Commit cdda3e7b authored by Guillaume Melquiond's avatar Guillaume Melquiond

Clean proofs a bit.

parent 72c2fac6
......@@ -198,7 +198,7 @@ apply Rle_trans with (1 := Rabs_triang _ _).
now apply Rplus_le_compat.
replace (2 * ((F2R b + F2R B) * F2R {| Fnum := 1; Fexp := - prec |} * M))%R
with ((F2R b + F2R B) * M * bpow radix2 (1 - prec))%R.
rewrite <- Rabs_pos_eq with (1 := H') at 2.
rewrite <- (Rabs_pos_eq _ H') at 2.
apply ulp_FLT_le.
rewrite Rabs_pos_eq with (1 := H').
apply Rle_trans with (1 := H).
......
......@@ -415,8 +415,7 @@ Theorem is_finite_Bopp :
forall opp_nan x,
is_finite (Bopp opp_nan x) = is_finite x.
Proof.
intros opp_nan [| | |] ; try easy.
intros s pl.
intros opp_nan [| |s pl|] ; try easy.
simpl.
now case opp_nan.
Qed.
......@@ -445,8 +444,7 @@ Theorem is_finite_Babs :
forall abs_nan x,
is_finite (Babs abs_nan x) = is_finite x.
Proof.
intros abs_nan [| | |] ; try easy.
intros s pl.
intros abs_nan [| |s pl|] ; try easy.
simpl.
now case abs_nan.
Qed.
......
......@@ -1673,7 +1673,7 @@ Qed.
(** Another well-used function for having the logarithm of a real number x to the base #&beta;# *)
Record ln_beta_prop x := {
ln_beta_val :> Z ;
_ : (x <> 0)%R -> (bpow (ln_beta_val - 1)%Z <= Rabs x < bpow ln_beta_val)%R
_ : (x <> 0)%R -> (bpow (ln_beta_val - 1)%Z <= Rabs x < bpow ln_beta_val)%R
}.
Definition ln_beta :
......
......@@ -136,9 +136,9 @@ Proof.
intros e He.
apply generic_format_bpow.
destruct (Zle_lt_or_eq _ _ He).
now apply valid_exp.
now apply valid_exp_.
rewrite <- H.
apply valid_exp_.
apply valid_exp.
rewrite H.
apply Zle_refl.
Qed.
......@@ -604,107 +604,6 @@ Qed.
Definition round x :=
F2R (Float beta (rnd (scaled_mantissa x)) (canonic_exp x)).
Theorem round_le_pos :
forall x y, (0 < x)%R -> (x <= y)%R -> (round x <= round y)%R.
Proof.
intros x y Hx Hxy.
unfold round, scaled_mantissa, canonic_exp.
destruct (ln_beta beta x) as (ex, Hex). simpl.
destruct (ln_beta beta y) as (ey, Hey). simpl.
specialize (Hex (Rgt_not_eq _ _ Hx)).
specialize (Hey (Rgt_not_eq _ _ (Rlt_le_trans _ _ _ Hx Hxy))).
rewrite Rabs_pos_eq in Hex.
2: now apply Rlt_le.
rewrite Rabs_pos_eq in Hey.
2: apply Rle_trans with (2:=Hxy); now apply Rlt_le.
assert (He: (ex <= ey)%Z).
cut (ex - 1 < ey)%Z. omega.
apply (lt_bpow beta).
apply Rle_lt_trans with (1 := proj1 Hex).
apply Rle_lt_trans with (1 := Hxy).
apply Hey.
destruct (Zle_or_lt ey (fexp ey)) as [Hy1|Hy1].
rewrite (proj2 (proj2 (valid_exp ey) Hy1) ex).
apply F2R_le_compat.
apply Zrnd_le.
apply Rmult_le_compat_r.
apply bpow_ge_0.
exact Hxy.
now apply Zle_trans with ey.
destruct (Zle_lt_or_eq _ _ He) as [He'|He'].
destruct (Zle_or_lt ey (fexp ex)) as [Hx2|Hx2].
rewrite (proj2 (proj2 (valid_exp ex) (Zle_trans _ _ _ He Hx2)) ey Hx2).
apply F2R_le_compat.
apply Zrnd_le.
apply Rmult_le_compat_r.
apply bpow_ge_0.
exact Hxy.
apply Rle_trans with (F2R (Float beta (rnd (bpow (ey - 1) * bpow (- fexp ey))) (fexp ey))).
rewrite <- bpow_plus.
rewrite <- (Z2R_Zpower beta (ey - 1 + -fexp ey)). 2: omega.
rewrite Zrnd_Z2R.
destruct (Zle_or_lt ex (fexp ex)) as [Hx1|Hx1].
apply Rle_trans with (F2R (Float beta 1 (fexp ex))).
apply F2R_le_compat.
rewrite <- (Zrnd_Z2R 1).
apply Zrnd_le.
apply Rlt_le.
exact (proj2 (mantissa_small_pos _ _ Hex Hx1)).
unfold F2R. simpl.
rewrite Z2R_Zpower. 2: omega.
rewrite <- bpow_plus, Rmult_1_l.
apply bpow_le.
omega.
apply Rle_trans with (F2R (Float beta (rnd (bpow ex * bpow (- fexp ex))) (fexp ex))).
apply F2R_le_compat.
apply Zrnd_le.
apply Rmult_le_compat_r.
apply bpow_ge_0.
apply Rlt_le.
apply Hex.
rewrite <- bpow_plus.
rewrite <- Z2R_Zpower. 2: omega.
rewrite Zrnd_Z2R.
unfold F2R. simpl.
rewrite 2!Z2R_Zpower ; try omega.
rewrite <- 2!bpow_plus.
apply bpow_le.
omega.
apply F2R_le_compat.
apply Zrnd_le.
apply Rmult_le_compat_r.
apply bpow_ge_0.
apply Hey.
rewrite He'.
apply F2R_le_compat.
apply Zrnd_le.
apply Rmult_le_compat_r.
apply bpow_ge_0.
exact Hxy.
Qed.
Theorem round_generic :
forall x,
generic_format x ->
round x = x.
Proof.
intros x Hx.
unfold round.
rewrite scaled_mantissa_generic with (1 := Hx).
rewrite Zrnd_Z2R.
now apply sym_eq.
Qed.
Theorem round_0 :
round 0 = R0.
Proof.
unfold round, scaled_mantissa.
rewrite Rmult_0_l.
fold (Z2R 0).
rewrite Zrnd_Z2R.
apply F2R_0.
Qed.
Theorem round_bounded_large_pos :
forall x ex,
(fexp ex < ex)%Z ->
......@@ -792,6 +691,74 @@ refine (let H := _ in conj (proj1 H) (Rlt_le _ _ (proj2 H))).
now apply mantissa_small_pos.
Qed.
Theorem round_le_pos :
forall x y, (0 < x)%R -> (x <= y)%R -> (round x <= round y)%R.
Proof.
intros x y Hx Hxy.
destruct (ln_beta beta x) as [ex Hex].
destruct (ln_beta beta y) as [ey Hey].
specialize (Hex (Rgt_not_eq _ _ Hx)).
specialize (Hey (Rgt_not_eq _ _ (Rlt_le_trans _ _ _ Hx Hxy))).
rewrite Rabs_pos_eq in Hex.
2: now apply Rlt_le.
rewrite Rabs_pos_eq in Hey.
2: apply Rle_trans with (2:=Hxy); now apply Rlt_le.
assert (He: (ex <= ey)%Z).
apply bpow_lt_bpow with beta.
apply Rle_lt_trans with (1 := proj1 Hex).
now apply Rle_lt_trans with y.
assert (Heq: fexp ex = fexp ey -> (round x <= round y)%R).
intros H.
unfold round, scaled_mantissa, canonic_exp.
rewrite ln_beta_unique_pos with (1 := Hex).
rewrite ln_beta_unique_pos with (1 := Hey).
rewrite H.
apply F2R_le_compat.
apply Zrnd_le.
apply Rmult_le_compat_r with (2 := Hxy).
apply bpow_ge_0.
destruct (Zle_or_lt ey (fexp ey)) as [Hy1|Hy1].
apply Heq.
apply valid_exp with (1 := Hy1).
now apply Zle_trans with ey.
destruct (Zle_lt_or_eq _ _ He) as [He'|He'].
