```* Step 1: DependencyPairs WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
activate(X) -> X
activate(n__from(X)) -> from(X)
activate(n__take(X1,X2)) -> take(X1,X2)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
sel(0(),cons(X,XS)) -> X
sel(s(N),cons(X,XS)) -> sel(N,activate(XS))
take(X1,X2) -> n__take(X1,X2)
take(0(),XS) -> nil()
take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS)))
- Signature:
- Obligation:
runtime complexity wrt. defined symbols {2nd,activate,from,head,sel,take} and constructors {0,cons,n__from
,n__take,nil,s}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
We add the following weak dependency pairs:

Strict DPs
activate#(X) -> c_2(X)
activate#(n__from(X)) -> c_3(from#(X))
activate#(n__take(X1,X2)) -> c_4(take#(X1,X2))
from#(X) -> c_5(X,X)
from#(X) -> c_6(X)
sel#(0(),cons(X,XS)) -> c_8(X)
sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)))
take#(X1,X2) -> c_10(X1,X2)
take#(0(),XS) -> c_11()
take#(s(N),cons(X,XS)) -> c_12(X,N,activate#(XS))
Weak DPs

and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
activate#(X) -> c_2(X)
activate#(n__from(X)) -> c_3(from#(X))
activate#(n__take(X1,X2)) -> c_4(take#(X1,X2))
from#(X) -> c_5(X,X)
from#(X) -> c_6(X)
sel#(0(),cons(X,XS)) -> c_8(X)
sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)))
take#(X1,X2) -> c_10(X1,X2)
take#(0(),XS) -> c_11()
take#(s(N),cons(X,XS)) -> c_12(X,N,activate#(XS))
- Strict TRS:
activate(X) -> X
activate(n__from(X)) -> from(X)
activate(n__take(X1,X2)) -> take(X1,X2)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
sel(0(),cons(X,XS)) -> X
sel(s(N),cons(X,XS)) -> sel(N,activate(XS))
take(X1,X2) -> n__take(X1,X2)
take(0(),XS) -> nil()
take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS)))
- Signature:
,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1,c_7/1,c_8/1,c_9/1,c_10/2,c_11/0
,c_12/3}
- Obligation:
runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel#,take#} and constructors {0,cons
,n__from,n__take,nil,s}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
activate(X) -> X
activate(n__from(X)) -> from(X)
activate(n__take(X1,X2)) -> take(X1,X2)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
take(X1,X2) -> n__take(X1,X2)
take(0(),XS) -> nil()
take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS)))
activate#(X) -> c_2(X)
activate#(n__from(X)) -> c_3(from#(X))
activate#(n__take(X1,X2)) -> c_4(take#(X1,X2))
from#(X) -> c_5(X,X)
from#(X) -> c_6(X)
sel#(0(),cons(X,XS)) -> c_8(X)
sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)))
take#(X1,X2) -> c_10(X1,X2)
take#(0(),XS) -> c_11()
take#(s(N),cons(X,XS)) -> c_12(X,N,activate#(XS))
* Step 3: PredecessorEstimation WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
activate#(X) -> c_2(X)
activate#(n__from(X)) -> c_3(from#(X))
activate#(n__take(X1,X2)) -> c_4(take#(X1,X2))
from#(X) -> c_5(X,X)
from#(X) -> c_6(X)
sel#(0(),cons(X,XS)) -> c_8(X)
sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)))
take#(X1,X2) -> c_10(X1,X2)
take#(0(),XS) -> c_11()
take#(s(N),cons(X,XS)) -> c_12(X,N,activate#(XS))
- Strict TRS:
activate(X) -> X
activate(n__from(X)) -> from(X)
activate(n__take(X1,X2)) -> take(X1,X2)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
take(X1,X2) -> n__take(X1,X2)
take(0(),XS) -> nil()
take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS)))
- Signature:
,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1,c_7/1,c_8/1,c_9/1,c_10/2,c_11/0
,c_12/3}
- Obligation:
runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel#,take#} and constructors {0,cons
,n__from,n__take,nil,s}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{11}
by application of
Pre({11}) = {2,4,5,6,7,8,10,12}.
