* 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) = [0]
p(2nd) = [0]
p(activate) = [1] x1 + [0]
p(cons) = [1] x2 + [1]
p(from) = [0]
p(n__from) = [0]
p(n__take) = [1] x2 + [0]
p(nil) = [0]
p(s) = [1] x1 + [0]
p(sel) = [0]
p(take) = [1] x2 + [0]
p(2nd#) = [5] x1 + [0]
p(activate#) = [1] x1 + [0]
p(from#) = [0]
p(head#) = [1] x1 + [0]
p(sel#) = [1] x2 + [0]
p(take#) = [1] x2 + [0]
p(c_1) = [1] x1 + [0]
p(c_2) = [1] x1 + [0]
p(c_3) = [1] x1 + [0]
p(c_4) = [1] x1 + [0]
p(c_5) = [0]
p(c_6) = [0]
p(c_7) = [0]
p(c_8) = [0]
p(c_9) = [1] x1 + [0]
p(c_10) = [1] x2 + [0]
p(c_11) = [0]
p(c_12) = [1] x3 + [0]

Following rules are strictly oriented:
2nd#(cons(X,XS)) = [5] XS + [5]
> [1] XS + [0]

head#(cons(X,XS)) = [1] XS + [1]
> [0]
= c_7(X)

sel#(0(),cons(X,XS)) = [1] XS + [1]
> [0]
= c_8(X)

sel#(s(N),cons(X,XS)) = [1] XS + [1]
> [1] XS + [0]
= c_9(sel#(N,activate(XS)))

take#(s(N),cons(X,XS)) = [1] XS + [1]
> [1] XS + [0]
= c_12(X,N,activate#(XS))

Following rules are (at-least) weakly oriented:
activate#(X) =  [1] X + [0]
>= [1] X + [0]
=  c_2(X)

activate#(n__from(X)) =  [0]
>= [0]
=  c_3(from#(X))

activate#(n__take(X1,X2)) =  [1] X2 + [0]
>= [1] X2 + [0]
=  c_4(take#(X1,X2))

from#(X) =  [0]
>= [0]
=  c_5(X,X)

from#(X) =  [0]
>= [0]
=  c_6(X)

take#(X1,X2) =  [1] X2 + [0]
>= [1] X2 + [0]
=  c_10(X1,X2)

take#(0(),XS) =  [1] XS + [0]
>= [0]
=  c_11()

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

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

activate(n__take(X1,X2)) =  [1] X2 + [0]
>= [1] X2 + [0]
=  take(X1,X2)

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

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

take(X1,X2) =  [1] X2 + [0]
>= [1] X2 + [0]
=  n__take(X1,X2)

take(0(),XS) =  [1] XS + [0]
>= [0]
=  nil()

take(s(N),cons(X,XS)) =  [1] XS + [1]
>= [1] XS + [1]
=  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) = [0]
p(2nd) = [0]
p(activate) = [1] x1 + [0]
p(cons) = [1] x2 + [6]
p(from) = [7]
p(n__from) = [0]
p(n__take) = [1] x2 + [0]
p(nil) = [0]
p(s) = [0]
p(sel) = [0]
p(take) = [1] x2 + [0]
p(2nd#) = [1] x1 + [2]
p(activate#) = [2] x1 + [5]
p(from#) = [0]
p(head#) = [1] x1 + [1]
p(sel#) = [1] x2 + [0]
p(take#) = [2] x2 + [3]
p(c_1) = [1] x1 + [7]
p(c_2) = [2] x1 + [4]
p(c_3) = [1] x1 + [0]
p(c_4) = [1] x1 + [0]
p(c_5) = [0]
p(c_6) = [0]
p(c_7) = [7]
p(c_8) = [0]
p(c_9) = [1] x1 + [0]
p(c_10) = [2] x2 + [0]
p(c_11) = [2]
p(c_12) = [1] x3 + [5]

Following rules are strictly oriented:
activate#(X) = [2] X + [5]
> [2] X + [4]
= c_2(X)

activate#(n__from(X)) = [5]
> [0]
= c_3(from#(X))

activate#(n__take(X1,X2)) = [2] X2 + [5]
> [2] X2 + [3]
= c_4(take#(X1,X2))

take#(X1,X2) = [2] X2 + [3]
> [2] X2 + [0]
= c_10(X1,X2)

from(X) = [7]
> [6]
= cons(X,n__from(s(X)))

from(X) = [7]
> [0]
= n__from(X)

