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:

from(X) → cons(X, n__from(s(X)))
head(cons(X, XS)) → X
2nd(cons(X, XS)) → head(activate(XS))
take(0, XS) → nil
take(s(N), cons(X, XS)) → cons(X, n__take(N, activate(XS)))
sel(0, cons(X, XS)) → X
sel(s(N), cons(X, XS)) → sel(N, activate(XS))
from(X) → n__from(X)
take(X1, X2) → n__take(X1, X2)
activate(n__from(X)) → from(X)
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
sel(s(N), cons(X, XS)) →⁺ sel(N, XS)
gives rise to a decreasing loop by considering the right hand sides subterm at position [].
The pumping substitution is [N / s(N), XS / cons(X, XS)].
The result substitution is [ ].

(2) BOUNDS(n^1, 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:

from(X) → cons(X, n__from(s(X)))
head(cons(X, XS)) → X
2nd(cons(X, XS)) → head(activate(XS))
take(0', XS) → nil
take(s(N), cons(X, XS)) → cons(X, n__take(N, activate(XS)))
sel(0', cons(X, XS)) → X
sel(s(N), cons(X, XS)) → sel(N, activate(XS))
from(X) → n__from(X)
take(X1, X2) → n__take(X1, X2)
activate(n__from(X)) → from(X)
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
from(X) → cons(X, n__from(s(X)))
head(cons(X, XS)) → X
2nd(cons(X, XS)) → head(activate(XS))
take(0', XS) → nil
take(s(N), cons(X, XS)) → cons(X, n__take(N, activate(XS)))
sel(0', cons(X, XS)) → X
sel(s(N), cons(X, XS)) → sel(N, activate(XS))
from(X) → n__from(X)
take(X1, X2) → n__take(X1, X2)
activate(n__from(X)) → from(X)
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

Types:
from :: s:0' → n__from:cons:nil:n__take
cons :: s:0' → n__from:cons:nil:n__take → n__from:cons:nil:n__take
n__from :: s:0' → n__from:cons:nil:n__take
s :: s:0' → s:0'
head :: n__from:cons:nil:n__take → s:0'
2nd :: n__from:cons:nil:n__take → s:0'
activate :: n__from:cons:nil:n__take → n__from:cons:nil:n__take
take :: s:0' → n__from:cons:nil:n__take → n__from:cons:nil:n__take
0' :: s:0'
nil :: n__from:cons:nil:n__take
n__take :: s:0' → n__from:cons:nil:n__take → n__from:cons:nil:n__take
sel :: s:0' → n__from:cons:nil:n__take → s:0'
hole_n__from:cons:nil:n__take1_0 :: n__from:cons:nil:n__take
hole_s:0'2_0 :: s:0'
gen_n__from:cons:nil:n__take3_0 :: Nat → n__from:cons:nil:n__take
gen_s:0'4_0 :: Nat → s:0'

(7) OrderProof (LOWER BOUND(ID) transformation)

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

They will be analysed ascendingly in the following order:
activate < sel

(8) Obligation:

TRS:
Rules:
from(X) → cons(X, n__from(s(X)))
head(cons(X, XS)) → X
2nd(cons(X, XS)) → head(activate(XS))
take(0', XS) → nil
take(s(N), cons(X, XS)) → cons(X, n__take(N, activate(XS)))
sel(0', cons(X, XS)) → X
sel(s(N), cons(X, XS)) → sel(N, activate(XS))
from(X) → n__from(X)
take(X1, X2) → n__take(X1, X2)
activate(n__from(X)) → from(X)
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

Types:
from :: s:0' → n__from:cons:nil:n__take
cons :: s:0' → n__from:cons:nil:n__take → n__from:cons:nil:n__take
n__from :: s:0' → n__from:cons:nil:n__take
s :: s:0' → s:0'
head :: n__from:cons:nil:n__take → s:0'
2nd :: n__from:cons:nil:n__take → s:0'
activate :: n__from:cons:nil:n__take → n__from:cons:nil:n__take
take :: s:0' → n__from:cons:nil:n__take → n__from:cons:nil:n__take
0' :: s:0'
nil :: n__from:cons:nil:n__take
n__take :: s:0' → n__from:cons:nil:n__take → n__from:cons:nil:n__take
sel :: s:0' → n__from:cons:nil:n__take → s:0'
hole_n__from:cons:nil:n__take1_0 :: n__from:cons:nil:n__take
hole_s:0'2_0 :: s:0'
gen_n__from:cons:nil:n__take3_0 :: Nat → n__from:cons:nil:n__take
gen_s:0'4_0 :: Nat → s:0'

