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:

natsFrom(N) → cons(N, n__natsFrom(n__s(N)))
fst(pair(XS, YS)) → XS
snd(pair(XS, YS)) → YS
splitAt(0, XS) → pair(nil, XS)
splitAt(s(N), cons(X, XS)) → u(splitAt(N, activate(XS)), N, X, activate(XS))
u(pair(YS, ZS), N, X, XS) → pair(cons(activate(X), YS), ZS)
head(cons(N, XS)) → N
tail(cons(N, XS)) → activate(XS)
sel(N, XS) → head(afterNth(N, XS))
take(N, XS) → fst(splitAt(N, XS))
afterNth(N, XS) → snd(splitAt(N, XS))
natsFrom(X) → n__natsFrom(X)
s(X) → n__s(X)
activate(n__natsFrom(X)) → natsFrom(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__natsFrom(X)) →⁺ cons(activate(X), n__natsFrom(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__natsFrom(X)].
The result substitution is [ ].

The rewrite sequence
activate(n__natsFrom(X)) →⁺ cons(activate(X), n__natsFrom(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__natsFrom(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:

natsFrom(N) → cons(N, n__natsFrom(n__s(N)))
fst(pair(XS, YS)) → XS
snd(pair(XS, YS)) → YS
splitAt(0', XS) → pair(nil, XS)
splitAt(s(N), cons(X, XS)) → u(splitAt(N, activate(XS)), N, X, activate(XS))
u(pair(YS, ZS), N, X, XS) → pair(cons(activate(X), YS), ZS)
head(cons(N, XS)) → N
tail(cons(N, XS)) → activate(XS)
sel(N, XS) → head(afterNth(N, XS))
take(N, XS) → fst(splitAt(N, XS))
afterNth(N, XS) → snd(splitAt(N, XS))
natsFrom(X) → n__natsFrom(X)
s(X) → n__s(X)
activate(n__natsFrom(X)) → natsFrom(activate(X))
activate(n__s(X)) → s(activate(X))
activate(X) → X

S is empty.
Rewrite Strategy: FULL

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

Sliced the following arguments:
u/1

(6) Obligation:

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

natsFrom(N) → cons(N, n__natsFrom(n__s(N)))
fst(pair(XS, YS)) → XS
snd(pair(XS, YS)) → YS
splitAt(0', XS) → pair(nil, XS)
splitAt(s(N), cons(X, XS)) → u(splitAt(N, activate(XS)), X, activate(XS))
u(pair(YS, ZS), X, XS) → pair(cons(activate(X), YS), ZS)
head(cons(N, XS)) → N
tail(cons(N, XS)) → activate(XS)
sel(N, XS) → head(afterNth(N, XS))
take(N, XS) → fst(splitAt(N, XS))
afterNth(N, XS) → snd(splitAt(N, XS))
natsFrom(X) → n__natsFrom(X)
s(X) → n__s(X)
activate(n__natsFrom(X)) → natsFrom(activate(X))
activate(n__s(X)) → s(activate(X))
activate(X) → X

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(8) Obligation:

TRS:
Rules:
natsFrom(N) → cons(N, n__natsFrom(n__s(N)))
fst(pair(XS, YS)) → XS
snd(pair(XS, YS)) → YS
splitAt(0', XS) → pair(nil, XS)
splitAt(s(N), cons(X, XS)) → u(splitAt(N, activate(XS)), X, activate(XS))
u(pair(YS, ZS), X, XS) → pair(cons(activate(X), YS), ZS)
head(cons(N, XS)) → N
tail(cons(N, XS)) → activate(XS)
sel(N, XS) → head(afterNth(N, XS))
take(N, XS) → fst(splitAt(N, XS))
afterNth(N, XS) → snd(splitAt(N, XS))
natsFrom(X) → n__natsFrom(X)
s(X) → n__s(X)
activate(n__natsFrom(X)) → natsFrom(activate(X))
activate(n__s(X)) → s(activate(X))
activate(X) → X

Types:
natsFrom :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
cons :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
n__natsFrom :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
n__s :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
fst :: pair → n__s:n__natsFrom:cons:0':nil
pair :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil → pair
snd :: pair → n__s:n__natsFrom:cons:0':nil
splitAt :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil → pair
0' :: n__s:n__natsFrom:cons:0':nil
nil :: n__s:n__natsFrom:cons:0':nil
s :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
u :: pair → n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil → pair
activate :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
head :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
tail :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
sel :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
afterNth :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
take :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
hole_n__s:n__natsFrom:cons:0':nil1_0 :: n__s:n__natsFrom:cons:0':nil
hole_pair2_0 :: pair
gen_n__s:n__natsFrom:cons:0':nil3_0 :: Nat → n__s:n__natsFrom:cons:0':nil

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

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

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

(10) Obligation:

TRS:
Rules:
natsFrom(N) → cons(N, n__natsFrom(n__s(N)))
fst(pair(XS, YS)) → XS
snd(pair(XS, YS)) → YS
splitAt(0', XS) → pair(nil, XS)
splitAt(s(N), cons(X, XS)) → u(splitAt(N, activate(XS)), X, activate(XS))
u(pair(YS, ZS), X, XS) → pair(cons(activate(X), YS), ZS)
head(cons(N, XS)) → N
tail(cons(N, XS)) → activate(XS)
sel(N, XS) → head(afterNth(N, XS))
take(N, XS) → fst(splitAt(N, XS))
afterNth(N, XS) → snd(splitAt(N, XS))
natsFrom(X) → n__natsFrom(X)
s(X) → n__s(X)
activate(n__natsFrom(X)) → natsFrom(activate(X))
activate(n__s(X)) → s(activate(X))
activate(X) → X