Following rules are (at-least) weakly oriented:
2nd#(cons(X,XS)) =  [1] XS + [8]
>= [1] XS + [8]

from#(X) =  [0]
>= [0]
=  c_5(X,X)

from#(X) =  [0]
>= [0]
=  c_6(X)

head#(cons(X,XS)) =  [1] XS + [7]
>= [7]
=  c_7(X)

sel#(0(),cons(X,XS)) =  [1] XS + [6]
>= [0]
=  c_8(X)

sel#(s(N),cons(X,XS)) =  [1] XS + [6]
>= [1] XS + [0]
=  c_9(sel#(N,activate(XS)))

take#(0(),XS) =  [2] XS + [3]
>= [2]
=  c_11()

take#(s(N),cons(X,XS)) =  [2] XS + [15]
>= [2] XS + [10]
=  c_12(X,N,activate#(XS))

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

activate(n__from(X)) =  [0]
>= [7]
=  from(X)

activate(n__take(X1,X2)) =  [1] X2 + [0]
>= [1] X2 + [0]
=  take(X1,X2)

take(X1,X2) =  [1] X2 + [0]
>= [1] X2 + [0]
=  n__take(X1,X2)

take(0(),XS) =  [1] XS + [0]
>= [0]
=  nil()

take(s(N),cons(X,XS)) =  [1] XS + [6]
>= [1] XS + [6]
=  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) = [0]
p(2nd) = [0]
p(activate) = [1] x1 + [3]
p(cons) = [1] x2 + [4]
p(from) = [4]
p(n__from) = [0]
p(n__take) = [1] x2 + [4]
p(nil) = [0]
p(s) = [0]
p(sel) = [0]
p(take) = [1] x2 + [0]
p(2nd#) = [2] x1 + [0]
p(activate#) = [0]
p(from#) = [0]
p(head#) = [1] x1 + [2]
p(sel#) = [1] x2 + [6]
p(take#) = [0]
p(c_1) = [1] x1 + [0]
p(c_2) = [0]
p(c_3) = [1] x1 + [0]
p(c_4) = [1] x1 + [0]
p(c_5) = [0]
p(c_6) = [2]
p(c_7) = [0]
p(c_8) = [1]
p(c_9) = [1] x1 + [1]
p(c_10) = [0]
p(c_11) = [0]
p(c_12) = [1] x3 + [0]

Following rules are strictly oriented:
activate(X) = [1] X + [3]
> [1] X + [0]
= X

activate(n__take(X1,X2)) = [1] X2 + [7]
> [1] X2 + [0]
= take(X1,X2)

Following rules are (at-least) weakly oriented:
2nd#(cons(X,XS)) =  [2] XS + [8]
>= [1] XS + [5]

activate#(X) =  [0]
>= [0]
=  c_2(X)

activate#(n__from(X)) =  [0]
>= [0]
=  c_3(from#(X))

activate#(n__take(X1,X2)) =  [0]
>= [0]
=  c_4(take#(X1,X2))

from#(X) =  [0]
>= [0]
=  c_5(X,X)

from#(X) =  [0]
>= [2]
=  c_6(X)

head#(cons(X,XS)) =  [1] XS + [6]
>= [0]
=  c_7(X)

sel#(0(),cons(X,XS)) =  [1] XS + [10]
>= [1]
=  c_8(X)

sel#(s(N),cons(X,XS)) =  [1] XS + [10]
>= [1] XS + [10]
=  c_9(sel#(N,activate(XS)))

take#(X1,X2) =  [0]
>= [0]
=  c_10(X1,X2)

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

take#(s(N),cons(X,XS)) =  [0]
>= [0]
=  c_12(X,N,activate#(XS))

activate(n__from(X)) =  [3]
>= [4]
=  from(X)

