(** Given a triple (mantissa, exponent, position), this function
    computes a triple with a canonic exponent, assuming the
    original triple had enough precision. *)

Definition truncate t :=
  let '(m, e, l) := t in
  let k := (fexp (Zdigits beta m + e) - e)%Z in
  if Zlt_bool 0 k then truncate_aux t k
  else t.

Theorem round_trunc_any_correct : forall x m e l, (0 <= x)%R -> inbetween_float beta m e x l -> (e <= fexp (Zdigits beta m + e))%Z \/ l = loc_Exact -> round beta fexp rnd x = let '(m', e', l') := truncate (m, e, l) in F2R (Float beta (choice m' l') e').