Update some additional Coq realizations now that notations are supported.

54f46f3b · Guillaume Melquiond · 47fb5a4c · 54f46f3b · 54f46f3b · 54f46f3b
Commit 54f46f3b authored Feb 25, 2012 by Guillaume Melquiond
5 changed files
--- a/Makefile.in
+++ b/Makefile.in
@@ -979,7 +979,7 @@ clean::
 	rm -f $(COQVO) $(COQVD) $(addsuffix .glob, $(COQLIBS_FILES))

 update-coq: bin/why3
-	#for f in $(COQLIBS_INT_FILES); do bin/why3 --realize -D drivers/coq-realize.drv -T int.$$f -o lib/coq/int/; done
+	for f in $(COQLIBS_INT_FILES); do bin/why3 --realize -D drivers/coq-realize.drv -T int.$$f -o lib/coq/int/; done
 	for f in $(COQLIBS_REAL_FILES); do bin/why3 --realize -D drivers/coq-realize.drv -T real.$$f -o lib/coq/real/; done
 	#for f in $(COQLIBS_FP_FILES); do bin/why3 --realize -D drivers/coq-realize.drv -T floating_point.$$f -o lib/coq/floating_point/; done


--- a/lib/coq/int/Abs.v
+++ b/lib/coq/int/Abs.v
@@ -2,19 +2,14 @@
 (* Beware! Only edit allowed sections below    *)
 Require Import ZArith.
 Require Import Rbase.
-(*Add Rec LoadPath "/home/guillaume/bin/why3/share/why3/theories".*)
-(*Add Rec LoadPath "/home/guillaume/bin/why3/share/why3/modules".*)
 Require int.Int.

+(* Why3 goal *)
 Notation abs := Zabs (only parsing).

-(* YOU MAY EDIT THE CONTEXT BELOW *)
-
-(* DO NOT EDIT BELOW *)
-
+(* Why3 goal *)
 Lemma abs_def : forall (x:Z), ((0%Z <= x)%Z -> ((abs x) = x)) /\
  ((~ (0%Z <= x)%Z) -> ((abs x) = (-x)%Z)).
-(* YOU MAY EDIT THE PROOF BELOW *)
 intros x.
 split ; intros H.
 now apply Zabs_eq.
@@ -24,29 +19,18 @@ contradict H.
 apply Zlt_le_weak.
 now apply Zgt_lt.
 Qed.
-(* DO NOT EDIT BELOW *)
-
-(* YOU MAY EDIT THE CONTEXT BELOW *)
-
-(* DO NOT EDIT BELOW *)

+(* Why3 goal *)
 Lemma Abs_le : forall (x:Z) (y:Z), ((abs x) <= y)%Z <-> (((-y)%Z <= x)%Z /\
  (x <= y)%Z).
-(* YOU MAY EDIT THE PROOF BELOW *)
 intros x y.
 zify.
 omega.
 Qed.
-(* DO NOT EDIT BELOW *)
-
-(* YOU MAY EDIT THE CONTEXT BELOW *)
-
-(* DO NOT EDIT BELOW *)

+(* Why3 goal *)
 Lemma Abs_pos : forall (x:Z), (0%Z <= (abs x))%Z.
-(* YOU MAY EDIT THE PROOF BELOW *)
 exact Zabs_pos.
 Qed.
-(* DO NOT EDIT BELOW *)


--- a/lib/coq/int/ComputerDivision.v
+++ b/lib/coq/int/ComputerDivision.v
@@ -3,34 +3,25 @@
 Require Import ZArith.
 Require Import Rbase.
 Require Import ZOdiv.
-(*Add Rec LoadPath "/home/guillaume/bin/why3/share/why3/theories".*)
-(*Add Rec LoadPath "/home/guillaume/bin/why3/share/why3/modules".*)
 Require int.Int.
 Require int.Abs.

+(* Why3 goal *)
 Notation div := ZOdiv (only parsing).

+(* Why3 goal *)
 Notation mod1 := ZOmod (only parsing).

