Runtime Complexity TRS:
The TRS R consists of the following rules:

a__primes → a__sieve(a__from(s(s(0))))
a__from(X) → cons(mark(X), from(s(X)))
a__head(cons(X, Y)) → mark(X)
a__tail(cons(X, Y)) → mark(Y)
a__if(true, X, Y) → mark(X)
a__if(false, X, Y) → mark(Y)
a__filter(s(s(X)), cons(Y, Z)) → a__if(divides(s(s(mark(X))), mark(Y)), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y))))
a__sieve(cons(X, Y)) → cons(mark(X), filter(X, sieve(Y)))
mark(primes) → a__primes
mark(sieve(X)) → a__sieve(mark(X))
mark(from(X)) → a__from(mark(X))
mark(head(X)) → a__head(mark(X))
mark(tail(X)) → a__tail(mark(X))
mark(if(X1, X2, X3)) → a__if(mark(X1), X2, X3)
mark(filter(X1, X2)) → a__filter(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(true) → true
mark(false) → false
mark(divides(X1, X2)) → divides(mark(X1), mark(X2))
a__primes → primes
a__sieve(X) → sieve(X)
a__from(X) → from(X)
a__head(X) → head(X)
a__tail(X) → tail(X)
a__if(X1, X2, X3) → if(X1, X2, X3)
a__filter(X1, X2) → filter(X1, X2)

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

Runtime Complexity TRS:
The TRS R consists of the following rules:

a__primes' → a__sieve'(a__from'(s'(s'(0'))))
a__from'(X) → cons'(mark'(X), from'(s'(X)))
a__head'(cons'(X, Y)) → mark'(X)
a__tail'(cons'(X, Y)) → mark'(Y)
a__if'(true', X, Y) → mark'(X)
a__if'(false', X, Y) → mark'(Y)
a__filter'(s'(s'(X)), cons'(Y, Z)) → a__if'(divides'(s'(s'(mark'(X))), mark'(Y)), filter'(s'(s'(X)), Z), cons'(Y, filter'(X, sieve'(Y))))
a__sieve'(cons'(X, Y)) → cons'(mark'(X), filter'(X, sieve'(Y)))
mark'(primes') → a__primes'
mark'(sieve'(X)) → a__sieve'(mark'(X))
mark'(from'(X)) → a__from'(mark'(X))
mark'(head'(X)) → a__head'(mark'(X))
mark'(tail'(X)) → a__tail'(mark'(X))
mark'(if'(X1, X2, X3)) → a__if'(mark'(X1), X2, X3)
mark'(filter'(X1, X2)) → a__filter'(mark'(X1), mark'(X2))
mark'(s'(X)) → s'(mark'(X))
mark'(0') → 0'
mark'(cons'(X1, X2)) → cons'(mark'(X1), X2)
mark'(true') → true'
mark'(false') → false'
mark'(divides'(X1, X2)) → divides'(mark'(X1), mark'(X2))
a__primes' → primes'
a__sieve'(X) → sieve'(X)
a__from'(X) → from'(X)
a__head'(X) → head'(X)
a__tail'(X) → tail'(X)
a__if'(X1, X2, X3) → if'(X1, X2, X3)
a__filter'(X1, X2) → filter'(X1, X2)

Rewrite Strategy: INNERMOST

Infered types.

Rules:
a__primes' → a__sieve'(a__from'(s'(s'(0'))))
a__from'(X) → cons'(mark'(X), from'(s'(X)))
a__head'(cons'(X, Y)) → mark'(X)
a__tail'(cons'(X, Y)) → mark'(Y)
a__if'(true', X, Y) → mark'(X)
a__if'(false', X, Y) → mark'(Y)
a__filter'(s'(s'(X)), cons'(Y, Z)) → a__if'(divides'(s'(s'(mark'(X))), mark'(Y)), filter'(s'(s'(X)), Z), cons'(Y, filter'(X, sieve'(Y))))
a__sieve'(cons'(X, Y)) → cons'(mark'(X), filter'(X, sieve'(Y)))
mark'(primes') → a__primes'
mark'(sieve'(X)) → a__sieve'(mark'(X))
mark'(from'(X)) → a__from'(mark'(X))
mark'(head'(X)) → a__head'(mark'(X))
mark'(tail'(X)) → a__tail'(mark'(X))
mark'(if'(X1, X2, X3)) → a__if'(mark'(X1), X2, X3)
mark'(filter'(X1, X2)) → a__filter'(mark'(X1), mark'(X2))
mark'(s'(X)) → s'(mark'(X))
mark'(0') → 0'
mark'(cons'(X1, X2)) → cons'(mark'(X1), X2)
mark'(true') → true'
mark'(false') → false'
mark'(divides'(X1, X2)) → divides'(mark'(X1), mark'(X2))
a__primes' → primes'
a__sieve'(X) → sieve'(X)
a__from'(X) → from'(X)
a__head'(X) → head'(X)
a__tail'(X) → tail'(X)
a__if'(X1, X2, X3) → if'(X1, X2, X3)
a__filter'(X1, X2) → filter'(X1, X2)

