* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1))
+ Considered Problem:
- Strict TRS:
2ndsneg(0(),Z) -> rnil()
2ndsneg(s(N),cons(X,n__cons(Y,Z))) -> rcons(negrecip(activate(Y)),2ndspos(N,activate(Z)))
2ndspos(0(),Z) -> rnil()
2ndspos(s(N),cons(X,n__cons(Y,Z))) -> rcons(posrecip(activate(Y)),2ndsneg(N,activate(Z)))
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)
pi(X) -> 2ndspos(X,from(0()))
plus(0(),Y) -> Y
plus(s(X),Y) -> s(plus(X,Y))
s(X) -> n__s(X)
square(X) -> times(X,X)
times(0(),Y) -> 0()
times(s(X),Y) -> plus(Y,times(X,Y))
- Signature:
{2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,s/1,square/1,times/2} / {0/0,n__cons/2,n__from/1
,n__s/1,negrecip/1,posrecip/1,rcons/2,rnil/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {2ndsneg,2ndspos,activate,cons,from,pi,plus,s,square
,times} and constructors {0,n__cons,n__from,n__s,negrecip,posrecip,rcons,rnil}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
2ndsneg(0(),Z) -> rnil()
2ndsneg(s(N),cons(X,n__cons(Y,Z))) -> rcons(negrecip(activate(Y)),2ndspos(N,activate(Z)))
2ndspos(0(),Z) -> rnil()
2ndspos(s(N),cons(X,n__cons(Y,Z))) -> rcons(posrecip(activate(Y)),2ndsneg(N,activate(Z)))
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)
pi(X) -> 2ndspos(X,from(0()))
plus(0(),Y) -> Y
plus(s(X),Y) -> s(plus(X,Y))
s(X) -> n__s(X)
square(X) -> times(X,X)
times(0(),Y) -> 0()
times(s(X),Y) -> plus(Y,times(X,Y))
- Signature:
{2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,s/1,square/1,times/2} / {0/0,n__cons/2,n__from/1
,n__s/1,negrecip/1,posrecip/1,rcons/2,rnil/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {2ndsneg,2ndspos,activate,cons,from,pi,plus,s,square
,times} and constructors {0,n__cons,n__from,n__s,negrecip,posrecip,rcons,rnil}
+ 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:
2ndsneg(0(),Z) -> rnil()
2ndsneg(s(N),cons(X,n__cons(Y,Z))) -> rcons(negrecip(activate(Y)),2ndspos(N,activate(Z)))
2ndspos(0(),Z) -> rnil()
2ndspos(s(N),cons(X,n__cons(Y,Z))) -> rcons(posrecip(activate(Y)),2ndsneg(N,activate(Z)))
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)
pi(X) -> 2ndspos(X,from(0()))
plus(0(),Y) -> Y
plus(s(X),Y) -> s(plus(X,Y))
s(X) -> n__s(X)
square(X) -> times(X,X)
times(0(),Y) -> 0()
times(s(X),Y) -> plus(Y,times(X,Y))
- Signature:
{2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,s/1,square/1,times/2} / {0/0,n__cons/2,n__from/1
,n__s/1,negrecip/1,posrecip/1,rcons/2,rnil/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {2ndsneg,2ndspos,activate,cons,from,pi,plus,s,square
,times} and constructors {0,n__cons,n__from,n__s,negrecip,posrecip,rcons,rnil}
+ Applied Processor:
InnermostRuleRemoval
+ Details:
Arguments of following rules are not normal-forms.
2ndsneg(s(N),cons(X,n__cons(Y,Z))) -> rcons(negrecip(activate(Y)),2ndspos(N,activate(Z)))
2ndspos(s(N),cons(X,n__cons(Y,Z))) -> rcons(posrecip(activate(Y)),2ndsneg(N,activate(Z)))
plus(s(X),Y) -> s(plus(X,Y))
times(s(X),Y) -> plus(Y,times(X,Y))
All above mentioned rules can be savely removed.