2: now apply Heq, f_equal.
apply Rle_trans with (bpow (ey - 1)).
2: now apply round_bounded_large_pos.
destruct (Zle_or_lt ex (fexp ex)) as [Hx1|Hx1].
destruct (round_bounded_small_pos _ _ Hx1 Hex) as [-> | ->].
apply bpow_ge_0.
apply bpow_le.
apply valid_exp, proj2 in Hx1.
specialize (Hx1 ey).
omega.
apply Rle_trans with (bpow ex).
now apply round_bounded_large_pos.
apply bpow_le.
now apply Z.lt_le_pred.
Qed.
Theorem round_generic :
forall x,
generic_format x ->
round x = x.
Proof.
intros x Hx.
unfold round.
rewrite scaled_mantissa_generic with (1 := Hx).
rewrite Zrnd_Z2R.
now apply sym_eq.
Qed.
Theorem round_0 :
round 0 = R0.
Proof.
unfold round, scaled_mantissa.
rewrite Rmult_0_l.
fold (Z2R 0).
rewrite Zrnd_Z2R.
apply F2R_0.
Qed.
Theorem exp_small_round_0_pos :
forall x ex,
......@@ -807,9 +774,6 @@ apply bpow_gt_0.
apply (round_bounded_large_pos); assumption.
Qed.
Theorem generic_format_round_pos :
forall x,
(0 < x)%R ->
......@@ -832,14 +796,11 @@ destruct (Rle_or_lt (bpow ex) (round x)) as [Hr|Hr].
rewrite <- (Rle_antisym _ _ Hr Hr2).
apply generic_format_bpow.
now apply valid_exp.
assert (Hr' := conj Hr1 Hr).
unfold generic_format, scaled_mantissa.
rewrite (canonic_exp_fexp_pos _ _ Hr').
unfold round, scaled_mantissa.
apply generic_format_F2R.
intros _.
rewrite (canonic_exp_fexp_pos (F2R _) _ (conj Hr1 Hr)).
rewrite (canonic_exp_fexp_pos _ _ Hex).
unfold F2R at 3. simpl.
rewrite Rmult_assoc, <- bpow_plus, Zplus_opp_r, Rmult_1_r.
now rewrite Ztrunc_Z2R.
now apply Zeq_le.
Qed.
End Fcore_generic_round_pos.
......
......@@ -275,15 +275,13 @@ Theorem Only_DN_or_UP :
F f -> (fd <= f <= fu)%R ->
f = fd \/ f = fu.
Proof.
intros F x fd fu f Hd Hu Hf ([Hdf|Hdf], Hfu).
2 : now left.
destruct Hfu.
2 : now right.
destruct (Rle_or_lt x f).
elim Rlt_not_le with (1 := H).
intros F x fd fu f Hd Hu Hf [Hdf Hfu].
destruct (Rle_or_lt x f) ; [right|left].
apply Rle_antisym with (1 := Hfu).
now apply Hu.
elim Rlt_not_le with (1 := Hdf).
apply Hd ; auto with real.
apply Rlt_le in H.
apply Rle_antisym with (2 := Hdf).
now apply Hd.
Qed.
Theorem Rnd_ZR_abs :
......
......@@ -158,8 +158,7 @@ rewrite ulp_neq_0.
unfold F2R; simpl.
apply Rmult_le_compat_r.
apply bpow_ge_0.
replace 1%R with (Z2R (Zsucc 0)) by reflexivity.
apply Z2R_le.
apply (Z2R_le (Zsucc 0)).
apply Zlt_le_succ.
apply F2R_gt_0_reg with beta (canonic_exp beta fexp x).
now rewrite <- Fx.
......@@ -206,6 +205,7 @@ Qed.
Theorem ulp_bpow :
forall e, ulp (bpow e) = bpow (fexp (e + 1)).
Proof.
intros e.
rewrite ulp_neq_0.
apply f_equal.
......
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