
=  c_9(X)

sel#(s(X),cons(Y,Z)) =   X +  Z + 
>=  X +  Z + 
=  c_10(sel#(X,activate(Z)))

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

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

first(s(X),cons(Y,Z)) =   Z + 
>=  Z + 
=  cons(Y,n__first(X,activate(Z)))

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 8: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
- 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_3(from#(X))
first#(X1,X2) -> c_4(X1,X2)
first#(0(),Z) -> c_5()
first#(s(X),cons(Y,Z)) -> c_6(Y,X,activate#(Z))
from#(X) -> c_7(X,X)
from#(X) -> c_8(X)
sel#(0(),cons(X,Z)) -> c_9(X)
sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)))
- 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(),Z) -> nil()
first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
- Signature:
{activate/1,first/2,from/1,sel/2,activate#/1,first#/2,from#/1,sel#/2} / {0/0,cons/2,n__first/2,n__from/1
,nil/0,s/1,c_1/1,c_2/1,c_3/1,c_4/2,c_5/0,c_6/3,c_7/2,c_8/1,c_9/1,c_10/1}
- Obligation:
runtime complexity wrt. defined symbols {activate#,first#,from#,sel#} and constructors {0,cons,n__first
,n__from,nil,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(activate) = {1},
uargs(cons) = {2},
uargs(first) = {2},
uargs(n__first) = {2},
uargs(sel#) = {2},
uargs(c_2) = {1},
uargs(c_3) = {1},
uargs(c_6) = {3},
uargs(c_10) = {1}

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 + 
p(sel) = 
p(activate#) =  x1 + 
p(first#) =  x2 + 
p(from#) = 
p(sel#) =  x2 + 
p(c_1) =  x1 + 
p(c_2) =  x1 + 
p(c_3) =  x1 + 
p(c_4) =  x2 + 
p(c_5) = 
p(c_6) =  x3 + 
p(c_7) = 
p(c_8) = 
p(c_9) = 
p(c_10) =  x1 + 

Following rules are strictly oriented:
activate#(n__first(X1,X2)) =  X2 + 
>  X2 + 
= c_2(first#(X1,X2))

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

activate#(n__from(X)) =  
>= 
=  c_3(from#(X))

first#(X1,X2) =   X2 + 
>=  X2 + 
=  c_4(X1,X2)

first#(0(),Z) =   Z + 
>= 
=  c_5()

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

from#(X) =  
>= 
=  c_7(X,X)

from#(X) =  
>= 
=  c_8(X)

sel#(0(),cons(X,Z)) =   Z + 
>= 
=  c_9(X)

sel#(s(X),cons(Y,Z)) =   Z + 
>=  Z + 
=  c_10(sel#(X,activate(Z)))

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

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

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

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

first(0(),Z) =   Z + 
>= 
=  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 9: 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__first(X1,X2)) -> c_2(first#(X1,X2))
activate#(n__from(X)) -> c_3(from#(X))
first#(X1,X2) -> c_4(X1,X2)
first#(0(),Z) -> c_5()
first#(s(X),cons(Y,Z)) -> c_6(Y,X,activate#(Z))
from#(X) -> c_7(X,X)
from#(X) -> c_8(X)
sel#(0(),cons(X,Z)) -> c_9(X)
sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)))
- 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(),Z) -> nil()
first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
- Signature:
{activate/1,first/2,from/1,sel/2,activate#/1,first#/2,from#/1,sel#/2} / {0/0,cons/2,n__first/2,n__from/1
,nil/0,s/1,c_1/1,c_2/1,c_3/1,c_4/2,c_5/0,c_6/3,c_7/2,c_8/1,c_9/1,c_10/1}
- Obligation:
runtime complexity wrt. defined symbols {activate#,first#,from#,sel#} and constructors {0,cons,n__first
,n__from,nil,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(activate) = {1},
uargs(cons) = {2},
uargs(first) = {2},
uargs(n__first) = {2},
uargs(sel#) = {2},
uargs(c_2) = {1},
uargs(c_3) = {1},
uargs(c_6) = {3},
uargs(c_10) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = 
p(activate) =  x1 + 
p(cons) =  x1 +  x2 + 
p(first) =  x2 + 
p(from) =  x1 + 
p(n__first) =  x2 + 
p(n__from) =  x1 + 
p(nil) = 
p(s) = 
p(sel) =  x2 + 
p(activate#) = 
p(first#) = 
p(from#) = 
p(sel#) =  x2 + 
p(c_1) = 
p(c_2) =  x1 + 
p(c_3) =  x1 + 
p(c_4) = 
p(c_5) = 
p(c_6) =  x3 + 
p(c_7) = 
p(c_8) = 
p(c_9) = 
p(c_10) =  x1 + 