Types:
natsFrom :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
cons :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
n__natsFrom :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
n__s :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
fst :: pair → n__s:n__natsFrom:cons:0':nil
pair :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil → pair
snd :: pair → n__s:n__natsFrom:cons:0':nil
splitAt :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil → pair
0' :: n__s:n__natsFrom:cons:0':nil
nil :: n__s:n__natsFrom:cons:0':nil
s :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
u :: pair → n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil → pair
activate :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
head :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
tail :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
sel :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
afterNth :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
take :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
hole_n__s:n__natsFrom:cons:0':nil1_0 :: n__s:n__natsFrom:cons:0':nil
hole_pair2_0 :: pair
gen_n__s:n__natsFrom:cons:0':nil3_0 :: Nat → n__s:n__natsFrom:cons:0':nil

Generator Equations:
gen_n__s:n__natsFrom:cons:0':nil3_0(0) ⇔ 0'
gen_n__s:n__natsFrom:cons:0':nil3_0(+(x, 1)) ⇔ cons(0', gen_n__s:n__natsFrom:cons:0':nil3_0(x))

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

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

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

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

(12) Obligation:

TRS:
Rules:
natsFrom(N) → cons(N, n__natsFrom(n__s(N)))
fst(pair(XS, YS)) → XS
snd(pair(XS, YS)) → YS
splitAt(0', XS) → pair(nil, XS)
splitAt(s(N), cons(X, XS)) → u(splitAt(N, activate(XS)), X, activate(XS))
u(pair(YS, ZS), X, XS) → pair(cons(activate(X), YS), ZS)
head(cons(N, XS)) → N
tail(cons(N, XS)) → activate(XS)
sel(N, XS) → head(afterNth(N, XS))
take(N, XS) → fst(splitAt(N, XS))
afterNth(N, XS) → snd(splitAt(N, XS))
natsFrom(X) → n__natsFrom(X)
s(X) → n__s(X)
activate(n__natsFrom(X)) → natsFrom(activate(X))
activate(n__s(X)) → s(activate(X))
activate(X) → X

Types:
natsFrom :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
cons :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
n__natsFrom :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
n__s :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
fst :: pair → n__s:n__natsFrom:cons:0':nil
pair :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil → pair
snd :: pair → n__s:n__natsFrom:cons:0':nil
splitAt :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil → pair
0' :: n__s:n__natsFrom:cons:0':nil
nil :: n__s:n__natsFrom:cons:0':nil
s :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
u :: pair → n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil → pair
activate :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
head :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
tail :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
sel :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
afterNth :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
take :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
hole_n__s:n__natsFrom:cons:0':nil1_0 :: n__s:n__natsFrom:cons:0':nil
hole_pair2_0 :: pair
gen_n__s:n__natsFrom:cons:0':nil3_0 :: Nat → n__s:n__natsFrom:cons:0':nil

Generator Equations:
gen_n__s:n__natsFrom:cons:0':nil3_0(0) ⇔ 0'
gen_n__s:n__natsFrom:cons:0':nil3_0(+(x, 1)) ⇔ cons(0', gen_n__s:n__natsFrom:cons:0':nil3_0(x))

The following defined symbols remain to be analysed:
splitAt

(13) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(14) Obligation:

TRS:
Rules:
natsFrom(N) → cons(N, n__natsFrom(n__s(N)))
fst(pair(XS, YS)) → XS
snd(pair(XS, YS)) → YS
splitAt(0', XS) → pair(nil, XS)
splitAt(s(N), cons(X, XS)) → u(splitAt(N, activate(XS)), X, activate(XS))
u(pair(YS, ZS), X, XS) → pair(cons(activate(X), YS), ZS)
head(cons(N, XS)) → N
tail(cons(N, XS)) → activate(XS)
sel(N, XS) → head(afterNth(N, XS))
take(N, XS) → fst(splitAt(N, XS))
afterNth(N, XS) → snd(splitAt(N, XS))
natsFrom(X) → n__natsFrom(X)
s(X) → n__s(X)
activate(n__natsFrom(X)) → natsFrom(activate(X))
activate(n__s(X)) → s(activate(X))
activate(X) → X

Types:
natsFrom :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
cons :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
n__natsFrom :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
n__s :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
fst :: pair → n__s:n__natsFrom:cons:0':nil
pair :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil → pair
snd :: pair → n__s:n__natsFrom:cons:0':nil
splitAt :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil → pair
0' :: n__s:n__natsFrom:cons:0':nil
nil :: n__s:n__natsFrom:cons:0':nil
s :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
u :: pair → n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil → pair
activate :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
head :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
tail :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
sel :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
afterNth :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
take :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
hole_n__s:n__natsFrom:cons:0':nil1_0 :: n__s:n__natsFrom:cons:0':nil
hole_pair2_0 :: pair
gen_n__s:n__natsFrom:cons:0':nil3_0 :: Nat → n__s:n__natsFrom:cons:0':nil

Generator Equations:
gen_n__s:n__natsFrom:cons:0':nil3_0(0) ⇔ 0'
gen_n__s:n__natsFrom:cons:0':nil3_0(+(x, 1)) ⇔ cons(0', gen_n__s:n__natsFrom:cons:0':nil3_0(x))

No more defined symbols left to analyse.