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:

first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z)))
from(X) → cons(X, n__from(s(X)))
first(X1, X2) → n__first(X1, X2)
from(X) → n__from(X)
activate(n__first(X1, X2)) → first(X1, X2)
activate(n__from(X)) → from(X)
activate(X) → X

Rewrite Strategy: FULL

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

Transformed TRS to relative TRS where S is empty.

(2) Obligation:

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

first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z)))
from(X) → cons(X, n__from(s(X)))
first(X1, X2) → n__first(X1, X2)
from(X) → n__from(X)
activate(n__first(X1, X2)) → first(X1, X2)
activate(n__from(X)) → from(X)
activate(X) → X

S is empty.
Rewrite Strategy: FULL

(3) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
cons/0
from/0
n__from/0

(4) Obligation:

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

first(0, X) → nil
first(s(X), cons(Z)) → cons(n__first(X, activate(Z)))
from → cons(n__from)
first(X1, X2) → n__first(X1, X2)
from → n__from
activate(n__first(X1, X2)) → first(X1, X2)
activate(n__from) → from
activate(X) → X

S is empty.
Rewrite Strategy: FULL

(5) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
activate(n__first(s(X7042_3), cons(Z7043_3))) →⁺ cons(n__first(X7042_3, activate(Z7043_3)))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,1].
The pumping substitution is [Z7043_3 / n__first(s(X7042_3), cons(Z7043_3))].
The result substitution is [ ].

(6) BOUNDS(n^1, INF)