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

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

take(X1,X2) =  [1] X2 + [0]
>= [1] X2 + [4]
=  n__take(X1,X2)

take(0(),XS) =  [1] XS + [0]
>= [0]
=  nil()

take(s(N),cons(X,XS)) =  [1] XS + [4]
>= [1] XS + [11]
=  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) = [0]
p(2nd) = [0]
p(activate) = [1] x1 + [2]
p(cons) = [1] x2 + [1]
p(from) = [1]
p(head) = [1] x1 + [2]
p(n__from) = [0]
p(n__take) = [1] x2 + [2]
p(nil) = [1]
p(s) = [1] x1 + [2]
p(sel) = [1] x1 + [0]
p(take) = [1] x2 + [0]
p(2nd#) = [1] x1 + [6]
p(activate#) = [4]
p(from#) = [0]
p(head#) = [1] x1 + [2]
p(sel#) = [1] x1 + [1] x2 + [4]
p(take#) = [4]
p(c_1) = [1] x1 + [3]
p(c_2) = [1]
p(c_3) = [1] x1 + [0]
p(c_4) = [1] x1 + [0]
p(c_5) = [1]
p(c_6) = [1]
p(c_7) = [0]
p(c_8) = [4]
p(c_9) = [1] x1 + [0]
p(c_10) = [0]
p(c_11) = [4]
p(c_12) = [1] x3 + [0]

Following rules are strictly oriented:
activate(n__from(X)) = [2]
> [1]
= from(X)

Following rules are (at-least) weakly oriented:
2nd#(cons(X,XS)) =  [1] XS + [7]
>= [1] XS + [7]

activate#(X) =  [4]
>= [1]
=  c_2(X)

activate#(n__from(X)) =  [4]
>= [0]
=  c_3(from#(X))

activate#(n__take(X1,X2)) =  [4]
>= [4]
=  c_4(take#(X1,X2))

from#(X) =  [0]
>= [1]
=  c_5(X,X)

from#(X) =  [0]
>= [1]
=  c_6(X)

head#(cons(X,XS)) =  [1] XS + [3]
>= [0]
=  c_7(X)

sel#(0(),cons(X,XS)) =  [1] XS + [5]
>= [4]
=  c_8(X)

sel#(s(N),cons(X,XS)) =  [1] N + [1] XS + [7]
>= [1] N + [1] XS + [6]
=  c_9(sel#(N,activate(XS)))

take#(X1,X2) =  [4]
>= [0]
=  c_10(X1,X2)

take#(0(),XS) =  [4]
>= [4]
=  c_11()

take#(s(N),cons(X,XS)) =  [4]
>= [4]
=  c_12(X,N,activate#(XS))

activate(X) =  [1] X + [2]
>= [1] X + [0]
=  X

activate(n__take(X1,X2)) =  [1] X2 + [4]
>= [1] X2 + [0]
=  take(X1,X2)

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

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

take(X1,X2) =  [1] X2 + [0]
>= [1] X2 + [2]
=  n__take(X1,X2)

take(0(),XS) =  [1] XS + [0]
>= [1]
=  nil()

take(s(N),cons(X,XS)) =  [1] XS + [1]
>= [1] XS + [5]
=  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) = [0]
p(2nd) = [4]
p(activate) = [1] x1 + [0]
p(cons) = [1] x2 + [0]
p(from) = [0]
p(n__from) = [0]
p(n__take) = [1] x2 + [3]
p(nil) = [0]
p(s) = [0]
p(sel) = [1] x1 + [4] x2 + [1]
p(take) = [1] x2 + [3]
p(2nd#) = [2] x1 + [0]
p(activate#) = [2] x1 + [1]
p(from#) = [1]
p(head#) = [1] x1 + [0]
p(sel#) = [1] x2 + [0]
p(take#) = [2] x2 + [7]
p(c_1) = [1] x1 + [0]
p(c_2) = [2] x1 + [1]
p(c_3) = [1] x1 + [0]
p(c_4) = [1] x1 + [0]
p(c_5) = [0]
p(c_6) = [0]
p(c_7) = [0]
p(c_8) = [0]
p(c_9) = [1] x1 + [0]
p(c_10) = [2] x2 + [7]
p(c_11) = [7]
p(c_12) = [1] x3 + [6]

Following rules are strictly oriented:
from#(X) = [1]
> [0]
= c_5(X,X)

from#(X) = [1]
> [0]
= c_6(X)

take(0(),XS) = [1] XS + [3]
> [0]
= nil()

