Clean trailing whitespaces.

examples/Average.v

@@ -78,7 +78,7 @@ apply Zle_trans with (1:=H3).
 apply Zle_succ.
 Qed.
 
-Lemma FLT_format_half: forall u, 
+Lemma FLT_format_half: forall u,
    format u -> bpow (prec+emin) <= Rabs u -> format (u/2).
 Proof with auto with typeclass_instances.
 intros u Fu H.
@@ -108,7 +108,7 @@ now apply IZR_lt.
 now apply Zlt_le_weak.
 Qed.
 
-Lemma FLT_round_half: forall z, bpow (prec+emin) <= Rabs z -> 
+Lemma FLT_round_half: forall z, bpow (prec+emin) <= Rabs z ->
    round_flt (z/2)= round_flt z /2.
 Proof with auto with typeclass_instances.
 intros z Hz.
@@ -252,7 +252,7 @@ apply bpow_le; omega.
 Qed.
 
 
-Lemma round_plus_small_id_aux: forall f h, format f -> (bpow (prec+emin) <= f) -> 0 < f 
+Lemma round_plus_small_id_aux: forall f h, format f -> (bpow (prec+emin) <= f) -> 0 < f
    -> Rabs h <= /4* ulp_flt f -> round_flt (f+h) = f.
 Proof with auto with typeclass_instances.
 intros f h Ff H1 H2 Hh.
@@ -309,8 +309,8 @@ apply mag_ge_bpow.
 replace (prec+emin+1-1)%Z with (prec+emin)%Z by ring.
 rewrite Rabs_right; try assumption.
 apply Rle_ge; now left.
-assert (T1:(ulp_flt (pred_flt f) = ulp_flt f) 
-     \/ ( ulp_flt (pred_flt f) = /2* ulp_flt f 
+assert (T1:(ulp_flt (pred_flt f) = ulp_flt f)
+     \/ ( ulp_flt (pred_flt f) = /2* ulp_flt f
                /\ f = bpow (mag radix2 f -1))).
 generalize H; rewrite pred_eq_pos; [idtac|now left].
 unfold pred_pos; case Req_bool_spec; intros K HH.
@@ -407,7 +407,7 @@ unfold cexp, FLT_exp.
 rewrite Z.max_l.
 reflexivity.
 omega.
-assert (T: (ulp_flt (pred_flt f) = ulp_flt f \/ 
+assert (T: (ulp_flt (pred_flt f) = ulp_flt f \/
               (ulp_flt (pred_flt f) = / 2 * ulp_flt f /\ - h < / 4 * ulp_flt f))
          \/ (ulp_flt (pred_flt f) = / 2 * ulp_flt f /\
               f = bpow (mag radix2 f - 1) /\
@@ -583,7 +583,7 @@ rewrite T3.
 field.
 Qed.
 
-Lemma round_plus_small_id: forall f h, format f -> (bpow (prec+emin) <= Rabs f)  
+Lemma round_plus_small_id: forall f h, format f -> (bpow (prec+emin) <= Rabs f)
    -> Rabs h <= /4* ulp_flt f -> round_flt (f+h) = f.
 intros f h Ff H1 H2.
 case (Rle_or_lt 0 f); intros V.
@@ -849,7 +849,7 @@ Variable emin prec : Z.
 Context { prec_gt_0_ : Prec_gt_0 prec }.
 
 Notation format := (generic_format radix2 (FLT_exp emin prec)).
-Notation round_flt :=(round radix2 (FLT_exp emin prec) ZnearestE). 
+Notation round_flt :=(round radix2 (FLT_exp emin prec) ZnearestE).
 Notation ulp_flt :=(ulp radix2 (FLT_exp emin prec)).
 Notation cexp := (cexp radix2 (FLT_exp emin prec)).
 
@@ -1092,7 +1092,7 @@ now apply FLT_format_double.
 Qed.
 
 
-Lemma avg_half_sub_no_underflow_aux_aux: forall z:Z, (0 < z)%Z -> 
+Lemma avg_half_sub_no_underflow_aux_aux: forall z:Z, (0 < z)%Z ->
     (ZnearestE (IZR z / 2) < z)%Z.
 Proof.
 intros z H1.
@@ -1186,7 +1186,7 @@ omega.
 Qed.
 
 
-Lemma avg_half_sub_no_underflow_aux2: forall u v, format u -> format v -> 
+Lemma avg_half_sub_no_underflow_aux2: forall u v, format u -> format v ->
     (0 <= u /\ 0 <= v) \/ (u <= 0 /\ v <= 0) ->
     u <= v ->
    (bpow emin) <= Rabs ((u+v)/2) -> avg_half_sub u v <> 0.
@@ -1241,7 +1241,7 @@ rewrite <- H1; assumption.
 lra.
 Qed.
 
