* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1))
+ Considered Problem:
- Strict TRS:
activate(X) -> X
activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
first(X1,X2) -> n__first(X1,X2)
first(0(),X) -> nil()
first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
s(X) -> n__s(X)
- Signature:
{activate/1,first/2,from/1,s/1} / {0/0,cons/2,n__first/2,n__from/1,n__s/1,nil/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate,first,from,s} and constructors {0,cons,n__first
,n__from,n__s,nil}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
activate(X) -> X
activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
first(X1,X2) -> n__first(X1,X2)
first(0(),X) -> nil()
first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
s(X) -> n__s(X)
- Signature:
{activate/1,first/2,from/1,s/1} / {0/0,cons/2,n__first/2,n__from/1,n__s/1,nil/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate,first,from,s} and constructors {0,cons,n__first
,n__from,n__s,nil}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
activate(x){x -> n__first(x,y)} =
activate(n__first(x,y)) ->^+ first(activate(x),activate(y))
= C[activate(x) = activate(x){}]
** Step 1.b:1: InnermostRuleRemoval WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
activate(X) -> X
activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
first(X1,X2) -> n__first(X1,X2)
first(0(),X) -> nil()
first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
s(X) -> n__s(X)
- Signature:
{activate/1,first/2,from/1,s/1} / {0/0,cons/2,n__first/2,n__from/1,n__s/1,nil/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate,first,from,s} and constructors {0,cons,n__first
,n__from,n__s,nil}
+ Applied Processor:
InnermostRuleRemoval
+ Details:
Arguments of following rules are not normal-forms.
first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
All above mentioned rules can be savely removed.
** Step 1.b:2: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
activate(X) -> X
activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
first(X1,X2) -> n__first(X1,X2)
first(0(),X) -> nil()
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
s(X) -> n__s(X)
- Signature:
{activate/1,first/2,from/1,s/1} / {0/0,cons/2,n__first/2,n__from/1,n__s/1,nil/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate,first,from,s} and constructors {0,cons,n__first
,n__from,n__s,nil}
+ 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(first) = {1,2},
uargs(from) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [3]
p(activate) = [2] x1 + [0]
p(cons) = [1] x2 + [0]
p(first) = [1] x1 + [1] x2 + [0]
p(from) = [1] x1 + [1]
p(n__first) = [1] x1 + [1] x2 + [0]
p(n__from) = [1] x1 + [0]
p(n__s) = [1] x1 + [0]
p(nil) = [0]
p(s) = [1] x1 + [1]
Following rules are strictly oriented:
first(0(),X) = [1] X + [3]
> [0]
= nil()
from(X) = [1] X + [1]
> [1] X + [0]
= cons(X,n__from(n__s(X)))
from(X) = [1] X + [1]
> [1] X + [0]
= n__from(X)
s(X) = [1] X + [1]
> [1] X + [0]
= n__s(X)
Following rules are (at-least) weakly oriented:
activate(X) = [2] X + [0]
>= [1] X + [0]
= X
activate(n__first(X1,X2)) = [2] X1 + [2] X2 + [0]
>= [2] X1 + [2] X2 + [0]
= first(activate(X1),activate(X2))
activate(n__from(X)) = [2] X + [0]
>= [2] X + [1]
= from(activate(X))
activate(n__s(X)) = [2] X + [0]
>= [2] X + [1]
= s(activate(X))
first(X1,X2) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= n__first(X1,X2)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:3: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
activate(X) -> X
activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
first(X1,X2) -> n__first(X1,X2)
- Weak TRS:
first(0(),X) -> nil()
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
s(X) -> n__s(X)
- Signature:
{activate/1,first/2,from/1,s/1} / {0/0,cons/2,n__first/2,n__from/1,n__s/1,nil/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate,first,from,s} and constructors {0,cons,n__first
,n__from,n__s,nil}
+ 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(first) = {1,2},
uargs(from) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(activate) = [10] x1 + [0]
p(cons) = [1] x2 + [0]
p(first) = [1] x1 + [1] x2 + [1]
p(from) = [1] x1 + [3]
p(n__first) = [1] x1 + [1] x2 + [0]
p(n__from) = [1] x1 + [3]
p(n__s) = [1] x1 + [0]
p(nil) = [1]
p(s) = [1] x1 + [0]
Following rules are strictly oriented:
activate(n__from(X)) = [10] X + [30]
> [10] X + [3]
= from(activate(X))
first(X1,X2) = [1] X1 + [1] X2 + [1]
> [1] X1 + [1] X2 + [0]
= n__first(X1,X2)
Following rules are (at-least) weakly oriented:
activate(X) = [10] X + [0]
>= [1] X + [0]
= X
activate(n__first(X1,X2)) = [10] X1 + [10] X2 + [0]
>= [10] X1 + [10] X2 + [1]
= first(activate(X1),activate(X2))
activate(n__s(X)) = [10] X + [0]
>= [10] X + [0]
= s(activate(X))
first(0(),X) = [1] X + [1]
>= [1]
= nil()
from(X) = [1] X + [3]
>= [1] X + [3]
= cons(X,n__from(n__s(X)))
from(X) = [1] X + [3]
>= [1] X + [3]
= n__from(X)
s(X) = [1] X + [0]
>= [1] X + [0]
= n__s(X)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:4: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
activate(X) -> X
activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
