The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(1)).

(0) Obligation:

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

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))))

Rewrite Strategy: INNERMOST

(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

filter(cons(z0), 0, z1) → cons(0)
filter(cons(z0), s(z1), z2) → cons(z0)
sieve(cons(0)) → cons(0)
sieve(cons(s(z0))) → cons(s(z0))
nats(z0) → cons(z0)
zprimes → sieve(nats(s(s(0))))

Tuples:

ZPRIMES → c5(SIEVE(nats(s(s(0)))), NATS(s(s(0))))

S tuples:

ZPRIMES → c5(SIEVE(nats(s(s(0)))), NATS(s(s(0))))

K tuples:none
Defined Rule Symbols:

filter, sieve, nats, zprimes

Defined Pair Symbols:

ZPRIMES

Compound Symbols:

c5

(3) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 1 of 1 dangling nodes:

ZPRIMES → c5(SIEVE(nats(s(s(0)))), NATS(s(s(0))))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

filter(cons(z0), 0, z1) → cons(0)
filter(cons(z0), s(z1), z2) → cons(z0)
sieve(cons(0)) → cons(0)
sieve(cons(s(z0))) → cons(s(z0))
nats(z0) → cons(z0)
zprimes → sieve(nats(s(s(0))))

Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

filter, sieve, nats, zprimes

Defined Pair Symbols:none

Compound Symbols:none

(5) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(6) BOUNDS(O(1), O(1))