Types:
a__primes' :: 0':s':from':cons':true':false':divides':filter':sieve':primes':head':tail':if'
a__sieve' :: 0':s':from':cons':true':false':divides':filter':sieve':primes':head':tail':if' → 0':s':from':cons':true':false':divides':filter':sieve':primes':head':tail':if'
a__from' :: 0':s':from':cons':true':false':divides':filter':sieve':primes':head':tail':if' → 0':s':from':cons':true':false':divides':filter':sieve':primes':head':tail':if'
s' :: 0':s':from':cons':true':false':divides':filter':sieve':primes':head':tail':if' → 0':s':from':cons':true':false':divides':filter':sieve':primes':head':tail':if'
0' :: 0':s':from':cons':true':false':divides':filter':sieve':primes':head':tail':if'
cons' :: 0':s':from':cons':true':false':divides':filter':sieve':primes':head':tail':if' → 0':s':from':cons':true':false':divides':filter':sieve':primes':head':tail':if' → 0':s':from':cons':true':false':divides':filter':sieve':primes':head':tail':if'
mark' :: 0':s':from':cons':true':false':divides':filter':sieve':primes':head':tail':if' → 0':s':from':cons':true':false':divides':filter':sieve':primes':head':tail':if'
from' :: 0':s':from':cons':true':false':divides':filter':sieve':primes':head':tail':if' → 0':s':from':cons':true':false':divides':filter':sieve':primes':head':tail':if'
a__head' :: 0':s':from':cons':true':false':divides':filter':sieve':primes':head':tail':if' → 0':s':from':cons':true':false':divides':filter':sieve':primes':head':tail':if'
a__tail' :: 0':s':from':cons':true':false':divides':filter':sieve':primes':head':tail':if' → 0':s':from':cons':true':false':divides':filter':sieve':primes':head':tail':if'
a__if' :: 0':s':from':cons':true':false':divides':filter':sieve':primes':head':tail':if' → 0':s':from':cons':true':false':divides':filter':sieve':primes':head':tail':if' → 0':s':from':cons':true':false':divides':filter':sieve':primes':head':tail':if' → 0':s':from':cons':true':false':divides':filter':sieve':primes':head':tail':if'
true' :: 0':s':from':cons':true':false':divides':filter':sieve':primes':head':tail':if'
false' :: 0':s':from':cons':true':false':divides':filter':sieve':primes':head':tail':if'
a__filter' :: 0':s':from':cons':true':false':divides':filter':sieve':primes':head':tail':if' → 0':s':from':cons':true':false':divides':filter':sieve':primes':head':tail':if' → 0':s':from':cons':true':false':divides':filter':sieve':primes':head':tail':if'
divides' :: 0':s':from':cons':true':false':divides':filter':sieve':primes':head':tail':if' → 0':s':from':cons':true':false':divides':filter':sieve':primes':head':tail':if' → 0':s':from':cons':true':false':divides':filter':sieve':primes':head':tail':if'
filter' :: 0':s':from':cons':true':false':divides':filter':sieve':primes':head':tail':if' → 0':s':from':cons':true':false':divides':filter':sieve':primes':head':tail':if' → 0':s':from':cons':true':false':divides':filter':sieve':primes':head':tail':if'
sieve' :: 0':s':from':cons':true':false':divides':filter':sieve':primes':head':tail':if' → 0':s':from':cons':true':false':divides':filter':sieve':primes':head':tail':if'
primes' :: 0':s':from':cons':true':false':divides':filter':sieve':primes':head':tail':if'
head' :: 0':s':from':cons':true':false':divides':filter':sieve':primes':head':tail':if' → 0':s':from':cons':true':false':divides':filter':sieve':primes':head':tail':if'
tail' :: 0':s':from':cons':true':false':divides':filter':sieve':primes':head':tail':if' → 0':s':from':cons':true':false':divides':filter':sieve':primes':head':tail':if'
if' :: 0':s':from':cons':true':false':divides':filter':sieve':primes':head':tail':if' → 0':s':from':cons':true':false':divides':filter':sieve':primes':head':tail':if' → 0':s':from':cons':true':false':divides':filter':sieve':primes':head':tail':if' → 0':s':from':cons':true':false':divides':filter':sieve':primes':head':tail':if'
_hole_0':s':from':cons':true':false':divides':filter':sieve':primes':head':tail':if'1 :: 0':s':from':cons':true':false':divides':filter':sieve':primes':head':tail':if'
_gen_0':s':from':cons':true':false':divides':filter':sieve':primes':head':tail':if'2 :: Nat → 0':s':from':cons':true':false':divides':filter':sieve':primes':head':tail':if'

