We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).
Strict Trs:
{ filter(cons(X), 0(), M) -> cons(0())
, filter(cons(X), s(N), M) -> cons(X)
, sieve(cons(0())) -> cons(0())
, sieve(cons(s(N))) -> cons(s(N))
, nats(N) -> cons(N)
, zprimes() -> sieve(nats(s(s(0())))) }
Obligation:
runtime complexity
Answer:
YES(O(1),O(1))
The input is overlay and right-linear. Switching to innermost
rewriting.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).
Strict Trs:
{ filter(cons(X), 0(), M) -> cons(0())
, filter(cons(X), s(N), M) -> cons(X)
, sieve(cons(0())) -> cons(0())
, sieve(cons(s(N))) -> cons(s(N))
, nats(N) -> cons(N)
, zprimes() -> sieve(nats(s(s(0())))) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(1))
We add the following weak dependency pairs:
Strict DPs:
{ filter^#(cons(X), 0(), M) -> c_1()
, filter^#(cons(X), s(N), M) -> c_2()
, sieve^#(cons(0())) -> c_3()
, sieve^#(cons(s(N))) -> c_4()
, nats^#(N) -> c_5()
, zprimes^#() -> c_6(sieve^#(nats(s(s(0()))))) }
and mark the set of starting terms.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).
Strict DPs:
{ filter^#(cons(X), 0(), M) -> c_1()
, filter^#(cons(X), s(N), M) -> c_2()
, sieve^#(cons(0())) -> c_3()
, sieve^#(cons(s(N))) -> c_4()
, nats^#(N) -> c_5()
, zprimes^#() -> c_6(sieve^#(nats(s(s(0()))))) }
Strict Trs:
{ filter(cons(X), 0(), M) -> cons(0())
, filter(cons(X), s(N), M) -> cons(X)
, sieve(cons(0())) -> cons(0())
, sieve(cons(s(N))) -> cons(s(N))
, nats(N) -> cons(N)
, zprimes() -> sieve(nats(s(s(0())))) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(1))
We replace rewrite rules by usable rules:
Strict Usable Rules: { nats(N) -> cons(N) }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).
Strict DPs:
{ filter^#(cons(X), 0(), M) -> c_1()
, filter^#(cons(X), s(N), M) -> c_2()
, sieve^#(cons(0())) -> c_3()
, sieve^#(cons(s(N))) -> c_4()
, nats^#(N) -> c_5()
, zprimes^#() -> c_6(sieve^#(nats(s(s(0()))))) }
Strict Trs: { nats(N) -> cons(N) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(1))
The weightgap principle applies (using the following constant
growth matrix-interpretation)
The following argument positions are usable:
Uargs(sieve^#) = {1}, Uargs(c_6) = {1}
TcT has computed the following constructor-restricted matrix
interpretation.
[cons](x1) = [0]
[0]
[0] = [0]
[0]
[s](x1) = [0]
[0]
[nats](x1) = [2]
[0]
[filter^#](x1, x2, x3) = [0]
[0]
[c_1] = [0]
[0]
[c_2] = [0]
[0]
[sieve^#](x1) = [2 0] x1 + [0]
[0 0] [0]
[c_3] = [0]
[0]
[c_4] = [0]
[0]
[nats^#](x1) = [0]
[0]
[c_5] = [0]
[0]
[zprimes^#] = [0]
[0]
[c_6](x1) = [1 0] x1 + [0]
[0 1] [0]
The order satisfies the following ordering constraints:
[nats(N)] = [2]
[0]
> [0]
[0]
= [cons(N)]
[filter^#(cons(X), 0(), M)] = [0]
[0]
>= [0]
[0]
= [c_1()]
[filter^#(cons(X), s(N), M)] = [0]
[0]
>= [0]
[0]
= [c_2()]
[sieve^#(cons(0()))] = [0]
[0]
>= [0]
[0]
= [c_3()]
[sieve^#(cons(s(N)))] = [0]
[0]
>= [0]
[0]
= [c_4()]
[nats^#(N)] = [0]
[0]
>= [0]
[0]
= [c_5()]
[zprimes^#()] = [0]
[0]
? [4]
[0]
= [c_6(sieve^#(nats(s(s(0())))))]
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).
Strict DPs:
{ filter^#(cons(X), 0(), M) -> c_1()
, filter^#(cons(X), s(N), M) -> c_2()
, sieve^#(cons(0())) -> c_3()
, sieve^#(cons(s(N))) -> c_4()
, nats^#(N) -> c_5()
, zprimes^#() -> c_6(sieve^#(nats(s(s(0()))))) }
Weak Trs: { nats(N) -> cons(N) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(1))
We estimate the number of application of {1,2,3,4,5} by
applications of Pre({1,2,3,4,5}) = {6}. Here rules are labeled as
follows:
DPs:
{ 1: filter^#(cons(X), 0(), M) -> c_1()
, 2: filter^#(cons(X), s(N), M) -> c_2()
, 3: sieve^#(cons(0())) -> c_3()
, 4: sieve^#(cons(s(N))) -> c_4()
, 5: nats^#(N) -> c_5()
, 6: zprimes^#() -> c_6(sieve^#(nats(s(s(0()))))) }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).
Strict DPs: { zprimes^#() -> c_6(sieve^#(nats(s(s(0()))))) }
Weak DPs:
{ filter^#(cons(X), 0(), M) -> c_1()
, filter^#(cons(X), s(N), M) -> c_2()
, sieve^#(cons(0())) -> c_3()
, sieve^#(cons(s(N))) -> c_4()
, nats^#(N) -> c_5() }
Weak Trs: { nats(N) -> cons(N) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(1))
We estimate the number of application of {1} by applications of
Pre({1}) = {}. Here rules are labeled as follows:
DPs:
{ 1: zprimes^#() -> c_6(sieve^#(nats(s(s(0())))))
, 2: filter^#(cons(X), 0(), M) -> c_1()
, 3: filter^#(cons(X), s(N), M) -> c_2()
, 4: sieve^#(cons(0())) -> c_3()
, 5: sieve^#(cons(s(N))) -> c_4()
, 6: nats^#(N) -> c_5() }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).
Weak DPs:
{ filter^#(cons(X), 0(), M) -> c_1()
, filter^#(cons(X), s(N), M) -> c_2()
, sieve^#(cons(0())) -> c_3()
, sieve^#(cons(s(N))) -> c_4()
, nats^#(N) -> c_5()
, zprimes^#() -> c_6(sieve^#(nats(s(s(0()))))) }
Weak Trs: { nats(N) -> cons(N) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(1))
The following weak DPs constitute a sub-graph of the DG that is
closed under successors. The DPs are removed.
{ filter^#(cons(X), 0(), M) -> c_1()
, filter^#(cons(X), s(N), M) -> c_2()
, sieve^#(cons(0())) -> c_3()
, sieve^#(cons(s(N))) -> c_4()
, nats^#(N) -> c_5()
, zprimes^#() -> c_6(sieve^#(nats(s(s(0()))))) }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).
Weak Trs: { nats(N) -> cons(N) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(1))
No rule is usable, rules are removed from the input problem.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).
Rules: Empty
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(1))
Empty rules are trivially bounded
Hurray, we answered YES(O(1),O(1))