(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:
Lexicographic Path Order [LPO].
Precedence:
0 > pair2 > activate1 > natsFrom1 > cons2
0 > pair2 > activate1 > natsFrom1 > nnatsFrom1
0 > pair2 > activate1 > natsFrom1 > ns1
0 > pair2 > activate1 > s1 > ns1
0 > nil
tail1 > activate1 > natsFrom1 > cons2
tail1 > activate1 > natsFrom1 > nnatsFrom1
tail1 > activate1 > natsFrom1 > ns1
tail1 > activate1 > s1 > ns1
sel2 > head1
sel2 > afterNth2 > snd1
sel2 > afterNth2 > splitAt2 > nil
sel2 > afterNth2 > splitAt2 > u4 > pair2 > activate1 > natsFrom1 > cons2
sel2 > afterNth2 > splitAt2 > u4 > pair2 > activate1 > natsFrom1 > nnatsFrom1
sel2 > afterNth2 > splitAt2 > u4 > pair2 > activate1 > natsFrom1 > ns1
sel2 > afterNth2 > splitAt2 > u4 > pair2 > activate1 > s1 > ns1
take2 > fst1
take2 > splitAt2 > nil
take2 > splitAt2 > u4 > pair2 > activate1 > natsFrom1 > cons2
take2 > splitAt2 > u4 > pair2 > activate1 > natsFrom1 > nnatsFrom1
take2 > splitAt2 > u4 > pair2 > activate1 > natsFrom1 > ns1
take2 > splitAt2 > u4 > pair2 > activate1 > s1 > ns1
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