MAYBE
MAYBE
TRS:
{
U12(mark(X1), X2) -> mark(U12(X1, X2)),
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)),
splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)),
splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)),
splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)),
active(U12(X1, X2)) -> U12(active(X1), X2),
active(U12(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)),
active(splitAt(X1, X2)) -> splitAt(X1, active(X2)),
active(splitAt(X1, X2)) -> splitAt(active(X1), X2),
active(splitAt(s(N), cons(X, XS))) -> mark(U11(tt(), N, X, XS)),
active(splitAt(0(), XS)) -> mark(pair(nil(), XS)),
active(U11(X1, X2, X3, X4)) -> U11(active(X1), X2, X3, X4),
active(U11(tt(), N, X, XS)) -> mark(U12(splitAt(N, XS), X)),
active(pair(X1, X2)) -> pair(X1, active(X2)),
active(pair(X1, X2)) -> pair(active(X1), X2),
active(cons(X1, X2)) -> cons(active(X1), X2),
active(snd(X)) -> snd(active(X)),
active(snd(pair(X, Y))) -> mark(Y),
active(afterNth(N, XS)) -> mark(snd(splitAt(N, XS))),
active(afterNth(X1, X2)) -> afterNth(X1, active(X2)),
active(afterNth(X1, X2)) -> afterNth(active(X1), X2),
active(and(X1, X2)) -> and(active(X1), X2),
active(and(tt(), X)) -> mark(X),
active(fst(X)) -> fst(active(X)),
active(fst(pair(X, Y))) -> mark(X),
active(head(X)) -> head(active(X)),
active(head(cons(N, XS))) -> mark(N),
active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N)))),
active(natsFrom(X)) -> natsFrom(active(X)),
active(s(X)) -> s(active(X)),
active(sel(N, XS)) -> mark(head(afterNth(N, XS))),
active(sel(X1, X2)) -> sel(X1, active(X2)),
active(sel(X1, X2)) -> sel(active(X1), X2),
active(tail(X)) -> tail(active(X)),
active(tail(cons(N, XS))) -> mark(XS),
active(take(N, XS)) -> mark(fst(splitAt(N, XS))),
active(take(X1, X2)) -> take(X1, active(X2)),
active(take(X1, X2)) -> take(active(X1), X2),
U11(mark(X1), X2, X3, X4) -> mark(U11(X1, X2, X3, X4)),
U11(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U11(X1, X2, X3, X4)),
pair(X1, mark(X2)) -> mark(pair(X1, X2)),
pair(mark(X1), X2) -> mark(pair(X1, X2)),
pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)),
cons(mark(X1), X2) -> mark(cons(X1, X2)),
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)),
snd(mark(X)) -> mark(snd(X)),
snd(ok(X)) -> ok(snd(X)),
afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2)),
afterNth(mark(X1), X2) -> mark(afterNth(X1, X2)),
afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)),
and(mark(X1), X2) -> mark(and(X1, X2)),
and(ok(X1), ok(X2)) -> ok(and(X1, X2)),
fst(mark(X)) -> mark(fst(X)),
fst(ok(X)) -> ok(fst(X)),
head(mark(X)) -> mark(head(X)),
head(ok(X)) -> ok(head(X)),
natsFrom(mark(X)) -> mark(natsFrom(X)),
natsFrom(ok(X)) -> ok(natsFrom(X)),
s(mark(X)) -> mark(s(X)),
s(ok(X)) -> ok(s(X)),
sel(X1, mark(X2)) -> mark(sel(X1, X2)),
sel(mark(X1), X2) -> mark(sel(X1, X2)),
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)),
tail(mark(X)) -> mark(tail(X)),
tail(ok(X)) -> ok(tail(X)),
take(X1, mark(X2)) -> mark(take(X1, X2)),
take(mark(X1), X2) -> mark(take(X1, X2)),
take(ok(X1), ok(X2)) -> ok(take(X1, X2)),
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)),
proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)),
proper(U11(X1, X2, X3, X4)) ->
U11(proper(X1), proper(X2), proper(X3), proper(X4)),
proper(tt()) -> ok(tt()),
proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)),
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)),
proper(snd(X)) -> snd(proper(X)),
proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)),
proper(and(X1, X2)) -> and(proper(X1), proper(X2)),
proper(fst(X)) -> fst(proper(X)),
proper(head(X)) -> head(proper(X)),
proper(natsFrom(X)) -> natsFrom(proper(X)),
proper(s(X)) -> s(proper(X)),
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)),
proper(nil()) -> ok(nil()),
proper(0()) -> ok(0()),
proper(tail(X)) -> tail(proper(X)),
proper(take(X1, X2)) -> take(proper(X1), proper(X2)),
top(mark(X)) -> top(proper(X)),
top(ok(X)) -> top(active(X))
}
DUP: We consider a duplicating system.