** Step 1.b:2: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
2ndsneg(0(),Z) -> rnil()
2ndspos(0(),Z) -> rnil()
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)
pi(X) -> 2ndspos(X,from(0()))
plus(0(),Y) -> Y
s(X) -> n__s(X)
square(X) -> times(X,X)
times(0(),Y) -> 0()
- Signature:
{2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,s/1,square/1,times/2} / {0/0,n__cons/2,n__from/1
,n__s/1,negrecip/1,posrecip/1,rcons/2,rnil/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {2ndsneg,2ndspos,activate,cons,from,pi,plus,s,square
,times} and constructors {0,n__cons,n__from,n__s,negrecip,posrecip,rcons,rnil}
+ 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(2ndspos) = {2},
uargs(cons) = {1},
uargs(from) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(2ndsneg) = [0]
p(2ndspos) = [1] x2 + [0]
p(activate) = [2] x1 + [0]
p(cons) = [1] x1 + [0]
p(from) = [1] x1 + [0]
p(n__cons) = [1] x1 + [1]
p(n__from) = [1] x1 + [0]
p(n__s) = [1] x1 + [0]
p(negrecip) = [1] x1 + [0]
p(pi) = [0]
p(plus) = [2] x2 + [0]
p(posrecip) = [1] x1 + [0]
p(rcons) = [1] x1 + [1] x2 + [0]
p(rnil) = [0]
p(s) = [1] x1 + [0]
p(square) = [0]
p(times) = [1]
Following rules are strictly oriented:
activate(n__cons(X1,X2)) = [2] X1 + [2]
> [2] X1 + [0]
= cons(activate(X1),X2)
times(0(),Y) = [1]
> [0]
= 0()
Following rules are (at-least) weakly oriented:
2ndsneg(0(),Z) = [0]
>= [0]
= rnil()
2ndspos(0(),Z) = [1] Z + [0]
>= [0]
= rnil()
activate(X) = [2] X + [0]
>= [1] X + [0]
= X
activate(n__from(X)) = [2] X + [0]
>= [2] X + [0]
= from(activate(X))
activate(n__s(X)) = [2] X + [0]
>= [2] X + [0]
= s(activate(X))
cons(X1,X2) = [1] X1 + [0]
>= [1] X1 + [1]
= 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)
pi(X) = [0]
>= [0]
= 2ndspos(X,from(0()))
plus(0(),Y) = [2] Y + [0]
>= [1] Y + [0]
= Y
s(X) = [1] X + [0]
>= [1] X + [0]
= n__s(X)
square(X) = [0]
>= [1]
= times(X,X)
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:
2ndsneg(0(),Z) -> rnil()
2ndspos(0(),Z) -> rnil()
activate(X) -> X
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)
pi(X) -> 2ndspos(X,from(0()))
plus(0(),Y) -> Y
s(X) -> n__s(X)
square(X) -> times(X,X)
- Weak TRS:
activate(n__cons(X1,X2)) -> cons(activate(X1),X2)
times(0(),Y) -> 0()
- Signature:
{2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,s/1,square/1,times/2} / {0/0,n__cons/2,n__from/1
,n__s/1,negrecip/1,posrecip/1,rcons/2,rnil/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {2ndsneg,2ndspos,activate,cons,from,pi,plus,s,square
,times} and constructors {0,n__cons,n__from,n__s,negrecip,posrecip,rcons,rnil}
+ 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(2ndspos) = {2},
uargs(cons) = {1},
uargs(from) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [1]
p(2ndsneg) = [0]
p(2ndspos) = [1] x2 + [0]
p(activate) = [10] x1 + [0]
p(cons) = [1] x1 + [0]
p(from) = [1] x1 + [3]
p(n__cons) = [1] x1 + [0]
p(n__from) = [1] x1 + [0]
p(n__s) = [1] x1 + [0]
p(negrecip) = [1] x1 + [0]
p(pi) = [0]
p(plus) = [2] x2 + [0]
p(posrecip) = [1] x1 + [0]
