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.