(** * Instances of the theorems above, for the usual rounding modes. *)

Definition round_DN_correct :=
  round_any_correct _ (fun m _ => m) inbetween_int_DN.Definition round_trunc_DN_correct :=
  round_trunc_any_correct _ (fun m _ => m) inbetween_int_DN.Definition round_sign_DN_correct :=
  round_sign_any_correct _ (fun s m l => cond_incr (round_sign_DN s l) m) inbetween_int_DN_sign.Definition round_trunc_sign_DN_correct :=
  round_trunc_sign_any_correct _ (fun s m l => cond_incr (round_sign_DN s l) m) inbetween_int_DN_sign.Definition round_UP_correct :=
  round_any_correct _ (fun m l => cond_incr (round_UP l) m) inbetween_int_UP.Definition round_trunc_UP_correct :=
  round_trunc_any_correct _ (fun m l => cond_incr (round_UP l) m) inbetween_int_UP.Definition round_sign_UP_correct :=
  round_sign_any_correct _ (fun s m l => cond_incr (round_sign_UP s l) m) inbetween_int_UP_sign.Definition round_trunc_sign_UP_correct :=
  round_trunc_sign_any_correct _ (fun s m l => cond_incr (round_sign_UP s l) m) inbetween_int_UP_sign.Definition round_ZR_correct :=
  round_any_correct _ (fun m l => cond_incr (round_ZR (Zlt_bool m 0) l) m) inbetween_int_ZR.Definition round_trunc_ZR_correct :=
  round_trunc_any_correct _ (fun m l => cond_incr (round_ZR (Zlt_bool m 0) l) m) inbetween_int_ZR.Definition round_sign_ZR_correct :=
  round_sign_any_correct _ (fun s m l => m) inbetween_int_ZR_sign.Definition round_trunc_sign_ZR_correct :=
  round_trunc_sign_any_correct _ (fun s m l => m) inbetween_int_ZR_sign.Definition round_NE_correct :=
  round_any_correct _ (fun m l => cond_incr (round_N (negb (Zeven m)) l) m) inbetween_int_NE.Definition round_trunc_NE_correct :=
  round_trunc_any_correct _ (fun m l => cond_incr (round_N (negb (Zeven m)) l) m) inbetween_int_NE.Definition round_sign_NE_correct :=
  round_sign_any_correct _ (fun s m l => cond_incr (round_N (negb (Zeven m)) l) m) inbetween_int_NE_sign.Definition round_trunc_sign_NE_correct :=
  round_trunc_sign_any_correct _ (fun s m l => cond_incr (round_N (negb (Zeven m)) l) m) inbetween_int_NE_sign.Definition round_NA_correct :=
  round_any_correct _ (fun m l => cond_incr (round_N (Zle_bool 0 m) l) m) inbetween_int_NA.Definition round_trunc_NA_correct :=
  round_trunc_any_correct _ (fun m l => cond_incr (round_N (Zle_bool 0 m) l) m) inbetween_int_NA.Definition round_sign_NA_correct :=
  round_sign_any_correct _ (fun s m l => cond_incr (round_N true l) m) inbetween_int_NA_sign.Definition round_trunc_sign_NA_correct :=
  round_trunc_sign_any_correct _ (fun s m l => cond_incr (round_N true l) m) inbetween_int_NA_sign.Definition truncate_FIX t :=
  let '(m, e, l) := t in
  let k := (emin - e)%Z in
  if Zlt_bool 0 k then
    let p := Zpower beta k in
    (Zdiv m p, (e + k)%Z, new_location p (Zmod m p) l)
  else t.

Theorem truncate_FIX_correct : forall x m e l, inbetween_float beta m e x l -> (e <= emin)%Z \/ l = loc_Exact -> let '(m', e', l') := truncate_FIX (m, e, l) in inbetween_float beta m' e' x l' /\ (e' = canonic_exp beta (FIX_exp emin) x \/ (l' = loc_Exact /\ generic_format beta (FIX_exp emin) x)).