-(* YOU MAY EDIT THE CONTEXT BELOW *)
-
-(* DO NOT EDIT BELOW *)
-
+(* Why3 goal *)
 Lemma Div_mod : forall (x:Z) (y:Z), (~ (y = 0%Z)) -> (x = ((y * (div x
  y))%Z + (mod1 x y))%Z).
-(* YOU MAY EDIT THE PROOF BELOW *)
 intros x y _.
 apply ZO_div_mod_eq.
 Qed.
-(* DO NOT EDIT BELOW *)
-
-(* YOU MAY EDIT THE CONTEXT BELOW *)
-
-(* DO NOT EDIT BELOW *)

+(* Why3 goal *)
 Lemma Div_bound : forall (x:Z) (y:Z), ((0%Z <= x)%Z /\ (0%Z <  y)%Z) ->
  ((0%Z <= (div x y))%Z /\ ((div x y) <= x)%Z).
-(* YOU MAY EDIT THE PROOF BELOW *)
 intros x y (Hx,Hy).
 split.
 apply ZO_div_pos with (1 := Hx).
@@ -44,15 +35,10 @@ apply ZO_div_lt with (1 := H').
 omega.
 now rewrite <- H', ZOdiv_0_l.
 Qed.
-(* DO NOT EDIT BELOW *)
-
-(* YOU MAY EDIT THE CONTEXT BELOW *)
-
-(* DO NOT EDIT BELOW *)

+(* Why3 goal *)
 Lemma Mod_bound : forall (x:Z) (y:Z), (~ (y = 0%Z)) ->
  (((-(Zabs y))%Z <  (mod1 x y))%Z /\ ((mod1 x y) <  (Zabs y))%Z).
-(* YOU MAY EDIT THE PROOF BELOW *)
 intros x y Zy.
 destruct (Zle_or_lt 0 x) as [Hx|Hx].
 refine ((fun H => conj (Zlt_le_trans _ 0 _ _ (proj1 H)) (proj2 H)) _).
@@ -63,66 +49,41 @@ clear -Zy ; zify ; omega.
 apply ZOmod_lt_neg with (2 := Zy).
 now apply Zlt_le_weak.
 Qed.
-(* DO NOT EDIT BELOW *)
-
-(* YOU MAY EDIT THE CONTEXT BELOW *)
-
-(* DO NOT EDIT BELOW *)

+(* Why3 goal *)
 Lemma Div_sign_pos : forall (x:Z) (y:Z), ((0%Z <= x)%Z /\ (0%Z <  y)%Z) ->
  (0%Z <= (div x y))%Z.
-(* YOU MAY EDIT THE PROOF BELOW *)
 intros x y (Hx, Hy).
 apply ZO_div_pos with (1 := Hx).
 now apply Zlt_le_weak.
 Qed.
-(* DO NOT EDIT BELOW *)
-
-(* YOU MAY EDIT THE CONTEXT BELOW *)
-
-(* DO NOT EDIT BELOW *)

+(* Why3 goal *)
 Lemma Div_sign_neg : forall (x:Z) (y:Z), ((x <= 0%Z)%Z /\ (0%Z <  y)%Z) ->
  ((div x y) <= 0%Z)%Z.
-(* YOU MAY EDIT THE PROOF BELOW *)
 intros x y (Hx, Hy).
 generalize (ZO_div_pos (-x) y).
 rewrite ZOdiv_opp_l.
 omega.
 Qed.
-(* DO NOT EDIT BELOW *)
-
-(* YOU MAY EDIT THE CONTEXT BELOW *)
-
-(* DO NOT EDIT BELOW *)

+(* Why3 goal *)
 Lemma Mod_sign_pos : forall (x:Z) (y:Z), ((0%Z <= x)%Z /\ ~ (y = 0%Z)) ->
  (0%Z <= (mod1 x y))%Z.
-(* YOU MAY EDIT THE PROOF BELOW *)
 intros x y (Hx, Zy).
 now apply ZOmod_lt_pos.
 Qed.
-(* DO NOT EDIT BELOW *)
-
-(* YOU MAY EDIT THE CONTEXT BELOW *)
-
-(* DO NOT EDIT BELOW *)

+(* Why3 goal *)
 Lemma Mod_sign_neg : forall (x:Z) (y:Z), ((x <= 0%Z)%Z /\ ~ (y = 0%Z)) ->
  ((mod1 x y) <= 0%Z)%Z.
-(* YOU MAY EDIT THE PROOF BELOW *)
 intros x y (Hx, Zy).
 now apply ZOmod_lt_neg.
 Qed.
-(* DO NOT EDIT BELOW *)
-
-(* YOU MAY EDIT THE CONTEXT BELOW *)
-
-(* DO NOT EDIT BELOW *)

