diff options
Diffstat (limited to 'theories/Bool/Bool.v')
| -rw-r--r-- | theories/Bool/Bool.v | 362 |
1 files changed, 211 insertions, 151 deletions
diff --git a/theories/Bool/Bool.v b/theories/Bool/Bool.v index 47b9fc83..7f54efa3 100644 --- a/theories/Bool/Bool.v +++ b/theories/Bool/Bool.v @@ -6,12 +6,18 @@ (* * GNU Lesser General Public License Version 2.1 *) (************************************************************************) -(*i $Id: Bool.v 10812 2008-04-17 16:42:37Z letouzey $ i*) +(*i $Id$ i*) (** The type [bool] is defined in the prelude as [Inductive bool : Set := true : bool | false : bool] *) +(** Most of the lemmas in this file are trivial after breaking all booleans *) + +Ltac destr_bool := + intros; destruct_all bool; simpl in *; trivial; try discriminate. + (** Interpretation of booleans as propositions *) + Definition Is_true (b:bool) := match b with | true => True @@ -33,42 +39,40 @@ Defined. Lemma diff_true_false : true <> false. Proof. - unfold not in |- *; intro contr; change (Is_true false) in |- *. - elim contr; simpl in |- *; trivial. + discriminate. Qed. Hint Resolve diff_true_false : bool v62. Lemma diff_false_true : false <> true. -Proof. - red in |- *; intros H; apply diff_true_false. - symmetry in |- *. -assumption. +Proof. + discriminate. Qed. Hint Resolve diff_false_true : bool v62. Hint Extern 1 (false <> true) => exact diff_false_true. Lemma eq_true_false_abs : forall b:bool, b = true -> b = false -> False. Proof. - intros b H; rewrite H; auto with bool. + destr_bool. Qed. Lemma not_true_is_false : forall b:bool, b <> true -> b = false. Proof. - destruct b. - intros. - red in H; elim H. - reflexivity. - intros abs. - reflexivity. + destr_bool; intuition. Qed. Lemma not_false_is_true : forall b:bool, b <> false -> b = true. Proof. - destruct b. - intros. - reflexivity. - intro H; red in H; elim H. - reflexivity. + destr_bool; intuition. +Qed. + +Lemma not_true_iff_false : forall b, b <> true <-> b = false. +Proof. + destr_bool; intuition. +Qed. + +Lemma not_false_iff_true : forall b, b <> false <-> b = true. +Proof. + destr_bool; intuition. Qed. (**********************) @@ -82,6 +86,11 @@ Definition leb (b1 b2:bool) := end. Hint Unfold leb: bool v62. +Lemma leb_implb : forall b1 b2, leb b1 b2 <-> implb b1 b2 = true. +Proof. + destr_bool; intuition. +Qed. + (* Infix "<=" := leb : bool_scope. *) (*************) @@ -99,37 +108,33 @@ Definition eqb (b1 b2:bool) : bool := Lemma eqb_subst : forall (P:bool -> Prop) (b1 b2:bool), eqb b1 b2 = true -> P b1 -> P b2. Proof. - unfold eqb in |- *. - intros P b1. - intros b2. - case b1. - case b2. - trivial with bool. - intros H. - inversion_clear H. - case b2. - intros H. - inversion_clear H. - trivial with bool. + destr_bool. Qed. Lemma eqb_reflx : forall b:bool, eqb b b = true. Proof. - intro b. - case b. - trivial with bool. - trivial with bool. + destr_bool. Qed. Lemma eqb_prop : forall a b:bool, eqb a b = true -> a = b. Proof. - destruct a; destruct b; simpl in |- *; intro; discriminate H || reflexivity. + destr_bool. +Qed. + +Lemma eqb_true_iff : forall a b:bool, eqb a b = true <-> a = b. +Proof. + destr_bool; intuition. +Qed. + +Lemma eqb_false_iff : forall a b:bool, eqb a b = false <-> a <> b. +Proof. + destr_bool; intuition. Qed. (************************) (** * A synonym of [if] on [bool] *) (************************) - + Definition ifb (b1 b2 b3:bool) : bool := match b1 with | true => b2 @@ -144,12 +149,12 @@ Open Scope bool_scope. Lemma negb_orb : forall b1 b2:bool, negb (b1 || b2) = negb b1 && negb b2. Proof. - destruct b1; destruct b2; simpl in |- *; reflexivity. + destr_bool. Qed. Lemma negb_andb : forall b1 b2:bool, negb (b1 && b2) = negb b1 || negb b2. Proof. - destruct b1; destruct b2; simpl in |- *; reflexivity. + destr_bool. Qed. (********************************) @@ -158,12 +163,12 @@ Qed. Lemma negb_involutive : forall b:bool, negb (negb b) = b. Proof. - destruct b; reflexivity. + destr_bool. Qed. Lemma negb_involutive_reverse : forall b:bool, b = negb (negb b). Proof. - destruct b; reflexivity. + destr_bool. Qed. Notation negb_elim := negb_involutive (only parsing). @@ -171,35 +176,39 @@ Notation negb_intro := negb_involutive_reverse (only parsing). Lemma negb_sym : forall b b':bool, b' = negb b -> b = negb b'. Proof. - destruct b; destruct b'; intros; simpl in |- *; trivial with bool. + destr_bool. Qed. Lemma no_fixpoint_negb : forall b:bool, negb b <> b. Proof. - destruct b; simpl in |- *; intro; apply diff_true_false; - auto with bool. + destr_bool. Qed. Lemma eqb_negb1 : forall b:bool, eqb (negb b) b = false. Proof. - destruct b. - trivial with bool. - trivial with bool. + destr_bool. Qed. - + Lemma eqb_negb2 : forall b:bool, eqb b (negb b) = false. Proof. - destruct b. - trivial with bool. - trivial with bool. + destr_bool. Qed. - Lemma if_negb : forall (A:Type) (b:bool) (x y:A), (if negb b then x else y) = (if b then y else x). Proof. - destruct b; trivial. + destr_bool. +Qed. + +Lemma negb_true_iff : forall b, negb b = true <-> b = false. +Proof. + destr_bool; intuition. +Qed. + +Lemma negb_false_iff : forall b, negb b = false <-> b = true. +Proof. + destr_bool; intuition. Qed. @@ -207,46 +216,60 @@ Qed. (** * Properties of [orb] *) (********************************) +Lemma orb_true_iff : + forall b1 b2, b1 || b2 = true <-> b1 = true \/ b2 = true. +Proof. + destr_bool; intuition. +Qed. + +Lemma orb_false_iff : + forall b1 b2, b1 || b2 = false <-> b1 = false /\ b2 = false. +Proof. + destr_bool; intuition. +Qed. + Lemma orb_true_elim : forall b1 b2:bool, b1 || b2 = true -> {b1 = true} + {b2 = true}. Proof. - destruct b1; simpl in |- *; auto with bool. + destruct b1; simpl; auto. Defined. Lemma orb_prop : forall a b:bool, a || b = true -> a = true \/ b = true. Proof. - destruct a; destruct b; simpl in |- *; try (intro H; discriminate H); - auto with bool. + intros; apply orb_true_iff; trivial. Qed. Lemma orb_true_intro : forall b1 b2:bool, b1 = true \/ b2 = true -> b1 || b2 = true. Proof. - destruct b1; auto with bool. - destruct 1; intros. - elim diff_true_false; auto with bool. - rewrite H; trivial with bool. + intros; apply orb_true_iff; trivial. Qed. Hint Resolve orb_true_intro: bool v62. Lemma orb_false_intro : forall b1 b2:bool, b1 = false -> b2 = false -> b1 || b2 = false. Proof. - intros b1 b2 H1 H2; rewrite H1; rewrite H2; trivial with bool. + intros. subst. reflexivity. Qed. Hint Resolve orb_false_intro: bool v62. +Lemma orb_false_elim : + forall b1 b2:bool, b1 || b2 = false -> b1 = false /\ b2 = false. +Proof. + intros. apply orb_false_iff; trivial. +Qed. + (** [true] is a zero for [orb] *) Lemma orb_true_r : forall b:bool, b || true = true. Proof. - auto with bool. + destr_bool. Qed. Hint Resolve orb_true_r: bool v62. Lemma orb_true_l : forall b:bool, true || b = true. Proof. - trivial with bool. + reflexivity. Qed. Notation orb_b_true := orb_true_r (only parsing). @@ -256,34 +279,24 @@ Notation orb_true_b := orb_true_l (only parsing). Lemma orb_false_r : forall b:bool, b || false = b. Proof. - destruct b; trivial with bool. + destr_bool. Qed. Hint Resolve orb_false_r: bool v62. Lemma orb_false_l : forall b:bool, false || b = b. Proof. - destruct b; trivial with bool. + destr_bool. Qed. Hint Resolve orb_false_l: bool v62. Notation orb_b_false := orb_false_r (only parsing). Notation orb_false_b := orb_false_l (only parsing). -Lemma orb_false_elim : - forall b1 b2:bool, b1 || b2 = false -> b1 = false /\ b2 = false. -Proof. - destruct b1. - intros; elim diff_true_false; auto with bool. - destruct