(CONDITIONTYPE ORIENTED)
(VAR x n h t m pid in_2 in_3 st_1 st_2 st_3)
(RULES
  fstsplit(0, x) -> nil
  fstsplit(s(n), nil) -> nil
  fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
  sndsplit(0, x) -> x
  sndsplit(s(n), nil) -> nil
  sndsplit(s(n), cons(h, t)) -> sndsplit(n, t)
  empty(nil) -> true
  empty(cons(h, t)) -> false
  leq(0, m) -> true
  leq(s(n), 0) -> false
  leq(s(n), s(m)) -> leq(n, m)
  length(nil) -> 0
  length(cons(h, t)) -> s(length(t))
  app(nil, x) -> x
  app(cons(h, t), x) -> cons(h, app(t, x))
  map_f(pid, nil) -> nil
  map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
  head(cons(h, t)) -> h
  tail(cons(h, t)) -> t
  ring(st_1, in_2, st_2, in_3, st_3, m) -> ring(sndsplit(m, st_1), cons(fstsplit(m, st_1), in_2), st_2, in_3, st_3, m) | empty(fstsplit(m, st_1)) == false
  ring(st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, in_2, sndsplit(m, st_2), cons(fstsplit(m, st_2), in_3), st_3, m) | leq(m, length(st_2)) == true, empty(fstsplit(m, st_2)) == false
  ring(st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, tail(in_2), sndsplit(m, app(map_f(two, head(in_2)), st_2)), cons(fstsplit(m, app(map_f(two, head(in_2)), st_2)), in_3), st_3, m) | leq(m, length(st_2)) == false, empty(fstsplit(m, app(map_f(two, head(in_2)), st_2))) == false
  ring(st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, tail(in_2), st_2, in_3, st_3, m) | empty(map_f(two, head(in_2))) == true
  ring(st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m) | leq(m, length(st_3)) == true, empty(fstsplit(m, st_3)) == false
  ring(st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m) | leq(m, length(st_3)) == false, empty(fstsplit(m, app(map_f(three, head(in_3)), st_3))) == false
  ring(st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, in_2, st_2, tail(in_3), st_3, m) | empty(map_f(three, head(in_3))) == true
)
(COMMENT
  \cite{GA01}, p. 61
  doi: 10.1007/s002000100063
)