Here rules are labelled as follows:
2: activate#(X) -> c_2(X)
3: activate#(n__from(X)) -> c_3(from#(X))
4: activate#(n__take(X1,X2)) -> c_4(take#(X1,X2))
5: from#(X) -> c_5(X,X)
6: from#(X) -> c_6(X)
8: sel#(0(),cons(X,XS)) -> c_8(X)
9: sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)))
10: take#(X1,X2) -> c_10(X1,X2)
11: take#(0(),XS) -> c_11()
12: take#(s(N),cons(X,XS)) -> c_12(X,N,activate#(XS))
* Step 4: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
activate#(X) -> c_2(X)
activate#(n__from(X)) -> c_3(from#(X))
activate#(n__take(X1,X2)) -> c_4(take#(X1,X2))
from#(X) -> c_5(X,X)
from#(X) -> c_6(X)
sel#(0(),cons(X,XS)) -> c_8(X)
sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)))
take#(X1,X2) -> c_10(X1,X2)
take#(s(N),cons(X,XS)) -> c_12(X,N,activate#(XS))
- Strict TRS:
activate(X) -> X
activate(n__from(X)) -> from(X)
activate(n__take(X1,X2)) -> take(X1,X2)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
take(X1,X2) -> n__take(X1,X2)
take(0(),XS) -> nil()
take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS)))
- Weak DPs:
take#(0(),XS) -> c_11()
- Signature:
,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1,c_7/1,c_8/1,c_9/1,c_10/2,c_11/0
,c_12/3}
- Obligation:
runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel#,take#} and constructors {0,cons
,n__from,n__take,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(n__take) = {2},
uargs(take) = {2},
uargs(sel#) = {2},
uargs(c_1) = {1},
uargs(c_3) = {1},
uargs(c_4) = {1},
uargs(c_9) = {1},
uargs(c_12) = {3}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = 
p(2nd) = 
p(activate) =  x1 + 
p(cons) =  x2 + 
p(from) = 
p(n__from) = 
p(n__take) =  x2 + 
p(nil) = 
p(s) =  x1 + 
p(sel) = 
p(take) =  x2 + 
p(2nd#) =  x1 + 
p(activate#) =  x1 + 
p(from#) = 
p(head#) =  x1 + 
p(sel#) =  x2 + 
p(take#) =  x2 + 
p(c_1) =  x1 + 
p(c_2) =  x1 + 
p(c_3) =  x1 + 
p(c_4) =  x1 + 
p(c_5) = 
p(c_6) = 
p(c_7) = 
p(c_8) = 
p(c_9) =  x1 + 
p(c_10) =  x2 + 
p(c_11) = 
p(c_12) =  x3 + 

Following rules are strictly oriented:
2nd#(cons(X,XS)) =  XS + 
>  XS + 

head#(cons(X,XS)) =  XS + 
> 
= c_7(X)

sel#(0(),cons(X,XS)) =  XS + 
> 
= c_8(X)

sel#(s(N),cons(X,XS)) =  XS + 
>  XS + 
= c_9(sel#(N,activate(XS)))

take#(s(N),cons(X,XS)) =  XS + 
>  XS + 
= c_12(X,N,activate#(XS))

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

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

activate#(n__take(X1,X2)) =   X2 + 
>=  X2 + 
=  c_4(take#(X1,X2))

from#(X) =  
>= 
=  c_5(X,X)

from#(X) =  
>= 
=  c_6(X)

take#(X1,X2) =   X2 + 
>=  X2 + 
=  c_10(X1,X2)

take#(0(),XS) =   XS + 
>= 
=  c_11()

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

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

activate(n__take(X1,X2)) =   X2 + 
>=  X2 + 
=  take(X1,X2)

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

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

take(X1,X2) =   X2 + 
>=  X2 + 
=  n__take(X1,X2)

take(0(),XS) =   XS + 
>= 
=  nil()

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

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_2(X)
activate#(n__from(X)) -> c_3(from#(X))
activate#(n__take(X1,X2)) -> c_4(take#(X1,X2))
from#(X) -> c_5(X,X)
from#(X) -> c_6(X)
take#(X1,X2) -> c_10(X1,X2)
- Strict TRS:
activate(X) -> X
activate(n__from(X)) -> from(X)
activate(n__take(X1,X2)) -> take(X1,X2)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