+(* Why3 goal *)
 Lemma Rounds_toward_zero : forall (x:Z) (y:Z), (~ (y = 0%Z)) ->
  ((Zabs ((div x y) * y)%Z) <= (Zabs x))%Z.
-(* YOU MAY EDIT THE PROOF BELOW *)
 intros x y Zy.
 rewrite Zmult_comm.
 zify.
@@ -130,57 +91,32 @@ generalize (ZO_mult_div_le x y).
 generalize (ZO_mult_div_ge x y).
 omega.
 Qed.
-(* DO NOT EDIT BELOW *)
-
-(* YOU MAY EDIT THE CONTEXT BELOW *)
-
-(* DO NOT EDIT BELOW *)

+(* Why3 goal *)
 Lemma Div_1 : forall (x:Z), ((div x 1%Z) = x).
-(* YOU MAY EDIT THE PROOF BELOW *)
 exact ZOdiv_1_r.
 Qed.
-(* DO NOT EDIT BELOW *)
-
-(* YOU MAY EDIT THE CONTEXT BELOW *)
-
-(* DO NOT EDIT BELOW *)

+(* Why3 goal *)
 Lemma Mod_1 : forall (x:Z), ((mod1 x 1%Z) = 0%Z).
-(* YOU MAY EDIT THE PROOF BELOW *)
 exact ZOmod_1_r.
 Qed.
-(* DO NOT EDIT BELOW *)
-
-(* YOU MAY EDIT THE CONTEXT BELOW *)
-
-(* DO NOT EDIT BELOW *)

+(* Why3 goal *)
 Lemma Div_inf : forall (x:Z) (y:Z), ((0%Z <= x)%Z /\ (x <  y)%Z) -> ((div x
  y) = 0%Z).
-(* YOU MAY EDIT THE PROOF BELOW *)
 exact ZOdiv_small.
 Qed.
-(* DO NOT EDIT BELOW *)
-
-(* YOU MAY EDIT THE CONTEXT BELOW *)
-
-(* DO NOT EDIT BELOW *)

+(* Why3 goal *)
 Lemma Mod_inf : forall (x:Z) (y:Z), ((0%Z <= x)%Z /\ (x <  y)%Z) -> ((mod1 x
  y) = x).
-(* YOU MAY EDIT THE PROOF BELOW *)
 exact ZOmod_small.
 Qed.
-(* DO NOT EDIT BELOW *)
-
-(* YOU MAY EDIT THE CONTEXT BELOW *)
-
-(* DO NOT EDIT BELOW *)

+(* Why3 goal *)
 Lemma Div_mult : forall (x:Z) (y:Z) (z:Z), ((0%Z <  x)%Z /\ ((0%Z <= y)%Z /\
  (0%Z <= z)%Z)) -> ((div ((x * y)%Z + z)%Z x) = (y + (div z x))%Z).
-(* YOU MAY EDIT THE PROOF BELOW *)
 intros x y z (Hx&Hy&Hz).
 rewrite (Zplus_comm y).
 rewrite <- ZO_div_plus.
@@ -192,15 +128,10 @@ now apply Zlt_le_weak.
 intros H.
 now rewrite H in Hx.
 Qed.
-(* DO NOT EDIT BELOW *)
-
-(* YOU MAY EDIT THE CONTEXT BELOW *)
-
-(* DO NOT EDIT BELOW *)

+(* Why3 goal *)
 Lemma Mod_mult : forall (x:Z) (y:Z) (z:Z), ((0%Z <  x)%Z /\ ((0%Z <= y)%Z /\
  (0%Z <= z)%Z)) -> ((mod1 ((x * y)%Z + z)%Z x) = (mod1 z x)).
-(* YOU MAY EDIT THE PROOF BELOW *)
 intros x y z (Hx&Hy&Hz).
 rewrite Zplus_comm, Zmult_comm.
 apply ZO_mod_plus.
@@ -209,6 +140,5 @@ apply Zplus_le_0_compat with (1 := Hz).
 apply Zmult_le_0_compat with (1 := Hy).
 now apply Zlt_le_weak.
 Qed.
-(* DO NOT EDIT BELOW *)


--- a/lib/coq/int/Int.v
+++ b/lib/coq/int/Int.v
@@ -2,11 +2,11 @@
 (* Beware! Only edit allowed sections below    *)
 Require Import ZArith.
 Require Import Rbase.
-(*Add Rec LoadPath "/home/guillaume/bin/why3/share/why3/theories".*)
-(*Add Rec LoadPath "/home/guillaume/bin/why3/share/why3/modules".*)

+(* Why3 goal *)
 Notation infix_ls := Zlt (only parsing).

