```* Step 1: DependencyPairs WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
2nd(cons(X,X1)) -> 2nd(cons1(X,activate(X1)))
2nd(cons1(X,cons(Y,Z))) -> Y
activate(X) -> X
activate(n__from(X)) -> from(X)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
- Signature:
{2nd/1,activate/1,from/1} / {cons/2,cons1/2,n__from/1,s/1}
- Obligation:
runtime complexity wrt. defined symbols {2nd,activate,from} and constructors {cons,cons1,n__from,s}
+ Applied Processor:
DependencyPairs {dpKind_ = WIDP}
+ Details:
We add the following weak dependency pairs:

Strict DPs
2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1))))
2nd#(cons1(X,cons(Y,Z))) -> c_2(Y)
activate#(X) -> c_3(X)
activate#(n__from(X)) -> c_4(from#(X))
from#(X) -> c_5(X,X)
from#(X) -> c_6(X)
Weak DPs

and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1))))
2nd#(cons1(X,cons(Y,Z))) -> c_2(Y)
activate#(X) -> c_3(X)
activate#(n__from(X)) -> c_4(from#(X))
from#(X) -> c_5(X,X)
from#(X) -> c_6(X)
- Strict TRS:
2nd(cons(X,X1)) -> 2nd(cons1(X,activate(X1)))
2nd(cons1(X,cons(Y,Z))) -> Y
activate(X) -> X
activate(n__from(X)) -> from(X)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
- Signature:
{2nd/1,activate/1,from/1,2nd#/1,activate#/1,from#/1} / {cons/2,cons1/2,n__from/1,s/1,c_1/1,c_2/1,c_3/1,c_4/1
,c_5/2,c_6/1}
- Obligation:
runtime complexity wrt. defined symbols {2nd#,activate#,from#} and constructors {cons,cons1,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)
2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1))))
2nd#(cons1(X,cons(Y,Z))) -> c_2(Y)
activate#(X) -> c_3(X)
activate#(n__from(X)) -> c_4(from#(X))
from#(X) -> c_5(X,X)
from#(X) -> c_6(X)
* Step 3: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1))))
2nd#(cons1(X,cons(Y,Z))) -> c_2(Y)
activate#(X) -> c_3(X)
activate#(n__from(X)) -> c_4(from#(X))
from#(X) -> c_5(X,X)
from#(X) -> c_6(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)
- Signature:
{2nd/1,activate/1,from/1,2nd#/1,activate#/1,from#/1} / {cons/2,cons1/2,n__from/1,s/1,c_1/1,c_2/1,c_3/1,c_4/1
,c_5/2,c_6/1}
- Obligation:
runtime complexity wrt. defined symbols {2nd#,activate#,from#} and constructors {cons,cons1,n__from,s}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnTrs}
+ Details:
The weightgap principle applies using the following constant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(cons1) = {2},
uargs(2nd#) = {1},
uargs(c_1) = {1},
uargs(c_4) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(2nd) = [0]
p(activate) = [1] x1 + [15]
p(cons) = [1] x2 + [5]
p(cons1) = [1] x2 + [0]
p(from) = [1] x1 + [6]
p(n__from) = [1] x1 + [0]
p(s) = [1] x1 + [0]
p(2nd#) = [1] x1 + [0]
p(activate#) = [0]
p(from#) = [0]
p(c_1) = [1] x1 + [0]
p(c_2) = [0]
p(c_3) = [0]
p(c_4) = [1] x1 + [0]
p(c_5) = [0]
p(c_6) = [0]

Following rules are strictly oriented:
2nd#(cons1(X,cons(Y,Z))) = [1] Z + [5]
> [0]
= c_2(Y)

activate(X) = [1] X + [15]
> [1] X + [0]
= X

activate(n__from(X)) = [1] X + [15]
> [1] X + [6]
= from(X)

from(X) = [1] X + [6]
> [1] X + [5]
= cons(X,n__from(s(X)))

from(X) = [1] X + [6]