Trs:
{
U12(mark(X1), X2) -> mark(U12(X1, X2)),
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)),
splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)),
splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)),
splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)),
active(U12(X1, X2)) -> U12(active(X1), X2),
active(U12(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)),
active(splitAt(X1, X2)) -> splitAt(X1, active(X2)),
active(splitAt(X1, X2)) -> splitAt(active(X1), X2),
active(splitAt(s(N), cons(X, XS))) -> mark(U11(tt(), N, X, XS)),
active(splitAt(0(), XS)) -> mark(pair(nil(), XS)),
active(U11(X1, X2, X3, X4)) -> U11(active(X1), X2, X3, X4),
active(U11(tt(), N, X, XS)) -> mark(U12(splitAt(N, XS), X)),
active(pair(X1, X2)) -> pair(X1, active(X2)),
active(pair(X1, X2)) -> pair(active(X1), X2),
active(cons(X1, X2)) -> cons(active(X1), X2),
active(snd(X)) -> snd(active(X)),
active(snd(pair(X, Y))) -> mark(Y),
active(afterNth(N, XS)) -> mark(snd(splitAt(N, XS))),
active(afterNth(X1, X2)) -> afterNth(X1, active(X2)),
active(afterNth(X1, X2)) -> afterNth(active(X1), X2),
active(and(X1, X2)) -> and(active(X1), X2),
active(and(tt(), X)) -> mark(X),
active(fst(X)) -> fst(active(X)),
active(fst(pair(X, Y))) -> mark(X),
active(head(X)) -> head(active(X)),
active(head(cons(N, XS))) -> mark(N),
active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N)))),
active(natsFrom(X)) -> natsFrom(active(X)),
active(s(X)) -> s(active(X)),
active(sel(N, XS)) -> mark(head(afterNth(N, XS))),
active(sel(X1, X2)) -> sel(X1, active(X2)),
active(sel(X1, X2)) -> sel(active(X1), X2),
active(tail(X)) -> tail(active(X)),
active(tail(cons(N, XS))) -> mark(XS),
active(take(N, XS)) -> mark(fst(splitAt(N, XS))),
active(take(X1, X2)) -> take(X1, active(X2)),
active(take(X1, X2)) -> take(active(X1), X2),
U11(mark(X1), X2, X3, X4) -> mark(U11(X1, X2, X3, X4)),
U11(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U11(X1, X2, X3, X4)),
pair(X1, mark(X2)) -> mark(pair(X1, X2)),
pair(mark(X1), X2) -> mark(pair(X1, X2)),
pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)),
cons(mark(X1), X2) -> mark(cons(X1, X2)),
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)),
snd(mark(X)) -> mark(snd(X)),
snd(ok(X)) -> ok(snd(X)),
afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2)),
afterNth(mark(X1), X2) -> mark(afterNth(X1, X2)),
afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)),
and(mark(X1), X2) -> mark(and(X1, X2)),
and(ok(X1), ok(X2)) -> ok(and(X1, X2)),
fst(mark(X)) -> mark(fst(X)),
fst(ok(X)) -> ok(fst(X)),
head(mark(X)) -> mark(head(X)),
head(ok(X)) -> ok(head(X)),
natsFrom(mark(X)) -> mark(natsFrom(X)),
natsFrom(ok(X)) -> ok(natsFrom(X)),
s(mark(X)) -> mark(s(X)),
s(ok(X)) -> ok(s(X)),
sel(X1, mark(X2)) -> mark(sel(X1, X2)),
sel(mark(X1), X2) -> mark(sel(X1, X2)),
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)),
tail(mark(X)) -> mark(tail(X)),
tail(ok(X)) -> ok(tail(X)),
take(X1, mark(X2)) -> mark(take(X1, X2)),
take(mark(X1), X2) -> mark(take(X1, X2)),
take(ok(X1), ok(X2)) -> ok(take(X1, X2)),
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)),
proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)),
proper(U11(X1, X2, X3, X4)) ->
U11(proper(X1), proper(X2), proper(X3), proper(X4)),
proper(tt()) -> ok(tt()),
proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)),
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)),
proper(snd(X)) -> snd(proper(X)),
proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)),
proper(and(X1, X2)) -> and(proper(X1), proper(X2)),
proper(fst(X)) -> fst(proper(X)),
proper(head(X)) -> head(proper(X)),
proper(natsFrom(X)) -> natsFrom(proper(X)),
proper(s(X)) -> s(proper(X)),
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)),
proper(nil()) -> ok(nil()),
proper(0()) -> ok(0()),
proper(tail(X)) -> tail(proper(X)),
proper(take(X1, X2)) -> take(proper(X1), proper(X2)),
top(mark(X)) -> top(proper(X)),
top(ok(X)) -> top(active(X))
}
Fail