The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).

(0) Obligation:

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

filter(cons(X, Y), 0, M) → cons(0, filter(Y, M, M))
filter(cons(X, Y), s(N), M) → cons(X, filter(Y, N, M))
sieve(cons(0, Y)) → cons(0, sieve(Y))
sieve(cons(s(N), Y)) → cons(s(N), sieve(filter(Y, N, N)))
nats(N) → cons(N, nats(s(N)))
zprimes → sieve(nats(s(s(0))))

Rewrite Strategy: FULL

(1) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(2) Obligation:

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

filter(cons(X, Y), 0', M) → cons(0', filter(Y, M, M))
filter(cons(X, Y), s(N), M) → cons(X, filter(Y, N, M))
sieve(cons(0', Y)) → cons(0', sieve(Y))
sieve(cons(s(N), Y)) → cons(s(N), sieve(filter(Y, N, N)))
nats(N) → cons(N, nats(s(N)))
zprimes → sieve(nats(s(s(0'))))

S is empty.
Rewrite Strategy: FULL

(3) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

Infered types.

(4) Obligation:

TRS:
Rules:
filter(cons(X, Y), 0', M) → cons(0', filter(Y, M, M))
filter(cons(X, Y), s(N), M) → cons(X, filter(Y, N, M))
sieve(cons(0', Y)) → cons(0', sieve(Y))
sieve(cons(s(N), Y)) → cons(s(N), sieve(filter(Y, N, N)))
nats(N) → cons(N, nats(s(N)))
zprimes → sieve(nats(s(s(0'))))

Types:
filter :: cons → 0':s → 0':s → cons
cons :: 0':s → cons → cons
0' :: 0':s
s :: 0':s → 0':s
sieve :: cons → cons
nats :: 0':s → cons
zprimes :: cons
hole_cons1_0 :: cons
hole_0':s2_0 :: 0':s
gen_cons3_0 :: Nat → cons
gen_0':s4_0 :: Nat → 0':s

(5) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
filter, sieve, nats

They will be analysed ascendingly in the following order:
filter < sieve

(6) Obligation:

TRS:
Rules:
filter(cons(X, Y), 0', M) → cons(0', filter(Y, M, M))
filter(cons(X, Y), s(N), M) → cons(X, filter(Y, N, M))
sieve(cons(0', Y)) → cons(0', sieve(Y))
sieve(cons(s(N), Y)) → cons(s(N), sieve(filter(Y, N, N)))
nats(N) → cons(N, nats(s(N)))
zprimes → sieve(nats(s(s(0'))))

Types:
filter :: cons → 0':s → 0':s → cons
cons :: 0':s → cons → cons
0' :: 0':s
s :: 0':s → 0':s
sieve :: cons → cons
nats :: 0':s → cons
zprimes :: cons
hole_cons1_0 :: cons
hole_0':s2_0 :: 0':s
gen_cons3_0 :: Nat → cons
gen_0':s4_0 :: Nat → 0':s

Generator Equations:
gen_cons3_0(0) ⇔ hole_cons1_0
gen_cons3_0(+(x, 1)) ⇔ cons(0', gen_cons3_0(x))
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

The following defined symbols remain to be analysed:
filter, sieve, nats

They will be analysed ascendingly in the following order:
filter < sieve

(7) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
filter(gen_cons3_0(+(1, n6_0)), gen_0':s4_0(0), gen_0':s4_0(0)) → *5_0, rt ∈ Ω(n6₀)

Induction Base:
filter(gen_cons3_0(+(1, 0)), gen_0':s4_0(0), gen_0':s4_0(0))

Induction Step:
filter(gen_cons3_0(+(1, +(n6_0, 1))), gen_0':s4_0(0), gen_0':s4_0(0)) →_R^Ω(1)
cons(0', filter(gen_cons3_0(+(1, n6_0)), gen_0':s4_0(0), gen_0':s4_0(0))) →_IH
cons(0', *5_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(8) Complex Obligation (BEST)

(9) Obligation:

TRS:
Rules:
filter(cons(X, Y), 0', M) → cons(0', filter(Y, M, M))
filter(cons(X, Y), s(N), M) → cons(X, filter(Y, N, M))
sieve(cons(0', Y)) → cons(0', sieve(Y))
sieve(cons(s(N), Y)) → cons(s(N), sieve(filter(Y, N, N)))
nats(N) → cons(N, nats(s(N)))
zprimes → sieve(nats(s(s(0'))))

Types:
filter :: cons → 0':s → 0':s → cons
cons :: 0':s → cons → cons
0' :: 0':s
s :: 0':s → 0':s
sieve :: cons → cons
nats :: 0':s → cons
zprimes :: cons
hole_cons1_0 :: cons
hole_0':s2_0 :: 0':s
gen_cons3_0 :: Nat → cons
gen_0':s4_0 :: Nat → 0':s

Lemmas:
filter(gen_cons3_0(+(1, n6_0)), gen_0':s4_0(0), gen_0':s4_0(0)) → *5_0, rt ∈ Ω(n6₀)

Generator Equations:
gen_cons3_0(0) ⇔ hole_cons1_0
gen_cons3_0(+(x, 1)) ⇔ cons(0', gen_cons3_0(x))
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

The following defined symbols remain to be analysed:
sieve, nats

(10) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
sieve(gen_cons3_0(+(1, n2748_0))) → *5_0, rt ∈ Ω(n2748₀)

Induction Base:
sieve(gen_cons3_0(+(1, 0)))

Induction Step:
sieve(gen_cons3_0(+(1, +(n2748_0, 1)))) →_R^Ω(1)
cons(0', sieve(gen_cons3_0(+(1, n2748_0)))) →_IH
cons(0', *5_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(11) Complex Obligation (BEST)

(12) Obligation:

TRS:
Rules:
filter(cons(X, Y), 0', M) → cons(0', filter(Y, M, M))
filter(cons(X, Y), s(N), M) → cons(X, filter(Y, N, M))
sieve(cons(0', Y)) → cons(0', sieve(Y))
sieve(cons(s(N), Y)) → cons(s(N), sieve(filter(Y, N, N)))
nats(N) → cons(N, nats(s(N)))
zprimes → sieve(nats(s(s(0'))))

Types:
filter :: cons → 0':s → 0':s → cons
cons :: 0':s → cons → cons
0' :: 0':s
s :: 0':s → 0':s
sieve :: cons → cons
nats :: 0':s → cons
zprimes :: cons
hole_cons1_0 :: cons
hole_0':s2_0 :: 0':s
gen_cons3_0 :: Nat → cons
gen_0':s4_0 :: Nat → 0':s

Lemmas:
filter(gen_cons3_0(+(1, n6_0)), gen_0':s4_0(0), gen_0':s4_0(0)) → *5_0, rt ∈ Ω(n6₀)
sieve(gen_cons3_0(+(1, n2748_0))) → *5_0, rt ∈ Ω(n2748₀)

Generator Equations:
gen_cons3_0(0) ⇔ hole_cons1_0
gen_cons3_0(+(x, 1)) ⇔ cons(0', gen_cons3_0(x))
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

The following defined symbols remain to be analysed:
nats

(13) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol nats.

(14) Obligation:

TRS:
Rules:
filter(cons(X, Y), 0', M) → cons(0', filter(Y, M, M))
filter(cons(X, Y), s(N), M) → cons(X, filter(Y, N, M))
sieve(cons(0', Y)) → cons(0', sieve(Y))
sieve(cons(s(N), Y)) → cons(s(N), sieve(filter(Y, N, N)))
nats(N) → cons(N, nats(s(N)))
zprimes → sieve(nats(s(s(0'))))

Types:
filter :: cons → 0':s → 0':s → cons
cons :: 0':s → cons → cons
0' :: 0':s
s :: 0':s → 0':s
sieve :: cons → cons
nats :: 0':s → cons
zprimes :: cons
hole_cons1_0 :: cons
hole_0':s2_0 :: 0':s
gen_cons3_0 :: Nat → cons
gen_0':s4_0 :: Nat → 0':s

