(** 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 truncate_correct_partial :
forall x m e l,
(0 < x)%R ->
inbetween_float beta m e x l ->
(e <= fexp (Zdigits beta m + e))%Z ->
let '(m', e', l') := truncate (m, e, l) in
inbetween_float beta m' e' x l' /\ e' = canonic_exp beta fexp x.