Heuristically decided to analyse the following defined symbols:
a__primes', a__sieve', a__from', mark', a__head', a__tail', a__if'

They will be analysed ascendingly in the following order:
a__primes' = a__sieve'
a__primes' = a__from'
a__primes' = mark'
a__primes' = a__head'
a__primes' = a__tail'
a__primes' = a__if'
a__sieve' = a__from'
a__sieve' = mark'
a__sieve' = a__head'
a__sieve' = a__tail'
a__sieve' = a__if'
a__from' = mark'
a__from' = a__head'
a__from' = a__tail'
a__from' = a__if'
mark' = a__head'
mark' = a__tail'
mark' = a__if'
a__head' = a__tail'
a__head' = a__if'
a__tail' = a__if'

Generator Equations:
_gen_0':s':from':cons':true':false':divides':filter':sieve':primes':head':tail':if'2(0) ⇔ 0'
_gen_0':s':from':cons':true':false':divides':filter':sieve':primes':head':tail':if'2(+(x, 1)) ⇔ s'(_gen_0':s':from':cons':true':false':divides':filter':sieve':primes':head':tail':if'2(x))

The following defined symbols remain to be analysed:
a__sieve', a__primes', a__from', mark', a__head', a__tail', a__if'

Could not prove a rewrite lemma for the defined symbol a__sieve'.

The following defined symbols remain to be analysed:
mark', a__primes', a__from', a__head', a__tail', a__if'

Proved the following rewrite lemma:
mark'(_gen_0':s':from':cons':true':false':divides':filter':sieve':primes':head':tail':if'2(_n16)) → _gen_0':s':from':cons':true':false':divides':filter':sieve':primes':head':tail':if'2(_n16), rt ∈ Ω(1 + _n16)

Induction Base:
mark'(_gen_0':s':from':cons':true':false':divides':filter':sieve':primes':head':tail':if'2(0)) →_R^Ω(1)
0'

Induction Step:
mark'(_gen_0':s':from':cons':true':false':divides':filter':sieve':primes':head':tail':if'2(+(_$n17, 1))) →_R^Ω(1)
s'(mark'(_gen_0':s':from':cons':true':false':divides':filter':sieve':primes':head':tail':if'2(_$n17))) →_IH
s'(_gen_0':s':from':cons':true':false':divides':filter':sieve':primes':head':tail':if'2(_$n17))

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Lemmas:
mark'(_gen_0':s':from':cons':true':false':divides':filter':sieve':primes':head':tail':if'2(_n16)) → _gen_0':s':from':cons':true':false':divides':filter':sieve':primes':head':tail':if'2(_n16), rt ∈ Ω(1 + _n16)

The following defined symbols remain to be analysed:
a__primes', a__sieve', a__from', a__head', a__tail', a__if'

Could not prove a rewrite lemma for the defined symbol a__primes'.

The following defined symbols remain to be analysed:
a__from', a__sieve', a__head', a__tail', a__if'

Could not prove a rewrite lemma for the defined symbol a__from'.

The following defined symbols remain to be analysed:
a__head', a__sieve', a__tail', a__if'

Could not prove a rewrite lemma for the defined symbol a__head'.

The following defined symbols remain to be analysed:
a__tail', a__sieve', a__if'

Could not prove a rewrite lemma for the defined symbol a__tail'.

The following defined symbols remain to be analysed:
a__if', a__sieve'

Could not prove a rewrite lemma for the defined symbol a__if'.

The following defined symbols remain to be analysed:
a__sieve'

Could not prove a rewrite lemma for the defined symbol a__sieve'.

No more defined symbols left to analyse.

The lowerbound Ω(n) was proven with the following lemma:
mark'(_gen_0':s':from':cons':true':false':divides':filter':sieve':primes':head':tail':if'2(_n16)) → _gen_0':s':from':cons':true':false':divides':filter':sieve':primes':head':tail':if'2(_n16), rt ∈ Ω(1 + _n16)