take(X1,X2) -> n__take(X1,X2)
take(0(),XS) -> nil()
take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS)))
- Weak DPs:
sel#(0(),cons(X,XS)) -> c_8(X)
sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)))
take#(0(),XS) -> c_11()
take#(s(N),cons(X,XS)) -> c_12(X,N,activate#(XS))
- Signature:
,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1,c_7/1,c_8/1,c_9/1,c_10/2,c_11/0
,c_12/3}
- Obligation:
runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel#,take#} and constructors {0,cons
,n__from,n__take,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(n__take) = {2},
uargs(take) = {2},
uargs(sel#) = {2},
uargs(c_1) = {1},
uargs(c_3) = {1},
uargs(c_4) = {1},
uargs(c_9) = {1},
uargs(c_12) = {3}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = 
p(2nd) = 
p(activate) =  x1 + 
p(cons) =  x2 + 
p(from) = 
p(n__from) = 
p(n__take) =  x2 + 
p(nil) = 
p(s) = 
p(sel) = 
p(take) =  x2 + 
p(2nd#) =  x1 + 
p(activate#) =  x1 + 
p(from#) = 
p(head#) =  x1 + 
p(sel#) =  x2 + 
p(take#) =  x2 + 
p(c_1) =  x1 + 
p(c_2) =  x1 + 
p(c_3) =  x1 + 
p(c_4) =  x1 + 
p(c_5) = 
p(c_6) = 
p(c_7) = 
p(c_8) = 
p(c_9) =  x1 + 
p(c_10) =  x2 + 
p(c_11) = 
p(c_12) =  x3 + 

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

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

activate#(n__take(X1,X2)) =  X2 + 
>  X2 + 
= c_4(take#(X1,X2))

take#(X1,X2) =  X2 + 
>  X2 + 
= c_10(X1,X2)

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

from(X) = 
> 
= n__from(X)

Following rules are (at-least) weakly oriented:
2nd#(cons(X,XS)) =   XS + 
>=  XS + 

from#(X) =  
>= 
=  c_5(X,X)

from#(X) =  
>= 
=  c_6(X)

head#(cons(X,XS)) =   XS + 
>= 
=  c_7(X)

sel#(0(),cons(X,XS)) =   XS + 
>= 
=  c_8(X)

sel#(s(N),cons(X,XS)) =   XS + 
>=  XS + 
=  c_9(sel#(N,activate(XS)))

take#(0(),XS) =   XS + 
>= 
=  c_11()

take#(s(N),cons(X,XS)) =   XS + 
>=  XS + 
=  c_12(X,N,activate#(XS))

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

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

activate(n__take(X1,X2)) =   X2 + 
>=  X2 + 
=  take(X1,X2)

take(X1,X2) =   X2 + 
>=  X2 + 
=  n__take(X1,X2)

take(0(),XS) =   XS + 
>= 
=  nil()

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

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:
from#(X) -> c_5(X,X)
from#(X) -> c_6(X)
- Strict TRS:
activate(X) -> X
activate(n__from(X)) -> from(X)
activate(n__take(X1,X2)) -> take(X1,X2)
take(X1,X2) -> n__take(X1,X2)
take(0(),XS) -> nil()
take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS)))
- Weak DPs:
activate#(X) -> c_2(X)
activate#(n__from(X)) -> c_3(from#(X))
activate#(n__take(X1,X2)) -> c_4(take#(X1,X2))
sel#(0(),cons(X,XS)) -> c_8(X)
sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)))
take#(X1,X2) -> c_10(X1,X2)
take#(0(),XS) -> c_11()
take#(s(N),cons(X,XS)) -> c_12(X,N,activate#(XS))
- Weak TRS:
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
- Signature:
,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1,c_7/1,c_8/1,c_9/1,c_10/2,c_11/0
,c_12/3}
- Obligation:
runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel#,take#} and constructors {0,cons
,n__from,n__take,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(n__take) = {2},
uargs(take) = {2},
uargs(sel#) = {2},
uargs(c_1) = {1},
uargs(c_3) = {1},
uargs(c_4) = {1},
uargs(c_9) = {1},
uargs(c_12) = {3}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = 
