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)))
after(0, XS) → XS
after(s(N), cons(X, XS)) → after(N, activate(XS))
from(X) → n__from(X)
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
after(s(N), cons(X, XS)) →⁺ after(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)))
after(0', XS) → XS
after(s(N), cons(X, XS)) → after(N, activate(XS))
from(X) → n__from(X)
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:
from/0
cons/0
n__from/0

(6) Obligation:

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

from → cons(n__from)
after(0', XS) → XS
after(s(N), cons(XS)) → after(N, activate(XS))
from → n__from
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:
from → cons(n__from)
after(0', XS) → XS
after(s(N), cons(XS)) → after(N, activate(XS))
from → n__from
activate(n__from) → from
activate(X) → X

Types:
from :: n__from:cons
cons :: n__from:cons → n__from:cons
n__from :: n__from:cons
after :: 0':s → n__from:cons → n__from:cons
0' :: 0':s
s :: 0':s → 0':s
activate :: n__from:cons → n__from:cons
hole_n__from:cons1_0 :: n__from:cons
hole_0':s2_0 :: 0':s
gen_n__from:cons3_0 :: Nat → n__from:cons
gen_0':s4_0 :: Nat → 0':s

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

Heuristically decided to analyse the following defined symbols:
after

(10) Obligation:

TRS:
Rules:
from → cons(n__from)
after(0', XS) → XS
after(s(N), cons(XS)) → after(N, activate(XS))
from → n__from
activate(n__from) → from
activate(X) → X

Types:
from :: n__from:cons
cons :: n__from:cons → n__from:cons
n__from :: n__from:cons
after :: 0':s → n__from:cons → n__from:cons
0' :: 0':s
s :: 0':s → 0':s
activate :: n__from:cons → n__from:cons
hole_n__from:cons1_0 :: n__from:cons
hole_0':s2_0 :: 0':s
gen_n__from:cons3_0 :: Nat → n__from:cons
gen_0':s4_0 :: Nat → 0':s

Generator Equations:
gen_n__from:cons3_0(0) ⇔ n__from
gen_n__from:cons3_0(+(x, 1)) ⇔ cons(gen_n__from:cons3_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:
after

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

Proved the following rewrite lemma:
after(gen_0':s4_0(n6_0), gen_n__from:cons3_0(1)) → gen_n__from:cons3_0(1), rt ∈ Ω(1 + n6₀)

Induction Base:
after(gen_0':s4_0(0), gen_n__from:cons3_0(1)) →_R^Ω(1)
gen_n__from:cons3_0(1)

Induction Step:
after(gen_0':s4_0(+(n6_0, 1)), gen_n__from:cons3_0(1)) →_R^Ω(1)
after(gen_0':s4_0(n6_0), activate(gen_n__from:cons3_0(0))) →_R^Ω(1)
after(gen_0':s4_0(n6_0), from) →_R^Ω(1)
after(gen_0':s4_0(n6_0), cons(n__from)) →_IH
gen_n__from:cons3_0(1)

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

(12) Complex Obligation (BEST)

(13) Obligation:

TRS:
Rules:
from → cons(n__from)
after(0', XS) → XS
after(s(N), cons(XS)) → after(N, activate(XS))
from → n__from
activate(n__from) → from
activate(X) → X

Types:
from :: n__from:cons
cons :: n__from:cons → n__from:cons
n__from :: n__from:cons
after :: 0':s → n__from:cons → n__from:cons
0' :: 0':s
s :: 0':s → 0':s
activate :: n__from:cons → n__from:cons
hole_n__from:cons1_0 :: n__from:cons
hole_0':s2_0 :: 0':s
gen_n__from:cons3_0 :: Nat → n__from:cons
gen_0':s4_0 :: Nat → 0':s

Lemmas:
after(gen_0':s4_0(n6_0), gen_n__from:cons3_0(1)) → gen_n__from:cons3_0(1), rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_n__from:cons3_0(0) ⇔ n__from
gen_n__from:cons3_0(+(x, 1)) ⇔ cons(gen_n__from:cons3_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.

(14) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
after(gen_0':s4_0(n6_0), gen_n__from:cons3_0(1)) → gen_n__from:cons3_0(1), rt ∈ Ω(1 + n6₀)

(15) BOUNDS(n^1, INF)

(16) Obligation:

TRS:
Rules:
from → cons(n__from)
after(0', XS) → XS
after(s(N), cons(XS)) → after(N, activate(XS))
from → n__from
activate(n__from) → from
activate(X) → X

Types:
from :: n__from:cons
cons :: n__from:cons → n__from:cons
n__from :: n__from:cons
after :: 0':s → n__from:cons → n__from:cons
0' :: 0':s
s :: 0':s → 0':s
activate :: n__from:cons → n__from:cons
hole_n__from:cons1_0 :: n__from:cons
hole_0':s2_0 :: 0':s
gen_n__from:cons3_0 :: Nat → n__from:cons
gen_0':s4_0 :: Nat → 0':s

Lemmas:
after(gen_0':s4_0(n6_0), gen_n__from:cons3_0(1)) → gen_n__from:cons3_0(1), rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_n__from:cons3_0(0) ⇔ n__from
gen_n__from:cons3_0(+(x, 1)) ⇔ cons(gen_n__from:cons3_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.

(17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
after(gen_0':s4_0(n6_0), gen_n__from:cons3_0(1)) → gen_n__from:cons3_0(1), rt ∈ Ω(1 + n6₀)

(18) BOUNDS(n^1, INF)