Definition canonic_exp x :=
fexp (ln_beta beta x).
Definition canonic (f : float beta) :=
Fexp f = canonic_exp (F2R f).
Definition scaled_mantissa x :=
(x * bpow (- canonic_exp x))%R.
Definition generic_format (x : R) :=
x = F2R (Float beta (Ztrunc (scaled_mantissa x)) (canonic_exp x)).

(* Specifications: *)
(** Basic facts *)
Theorem generic_format_0 :
generic_format 0.
Theorem canonic_exp_opp :
forall x,
canonic_exp (-x) = canonic_exp x.
Theorem canonic_exp_abs :
forall x,
canonic_exp (Rabs x) = canonic_exp x.
Theorem generic_format_bpow :
forall e, (fexp (e + 1) <= e)%Z ->
generic_format (bpow e).
Theorem generic_format_bpow' :
forall e, (fexp e <= e)%Z ->
generic_format (bpow e).
Theorem generic_format_F2R :
forall m e,
( m <> 0 -> canonic_exp (F2R (Float beta m e)) <= e )%Z ->
generic_format (F2R (Float beta m e)).
Lemma generic_format_F2R': forall (x:R) (f:float beta),
F2R f = x -> ((x <> 0)%R ->
(canonic_exp x <= Fexp f)%Z) ->
generic_format x.
Theorem canonic_opp :
forall m e,
canonic (Float beta m e) ->
canonic (Float beta (-m) e).
Theorem canonic_abs :
forall m e,
canonic (Float beta m e) ->
canonic (Float beta (Zabs m) e).
Theorem canonic_0: canonic (Float beta 0 (fexp (ln_beta beta 0%R))).
Theorem canonic_unicity :
forall f1 f2,
canonic f1 ->
canonic f2 ->
F2R f1 = F2R f2 ->
f1 = f2.
Theorem scaled_mantissa_generic :
forall x,
generic_format x ->
scaled_mantissa x = Z2R (Ztrunc (scaled_mantissa x)).
Theorem scaled_mantissa_mult_bpow :
forall x,
(scaled_mantissa x * bpow (canonic_exp x))%R = x.
Theorem scaled_mantissa_0 :
scaled_mantissa 0 = 0%R.
Theorem scaled_mantissa_opp :
forall x,
scaled_mantissa (-x) = (-scaled_mantissa x)%R.
Theorem scaled_mantissa_abs :
forall x,
scaled_mantissa (Rabs x) = Rabs (scaled_mantissa x).
Theorem generic_format_opp :
forall x, generic_format x -> generic_format (-x).
Theorem generic_format_abs :
forall x, generic_format x -> generic_format (Rabs x).
Theorem generic_format_abs_inv :
forall x, generic_format (Rabs x) -> generic_format x.
Theorem canonic_exp_fexp :
forall x ex,
(bpow (ex - 1) <= Rabs x < bpow ex)%R ->
canonic_exp x = fexp ex.
Theorem canonic_exp_fexp_pos :
forall x ex,
(bpow (ex - 1) <= x < bpow ex)%R ->
canonic_exp x = fexp ex.
(** Properties when the real number is "small" (kind of subnormal) *)
Theorem mantissa_small_pos :
forall x ex,
(bpow (ex - 1) <= x < bpow ex)%R ->
(ex <= fexp ex)%Z ->
(0 < x * bpow (- fexp ex) < 1)%R.
Theorem scaled_mantissa_small :
forall x ex,
(Rabs x < bpow ex)%R ->
(ex <= fexp ex)%Z ->
(Rabs (scaled_mantissa x) < 1)%R.
Theorem abs_scaled_mantissa_lt_bpow :
forall x,
(Rabs (scaled_mantissa x) < bpow (ln_beta beta x - canonic_exp x))%R.
Theorem ln_beta_generic_gt :
forall x, (x <> 0)%R ->
generic_format x ->
(canonic_exp x < ln_beta beta x)%Z.
Theorem mantissa_DN_small_pos :
forall x ex,
(bpow (ex - 1) <= x < bpow ex)%R ->
(ex <= fexp ex)%Z ->
Zfloor (x * bpow (- fexp ex)) = Z0.
Theorem mantissa_UP_small_pos :
forall x ex,
(bpow (ex - 1) <= x < bpow ex)%R ->
(ex <= fexp ex)%Z ->
Zceil (x * bpow (- fexp ex)) = 1%Z.
(** Generic facts about any format *)
Theorem generic_format_discrete :
forall x m,
let e := canonic_exp x in
(F2R (Float beta m e) < x < F2R (Float beta (m + 1) e))%R ->
~ generic_format x.
Theorem generic_format_canonic :
forall f, canonic f ->
generic_format (F2R f).
Theorem generic_format_ge_bpow :
forall emin,
( forall e, (emin <= fexp e)%Z ) ->
forall x,
(0 < x)%R ->
generic_format x ->
(bpow emin <= x)%R.
