*** 1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
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))
Weak DP Rules:
Weak TRS Rules:
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
basic terms: {2ndsneg,2ndspos,activate,cons,from,pi,plus,s,square,times}/{0,n__cons,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,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.
*** 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__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()
Weak DP Rules:
Weak TRS Rules:
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
basic terms: {2ndsneg,2ndspos,activate,cons,from,pi,plus,s,square,times}/{0,n__cons,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(cons) = {1},
uargs(from) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{2ndsneg,2ndspos,activate,cons,from,pi,plus,s,square,times}
TcT has computed the following interpretation:
p(0) = [0]
p(2ndsneg) = [10]
p(2ndspos) = [4] x2 + [4]
p(activate) = [2] x1 + [0]
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 + [3]
p(negrecip) = [0]
p(pi) = [13]
p(plus) = [4] x1 + [2] x2 + [2]
p(posrecip) = [1]
p(rcons) = [1] x2 + [2]
p(rnil) = [0]
p(s) = [1] x1 + [4]
p(square) = [10] x1 + [1]
p(times) = [4] x1 + [6] x2 + [0]
Following rules are strictly oriented:
2ndsneg(0(),Z) = [10]
> [0]
= rnil()
2ndspos(0(),Z) = [4] Z + [4]
> [0]
= rnil()
activate(n__cons(X1,X2)) = [2] X1 + [4]
> [2] X1 + [2]
= cons(activate(X1),X2)
activate(n__from(X)) = [2] X + [4]
> [2] X + [2]
= from(activate(X))
activate(n__s(X)) = [2] X + [6]
> [2] X + [4]
= s(activate(X))
pi(X) = [13]
> [12]
= 2ndspos(X,from(0()))
plus(0(),Y) = [2] Y + [2]
> [1] Y + [0]
= Y
s(X) = [1] X + [4]
> [1] X + [3]
= n__s(X)
square(X) = [10] X + [1]
> [10] X + [0]
= times(X,X)
Following rules are (at-least) weakly oriented:
activate(X) = [2] X + [0]
>= [1] X + [0]
= 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)
times(0(),Y) = [6] Y + [0]
>= [0]
= 0()
*** 1.1.1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
activate(X) -> X
cons(X1,X2) -> n__cons(X1,X2)
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
times(0(),Y) -> 0()
Weak DP Rules:
Weak TRS Rules:
2ndsneg(0(),Z) -> rnil()
2ndspos(0(),Z) -> rnil()
activate(n__cons(X1,X2)) -> cons(activate(X1),X2)
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
pi(X) -> 2ndspos(X,from(0()))
plus(0(),Y) -> Y
s(X) -> n__s(X)
square(X) -> times(X,X)
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
basic terms: {2ndsneg,2ndspos,activate,cons,from,pi,plus,s,square,times}/{0,n__cons,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(cons) = {1},
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) = [14] 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 + [0]
p(square) = [1]
p(times) = [1]
Following rules are strictly oriented:
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) = [14] X + [0]
>= [1] X + [0]
= X
activate(n__cons(X1,X2)) = [14] X1 + [0]
>= [14] X1 + [0]
= cons(activate(X1),X2)
activate(n__from(X)) = [14] X + [0]
>= [14] X + [0]
= from(activate(X))
activate(n__s(X)) = [14] X + [0]
>= [14] X + [0]
= 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
s(X) = [1] X + [0]
>= [1] X + [0]
= n__s(X)
square(X) = [1]
>= [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(X) -> X
cons(X1,X2) -> n__cons(X1,X2)
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
Weak DP Rules:
Weak TRS Rules:
2ndsneg(0(),Z) -> rnil()
2ndspos(0(),Z) -> rnil()
activate(n__cons(X1,X2)) -> cons(activate(X1),X2)
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(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
basic terms: {2ndsneg,2ndspos,activate,cons,from,pi,plus,s,square,times}/{0,n__cons,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(cons) = {1},
uargs(from) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(2ndsneg) = [1]
p(2ndspos) = [1] x2 + [0]
p(activate) = [1] x1 + [11]
p(cons) = [1] x1 + [0]
p(from) = [1] x1 + [0]
p(n__cons) = [1] x1 + [0]
p(n__from) = [1] x1 + [2]
p(n__s) = [1] x1 + [4]
p(negrecip) = [1] x1 + [2]
p(pi) = [0]
p(plus) = [2] x2 + [0]
p(posrecip) = [1] x1 + [0]
p(rcons) = [1] x1 + [0]
p(rnil) = [0]
p(s) = [1] x1 + [4]
p(square) = [0]
p(times) = [0]
Following rules are strictly oriented:
activate(X) = [1] X + [11]
> [1] X + [0]
= X
Following rules are (at-least) weakly oriented:
2ndsneg(0(),Z) = [1]
>= [0]
= rnil()
2ndspos(0(),Z) = [1] Z + [0]
>= [0]
= rnil()
activate(n__cons(X1,X2)) = [1] X1 + [11]
>= [1] X1 + [11]
= cons(activate(X1),X2)
activate(n__from(X)) = [1] X + [13]
>= [1] X + [11]
= from(activate(X))
activate(n__s(X)) = [1] X + [15]
>= [1] X + [15]
= 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 + [2]
= 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 + [4]
>= [1] X + [4]
= 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.
*** 1.1.1.1.1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
cons(X1,X2) -> n__cons(X1,X2)
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
Weak DP Rules:
Weak TRS Rules:
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))
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
basic terms: {2ndsneg,2ndspos,activate,cons,from,pi,plus,s,square,times}/{0,n__cons,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(cons) = {1},
uargs(from) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [2]
p(2ndsneg) = [6] x1 + [5]
p(2ndspos) = [1] x2 + [9]
p(activate) = [2] x1 + [3]
p(cons) = [1] x1 + [14]
p(from) = [1] x1 + [0]
p(n__cons) = [1] x1 + [13]
p(n__from) = [1] x1 + [0]
p(n__s) = [1] x1 + [1]
p(negrecip) = [1] x1 + [0]
p(pi) = [11]
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) = [9] x1 + [0]
p(times) = [8] x1 + [0]
Following rules are strictly oriented:
cons(X1,X2) = [1] X1 + [14]
> [1] X1 + [13]
= n__cons(X1,X2)
Following rules are (at-least) weakly oriented:
2ndsneg(0(),Z) = [17]
>= [0]
= rnil()
2ndspos(0(),Z) = [1] Z + [9]
>= [0]
= rnil()
activate(X) = [2] X + [3]
>= [1] X + [0]
= X
activate(n__cons(X1,X2)) = [2] X1 + [29]
>= [2] X1 + [17]
= cons(activate(X1),X2)
activate(n__from(X)) = [2] X + [3]
>= [2] X + [3]
= from(activate(X))
activate(n__s(X)) = [2] X + [5]
>= [2] X + [4]
= s(activate(X))
from(X) = [1] X + [0]
>= [1] X + [14]
= cons(X,n__from(n__s(X)))
from(X) = [1] X + [0]
>= [1] X + [0]
= n__from(X)
pi(X) = [11]
>= [11]
= 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) = [9] X + [0]
>= [8] X + [0]
= times(X,X)
times(0(),Y) = [16]
>= [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(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
Weak DP Rules:
Weak TRS Rules:
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)
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
basic terms: {2ndsneg,2ndspos,activate,cons,from,pi,plus,s,square,times}/{0,n__cons,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(cons) = {1},
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) = [3] x1 + [0]
p(cons) = [1] x1 + [0]
p(from) = [1] x1 + [3]
p(n__cons) = [1] x1 + [0]
p(n__from) = [1] x1 + [1]
p(n__s) = [1] x1 + [0]
p(negrecip) = [1] x1 + [0]
p(pi) = [3]
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:
from(X) = [1] X + [3]
> [1] X + [0]
= cons(X,n__from(n__s(X)))
from(X) = [1] X + [3]
> [1] X + [1]
= 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) = [3] X + [0]
>= [1] X + [0]
= X
activate(n__cons(X1,X2)) = [3] X1 + [0]
>= [3] X1 + [0]
= cons(activate(X1),X2)
activate(n__from(X)) = [3] X + [3]
>= [3] X + [3]
= from(activate(X))
activate(n__s(X)) = [3] X + [0]
>= [3] X + [0]
= s(activate(X))
cons(X1,X2) = [1] X1 + [0]
>= [1] X1 + [0]
= n__cons(X1,X2)
pi(X) = [3]
>= [3]
= 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.
*** 1.1.1.1.1.1.1 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__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
basic terms: {2ndsneg,2ndspos,activate,cons,from,pi,plus,s,square,times}/{0,n__cons,n__from,n__s,negrecip,posrecip,rcons,rnil}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).