Commit b4ed2dbb authored by Guillaume Melquiond's avatar Guillaume Melquiond

Rename theorems and clean proofs.

parent b13beb9a
......@@ -19,6 +19,7 @@ COPYING file for more details.
(** * Error of the rounded-to-nearest addition is representable. *)
Require Import Psatz.
Require Import Fcore_Raux.
Require Import Fcore_defs.
Require Import Fcore_float_prop.
......@@ -269,7 +270,7 @@ Qed.
End Fprop_plus_FLT.
Section Fprop_plus_multi.
Section Fprop_plus_mult_ulp.
Variable beta : radix.
Notation bpow e := (bpow beta e).
......@@ -280,12 +281,12 @@ Context { monotone_exp : Monotone_exp fexp }.
Variable rnd : R -> Z.
Context { valid_rnd : Valid_rnd rnd }.
Notation format := (generic_format beta fexp).
Notation cexp := (canonic_exp beta fexp).
Lemma ex_shift: forall x e, format x -> (e <= cexp x)%Z ->
exists m, (x = Z2R m*bpow e)%R.
Lemma ex_shift :
forall x e, format x -> (e <= cexp x)%Z ->
exists m, (x = Z2R m * bpow e)%R.
Proof with auto with typeclass_instances.
intros x e Fx He.
exists (Ztrunc (scaled_mantissa beta fexp x)*Zpower beta (cexp x -e))%Z.
......@@ -297,34 +298,19 @@ rewrite Z2R_Zpower.
rewrite <- bpow_plus; f_equal; ring.
Qed.
Lemma ln_beta_minus1:
forall z, (z<>0)%R -> (ln_beta beta z -1 = ln_beta beta (z / Z2R beta))%Z.
Proof with auto with typeclass_instances.
intros z Hz; apply sym_eq, ln_beta_unique.
destruct (ln_beta beta z) as (e,He); simpl.
replace (z / Z2R beta)%R with (z*bpow (-1))%R.
rewrite Rabs_mult, (Rabs_right (bpow _)); try split.
apply Rmult_le_reg_r with (bpow 1).
apply bpow_gt_0.
rewrite Rmult_assoc, <- 2!bpow_plus.
rewrite Rmult_1_r.
apply Rle_trans with (2:=proj1 (He Hz)).
apply bpow_le; omega.
apply Rmult_lt_reg_r with (bpow 1).
apply bpow_gt_0.
rewrite Rmult_assoc, <- 2!bpow_plus.
rewrite Rmult_1_r.
apply Rlt_le_trans with (1:=proj2 (He Hz)).
apply bpow_le; omega.
apply Rle_ge, bpow_ge_0.
simpl; unfold Rdiv; f_equal; f_equal; f_equal.
unfold Z.pow_pos; simpl; ring.
Lemma ln_beta_minus1 :
forall z, z <> 0%R ->
(ln_beta beta z - 1)%Z = ln_beta beta (z / Z2R beta).
Proof.
intros z Hz.
unfold Zminus.
rewrite <- ln_beta_mult_bpow with (1 := Hz).
now rewrite bpow_opp, bpow_1.
Qed.
Lemma rnd_plus_mutiple:
forall x y, format x -> format y -> (x <> 0)%R ->
exists m,
(round beta fexp rnd (x+y) = Z2R m * ulp beta fexp (x/Z2R beta))%R.
Theorem round_plus_mult_ulp :
forall x y, format x -> format y -> (x <> 0)%R ->
exists m, (round beta fexp rnd (x+y) = Z2R m * ulp beta fexp (x/Z2R beta))%R.
Proof with auto with typeclass_instances.
intros x y Fx Fy Zx.
case (Zle_or_lt (ln_beta beta (x/Z2R beta)) (ln_beta beta y)); intros H1.
......@@ -350,22 +336,22 @@ destruct (ex_shift (round beta fexp rnd (x + y)) (cexp (x/Z2R beta))) as (n,Hn).
apply generic_format_round...
apply Zle_trans with (cexp (x+y)).
apply monotone_exp.
rewrite <- ln_beta_minus1; try assumption.
rewrite <- ln_beta_minus1 by easy.
rewrite <- (ln_beta_abs beta (x+y)).
(* . *)
assert (U:(Rabs (x+y) = Rabs x + Rabs y)%R \/ (y <> 0 /\ Rabs (x+y)=Rabs x - Rabs y)%R).
assert (V: forall x y, (Rabs y <= Rabs x)%R ->
(Rabs (x+y) = Rabs x + Rabs y)%R \/ (y <> 0 /\ Rabs (x+y)=Rabs x - Rabs y)%R).
assert (U: (Rabs (x+y) = Rabs x + Rabs y)%R \/ (y <> 0 /\ Rabs (x+y) = Rabs x - Rabs y)%R).
assert (V: forall x y, (Rabs y <= Rabs x)%R ->
(Rabs (x+y) = Rabs x + Rabs y)%R \/ (y <> 0 /\ Rabs (x+y) = Rabs x - Rabs y)%R).
clear; intros x y.
case (Rle_or_lt 0 y); intros Hy.
case Hy; intros Hy'.
case (Rle_or_lt 0 x); intros Hx.
intros _; rewrite (Rabs_right y); [idtac|now apply Rle_ge].
rewrite (Rabs_right x); [idtac|now apply Rle_ge].
left; apply Rabs_right.
apply Rle_ge; apply Rplus_le_le_0_compat; assumption.
rewrite (Rabs_right y); [idtac|now apply Rle_ge].
rewrite (Rabs_left x); [idtac|assumption].
intros _; rewrite (Rabs_pos_eq y) by easy.
rewrite (Rabs_pos_eq x) by easy.
left; apply Rabs_pos_eq.
now apply Rplus_le_le_0_compat.
rewrite (Rabs_pos_eq y) by easy.
rewrite (Rabs_left x) by easy.
intros H; right; split.
now apply Rgt_not_eq.
rewrite Rabs_left1.
......@@ -374,23 +360,19 @@ apply Rplus_le_reg_l with (-x)%R; ring_simplify; assumption.
intros _; left.
now rewrite <- Hy', Rabs_R0, 2!Rplus_0_r.
case (Rle_or_lt 0 x); intros Hx.
rewrite (Rabs_left y); [idtac|assumption].
rewrite (Rabs_right x); [idtac|now apply Rle_ge].
rewrite (Rabs_left y) by easy.
rewrite (Rabs_pos_eq x) by easy.
intros H; right; split.
apply sym_not_eq; now apply Rgt_not_eq.
rewrite Rabs_right.
now apply Rlt_not_eq.
rewrite Rabs_pos_eq.
ring.
apply Rle_ge; apply Rplus_le_reg_l with (-y)%R; ring_simplify; assumption.
apply Rplus_le_reg_l with (-y)%R; ring_simplify; assumption.
intros _; left.
rewrite (Rabs_left y); [idtac|assumption].
rewrite (Rabs_left x); [idtac|assumption].
rewrite (Rabs_left y) by easy.
rewrite (Rabs_left x) by easy.
rewrite Rabs_left1.
ring.
rewrite <- (Ropp_involutive (x+y)), <- Ropp_0.
apply Ropp_le_contravar; rewrite Ropp_plus_distr.
apply Rplus_le_le_0_compat.
rewrite <- Ropp_0; apply Ropp_le_contravar; now left.
rewrite <- Ropp_0; apply Ropp_le_contravar; now left.
lra.
apply V; left.
apply ln_beta_lt_pos with beta.
now apply Rabs_pos_lt.
......@@ -398,7 +380,8 @@ rewrite <- ln_beta_minus1 in H1; try assumption.
rewrite 2!ln_beta_abs; omega.
(* . *)
destruct U as [U|U].
rewrite U; apply Zle_trans with (ln_beta beta x);[omega|idtac].
rewrite U; apply Zle_trans with (ln_beta beta x).
omega.
rewrite <- ln_beta_abs.
apply ln_beta_le.
now apply Rabs_pos_lt.
......@@ -410,18 +393,17 @@ apply ln_beta_minus_lb.
now apply Rabs_pos_lt.
now apply Rabs_pos_lt.
rewrite 2!ln_beta_abs.
assert (ln_beta beta y < ln_beta beta x -1)%Z;[idtac|omega].
assert (ln_beta beta y < ln_beta beta x - 1)%Z.
now rewrite (ln_beta_minus1 x Zx).
omega.
apply canonic_exp_round_ge...
intros K.
absurd (x+y=0)%R.
intros K'.
apply round_plus_eq_zero in K...
contradict H1; apply Zle_not_lt.
rewrite <- (ln_beta_minus1 x Zx).
replace y with (-x)%R.
rewrite ln_beta_opp; omega.
apply Rplus_eq_reg_l with x; rewrite K'; ring.
apply round_plus_eq_zero with (6:=K)...
lra.
exists n.
rewrite ulp_neq_0.
assumption.
......@@ -431,11 +413,14 @@ apply Rgt_not_eq.
apply radix_pos.
Qed.
Lemma rnd_0_or_ge: Exp_not_FTZ fexp -> forall x y, format x -> format y ->
(round beta fexp rnd (x+y) = 0)%R \/
(ulp beta fexp (x/Z2R beta) <= Rabs (round beta fexp rnd (x+y)))%R.
Context {exp_not_FTZ : Exp_not_FTZ fexp}.
Theorem round_plus_ge_ulp :
forall x y, format x -> format y ->
round beta fexp rnd (x+y) = 0%R \/
(ulp beta fexp (x/Z2R beta) <= Rabs (round beta fexp rnd (x+y)))%R.
Proof with auto with typeclass_instances.
intros exp_not_FTZ x y Fx Fy.
intros x y Fx Fy.
case (Req_dec x 0); intros Zx.
(* *)
rewrite Zx, Rplus_0_l.
......@@ -443,8 +428,7 @@ rewrite round_generic...
unfold Rdiv; rewrite Rmult_0_l.
rewrite Fy at 2.
unfold F2R; simpl; rewrite Rabs_mult.
rewrite (Rabs_right (bpow _)).
2: apply Rle_ge, bpow_ge_0.
rewrite (Rabs_pos_eq (bpow _)) by apply bpow_ge_0.
case (Z.eq_dec (Ztrunc (scaled_mantissa beta fexp y)) 0); intros Hm.
left.
rewrite Fy, Hm; unfold F2R; simpl; ring.
......@@ -455,37 +439,35 @@ rewrite <- ulp_neq_0.
apply ulp_ge_ulp_0...
intros K; apply Hm.
rewrite K, scaled_mantissa_0.
replace 0%R with (Z2R 0) by reflexivity.
apply Ztrunc_Z2R.
apply (Ztrunc_Z2R 0).
apply Rmult_le_compat_r.
apply bpow_ge_0.
rewrite <- Z2R_abs.
replace 1%R with (Z2R 1) by reflexivity.
apply Z2R_le.
assert (0 < Z.abs (Ztrunc (scaled_mantissa beta fexp y)))%Z;[|omega].
apply (Z2R_le 1).
apply (Zlt_le_succ 0).
now apply Z.abs_pos.
(* *)
destruct (rnd_plus_mutiple x y Fx Fy Zx) as (m,Hm).
destruct (round_plus_mult_ulp x y Fx Fy Zx) as (m,Hm).
case (Z.eq_dec m 0); intros Zm.
left.
rewrite Hm, Zm; simpl; ring.
right.
rewrite Hm, Rabs_mult.
rewrite (Rabs_right (ulp _ _ _)).
2: apply Rle_ge, ulp_ge_0.
rewrite (Rabs_pos_eq (ulp _ _ _)) by apply ulp_ge_0.
apply Rle_trans with (1*ulp beta fexp (x/Z2R beta))%R.
right; ring.
apply Rmult_le_compat_r.
apply ulp_ge_0.
rewrite <- Z2R_abs.
replace 1%R with (Z2R 1) by reflexivity.
apply Z2R_le.
assert (0 < Z.abs m)%Z;[|omega].
apply (Z2R_le 1).
apply (Zlt_le_succ 0).
now apply Z.abs_pos.
Qed.
End Fprop_plus_multi.
Section Fprop_plus_multii.
End Fprop_plus_mult_ulp.