+(* Why3 assumption *)
 Definition infix_lseq(x:Z) (y:Z): Prop := (infix_ls x y) \/ (x = y).

 Lemma infix_lseq_Zle :
@@ -17,156 +17,96 @@ apply iff_Symmetric.
 apply Zle_lt_or_eq_iff.
 Qed.

+(* Why3 goal *)
 Notation infix_pl := Zplus (only parsing).

+(* Why3 goal *)
 Notation prefix_mn := Zopp (only parsing).

+(* Why3 goal *)
 Notation infix_as := Zmult (only parsing).

-(* YOU MAY EDIT THE CONTEXT BELOW *)
-
-(* DO NOT EDIT BELOW *)
-
+(* Why3 goal *)
 Lemma Unit_def : forall (x:Z), ((infix_pl x 0%Z) = x).
-(* YOU MAY EDIT THE PROOF BELOW *)
 exact Zplus_0_r.
 Qed.
-(* DO NOT EDIT BELOW *)
-
-(* YOU MAY EDIT THE CONTEXT BELOW *)
-
-(* DO NOT EDIT BELOW *)

+(* Why3 goal *)
 Lemma Assoc : forall (x:Z) (y:Z) (z:Z), ((infix_pl (infix_pl x y)
  z) = (infix_pl x (infix_pl y z))).
-(* YOU MAY EDIT THE PROOF BELOW *)
 intros x y z.
 apply sym_eq.
 apply Zplus_assoc.
 Qed.
-(* DO NOT EDIT BELOW *)
-
-(* YOU MAY EDIT THE CONTEXT BELOW *)
-
-(* DO NOT EDIT BELOW *)

+(* Why3 goal *)
 Lemma Inv_def : forall (x:Z), ((infix_pl x (prefix_mn x)) = 0%Z).
-(* YOU MAY EDIT THE PROOF BELOW *)
 exact Zplus_opp_r.
 Qed.
-(* DO NOT EDIT BELOW *)
-
-(* YOU MAY EDIT THE CONTEXT BELOW *)
-
-(* DO NOT EDIT BELOW *)

+(* Why3 goal *)
 Lemma Comm : forall (x:Z) (y:Z), ((infix_pl x y) = (infix_pl y x)).
-(* YOU MAY EDIT THE PROOF BELOW *)
 exact Zplus_comm.
 Qed.
-(* DO NOT EDIT BELOW *)
-
-(* YOU MAY EDIT THE CONTEXT BELOW *)
-
-(* DO NOT EDIT BELOW *)

+(* Why3 goal *)
 Lemma Assoc1 : forall (x:Z) (y:Z) (z:Z), ((infix_as (infix_as x y)
  z) = (infix_as x (infix_as y z))).
-(* YOU MAY EDIT THE PROOF BELOW *)
 intros x y z.
 apply sym_eq.
 apply Zmult_assoc.
 Qed.
-(* DO NOT EDIT BELOW *)
-
-(* YOU MAY EDIT THE CONTEXT BELOW *)
-
-(* DO NOT EDIT BELOW *)

+(* Why3 goal *)
 Lemma Mul_distr : forall (x:Z) (y:Z) (z:Z), ((infix_as x (infix_pl y
  z)) = (infix_pl (infix_as x y) (infix_as x z))).
-(* YOU MAY EDIT THE PROOF BELOW *)
 exact Zmult_plus_distr_r.
 Qed.
-(* DO NOT EDIT BELOW *)

+(* Why3 assumption *)
 Definition infix_mn(x:Z) (y:Z): Z := (infix_pl x (prefix_mn y)).

