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:

from(X) → cons(X, n__from(n__s(X)))
sel(0, cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, activate(Z))
from(X) → n__from(X)
s(X) → n__s(X)
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(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:

from(X) → cons(X, n__from(n__s(X)))
sel(0', cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, activate(Z))
from(X) → n__from(X)
s(X) → n__s(X)
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(activate(X))
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(n__s(X)))
sel(0', cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, activate(Z))
from(X) → n__from(X)
s(X) → n__s(X)
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(activate(X))
activate(X) → X

Types:
from :: n__s:n__from:cons:0' → n__s:n__from:cons:0'
cons :: n__s:n__from:cons:0' → n__s:n__from:cons:0' → n__s:n__from:cons:0'
n__from :: n__s:n__from:cons:0' → n__s:n__from:cons:0'
n__s :: n__s:n__from:cons:0' → n__s:n__from:cons:0'
sel :: n__s:n__from:cons:0' → n__s:n__from:cons:0' → n__s:n__from:cons:0'
0' :: n__s:n__from:cons:0'
s :: n__s:n__from:cons:0' → n__s:n__from:cons:0'
activate :: n__s:n__from:cons:0' → n__s:n__from:cons:0'
hole_n__s:n__from:cons:0'1_0 :: n__s:n__from:cons:0'
gen_n__s:n__from:cons:0'2_0 :: Nat → n__s:n__from:cons:0'

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

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

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

(8) Obligation:

TRS:
Rules:
from(X) → cons(X, n__from(n__s(X)))
sel(0', cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, activate(Z))
from(X) → n__from(X)
s(X) → n__s(X)
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(activate(X))
activate(X) → X

Types:
from :: n__s:n__from:cons:0' → n__s:n__from:cons:0'
cons :: n__s:n__from:cons:0' → n__s:n__from:cons:0' → n__s:n__from:cons:0'
n__from :: n__s:n__from:cons:0' → n__s:n__from:cons:0'
n__s :: n__s:n__from:cons:0' → n__s:n__from:cons:0'
sel :: n__s:n__from:cons:0' → n__s:n__from:cons:0' → n__s:n__from:cons:0'
0' :: n__s:n__from:cons:0'
s :: n__s:n__from:cons:0' → n__s:n__from:cons:0'
activate :: n__s:n__from:cons:0' → n__s:n__from:cons:0'
hole_n__s:n__from:cons:0'1_0 :: n__s:n__from:cons:0'
gen_n__s:n__from:cons:0'2_0 :: Nat → n__s:n__from:cons:0'

Generator Equations:
gen_n__s:n__from:cons:0'2_0(0) ⇔ 0'
gen_n__s:n__from:cons:0'2_0(+(x, 1)) ⇔ cons(0', gen_n__s:n__from:cons:0'2_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(n__s(X)))
sel(0', cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, activate(Z))
from(X) → n__from(X)
s(X) → n__s(X)
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(activate(X))
activate(X) → X

Types:
from :: n__s:n__from:cons:0' → n__s:n__from:cons:0'
cons :: n__s:n__from:cons:0' → n__s:n__from:cons:0' → n__s:n__from:cons:0'
n__from :: n__s:n__from:cons:0' → n__s:n__from:cons:0'
n__s :: n__s:n__from:cons:0' → n__s:n__from:cons:0'
sel :: n__s:n__from:cons:0' → n__s:n__from:cons:0' → n__s:n__from:cons:0'
0' :: n__s:n__from:cons:0'
s :: n__s:n__from:cons:0' → n__s:n__from:cons:0'
activate :: n__s:n__from:cons:0' → n__s:n__from:cons:0'
hole_n__s:n__from:cons:0'1_0 :: n__s:n__from:cons:0'
gen_n__s:n__from:cons:0'2_0 :: Nat → n__s:n__from:cons:0'

Generator Equations:
gen_n__s:n__from:cons:0'2_0(0) ⇔ 0'
gen_n__s:n__from:cons:0'2_0(+(x, 1)) ⇔ cons(0', gen_n__s:n__from:cons:0'2_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(n__s(X)))
sel(0', cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, activate(Z))
from(X) → n__from(X)
s(X) → n__s(X)
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(activate(X))
activate(X) → X

Types:
from :: n__s:n__from:cons:0' → n__s:n__from:cons:0'
cons :: n__s:n__from:cons:0' → n__s:n__from:cons:0' → n__s:n__from:cons:0'
n__from :: n__s:n__from:cons:0' → n__s:n__from:cons:0'
n__s :: n__s:n__from:cons:0' → n__s:n__from:cons:0'
sel :: n__s:n__from:cons:0' → n__s:n__from:cons:0' → n__s:n__from:cons:0'
0' :: n__s:n__from:cons:0'
s :: n__s:n__from:cons:0' → n__s:n__from:cons:0'
activate :: n__s:n__from:cons:0' → n__s:n__from:cons:0'
hole_n__s:n__from:cons:0'1_0 :: n__s:n__from:cons:0'
gen_n__s:n__from:cons:0'2_0 :: Nat → n__s:n__from:cons:0'

Generator Equations:
gen_n__s:n__from:cons:0'2_0(0) ⇔ 0'
gen_n__s:n__from:cons:0'2_0(+(x, 1)) ⇔ cons(0', gen_n__s:n__from:cons:0'2_0(x))

No more defined symbols left to analyse.