The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(2^n, INF).

(0) Obligation:

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

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

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(2ⁿ):
The rewrite sequence
activate(n__from(X)) →⁺ cons(activate(X), n__from(n__s(activate(X))))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X / n__from(X)].
The result substitution is [ ].

The rewrite sequence
activate(n__from(X)) →⁺ cons(activate(X), n__from(n__s(activate(X))))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0,0].
The pumping substitution is [X / n__from(X)].
The result substitution is [ ].

(2) BOUNDS(2^n, INF)

(3) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(4) Obligation:

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

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

S is empty.
Rewrite Strategy: FULL

(5) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
divides/1

(6) Obligation:

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

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))), 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

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(8) Obligation:

TRS:
Rules:
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))), 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

Types:
primes :: 0':n__s:n__from:n__filter:n__sieve:n__cons
sieve :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
from :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
s :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
0' :: 0':n__s:n__from:n__filter:n__sieve:n__cons
cons :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
n__from :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
n__s :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
head :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
tail :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
activate :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
if :: true:false:divides → 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
true :: true:false:divides
false :: true:false:divides
filter :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
divides :: 0':n__s:n__from:n__filter:n__sieve:n__cons → true:false:divides
n__filter :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
n__cons :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
n__sieve :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
hole_0':n__s:n__from:n__filter:n__sieve:n__cons1_0 :: 0':n__s:n__from:n__filter:n__sieve:n__cons
hole_true:false:divides2_0 :: true:false:divides
gen_0':n__s:n__from:n__filter:n__sieve:n__cons3_0 :: Nat → 0':n__s:n__from:n__filter:n__sieve:n__cons

(9) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
sieve, activate

They will be analysed ascendingly in the following order:
sieve = activate

(10) Obligation:

TRS:
Rules:
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))), 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

Types:
primes :: 0':n__s:n__from:n__filter:n__sieve:n__cons
sieve :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
from :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
s :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
0' :: 0':n__s:n__from:n__filter:n__sieve:n__cons
cons :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
n__from :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
n__s :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
head :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
tail :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
activate :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
if :: true:false:divides → 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
true :: true:false:divides
false :: true:false:divides
filter :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
divides :: 0':n__s:n__from:n__filter:n__sieve:n__cons → true:false:divides
n__filter :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
n__cons :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
n__sieve :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
hole_0':n__s:n__from:n__filter:n__sieve:n__cons1_0 :: 0':n__s:n__from:n__filter:n__sieve:n__cons
hole_true:false:divides2_0 :: true:false:divides
gen_0':n__s:n__from:n__filter:n__sieve:n__cons3_0 :: Nat → 0':n__s:n__from:n__filter:n__sieve:n__cons

Generator Equations:
gen_0':n__s:n__from:n__filter:n__sieve:n__cons3_0(0) ⇔ 0'
gen_0':n__s:n__from:n__filter:n__sieve:n__cons3_0(+(x, 1)) ⇔ n__from(gen_0':n__s:n__from:n__filter:n__sieve:n__cons3_0(x))

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

They will be analysed ascendingly in the following order:
sieve = activate

(11) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
activate(gen_0':n__s:n__from:n__filter:n__sieve:n__cons3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

Induction Base:
activate(gen_0':n__s:n__from:n__filter:n__sieve:n__cons3_0(+(1, 0)))

Induction Step:
activate(gen_0':n__s:n__from:n__filter:n__sieve:n__cons3_0(+(1, +(n5_0, 1)))) →_R^Ω(1)
from(activate(gen_0':n__s:n__from:n__filter:n__sieve:n__cons3_0(+(1, n5_0)))) →_IH
from(*4_0)

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

(12) Complex Obligation (BEST)

(13) Obligation:

TRS:
Rules:
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))), 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

Types:
primes :: 0':n__s:n__from:n__filter:n__sieve:n__cons
sieve :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
from :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
s :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
0' :: 0':n__s:n__from:n__filter:n__sieve:n__cons
cons :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
n__from :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
n__s :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
head :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
tail :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
activate :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
if :: true:false:divides → 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
true :: true:false:divides
false :: true:false:divides
filter :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
divides :: 0':n__s:n__from:n__filter:n__sieve:n__cons → true:false:divides
n__filter :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
n__cons :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
n__sieve :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
hole_0':n__s:n__from:n__filter:n__sieve:n__cons1_0 :: 0':n__s:n__from:n__filter:n__sieve:n__cons
hole_true:false:divides2_0 :: true:false:divides
gen_0':n__s:n__from:n__filter:n__sieve:n__cons3_0 :: Nat → 0':n__s:n__from:n__filter:n__sieve:n__cons

