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) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
activate(n__first(s(X10682_3), cons(Y10683_3, Z10684_3))) →⁺ cons(Y10683_3, n__first(X10682_3, activate(Z10684_3)))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1,1].
The pumping substitution is [Z10684_3 / n__first(s(X10682_3), cons(Y10683_3, Z10684_3))].
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:

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

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

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

(6) 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

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

Infered types.

(8) Obligation:

TRS:
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

Types:
first :: 0':s → nil:cons:n__first:n__from → nil:cons:n__first:n__from
0' :: 0':s
nil :: nil:cons:n__first:n__from
s :: 0':s → 0':s
cons :: nil:cons:n__first:n__from → nil:cons:n__first:n__from
n__first :: 0':s → nil:cons:n__first:n__from → nil:cons:n__first:n__from
activate :: nil:cons:n__first:n__from → nil:cons:n__first:n__from
from :: nil:cons:n__first:n__from
n__from :: nil:cons:n__first:n__from
hole_nil:cons:n__first:n__from1_0 :: nil:cons:n__first:n__from
hole_0':s2_0 :: 0':s
gen_nil:cons:n__first:n__from3_0 :: Nat → nil:cons:n__first:n__from
gen_0':s4_0 :: Nat → 0':s

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

Heuristically decided to analyse the following defined symbols:
activate

(10) Obligation:

TRS:
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

Types:
first :: 0':s → nil:cons:n__first:n__from → nil:cons:n__first:n__from
0' :: 0':s
nil :: nil:cons:n__first:n__from
s :: 0':s → 0':s
cons :: nil:cons:n__first:n__from → nil:cons:n__first:n__from
n__first :: 0':s → nil:cons:n__first:n__from → nil:cons:n__first:n__from
activate :: nil:cons:n__first:n__from → nil:cons:n__first:n__from
from :: nil:cons:n__first:n__from
n__from :: nil:cons:n__first:n__from
hole_nil:cons:n__first:n__from1_0 :: nil:cons:n__first:n__from
hole_0':s2_0 :: 0':s
gen_nil:cons:n__first:n__from3_0 :: Nat → nil:cons:n__first:n__from
gen_0':s4_0 :: Nat → 0':s

Generator Equations:
gen_nil:cons:n__first:n__from3_0(0) ⇔ n__from
gen_nil:cons:n__first:n__from3_0(+(x, 1)) ⇔ cons(gen_nil:cons:n__first:n__from3_0(x))
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

The following defined symbols remain to be analysed:
activate

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

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

(12) Obligation:

TRS:
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

Types:
first :: 0':s → nil:cons:n__first:n__from → nil:cons:n__first:n__from
0' :: 0':s
nil :: nil:cons:n__first:n__from
s :: 0':s → 0':s
cons :: nil:cons:n__first:n__from → nil:cons:n__first:n__from
n__first :: 0':s → nil:cons:n__first:n__from → nil:cons:n__first:n__from
activate :: nil:cons:n__first:n__from → nil:cons:n__first:n__from
from :: nil:cons:n__first:n__from
n__from :: nil:cons:n__first:n__from
hole_nil:cons:n__first:n__from1_0 :: nil:cons:n__first:n__from
hole_0':s2_0 :: 0':s
gen_nil:cons:n__first:n__from3_0 :: Nat → nil:cons:n__first:n__from
gen_0':s4_0 :: Nat → 0':s

Generator Equations:
gen_nil:cons:n__first:n__from3_0(0) ⇔ n__from
gen_nil:cons:n__first:n__from3_0(+(x, 1)) ⇔ cons(gen_nil:cons:n__first:n__from3_0(x))
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

No more defined symbols left to analyse.