b2. - intros; elim diff_true_false; auto with bool. - auto with bool. -Qed. - (** Complementation *) Lemma orb_negb_r : forall b:bool, b || negb b = true. Proof. - destruct b; reflexivity. + destr_bool. Qed. Hint Resolve orb_negb_r: bool v62. @@ -293,14 +306,14 @@ Notation orb_neg_b := orb_negb_r (only parsing). Lemma orb_comm : forall b1 b2:bool, b1 || b2 = b2 || b1. Proof. - destruct b1; destruct b2; reflexivity. + destr_bool. Qed. (** Associativity *) Lemma orb_assoc : forall b1 b2 b3:bool, b1 || (b2 || b3) = b1 || b2 || b3. Proof. - destruct b1; destruct b2; destruct b3; reflexivity. + destr_bool. Qed. Hint Resolve orb_comm orb_assoc: bool v62. @@ -308,38 +321,44 @@ Hint Resolve orb_comm orb_assoc: bool v62. (** * Properties of [andb] *) (*******************************) -Lemma andb_true_iff : +Lemma andb_true_iff : forall b1 b2:bool, b1 && b2 = true <-> b1 = true /\ b2 = true. Proof. - destruct b1; destruct b2; intuition. + destr_bool; intuition. +Qed. + +Lemma andb_false_iff : + forall b1 b2:bool, b1 && b2 = false <-> b1 = false \/ b2 = false. +Proof. + destr_bool; intuition. Qed. Lemma andb_true_eq : forall a b:bool, true = a && b -> true = a /\ true = b. Proof. - destruct a; destruct b; auto. + destr_bool. auto. Defined. Lemma andb_false_intro1 : forall b1 b2:bool, b1 = false -> b1 && b2 = false. Proof. - destruct b1; destruct b2; simpl in |- *; tauto || auto with bool. + intros. apply andb_false_iff. auto. Qed. Lemma andb_false_intro2 : forall b1 b2:bool, b2 = false -> b1 && b2 = false. Proof. - destruct b1; destruct b2; simpl in |- *; tauto || auto with bool. + intros. apply andb_false_iff. auto. Qed. (** [false] is a zero for [andb] *) Lemma andb_false_r : forall b:bool, b && false = false. Proof. - destruct b; auto with bool. + destr_bool. Qed. Lemma andb_false_l : forall b:bool, false && b = false. Proof. - trivial with bool. + reflexivity. Qed. Notation andb_b_false := andb_false_r (only parsing). @@ -349,12 +368,12 @@ Notation andb_false_b := andb_false_l (only parsing). Lemma andb_true_r : forall b:bool, b && true = b. Proof. - destruct b; auto with bool. + destr_bool. Qed. Lemma andb_true_l : forall b:bool, true && b = b. Proof. - trivial with bool. + reflexivity. Qed. Notation andb_b_true := andb_true_r (only parsing). @@ -363,7 +382,7 @@ Notation andb_true_b := andb_true_l (only parsing). Lemma andb_false_elim : forall b1 b2:bool, b1 && b2 = false -> {b1 = false} + {b2 = false}. Proof. - destruct b1; simpl in |- *; auto with bool. + destruct b1; simpl; auto. Defined. Hint Resolve andb_false_elim: bool v62. @@ -371,8 +390,8 @@ Hint Resolve andb_false_elim: bool v62. Lemma andb_negb_r : forall b:bool, b && negb b = false. Proof. - destruct b; reflexivity. -Qed. + destr_bool. +Qed. Hint Resolve andb_negb_r: bool v62. Notation andb_neg_b := andb_negb_r (only parsing). @@ -381,14 +400,14 @@ Notation andb_neg_b := andb_negb_r (only parsing). Lemma andb_comm : forall b1 b2:bool, b1 && b2 = b2 && b1. Proof. - destruct b1; destruct b2; reflexivity. + destr_bool. Qed. (** Associativity *) Lemma andb_assoc : forall b1 b2 b3:bool, b1 && (b2 && b3) = b1 && b2 && b3. Proof. - destruct b1; destruct b2; destruct b3; reflexivity. + destr_bool. Qed. Hint Resolve andb_comm andb_assoc: bool v62. @@ -402,25 +421,25 @@ Hint Resolve andb_comm andb_assoc: bool v62. Lemma andb_orb_distrib_r : forall b1 b2 b3:bool, b1 && (b2 || b3) = b1 && b2 || b1 && b3. Proof. - destruct b1; destruct b2; destruct b3; reflexivity. + destr_bool. Qed. Lemma andb_orb_distrib_l : forall b1 b2 b3:bool, (b1 || b2) && b3 = b1 && b3 || b2 && b3. Proof. - destruct b1; destruct b2; destruct b3; reflexivity. + destr_bool. Qed. Lemma orb_andb_distrib_r : forall b1 b2 b3:bool, b1 || b2 && b3 = (b1 || b2) && (b1 || b3). Proof. - destruct b1; destruct