Lemmas:
activate(gen_0':n__s:n__from:n__filter:n__sieve:n__cons3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

Generator Equations:
gen_0':n__s:n__from:n__filter:n__sieve:n__cons3_0(0) ⇔ 0'
gen_0':n__s:n__from:n__filter:n__sieve:n__cons3_0(+(x, 1)) ⇔ n__from(gen_0':n__s:n__from:n__filter:n__sieve:n__cons3_0(x))

The following defined symbols remain to be analysed:
sieve

They will be analysed ascendingly in the following order:
sieve = activate

(14) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(15) Obligation:

TRS:
Rules:
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))), 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

Types:
primes :: 0':n__s:n__from:n__filter:n__sieve:n__cons
sieve :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
from :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
s :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
0' :: 0':n__s:n__from:n__filter:n__sieve:n__cons
cons :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
n__from :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
n__s :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
head :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
tail :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
activate :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
if :: true:false:divides → 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
true :: true:false:divides
false :: true:false:divides
filter :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
divides :: 0':n__s:n__from:n__filter:n__sieve:n__cons → true:false:divides
n__filter :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
n__cons :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
n__sieve :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
hole_0':n__s:n__from:n__filter:n__sieve:n__cons1_0 :: 0':n__s:n__from:n__filter:n__sieve:n__cons
hole_true:false:divides2_0 :: true:false:divides
gen_0':n__s:n__from:n__filter:n__sieve:n__cons3_0 :: Nat → 0':n__s:n__from:n__filter:n__sieve:n__cons

Lemmas:
activate(gen_0':n__s:n__from:n__filter:n__sieve:n__cons3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

Generator Equations:
gen_0':n__s:n__from:n__filter:n__sieve:n__cons3_0(0) ⇔ 0'
gen_0':n__s:n__from:n__filter:n__sieve:n__cons3_0(+(x, 1)) ⇔ n__from(gen_0':n__s:n__from:n__filter:n__sieve:n__cons3_0(x))

No more defined symbols left to analyse.

(16) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
activate(gen_0':n__s:n__from:n__filter:n__sieve:n__cons3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

(17) BOUNDS(n^1, INF)

(18) Obligation:

TRS:
Rules:
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))), 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

Types:
primes :: 0':n__s:n__from:n__filter:n__sieve:n__cons
sieve :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
from :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
s :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
0' :: 0':n__s:n__from:n__filter:n__sieve:n__cons
cons :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
n__from :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
n__s :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
head :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
tail :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
activate :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
if :: true:false:divides → 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
true :: true:false:divides
false :: true:false:divides
filter :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
divides :: 0':n__s:n__from:n__filter:n__sieve:n__cons → true:false:divides
n__filter :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
n__cons :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
n__sieve :: 0':n__s:n__from:n__filter:n__sieve:n__cons → 0':n__s:n__from:n__filter:n__sieve:n__cons
hole_0':n__s:n__from:n__filter:n__sieve:n__cons1_0 :: 0':n__s:n__from:n__filter:n__sieve:n__cons
hole_true:false:divides2_0 :: true:false:divides
gen_0':n__s:n__from:n__filter:n__sieve:n__cons3_0 :: Nat → 0':n__s:n__from:n__filter:n__sieve:n__cons

Lemmas:
activate(gen_0':n__s:n__from:n__filter:n__sieve:n__cons3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

Generator Equations:
gen_0':n__s:n__from:n__filter:n__sieve:n__cons3_0(0) ⇔ 0'
gen_0':n__s:n__from:n__filter:n__sieve:n__cons3_0(+(x, 1)) ⇔ n__from(gen_0':n__s:n__from:n__filter:n__sieve:n__cons3_0(x))

No more defined symbols left to analyse.

(19) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
activate(gen_0':n__s:n__from:n__filter:n__sieve:n__cons3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

(20) BOUNDS(n^1, INF)