*** 1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
2ndsneg(0(),Z) -> rnil()
2ndsneg(s(N),cons(X,Z)) -> 2ndsneg(s(N),cons2(X,activate(Z)))
2ndsneg(s(N),cons2(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,activate(Z)))
2ndspos(0(),Z) -> rnil()
2ndspos(s(N),cons(X,Z)) -> 2ndspos(s(N),cons2(X,activate(Z)))
2ndspos(s(N),cons2(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,activate(Z)))
activate(X) -> X
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
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))
Weak DP Rules:
Weak TRS Rules:
Signature:
{2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,s/1,square/1,times/2} / {0/0,cons/2,cons2/2,n__from/1,n__s/1,negrecip/1,posrecip/1,rcons/2,rnil/0}
Obligation:
Innermost
basic terms: {2ndsneg,2ndspos,activate,from,pi,plus,s,square,times}/{0,cons,cons2,n__from,n__s,negrecip,posrecip,rcons,rnil}
Applied Processor:
InnermostRuleRemoval
Proof:
Arguments of following rules are not normal-forms.
2ndsneg(s(N),cons(X,Z)) -> 2ndsneg(s(N),cons2(X,activate(Z)))
2ndsneg(s(N),cons2(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,activate(Z)))
2ndspos(s(N),cons(X,Z)) -> 2ndspos(s(N),cons2(X,activate(Z)))
2ndspos(s(N),cons2(X,cons(Y,Z))) -> rcons(posrecip(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.
*** 1.1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
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))
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()
Weak DP Rules:
Weak TRS Rules:
Signature:
{2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,s/1,square/1,times/2} / {0/0,cons/2,cons2/2,n__from/1,n__s/1,negrecip/1,posrecip/1,rcons/2,rnil/0}
Obligation:
Innermost
basic terms: {2ndsneg,2ndspos,activate,from,pi,plus,s,square,times}/{0,cons,cons2,n__from,n__s,negrecip,posrecip,rcons,rnil}
Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
Proof:
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(from) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(2ndsneg) = [0]
p(2ndspos) = [1] x2 + [0]
p(activate) = [4] x1 + [2]
p(cons) = [0]
p(cons2) = [1] x1 + [1] x2 + [0]
p(from) = [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) = [0]
p(times) = [0]
Following rules are strictly oriented:
activate(X) = [4] X + [2]
> [1] X + [0]
= X
Following rules are (at-least) weakly oriented:
2ndsneg(0(),Z) = [0]
>= [0]
= rnil()
2ndspos(0(),Z) = [1] Z + [0]
>= [0]
= rnil()
activate(n__from(X)) = [4] X + [2]
>= [4] X + [2]
= from(activate(X))
activate(n__s(X)) = [4] X + [2]
>= [4] X + [2]
= s(activate(X))
from(X) = [1] X + [0]
>= [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]
>= [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.
*** 1.1.1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
2ndsneg(0(),Z) -> rnil()
2ndspos(0(),Z) -> rnil()
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
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()
Weak DP Rules:
Weak TRS Rules:
activate(X) -> X
Signature:
{2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,s/1,square/1,times/2} / {0/0,cons/2,cons2/2,n__from/1,n__s/1,negrecip/1,posrecip/1,rcons/2,rnil/0}
Obligation:
Innermost
basic terms: {2ndsneg,2ndspos,activate,from,pi,plus,s,square,times}/{0,cons,cons2,n__from,n__s,negrecip,posrecip,rcons,rnil}
Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
Proof:
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(from) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [4]
p(2ndsneg) = [3] x1 + [0]
p(2ndspos) = [1] x1 + [1] x2 + [0]
p(activate) = [2] x1 + [1]
p(cons) = [0]
p(cons2) = [1] x1 + [1] x2 + [0]
p(from) = [1] x1 + [4]
p(n__from) = [1] x1 + [0]
p(n__s) = [1] x1 + [0]
p(negrecip) = [1] x1 + [0]
p(pi) = [1] x1 + [0]
p(plus) = [6] x1 + [2] x2 + [0]
p(posrecip) = [1] x1 + [0]
p(rcons) = [2]
p(rnil) = [1]
p(s) = [1] x1 + [15]
p(square) = [8] x1 + [0]
p(times) = [1] x1 + [1]
Following rules are strictly oriented:
2ndsneg(0(),Z) = [12]
> [1]
= rnil()
2ndspos(0(),Z) = [1] Z + [4]
> [1]
= rnil()
from(X) = [1] X + [4]
> [0]
= cons(X,n__from(n__s(X)))
from(X) = [1] X + [4]
> [1] X + [0]
= n__from(X)
plus(0(),Y) = [2] Y + [24]
> [1] Y + [0]
= Y
s(X) = [1] X + [15]
> [1] X + [0]
= n__s(X)
times(0(),Y) = [5]
> [4]
= 0()
Following rules are (at-least) weakly oriented:
activate(X) = [2] X + [1]
>= [1] X + [0]
= X
activate(n__from(X)) = [2] X + [1]
>= [2] X + [5]
= from(activate(X))
activate(n__s(X)) = [2] X + [1]
>= [2] X + [16]
= s(activate(X))
pi(X) = [1] X + [0]
>= [1] X + [8]
= 2ndspos(X,from(0()))
square(X) = [8] X + [0]
>= [1] X + [1]
= times(X,X)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
pi(X) -> 2ndspos(X,from(0()))
square(X) -> times(X,X)
Weak DP Rules:
Weak TRS Rules:
2ndsneg(0(),Z) -> rnil()
2ndspos(0(),Z) -> rnil()
activate(X) -> X
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
plus(0(),Y) -> Y
s(X) -> n__s(X)
times(0(),Y) -> 0()
Signature:
{2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,s/1,square/1,times/2} / {0/0,cons/2,cons2/2,n__from/1,n__s/1,negrecip/1,posrecip/1,rcons/2,rnil/0}
Obligation:
Innermost
basic terms: {2ndsneg,2ndspos,activate,from,pi,plus,s,square,times}/{0,cons,cons2,n__from,n__s,negrecip,posrecip,rcons,rnil}
Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules, greedy = NoGreedy}
Proof:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(2ndspos) = {2},
uargs(from) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{2ndsneg,2ndspos,activate,from,pi,plus,s,square,times}
TcT has computed the following interpretation:
p(0) = [1]
p(2ndsneg) = [2] x2 + [5]
p(2ndspos) = [5] x1 + [4] x2 + [3]
p(activate) = [1] x1 + [1]
p(cons) = [1] x1 + [0]
p(cons2) = [1] x2 + [1]
p(from) = [1] x1 + [0]
p(n__from) = [1] x1 + [0]
p(n__s) = [1] x1 + [4]
p(negrecip) = [1] x1 + [1]
p(pi) = [5] x1 + [12]
p(plus) = [8] x1 + [4] x2 + [12]
p(posrecip) = [1]
p(rcons) = [1] x2 + [1]
p(rnil) = [5]
p(s) = [1] x1 + [4]
p(square) = [2] x1 + [11]
p(times) = [2] x1 + [9]
Following rules are strictly oriented:
pi(X) = [5] X + [12]
> [5] X + [7]
= 2ndspos(X,from(0()))
square(X) = [2] X + [11]
> [2] X + [9]
= times(X,X)
Following rules are (at-least) weakly oriented:
2ndsneg(0(),Z) = [2] Z + [5]
>= [5]
= rnil()
2ndspos(0(),Z) = [4] Z + [8]
>= [5]
= rnil()
activate(X) = [1] X + [1]
>= [1] X + [0]
= X
activate(n__from(X)) = [1] X + [1]
>= [1] X + [1]
= from(activate(X))
activate(n__s(X)) = [1] X + [5]
>= [1] X + [5]
= s(activate(X))
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) = [4] Y + [20]
>= [1] Y + [0]
= Y
s(X) = [1] X + [4]
>= [1] X + [4]
= n__s(X)
times(0(),Y) = [11]
>= [1]
= 0()
*** 1.1.1.1.1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
Weak DP Rules:
Weak TRS Rules:
2ndsneg(0(),Z) -> rnil()
2ndspos(0(),Z) -> rnil()
activate(X) -> X
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,from/1,pi/1,plus/2,s/1,square/1,times/2} / {0/0,cons/2,cons2/2,n__from/1,n__s/1,negrecip/1,posrecip/1,rcons/2,rnil/0}
Obligation:
Innermost
basic terms: {2ndsneg,2ndspos,activate,from,pi,plus,s,square,times}/{0,cons,cons2,n__from,n__s,negrecip,posrecip,rcons,rnil}
Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
Proof:
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(from) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [2]
p(2ndsneg) = [2]
p(2ndspos) = [5] x1 + [1] x2 + [0]
p(activate) = [8] x1 + [1]
p(cons) = [2]
p(cons2) = [1] x1 + [0]
p(from) = [1] x1 + [2]
p(n__from) = [1] x1 + [2]
p(n__s) = [1] x1 + [1]
p(negrecip) = [1]
p(pi) = [5] x1 + [7]
p(plus) = [1] x2 + [0]
p(posrecip) = [1] x1 + [0]
p(rcons) = [1] x1 + [1] x2 + [0]
p(rnil) = [1]
p(s) = [1] x1 + [1]
p(square) = [12] x1 + [8]
p(times) = [12] x2 + [2]
Following rules are strictly oriented:
activate(n__from(X)) = [8] X + [17]
> [8] X + [3]
= from(activate(X))
activate(n__s(X)) = [8] X + [9]
> [8] X + [2]
= s(activate(X))
Following rules are (at-least) weakly oriented:
2ndsneg(0(),Z) = [2]
>= [1]
= rnil()
2ndspos(0(),Z) = [1] Z + [10]
>= [1]
= rnil()
activate(X) = [8] X + [1]
>= [1] X + [0]
= X
from(X) = [1] X + [2]
>= [2]
= cons(X,n__from(n__s(X)))
from(X) = [1] X + [2]
>= [1] X + [2]
= n__from(X)
pi(X) = [5] X + [7]
>= [5] X + [4]
= 2ndspos(X,from(0()))
plus(0(),Y) = [1] Y + [0]
>= [1] Y + [0]
= Y
s(X) = [1] X + [1]
>= [1] X + [1]
= n__s(X)
square(X) = [12] X + [8]
>= [12] X + [2]
= times(X,X)
times(0(),Y) = [12] Y + [2]
>= [2]
= 0()
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1.1.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
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))
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,from/1,pi/1,plus/2,s/1,square/1,times/2} / {0/0,cons/2,cons2/2,n__from/1,n__s/1,negrecip/1,posrecip/1,rcons/2,rnil/0}
Obligation:
Innermost
basic terms: {2ndsneg,2ndspos,activate,from,pi,plus,s,square,times}/{0,cons,cons2,n__from,n__s,negrecip,posrecip,rcons,rnil}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).