> [1] X + [0]
= n__from(X)

Following rules are (at-least) weakly oriented:
2nd#(cons(X,X1)) =  [1] X1 + [5]
>= [1] X1 + [15]
=  c_1(2nd#(cons1(X,activate(X1))))

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

activate#(n__from(X)) =  [0]
>= [0]
=  c_4(from#(X))

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

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

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 4: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1))))
activate#(X) -> c_3(X)
activate#(n__from(X)) -> c_4(from#(X))
from#(X) -> c_5(X,X)
from#(X) -> c_6(X)
- Weak DPs:
2nd#(cons1(X,cons(Y,Z))) -> c_2(Y)
- 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:
{2nd/1,activate/1,from/1,2nd#/1,activate#/1,from#/1} / {cons/2,cons1/2,n__from/1,s/1,c_1/1,c_2/1,c_3/1,c_4/1
,c_5/2,c_6/1}
- Obligation:
runtime complexity wrt. defined symbols {2nd#,activate#,from#} and constructors {cons,cons1,n__from,s}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
2: activate#(X) -> c_3(X)
3: activate#(n__from(X)) -> c_4(from#(X))
4: from#(X) -> c_5(X,X)
5: from#(X) -> c_6(X)

The strictly oriented rules are moved into the weak component.
** Step 4.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1))))
activate#(X) -> c_3(X)
activate#(n__from(X)) -> c_4(from#(X))
from#(X) -> c_5(X,X)
from#(X) -> c_6(X)
- Weak DPs:
2nd#(cons1(X,cons(Y,Z))) -> c_2(Y)
- 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:
{2nd/1,activate/1,from/1,2nd#/1,activate#/1,from#/1} / {cons/2,cons1/2,n__from/1,s/1,c_1/1,c_2/1,c_3/1,c_4/1
,c_5/2,c_6/1}
- Obligation:
runtime complexity wrt. defined symbols {2nd#,activate#,from#} and constructors {cons,cons1,n__from,s}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_1) = {1},
uargs(c_4) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(2nd) = [0]
p(activate) = [8] x1 + [4]
p(cons) = [1] x1 + [12]
p(cons1) = [1] x1 + [1] x2 + [0]
p(from) = [8] x1 + [12]
p(n__from) = [1] x1 + [1]
p(s) = [1] x1 + [1]
p(2nd#) = [0]
p(activate#) = [2] x1 + [8]
p(from#) = [2] x1 + [7]
p(c_1) = [2] x1 + [0]
p(c_2) = [0]
p(c_3) = [1] x1 + [4]
p(c_4) = [1] x1 + [0]
p(c_5) = [1] x1 + [0]
p(c_6) = [2] x1 + [5]

Following rules are strictly oriented:
activate#(X) = [2] X + [8]
> [1] X + [4]
= c_3(X)

activate#(n__from(X)) = [2] X + [10]
> [2] X + [7]
= c_4(from#(X))

from#(X) = [2] X + [7]
> [1] X + [0]
= c_5(X,X)

from#(X) = [2] X + [7]
> [2] X + [5]
= c_6(X)

Following rules are (at-least) weakly oriented:
2nd#(cons(X,X1)) =  [0]
>= [0]
=  c_1(2nd#(cons1(X,activate(X1))))

2nd#(cons1(X,cons(Y,Z))) =  [0]
>= [0]
=  c_2(Y)

activate(X) =  [8] X + [4]
>= [1] X + [0]
=  X

activate(n__from(X)) =  [8] X + [12]
>= [8] X + [12]
=  from(X)

from(X) =  [8] X + [12]
>= [1] X + [12]
=  cons(X,n__from(s(X)))

from(X) =  [8] X + [12]
>= [1] X + [1]
=  n__from(X)

** Step 4.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1))))
- Weak DPs:
2nd#(cons1(X,cons(Y,Z))) -> c_2(Y)
activate#(X) -> c_3(X)
activate#(n__from(X)) -> c_4(from#(X))
from#(X) -> c_5(X,X)
from#(X) -> c_6(X)
- 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:
{2nd/1,activate/1,from/1,2nd#/1,activate#/1,from#/1} / {cons/2,cons1/2,n__from/1,s/1,c_1/1,c_2/1,c_3/1,c_4/1
,c_5/2,c_6/1}
- Obligation:
runtime complexity wrt. defined symbols {2nd#,activate#,from#} and constructors {cons,cons1,n__from,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()

** Step 4.b:1: PredecessorEstimationCP WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1))))
- Weak DPs:
2nd#(cons1(X,cons(Y,Z))) -> c_2(Y)
activate#(X) -> c_3(X)
activate#(n__from(X)) -> c_4(from#(X))
from#(X) -> c_5(X,X)
from#(X) -> c_6(X)
- 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:
{2nd/1,activate/1,from/1,2nd#/1,activate#/1,from#/1} / {cons/2,cons1/2,n__from/1,s/1,c_1/1,c_2/1,c_3/1,c_4/1
,c_5/2,c_6/1}
- Obligation:
runtime complexity wrt. defined symbols {2nd#,activate#,from#} and constructors {cons,cons1,n__from,s}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
1: 2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1))))

The strictly oriented rules are moved into the weak component.
*** Step 4.b:1.a:1: NaturalMI WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1))))
- Weak DPs:
2nd#(cons1(X,cons(Y,Z))) -> c_2(Y)
activate#(X) -> c_3(X)
activate#(n__from(X)) -> c_4(from#(X))
from#(X) -> c_5(X,X)
from#(X) -> c_6(X)
- 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:
{2nd/1,activate/1,from/1,2nd#/1,activate#/1,from#/1} / {cons/2,cons1/2,n__from/1,s/1,c_1/1,c_2/1,c_3/1,c_4/1
,c_5/2,c_6/1}
- Obligation:
runtime complexity wrt. defined symbols {2nd#,activate#,from#} and constructors {cons,cons1,n__from,s}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 0 non-zero interpretation-entries in the diagonal of the component-wise maxima):
The following argument positions are considered usable:
uargs(c_1) = {1},
uargs(c_4) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(2nd) = [0]
p(activate) = [1] x1 + [5]
p(cons) = [8]
p(cons1) = [0]
p(from) = [8]
p(n__from) = [3]
p(s) = [1]
p(2nd#) = [2] x1 + [0]
p(activate#) = [6] x1 + [2]
p(from#) = [0]
p(c_1) = [2] x1 + [15]
p(c_2) = [0]
p(c_3) = [1] x1 + [1]
p(c_4) = [2] x1 + [8]
p(c_5) = [0]
p(c_6) = [0]

Following rules are strictly oriented:
2nd#(cons(X,X1)) = [16]
> [15]
= c_1(2nd#(cons1(X,activate(X1))))

Following rules are (at-least) weakly oriented:
2nd#(cons1(X,cons(Y,Z))) =  [0]
>= [0]
=  c_2(Y)

activate#(X) =  [6] X + [2]
>= [1] X + [1]
=  c_3(X)

activate#(n__from(X)) =  [20]
>= [8]
=  c_4(from#(X))

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

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

activate(X) =  [1] X + [5]
>= [1] X + [0]
=  X

activate(n__from(X)) =  [8]
>= [8]
=  from(X)

from(X) =  [8]
>= [8]
=  cons(X,n__from(s(X)))

from(X) =  [8]
>= [3]
=  n__from(X)

*** Step 4.b:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1))))
2nd#(cons1(X,cons(Y,Z))) -> c_2(Y)
activate#(X) -> c_3(X)
activate#(n__from(X)) -> c_4(from#(X))
from#(X) -> c_5(X,X)
from#(X) -> c_6(X)
- 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:
{2nd/1,activate/1,from/1,2nd#/1,activate#/1,from#/1} / {cons/2,cons1/2,n__from/1,s/1,c_1/1,c_2/1,c_3/1,c_4/1
,c_5/2,c_6/1}
- Obligation:
runtime complexity wrt. defined symbols {2nd#,activate#,from#} and constructors {cons,cons1,n__from,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()

*** Step 4.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1))))
2nd#(cons1(X,cons(Y,Z))) -> c_2(Y)
activate#(X) -> c_3(X)
activate#(n__from(X)) -> c_4(from#(X))
from#(X) -> c_5(X,X)
from#(X) -> c_6(X)
- 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:
{2nd/1,activate/1,from/1,2nd#/1,activate#/1,from#/1} / {cons/2,cons1/2,n__from/1,s/1,c_1/1,c_2/1,c_3/1,c_4/1
,c_5/2,c_6/1}
- Obligation:
runtime complexity wrt. defined symbols {2nd#,activate#,from#} and constructors {cons,cons1,n__from,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1))))
-->_1 2nd#(cons1(X,cons(Y,Z))) -> c_2(Y):2

2:W:2nd#(cons1(X,cons(Y,Z))) -> c_2(Y)
-->_1 from#(X) -> c_6(X):6
-->_1 from#(X) -> c_5(X,X):5
-->_1 activate#(n__from(X)) -> c_4(from#(X)):4
-->_1 activate#(X) -> c_3(X):3
-->_1 2nd#(cons1(X,cons(Y,Z))) -> c_2(Y):2
-->_1 2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1)))):1

3:W:activate#(X) -> c_3(X)
-->_1 from#(X) -> c_6(X):6
-->_1 from#(X) -> c_5(X,X):5
-->_1 activate#(n__from(X)) -> c_4(from#(X)):4
-->_1 activate#(X) -> c_3(X):3
-->_1 2nd#(cons1(X,cons(Y,Z))) -> c_2(Y):2
-->_1 2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1)))):1

4:W:activate#(n__from(X)) -> c_4(from#(X))
-->_1 from#(X) -> c_6(X):6
-->_1 from#(X) -> c_5(X,X):5

5:W:from#(X) -> c_5(X,X)
-->_2 from#(X) -> c_6(X):6
-->_1 from#(X) -> c_6(X):6
-->_2 from#(X) -> c_5(X,X):5
-->_1 from#(X) -> c_5(X,X):5
-->_2 activate#(n__from(X)) -> c_4(from#(X)):4
-->_1 activate#(n__from(X)) -> c_4(from#(X)):4
-->_2 activate#(X) -> c_3(X):3
-->_1 activate#(X) -> c_3(X):3
-->_2 2nd#(cons1(X,cons(Y,Z))) -> c_2(Y):2
-->_1 2nd#(cons1(X,cons(Y,Z))) -> c_2(Y):2
-->_2 2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1)))):1
-->_1 2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1)))):1

6:W:from#(X) -> c_6(X)
-->_1 from#(X) -> c_6(X):6
-->_1 from#(X) -> c_5(X,X):5
-->_1 activate#(n__from(X)) -> c_4(from#(X)):4
-->_1 activate#(X) -> c_3(X):3
-->_1 2nd#(cons1(X,cons(Y,Z))) -> c_2(Y):2
-->_1 2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1)))):1

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: 2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1))))
6: from#(X) -> c_6(X)
5: from#(X) -> c_5(X,X)
4: activate#(n__from(X)) -> c_4(from#(X))
3: activate#(X) -> c_3(X)
2: 2nd#(cons1(X,cons(Y,Z))) -> c_2(Y)
*** Step 4.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- 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:
{2nd/1,activate/1,from/1,2nd#/1,activate#/1,from#/1} / {cons/2,cons1/2,n__from/1,s/1,c_1/1,c_2/1,c_3/1,c_4/1
,c_5/2,c_6/1}
- Obligation:
runtime complexity wrt. defined symbols {2nd#,activate#,from#} and constructors {cons,cons1,n__from,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

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