Generator Equations:
gen_n__from:cons:nil:n__take3_0(0) ⇔ n__from(0')
gen_n__from:cons:nil:n__take3_0(+(x, 1)) ⇔ cons(0', gen_n__from:cons:nil:n__take3_0(x))
gen_s:0'4_0(0) ⇔ 0'
gen_s:0'4_0(+(x, 1)) ⇔ s(gen_s:0'4_0(x))

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

They will be analysed ascendingly in the following order:
activate < sel

(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(10) Obligation:

TRS:
Rules:
from(X) → cons(X, n__from(s(X)))
head(cons(X, XS)) → X
2nd(cons(X, XS)) → head(activate(XS))
take(0', XS) → nil
take(s(N), cons(X, XS)) → cons(X, n__take(N, activate(XS)))
sel(0', cons(X, XS)) → X
sel(s(N), cons(X, XS)) → sel(N, activate(XS))
from(X) → n__from(X)
take(X1, X2) → n__take(X1, X2)
activate(n__from(X)) → from(X)
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

Types:
from :: s:0' → n__from:cons:nil:n__take
cons :: s:0' → n__from:cons:nil:n__take → n__from:cons:nil:n__take
n__from :: s:0' → n__from:cons:nil:n__take
s :: s:0' → s:0'
head :: n__from:cons:nil:n__take → s:0'
2nd :: n__from:cons:nil:n__take → s:0'
activate :: n__from:cons:nil:n__take → n__from:cons:nil:n__take
take :: s:0' → n__from:cons:nil:n__take → n__from:cons:nil:n__take
0' :: s:0'
nil :: n__from:cons:nil:n__take
n__take :: s:0' → n__from:cons:nil:n__take → n__from:cons:nil:n__take
sel :: s:0' → n__from:cons:nil:n__take → s:0'
hole_n__from:cons:nil:n__take1_0 :: n__from:cons:nil:n__take
hole_s:0'2_0 :: s:0'
gen_n__from:cons:nil:n__take3_0 :: Nat → n__from:cons:nil:n__take
gen_s:0'4_0 :: Nat → s:0'

Generator Equations:
gen_n__from:cons:nil:n__take3_0(0) ⇔ n__from(0')
gen_n__from:cons:nil:n__take3_0(+(x, 1)) ⇔ cons(0', gen_n__from:cons:nil:n__take3_0(x))
gen_s:0'4_0(0) ⇔ 0'
gen_s:0'4_0(+(x, 1)) ⇔ s(gen_s:0'4_0(x))

The following defined symbols remain to be analysed:
sel

(11) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(12) Obligation:

TRS:
Rules:
from(X) → cons(X, n__from(s(X)))
head(cons(X, XS)) → X
2nd(cons(X, XS)) → head(activate(XS))
take(0', XS) → nil
take(s(N), cons(X, XS)) → cons(X, n__take(N, activate(XS)))
sel(0', cons(X, XS)) → X
sel(s(N), cons(X, XS)) → sel(N, activate(XS))
from(X) → n__from(X)
take(X1, X2) → n__take(X1, X2)
activate(n__from(X)) → from(X)
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

Types:
from :: s:0' → n__from:cons:nil:n__take
cons :: s:0' → n__from:cons:nil:n__take → n__from:cons:nil:n__take
n__from :: s:0' → n__from:cons:nil:n__take
s :: s:0' → s:0'
head :: n__from:cons:nil:n__take → s:0'
2nd :: n__from:cons:nil:n__take → s:0'
activate :: n__from:cons:nil:n__take → n__from:cons:nil:n__take
take :: s:0' → n__from:cons:nil:n__take → n__from:cons:nil:n__take
0' :: s:0'
nil :: n__from:cons:nil:n__take
n__take :: s:0' → n__from:cons:nil:n__take → n__from:cons:nil:n__take
sel :: s:0' → n__from:cons:nil:n__take → s:0'
hole_n__from:cons:nil:n__take1_0 :: n__from:cons:nil:n__take
hole_s:0'2_0 :: s:0'
gen_n__from:cons:nil:n__take3_0 :: Nat → n__from:cons:nil:n__take
gen_s:0'4_0 :: Nat → s:0'

Generator Equations:
gen_n__from:cons:nil:n__take3_0(0) ⇔ n__from(0')
gen_n__from:cons:nil:n__take3_0(+(x, 1)) ⇔ cons(0', gen_n__from:cons:nil:n__take3_0(x))
gen_s:0'4_0(0) ⇔ 0'
gen_s:0'4_0(+(x, 1)) ⇔ s(gen_s:0'4_0(x))

No more defined symbols left to analyse.