-Lemma avg_half_sub_no_underflow_aux3: forall u v, format u -> format v -> 
+Lemma avg_half_sub_no_underflow_aux3: forall u v, format u -> format v ->
     (0 <= u /\ 0 <= v) \/ (u <= 0 /\ v <= 0) ->
    (bpow emin) <= Rabs ((u+v)/2) -> avg_half_sub u v <> 0.
 Proof with auto with typeclass_instances.
@@ -1266,7 +1266,7 @@ unfold Rdiv; ring.
 Qed.
 
 
-Lemma avg_half_sub_no_underflow: 
+Lemma avg_half_sub_no_underflow:
   (0 <= x /\ 0 <= y) \/ (x <= 0 /\ y <= 0) ->
   (bpow emin) <= Rabs a -> av <> 0.
 Proof with auto with typeclass_instances.
@@ -1297,7 +1297,7 @@ change (bpow (-(1))) with (/2).
 field.
 assert (Z.abs (nu+nv) = 1)%Z.
 assert (0 < Z.abs (nu+nv) < 2)%Z;[idtac|omega].
-split; apply lt_IZR; simpl; rewrite abs_IZR; 
+split; apply lt_IZR; simpl; rewrite abs_IZR;
  apply Rmult_lt_reg_l with (bpow (emin-1)); try apply bpow_gt_0.
 rewrite Rmult_0_r.
 apply Rlt_le_trans with (1:=H1).
@@ -1637,7 +1637,7 @@ Qed.
 
 
 
-(* tight example x=1/2 and y=2^p-1: error is 5/4 ulp *) 
+(* tight example x=1/2 and y=2^p-1: error is 5/4 ulp *)
 
 Lemma avg_half_sub_correct: (0 <= x /\ 0 <= y) \/ (x <= 0 /\ y <= 0) ->
      Rabs (av-a) <= 3/2 * ulp_flt a.
@@ -1684,17 +1684,17 @@ Variable emin prec : Z.
 Context { prec_gt_0_ : Prec_gt_0 prec }.
 
 Notation format := (generic_format radix2 (FLT_exp emin prec)).
-Notation round_flt :=(round radix2 (FLT_exp emin prec) ZnearestE). 
+Notation round_flt :=(round radix2 (FLT_exp emin prec) ZnearestE).
 Notation ulp_flt :=(ulp radix2 (FLT_exp emin prec)).
 Notation cexp := (cexp radix2 (FLT_exp emin prec)).
 
-Definition average (x y : R) := 
+Definition average (x y : R) :=
    let samesign :=  match (Rle_bool 0 x), (Rle_bool 0 y) with
         true  , true   => true
       | false , false => true
       | _,_ => false
    end in
-     if samesign then 
+     if samesign then
        match (Rle_bool (Rabs x) (Rabs y)) with
             true => avg_half_sub emin prec x y
           | false => avg_half_sub emin prec y x
@@ -1724,11 +1724,11 @@ intros; replace u with v; trivial; auto with real.
 intros H1 H2; contradict H1; auto with real.
 Qed.
 
-Lemma average_symmetry_Ropp: forall u v, format u -> format v -> 
+Lemma average_symmetry_Ropp: forall u v, format u -> format v ->
   average (-u) (-v) = - average u v.
 Proof with auto with typeclass_instances.
 (* first: nonnegative u *)
-assert (forall u v, 0 <= u -> format u -> format v -> 
+assert (forall u v, 0 <= u -> format u -> format v ->
   average (-u) (-v) = - average u v).
 intros u v Hu Fu Fv; unfold average.
 rewrite 2!Rabs_Ropp.

examples/Double_rounding_odd_radix.v

@@ -567,7 +567,7 @@ destruct (Rle_or_lt x (midp beta fexp1 x)) as [H1|H1].
       as [H2|H2].
     * (* midp fexp1 x + / 2 * ulp beta fexp2 x < x *)
       now apply round_round_gt_mid_further_place; [| | |omega| |rewrite Hm].
-    * (* x <= midp fexp1 x + / 2 * ulp beta fexp2 x *)    
+    * (* x <= midp fexp1 x + / 2 * ulp beta fexp2 x *)
       now apply neq_midpoint_beta_odd_aux2; [| | | |split].
 Qed.
 
