(** As noticed by P. Casteran, asking for the general [SetoidList.transpose]
      here is too restrictive. Think for instance of [f] being [M.add] :
      in general, [M.add k e (M.add k e' m)] is not equivalent to
      [M.add k e' (M.add k e m)]. Fortunately, we will never encounter this
      situation during a real [fold], since the keys received by this [fold]
      are unique. Hence we can ask the transposition property to hold only
      for non-equal keys.

      This idea could be push slightly further, by asking the transposition
      property to hold only for (non-equal) keys living in the map given to
      [fold]. Please contact us if you need such a version.

      FSets could also benefit from a restricted [transpose], but for this
      case the gain is unclear. *)

  Definition transpose_neqkey :=
    forall k k' e e' a, ~E.eq k k' ->
      eqA (f k e (f k' e' a)) (f k' e' (f k e a)).

Lemma cardinal_2 : forall m m' x e, ~ In x m -> Add x e m m' -> cardinal m' = S (cardinal m).