We consider the following Problem:
Strict Trs:
{ primes() -> sieve(from(s(s(0()))))
, from(X) -> cons(X, n__from(n__s(X)))
, head(cons(X, Y)) -> X
, tail(cons(X, Y)) -> activate(Y)
, if(true(), X, Y) -> activate(X)
, if(false(), X, Y) -> activate(Y)
, 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))))
, sieve(cons(X, Y)) -> cons(X, n__filter(X, n__sieve(activate(Y))))
, from(X) -> n__from(X)
, s(X) -> n__s(X)
, filter(X1, X2) -> n__filter(X1, X2)
, cons(X1, X2) -> n__cons(X1, X2)
, sieve(X) -> n__sieve(X)
, activate(n__from(X)) -> from(activate(X))
, activate(n__s(X)) -> s(activate(X))
, activate(n__filter(X1, X2)) -> filter(activate(X1), activate(X2))
, activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
, activate(n__sieve(X)) -> sieve(activate(X))
, activate(X) -> X}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
Arguments of following rules are not normal-forms:
{ head(cons(X, Y)) -> X
, tail(cons(X, Y)) -> activate(Y)
, 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))))
, sieve(cons(X, Y)) ->
cons(X, n__filter(X, n__sieve(activate(Y))))}
All above mentioned rules can be savely removed.
We consider the following Problem:
Strict Trs:
{ primes() -> sieve(from(s(s(0()))))
, from(X) -> cons(X, n__from(n__s(X)))
, if(true(), X, Y) -> activate(X)
, if(false(), X, Y) -> activate(Y)
, from(X) -> n__from(X)
, s(X) -> n__s(X)
, filter(X1, X2) -> n__filter(X1, X2)
, cons(X1, X2) -> n__cons(X1, X2)
, sieve(X) -> n__sieve(X)
, activate(n__from(X)) -> from(activate(X))
, activate(n__s(X)) -> s(activate(X))
, activate(n__filter(X1, X2)) -> filter(activate(X1), activate(X2))
, activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
, activate(n__sieve(X)) -> sieve(activate(X))
, activate(X) -> X}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component:
{ primes() -> sieve(from(s(s(0()))))
, filter(X1, X2) -> n__filter(X1, X2)
, cons(X1, X2) -> n__cons(X1, X2)
, sieve(X) -> n__sieve(X)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(sieve) = {1}, Uargs(from) = {1}, Uargs(s) = {1},
Uargs(cons) = {1}, Uargs(n__from) = {}, Uargs(n__s) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(activate) = {},
Uargs(if) = {}, Uargs(filter) = {1, 2}, Uargs(divides) = {},
Uargs(n__filter) = {}, Uargs(n__cons) = {}, Uargs(n__sieve) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
primes() = [2]
[2]
sieve(x1) = [1 0] x1 + [1]
[0 0] [1]
from(x1) = [1 2] x1 + [0]
[0 0] [1]
s(x1) = [1 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
cons(x1, x2) = [1 1] x1 + [0 0] x2 + [1]
[0 0] [0 0] [1]
n__from(x1) = [1 0] x1 + [0]
[0 0] [0]
n__s(x1) = [1 0] x1 + [0]
[0 0] [0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
activate(x1) = [1 0] x1 + [1]
[0 0] [0]
if(x1, x2, x3) = [0 0] x1 + [1 0] x2 + [0 0] x3 + [1]
[0 0] [0 0] [0 0] [1]
true() = [0]
[0]
false() = [0]
[0]
filter(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 0] [0 0] [1]
divides(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
n__filter(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
n__cons(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
n__sieve(x1) = [1 0] x1 + [0]
[0 0] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ from(X) -> cons(X, n__from(n__s(X)))
, if(true(), X, Y) -> activate(X)
, if(false(), X, Y) -> activate(Y)