-(* YOU MAY EDIT THE CONTEXT BELOW *)
-
-(* DO NOT EDIT BELOW *)
-
+(* Why3 goal *)
 Lemma Comm1 : forall (x:Z) (y:Z), ((infix_as x y) = (infix_as y x)).
-(* YOU MAY EDIT THE PROOF BELOW *)
 exact Zmult_comm.
 Qed.
-(* DO NOT EDIT BELOW *)
-
-(* YOU MAY EDIT THE CONTEXT BELOW *)
-
-(* DO NOT EDIT BELOW *)

+(* Why3 goal *)
 Lemma Unitary : forall (x:Z), ((infix_as 1%Z x) = x).
-(* YOU MAY EDIT THE PROOF BELOW *)
 exact Zmult_1_l.
 Qed.
-(* DO NOT EDIT BELOW *)
-
-(* YOU MAY EDIT THE CONTEXT BELOW *)
-
-(* DO NOT EDIT BELOW *)

+(* Why3 goal *)
 Lemma NonTrivialRing : ~ (0%Z = 1%Z).
-(* YOU MAY EDIT THE PROOF BELOW *)
 discriminate.
 Qed.
-(* DO NOT EDIT BELOW *)
-
-(* YOU MAY EDIT THE CONTEXT BELOW *)
-
-(* DO NOT EDIT BELOW *)

+(* Why3 goal *)
 Lemma Refl : forall (x:Z), (infix_lseq x x).
-(* YOU MAY EDIT THE PROOF BELOW *)
 intros x.
 apply infix_lseq_Zle.
 apply Zle_refl.
 Qed.
-(* DO NOT EDIT BELOW *)
-
-(* YOU MAY EDIT THE CONTEXT BELOW *)
-
-(* DO NOT EDIT BELOW *)

+(* Why3 goal *)
 Lemma Trans : forall (x:Z) (y:Z) (z:Z), (infix_lseq x y) -> ((infix_lseq y
  z) -> (infix_lseq x z)).
-(* YOU MAY EDIT THE PROOF BELOW *)
 intros x y z H1 H2.
 apply infix_lseq_Zle.
 apply Zle_trans with y ;
  now apply infix_lseq_Zle.
 Qed.
-(* DO NOT EDIT BELOW *)
-
-(* YOU MAY EDIT THE CONTEXT BELOW *)
-
-(* DO NOT EDIT BELOW *)

+(* Why3 goal *)
 Lemma Antisymm : forall (x:Z) (y:Z), (infix_lseq x y) -> ((infix_lseq y x) ->
  (x = y)).
-(* YOU MAY EDIT THE PROOF BELOW *)
 intros x y H1 H2.
 apply Zle_antisym ;
  now apply infix_lseq_Zle.
 Qed.
-(* DO NOT EDIT BELOW *)
-
-(* YOU MAY EDIT THE CONTEXT BELOW *)
-
-(* DO NOT EDIT BELOW *)

+(* Why3 goal *)
 Lemma Total : forall (x:Z) (y:Z), (infix_lseq x y) \/ (infix_lseq y x).
-(* YOU MAY EDIT THE PROOF BELOW *)
 intros x y.
 destruct (Zle_or_lt x y) as [H|H].
 left.
@@ -175,34 +115,28 @@ right.
 apply infix_lseq_Zle.
 now apply Zlt_le_weak.
 Qed.
-(* DO NOT EDIT BELOW *)
-
-(* YOU MAY EDIT THE CONTEXT BELOW *)

-(* DO NOT EDIT BELOW *)
+(* Why3 goal *)
+Lemma ZeroLessOne : (infix_lseq 0%Z 1%Z).
+now left.
+Qed.

+(* Why3 goal *)
 Lemma CompatOrderAdd : forall (x:Z) (y:Z) (z:Z), (infix_lseq x y) ->
  (infix_lseq (infix_pl x z) (infix_pl y z)).
-(* YOU MAY EDIT THE PROOF BELOW *)
 intros x y z H.
 apply infix_lseq_Zle.
 apply Zplus_le_compat_r.
 now apply infix_lseq_Zle.
 Qed.
-(* DO NOT EDIT BELOW *)
-
-(* YOU MAY EDIT THE CONTEXT BELOW *)
-
-(* DO NOT EDIT BELOW *)