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

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

activate#(n__first(X1,X2)) =  
>= 
=  c_2(first#(X1,X2))

activate#(n__from(X)) =  
>= 
=  c_3(from#(X))

first#(X1,X2) =  
>= 
=  c_4(X1,X2)

first#(0(),Z) =  
>= 
=  c_5()

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

from#(X) =  
>= 
=  c_7(X,X)

from#(X) =  
>= 
=  c_8(X)

sel#(0(),cons(X,Z)) =   X +  Z + 
>= 
=  c_9(X)

sel#(s(X),cons(Y,Z)) =   Y +  Z + 
>=  Z + 
=  c_10(sel#(X,activate(Z)))

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

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

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

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

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

first(s(X),cons(Y,Z)) =   Y +  Z + 
>=  Y +  Z + 
=  cons(Y,n__first(X,activate(Z)))

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

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 10: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
from(X) -> cons(X,n__from(s(X)))
- Weak DPs:
activate#(X) -> c_1(X)
activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
activate#(n__from(X)) -> c_3(from#(X))
first#(X1,X2) -> c_4(X1,X2)
first#(0(),Z) -> c_5()
first#(s(X),cons(Y,Z)) -> c_6(Y,X,activate#(Z))
from#(X) -> c_7(X,X)
from#(X) -> c_8(X)
sel#(0(),cons(X,Z)) -> c_9(X)
sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)))
- 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(),Z) -> nil()
first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
from(X) -> n__from(X)
- Signature:
{activate/1,first/2,from/1,sel/2,activate#/1,first#/2,from#/1,sel#/2} / {0/0,cons/2,n__first/2,n__from/1
,nil/0,s/1,c_1/1,c_2/1,c_3/1,c_4/2,c_5/0,c_6/3,c_7/2,c_8/1,c_9/1,c_10/1}
- Obligation:
runtime complexity wrt. defined symbols {activate#,first#,from#,sel#} and constructors {0,cons,n__first
,n__from,nil,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(activate) = {1},
uargs(cons) = {2},
uargs(first) = {2},
uargs(n__first) = {2},
uargs(sel#) = {2},
uargs(c_2) = {1},
uargs(c_3) = {1},
uargs(c_6) = {3},
uargs(c_10) = {1}

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 + 
p(sel) = 
p(activate#) = 
p(first#) = 
p(from#) = 
p(sel#) =  x1 +  x2 + 
p(c_1) = 
p(c_2) =  x1 + 
p(c_3) =  x1 + 
p(c_4) = 
p(c_5) = 
p(c_6) =  x3 + 
p(c_7) = 
p(c_8) = 
p(c_9) = 
p(c_10) =  x1 + 

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

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

activate#(n__first(X1,X2)) =  
>= 
=  c_2(first#(X1,X2))

activate#(n__from(X)) =  
>= 
=  c_3(from#(X))

first#(X1,X2) =  
>= 
=  c_4(X1,X2)

first#(0(),Z) =  
>= 
=  c_5()

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

from#(X) =  
>= 
=  c_7(X,X)

from#(X) =  
>= 
=  c_8(X)

sel#(0(),cons(X,Z)) =   Z + 
>= 
=  c_9(X)

sel#(s(X),cons(Y,Z)) =   X +  Z + 
>=  X +  Z + 
=  c_10(sel#(X,activate(Z)))

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(),Z) =   Z + 
>= 
=  nil()

first(s(X),cons(Y,Z)) =   X +  Z + 
>=  X +  Z + 
=  cons(Y,n__first(X,activate(Z)))

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

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 11: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
activate#(X) -> c_1(X)
activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
activate#(n__from(X)) -> c_3(from#(X))
first#(X1,X2) -> c_4(X1,X2)
first#(0(),Z) -> c_5()
first#(s(X),cons(Y,Z)) -> c_6(Y,X,activate#(Z))
from#(X) -> c_7(X,X)
from#(X) -> c_8(X)
sel#(0(),cons(X,Z)) -> c_9(X)
sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)))
- 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(),Z) -> 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,sel/2,activate#/1,first#/2,from#/1,sel#/2} / {0/0,cons/2,n__first/2,n__from/1
,nil/0,s/1,c_1/1,c_2/1,c_3/1,c_4/2,c_5/0,c_6/3,c_7/2,c_8/1,c_9/1,c_10/1}
- Obligation:
runtime complexity wrt. defined symbols {activate#,first#,from#,sel#} 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))
```