Avoid using R0 and R1.

src/Appli/Fappli_IEEE.v

@@ -39,7 +39,7 @@ Inductive full_float :=
 Definition FF2R beta x :=
   match x with
   | F754_finite s m e => F2R (Float beta (cond_Zopp s (Zpos m)) e)
-  | _ => R0
+  | _ => 0%R
   end.
 
 End AnyRadix.
@@ -104,7 +104,7 @@ Definition B2FF x :=
 Definition B2R f :=
   match f with
   | B754_finite s m e _ => F2R (Float radix2 (cond_Zopp s (Zpos m)) e)
-  | _ => R0
+  | _ => 0%R
   end.
 
 Theorem FF2R_B2FF :
@@ -1714,7 +1714,7 @@ Definition Bdiv div_nan m x y :=
 
 Theorem Bdiv_correct :
   forall div_nan m x y,
-  B2R y <> R0 ->
+  B2R y <> 0%R ->
   if Rlt_bool (Rabs (round radix2 fexp (round_mode m) (B2R x / B2R y))) (bpow radix2 emax) then
     B2R (Bdiv div_nan m x y) = round radix2 fexp (round_mode m) (B2R x / B2R y) /\
     is_finite (Bdiv div_nan m x y) = is_finite x /\

src/Core/Fcore_FLT.v

@@ -187,7 +187,7 @@ Qed.
 
 (** Links between FLT and FIX (underflow) *)
 Theorem canonic_exp_FLT_FIX :
-  forall x, x <> R0 ->
+  forall x, x <> 0%R ->
   (Rabs x < bpow (emin + prec))%R ->
   canonic_exp beta FLT_exp x = canonic_exp beta (FIX_exp emin) x.
 Proof.

src/Core/Fcore_FTZ.v

@@ -217,7 +217,7 @@ Variable rnd : R -> Z.
 Context { valid_rnd : Valid_rnd rnd }.
 
 Definition Zrnd_FTZ x :=
-  if Rle_bool R1 (Rabs x) then rnd x else Z0.
+  if Rle_bool 1 (Rabs x) then rnd x else Z0.
 
 Global Instance valid_rnd_FTZ : Valid_rnd Zrnd_FTZ.
 Proof with auto with typeclass_instances.
@@ -270,7 +270,7 @@ Proof.
 intros x Hx.
 unfold round, scaled_mantissa, canonic_exp.
 destruct (ln_beta beta x) as (ex, He). simpl.
-assert (Hx0: x <> R0).
+assert (Hx0: x <> 0%R).
 intros Hx0.
 apply Rle_not_lt with (1 := Hx).
 rewrite Hx0, Rabs_R0.
@@ -286,7 +286,7 @@ rewrite Rle_bool_true.
 apply refl_equal.
 rewrite Rabs_mult.
 rewrite (Rabs_pos_eq (bpow (- FLX_exp prec ex))).
-change R1 with (bpow 0).
+change 1%R with (bpow 0).
 rewrite <- (Zplus_opp_r (FLX_exp prec ex)).
 rewrite bpow_plus.
 apply Rmult_le_compat_r.
@@ -320,7 +320,7 @@ rewrite Rle_bool_false.
 apply F2R_0.
 rewrite Rabs_mult.
 rewrite (Rabs_pos_eq (bpow (- FTZ_exp ex))).
-change R1 with (bpow 0).
+change 1%R with (bpow 0).
 rewrite <- (Zplus_opp_r (FTZ_exp ex)).
 rewrite bpow_plus.
 apply Rmult_lt_compat_r.

src/Core/Fcore_Raux.v

@@ -1456,7 +1456,7 @@ Definition bpow e :=
   match e with
   | Zpos p => Z2R (Zpower_pos r p)
   | Zneg p => Rinv (Z2R (Zpower_pos r p))
-  | Z0 => R1
+  | Z0 => 1%R
   end.
 
 Theorem Z2R_Zpower_pos :
@@ -1875,7 +1875,7 @@ apply bpow_ge_0.
 Qed.
 
 Theorem ln_beta_mult_bpow :
-  forall x e, x <> R0 ->
+  forall x e, x <> 0%R ->
   (ln_beta (x * bpow e) = ln_beta x + e :>Z)%Z.
 Proof.
 intros x e Zx.
@@ -1898,7 +1898,7 @@ Qed.
 
 Theorem ln_beta_le_bpow :
   forall x e,
-  x <> R0 ->
+  x <> 0%R ->
   (Rabs x < bpow e)%R ->
   (ln_beta x <= e)%Z.
 Proof.

src/Core/Fcore_float_prop.v

@@ -136,7 +136,7 @@ Qed.
 (** Sign facts *)
 Theorem F2R_0 :
   forall e : Z,
-  F2R (Float beta 0 e) = R0.
+  F2R (Float beta 0 e) = 0%R.
 Proof.
 intros e.
 unfold F2R. simpl.
@@ -145,7 +145,7 @@ Qed.
 
 Theorem F2R_eq_0_reg :
   forall m e : Z,
-  F2R (Float beta m e) = R0 ->
+  F2R (Float beta m e) = 0%R ->
   m = Z0.
 Proof.
 intros m e H.

src/Core/Fcore_generic_fmt.v

@@ -252,7 +252,7 @@ apply Rmult_1_r.
 Qed.
 
 Theorem scaled_mantissa_0 :
-  scaled_mantissa 0 = R0.
+  scaled_mantissa 0 = 0%R.
 Proof.
 apply Rmult_0_l.
 Qed.
@@ -667,7 +667,7 @@ Theorem round_bounded_small_pos :
   forall x ex,
   (ex <= fexp ex)%Z ->
   (bpow (ex - 1) <= x < bpow ex)%R ->
-  round x = R0 \/ round x = bpow (fexp ex).
+  round x = 0%R \/ round x = bpow (fexp ex).
 Proof.
 intros x ex He Hx.
 unfold round, scaled_mantissa.
@@ -751,7 +751,7 @@ now apply sym_eq.
 Qed.
 
 Theorem round_0 :
-  round 0 = R0.
+  round 0 = 0%R.
 Proof.
 unfold round, scaled_mantissa.
 rewrite Rmult_0_l.
@@ -762,8 +762,8 @@ Qed.
 
 Theorem exp_small_round_0_pos :
   forall x ex,
- (bpow (ex - 1) <= x < bpow ex)%R ->
-   round x =R0 -> (ex <= fexp ex)%Z .
+  (bpow (ex - 1) <= x < bpow ex)%R ->
+  round x = 0%R -> (ex <= fexp ex)%Z .
 Proof.
 intros x ex H H1.
 case (Zle_or_lt ex (fexp ex)); trivial; intros V.
@@ -771,7 +771,7 @@ contradict H1.
 apply Rgt_not_eq.
 apply Rlt_le_trans with (bpow (ex-1)).
 apply bpow_gt_0.
-apply  (round_bounded_large_pos); assumption.
+apply (round_bounded_large_pos); assumption.
 Qed.
 
 Theorem generic_format_round_pos :
@@ -931,7 +931,7 @@ rewrite <- Ropp_0.
 now apply Ropp_lt_contravar.
 now apply Ropp_le_contravar.
 (* . 0 <= y *)
-apply Rle_trans with R0.
+apply Rle_trans with 0%R.
 apply F2R_le_0_compat. simpl.
 rewrite <- (Zrnd_Z2R rnd 0).
 apply Zrnd_le...
@@ -1020,7 +1020,7 @@ Qed.
 Theorem exp_small_round_0 :
   forall rnd {Hr : Valid_rnd rnd} x ex,
   (bpow (ex - 1) <= Rabs x < bpow ex)%R ->
-   round rnd x =R0 -> (ex <= fexp ex)%Z .
+   round rnd x = 0%R -> (ex <= fexp ex)%Z .
 Proof.
 intros rnd Hr x ex H1 H2.
 generalize Rabs_R0.
@@ -1296,7 +1296,7 @@ Theorem round_DN_small_pos :
   forall x ex,
   (bpow (ex - 1) <= x < bpow ex)%R ->
   (ex <= fexp ex)%Z ->
-  round Zfloor x = R0.
+  round Zfloor x = 0%R.
 Proof.
 intros x ex Hx He.
 rewrite <- (F2R_0 beta (canonic_exp x)).
@@ -1552,7 +1552,7 @@ Qed.
 
 Lemma canonic_exp_le_bpow :
   forall (x : R) (e : Z),
-  x <> R0 ->
+  x <> 0%R ->
   (Rabs x < bpow e)%R ->
   (canonic_exp x <= fexp e)%Z.
 Proof.
@@ -1578,7 +1578,7 @@ Context { valid_rnd : Valid_rnd rnd }.
 
 Theorem ln_beta_round_ge :
   forall x,
-  round rnd x <> R0 ->
+  round rnd x <> 0%R ->
   (ln_beta beta x <= ln_beta beta (round rnd x))%Z.
 Proof with auto with typeclass_instances.
 intros x.
@@ -1597,7 +1597,7 @@ Qed.
 