p(2nd) = 
p(activate) =  x1 + 
p(cons) =  x2 + 
p(from) = 
p(n__from) = 
p(n__take) =  x2 + 
p(nil) = 
p(s) = 
p(sel) = 
p(take) =  x2 + 
p(2nd#) =  x1 + 
p(activate#) = 
p(from#) = 
p(head#) =  x1 + 
p(sel#) =  x2 + 
p(take#) = 
p(c_1) =  x1 + 
p(c_2) = 
p(c_3) =  x1 + 
p(c_4) =  x1 + 
p(c_5) = 
p(c_6) = 
p(c_7) = 
p(c_8) = 
p(c_9) =  x1 + 
p(c_10) = 
p(c_11) = 
p(c_12) =  x3 + 

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

activate(n__take(X1,X2)) =  X2 + 
>  X2 + 
= take(X1,X2)

Following rules are (at-least) weakly oriented:
2nd#(cons(X,XS)) =   XS + 
>=  XS + 

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

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

activate#(n__take(X1,X2)) =  
>= 
=  c_4(take#(X1,X2))

from#(X) =  
>= 
=  c_5(X,X)

from#(X) =  
>= 
=  c_6(X)

head#(cons(X,XS)) =   XS + 
>= 
=  c_7(X)

sel#(0(),cons(X,XS)) =   XS + 
>= 
=  c_8(X)

sel#(s(N),cons(X,XS)) =   XS + 
>=  XS + 
=  c_9(sel#(N,activate(XS)))

take#(X1,X2) =  
>= 
=  c_10(X1,X2)

take#(0(),XS) =  
>= 
=  c_11()

take#(s(N),cons(X,XS)) =  
>= 
=  c_12(X,N,activate#(XS))

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

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

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

take(X1,X2) =   X2 + 
>=  X2 + 
=  n__take(X1,X2)

take(0(),XS) =   XS + 
>= 
=  nil()

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

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:
from#(X) -> c_5(X,X)
from#(X) -> c_6(X)
- Strict TRS:
activate(n__from(X)) -> from(X)
take(X1,X2) -> n__take(X1,X2)
take(0(),XS) -> nil()
take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS)))
- Weak DPs:
activate#(X) -> c_2(X)
activate#(n__from(X)) -> c_3(from#(X))
activate#(n__take(X1,X2)) -> c_4(take#(X1,X2))
sel#(0(),cons(X,XS)) -> c_8(X)
sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)))
take#(X1,X2) -> c_10(X1,X2)
take#(0(),XS) -> c_11()
take#(s(N),cons(X,XS)) -> c_12(X,N,activate#(XS))
- Weak TRS:
activate(X) -> X
activate(n__take(X1,X2)) -> take(X1,X2)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
- Signature:
,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1,c_7/1,c_8/1,c_9/1,c_10/2,c_11/0
,c_12/3}
- Obligation:
runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel#,take#} and constructors {0,cons
,n__from,n__take,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(n__take) = {2},
uargs(take) = {2},
uargs(sel#) = {2},
uargs(c_1) = {1},
uargs(c_3) = {1},
uargs(c_4) = {1},
uargs(c_9) = {1},
uargs(c_12) = {3}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = 
p(2nd) = 
p(activate) =  x1 + 
p(cons) =  x2 + 
p(from) = 
p(head) =  x1 + 
p(n__from) = 
p(n__take) =  x2 + 
p(nil) = 
p(s) =  x1 + 
p(sel) =  x1 + 
p(take) =  x2 + 
p(2nd#) =  x1 + 
p(activate#) = 
p(from#) = 
p(head#) =  x1 + 
p(sel#) =  x1 +  x2 + 
p(take#) = 
p(c_1) =  x1 + 
p(c_2) = 
p(c_3) =  x1 + 
p(c_4) =  x1 + 
p(c_5) = 
p(c_6) = 
p(c_7) = 
p(c_8) = 
p(c_9) =  x1 + 
p(c_10) = 
p(c_11) = 
p(c_12) =  x3 + 

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

Following rules are (at-least) weakly oriented:
2nd#(cons(X,XS)) =   XS + 
>=  XS + 

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

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

activate#(n__take(X1,X2)) =  
>= 
=  c_4(take#(X1,X2))

from#(X) =  
>= 
=  c_5(X,X)

from#(X) =  
>= 
=  c_6(X)

head#(cons(X,XS)) =   XS + 
>= 