Section Fprop_plus_ge_ulp.
Variable beta : radix.
Notation bpow e := (bpow beta e).
......@@ -494,23 +476,23 @@ Context { valid_rnd : Valid_rnd rnd }.
Variable emin prec : Z.
Context { prec_gt_0_ : Prec_gt_0 prec }.
Lemma rnd_0_or_ge_FLT: forall x y e,
generic_format beta (FLT_exp emin prec) x -> generic_format beta (FLT_exp emin prec) y ->
(bpow e <= Rabs x)%R ->
(round beta (FLT_exp emin prec) rnd (x+y) = 0)%R \/
(bpow (e - prec) <= Rabs (round beta (FLT_exp emin prec) rnd (x+y)))%R.
Theorem round_plus_ge_ulp_FLT : forall x y e,
generic_format beta (FLT_exp emin prec) x -> generic_format beta (FLT_exp emin prec) y ->
(bpow e <= Rabs x)%R ->
round beta (FLT_exp emin prec) rnd (x+y) = 0%R \/
(bpow (e - prec) <= Rabs (round beta (FLT_exp emin prec) rnd (x+y)))%R.
Proof with auto with typeclass_instances.
intros x y e Fx Fy He.
assert (Zx:(x <> 0)%R).
intros K; contradict He.
apply Rlt_not_le; rewrite K, Rabs_R0.
apply bpow_gt_0.
case rnd_0_or_ge with beta (FLT_exp emin prec) rnd x y...
assert (Zx: x <> 0%R).
contradict He.
apply Rlt_not_le; rewrite He, Rabs_R0.
apply bpow_gt_0.
case round_plus_ge_ulp with beta (FLT_exp emin prec) rnd x y...
intros H; right.
apply Rle_trans with (2:=H).
rewrite ulp_neq_0.
unfold canonic_exp.
rewrite <- ln_beta_minus1; try assumption.
rewrite <- ln_beta_minus1 by easy.
unfold FLT_exp; apply bpow_le.
apply Zle_trans with (2:=Z.le_max_l _ _).
destruct (ln_beta beta x) as (n,Hn); simpl.
......@@ -524,24 +506,23 @@ apply Rgt_not_eq.
apply radix_pos.
Qed.
Lemma rnd_0_or_ge_FLX: forall x y e,
generic_format beta (FLX_exp prec) x -> generic_format beta (FLX_exp prec) y ->
(bpow e <= Rabs x)%R ->
(round beta (FLX_exp prec) rnd (x+y) = 0)%R \/
(bpow (e - prec) <= Rabs (round beta (FLX_exp prec) rnd (x+y)))%R.
Theorem round_plus_ge_ulp_FLX : forall x y e,
generic_format beta (FLX_exp prec) x -> generic_format beta (FLX_exp prec) y ->
(bpow e <= Rabs x)%R ->
round beta (FLX_exp prec) rnd (x+y) = 0%R \/
(bpow (e - prec) <= Rabs (round beta (FLX_exp prec) rnd (x+y)))%R.
Proof with auto with typeclass_instances.
intros x y e Fx Fy He.
assert (Zx:(x <> 0)%R).
intros K; contradict He.
apply Rlt_not_le; rewrite K, Rabs_R0.
apply bpow_gt_0.
case rnd_0_or_ge with beta (FLX_exp prec) rnd x y...
assert (Zx: x <> 0%R).
contradict He.
apply Rlt_not_le; rewrite He, Rabs_R0.
apply bpow_gt_0.
case round_plus_ge_ulp with beta (FLX_exp prec) rnd x y...
intros H; right.
apply Rle_trans with (2:=H).
rewrite ulp_neq_0.
unfold canonic_exp.
rewrite <- ln_beta_minus1; try assumption.
rewrite <- ln_beta_minus1 by easy.
unfold FLX_exp; apply bpow_le.
destruct (ln_beta beta x) as (n,Hn); simpl.
assert (e < n)%Z; try omega.
......@@ -554,4 +535,4 @@ apply Rgt_not_eq.
apply radix_pos.
Qed.
End Fprop_plus_multii.
\ No newline at end of file
End Fprop_plus_ge_ulp.
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