, from(X) -> n__from(X)
, s(X) -> n__s(X)
, activate(n__from(X)) -> from(activate(X))
, activate(n__s(X)) -> s(activate(X))
, activate(n__filter(X1, X2)) -> filter(activate(X1), activate(X2))
, activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
, activate(n__sieve(X)) -> sieve(activate(X))
, activate(X) -> X}
Weak Trs:
{ primes() -> sieve(from(s(s(0()))))
, filter(X1, X2) -> n__filter(X1, X2)
, cons(X1, X2) -> n__cons(X1, X2)
, sieve(X) -> n__sieve(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component:
{ from(X) -> cons(X, n__from(n__s(X)))
, from(X) -> n__from(X)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(sieve) = {1}, Uargs(from) = {1}, Uargs(s) = {1},
Uargs(cons) = {1}, Uargs(n__from) = {}, Uargs(n__s) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(activate) = {},
Uargs(if) = {}, Uargs(filter) = {1, 2}, Uargs(divides) = {},
Uargs(n__filter) = {}, Uargs(n__cons) = {}, Uargs(n__sieve) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
primes() = [3]
[2]
sieve(x1) = [1 0] x1 + [0]
[0 0] [1]
from(x1) = [1 2] x1 + [2]
[0 0] [2]
s(x1) = [1 0] x1 + [0]
[0 0] [0]
0() = [1]
[0]
cons(x1, x2) = [1 1] x1 + [0 0] x2 + [1]
[0 0] [0 0] [1]
n__from(x1) = [1 0] x1 + [0]
[0 0] [0]
n__s(x1) = [1 0] x1 + [0]
[0 0] [0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
activate(x1) = [1 0] x1 + [1]
[0 0] [0]
if(x1, x2, x3) = [0 0] x1 + [1 0] x2 + [0 0] x3 + [1]
[0 0] [0 0] [0 0] [1]
true() = [0]
[0]
false() = [0]
[0]
filter(x1, x2) = [1 0] x1 + [1 0] x2 + [3]
[0 0] [0 0] [1]
divides(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
n__filter(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
n__cons(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
n__sieve(x1) = [1 0] x1 + [0]
[0 0] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ if(true(), X, Y) -> activate(X)
, if(false(), X, Y) -> activate(Y)
, s(X) -> n__s(X)
, activate(n__from(X)) -> from(activate(X))
, activate(n__s(X)) -> s(activate(X))
, activate(n__filter(X1, X2)) -> filter(activate(X1), activate(X2))
, activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
, activate(n__sieve(X)) -> sieve(activate(X))
, activate(X) -> X}
Weak Trs:
{ from(X) -> cons(X, n__from(n__s(X)))
, from(X) -> n__from(X)
, primes() -> sieve(from(s(s(0()))))
, filter(X1, X2) -> n__filter(X1, X2)
, cons(X1, X2) -> n__cons(X1, X2)
, sieve(X) -> n__sieve(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component:
{ s(X) -> n__s(X)
, activate(X) -> X}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(sieve) = {1}, Uargs(from) = {1}, Uargs(s) = {1},
Uargs(cons) = {1}, Uargs(n__from) = {}, Uargs(n__s) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(activate) = {},
Uargs(if) = {}, Uargs(filter) = {1, 2}, Uargs(divides) = {},
Uargs(n__filter) = {}, Uargs(n__cons) = {}, Uargs(n__sieve) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
primes() = [3]
[2]
sieve(x1) = [1 0] x1 + [0]
[0 0] [1]
from(x1) = [1 0] x1 + [0]
[0 0] [1]
s(x1) = [1 1] x1 + [1]
[0 0] [1]
0() = [0]
[0]
cons(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [1]
n__from(x1) = [1 0] x1 + [0]
[0 0] [0]
n__s(x1) = [1 1] x1 + [0]
[0 0] [0]
head(x1) = [1 0] x1 + [0]
[0 1] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
activate(x1) = [1 0] x1 + [1]
[0 1] [1]