p(rcons) = [1] x1 + [1] x2 + [0]
p(rnil) = [0]
p(s) = [1] x1 + [0]
p(square) = [1] x1 + [1]
p(times) = [1] x1 + [0]
Following rules are strictly oriented:
from(X) = [1] X + [3]
> [1] X + [0]
= cons(X,n__from(n__s(X)))
from(X) = [1] X + [3]
> [1] X + [0]
= n__from(X)
square(X) = [1] X + [1]
> [1] X + [0]
= times(X,X)
Following rules are (at-least) weakly oriented:
2ndsneg(0(),Z) = [0]
>= [0]
= rnil()
2ndspos(0(),Z) = [1] Z + [0]
>= [0]
= rnil()
activate(X) = [10] X + [0]
>= [1] X + [0]
= X
activate(n__cons(X1,X2)) = [10] X1 + [0]
>= [10] X1 + [0]
= cons(activate(X1),X2)
activate(n__from(X)) = [10] X + [0]
>= [10] X + [3]
= from(activate(X))
activate(n__s(X)) = [10] X + [0]
>= [10] X + [0]
= s(activate(X))
cons(X1,X2) = [1] X1 + [0]
>= [1] X1 + [0]
= n__cons(X1,X2)
pi(X) = [0]
>= [4]
= 2ndspos(X,from(0()))
plus(0(),Y) = [2] Y + [0]
>= [1] Y + [0]
= Y
s(X) = [1] X + [0]
>= [1] X + [0]
= n__s(X)
times(0(),Y) = [1]
>= [1]
= 0()
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:
2ndsneg(0(),Z) -> rnil()
2ndspos(0(),Z) -> rnil()
activate(X) -> X
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
cons(X1,X2) -> n__cons(X1,X2)
pi(X) -> 2ndspos(X,from(0()))
plus(0(),Y) -> Y
s(X) -> n__s(X)
- Weak TRS:
activate(n__cons(X1,X2)) -> cons(activate(X1),X2)
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
square(X) -> times(X,X)
times(0(),Y) -> 0()
- Signature:
{2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,s/1,square/1,times/2} / {0/0,n__cons/2,n__from/1
,n__s/1,negrecip/1,posrecip/1,rcons/2,rnil/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {2ndsneg,2ndspos,activate,cons,from,pi,plus,s,square
,times} and constructors {0,n__cons,n__from,n__s,negrecip,posrecip,rcons,rnil}
+ 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(2ndspos) = {2},
uargs(cons) = {1},
uargs(from) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(2ndsneg) = [0]
p(2ndspos) = [1] x2 + [0]
p(activate) = [10] x1 + [0]
p(cons) = [1] x1 + [0]
p(from) = [1] x1 + [0]
p(n__cons) = [1] x1 + [0]
p(n__from) = [1] x1 + [0]
p(n__s) = [1] x1 + [0]
p(negrecip) = [1] x1 + [0]
p(pi) = [0]
p(plus) = [2] x2 + [0]
p(posrecip) = [1] x1 + [0]
p(rcons) = [1] x1 + [1] x2 + [0]
p(rnil) = [0]
p(s) = [1] x1 + [1]
p(square) = [0]
p(times) = [0]
Following rules are strictly oriented:
s(X) = [1] X + [1]
> [1] X + [0]
= n__s(X)
Following rules are (at-least) weakly oriented:
2ndsneg(0(),Z) = [0]
>= [0]
= rnil()
2ndspos(0(),Z) = [1] Z + [0]
>= [0]
= rnil()
activate(X) = [10] X + [0]
>= [1] X + [0]
= X
activate(n__cons(X1,X2)) = [10] X1 + [0]
>= [10] X1 + [0]
= cons(activate(X1),X2)
activate(n__from(X)) = [10] X + [0]
>= [10] X + [0]
= from(activate(X))
activate(n__s(X)) = [10] X + [0]
>= [10] X + [1]
= 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)
pi(X) = [0]
>= [0]
= 2ndspos(X,from(0()))
plus(0(),Y) = [2] Y + [0]
>= [1] Y + [0]
= Y
square(X) = [0]
>= [0]
= times(X,X)
times(0(),Y) = [0]
>= [0]
= 0()
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:
2ndsneg(0(),Z) -> rnil()
2ndspos(0(),Z) -> rnil()
activate(X) -> X
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
cons(X1,X2) -> n__cons(X1,X2)
pi(X) -> 2ndspos(X,from(0()))
plus(0(),Y) -> Y
- Weak TRS:
activate(n__cons(X1,X2)) -> cons(activate(X1),X2)
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
s(X) -> n__s(X)
square(X) -> times(X,X)
times(0(),Y) -> 0()
- Signature:
{2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,s/1,square/1,times/2} / {0/0,n__cons/2,n__from/1
,n__s/1,negrecip/1,posrecip/1,rcons/2,rnil/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {2ndsneg,2ndspos,activate,cons,from,pi,plus,s,square
,times} and constructors {0,n__cons,n__from,n__s,negrecip,posrecip,rcons,rnil}
+ 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(2ndspos) = {2},
uargs(cons) = {1},
uargs(from) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [2]
p(2ndsneg) = [0]
p(2ndspos) = [2] x1 + [1] x2 + [0]
p(activate) = [8] x1 + [0]
p(cons) = [1] x1 + [6]
p(from) = [1] x1 + [6]
p(n__cons) = [1] x1 + [1]
p(n__from) = [1] x1 + [0]
p(n__s) = [1] x1 + [1]
p(negrecip) = [1] x1 + [0]
p(pi) = [2] x1 + [0]
p(plus) = [9] x1 + [2] x2 + [0]
p(posrecip) = [1] x1 + [0]
p(rcons) = [1] x1 + [1] x2 + [0]
p(rnil) = [0]
p(s) = [1] x1 + [1]
p(square) = [2]
p(times) = [2]
Following rules are strictly oriented:
2ndspos(0(),Z) = [1] Z + [4]
> [0]
= rnil()
activate(n__s(X)) = [8] X + [8]
> [8] X + [1]
= s(activate(X))
cons(X1,X2) = [1] X1 + [6]
> [1] X1 + [1]
= n__cons(X1,X2)
plus(0(),Y) = [2] Y + [18]
> [1] Y + [0]
= Y
Following rules are (at-least) weakly oriented:
2ndsneg(0(),Z) = [0]
>= [0]
= rnil()
activate(X) = [8] X + [0]
>= [1] X + [0]
= X
activate(n__cons(X1,X2)) = [8] X1 + [8]
>= [8] X1 + [6]
= cons(activate(X1),X2)
activate(n__from(X)) = [8] X + [0]
>= [8] X + [6]
= from(activate(X))
from(X) = [1] X + [6]
>= [1] X + [6]
= cons(X,n__from(n__s(X)))
from(X) = [1] X + [6]
>= [1] X + [0]
= n__from(X)
pi(X) = [2] X + [0]
>= [2] X + [8]
= 2ndspos(X,from(0()))
s(X) = [1] X + [1]
>= [1] X + [1]
= n__s(X)
square(X) = [2]
>= [2]
= times(X,X)
times(0(),Y) = [2]
>= [2]
= 0()
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:
2ndsneg(0(),Z) -> rnil()
activate(X) -> X
activate(n__from(X)) -> from(activate(X))
pi(X) -> 2ndspos(X,from(0()))
- Weak TRS:
2ndspos(0(),Z) -> rnil()
activate(n__cons(X1,X2)) -> cons(activate(X1),X2)
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)
plus(0(),Y) -> Y
s(X) -> n__s(X)
square(X) -> times(X,X)
times(0(),Y) -> 0()
- Signature:
{2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,s/1,square/1,times/2} / {0/0,n__cons/2,n__from/1
,n__s/1,negrecip/1,posrecip/1,rcons/2,rnil/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {2ndsneg,2ndspos,activate,cons,from,pi,plus,s,square
,times} and constructors {0,n__cons,n__from,n__s,negrecip,posrecip,rcons,rnil}
+ 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(2ndspos) = {2},
uargs(cons) = {1},
uargs(from) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(2ndsneg) = [1]
p(2ndspos) = [1] x2 + [0]
p(activate) = [1] x1 + [0]
p(cons) = [1] x1 + [0]
p(from) = [1] x1 + [0]
p(n__cons) = [1] x1 + [0]
p(n__from) = [1] x1 + [0]
p(n__s) = [1] x1 + [4]
p(negrecip) = [1] x1 + [0]
p(pi) = [2]
p(plus) = [1] x2 + [0]
p(posrecip) = [1]
p(rcons) = [1] x1 + [8]
p(rnil) = [0]
p(s) = [1] x1 + [4]
p(square) = [1] x1 + [2]
p(times) = [2]
Following rules are strictly oriented:
2ndsneg(0(),Z) = [1]
> [0]
= rnil()
pi(X) = [2]
> [0]
= 2ndspos(X,from(0()))
Following rules are (at-least) weakly oriented:
2ndspos(0(),Z) = [1] Z + [0]
>= [0]
= rnil()
activate(X) = [1] X + [0]
>= [1] X + [0]
= X
activate(n__cons(X1,X2)) = [1] X1 + [0]
>= [1] X1 + [0]
= cons(activate(X1),X2)
activate(n__from(X)) = [1] X + [0]
>= [1] X + [0]
= from(activate(X))
activate(n__s(X)) = [1] X + [4]
>= [1] X + [4]
= 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)
plus(0(),Y) = [1] Y + [0]
>= [1] Y + [0]
= Y
s(X) = [1] X + [4]
>= [1] X + [4]
= n__s(X)
square(X) = [1] X + [2]
>= [2]
= times(X,X)
times(0(),Y) = [2]
>= [0]
= 0()
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:7: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
activate(X) -> X
activate(n__from(X)) -> from(activate(X))
- Weak TRS:
2ndsneg(0(),Z) -> rnil()
2ndspos(0(),Z) -> rnil()
activate(n__cons(X1,X2)) -> cons(activate(X1),X2)
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)
pi(X) -> 2ndspos(X,from(0()))
plus(0(),Y) -> Y
s(X) -> n__s(X)
square(X) -> times(X,X)
times(0(),Y) -> 0()
- Signature:
{2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,s/1,square/1,times/2} / {0/0,n__cons/2,n__from/1
,n__s/1,negrecip/1,posrecip/1,rcons/2,rnil/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {2ndsneg,2ndspos,activate,cons,from,pi,plus,s,square
,times} and constructors {0,n__cons,n__from,n__s,negrecip,posrecip,rcons,rnil}
+ 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(2ndspos) = {2},
uargs(cons) = {1},
uargs(from) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(2ndsneg) = [0]
p(2ndspos) = [1] x2 + [6]
p(activate) = [8] x1 + [6]
p(cons) = [1] x1 + [0]
p(from) = [1] x1 + [9]
p(n__cons) = [1] x1 + [0]
p(n__from) = [1] x1 + [0]
p(n__s) = [1] x1 + [0]
p(negrecip) = [0]
p(pi) = [1] x1 + [15]
p(plus) = [2] x2 + [4]
p(posrecip) = [1] x1 + [4]
p(rcons) = [0]
p(rnil) = [0]
p(s) = [1] x1 + [0]
p(square) = [8] x1 + [1]
p(times) = [0]
Following rules are strictly oriented:
activate(X) = [8] X + [6]
> [1] X + [0]
= X
Following rules are (at-least) weakly oriented:
2ndsneg(0(),Z) = [0]
>= [0]
= rnil()
2ndspos(0(),Z) = [1] Z + [6]
>= [0]
= rnil()
activate(n__cons(X1,X2)) = [8] X1 + [6]
>= [8] X1 + [6]
= cons(activate(X1),X2)
activate(n__from(X)) = [8] X + [6]
>= [8] X + [15]
= from(activate(X))
activate(n__s(X)) = [8] X + [6]
>= [8] X + [6]
= s(activate(X))
cons(X1,X2) = [1] X1 + [0]
>= [1] X1 + [0]
= n__cons(X1,X2)
from(X) = [1] X + [9]
>= [1] X + [0]
= cons(X,n__from(n__s(X)))
from(X) = [1] X + [9]
>= [1] X + [0]
= n__from(X)
pi(X) = [1] X + [15]
>= [15]
= 2ndspos(X,from(0()))
plus(0(),Y) = [2] Y + [4]
>= [1] Y + [0]
= Y
s(X) = [1] X + [0]
>= [1] X + [0]
= n__s(X)