=  c_7(X)

sel#(0(),cons(X,XS)) =   XS + 
>= 
=  c_8(X)

sel#(s(N),cons(X,XS)) =   N +  XS + 
>=  N +  XS + 
=  c_9(sel#(N,activate(XS)))

take#(X1,X2) =  
>= 
=  c_10(X1,X2)

take#(0(),XS) =  
>= 
=  c_11()

take#(s(N),cons(X,XS)) =  
>= 
=  c_12(X,N,activate#(XS))

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

activate(n__take(X1,X2)) =   X2 + 
>=  X2 + 
=  take(X1,X2)

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

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

take(X1,X2) =   X2 + 
>=  X2 + 
=  n__take(X1,X2)

take(0(),XS) =   XS + 
>= 
=  nil()

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

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:
from#(X) -> c_5(X,X)
from#(X) -> c_6(X)
- Strict TRS:
take(X1,X2) -> n__take(X1,X2)
take(0(),XS) -> nil()
take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS)))
- Weak DPs:
activate#(X) -> c_2(X)
activate#(n__from(X)) -> c_3(from#(X))
activate#(n__take(X1,X2)) -> c_4(take#(X1,X2))
sel#(0(),cons(X,XS)) -> c_8(X)
sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)))
take#(X1,X2) -> c_10(X1,X2)
take#(0(),XS) -> c_11()
take#(s(N),cons(X,XS)) -> c_12(X,N,activate#(XS))
- Weak TRS:
activate(X) -> X
activate(n__from(X)) -> from(X)
activate(n__take(X1,X2)) -> take(X1,X2)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
- Signature:
,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1,c_7/1,c_8/1,c_9/1,c_10/2,c_11/0
,c_12/3}
- Obligation:
runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel#,take#} and constructors {0,cons
,n__from,n__take,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(n__take) = {2},
uargs(take) = {2},
uargs(sel#) = {2},
uargs(c_1) = {1},
uargs(c_3) = {1},
uargs(c_4) = {1},
uargs(c_9) = {1},
uargs(c_12) = {3}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = 
p(2nd) = 
p(activate) =  x1 + 
p(cons) =  x2 + 
p(from) = 
p(n__from) = 
p(n__take) =  x2 + 
p(nil) = 
p(s) = 
p(sel) =  x1 +  x2 + 
p(take) =  x2 + 
p(2nd#) =  x1 + 
p(activate#) =  x1 + 
p(from#) = 
p(head#) =  x1 + 
p(sel#) =  x2 + 
p(take#) =  x2 + 
p(c_1) =  x1 + 
p(c_2) =  x1 + 
p(c_3) =  x1 + 
p(c_4) =  x1 + 
p(c_5) = 
p(c_6) = 
p(c_7) = 
p(c_8) = 
p(c_9) =  x1 + 
p(c_10) =  x2 + 
p(c_11) = 
p(c_12) =  x3 + 

Following rules are strictly oriented:
from#(X) = 
> 
= c_5(X,X)

from#(X) = 
> 
= c_6(X)

take(0(),XS) =  XS + 
> 
= nil()

Following rules are (at-least) weakly oriented:
2nd#(cons(X,XS)) =   XS + 
>=  XS + 

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

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

activate#(n__take(X1,X2)) =   X2 + 
>=  X2 + 
=  c_4(take#(X1,X2))

head#(cons(X,XS)) =   XS + 
>= 
=  c_7(X)

sel#(0(),cons(X,XS)) =   XS + 
>= 
=  c_8(X)

sel#(s(N),cons(X,XS)) =   XS + 
>=  XS + 
=  c_9(sel#(N,activate(XS)))

take#(X1,X2) =   X2 + 
>=  X2 + 
=  c_10(X1,X2)

take#(0(),XS) =   XS + 
>= 
=  c_11()

take#(s(N),cons(X,XS)) =   XS + 
>=  XS + 
=  c_12(X,N,activate#(XS))

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

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

activate(n__take(X1,X2)) =   X2 + 
>=  X2 + 
=  take(X1,X2)

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

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

take(X1,X2) =   X2 + 
>=  X2 + 
=  n__take(X1,X2)

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

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:
take(X1,X2) -> n__take(X1,X2)