if(x1, x2, x3) = [0 0] x1 + [1 0] x2 + [0 0] x3 + [1]
[0 0] [0 0] [0 0] [1]
true() = [0]
[0]
false() = [0]
[0]
filter(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 0] [0 0] [1]
divides(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
n__filter(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
n__cons(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
n__sieve(x1) = [1 0] x1 + [0]
[0 0] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ if(true(), X, Y) -> activate(X)
, if(false(), X, Y) -> activate(Y)
, activate(n__from(X)) -> from(activate(X))
, activate(n__s(X)) -> s(activate(X))
, activate(n__filter(X1, X2)) -> filter(activate(X1), activate(X2))
, activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
, activate(n__sieve(X)) -> sieve(activate(X))}
Weak Trs:
{ s(X) -> n__s(X)
, activate(X) -> X
, from(X) -> cons(X, n__from(n__s(X)))
, from(X) -> n__from(X)
, primes() -> sieve(from(s(s(0()))))
, filter(X1, X2) -> n__filter(X1, X2)
, cons(X1, X2) -> n__cons(X1, X2)
, sieve(X) -> n__sieve(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {if(false(), X, Y) -> activate(Y)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(sieve) = {1}, Uargs(from) = {1}, Uargs(s) = {1},
Uargs(cons) = {1}, Uargs(n__from) = {}, Uargs(n__s) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(activate) = {},
Uargs(if) = {}, Uargs(filter) = {1, 2}, Uargs(divides) = {},
Uargs(n__filter) = {}, Uargs(n__cons) = {}, Uargs(n__sieve) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
primes() = [2]
[3]
sieve(x1) = [1 0] x1 + [0]
[0 1] [1]
from(x1) = [1 0] x1 + [1]
[0 0] [2]
s(x1) = [1 1] x1 + [0]
[0 0] [0]
0() = [0]
[0]
cons(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [1]
n__from(x1) = [1 0] x1 + [0]
[0 0] [0]
n__s(x1) = [1 1] x1 + [0]
[0 0] [0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
activate(x1) = [1 0] x1 + [0]
[0 1] [0]
if(x1, x2, x3) = [0 0] x1 + [1 0] x2 + [1 0] x3 + [1]
[0 0] [0 0] [0 1] [1]
true() = [0]
[0]
false() = [0]
[0]
filter(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 1] [1]
divides(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
n__filter(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 0] [0]
n__cons(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
n__sieve(x1) = [1 0] x1 + [0]
[0 0] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ if(true(), X, Y) -> activate(X)
, activate(n__from(X)) -> from(activate(X))
, activate(n__s(X)) -> s(activate(X))
, activate(n__filter(X1, X2)) -> filter(activate(X1), activate(X2))
, activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
, activate(n__sieve(X)) -> sieve(activate(X))}
Weak Trs:
{ if(false(), X, Y) -> activate(Y)
, s(X) -> n__s(X)
, activate(X) -> X
, from(X) -> cons(X, n__from(n__s(X)))
, from(X) -> n__from(X)
, primes() -> sieve(from(s(s(0()))))
, filter(X1, X2) -> n__filter(X1, X2)
, cons(X1, X2) -> n__cons(X1, X2)
, sieve(X) -> n__sieve(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {if(true(), X, Y) -> activate(X)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(sieve) = {1}, Uargs(from) = {1}, Uargs(s) = {1},
Uargs(cons) = {1}, Uargs(n__from) = {}, Uargs(n__s) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(activate) = {},
Uargs(if) = {}, Uargs(filter) = {1, 2}, Uargs(divides) = {},