Theorem abs_lt_bpow_prec:
forall prec,
(forall e, (e - prec <= fexp e)%Z) ->
forall x,
(Rabs x < bpow (prec + canonic_exp x))%R.
(** Basic facts *)
Theorem generic_format_0 :
generic_format 0.
Theorem canonic_exp_opp :
forall x,
canonic_exp (-x) = canonic_exp x.
Theorem canonic_exp_abs :
forall x,
canonic_exp (Rabs x) = canonic_exp x.
Theorem generic_format_bpow :
forall e, (fexp (e + 1) <= e)%Z ->
generic_format (bpow e).
Theorem generic_format_bpow' :
forall e, (fexp e <= e)%Z ->
generic_format (bpow e).
Theorem generic_format_F2R :
forall m e,
( m <> 0 -> canonic_exp (F2R (Float beta m e)) <= e )%Z ->
generic_format (F2R (Float beta m e)).
Lemma generic_format_F2R': forall (x:R) (f:float beta),
F2R f = x -> ((x <> 0)%R ->
(canonic_exp x <= Fexp f)%Z) ->
generic_format x.
Theorem canonic_opp :
forall m e,
canonic (Float beta m e) ->
canonic (Float beta (-m) e).
Theorem canonic_abs :
forall m e,
canonic (Float beta m e) ->
canonic (Float beta (Zabs m) e).
Theorem canonic_0: canonic (Float beta 0 (fexp (ln_beta beta 0%R))).
Theorem canonic_unicity :
forall f1 f2,
canonic f1 ->
canonic f2 ->
F2R f1 = F2R f2 ->
f1 = f2.
Theorem scaled_mantissa_generic :
forall x,
generic_format x ->
scaled_mantissa x = Z2R (Ztrunc (scaled_mantissa x)).
Theorem scaled_mantissa_mult_bpow :
forall x,
(scaled_mantissa x * bpow (canonic_exp x))%R = x.
Theorem scaled_mantissa_0 :
scaled_mantissa 0 = 0%R.
Theorem scaled_mantissa_opp :
forall x,
scaled_mantissa (-x) = (-scaled_mantissa x)%R.
Theorem scaled_mantissa_abs :
forall x,
scaled_mantissa (Rabs x) = Rabs (scaled_mantissa x).
Theorem generic_format_opp :
forall x, generic_format x -> generic_format (-x).
Theorem generic_format_abs :
forall x, generic_format x -> generic_format (Rabs x).
Theorem generic_format_abs_inv :
forall x, generic_format (Rabs x) -> generic_format x.
Theorem canonic_exp_fexp :
forall x ex,
(bpow (ex - 1) <= Rabs x < bpow ex)%R ->
canonic_exp x = fexp ex.
Theorem canonic_exp_fexp_pos :
forall x ex,
(bpow (ex - 1) <= x < bpow ex)%R ->
canonic_exp x = fexp ex.
(** Properties when the real number is "small" (kind of subnormal) *)
Theorem mantissa_small_pos :
forall x ex,
(bpow (ex - 1) <= x < bpow ex)%R ->
(ex <= fexp ex)%Z ->
(0 < x * bpow (- fexp ex) < 1)%R.
Theorem scaled_mantissa_small :
forall x ex,
(Rabs x < bpow ex)%R ->
(ex <= fexp ex)%Z ->
(Rabs (scaled_mantissa x) < 1)%R.
Theorem abs_scaled_mantissa_lt_bpow :
forall x,
(Rabs (scaled_mantissa x) < bpow (ln_beta beta x - canonic_exp x))%R.
Theorem ln_beta_generic_gt :
forall x, (x <> 0)%R ->
generic_format x ->
(canonic_exp x < ln_beta beta x)%Z.
Theorem mantissa_DN_small_pos :
forall x ex,
(bpow (ex - 1) <= x < bpow ex)%R ->
(ex <= fexp ex)%Z ->
Zfloor (x * bpow (- fexp ex)) = Z0.
Theorem mantissa_UP_small_pos :
forall x ex,
(bpow (ex - 1) <= x < bpow ex)%R ->
(ex <= fexp ex)%Z ->
Zceil (x * bpow (- fexp ex)) = 1%Z.
(** Generic facts about any format *)
Theorem generic_format_discrete :
forall x m,
let e := canonic_exp x in
(F2R (Float beta m e) < x < F2R (Float beta (m + 1) e))%R ->
~ generic_format x.
Theorem generic_format_canonic :
forall f, canonic f ->
generic_format (F2R f).
Theorem generic_format_ge_bpow :
forall emin,
( forall e, (emin <= fexp e)%Z ) ->
forall x,
(0 < x)%R ->
generic_format x ->
(bpow emin <= x)%R.
Theorem abs_lt_bpow_prec:
forall prec,
(forall e, (e - prec <= fexp e)%Z) ->
forall x,
(Rabs x < bpow (prec + canonic_exp x))%R.