(0) Obligation:

Q restricted rewrite system:
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

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Recursive path order with status [RPO].
Quasi-Precedence:
tail1 > activate1 > natsFrom1 > cons2 > [ns1, snd1]
tail1 > activate1 > natsFrom1 > nnatsFrom1 > [ns1, snd1]
tail1 > activate1 > s1 > [ns1, snd1]
sel2 > head1 > [ns1, snd1]
sel2 > afterNth2 > [splitAt2, u4] > pair2 > cons2 > [ns1, snd1]
sel2 > afterNth2 > [splitAt2, u4] > [0, nil] > [ns1, snd1]
sel2 > afterNth2 > [splitAt2, u4] > activate1 > natsFrom1 > cons2 > [ns1, snd1]
sel2 > afterNth2 > [splitAt2, u4] > activate1 > natsFrom1 > nnatsFrom1 > [ns1, snd1]
sel2 > afterNth2 > [splitAt2, u4] > activate1 > s1 > [ns1, snd1]
take2 > fst1 > [ns1, snd1]
take2 > [splitAt2, u4] > pair2 > cons2 > [ns1, snd1]
take2 > [splitAt2, u4] > [0, nil] > [ns1, snd1]
take2 > [splitAt2, u4] > activate1 > natsFrom1 > cons2 > [ns1, snd1]
take2 > [splitAt2, u4] > activate1 > natsFrom1 > nnatsFrom1 > [ns1, snd1]
take2 > [splitAt2, u4] > activate1 > s1 > [ns1, snd1]

Status:
sel2: [2,1]
tail1: multiset
afterNth2: multiset
snd1: [1]
head1: multiset
nnatsFrom1: [1]
activate1: [1]
ns1: multiset
take2: multiset
splitAt2: [1,2]
0: multiset
cons2: multiset
u4: [2,3,1,4]
fst1: [1]
pair2: multiset
s1: [1]
natsFrom1: multiset
nil: multiset

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(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


(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