square(X) = [8] X + [1]
>= [0]
= times(X,X)
times(0(),Y) = [0]
>= [0]
= 0()
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:8: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
activate(n__from(X)) -> from(activate(X))
- Weak TRS:
2ndsneg(0(),Z) -> rnil()
2ndspos(0(),Z) -> rnil()
activate(X) -> X
activate(n__cons(X1,X2)) -> cons(activate(X1),X2)
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)
pi(X) -> 2ndspos(X,from(0()))
plus(0(),Y) -> Y
s(X) -> n__s(X)
square(X) -> times(X,X)
times(0(),Y) -> 0()
- Signature:
{2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,s/1,square/1,times/2} / {0/0,n__cons/2,n__from/1
,n__s/1,negrecip/1,posrecip/1,rcons/2,rnil/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {2ndsneg,2ndspos,activate,cons,from,pi,plus,s,square
,times} and constructors {0,n__cons,n__from,n__s,negrecip,posrecip,rcons,rnil}
+ 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(2ndspos) = {2},
uargs(cons) = {1},
uargs(from) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [7]
p(2ndsneg) = [3] x1 + [3]
p(2ndspos) = [1] x1 + [1] x2 + [4]
p(activate) = [8] x1 + [14]
p(cons) = [1] x1 + [2]
p(from) = [1] x1 + [2]
p(n__cons) = [1] x1 + [2]
p(n__from) = [1] x1 + [2]
p(n__s) = [1] x1 + [1]
p(negrecip) = [1]
p(pi) = [6] x1 + [13]
p(plus) = [1] x1 + [4] x2 + [0]
p(posrecip) = [1] x1 + [1]
p(rcons) = [1]
p(rnil) = [2]
p(s) = [1] x1 + [8]
p(square) = [9] x1 + [0]
p(times) = [3] x1 + [5] x2 + [0]
Following rules are strictly oriented:
activate(n__from(X)) = [8] X + [30]
> [8] X + [16]
= from(activate(X))
Following rules are (at-least) weakly oriented:
2ndsneg(0(),Z) = [24]
>= [2]
= rnil()
2ndspos(0(),Z) = [1] Z + [11]
>= [2]
= rnil()
activate(X) = [8] X + [14]
>= [1] X + [0]
= X
activate(n__cons(X1,X2)) = [8] X1 + [30]
>= [8] X1 + [16]
= cons(activate(X1),X2)
activate(n__s(X)) = [8] X + [22]
>= [8] X + [22]
= s(activate(X))
cons(X1,X2) = [1] X1 + [2]
>= [1] X1 + [2]
= n__cons(X1,X2)
from(X) = [1] X + [2]
>= [1] X + [2]
= cons(X,n__from(n__s(X)))
from(X) = [1] X + [2]
>= [1] X + [2]
= n__from(X)
pi(X) = [6] X + [13]
>= [1] X + [13]
= 2ndspos(X,from(0()))
plus(0(),Y) = [4] Y + [7]
>= [1] Y + [0]
= Y
s(X) = [1] X + [8]
>= [1] X + [1]
= n__s(X)
square(X) = [9] X + [0]
>= [8] X + [0]
= times(X,X)
times(0(),Y) = [5] Y + [21]
>= [7]
= 0()
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:9: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
2ndsneg(0(),Z) -> rnil()
2ndspos(0(),Z) -> rnil()
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)
pi(X) -> 2ndspos(X,from(0()))
plus(0(),Y) -> Y
s(X) -> n__s(X)
square(X) -> times(X,X)
times(0(),Y) -> 0()
- Signature:
{2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,s/1,square/1,times/2} / {0/0,n__cons/2,n__from/1
,n__s/1,negrecip/1,posrecip/1,rcons/2,rnil/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {2ndsneg,2ndspos,activate,cons,from,pi,plus,s,square
,times} and constructors {0,n__cons,n__from,n__s,negrecip,posrecip,rcons,rnil}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(Omega(n^1),O(n^1))