- Weak TRS:
activate(n__from(X)) -> from(activate(X))
first(X1,X2) -> n__first(X1,X2)
first(0(),X) -> nil()
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
s(X) -> n__s(X)
- Signature:
{activate/1,first/2,from/1,s/1} / {0/0,cons/2,n__first/2,n__from/1,n__s/1,nil/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate,first,from,s} and constructors {0,cons,n__first
,n__from,n__s,nil}
+ 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(first) = {1,2},
uargs(from) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(activate) = [3] x1 + [0]
p(cons) = [1] x2 + [1]
p(first) = [1] x1 + [1] x2 + [0]
p(from) = [1] x1 + [3]
p(n__first) = [1] x1 + [1] x2 + [0]
p(n__from) = [1] x1 + [1]
p(n__s) = [1] x1 + [1]
p(nil) = [0]
p(s) = [1] x1 + [1]
Following rules are strictly oriented:
activate(n__s(X)) = [3] X + [3]
> [3] X + [1]
= s(activate(X))
Following rules are (at-least) weakly oriented:
activate(X) = [3] X + [0]
>= [1] X + [0]
= X
activate(n__first(X1,X2)) = [3] X1 + [3] X2 + [0]
>= [3] X1 + [3] X2 + [0]
= first(activate(X1),activate(X2))
activate(n__from(X)) = [3] X + [3]
>= [3] X + [3]
= from(activate(X))
first(X1,X2) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= n__first(X1,X2)
first(0(),X) = [1] X + [0]
>= [0]
= nil()
from(X) = [1] X + [3]
>= [1] X + [3]
= cons(X,n__from(n__s(X)))
from(X) = [1] X + [3]
>= [1] X + [1]
= n__from(X)
s(X) = [1] X + [1]
>= [1] X + [1]
= n__s(X)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:5: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
activate(X) -> X
activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
- Weak TRS:
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
first(X1,X2) -> n__first(X1,X2)
first(0(),X) -> nil()
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
s(X) -> n__s(X)
- Signature:
{activate/1,first/2,from/1,s/1} / {0/0,cons/2,n__first/2,n__from/1,n__s/1,nil/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate,first,from,s} and constructors {0,cons,n__first
,n__from,n__s,nil}
+ 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(first) = {1,2},
uargs(from) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [3]
p(activate) = [11] x1 + [0]
p(cons) = [1] x2 + [10]
p(first) = [1] x1 + [1] x2 + [2]
p(from) = [1] x1 + [11]
p(n__first) = [1] x1 + [1] x2 + [1]
p(n__from) = [1] x1 + [1]
p(n__s) = [1] x1 + [0]
p(nil) = [5]
p(s) = [1] x1 + [0]
Following rules are strictly oriented:
activate(n__first(X1,X2)) = [11] X1 + [11] X2 + [11]
> [11] X1 + [11] X2 + [2]
= first(activate(X1),activate(X2))
Following rules are (at-least) weakly oriented:
activate(X) = [11] X + [0]
>= [1] X + [0]
= X
activate(n__from(X)) = [11] X + [11]
>= [11] X + [11]
= from(activate(X))
activate(n__s(X)) = [11] X + [0]
>= [11] X + [0]
= s(activate(X))
first(X1,X2) = [1] X1 + [1] X2 + [2]
>= [1] X1 + [1] X2 + [1]
= n__first(X1,X2)
first(0(),X) = [1] X + [5]
>= [5]
= nil()
from(X) = [1] X + [11]
>= [1] X + [11]
= cons(X,n__from(n__s(X)))
from(X) = [1] X + [11]
>= [1] X + [1]
= n__from(X)
s(X) = [1] X + [0]
>= [1] X + [0]
= n__s(X)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:6: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
activate(X) -> X
- Weak TRS:
activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
first(X1,X2) -> n__first(X1,X2)
first(0(),X) -> nil()
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
s(X) -> n__s(X)
- Signature:
{activate/1,first/2,from/1,s/1} / {0/0,cons/2,n__first/2,n__from/1,n__s/1,nil/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate,first,from,s} and constructors {0,cons,n__first
,n__from,n__s,nil}
+ 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(first) = {1,2},
uargs(from) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [2]
p(activate) = [3] x1 + [1]
p(cons) = [1] x2 + [0]
p(first) = [1] x1 + [1] x2 + [2]
p(from) = [1] x1 + [0]
p(n__first) = [1] x1 + [1] x2 + [1]
p(n__from) = [1] x1 + [0]
p(n__s) = [1] x1 + [0]
p(nil) = [4]
p(s) = [1] x1 + [0]
Following rules are strictly oriented:
activate(X) = [3] X + [1]
> [1] X + [0]
= X
Following rules are (at-least) weakly oriented:
activate(n__first(X1,X2)) = [3] X1 + [3] X2 + [4]
>= [3] X1 + [3] X2 + [4]
= first(activate(X1),activate(X2))
activate(n__from(X)) = [3] X + [1]
>= [3] X + [1]
= from(activate(X))
activate(n__s(X)) = [3] X + [1]
>= [3] X + [1]
= s(activate(X))
first(X1,X2) = [1] X1 + [1] X2 + [2]
>= [1] X1 + [1] X2 + [1]
= n__first(X1,X2)
first(0(),X) = [1] X + [4]
>= [4]
= nil()
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 + [0]
>= [1] X + [0]
= n__s(X)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:7: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
activate(X) -> X
activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
first(X1,X2) -> n__first(X1,X2)
first(0(),X) -> nil()
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
s(X) -> n__s(X)
- Signature:
{activate/1,first/2,from/1,s/1} / {0/0,cons/2,n__first/2,n__from/1,n__s/1,nil/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate,first,from,s} and constructors {0,cons,n__first
,n__from,n__s,nil}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(Omega(n^1),O(n^1))