Simplified proofs.

src/Flocq_ulp.v

@@ -80,9 +80,7 @@ destruct (Rdichotomy x 0) as [Hx2|Hx2].
 intros Hx.
 elim Fx.
 rewrite Hx.
-exists (Float beta 0 _) ; repeat split.
-unfold F2R. simpl.
-now rewrite Rmult_0_l.
+apply generic_format_0.
 (* negative *)
 assert (Hu2 : Rnd_DN_pt F (-x) (-xu)).
 apply Rnd_UP_DN_pt_sym.
@@ -119,9 +117,8 @@ Theorem ulp_error :
 Proof.
 intros rnd Hrnd x.
 assert (Hs := generic_format_satisfies_any beta _ prop_exp).
-destruct (rounding_fun_of_pred _ (satisfies_any_imp_DN F Hs)) as (rndd, Hd).
-specialize (Hd x).
-destruct (Rle_lt_or_eq_dec (rndd x) x) as [Hxd|Hxd].
+destruct (proj1 (satisfies_any_imp_DN F Hs) x) as (d, Hd).
+destruct (Rle_lt_or_eq_dec d x) as [Hxd|Hxd].
 (* x <> rnd x *)
 apply Hd.
 assert (Fx : ~F x).
@@ -130,22 +127,21 @@ apply Rlt_not_le with (1 := Hxd).
 apply Req_le.
 apply sym_eq.
 now apply Rnd_DN_pt_idempotent with (1 := Hd).
-destruct (rounding_fun_of_pred _ (satisfies_any_imp_UP F Hs)) as (rndu, Hu).
-specialize (Hu x).
-assert (Hxu : (x < rndu x)%R).
-apply Rnot_le_lt.
-intros Hxu.
-apply Fx.
-rewrite Rle_antisym with (2 := Hxu).
+destruct (proj1 (satisfies_any_imp_UP F Hs) x) as (u, Hu).
+assert (Hxu : (x < u)%R).
+destruct (Rle_lt_or_eq_dec x u) as [Hxu|Hxu].
 apply Hu.
+exact Hxu.
+elim Fx.
+rewrite Hxu.
 apply Hu.
 rewrite (ulp_pred_succ_pt _ _ _ Fx Hd Hu) in Hxu, Hu.
 destruct (Rnd_DN_or_UP_pt _ _ Hrnd _ _ _ Hd Hu) as [Hr|Hr] ;
   rewrite Hr ; clear Hr.
 rewrite <- Ropp_minus_distr.
 rewrite Rabs_Ropp, Rabs_pos_eq.
-apply Rplus_lt_reg_r with (rndd x).
-now replace (rndd x + (x - rndd x))%R with x by ring.
+apply Rplus_lt_reg_r with d.
+now replace (d + (x - d))%R with x by ring.
 apply Rle_0_minus.
 apply Hd.
 rewrite Rabs_pos_eq.
@@ -172,9 +168,8 @@ Theorem ulp_half_error_pt :
 Proof.
 intros x xr Hxr.
 assert (Hs := generic_format_satisfies_any beta _ prop_exp).
-destruct (rounding_fun_of_pred _ (satisfies_any_imp_DN F Hs)) as (rndd, Hd).
-specialize (Hd x).
-destruct (Rle_lt_or_eq_dec (rndd x) x) as [Hxd|Hxd].
+destruct (proj1 (satisfies_any_imp_DN F Hs) x) as (d, Hd).
+destruct (Rle_lt_or_eq_dec d x) as [Hxd|Hxd].
 (* x <> rnd x *)
 apply Hd.
 assert (Fx : ~F x).
@@ -183,21 +178,20 @@ apply Rlt_not_le with (1 := Hxd).
 apply Req_le.
 apply sym_eq.
 now apply Rnd_DN_pt_idempotent with (1 := Hd).
-destruct (rounding_fun_of_pred _ (satisfies_any_imp_UP F Hs)) as (rndu, Hu).
-specialize (Hu x).
+destruct (proj1 (satisfies_any_imp_UP F Hs) x) as (u, Hu).
 rewrite (ulp_pred_succ_pt _ _ _ Fx Hd Hu) in Hu.
 destruct Hxr as (Hr1, Hr2).
-assert (Hdx : (Rabs (rndd x - x) = x - rndd x)%R).
+assert (Hdx : (Rabs (d - x) = x - d)%R).
 rewrite <- Ropp_minus_distr.
 rewrite Rabs_Ropp.
 apply Rabs_pos_eq.
 apply Rle_0_minus.
 apply Hd.
-assert (Hux : (Rabs (rndd x + ulp x - x) = rndd x + ulp x - x)%R).
+assert (Hux : (Rabs (d + ulp x - x) = d + ulp x - x)%R).
 apply Rabs_pos_eq.
 apply Rle_0_minus.
 apply Hu.
-destruct (Rle_or_lt (x - rndd x) (rndd x + ulp x - x)) as [H|H].
+destruct (Rle_or_lt (x - d) (d + ulp x - x)) as [H|H].
 (* . rnd(x) = rndd(x) *)
 apply Rle_trans with (1 := Hr2 _ (proj1 Hd)).
 rewrite Hdx.
@@ -205,7 +199,7 @@ apply Rmult_le_reg_l with 2%R.
 now apply (Z2R_lt 0 2).
 rewrite Rmult_plus_distr_r.
 rewrite Rmult_1_l.
-apply Rle_trans with (1 := Rplus_le_compat_l (x - rndd x) _ _ H).
+apply Rle_trans with (1 := Rplus_le_compat_l (x - d) _ _ H).
 field_simplify.
 apply Rle_refl.
 (* . rnd(x) = rndu(x) *)
@@ -216,7 +210,7 @@ now apply (Z2R_lt 0 2).
 rewrite Rmult_plus_distr_r.
 rewrite Rmult_1_l.
 apply Rlt_le.
-apply Rlt_le_trans with (1 := Rplus_lt_compat_l (rndd x + ulp x - x) _ _ H).
+apply Rlt_le_trans with (1 := Rplus_lt_compat_l (d + ulp x - x) _ _ H).
 field_simplify.
 apply Rle_refl.
 (* x = rnd x *)