+(* Why3 goal *)
 Lemma CompatOrderMult : forall (x:Z) (y:Z) (z:Z), (infix_lseq x y) ->
  ((infix_lseq 0%Z z) -> (infix_lseq (infix_as x z) (infix_as y z))).
-(* YOU MAY EDIT THE PROOF BELOW *)
 intros x y z H1 H2.
 apply infix_lseq_Zle.
 apply Zmult_le_compat_r ;
  now apply infix_lseq_Zle.
 Qed.
-(* DO NOT EDIT BELOW *)


--- a/lib/coq/int/MinMax.v
+++ b/lib/coq/int/MinMax.v
@@ -2,33 +2,24 @@
 (* Beware! Only edit allowed sections below    *)
 Require Import ZArith.
 Require Import Rbase.
-(*Add Rec LoadPath "/home/guillaume/bin/why3/share/why3/theories".*)
-(*Add Rec LoadPath "/home/guillaume/bin/why3/share/why3/modules".*)
 Require int.Int.

+(* Why3 goal *)
 Notation min := Zmin (only parsing).

+(* Why3 goal *)
 Notation max := Zmax (only parsing).

-(* YOU MAY EDIT THE CONTEXT BELOW *)
-
-(* DO NOT EDIT BELOW *)
-
+(* Why3 goal *)
 Lemma Max_is_ge : forall (x:Z) (y:Z), (x <= (max x y))%Z /\ (y <= (max x
  y))%Z.
-(* YOU MAY EDIT THE PROOF BELOW *)
 split.
 apply Zle_max_l.
 apply Zle_max_r.
 Qed.
-(* DO NOT EDIT BELOW *)
-
-(* YOU MAY EDIT THE CONTEXT BELOW *)
-
-(* DO NOT EDIT BELOW *)

+(* Why3 goal *)
 Lemma Max_is_some : forall (x:Z) (y:Z), ((max x y) = x) \/ ((max x y) = y).
-(* YOU MAY EDIT THE PROOF BELOW *)
 intros x y.
 unfold Zmax.
 case Zcompare.
@@ -36,27 +27,17 @@ now left.
 now right.
 now left.
 Qed.
-(* DO NOT EDIT BELOW *)
-
-(* YOU MAY EDIT THE CONTEXT BELOW *)
-
-(* DO NOT EDIT BELOW *)

+(* Why3 goal *)
 Lemma Min_is_le : forall (x:Z) (y:Z), ((min x y) <= x)%Z /\ ((min x
  y) <= y)%Z.
-(* YOU MAY EDIT THE PROOF BELOW *)
 split.
 apply Zle_min_l.
 apply Zle_min_r.
 Qed.
-(* DO NOT EDIT BELOW *)
-
-(* YOU MAY EDIT THE CONTEXT BELOW *)
-
-(* DO NOT EDIT BELOW *)

+(* Why3 goal *)
 Lemma Min_is_some : forall (x:Z) (y:Z), ((min x y) = x) \/ ((min x y) = y).
-(* YOU MAY EDIT THE PROOF BELOW *)
 intros x y.
 unfold Zmin.
 case Zcompare.
@@ -64,68 +45,37 @@ now left.
 now left.
 now right.
 Qed.
-(* DO NOT EDIT BELOW *)
-
-(* YOU MAY EDIT THE CONTEXT BELOW *)
-
-(* DO NOT EDIT BELOW *)

+(* Why3 goal *)
 Lemma Max_x : forall (x:Z) (y:Z), (y <= x)%Z -> ((max x y) = x).
-(* YOU MAY EDIT THE PROOF BELOW *)
 exact Zmax_l.
 Qed.
-(* DO NOT EDIT BELOW *)
-
-(* YOU MAY EDIT THE CONTEXT BELOW *)
-
-(* DO NOT EDIT BELOW *)

+(* Why3 goal *)
 Lemma Max_y : forall (x:Z) (y:Z), (x <= y)%Z -> ((max x y) = y).
-(* YOU MAY EDIT THE PROOF BELOW *)
 exact Zmax_r.
 Qed.
-(* DO NOT EDIT BELOW *)
-
-(* YOU MAY EDIT THE CONTEXT BELOW *)
-
-(* DO NOT EDIT BELOW *)

+(* Why3 goal *)
 Lemma Min_x : forall (x:Z) (y:Z), (x <= y)%Z -> ((min x y) = x).
-(* YOU MAY EDIT THE PROOF BELOW *)
 exact Zmin_l.
 Qed.