 Theorem canonic_exp_round_ge :
   forall x,
-  round rnd x <> R0 ->
+  round rnd x <> 0%R ->
   (canonic_exp x <= canonic_exp (round rnd x))%Z.
 Proof with auto with typeclass_instances.
 intros x Zr.
@@ -1807,7 +1807,7 @@ rewrite Zceil_floor_neq.
 rewrite Z2R_plus.
 simpl.
 apply Ropp_lt_cancel.
-apply Rplus_lt_reg_l with R1.
+apply Rplus_lt_reg_l with 1%R.
 replace (1 + -/2)%R with (/2)%R by field.
 now replace (1 + - (Z2R (Zfloor x) + 1 - x))%R with (x - Z2R (Zfloor x))%R by ring.
 apply Rlt_not_eq.
@@ -1866,7 +1866,7 @@ rewrite Z2R_abs, Z2R_minus.
 replace (Z2R (Znearest x) - Z2R n)%R with (- (x - Z2R (Znearest x)) + (x - Z2R n))%R by ring.
 apply Rle_lt_trans with (1 := Rabs_triang _ _).
 simpl.
-replace R1 with (/2 + /2)%R by field.
+replace 1%R with (/2 + /2)%R by field.
 apply Rplus_le_lt_compat with (2 := Hd).
 rewrite Rabs_Ropp.
 apply Znearest_N.

src/Core/Fcore_ulp.v

@@ -1514,7 +1514,7 @@ now apply pred_pos_plus_ulp.
 Qed.
 
 Theorem pred_ulp_0 :
-  pred (ulp 0) = R0.
+  pred (ulp 0) = 0%R.
 Proof.
 rewrite pred_eq_pos.
 2: apply ulp_ge_0.

src/Prop/Fprop_div_sqrt_error.v

@@ -103,7 +103,7 @@ apply Rlt_le_trans with (Rabs x * 1)%R.
 apply Rmult_lt_compat_l.
 now apply Rabs_pos_lt.
 apply Rlt_le_trans with (1 := Heps1).
-change R1 with (bpow 0).
+change 1%R with (bpow 0).
 apply bpow_le.
 generalize (prec_gt_0 prec).
 clear ; omega.

src/Prop/Fprop_plus_error.v

@@ -158,7 +158,7 @@ Lemma round_plus_eq_zero_aux :
   (canonic_exp beta fexp x <= canonic_exp beta fexp y)%Z ->
   format x -> format y ->
   (0 <= x + y)%R ->
-  round beta fexp rnd (x + y) = R0 ->
+  round beta fexp rnd (x + y) = 0%R ->
   (x + y = 0)%R.
 Proof with auto with typeclass_instances.
 intros x y He Hx Hy Hp Hxy.
@@ -203,11 +203,11 @@ Context { valid_rnd : Valid_rnd rnd }.
 Theorem round_plus_eq_zero :
   forall x y,
   format x -> format y ->
-  round beta fexp rnd (x + y) = R0 ->
+  round beta fexp rnd (x + y) = 0%R ->
   (x + y = 0)%R.
 Proof with auto with typeclass_instances.
 intros x y Hx Hy.
-destruct (Rle_or_lt R0 (x + y)) as [H1|H1].
+destruct (Rle_or_lt 0 (x + y)) as [H1|H1].
 (* . *)
 revert H1.
 destruct (Zle_or_lt (canonic_exp beta fexp x) (canonic_exp beta fexp y)) as [H2|H2].

src/Prop/Fprop_relative.v

@@ -44,7 +44,7 @@ Proof with auto with typeclass_instances.
 intros x b Hb0 Hxb.
 destruct (Req_dec x 0) as [Hx0|Hx0].
 (* *)
-exists R0.
+exists 0%R.
 split.
 now rewrite Rabs_R0.
 rewrite Hx0, Rmult_0_l.
@@ -71,7 +71,7 @@ Proof with auto with typeclass_instances.
 intros x b Hb0 Hxb.
 destruct (Req_dec x 0) as [Hx0|Hx0].
 (* *)
-exists R0.
+exists 0%R.
 split.
 now rewrite Rabs_R0.
 rewrite Hx0, Rmult_0_l.
@@ -650,7 +650,7 @@ destruct (Rtotal_order x 0) as [Nx|[Zx|Px]].
   { apply (Rmult_le_reg_l 2 _ _ Rlt_0_2).
     rewrite Rmult_0_r, Rinv_r; [exact Rle_0_1|].
     apply Rgt_not_eq, Rlt_gt, Rlt_0_2. }
-  exists R0; exists R0; rewrite Zx; split; [|split; [|split]].
+  exists 0%R; exists 0%R; rewrite Zx; split; [|split; [|split]].
   { now rewrite Rabs_R0; apply Rmult_le_pos; [|apply bpow_ge_0]. }
   { now rewrite Rabs_R0; apply Rmult_le_pos; [|apply bpow_ge_0]. }
   { now rewrite Rmult_0_l. }