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:

U11(tt, N, X, XS) → U12(splitAt(activate(N), activate(XS)), activate(X))
U12(pair(YS, ZS), X) → pair(cons(activate(X), YS), ZS)
afterNth(N, XS) → snd(splitAt(N, XS))
and(tt, X) → activate(X)
fst(pair(X, Y)) → X
head(cons(N, XS)) → N
natsFrom(N) → cons(N, n__natsFrom(n__s(N)))
sel(N, XS) → head(afterNth(N, XS))
snd(pair(X, Y)) → Y
splitAt(0, XS) → pair(nil, XS)
splitAt(s(N), cons(X, XS)) → U11(tt, N, X, activate(XS))
tail(cons(N, XS)) → activate(XS)
take(N, XS) → fst(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:

U11(tt, N, X, XS) → U12(splitAt(activate(N), activate(XS)), activate(X))
U12(pair(YS, ZS), X) → pair(cons(activate(X), YS), ZS)
afterNth(N, XS) → snd(splitAt(N, XS))
and(tt, X) → activate(X)
fst(pair(X, Y)) → X
head(cons(N, XS)) → N
natsFrom(N) → cons(N, n__natsFrom(n__s(N)))
sel(N, XS) → head(afterNth(N, XS))
snd(pair(X, Y)) → Y
splitAt(0', XS) → pair(nil, XS)
splitAt(s(N), cons(X, XS)) → U11(tt, N, X, activate(XS))
tail(cons(N, XS)) → activate(XS)
take(N, XS) → fst(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) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

Infered types.

(6) Obligation:

TRS:
Rules:
U11(tt, N, X, XS) → U12(splitAt(activate(N), activate(XS)), activate(X))
U12(pair(YS, ZS), X) → pair(cons(activate(X), YS), ZS)
afterNth(N, XS) → snd(splitAt(N, XS))
and(tt, X) → activate(X)
fst(pair(X, Y)) → X
head(cons(N, XS)) → N
natsFrom(N) → cons(N, n__natsFrom(n__s(N)))
sel(N, XS) → head(afterNth(N, XS))
snd(pair(X, Y)) → Y
splitAt(0', XS) → pair(nil, XS)
splitAt(s(N), cons(X, XS)) → U11(tt, N, X, activate(XS))
tail(cons(N, XS)) → activate(XS)
take(N, XS) → fst(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:
U11 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → pair
tt :: tt
U12 :: pair → cons:n__s:n__natsFrom:0':nil → pair
splitAt :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → pair
activate :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
pair :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → pair
cons :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
afterNth :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
snd :: pair → cons:n__s:n__natsFrom:0':nil
and :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
fst :: pair → cons:n__s:n__natsFrom:0':nil
head :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
natsFrom :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
n__natsFrom :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
n__s :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
sel :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
0' :: cons:n__s:n__natsFrom:0':nil
nil :: cons:n__s:n__natsFrom:0':nil
s :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
tail :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
take :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
hole_pair1_0 :: pair
hole_tt2_0 :: tt
hole_cons:n__s:n__natsFrom:0':nil3_0 :: cons:n__s:n__natsFrom:0':nil
gen_cons:n__s:n__natsFrom:0':nil4_0 :: Nat → cons:n__s:n__natsFrom:0':nil

(7) 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

(8) Obligation:

TRS:
Rules:
U11(tt, N, X, XS) → U12(splitAt(activate(N), activate(XS)), activate(X))
U12(pair(YS, ZS), X) → pair(cons(activate(X), YS), ZS)
afterNth(N, XS) → snd(splitAt(N, XS))
and(tt, X) → activate(X)
fst(pair(X, Y)) → X
head(cons(N, XS)) → N
natsFrom(N) → cons(N, n__natsFrom(n__s(N)))
sel(N, XS) → head(afterNth(N, XS))
snd(pair(X, Y)) → Y
splitAt(0', XS) → pair(nil, XS)
splitAt(s(N), cons(X, XS)) → U11(tt, N, X, activate(XS))
tail(cons(N, XS)) → activate(XS)
take(N, XS) → fst(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:
U11 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → pair
tt :: tt
U12 :: pair → cons:n__s:n__natsFrom:0':nil → pair
splitAt :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → pair
activate :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
pair :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → pair
cons :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
afterNth :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
snd :: pair → cons:n__s:n__natsFrom:0':nil
and :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
fst :: pair → cons:n__s:n__natsFrom:0':nil
head :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
natsFrom :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
n__natsFrom :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
n__s :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
sel :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
0' :: cons:n__s:n__natsFrom:0':nil
nil :: cons:n__s:n__natsFrom:0':nil
s :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
tail :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
take :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
hole_pair1_0 :: pair
hole_tt2_0 :: tt
hole_cons:n__s:n__natsFrom:0':nil3_0 :: cons:n__s:n__natsFrom:0':nil
gen_cons:n__s:n__natsFrom:0':nil4_0 :: Nat → cons:n__s:n__natsFrom:0':nil