Following rules are (at-least) weakly oriented:
2nd#(cons(X,XS)) =  [2] XS + [0]
>= [1] XS + [0]

activate#(X) =  [2] X + [1]
>= [2] X + [1]
=  c_2(X)

activate#(n__from(X)) =  [1]
>= [1]
=  c_3(from#(X))

activate#(n__take(X1,X2)) =  [2] X2 + [7]
>= [2] X2 + [7]
=  c_4(take#(X1,X2))

head#(cons(X,XS)) =  [1] XS + [0]
>= [0]
=  c_7(X)

sel#(0(),cons(X,XS)) =  [1] XS + [0]
>= [0]
=  c_8(X)

sel#(s(N),cons(X,XS)) =  [1] XS + [0]
>= [1] XS + [0]
=  c_9(sel#(N,activate(XS)))

take#(X1,X2) =  [2] X2 + [7]
>= [2] X2 + [7]
=  c_10(X1,X2)

take#(0(),XS) =  [2] XS + [7]
>= [7]
=  c_11()

take#(s(N),cons(X,XS)) =  [2] XS + [7]
>= [2] XS + [7]
=  c_12(X,N,activate#(XS))

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

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

activate(n__take(X1,X2)) =  [1] X2 + [3]
>= [1] X2 + [3]
=  take(X1,X2)

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

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

take(X1,X2) =  [1] X2 + [3]
>= [1] X2 + [3]
=  n__take(X1,X2)

take(s(N),cons(X,XS)) =  [1] XS + [3]
>= [1] XS + [3]
=  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) = [1]
p(2nd) = [0]
p(activate) = [1] x1 + [2]
p(cons) = [1] x2 + [0]
p(from) = [2]
p(n__from) = [0]
p(n__take) = [1] x1 + [1] x2 + [3]
p(nil) = [0]
p(s) = [1] x1 + [2]
p(sel) = [1] x2 + [1]
p(take) = [1] x1 + [1] x2 + [4]
p(2nd#) = [2] x1 + [5]
p(activate#) = [4] x1 + [2]
p(from#) = [0]
p(head#) = [1] x1 + [2]
p(sel#) = [4] x1 + [1] x2 + [0]
p(take#) = [1] x1 + [4] x2 + [0]
p(c_1) = [1] x1 + [0]
p(c_2) = [0]
p(c_3) = [1] x1 + [0]
p(c_4) = [1] x1 + [0]
p(c_5) = [0]
p(c_6) = [0]
p(c_7) = [0]
p(c_8) = [1]
p(c_9) = [1] x1 + [0]
p(c_10) = [0]
p(c_11) = [0]
p(c_12) = [1] x3 + [0]

Following rules are strictly oriented:
take(X1,X2) = [1] X1 + [1] X2 + [4]
> [1] X1 + [1] X2 + [3]
= n__take(X1,X2)

take(s(N),cons(X,XS)) = [1] N + [1] XS + [6]
> [1] N + [1] XS + [5]
= cons(X,n__take(N,activate(XS)))

Following rules are (at-least) weakly oriented:
2nd#(cons(X,XS)) =  [2] XS + [5]
>= [1] XS + [4]

activate#(X) =  [4] X + [2]
>= [0]
=  c_2(X)

activate#(n__from(X)) =  [2]
>= [0]
=  c_3(from#(X))

activate#(n__take(X1,X2)) =  [4] X1 + [4] X2 + [14]
>= [1] X1 + [4] X2 + [0]
=  c_4(take#(X1,X2))

from#(X) =  [0]
>= [0]
=  c_5(X,X)

from#(X) =  [0]
>= [0]
=  c_6(X)

head#(cons(X,XS)) =  [1] XS + [2]
>= [0]
=  c_7(X)

sel#(0(),cons(X,XS)) =  [1] XS + [4]
>= [1]
=  c_8(X)

sel#(s(N),cons(X,XS)) =  [4] N + [1] XS + [8]
>= [4] N + [1] XS + [2]
=  c_9(sel#(N,activate(XS)))

take#(X1,X2) =  [1] X1 + [4] X2 + [0]
>= [0]
=  c_10(X1,X2)

take#(0(),XS) =  [4] XS + [1]
>= [0]
=  c_11()

take#(s(N),cons(X,XS)) =  [1] N + [4] XS + [2]
>= [4] XS + [2]
=  c_12(X,N,activate#(XS))

activate(X) =  [1] X + [2]
>= [1] X + [0]
=  X

activate(n__from(X)) =  [2]
>= [2]
=  from(X)

activate(n__take(X1,X2)) =  [1] X1 + [1] X2 + [5]
>= [1] X1 + [1] X2 + [4]
=  take(X1,X2)

from(X) =  [2]
>= [0]
=  cons(X,n__from(s(X)))

from(X) =  [2]
>= [0]
=  n__from(X)

take(0(),XS) =  [1] XS + [5]
>= [0]
=  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: