* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1))
+ Considered Problem:
- Strict TRS:
2nd(cons(X,n__cons(Y,Z))) -> activate(Y)
activate(X) -> X
activate(n__cons(X1,X2)) -> cons(activate(X1),X2)
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
cons(X1,X2) -> n__cons(X1,X2)
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
s(X) -> n__s(X)
- Signature:
{2nd/1,activate/1,cons/2,from/1,s/1} / {n__cons/2,n__from/1,n__s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {2nd,activate,cons,from,s} and constructors {n__cons
,n__from,n__s}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
2nd(cons(X,n__cons(Y,Z))) -> activate(Y)
activate(X) -> X
activate(n__cons(X1,X2)) -> cons(activate(X1),X2)
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
cons(X1,X2) -> n__cons(X1,X2)
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
s(X) -> n__s(X)
- Signature:
{2nd/1,activate/1,cons/2,from/1,s/1} / {n__cons/2,n__from/1,n__s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {2nd,activate,cons,from,s} and constructors {n__cons
,n__from,n__s}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
activate(x){x -> n__cons(x,y)} =
activate(n__cons(x,y)) ->^+ cons(activate(x),y)
= C[activate(x) = activate(x){}]
** Step 1.b:1: InnermostRuleRemoval WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
2nd(cons(X,n__cons(Y,Z))) -> activate(Y)
activate(X) -> X
activate(n__cons(X1,X2)) -> cons(activate(X1),X2)
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
cons(X1,X2) -> n__cons(X1,X2)
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
s(X) -> n__s(X)
- Signature:
{2nd/1,activate/1,cons/2,from/1,s/1} / {n__cons/2,n__from/1,n__s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {2nd,activate,cons,from,s} and constructors {n__cons
,n__from,n__s}
+ Applied Processor:
InnermostRuleRemoval
+ Details:
Arguments of following rules are not normal-forms.
2nd(cons(X,n__cons(Y,Z))) -> activate(Y)
All above mentioned rules can be savely removed.
** Step 1.b:2: MI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
activate(X) -> X
activate(n__cons(X1,X2)) -> cons(activate(X1),X2)
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
cons(X1,X2) -> n__cons(X1,X2)
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
s(X) -> n__s(X)
- Signature:
{2nd/1,activate/1,cons/2,from/1,s/1} / {n__cons/2,n__from/1,n__s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {2nd,activate,cons,from,s} and constructors {n__cons
,n__from,n__s}
+ Applied Processor:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 1, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity Nothing)):
The following argument positions are considered usable:
uargs(cons) = {1},
uargs(from) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{2nd,activate,cons,from,s}
TcT has computed the following interpretation:
p(2nd) = [1] x_1 + [2]
p(activate) = [8] x_1 + [0]
p(cons) = [1] x_1 + [11]
p(from) = [1] x_1 + [13]
p(n__cons) = [1] x_1 + [2]
p(n__from) = [1] x_1 + [2]
p(n__s) = [1] x_1 + [1]
p(s) = [1] x_1 + [1]
Following rules are strictly oriented:
activate(n__cons(X1,X2)) = [8] X1 + [16]
> [8] X1 + [11]
= cons(activate(X1),X2)
activate(n__from(X)) = [8] X + [16]
> [8] X + [13]
= from(activate(X))
activate(n__s(X)) = [8] X + [8]
> [8] X + [1]
= s(activate(X))
cons(X1,X2) = [1] X1 + [11]
> [1] X1 + [2]
= n__cons(X1,X2)
from(X) = [1] X + [13]
> [1] X + [11]
= cons(X,n__from(n__s(X)))
from(X) = [1] X + [13]
> [1] X + [2]
= n__from(X)
Following rules are (at-least) weakly oriented:
activate(X) = [8] X + [0]
>= [1] X + [0]
= X
s(X) = [1] X + [1]
>= [1] X + [1]
= n__s(X)
** Step 1.b:3: MI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
activate(X) -> X
s(X) -> n__s(X)
- Weak TRS:
activate(n__cons(X1,X2)) -> cons(activate(X1),X2)
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
cons(X1,X2) -> n__cons(X1,X2)
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
- Signature:
{2nd/1,activate/1,cons/2,from/1,s/1} / {n__cons/2,n__from/1,n__s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {2nd,activate,cons,from,s} and constructors {n__cons
,n__from,n__s}
+ Applied Processor:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 1, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity Nothing)):
The following argument positions are considered usable:
uargs(cons) = {1},
uargs(from) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{2nd,activate,cons,from,s}
TcT has computed the following interpretation:
p(2nd) = [2] x_1 + [0]
p(activate) = [1] x_1 + [8]
p(cons) = [1] x_1 + [0]
p(from) = [1] x_1 + [0]
p(n__cons) = [1] x_1 + [0]
p(n__from) = [1] x_1 + [0]
p(n__s) = [1] x_1 + [8]
p(s) = [1] x_1 + [8]
Following rules are strictly oriented:
activate(X) = [1] X + [8]
> [1] X + [0]
= X
Following rules are (at-least) weakly oriented:
activate(n__cons(X1,X2)) = [1] X1 + [8]
>= [1] X1 + [8]
= cons(activate(X1),X2)
activate(n__from(X)) = [1] X + [8]
>= [1] X + [8]
= from(activate(X))
activate(n__s(X)) = [1] X + [16]
>= [1] X + [16]
= s(activate(X))
cons(X1,X2) = [1] X1 + [0]
>= [1] X1 + [0]
= n__cons(X1,X2)
from(X) = [1] X + [0]
>= [1] X + [0]
= cons(X,n__from(n__s(X)))
from(X) = [1] X + [0]
>= [1] X + [0]
= n__from(X)
s(X) = [1] X + [8]
>= [1] X + [8]
= n__s(X)
** Step 1.b:4: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
s(X) -> n__s(X)
- Weak TRS:
activate(X) -> X
activate(n__cons(X1,X2)) -> cons(activate(X1),X2)
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
cons(X1,X2) -> n__cons(X1,X2)
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
- Signature:
{2nd/1,activate/1,cons/2,from/1,s/1} / {n__cons/2,n__from/1,n__s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {2nd,activate,cons,from,s} and constructors {n__cons
,n__from,n__s}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(cons) = {1},
uargs(from) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(2nd) = [0]
p(activate) = [2] x1 + [0]
p(cons) = [1] x1 + [8]
p(from) = [1] x1 + [8]
p(n__cons) = [1] x1 + [6]
p(n__from) = [1] x1 + [4]
p(n__s) = [1] x1 + [4]
p(s) = [1] x1 + [6]
Following rules are strictly oriented:
s(X) = [1] X + [6]
> [1] X + [4]
= n__s(X)
Following rules are (at-least) weakly oriented:
activate(X) = [2] X + [0]
>= [1] X + [0]
= X
activate(n__cons(X1,X2)) = [2] X1 + [12]
>= [2] X1 + [8]
= cons(activate(X1),X2)
activate(n__from(X)) = [2] X + [8]
>= [2] X + [8]
= from(activate(X))
activate(n__s(X)) = [2] X + [8]
>= [2] X + [6]
= s(activate(X))
cons(X1,X2) = [1] X1 + [8]
>= [1] X1 + [6]
= n__cons(X1,X2)
from(X) = [1] X + [8]
>= [1] X + [8]
= cons(X,n__from(n__s(X)))
from(X) = [1] X + [8]
>= [1] X + [4]
= n__from(X)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:5: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
activate(X) -> X
activate(n__cons(X1,X2)) -> cons(activate(X1),X2)
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
cons(X1,X2) -> n__cons(X1,X2)
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
s(X) -> n__s(X)
- Signature:
{2nd/1,activate/1,cons/2,from/1,s/1} / {n__cons/2,n__from/1,n__s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {2nd,activate,cons,from,s} and constructors {n__cons
,n__from,n__s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(Omega(n^1),O(n^1))