Generator Equations:
gen_cons:n__s:n__natsFrom:0':nil4_0(0) ⇔ 0'
gen_cons:n__s:n__natsFrom:0':nil4_0(+(x, 1)) ⇔ cons(0', gen_cons:n__s:n__natsFrom:0':nil4_0(x))

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

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

(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(10) Obligation:

TRS:
Rules:
U11(tt, N, X, XS) → U12(splitAt(activate(N), activate(XS)), activate(X))
U12(pair(YS, ZS), X) → pair(cons(activate(X), YS), ZS)
afterNth(N, XS) → snd(splitAt(N, XS))
and(tt, X) → activate(X)
fst(pair(X, Y)) → X
head(cons(N, XS)) → N
natsFrom(N) → cons(N, n__natsFrom(n__s(N)))
sel(N, XS) → head(afterNth(N, XS))
snd(pair(X, Y)) → Y
splitAt(0', XS) → pair(nil, XS)
splitAt(s(N), cons(X, XS)) → U11(tt, N, X, activate(XS))
tail(cons(N, XS)) → activate(XS)
take(N, XS) → fst(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:
U11 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → pair
tt :: tt
U12 :: pair → cons:n__s:n__natsFrom:0':nil → pair
splitAt :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → pair
activate :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
pair :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → pair
cons :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
afterNth :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
snd :: pair → cons:n__s:n__natsFrom:0':nil
and :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
fst :: pair → cons:n__s:n__natsFrom:0':nil
head :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
natsFrom :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
n__natsFrom :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
n__s :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
sel :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
0' :: cons:n__s:n__natsFrom:0':nil
nil :: cons:n__s:n__natsFrom:0':nil
s :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
tail :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
take :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
hole_pair1_0 :: pair
hole_tt2_0 :: tt
hole_cons:n__s:n__natsFrom:0':nil3_0 :: cons:n__s:n__natsFrom:0':nil
gen_cons:n__s:n__natsFrom:0':nil4_0 :: Nat → cons:n__s:n__natsFrom:0':nil

Generator Equations:
gen_cons:n__s:n__natsFrom:0':nil4_0(0) ⇔ 0'
gen_cons:n__s:n__natsFrom:0':nil4_0(+(x, 1)) ⇔ cons(0', gen_cons:n__s:n__natsFrom:0':nil4_0(x))

The following defined symbols remain to be analysed:
splitAt

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

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

(12) Obligation:

TRS:
Rules:
U11(tt, N, X, XS) → U12(splitAt(activate(N), activate(XS)), activate(X))
U12(pair(YS, ZS), X) → pair(cons(activate(X), YS), ZS)
afterNth(N, XS) → snd(splitAt(N, XS))
and(tt, X) → activate(X)
fst(pair(X, Y)) → X
head(cons(N, XS)) → N
natsFrom(N) → cons(N, n__natsFrom(n__s(N)))
sel(N, XS) → head(afterNth(N, XS))
snd(pair(X, Y)) → Y
splitAt(0', XS) → pair(nil, XS)
splitAt(s(N), cons(X, XS)) → U11(tt, N, X, activate(XS))
tail(cons(N, XS)) → activate(XS)
take(N, XS) → fst(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:
U11 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → pair
tt :: tt
U12 :: pair → cons:n__s:n__natsFrom:0':nil → pair
splitAt :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → pair
activate :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
pair :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → pair
cons :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
afterNth :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
snd :: pair → cons:n__s:n__natsFrom:0':nil
and :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
fst :: pair → cons:n__s:n__natsFrom:0':nil
head :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
natsFrom :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
n__natsFrom :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
n__s :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
sel :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
0' :: cons:n__s:n__natsFrom:0':nil
nil :: cons:n__s:n__natsFrom:0':nil
s :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
tail :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
take :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
hole_pair1_0 :: pair
hole_tt2_0 :: tt
hole_cons:n__s:n__natsFrom:0':nil3_0 :: cons:n__s:n__natsFrom:0':nil
gen_cons:n__s:n__natsFrom:0':nil4_0 :: Nat → cons:n__s:n__natsFrom:0':nil

Generator Equations:
gen_cons:n__s:n__natsFrom:0':nil4_0(0) ⇔ 0'
gen_cons:n__s:n__natsFrom:0':nil4_0(+(x, 1)) ⇔ cons(0', gen_cons:n__s:n__natsFrom:0':nil4_0(x))

No more defined symbols left to analyse.