Lemmas:
filter(gen_cons3_0(+(1, n6_0)), gen_0':s4_0(0), gen_0':s4_0(0)) → *5_0, rt ∈ Ω(n6₀)
sieve(gen_cons3_0(+(1, n2748_0))) → *5_0, rt ∈ Ω(n2748₀)

Generator Equations:
gen_cons3_0(0) ⇔ hole_cons1_0
gen_cons3_0(+(x, 1)) ⇔ cons(0', gen_cons3_0(x))
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

No more defined symbols left to analyse.

(15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
filter(gen_cons3_0(+(1, n6_0)), gen_0':s4_0(0), gen_0':s4_0(0)) → *5_0, rt ∈ Ω(n6₀)

(16) BOUNDS(n^1, INF)

(17) Obligation:

TRS:
Rules:
filter(cons(X, Y), 0', M) → cons(0', filter(Y, M, M))
filter(cons(X, Y), s(N), M) → cons(X, filter(Y, N, M))
sieve(cons(0', Y)) → cons(0', sieve(Y))
sieve(cons(s(N), Y)) → cons(s(N), sieve(filter(Y, N, N)))
nats(N) → cons(N, nats(s(N)))
zprimes → sieve(nats(s(s(0'))))

Types:
filter :: cons → 0':s → 0':s → cons
cons :: 0':s → cons → cons
0' :: 0':s
s :: 0':s → 0':s
sieve :: cons → cons
nats :: 0':s → cons
zprimes :: cons
hole_cons1_0 :: cons
hole_0':s2_0 :: 0':s
gen_cons3_0 :: Nat → cons
gen_0':s4_0 :: Nat → 0':s

Lemmas:
filter(gen_cons3_0(+(1, n6_0)), gen_0':s4_0(0), gen_0':s4_0(0)) → *5_0, rt ∈ Ω(n6₀)
sieve(gen_cons3_0(+(1, n2748_0))) → *5_0, rt ∈ Ω(n2748₀)

Generator Equations:
gen_cons3_0(0) ⇔ hole_cons1_0
gen_cons3_0(+(x, 1)) ⇔ cons(0', gen_cons3_0(x))
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

No more defined symbols left to analyse.

(18) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
filter(gen_cons3_0(+(1, n6_0)), gen_0':s4_0(0), gen_0':s4_0(0)) → *5_0, rt ∈ Ω(n6₀)

(19) BOUNDS(n^1, INF)

(20) Obligation:

TRS:
Rules:
filter(cons(X, Y), 0', M) → cons(0', filter(Y, M, M))
filter(cons(X, Y), s(N), M) → cons(X, filter(Y, N, M))
sieve(cons(0', Y)) → cons(0', sieve(Y))
sieve(cons(s(N), Y)) → cons(s(N), sieve(filter(Y, N, N)))
nats(N) → cons(N, nats(s(N)))
zprimes → sieve(nats(s(s(0'))))

Types:
filter :: cons → 0':s → 0':s → cons
cons :: 0':s → cons → cons
0' :: 0':s
s :: 0':s → 0':s
sieve :: cons → cons
nats :: 0':s → cons
zprimes :: cons
hole_cons1_0 :: cons
hole_0':s2_0 :: 0':s
gen_cons3_0 :: Nat → cons
gen_0':s4_0 :: Nat → 0':s

Lemmas:
filter(gen_cons3_0(+(1, n6_0)), gen_0':s4_0(0), gen_0':s4_0(0)) → *5_0, rt ∈ Ω(n6₀)

Generator Equations:
gen_cons3_0(0) ⇔ hole_cons1_0
gen_cons3_0(+(x, 1)) ⇔ cons(0', gen_cons3_0(x))
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

No more defined symbols left to analyse.

(21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
filter(gen_cons3_0(+(1, n6_0)), gen_0':s4_0(0), gen_0':s4_0(0)) → *5_0, rt ∈ Ω(n6₀)

(22) BOUNDS(n^1, INF)