Uargs(n__filter) = {}, Uargs(n__cons) = {}, Uargs(n__sieve) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
primes() = [0]
[3]
sieve(x1) = [1 0] x1 + [0]
[0 1] [1]
from(x1) = [1 0] x1 + [0]
[0 0] [2]
s(x1) = [1 1] x1 + [0]
[0 0] [0]
0() = [0]
[0]
cons(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [1]
n__from(x1) = [1 0] x1 + [0]
[0 0] [1]
n__s(x1) = [1 1] x1 + [0]
[0 0] [0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
activate(x1) = [1 0] x1 + [0]
[0 1] [0]
if(x1, x2, x3) = [0 0] x1 + [1 0] x2 + [1 0] x3 + [1]
[0 0] [0 1] [0 1] [1]
true() = [0]
[0]
false() = [0]
[0]
filter(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 0] [0 1] [1]
divides(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
n__filter(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 0] [0 1] [1]
n__cons(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
n__sieve(x1) = [1 0] x1 + [0]
[0 0] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ activate(n__from(X)) -> from(activate(X))
, activate(n__s(X)) -> s(activate(X))
, activate(n__filter(X1, X2)) -> filter(activate(X1), activate(X2))
, activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
, activate(n__sieve(X)) -> sieve(activate(X))}
Weak Trs:
{ if(true(), X, Y) -> activate(X)
, if(false(), X, Y) -> activate(Y)
, s(X) -> n__s(X)
, activate(X) -> X
, from(X) -> cons(X, n__from(n__s(X)))
, from(X) -> n__from(X)
, primes() -> sieve(from(s(s(0()))))
, filter(X1, X2) -> n__filter(X1, X2)
, cons(X1, X2) -> n__cons(X1, X2)
, sieve(X) -> n__sieve(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict Trs:
{ activate(n__from(X)) -> from(activate(X))
, activate(n__s(X)) -> s(activate(X))
, activate(n__filter(X1, X2)) -> filter(activate(X1), activate(X2))
, activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
, activate(n__sieve(X)) -> sieve(activate(X))}
Weak Trs:
{ if(true(), X, Y) -> activate(X)
, if(false(), X, Y) -> activate(Y)
, s(X) -> n__s(X)
, activate(X) -> X
, from(X) -> cons(X, n__from(n__s(X)))
, from(X) -> n__from(X)
, primes() -> sieve(from(s(s(0()))))
, filter(X1, X2) -> n__filter(X1, X2)
, cons(X1, X2) -> n__cons(X1, X2)
, sieve(X) -> n__sieve(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The following argument positions are usable:
Uargs(sieve) = {1}, Uargs(from) = {1}, Uargs(s) = {1},
Uargs(cons) = {1}, Uargs(n__from) = {}, Uargs(n__s) = {},
Uargs(activate) = {}, Uargs(if) = {}, Uargs(filter) = {1, 2},
Uargs(n__filter) = {}, Uargs(n__cons) = {}, Uargs(n__sieve) = {}
We have the following constructor-based EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
primes() = [3]
[2]
sieve(x1) = [1 0] x1 + [1]
[0 1] [0]
from(x1) = [1 2] x1 + [0]
[0 0] [2]
s(x1) = [1 3] x1 + [1]
[0 0] [0]
0() = [0]
[0]
cons(x1, x2) = [1 1] x1 + [0 0] x2 + [0]
[0 0] [0 0] [1]
n__from(x1) = [1 2] x1 + [0]
[0 0] [2]
n__s(x1) = [1 3] x1 + [1]
[0 0] [0]
activate(x1) = [3 2] x1 + [0]
[0 1] [0]
if(x1, x2, x3) = [2 2] x1 + [3 2] x2 + [3 2] x3 + [1]
[1 3] [0 2] [0 2] [0]
true() = [2]
[2]
false() = [0]
[0]
filter(x1, x2) = [1 0] x1 + [1 3] x2 + [1]
[0 1] [0 0] [3]
n__filter(x1, x2) = [1 0] x1 + [1 3] x2 + [1]
[0 1] [0 0] [3]
n__cons(x1, x2) = [1 1] x1 + [0 0] x2 + [0]
[0 0] [0 0] [1]
n__sieve(x1) = [1 0] x1 + [1]
[0 1] [0]
Hurray, we answered YES(?,O(n^1))