@@ -1433,7 +1433,7 @@ destruct (Req_dec r 0) as [Zr|Nzr].
     apply (Zodd_not_Zeven 1); [now simpl|].
     rewrite <- H.
     now apply Zeven_mult_Zeven_l. }
-(* 0 < r *)    
+(* 0 < r *)
 assert (Pr : 0 < r) by lra.
 destruct (Zle_or_lt exx (2 * exsx)) as [H1|H1].
 - (* exx <= 2 * exsx *)
@@ -1842,7 +1842,7 @@ destruct (Req_dec rd 0) as [Zrd|Nzrd].
       apply Rplus_le_le_0_compat.
       + now apply Rlt_le.
       + apply Rmult_le_pos; [lra|].
-        apply bpow_ge_0. 
+        apply bpow_ge_0.
     - apply (Rmult_lt_reg_r (bpow (mag x))); [now apply bpow_gt_0|].
       rewrite Rmult_1_l; bpow_simplify.
       apply Rlt_le_trans with (/ 2 * (2 * u)).
@@ -1900,7 +1900,7 @@ destruct (Req_dec rd 0) as [Zrd|Nzrd].
   { rewrite Hlr.
     rewrite <- Ztrunc_floor.
     - apply generic_format_round; [exact Vfexp1|].
-      apply valid_rnd_DN. 
+      apply valid_rnd_DN.
     - now apply Rmult_le_pos; [apply Rlt_le|apply bpow_ge_0]. }
   change (bpow (fexp1 _)) with u in Hx2_3.
   rewrite <- Xmid' in Hx2_3.
@@ -2034,7 +2034,7 @@ destruct (Req_dec rd 0) as [Zrd|Nzrd].
     apply Rmult_lt_0_compat; [lra|apply bpow_gt_0].
   + rewrite plus_IZR; apply Rplus_le_compat_l; simpl.
     unfold u, ulp, cexp; fold f1; bpow_simplify; lra.
-Qed.      
+Qed.
 
 Lemma round_round_eq_mid_beta_odd_rna_aux2 :
   forall (fexp1 fexp2 : Z -> Z),

examples/Sqrt_sqr.v

@@ -1117,7 +1117,7 @@ Qed.
 
 
 
-Lemma round_flx_sqr_sqrt_snd_deg: 
+Lemma round_flx_sqr_sqrt_snd_deg:
   (((radix_val beta=5)%Z /\ (3 < prec)%Z) \/ (5 < radix_val beta)%Z) ->
     sqrt (IZR beta) + / 4 * bpow (3 - prec) <=
       IZR beta * (2 - bpow (1 - prec)) / (2 * (2 + bpow (1 - prec))).
@@ -1821,7 +1821,7 @@ Proof with auto with typeclass_instances.
 intros x y Fx.
 assert (Y:Valid_exp (FLX_exp prec)).
 apply FLX_exp_valid; unfold Prec_gt_0; omega.
-assert (H: ((radix_val beta=5)%Z -> (3 < prec)%Z) \/ 
+assert (H: ((radix_val beta=5)%Z -> (3 < prec)%Z) \/
     ((radix_val beta=5)%Z /\ (prec=3)%Z)).
 case (Zle_lt_or_eq 3 prec); omega.
 case H; intros H'; clear H.

examples/Triangle.v

@@ -98,7 +98,7 @@ apply round_NE_pt...
 Qed.
 
 
-Lemma FLXN_le_exp: forall f1 f2:float beta, 
+Lemma FLXN_le_exp: forall f1 f2:float beta,
 ((Zpower beta (prec - 1) <= Zabs (Fnum f1) < Zpower beta prec)%Z) ->
 ((Zpower beta (prec - 1) <= Zabs (Fnum f2) < Zpower beta prec))%Z ->
 0 <= F2R f1 -> F2R f1 <= F2R f2 ->  (Fexp f1 <= Fexp f2)%Z.
@@ -215,8 +215,8 @@ Qed.
 
 Lemma t4_exact_aux: forall (f:float beta) g,
   (Z.abs (Fnum f) < Zpower beta prec)%Z
-   -> (0 <= g <= F2R f)%R 
-   -> (exists n:Z, (g=IZR n*bpow (Fexp f))%R) 
+   -> (0 <= g <= F2R f)%R
+   -> (exists n:Z, (g=IZR n*bpow (Fexp f))%R)
    -> format g.
 Proof with auto with typeclass_instances.
 intros f g Hf (Hg1,Hg2) (n,Hg3).
@@ -271,7 +271,7 @@ apply FLXN_format_generic in Fa...
 destruct Fa as [fa Hfa1 Hfa2].
 apply FLXN_format_generic in Fb...
 destruct Fb as [fb Hfb1 Hfb2].
-exists (Fnum fc -(Fnum fa*Zpower beta (Fexp fa-Fexp fc) 
+exists (Fnum fc -(Fnum fa*Zpower beta (Fexp fa-Fexp fc)
 -Fnum fb*Zpower beta (Fexp fb-Fexp fc)))%Z.
 rewrite Hfa1, Hfb1, Hfc1; unfold F2R; simpl.
 rewrite 2!minus_IZR.
@@ -399,14 +399,14 @@ now apply Rplus_le_le_0_compat.
 Qed.
 
 
-Lemma err_add2: forall x x2 y2 e2, err x2 y2 e2 
-  -> 0 <= e2 -> 0 <= y2 <= x 
+Lemma err_add2: forall x x2 y2 e2, err x2 y2 e2
+  -> 0 <= e2 -> 0 <= y2 <= x
   -> err (round_flx (x+x2)) (x+y2) (eps*(1+e2)+e2/2).
 Proof with auto with typeclass_instances.
 intros x x2 y2 e2 H2 H (Y1,Y2).
 replace (round_flx (x+x2) - (x+y2)) with ((round_flx (x+x2)-(x+x2))+(x2 - y2)) by ring.
 apply Rle_trans with (1:=Rabs_triang _ _).
-rewrite Rmult_plus_distr_r. 
+rewrite Rmult_plus_distr_r.
 apply Rplus_le_compat.
 apply Rle_trans with (eps*Rabs (x+x2)).
 apply err_init.
@@ -415,7 +415,7 @@ apply Rmult_le_compat_l.
 apply epsPos.
 replace (x+x2) with ((x + y2) +(x2-y2)) by ring.
 apply Rle_trans with (1:=Rabs_triang _ _).
-rewrite Rmult_plus_distr_r. 
+rewrite Rmult_plus_distr_r.
 rewrite Rmult_1_l.
 apply Rplus_le_compat_l.
 apply Rle_trans with (1:=H2).
@@ -445,7 +445,7 @@ Qed.
 
 
 
-Lemma err_mult: forall x1 y1 e1 x2 y2 e2, err x1 y1 e1 -> err x2 y2 e2 
+Lemma err_mult: forall x1 y1 e1 x2 y2 e2, err x1 y1 e1 -> err x2 y2 e2
   -> err (round_flx (x1*x2)) (y1*y2) (eps+(1+eps)*(e1+e2+e1*e2)).
 Proof with auto with typeclass_instances.
 intros x1 y1 e1 x2 y2 e2 H1 H2.
@@ -513,7 +513,7 @@ Qed.
 
 
 
-Lemma sqrt_var_maj_2: forall h : R, Rabs h <= /2 -> 
+Lemma sqrt_var_maj_2: forall h : R, Rabs h <= /2 ->
   Rabs (sqrt (1 + h) - 1) <= Rabs h / 2 + (Rabs h) * (Rabs h) / 4.
 Proof.
 intros h H1.
@@ -560,7 +560,7 @@ Qed.
 
 
 
-Lemma err_sqrt: forall x y e, 0 <= y -> e <= /2 -> err x y e 
+Lemma err_sqrt: forall x y e, 0 <= y -> e <= /2 -> err x y e
   -> err (round_flx (sqrt x)) (sqrt y) (eps+(1+eps)*(/2*e+/4*e*e)).
 Proof with auto with typeclass_instances.
 intros x y e Hy He H.
@@ -828,7 +828,7 @@ left; apply Rinv_0_lt_compat, Rmult_lt_0_compat; apply Rle_lt_0_plus_1; apply Rl
 apply sqrt_pos.
 Qed.
 
-Lemma Delta_correct_2 : radix_val beta=2%Z -> 
+Lemma Delta_correct_2 : radix_val beta=2%Z ->
     (Rabs (Delta - E_Delta) <= (19/4*eps+33*eps*eps) * E_Delta).
 Proof with auto with typeclass_instances.
 intros Hradix.
@@ -1125,8 +1125,8 @@ Qed.
 
 Lemma t4_exact_aux_: forall (f:float beta) g,
   (Z.abs (Fnum f) < Zpower beta prec)%Z
-   -> (0 <= g <= F2R f)%R 
-   -> (exists n:Z, (g=IZR n*bpow (Fexp f))%R) 
+   -> (0 <= g <= F2R f)%R
+   -> (exists n:Z, (g=IZR n*bpow (Fexp f))%R)
    -> (emin <= Fexp f)%Z
    -> format g.
 Proof with auto with typeclass_instances.
@@ -1189,8 +1189,8 @@ destruct Fa as [fa Hfa1 Hfa2].
 apply generic_format_FLX_FLT in Fb.
 apply FLXN_format_generic in Fb...
 destruct Fb as [fb Hfb1 Hfb2].
-exists (Fnum fc -(Fnum fa*Zpower beta (Fexp fa-Fexp fc) 
--Fnum fb*Zpower beta (Fexp fb-Fexp fc)))%Z. 
+exists (Fnum fc -(Fnum fa*Zpower beta (Fexp fa-Fexp fc)
+-Fnum fb*Zpower beta (Fexp fb-Fexp fc)))%Z.
 rewrite Hfa1, Hfb1, Hfc1; unfold F2R; simpl.
 rewrite 2!minus_IZR.
 rewrite 2!mult_IZR.
@@ -1275,7 +1275,7 @@ destruct Fa as [fa Hfa1 Hfa2 Hfa3].
 apply FLT_format_generic in Fb...
 destruct Fb as [fb Hfb1 Hfb2 Hfb3].
 rewrite Hgc2.
-exists (Fnum gc -(Fnum fa*Zpower beta (Fexp fa-emin) 
+exists (Fnum gc -(Fnum fa*Zpower beta (Fexp fa-emin)
 -Fnum fb*Zpower beta (Fexp fb -emin)))%Z.
 rewrite Hfa1, Hfb1, Hgc1; unfold F2R; simpl.
 rewrite Hgc2, 2!minus_IZR.
@@ -1337,7 +1337,7 @@ Qed.
 Notation err x y e := (Rabs (x - y) <= e * Rabs y).
 Notation eps :=(/2*bpow (1-prec)).
 
-Lemma err_add_no_err: forall x1 x2, 
+Lemma err_add_no_err: forall x1 x2,
     format x1 -> format x2
   -> err (round_flt (x1+x2)) (x1+x2) eps.
 Proof with auto with typeclass_instances.
@@ -1422,15 +1422,15 @@ repeat rewrite Rabs_right; try reflexivity; apply Rle_ge; try assumption.
 now apply Rplus_le_le_0_compat.
 Qed.
 
-Lemma err_add2_: forall x x2 y2 e2, err x2 y2 e2 
+Lemma err_add2_: forall x x2 y2 e2, err x2 y2 e2
   -> format x -> format x2
-  -> 0 <= e2 -> 0 <= y2 <= x 
+  -> 0 <= e2 -> 0 <= y2 <= x
   -> err (round_flt (x+x2)) (x+y2) (eps*(1+e2)+e2/2).
 Proof with auto with typeclass_instances.
 intros x x2 y2 e2 H2 Z1 Z2 H (Y1,Y2).
 replace (round_flt (x+x2) - (x+y2)) with ((round_flt (x+x2)-(x+x2))+(x2 - y2)) by ring.
 apply Rle_trans with (1:=Rabs_triang _ _).
-rewrite Rmult_plus_distr_r. 
+rewrite Rmult_plus_distr_r.
 apply Rplus_le_compat.
 apply Rle_trans with (eps*Rabs (x+x2)).
 now apply err_add_no_err.
@@ -1439,7 +1439,7 @@ apply Rmult_le_compat_l.
 apply epsPos.
 replace (x+x2) with ((x + y2) +(x2-y2)) by ring.
 apply Rle_trans with (1:=Rabs_triang _ _).
-rewrite Rmult_plus_distr_r. 
+rewrite Rmult_plus_distr_r.
 rewrite Rmult_1_l.
 apply Rplus_le_compat_l.
 apply Rle_trans with (1:=H2).
@@ -1463,8 +1463,8 @@ rewrite <- Rmult_assoc, Rinv_r.
 rewrite 2!Rabs_pos_eq ; lra.
 Qed.
 
-Lemma err_mult_aux: forall x1 y1 e1 x2 y2 e2, format x1 -> format x2 -> err x1 y1 e1 -> err x2 y2 e2 
-  -> err (round_flt (x1*x2)) (y1*y2) (eps+(1+eps)*(e1+e2+e1*e2)) 
+Lemma err_mult_aux: forall x1 y1 e1 x2 y2 e2, format x1 -> format x2 -> err x1 y1 e1 -> err x2 y2 e2
+  -> err (round_flt (x1*x2)) (y1*y2) (eps+(1+eps)*(e1+e2+e1*e2))
        \/ (Rabs (round_flt (x1*x2)) <= bpow (emin+prec-1)).
 Proof with auto with typeclass_instances.
 intros x1 y1 e1 x2 y2 e2 Hx1 Hx2 H1 H2.
@@ -1525,7 +1525,7 @@ apply Rmult_le_compat; try assumption; apply Rabs_pos.
 rewrite Rabs_mult; right; ring.
 Qed.
 
-Lemma err_mult_: forall x1 y1 e1 x2 y2 e2, format x1 -> format x2 -> err x1 y1 e1 -> err x2 y2 e2 
+Lemma err_mult_: forall x1 y1 e1 x2 y2 e2, format x1 -> format x2 -> err x1 y1 e1 -> err x2 y2 e2
   -> (bpow (emin+prec-1) < Rabs (round_flt (x1*x2)))
   -> err (round_flt (x1*x2)) (y1*y2) (eps+(1+eps)*(e1+e2+e1*e2)).
 Proof.
@@ -1580,7 +1580,7 @@ Qed.
 
 
 
-Lemma err_sqrt_: forall x y e, 0 <= y -> e <= /2 -> err x y e -> 
+Lemma err_sqrt_: forall x y e, 0 <= y -> e <= /2 -> err x y e ->
      bpow (emin+prec-1) < round_flt (sqrt x)
   -> err (round_flt (sqrt x)) (sqrt y) (eps+(1+eps)*(/2*e+/4*e*e)).
 Proof with auto with typeclass_instances.
@@ -1688,7 +1688,7 @@ apply Rle_ge, Hy.
 Qed.
 
 
-Lemma subnormal_aux: forall x y, format x -> (Rabs x <= 1 -> Rabs y <= 1) -> bpow (emin+prec-1) < Rabs (round_flt (x*y)) 
+Lemma subnormal_aux: forall x y, format x -> (Rabs x <= 1 -> Rabs y <= 1) -> bpow (emin+prec-1) < Rabs (round_flt (x*y))
    -> bpow (emin+prec-1) < Rabs x.
 Proof with auto with typeclass_instances.
 intros x y Fx H1 H2.
@@ -1874,7 +1874,7 @@ Qed.
 (* argh, would be simpler in radix 2  Delta = /4 * round_flx (sqrt M) *)
 
 
-Lemma Delta_correct_ : 
+Lemma Delta_correct_ :
   /4 * bpow (Zceil ((IZR (emin+prec-1))/2)) < Delta  ->
    (Rabs (Delta - E_Delta) <= (23/4*eps+38*eps*eps) * E_Delta).
 Proof with auto with typeclass_instances.
@@ -2024,7 +2024,7 @@ apply sqrt_pos.
 Qed.
 
 
-Lemma Delta_correct_2_ : radix_val beta=2%Z -> 
+Lemma Delta_correct_2_ : radix_val beta=2%Z ->
   bpow (Zceil ((IZR (emin+prec-1))/2) -2) < Delta  ->
   (Rabs (Delta - E_Delta) <= (19/4*eps+33*eps*eps) * E_Delta).
 Proof with auto with typeclass_instances.

src/Prop/Plus_error.v

@@ -322,7 +322,7 @@ rewrite <- mag_minus1; try assumption.
 rewrite <- (mag_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 -> 
+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.

src/Prop/Sterbenz.v

@@ -88,7 +88,7 @@ apply generic_format_abs_inv.
 rewrite Hxy.
 apply generic_format_bpow.
 apply valid_exp.
-case (Zmin_spec (mag beta x) (mag beta y)); intros (H1,H2); 
+case (Zmin_spec (mag beta x) (mag beta y)); intros (H1,H2);
    rewrite H2; now apply mag_generic_gt.
 Qed.
 

src/Translate/Ftranslate_flocq2Pff.v

@@ -162,7 +162,7 @@ rewrite Zabs2N.id_abs.
 now apply Z.abs_neq.
 Qed.
 
-Lemma make_bound_p: 
+Lemma make_bound_p:
         Zpos (vNum (make_bound))=(Zpower_nat beta (Zabs_nat p)).
 Proof.
 unfold make_bound, vNum; simpl.
@@ -175,7 +175,7 @@ omega.
 Qed.
 
 
-Lemma FtoR_F2R: forall (f:Pff.float) (g: float beta), Pff.Fnum f = Fnum g -> Pff.Fexp f = Fexp g -> 
+Lemma FtoR_F2R: forall (f:Pff.float) (g: float beta), Pff.Fnum f = Fnum g -> Pff.Fexp f = Fexp g ->
   FtoR beta f = F2R g.
 Proof.
 intros f g H1 H2; unfold FtoR, F2R.
@@ -238,7 +238,7 @@ set (ex := cexp beta (FLT_exp (-dExp b) p) x).
 set (mx := Ztrunc (scaled_mantissa beta (FLT_exp (-dExp b) p) x)).
 intros Hx; repeat split ; simpl.
 apply lt_IZR.
-rewrite pGivesBound, IZR_Zpower_nat. 
+rewrite pGivesBound, IZR_Zpower_nat.
 apply Rmult_lt_reg_r with (bpow beta ex).
 apply bpow_gt_0.
 rewrite <- bpow_plus.
@@ -501,7 +501,7 @@ apply pff_round_NE_is_round.
 Qed.
 