take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS)))
- Weak DPs:
activate#(X) -> c_2(X)
activate#(n__from(X)) -> c_3(from#(X))
activate#(n__take(X1,X2)) -> c_4(take#(X1,X2))
from#(X) -> c_5(X,X)
from#(X) -> c_6(X)
sel#(0(),cons(X,XS)) -> c_8(X)
sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)))
take#(X1,X2) -> c_10(X1,X2)
take#(0(),XS) -> c_11()
take#(s(N),cons(X,XS)) -> c_12(X,N,activate#(XS))
- Weak TRS:
activate(X) -> X
activate(n__from(X)) -> from(X)
activate(n__take(X1,X2)) -> take(X1,X2)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
take(0(),XS) -> nil()
- Signature:
,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1,c_7/1,c_8/1,c_9/1,c_10/2,c_11/0
,c_12/3}
- Obligation:
runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel#,take#} and constructors {0,cons
,n__from,n__take,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(n__take) = {2},
uargs(take) = {2},
uargs(sel#) = {2},
uargs(c_1) = {1},
uargs(c_3) = {1},
uargs(c_4) = {1},
uargs(c_9) = {1},
uargs(c_12) = {3}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = 
p(2nd) = 
p(activate) =  x1 + 
p(cons) =  x2 + 
p(from) = 
p(n__from) = 
p(n__take) =  x1 +  x2 + 
p(nil) = 
p(s) =  x1 + 
p(sel) =  x2 + 
p(take) =  x1 +  x2 + 
p(2nd#) =  x1 + 
p(activate#) =  x1 + 
p(from#) = 
p(head#) =  x1 + 
p(sel#) =  x1 +  x2 + 
p(take#) =  x1 +  x2 + 
p(c_1) =  x1 + 
p(c_2) = 
p(c_3) =  x1 + 
p(c_4) =  x1 + 
p(c_5) = 
p(c_6) = 
p(c_7) = 
p(c_8) = 
p(c_9) =  x1 + 
p(c_10) = 
p(c_11) = 
p(c_12) =  x3 + 

Following rules are strictly oriented:
take(X1,X2) =  X1 +  X2 + 
>  X1 +  X2 + 
= n__take(X1,X2)

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

Following rules are (at-least) weakly oriented:
2nd#(cons(X,XS)) =   XS + 
>=  XS + 

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

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

activate#(n__take(X1,X2)) =   X1 +  X2 + 
>=  X1 +  X2 + 
=  c_4(take#(X1,X2))

from#(X) =  
>= 
=  c_5(X,X)

from#(X) =  
>= 
=  c_6(X)

head#(cons(X,XS)) =   XS + 
>= 
=  c_7(X)

sel#(0(),cons(X,XS)) =   XS + 
>= 
=  c_8(X)

sel#(s(N),cons(X,XS)) =   N +  XS + 
>=  N +  XS + 
=  c_9(sel#(N,activate(XS)))

take#(X1,X2) =   X1 +  X2 + 
>= 
=  c_10(X1,X2)

take#(0(),XS) =   XS + 
>= 
=  c_11()

take#(s(N),cons(X,XS)) =   N +  XS + 
>=  XS + 
=  c_12(X,N,activate#(XS))

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

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

activate(n__take(X1,X2)) =   X1 +  X2 + 
>=  X1 +  X2 + 
=  take(X1,X2)

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

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

take(0(),XS) =   XS + 
>= 
=  nil()

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 10: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
activate#(X) -> c_2(X)
activate#(n__from(X)) -> c_3(from#(X))
activate#(n__take(X1,X2)) -> c_4(take#(X1,X2))
from#(X) -> c_5(X,X)
from#(X) -> c_6(X)
sel#(0(),cons(X,XS)) -> c_8(X)
sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)))
take#(X1,X2) -> c_10(X1,X2)
take#(0(),XS) -> c_11()
take#(s(N),cons(X,XS)) -> c_12(X,N,activate#(XS))
- Weak TRS:
activate(X) -> X
activate(n__from(X)) -> from(X)
activate(n__take(X1,X2)) -> take(X1,X2)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
take(X1,X2) -> n__take(X1,X2)
take(0(),XS) -> nil()
take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS)))
- Signature: