*** 1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
activate(X) -> X
activate(n__cons(X1,X2)) -> cons(activate(X1),X2)
activate(n__filter(X1,X2)) -> filter(activate(X1),activate(X2))
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
activate(n__sieve(X)) -> sieve(activate(X))
cons(X1,X2) -> n__cons(X1,X2)
filter(X1,X2) -> n__filter(X1,X2)
filter(s(s(X)),cons(Y,Z)) -> if(divides(s(s(X)),Y),n__filter(n__s(n__s(X)),activate(Z)),n__cons(Y,n__filter(X,n__sieve(Y))))
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
head(cons(X,Y)) -> X
if(false(),X,Y) -> activate(Y)
if(true(),X,Y) -> activate(X)
primes() -> sieve(from(s(s(0()))))
s(X) -> n__s(X)
sieve(X) -> n__sieve(X)
sieve(cons(X,Y)) -> cons(X,n__filter(X,n__sieve(activate(Y))))
tail(cons(X,Y)) -> activate(Y)
Weak DP Rules:
Weak TRS Rules:
Signature:
{activate/1,cons/2,filter/2,from/1,head/1,if/3,primes/0,s/1,sieve/1,tail/1} / {0/0,divides/2,false/0,n__cons/2,n__filter/2,n__from/1,n__s/1,n__sieve/1,true/0}
Obligation:
Innermost
basic terms: {activate,cons,filter,from,head,if,primes,s,sieve,tail}/{0,divides,false,n__cons,n__filter,n__from,n__s,n__sieve,true}
Applied Processor:
InnermostRuleRemoval
Proof:
Arguments of following rules are not normal-forms.
filter(s(s(X)),cons(Y,Z)) -> if(divides(s(s(X)),Y),n__filter(n__s(n__s(X)),activate(Z)),n__cons(Y,n__filter(X,n__sieve(Y))))
head(cons(X,Y)) -> X
sieve(cons(X,Y)) -> cons(X,n__filter(X,n__sieve(activate(Y))))
tail(cons(X,Y)) -> activate(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:
activate(X) -> X
activate(n__cons(X1,X2)) -> cons(activate(X1),X2)
activate(n__filter(X1,X2)) -> filter(activate(X1),activate(X2))
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
activate(n__sieve(X)) -> sieve(activate(X))
cons(X1,X2) -> n__cons(X1,X2)
filter(X1,X2) -> n__filter(X1,X2)
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
if(false(),X,Y) -> activate(Y)
if(true(),X,Y) -> activate(X)
primes() -> sieve(from(s(s(0()))))
s(X) -> n__s(X)
sieve(X) -> n__sieve(X)
Weak DP Rules:
Weak TRS Rules:
Signature:
{activate/1,cons/2,filter/2,from/1,head/1,if/3,primes/0,s/1,sieve/1,tail/1} / {0/0,divides/2,false/0,n__cons/2,n__filter/2,n__from/1,n__s/1,n__sieve/1,true/0}
Obligation:
Innermost
basic terms: {activate,cons,filter,from,head,if,primes,s,sieve,tail}/{0,divides,false,n__cons,n__filter,n__from,n__s,n__sieve,true}
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(cons) = {1},
uargs(filter) = {1,2},
uargs(from) = {1},
uargs(s) = {1},
uargs(sieve) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(activate) = [7] x1 + [0]
p(cons) = [1] x1 + [5]
p(divides) = [1] x1 + [1] x2 + [0]
p(false) = [3]
p(filter) = [1] x1 + [1] x2 + [0]
p(from) = [1] x1 + [5]
p(head) = [0]
p(if) = [9] x1 + [7] x2 + [7] x3 + [1]
p(n__cons) = [1] x1 + [3]
p(n__filter) = [1] x1 + [1] x2 + [0]
p(n__from) = [1] x1 + [1]
p(n__s) = [1] x1 + [0]
p(n__sieve) = [1] x1 + [0]
p(primes) = [0]
p(s) = [1] x1 + [0]
p(sieve) = [1] x1 + [1]
p(tail) = [0]
p(true) = [0]
Following rules are strictly oriented:
activate(n__cons(X1,X2)) = [7] X1 + [21]
> [7] X1 + [5]
= cons(activate(X1),X2)
activate(n__from(X)) = [7] X + [7]
> [7] X + [5]
= from(activate(X))
cons(X1,X2) = [1] X1 + [5]
> [1] X1 + [3]
= n__cons(X1,X2)
from(X) = [1] X + [5]
> [1] X + [1]
= n__from(X)
if(false(),X,Y) = [7] X + [7] Y + [28]
> [7] Y + [0]
= activate(Y)
if(true(),X,Y) = [7] X + [7] Y + [1]
> [7] X + [0]
= activate(X)
sieve(X) = [1] X + [1]
> [1] X + [0]
= n__sieve(X)
Following rules are (at-least) weakly oriented:
activate(X) = [7] X + [0]
>= [1] X + [0]
= X
activate(n__filter(X1,X2)) = [7] X1 + [7] X2 + [0]
>= [7] X1 + [7] X2 + [0]
= filter(activate(X1)
,activate(X2))
activate(n__s(X)) = [7] X + [0]
>= [7] X + [0]
= s(activate(X))
activate(n__sieve(X)) = [7] X + [0]
>= [7] X + [1]
= sieve(activate(X))
filter(X1,X2) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= n__filter(X1,X2)
from(X) = [1] X + [5]
>= [1] X + [5]
= cons(X,n__from(n__s(X)))
primes() = [0]
>= [6]
= sieve(from(s(s(0()))))
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.
*** 1.1.1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
activate(X) -> X
activate(n__filter(X1,X2)) -> filter(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
activate(n__sieve(X)) -> sieve(activate(X))
filter(X1,X2) -> n__filter(X1,X2)
from(X) -> cons(X,n__from(n__s(X)))
primes() -> sieve(from(s(s(0()))))
s(X) -> n__s(X)
Weak DP Rules:
Weak TRS Rules:
activate(n__cons(X1,X2)) -> cons(activate(X1),X2)
activate(n__from(X)) -> from(activate(X))
cons(X1,X2) -> n__cons(X1,X2)
from(X) -> n__from(X)
if(false(),X,Y) -> activate(Y)
if(true(),X,Y) -> activate(X)
sieve(X) -> n__sieve(X)
Signature:
{activate/1,cons/2,filter/2,from/1,head/1,if/3,primes/0,s/1,sieve/1,tail/1} / {0/0,divides/2,false/0,n__cons/2,n__filter/2,n__from/1,n__s/1,n__sieve/1,true/0}
Obligation:
Innermost
basic terms: {activate,cons,filter,from,head,if,primes,s,sieve,tail}/{0,divides,false,n__cons,n__filter,n__from,n__s,n__sieve,true}
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(cons) = {1},
uargs(filter) = {1,2},
uargs(from) = {1},
uargs(s) = {1},
uargs(sieve) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(activate) = [1] x1 + [1]
p(cons) = [1] x1 + [1]
p(divides) = [1] x1 + [1] x2 + [0]
p(false) = [1]
p(filter) = [1] x1 + [1] x2 + [0]
p(from) = [1] x1 + [2]
p(head) = [0]
p(if) = [2] x2 + [1] x3 + [1]
p(n__cons) = [1] x1 + [1]
p(n__filter) = [1] x1 + [1] x2 + [0]
p(n__from) = [1] x1 + [2]
p(n__s) = [1] x1 + [0]
p(n__sieve) = [1] x1 + [0]
p(primes) = [0]
p(s) = [1] x1 + [0]
p(sieve) = [1] x1 + [0]
p(tail) = [0]
p(true) = [4]
Following rules are strictly oriented:
activate(X) = [1] X + [1]
> [1] X + [0]
= X
from(X) = [1] X + [2]
> [1] X + [1]
= cons(X,n__from(n__s(X)))
Following rules are (at-least) weakly oriented:
activate(n__cons(X1,X2)) = [1] X1 + [2]
>= [1] X1 + [2]
= cons(activate(X1),X2)
activate(n__filter(X1,X2)) = [1] X1 + [1] X2 + [1]
>= [1] X1 + [1] X2 + [2]
= filter(activate(X1)
,activate(X2))
activate(n__from(X)) = [1] X + [3]
>= [1] X + [3]
= from(activate(X))
activate(n__s(X)) = [1] X + [1]
>= [1] X + [1]
= s(activate(X))
activate(n__sieve(X)) = [1] X + [1]
>= [1] X + [1]
= sieve(activate(X))
cons(X1,X2) = [1] X1 + [1]
>= [1] X1 + [1]
= n__cons(X1,X2)
filter(X1,X2) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= n__filter(X1,X2)
from(X) = [1] X + [2]
>= [1] X + [2]
= n__from(X)
if(false(),X,Y) = [2] X + [1] Y + [1]
>= [1] Y + [1]
= activate(Y)
if(true(),X,Y) = [2] X + [1] Y + [1]
>= [1] X + [1]
= activate(X)
primes() = [0]
>= [2]
= sieve(from(s(s(0()))))
s(X) = [1] X + [0]
>= [1] X + [0]
= n__s(X)
sieve(X) = [1] X + [0]
>= [1] X + [0]
= n__sieve(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__filter(X1,X2)) -> filter(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
activate(n__sieve(X)) -> sieve(activate(X))
filter(X1,X2) -> n__filter(X1,X2)
primes() -> sieve(from(s(s(0()))))
s(X) -> n__s(X)
Weak DP Rules:
Weak TRS Rules:
activate(X) -> X
activate(n__cons(X1,X2)) -> cons(activate(X1),X2)
activate(n__from(X)) -> from(activate(X))
cons(X1,X2) -> n__cons(X1,X2)
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
if(false(),X,Y) -> activate(Y)
if(true(),X,Y) -> activate(X)
sieve(X) -> n__sieve(X)
Signature:
{activate/1,cons/2,filter/2,from/1,head/1,if/3,primes/0,s/1,sieve/1,tail/1} / {0/0,divides/2,false/0,n__cons/2,n__filter/2,n__from/1,n__s/1,n__sieve/1,true/0}
Obligation:
Innermost
basic terms: {activate,cons,filter,from,head,if,primes,s,sieve,tail}/{0,divides,false,n__cons,n__filter,n__from,n__s,n__sieve,true}
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(cons) = {1},
uargs(filter) = {1,2},
uargs(from) = {1},
uargs(s) = {1},
uargs(sieve) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [5]
p(activate) = [1] x1 + [0]
p(cons) = [1] x1 + [0]
p(divides) = [1]
p(false) = [0]
p(filter) = [1] x1 + [1] x2 + [2]
p(from) = [1] x1 + [0]
p(head) = [1] x1 + [1]
p(if) = [4] x1 + [2] x2 + [2] x3 + [4]
p(n__cons) = [1] x1 + [0]
p(n__filter) = [1] x1 + [1] x2 + [0]
p(n__from) = [1] x1 + [0]
p(n__s) = [1] x1 + [0]
p(n__sieve) = [1] x1 + [2]
p(primes) = [1]
p(s) = [1] x1 + [0]
p(sieve) = [1] x1 + [3]
p(tail) = [2]
p(true) = [2]
Following rules are strictly oriented:
filter(X1,X2) = [1] X1 + [1] X2 + [2]
> [1] X1 + [1] X2 + [0]
= n__filter(X1,X2)
Following rules are (at-least) weakly oriented:
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__filter(X1,X2)) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [2]
= filter(activate(X1)
,activate(X2))
activate(n__from(X)) = [1] X + [0]
>= [1] X + [0]
= from(activate(X))
activate(n__s(X)) = [1] X + [0]
>= [1] X + [0]
= s(activate(X))
activate(n__sieve(X)) = [1] X + [2]
>= [1] X + [3]
= sieve(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)
if(false(),X,Y) = [2] X + [2] Y + [4]
>= [1] Y + [0]
= activate(Y)
if(true(),X,Y) = [2] X + [2] Y + [12]
>= [1] X + [0]
= activate(X)
primes() = [1]
>= [8]
= sieve(from(s(s(0()))))
s(X) = [1] X + [0]
>= [1] X + [0]
= n__s(X)
sieve(X) = [1] X + [3]
>= [1] X + [2]
= n__sieve(X)
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:
activate(n__filter(X1,X2)) -> filter(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
activate(n__sieve(X)) -> sieve(activate(X))
primes() -> sieve(from(s(s(0()))))
s(X) -> n__s(X)
Weak DP Rules:
Weak TRS Rules:
activate(X) -> X
activate(n__cons(X1,X2)) -> cons(activate(X1),X2)
activate(n__from(X)) -> from(activate(X))
cons(X1,X2) -> n__cons(X1,X2)
filter(X1,X2) -> n__filter(X1,X2)
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
if(false(),X,Y) -> activate(Y)
if(true(),X,Y) -> activate(X)
sieve(X) -> n__sieve(X)
Signature:
{activate/1,cons/2,filter/2,from/1,head/1,if/3,primes/0,s/1,sieve/1,tail/1} / {0/0,divides/2,false/0,n__cons/2,n__filter/2,n__from/1,n__s/1,n__sieve/1,true/0}
Obligation:
Innermost
basic terms: {activate,cons,filter,from,head,if,primes,s,sieve,tail}/{0,divides,false,n__cons,n__filter,n__from,n__s,n__sieve,true}
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(cons) = {1},
uargs(filter) = {1,2},
uargs(from) = {1},
uargs(s) = {1},
uargs(sieve) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(activate) = [5] x1 + [0]
p(cons) = [1] x1 + [0]
p(divides) = [1] x1 + [1] x2 + [0]
p(false) = [0]
p(filter) = [1] x1 + [1] x2 + [0]
p(from) = [1] x1 + [0]
p(head) = [0]
p(if) = [5] x2 + [5] x3 + [0]
p(n__cons) = [1] x1 + [0]
p(n__filter) = [1] x1 + [1] x2 + [0]
p(n__from) = [1] x1 + [0]
p(n__s) = [1] x1 + [2]
p(n__sieve) = [1] x1 + [0]
p(primes) = [0]
p(s) = [1] x1 + [7]
p(sieve) = [1] x1 + [0]
p(tail) = [0]
p(true) = [0]
Following rules are strictly oriented:
activate(n__s(X)) = [5] X + [10]
> [5] X + [7]
= s(activate(X))
s(X) = [1] X + [7]
> [1] X + [2]
= n__s(X)
Following rules are (at-least) weakly oriented:
activate(X) = [5] X + [0]
>= [1] X + [0]
= X
activate(n__cons(X1,X2)) = [5] X1 + [0]
>= [5] X1 + [0]
= cons(activate(X1),X2)
activate(n__filter(X1,X2)) = [5] X1 + [5] X2 + [0]
>= [5] X1 + [5] X2 + [0]
= filter(activate(X1)
,activate(X2))
activate(n__from(X)) = [5] X + [0]
>= [5] X + [0]
= from(activate(X))
activate(n__sieve(X)) = [5] X + [0]
>= [5] X + [0]
= sieve(activate(X))
cons(X1,X2) = [1] X1 + [0]
>= [1] X1 + [0]
= n__cons(X1,X2)
filter(X1,X2) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= n__filter(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)
if(false(),X,Y) = [5] X + [5] Y + [0]
>= [5] Y + [0]
= activate(Y)
if(true(),X,Y) = [5] X + [5] Y + [0]
>= [5] X + [0]
= activate(X)
primes() = [0]
>= [14]
= sieve(from(s(s(0()))))
sieve(X) = [1] X + [0]
>= [1] X + [0]
= n__sieve(X)
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:
activate(n__filter(X1,X2)) -> filter(activate(X1),activate(X2))
activate(n__sieve(X)) -> sieve(activate(X))
primes() -> sieve(from(s(s(0()))))
Weak DP Rules:
Weak TRS Rules:
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)
filter(X1,X2) -> n__filter(X1,X2)
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
if(false(),X,Y) -> activate(Y)
if(true(),X,Y) -> activate(X)
s(X) -> n__s(X)
sieve(X) -> n__sieve(X)
Signature:
{activate/1,cons/2,filter/2,from/1,head/1,if/3,primes/0,s/1,sieve/1,tail/1} / {0/0,divides/2,false/0,n__cons/2,n__filter/2,n__from/1,n__s/1,n__sieve/1,true/0}
Obligation:
Innermost
basic terms: {activate,cons,filter,from,head,if,primes,s,sieve,tail}/{0,divides,false,n__cons,n__filter,n__from,n__s,n__sieve,true}
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(cons) = {1},
uargs(filter) = {1,2},
uargs(from) = {1},
uargs(s) = {1},
uargs(sieve) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [2]
p(activate) = [1] x1 + [0]
p(cons) = [1] x1 + [2]
p(divides) = [1] x1 + [1] x2 + [0]
p(false) = [0]
p(filter) = [1] x1 + [1] x2 + [4]
p(from) = [1] x1 + [2]
p(head) = [0]
p(if) = [1] x2 + [1] x3 + [0]
p(n__cons) = [1] x1 + [2]
p(n__filter) = [1] x1 + [1] x2 + [4]
p(n__from) = [1] x1 + [2]
p(n__s) = [1] x1 + [0]
p(n__sieve) = [1] x1 + [0]
p(primes) = [5]
p(s) = [1] x1 + [0]
p(sieve) = [1] x1 + [0]
p(tail) = [0]
p(true) = [0]
Following rules are strictly oriented:
primes() = [5]
> [4]
= sieve(from(s(s(0()))))
Following rules are (at-least) weakly oriented:
activate(X) = [1] X + [0]
>= [1] X + [0]
= X
activate(n__cons(X1,X2)) = [1] X1 + [2]
>= [1] X1 + [2]
= cons(activate(X1),X2)
activate(n__filter(X1,X2)) = [1] X1 + [1] X2 + [4]
>= [1] X1 + [1] X2 + [4]
= filter(activate(X1)
,activate(X2))
activate(n__from(X)) = [1] X + [2]
>= [1] X + [2]
= from(activate(X))
activate(n__s(X)) = [1] X + [0]
>= [1] X + [0]
= s(activate(X))
activate(n__sieve(X)) = [1] X + [0]
>= [1] X + [0]
= sieve(activate(X))
cons(X1,X2) = [1] X1 + [2]
>= [1] X1 + [2]
= n__cons(X1,X2)
filter(X1,X2) = [1] X1 + [1] X2 + [4]
>= [1] X1 + [1] X2 + [4]
= n__filter(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)
if(false(),X,Y) = [1] X + [1] Y + [0]
>= [1] Y + [0]
= activate(Y)
if(true(),X,Y) = [1] X + [1] Y + [0]
>= [1] X + [0]
= activate(X)
s(X) = [1] X + [0]
>= [1] X + [0]
= n__s(X)
sieve(X) = [1] X + [0]
>= [1] X + [0]
= n__sieve(X)
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(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
activate(n__filter(X1,X2)) -> filter(activate(X1),activate(X2))
activate(n__sieve(X)) -> sieve(activate(X))
Weak DP Rules:
Weak TRS Rules:
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)
filter(X1,X2) -> n__filter(X1,X2)
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
if(false(),X,Y) -> activate(Y)
if(true(),X,Y) -> activate(X)
primes() -> sieve(from(s(s(0()))))
s(X) -> n__s(X)
sieve(X) -> n__sieve(X)
Signature:
{activate/1,cons/2,filter/2,from/1,head/1,if/3,primes/0,s/1,sieve/1,tail/1} / {0/0,divides/2,false/0,n__cons/2,n__filter/2,n__from/1,n__s/1,n__sieve/1,true/0}
Obligation:
Innermost
basic terms: {activate,cons,filter,from,head,if,primes,s,sieve,tail}/{0,divides,false,n__cons,n__filter,n__from,n__s,n__sieve,true}
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(cons) = {1},
uargs(filter) = {1,2},
uargs(from) = {1},
uargs(s) = {1},
uargs(sieve) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [2]
p(activate) = [5] x1 + [0]
p(cons) = [1] x1 + [0]
p(divides) = [1] x1 + [1] x2 + [0]
p(false) = [0]
p(filter) = [1] x1 + [1] x2 + [2]
p(from) = [1] x1 + [0]
p(head) = [0]
p(if) = [6] x2 + [5] x3 + [0]
p(n__cons) = [1] x1 + [0]
p(n__filter) = [1] x1 + [1] x2 + [0]
p(n__from) = [1] x1 + [0]
p(n__s) = [1] x1 + [0]
p(n__sieve) = [1] x1 + [1]
p(primes) = [7]
p(s) = [1] x1 + [0]
p(sieve) = [1] x1 + [4]
p(tail) = [4] x1 + [1]
p(true) = [0]
Following rules are strictly oriented:
activate(n__sieve(X)) = [5] X + [5]
> [5] X + [4]
= sieve(activate(X))
Following rules are (at-least) weakly oriented:
activate(X) = [5] X + [0]
>= [1] X + [0]
= X
activate(n__cons(X1,X2)) = [5] X1 + [0]
>= [5] X1 + [0]
= cons(activate(X1),X2)
activate(n__filter(X1,X2)) = [5] X1 + [5] X2 + [0]
>= [5] X1 + [5] X2 + [2]
= filter(activate(X1)
,activate(X2))
activate(n__from(X)) = [5] X + [0]
>= [5] X + [0]
= from(activate(X))
activate(n__s(X)) = [5] X + [0]
>= [5] X + [0]
= s(activate(X))
cons(X1,X2) = [1] X1 + [0]
>= [1] X1 + [0]
= n__cons(X1,X2)
filter(X1,X2) = [1] X1 + [1] X2 + [2]
>= [1] X1 + [1] X2 + [0]
= n__filter(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)
if(false(),X,Y) = [6] X + [5] Y + [0]
>= [5] Y + [0]
= activate(Y)
if(true(),X,Y) = [6] X + [5] Y + [0]
>= [5] X + [0]
= activate(X)
primes() = [7]
>= [6]
= sieve(from(s(s(0()))))
s(X) = [1] X + [0]
>= [1] X + [0]
= n__s(X)
sieve(X) = [1] X + [4]
>= [1] X + [1]
= n__sieve(X)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1.1.1.1.1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
activate(n__filter(X1,X2)) -> filter(activate(X1),activate(X2))
Weak DP Rules:
Weak TRS Rules:
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))
activate(n__sieve(X)) -> sieve(activate(X))
cons(X1,X2) -> n__cons(X1,X2)
filter(X1,X2) -> n__filter(X1,X2)
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
if(false(),X,Y) -> activate(Y)
if(true(),X,Y) -> activate(X)
primes() -> sieve(from(s(s(0()))))
s(X) -> n__s(X)
sieve(X) -> n__sieve(X)
Signature:
{activate/1,cons/2,filter/2,from/1,head/1,if/3,primes/0,s/1,sieve/1,tail/1} / {0/0,divides/2,false/0,n__cons/2,n__filter/2,n__from/1,n__s/1,n__sieve/1,true/0}
Obligation:
Innermost
basic terms: {activate,cons,filter,from,head,if,primes,s,sieve,tail}/{0,divides,false,n__cons,n__filter,n__from,n__s,n__sieve,true}
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(cons) = {1},
uargs(filter) = {1,2},
uargs(from) = {1},
uargs(s) = {1},
uargs(sieve) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [4]
p(activate) = [3] x1 + [0]
p(cons) = [1] x1 + [0]
p(divides) = [0]
p(false) = [4]
p(filter) = [1] x1 + [1] x2 + [2]
p(from) = [1] x1 + [3]
p(head) = [2] x1 + [0]
p(if) = [1] x1 + [5] x2 + [3] x3 + [0]
p(n__cons) = [1] x1 + [0]
p(n__filter) = [1] x1 + [1] x2 + [1]
p(n__from) = [1] x1 + [1]
p(n__s) = [1] x1 + [0]
p(n__sieve) = [1] x1 + [0]
p(primes) = [7]
p(s) = [1] x1 + [0]
p(sieve) = [1] x1 + [0]
p(tail) = [0]
p(true) = [5]
Following rules are strictly oriented:
activate(n__filter(X1,X2)) = [3] X1 + [3] X2 + [3]
> [3] X1 + [3] X2 + [2]
= filter(activate(X1)
,activate(X2))
Following rules are (at-least) weakly oriented:
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))
activate(n__sieve(X)) = [3] X + [0]
>= [3] X + [0]
= sieve(activate(X))
cons(X1,X2) = [1] X1 + [0]
>= [1] X1 + [0]
= n__cons(X1,X2)
filter(X1,X2) = [1] X1 + [1] X2 + [2]
>= [1] X1 + [1] X2 + [1]
= n__filter(X1,X2)
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)
if(false(),X,Y) = [5] X + [3] Y + [4]
>= [3] Y + [0]
= activate(Y)
if(true(),X,Y) = [5] X + [3] Y + [5]
>= [3] X + [0]
= activate(X)
primes() = [7]
>= [7]
= sieve(from(s(s(0()))))
s(X) = [1] X + [0]
>= [1] X + [0]
= n__s(X)
sieve(X) = [1] X + [0]
>= [1] X + [0]
= n__sieve(X)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1.1.1.1.1.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
activate(X) -> X
activate(n__cons(X1,X2)) -> cons(activate(X1),X2)
activate(n__filter(X1,X2)) -> filter(activate(X1),activate(X2))
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
activate(n__sieve(X)) -> sieve(activate(X))
cons(X1,X2) -> n__cons(X1,X2)
filter(X1,X2) -> n__filter(X1,X2)
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
if(false(),X,Y) -> activate(Y)
if(true(),X,Y) -> activate(X)
primes() -> sieve(from(s(s(0()))))
s(X) -> n__s(X)
sieve(X) -> n__sieve(X)
Signature:
{activate/1,cons/2,filter/2,from/1,head/1,if/3,primes/0,s/1,sieve/1,tail/1} / {0/0,divides/2,false/0,n__cons/2,n__filter/2,n__from/1,n__s/1,n__sieve/1,true/0}
Obligation:
Innermost
basic terms: {activate,cons,filter,from,head,if,primes,s,sieve,tail}/{0,divides,false,n__cons,n__filter,n__from,n__s,n__sieve,true}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).