 Lemma pff_round_UP_is_round: forall (r:R),
-  FtoR beta (RND_Max b beta (Z.abs_nat p) r) 
+  FtoR beta (RND_Max b beta (Z.abs_nat p) r)
              = round beta (FLT_exp (- dExp b) p) Zceil r.
 Proof with auto with typeclass_instances.
 intros r.
@@ -530,7 +530,7 @@ Qed.
 
 
 Lemma pff_round_DN_is_round: forall (r:R),
-  FtoR beta (RND_Min b beta (Z.abs_nat p) r) 
+  FtoR beta (RND_Min b beta (Z.abs_nat p) r)
              = round beta (FLT_exp (- dExp b) p) Zfloor r.
 Proof with auto with typeclass_instances.
 intros r.
@@ -627,7 +627,7 @@ apply pff_round_N_is_round.
 Qed.
 
 Lemma pff_round_is_round_N: forall r f, Closest b beta r f ->
-    exists (choice:Z->bool), 
+    exists (choice:Z->bool),
       FtoR beta f = round beta (FLT_exp (-dExp b) p) (Znearest choice) r.
 Proof with auto with typeclass_instances.
 intros r f Hf.

src/Translate/Missing_theos.v

@@ -77,11 +77,11 @@ destruct (format_is_pff_format radix2 (make_bound radix2 prec emin)
 rewrite make_bound_Emin; try assumption.
 replace (--emin)%Z with emin by omega; assumption.
 (* *)
-pose (Iplus := fun (f g:Pff.float) => RND_Closest 
-        (make_bound radix2 prec emin) radix2 (Zabs_nat prec) choice 
+pose (Iplus := fun (f g:Pff.float) => RND_Closest
+        (make_bound radix2 prec emin) radix2 (Zabs_nat prec) choice
          (FtoR radix2 f + FtoR radix2 g)).
-pose (Iminus := fun (f g:Pff.float) => RND_Closest 
-        (make_bound radix2 prec emin) radix2 (Zabs_nat prec) choice 
+pose (Iminus := fun (f g:Pff.float) => RND_Closest
+        (make_bound radix2 prec emin) radix2 (Zabs_nat prec) choice
          (FtoR radix2 f - FtoR radix2 g)).
 assert (H1: forall x y, FtoR 2 (Iplus x y) = round_flt (FtoR 2 x + FtoR 2 y)).
 clear -prec_gt_0_ precisionNotZero emin_neg; intros x y.
@@ -211,11 +211,11 @@ destruct (format_is_pff_format radix2 (make_bound radix2 prec emin)
 rewrite make_bound_Emin; try assumption.
 replace (--emin)%Z with emin by omega; assumption.
 (* *)
-pose (Iplus := fun (f g:Pff.float) => RND_Closest 
-        (make_bound radix2 prec emin) radix2 (Zabs_nat prec) choice 
+pose (Iplus := fun (f g:Pff.float) => RND_Closest
+        (make_bound radix2 prec emin) radix2 (Zabs_nat prec) choice
          (FtoR radix2 f + FtoR radix2 g)).
-pose (Iminus := fun (f g:Pff.float) => RND_Closest 
-        (make_bound radix2 prec emin) radix2 (Zabs_nat prec) choice 
+pose (Iminus := fun (f g:Pff.float) => RND_Closest
+        (make_bound radix2 prec emin) radix2 (Zabs_nat prec) choice
          (FtoR radix2 f - FtoR radix2 g)).
 assert (H1: forall x y, FtoR 2 (Iplus x y) = round_flt (FtoR 2 x + FtoR 2 y)).
 clear -prec_gt_0_ precisionNotZero emin_neg; intros x y.
@@ -379,7 +379,7 @@ assumption.
 Qed.
 
