(0) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

natsFrom(N) → cons(N, n__natsFrom(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)
activate(n__natsFrom(X)) → natsFrom(X)
activate(X) → X

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Lexicographic path order with status [LPO].
Quasi-Precedence:
0 > [pair2, activate1] > natsFrom1 > cons2
0 > [pair2, activate1] > natsFrom1 > nnatsFrom1
0 > [pair2, activate1] > natsFrom1 > s1
0 > nil
tail1 > [pair2, activate1] > natsFrom1 > cons2
tail1 > [pair2, activate1] > natsFrom1 > nnatsFrom1
tail1 > [pair2, activate1] > natsFrom1 > s1
sel2 > head1
sel2 > afterNth2 > snd1
sel2 > afterNth2 > [splitAt2, u4] > [pair2, activate1] > natsFrom1 > cons2
sel2 > afterNth2 > [splitAt2, u4] > [pair2, activate1] > natsFrom1 > nnatsFrom1
sel2 > afterNth2 > [splitAt2, u4] > [pair2, activate1] > natsFrom1 > s1
sel2 > afterNth2 > [splitAt2, u4] > nil
take2 > fst1
take2 > [splitAt2, u4] > [pair2, activate1] > natsFrom1 > cons2
take2 > [splitAt2, u4] > [pair2, activate1] > natsFrom1 > nnatsFrom1
take2 > [splitAt2, u4] > [pair2, activate1] > natsFrom1 > s1
take2 > [splitAt2, u4] > nil

Status:
sel2: [1,2]
tail1: [1]
afterNth2: [2,1]
snd1: [1]
head1: [1]
nnatsFrom1: [1]
activate1: [1]
take2: [2,1]
splitAt2: [1,2]
0: []
cons2: [2,1]
u4: [2,1,4,3]
fst1: [1]
pair2: [2,1]
s1: [1]
natsFrom1: [1]
nil: []

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

natsFrom(N) → cons(N, n__natsFrom(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)
activate(n__natsFrom(X)) → natsFrom(X)
activate(X) → X


(2) Obligation:

Q restricted rewrite system:
R is empty.
Q is empty.

(3) RisEmptyProof (EQUIVALENT transformation)

The TRS R is empty. Hence, termination is trivially proven.

(4) TRUE