* Step 1: Sum WORST_CASE(Omega(n^1),O(n^2))
+ 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(X1,X2)
activate(n__from(X)) -> from(X)
cons(X1,X2) -> n__cons(X1,X2)
from(X) -> cons(X,n__from(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))
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,square/1,times/2} / {0/0,n__cons/2,n__from/1
,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {2ndsneg,2ndspos,activate,cons,from,pi,plus,square
,times} and constructors {0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s}
+ 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(X1,X2)
activate(n__from(X)) -> from(X)
cons(X1,X2) -> n__cons(X1,X2)
from(X) -> cons(X,n__from(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))
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,square/1,times/2} / {0/0,n__cons/2,n__from/1
,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {2ndsneg,2ndspos,activate,cons,from,pi,plus,square
,times} and constructors {0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
plus(x,y){x -> s(x)} =
plus(s(x),y) ->^+ s(plus(x,y))
= C[plus(x,y) = plus(x,y){}]
** Step 1.b:1: InnermostRuleRemoval WORST_CASE(?,O(n^2))
+ 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(X1,X2)
activate(n__from(X)) -> from(X)
cons(X1,X2) -> n__cons(X1,X2)
from(X) -> cons(X,n__from(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))
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,square/1,times/2} / {0/0,n__cons/2,n__from/1
,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {2ndsneg,2ndspos,activate,cons,from,pi,plus,square
,times} and constructors {0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s}
+ 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)))
All above mentioned rules can be savely removed.
** Step 1.b:2: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
2ndsneg(0(),Z) -> rnil()
2ndspos(0(),Z) -> rnil()
activate(X) -> X
activate(n__cons(X1,X2)) -> cons(X1,X2)
activate(n__from(X)) -> from(X)
cons(X1,X2) -> n__cons(X1,X2)
from(X) -> cons(X,n__from(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))
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,square/1,times/2} / {0/0,n__cons/2,n__from/1
,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {2ndsneg,2ndspos,activate,cons,from,pi,plus,square
,times} and constructors {0,n__cons,n__from,negrecip,posrecip,rcons,rnil,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(2ndspos) = {2},
uargs(plus) = {2},
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) = [8] x1 + [3]
p(cons) = [1]
p(from) = [8] x1 + [0]
p(n__cons) = [0]
p(n__from) = [1] x1 + [2]
p(negrecip) = [1] x1 + [0]
p(pi) = [0]
p(plus) = [1] x2 + [1]
p(posrecip) = [1] x1 + [0]
p(rcons) = [1] x1 + [1] x2 + [0]
p(rnil) = [0]
p(s) = [1] x1 + [0]
p(square) = [1] x1 + [0]
p(times) = [1] x1 + [9]
Following rules are strictly oriented:
activate(X) = [8] X + [3]
> [1] X + [0]
= X
activate(n__cons(X1,X2)) = [3]
> [1]
= cons(X1,X2)
activate(n__from(X)) = [8] X + [19]
> [8] X + [0]
= from(X)
cons(X1,X2) = [1]
> [0]
= n__cons(X1,X2)
plus(0(),Y) = [1] Y + [1]
> [1] Y + [0]
= Y
times(0(),Y) = [9]
> [0]
= 0()
Following rules are (at-least) weakly oriented:
2ndsneg(0(),Z) = [0]
>= [0]
= rnil()
2ndspos(0(),Z) = [1] Z + [0]
>= [0]
= rnil()
from(X) = [8] X + [0]
>= [1]
= cons(X,n__from(s(X)))
from(X) = [8] X + [0]
>= [1] X + [2]
= n__from(X)
pi(X) = [0]
>= [0]
= 2ndspos(X,from(0()))
plus(s(X),Y) = [1] Y + [1]
>= [1] Y + [1]
= s(plus(X,Y))
square(X) = [1] X + [0]
>= [1] X + [9]
= times(X,X)
times(s(X),Y) = [1] X + [9]
>= [1] X + [10]
= plus(Y,times(X,Y))
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^2))
+ Considered Problem:
- Strict TRS:
2ndsneg(0(),Z) -> rnil()
2ndspos(0(),Z) -> rnil()
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
pi(X) -> 2ndspos(X,from(0()))
plus(s(X),Y) -> s(plus(X,Y))
square(X) -> times(X,X)
times(s(X),Y) -> plus(Y,times(X,Y))
- Weak TRS:
activate(X) -> X
activate(n__cons(X1,X2)) -> cons(X1,X2)
activate(n__from(X)) -> from(X)
cons(X1,X2) -> n__cons(X1,X2)
plus(0(),Y) -> Y
times(0(),Y) -> 0()
- Signature:
{2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,square/1,times/2} / {0/0,n__cons/2,n__from/1
,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {2ndsneg,2ndspos,activate,cons,from,pi,plus,square
,times} and constructors {0,n__cons,n__from,negrecip,posrecip,rcons,rnil,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(2ndspos) = {2},
uargs(plus) = {2},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [4]
p(2ndsneg) = [0]
p(2ndspos) = [1] x2 + [0]
p(activate) = [6] x1 + [0]
p(cons) = [4] x1 + [0]
p(from) = [4] x1 + [10]
p(n__cons) = [1] x1 + [0]
p(n__from) = [1] x1 + [4]
p(negrecip) = [1] x1 + [0]
p(pi) = [4] x1 + [0]
p(plus) = [1] 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 + [0]
p(times) = [12]
Following rules are strictly oriented:
from(X) = [4] X + [10]
> [4] X + [0]
= cons(X,n__from(s(X)))
from(X) = [4] X + [10]
> [1] X + [4]
= n__from(X)
Following rules are (at-least) weakly oriented:
2ndsneg(0(),Z) = [0]
>= [0]
= rnil()
2ndspos(0(),Z) = [1] Z + [0]
>= [0]
= rnil()
activate(X) = [6] X + [0]
>= [1] X + [0]
= X
activate(n__cons(X1,X2)) = [6] X1 + [0]
>= [4] X1 + [0]
= cons(X1,X2)
activate(n__from(X)) = [6] X + [24]
>= [4] X + [10]
= from(X)
cons(X1,X2) = [4] X1 + [0]
>= [1] X1 + [0]
= n__cons(X1,X2)
pi(X) = [4] X + [0]
>= [26]
= 2ndspos(X,from(0()))
plus(0(),Y) = [1] Y + [0]
>= [1] Y + [0]
= Y
plus(s(X),Y) = [1] Y + [0]
>= [1] Y + [0]
= s(plus(X,Y))
square(X) = [1] X + [0]
>= [12]
= times(X,X)
times(0(),Y) = [12]
>= [4]
= 0()
times(s(X),Y) = [12]
>= [12]
= plus(Y,times(X,Y))
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^2))
+ Considered Problem:
- Strict TRS:
2ndsneg(0(),Z) -> rnil()
2ndspos(0(),Z) -> rnil()
pi(X) -> 2ndspos(X,from(0()))
plus(s(X),Y) -> s(plus(X,Y))
square(X) -> times(X,X)
times(s(X),Y) -> plus(Y,times(X,Y))
- Weak TRS:
activate(X) -> X
activate(n__cons(X1,X2)) -> cons(X1,X2)
activate(n__from(X)) -> from(X)
cons(X1,X2) -> n__cons(X1,X2)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
plus(0(),Y) -> Y
times(0(),Y) -> 0()
- Signature:
{2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,square/1,times/2} / {0/0,n__cons/2,n__from/1
,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {2ndsneg,2ndspos,activate,cons,from,pi,plus,square
,times} and constructors {0,n__cons,n__from,negrecip,posrecip,rcons,rnil,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(2ndspos) = {2},
uargs(plus) = {2},
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] x2 + [0]
p(from) = [0]
p(n__cons) = [1] x2 + [0]
p(n__from) = [0]
p(negrecip) = [1] x1 + [0]
p(pi) = [0]
p(plus) = [1] x2 + [0]
p(posrecip) = [1] x1 + [0]
p(rcons) = [1] x1 + [1] x2 + [0]
p(rnil) = [0]
p(s) = [1] x1 + [9]
p(square) = [2] x1 + [0]
p(times) = [2] x1 + [0]
Following rules are strictly oriented:
2ndsneg(0(),Z) = [1]
> [0]
= rnil()
times(s(X),Y) = [2] X + [18]
> [2] X + [0]
= plus(Y,times(X,Y))
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] X2 + [0]
>= [1] X2 + [0]
= cons(X1,X2)
activate(n__from(X)) = [0]
>= [0]
= from(X)
cons(X1,X2) = [1] X2 + [0]
>= [1] X2 + [0]
= n__cons(X1,X2)
from(X) = [0]
>= [0]
= cons(X,n__from(s(X)))
from(X) = [0]
>= [0]
= n__from(X)
pi(X) = [0]
>= [0]
= 2ndspos(X,from(0()))
plus(0(),Y) = [1] Y + [0]
>= [1] Y + [0]
= Y
plus(s(X),Y) = [1] Y + [0]
>= [1] Y + [9]
= s(plus(X,Y))
square(X) = [2] X + [0]
>= [2] X + [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^2))
+ Considered Problem:
- Strict TRS:
2ndspos(0(),Z) -> rnil()
pi(X) -> 2ndspos(X,from(0()))
plus(s(X),Y) -> s(plus(X,Y))
square(X) -> times(X,X)
- Weak TRS:
2ndsneg(0(),Z) -> rnil()
activate(X) -> X
activate(n__cons(X1,X2)) -> cons(X1,X2)
activate(n__from(X)) -> from(X)
cons(X1,X2) -> n__cons(X1,X2)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
plus(0(),Y) -> Y
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,square/1,times/2} / {0/0,n__cons/2,n__from/1
,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {2ndsneg,2ndspos,activate,cons,from,pi,plus,square
,times} and constructors {0,n__cons,n__from,negrecip,posrecip,rcons,rnil,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(2ndspos) = {2},
uargs(plus) = {2},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [3]
p(2ndsneg) = [4] x1 + [0]
p(2ndspos) = [1] x2 + [0]
p(activate) = [1] x1 + [4]
p(cons) = [2]
p(from) = [1] x1 + [4]
p(n__cons) = [0]
p(n__from) = [1] x1 + [0]
p(negrecip) = [1]
p(pi) = [2] x1 + [0]
p(plus) = [1] x2 + [6]
p(posrecip) = [1]
p(rcons) = [1] x1 + [1] x2 + [2]
p(rnil) = [0]
p(s) = [1] x1 + [5]
p(square) = [4] x1 + [4]
p(times) = [3] x1 + [1] x2 + [0]
Following rules are strictly oriented:
square(X) = [4] X + [4]
> [4] X + [0]
= times(X,X)
Following rules are (at-least) weakly oriented:
2ndsneg(0(),Z) = [12]
>= [0]
= rnil()
2ndspos(0(),Z) = [1] Z + [0]
>= [0]
= rnil()
activate(X) = [1] X + [4]
>= [1] X + [0]
= X
activate(n__cons(X1,X2)) = [4]
>= [2]
= cons(X1,X2)
activate(n__from(X)) = [1] X + [4]
>= [1] X + [4]
= from(X)
cons(X1,X2) = [2]
>= [0]
= n__cons(X1,X2)
from(X) = [1] X + [4]
>= [2]
= cons(X,n__from(s(X)))
from(X) = [1] X + [4]
>= [1] X + [0]
= n__from(X)
pi(X) = [2] X + [0]
>= [7]
= 2ndspos(X,from(0()))
plus(0(),Y) = [1] Y + [6]
>= [1] Y + [0]
= Y
plus(s(X),Y) = [1] Y + [6]
>= [1] Y + [11]
= s(plus(X,Y))
times(0(),Y) = [1] Y + [9]
>= [3]
= 0()
times(s(X),Y) = [3] X + [1] Y + [15]
>= [3] X + [1] Y + [6]
= plus(Y,times(X,Y))
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^2))
+ Considered Problem:
- Strict TRS:
2ndspos(0(),Z) -> rnil()
pi(X) -> 2ndspos(X,from(0()))
plus(s(X),Y) -> s(plus(X,Y))
- Weak TRS:
2ndsneg(0(),Z) -> rnil()
activate(X) -> X
activate(n__cons(X1,X2)) -> cons(X1,X2)
activate(n__from(X)) -> from(X)
cons(X1,X2) -> n__cons(X1,X2)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
plus(0(),Y) -> Y
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,square/1,times/2} / {0/0,n__cons/2,n__from/1
,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {2ndsneg,2ndspos,activate,cons,from,pi,plus,square
,times} and constructors {0,n__cons,n__from,negrecip,posrecip,rcons,rnil,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(2ndspos) = {2},
uargs(plus) = {2},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [4]
p(2ndsneg) = [1] x1 + [7]
p(2ndspos) = [1] x2 + [1]
p(activate) = [2] x1 + [4]
p(cons) = [1] x2 + [0]
p(from) = [1]
p(n__cons) = [1] x2 + [0]
p(n__from) = [1]
p(negrecip) = [1] x1 + [0]
p(pi) = [2]
p(plus) = [1] x2 + [0]
p(posrecip) = [1] x1 + [0]
p(rcons) = [1] x1 + [1] x2 + [0]
p(rnil) = [0]
p(s) = [1] x1 + [1]
p(square) = [7] x1 + [4]
p(times) = [2] x1 + [4] x2 + [2]
Following rules are strictly oriented:
2ndspos(0(),Z) = [1] Z + [1]
> [0]
= rnil()
Following rules are (at-least) weakly oriented:
2ndsneg(0(),Z) = [11]
>= [0]
= rnil()
activate(X) = [2] X + [4]
>= [1] X + [0]
= X
activate(n__cons(X1,X2)) = [2] X2 + [4]
>= [1] X2 + [0]
= cons(X1,X2)
activate(n__from(X)) = [6]
>= [1]
= from(X)
cons(X1,X2) = [1] X2 + [0]
>= [1] X2 + [0]
= n__cons(X1,X2)
from(X) = [1]
>= [1]
= cons(X,n__from(s(X)))
from(X) = [1]
>= [1]
= n__from(X)
pi(X) = [2]
>= [2]
= 2ndspos(X,from(0()))
plus(0(),Y) = [1] Y + [0]
>= [1] Y + [0]
= Y
plus(s(X),Y) = [1] Y + [0]
>= [1] Y + [1]
= s(plus(X,Y))
square(X) = [7] X + [4]
>= [6] X + [2]
= times(X,X)
times(0(),Y) = [4] Y + [10]
>= [4]
= 0()
times(s(X),Y) = [2] X + [4] Y + [4]
>= [2] X + [4] Y + [2]
= plus(Y,times(X,Y))
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^2))
+ Considered Problem:
- Strict TRS:
pi(X) -> 2ndspos(X,from(0()))
plus(s(X),Y) -> s(plus(X,Y))
- Weak TRS:
2ndsneg(0(),Z) -> rnil()
2ndspos(0(),Z) -> rnil()
activate(X) -> X
activate(n__cons(X1,X2)) -> cons(X1,X2)
activate(n__from(X)) -> from(X)
cons(X1,X2) -> n__cons(X1,X2)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
plus(0(),Y) -> Y
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,square/1,times/2} / {0/0,n__cons/2,n__from/1
,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {2ndsneg,2ndspos,activate,cons,from,pi,plus,square
,times} and constructors {0,n__cons,n__from,negrecip,posrecip,rcons,rnil,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(2ndspos) = {2},
uargs(plus) = {2},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [1]
p(2ndsneg) = [2] x1 + [0]
p(2ndspos) = [1] x2 + [1]
p(activate) = [2] x1 + [6]
p(cons) = [1] x1 + [0]
p(from) = [1] x1 + [0]
p(n__cons) = [1] x1 + [0]
p(n__from) = [1] x1 + [0]
p(negrecip) = [1] x1 + [0]
p(pi) = [3]
p(plus) = [1] x2 + [0]
p(posrecip) = [1] x1 + [0]
p(rcons) = [1] x1 + [1] x2 + [0]
p(rnil) = [0]
p(s) = [1] x1 + [1]
p(square) = [5] x1 + [1]
p(times) = [4] x2 + [1]
Following rules are strictly oriented:
pi(X) = [3]
> [2]
= 2ndspos(X,from(0()))
Following rules are (at-least) weakly oriented:
2ndsneg(0(),Z) = [2]
>= [0]
= rnil()
2ndspos(0(),Z) = [1] Z + [1]
>= [0]
= rnil()
activate(X) = [2] X + [6]
>= [1] X + [0]
= X
activate(n__cons(X1,X2)) = [2] X1 + [6]
>= [1] X1 + [0]
= cons(X1,X2)
activate(n__from(X)) = [2] X + [6]
>= [1] X + [0]
= from(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(s(X)))
from(X) = [1] X + [0]
>= [1] X + [0]
= n__from(X)
plus(0(),Y) = [1] Y + [0]
>= [1] Y + [0]
= Y
plus(s(X),Y) = [1] Y + [0]
>= [1] Y + [1]
= s(plus(X,Y))
square(X) = [5] X + [1]
>= [4] X + [1]
= times(X,X)
times(0(),Y) = [4] Y + [1]
>= [1]
= 0()
times(s(X),Y) = [4] Y + [1]
>= [4] Y + [1]
= plus(Y,times(X,Y))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:8: NaturalPI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
plus(s(X),Y) -> s(plus(X,Y))
- Weak TRS:
2ndsneg(0(),Z) -> rnil()
2ndspos(0(),Z) -> rnil()
activate(X) -> X
activate(n__cons(X1,X2)) -> cons(X1,X2)
activate(n__from(X)) -> from(X)
cons(X1,X2) -> n__cons(X1,X2)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
pi(X) -> 2ndspos(X,from(0()))
plus(0(),Y) -> Y
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,square/1,times/2} / {0/0,n__cons/2,n__from/1
,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {2ndsneg,2ndspos,activate,cons,from,pi,plus,square
,times} and constructors {0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s}
+ Applied Processor:
NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a polynomial interpretation of kind constructor-based(mixed(2)):
The following argument positions are considered usable:
uargs(2ndspos) = {2},
uargs(plus) = {2},
uargs(s) = {1}
Following symbols are considered usable:
{2ndsneg,2ndspos,activate,cons,from,pi,plus,square,times}
TcT has computed the following interpretation:
p(0) = 0
p(2ndsneg) = x1*x2 + 3*x2
p(2ndspos) = x2
p(activate) = 1 + 4*x1 + 4*x1^2
p(cons) = 0
p(from) = 0
p(n__cons) = 0
p(n__from) = 0
p(negrecip) = 1
p(pi) = 3 + x1
p(plus) = 2*x1 + x2
p(posrecip) = 1
p(rcons) = 0
p(rnil) = 0
p(s) = 1 + x1
p(square) = 2*x1 + 4*x1^2
p(times) = 2*x1*x2
Following rules are strictly oriented:
plus(s(X),Y) = 2 + 2*X + Y
> 1 + 2*X + Y
= s(plus(X,Y))
Following rules are (at-least) weakly oriented:
2ndsneg(0(),Z) = 3*Z
>= 0
= rnil()
2ndspos(0(),Z) = Z
>= 0
= rnil()
activate(X) = 1 + 4*X + 4*X^2
>= X
= X
activate(n__cons(X1,X2)) = 1
>= 0
= cons(X1,X2)
activate(n__from(X)) = 1
>= 0
= from(X)
cons(X1,X2) = 0
>= 0
= n__cons(X1,X2)
from(X) = 0
>= 0
= cons(X,n__from(s(X)))
from(X) = 0
>= 0
= n__from(X)
pi(X) = 3 + X
>= 0
= 2ndspos(X,from(0()))
plus(0(),Y) = Y
>= Y
= Y
square(X) = 2*X + 4*X^2
>= 2*X^2
= times(X,X)
times(0(),Y) = 0
>= 0
= 0()
times(s(X),Y) = 2*X*Y + 2*Y
>= 2*X*Y + 2*Y
= plus(Y,times(X,Y))
** 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(X1,X2)
activate(n__from(X)) -> from(X)
cons(X1,X2) -> n__cons(X1,X2)
from(X) -> cons(X,n__from(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))
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,square/1,times/2} / {0/0,n__cons/2,n__from/1
,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {2ndsneg,2ndspos,activate,cons,from,pi,plus,square
,times} and constructors {0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(Omega(n^1),O(n^2))