 
-Theorem Veltkamp_Even: 
+Theorem Veltkamp_Even:
   (choice = fun z => negb (Z.even z)) ->
    hx = round_flt_s x.
 Proof with auto with typeclass_instances.
@@ -391,25 +391,25 @@ destruct (format_is_pff_format beta (make_bound beta prec emin)
 rewrite make_bound_Emin; try assumption.
 replace (--emin)%Z with emin by omega; assumption.
 destruct (round_NE_is_pff_round beta (make_bound beta prec emin)
-   prec (make_bound_p beta prec emin precisionNotZero) precisionNotZero 
+   prec (make_bound_p beta prec emin precisionNotZero) precisionNotZero
   (x*(bpow s+1)))
   as (fp,(Hfp, (Hfp',Hfp''))).
 rewrite make_bound_Emin in Hfp''; try assumption.
 replace (--emin)%Z with emin in Hfp'' by omega.
 destruct (round_NE_is_pff_round beta (make_bound beta prec emin)
-   prec (make_bound_p beta prec emin precisionNotZero) precisionNotZero 
+   prec (make_bound_p beta prec emin precisionNotZero) precisionNotZero
   (x-p))
   as (fq,(Hfq, (Hfq',Hfq''))).
 rewrite make_bound_Emin in Hfq''; try assumption.
 replace (--emin)%Z with emin in Hfq'' by omega.
 destruct (round_NE_is_pff_round beta (make_bound beta prec emin)
-   prec (make_bound_p beta prec emin precisionNotZero) precisionNotZero 
+   prec (make_bound_p beta prec emin precisionNotZero) precisionNotZero
   (q+p))
   as (fhx,(Hfhx, (Hfhx',Hfhx''))).
 rewrite make_bound_Emin in Hfhx''; try assumption.
 replace (--emin)%Z with emin in Hfhx'' by omega.
 (* *)
-destruct VeltkampEven with beta (make_bound beta prec emin) (Zabs_nat s) 
+destruct VeltkampEven with beta (make_bound beta prec emin) (Zabs_nat s)
    (Zabs_nat prec) fx fp fq fhx as (hx', (H1,H2)); try assumption.
 apply radix_gt_1.
 apply make_bound_p; omega.
@@ -424,9 +424,9 @@ rewrite Hfx, Hfp'', <- Hchoice; assumption.
 rewrite Hfp'', Hfq'', <- Hchoice; assumption.
 (* *)
 unfold hx; rewrite Hchoice, <- Hfhx'', <- H1.
-apply trans_eq with (FtoR beta (RND_EvenClosest 
+apply trans_eq with (FtoR beta (RND_EvenClosest
  (make_bound beta (prec-s) emin) beta (Zabs_nat (prec-s)) x)).
-generalize (EvenClosestUniqueP (make_bound beta (prec-s) emin) beta 
+generalize (EvenClosestUniqueP (make_bound beta (prec-s) emin) beta
    (Zabs_nat (prec-s))); unfold UniqueP; intros T.
 apply T with x; clear T.
 apply radix_gt_1.
@@ -485,25 +485,25 @@ destruct (format_is_pff_format beta (make_bound beta prec emin)
 rewrite make_bound_Emin; try assumption.
 replace (--emin)%Z with emin by omega; assumption.
 destruct (round_N_is_pff_round beta (make_bound beta prec emin)
-   prec (make_bound_p beta prec emin precisionNotZero) precisionNotZero 
+   prec (make_bound_p beta prec emin precisionNotZero) precisionNotZero
   choice (x*(bpow s+1)))
   as (fp,(Hfp, (Hfp',Hfp''))).
 rewrite make_bound_Emin in Hfp''; try assumption.
 replace (--emin)%Z with emin in Hfp'' by omega.
 destruct (round_N_is_pff_round beta (make_bound beta prec emin)
-   prec (make_bound_p beta prec emin precisionNotZero) precisionNotZero 
+   prec (make_bound_p beta prec emin precisionNotZero) precisionNotZero
   choice (x-p))
   as (fq,(Hfq, (Hfq',Hfq''))).
 rewrite make_bound_Emin in Hfq''; try assumption.
 replace (--emin)%Z with emin in Hfq'' by omega.
 destruct (round_N_is_pff_round beta (make_bound beta prec emin)
-   prec (make_bound_p beta prec emin precisionNotZero) precisionNotZero 
+   prec (make_bound_p beta prec emin precisionNotZero) precisionNotZero
   choice (q+p))
   as (fhx,(Hfhx, (Hfhx',Hfhx''))).
 rewrite make_bound_Emin in Hfhx''; try assumption.
 replace (--emin)%Z with emin in Hfhx'' by omega.
 (* *)
-destruct Veltkamp with beta (make_bound beta prec emin) (Zabs_nat s) 
+destruct Veltkamp with beta (make_bound beta prec emin) (Zabs_nat s)
    (Zabs_nat prec) fx fp fq fhx as (H1,(hx', (H2,(H3,H4)))); try assumption.
 apply radix_gt_1.
 apply make_bound_p; omega.
@@ -558,31 +558,31 @@ destruct (format_is_pff_format beta (make_bound beta prec emin)
 rewrite make_bound_Emin; try assumption.
 replace (--emin)%Z with emin by omega; assumption.
 destruct (round_N_is_pff_round beta (make_bound beta prec emin)
-   prec (make_bound_p beta prec emin precisionNotZero) precisionNotZero 
+   prec (make_bound_p beta prec emin precisionNotZero) precisionNotZero
   choice