b2; destruct b3; reflexivity. + destr_bool. Qed. Lemma orb_andb_distrib_l : forall b1 b2 b3:bool, b1 && b2 || b3 = (b1 || b3) && (b2 || b3). Proof. - destruct b1; destruct b2; destruct b3; reflexivity. + destr_bool. Qed. (* Compatibility *) @@ -433,12 +452,12 @@ Notation demorgan4 := orb_andb_distrib_l (only parsing). Lemma absoption_andb : forall b1 b2:bool, b1 && (b1 || b2) = b1. Proof. - destruct b1; destruct b2; simpl in |- *; reflexivity. + destr_bool. Qed. Lemma absoption_orb : forall b1 b2:bool, b1 || b1 && b2 = b1. Proof. - destruct b1; destruct b2; simpl in |- *; reflexivity. + destr_bool. Qed. (*********************************) @@ -449,12 +468,12 @@ Qed. Lemma xorb_false_r : forall b:bool, xorb b false = b. Proof. - destruct b; trivial. + destr_bool. Qed. Lemma xorb_false_l : forall b:bool, xorb false b = b. Proof. - destruct b; trivial. + destr_bool. Qed. Notation xorb_false := xorb_false_r (only parsing). @@ -464,12 +483,12 @@ Notation false_xorb := xorb_false_l (only parsing). Lemma xorb_true_r : forall b:bool, xorb b true = negb b. Proof. - trivial. + reflexivity. Qed. Lemma xorb_true_l : forall b:bool, xorb true b = negb b. Proof. - destruct b; trivial. + reflexivity. Qed. Notation xorb_true := xorb_true_r (only parsing). @@ -479,14 +498,14 @@ Notation true_xorb := xorb_true_l (only parsing). Lemma xorb_nilpotent : forall b:bool, xorb b b = false. Proof. - destruct b; trivial. + destr_bool. Qed. (** Commutativity *) Lemma xorb_comm : forall b b':bool, xorb b b' = xorb b' b. Proof. - destruct b; destruct b'; trivial. + destr_bool. Qed. (** Associativity *) @@ -494,61 +513,64 @@ Qed. Lemma xorb_assoc_reverse : forall b b' b'':bool, xorb (xorb b b') b'' = xorb b (xorb b' b''). Proof. - destruct b; destruct b'; destruct b''; trivial. + destr_bool. Qed. Notation xorb_assoc := xorb_assoc_reverse (only parsing). (* Compatibility *) Lemma xorb_eq : forall b b':bool, xorb b b' = false -> b = b'. Proof. - destruct b; destruct b'; trivial. - unfold xorb in |- *. intros. rewrite H. reflexivity. + destr_bool. Qed. Lemma xorb_move_l_r_1 : forall b b' b'':bool, xorb b b' = b'' -> b' = xorb b b''. Proof. - intros. rewrite <- (false_xorb b'). rewrite <- (xorb_nilpotent b). rewrite xorb_assoc. - rewrite H. reflexivity. + destr_bool. Qed. Lemma xorb_move_l_r_2 : forall b b' b'':bool, xorb b b' = b'' -> b = xorb b'' b'. Proof. - intros. rewrite xorb_comm in H. rewrite (xorb_move_l_r_1 b' b b'' H). apply xorb_comm. + destr_bool. Qed. Lemma xorb_move_r_l_1 : forall b b' b'':bool, b = xorb b' b'' -> xorb b' b = b''. Proof. - intros. rewrite H. rewrite <- xorb_assoc. rewrite xorb_nilpotent. apply false_xorb. + destr_bool. Qed. Lemma xorb_move_r_l_2 : forall b b' b'':bool, b = xorb b' b'' -> xorb b b'' = b'. Proof. - intros. rewrite H. rewrite xorb_assoc. rewrite xorb_nilpotent. apply xorb_false. + destr_bool. Qed. (** Lemmas about the [b = true] embedding of [bool] to [Prop] *) -Lemma eq_true_iff_eq : forall b1 b2, (b1 = true <-> b2 = true) -> b1 = b2. -Proof. - intros b1 b2; case b1; case b2; intuition. +Lemma eq_iff_eq_true : forall b1 b2, b1 = b2 <-> (b1 = true <-> b2 = true). +Proof. + destr_bool; intuition. +Qed. + +Lemma eq_true_iff_eq : forall b1 b2, (b1 = true <-> b2 = true) -> b1 = b2. +Proof. + apply eq_iff_eq_true. Qed. Notation bool_1 := eq_true_iff_eq (only parsing). (* Compatibility *) Lemma eq_true_negb_classical : forall b:bool, negb b <> true -> b = true. Proof. - destruct b; intuition. + destr_bool; intuition. Qed. Notation bool_3 := eq_true_negb_classical (only parsing). (* Compatibility *) -Lemma eq_true_not_negb : forall b:bool, b <> true -> negb b = true. +Lemma eq_true_not_negb : forall b:bool, b <> true -> negb b = true. Proof. - destruct b; intuition. + destr_bool; intuition. Qed. Notation bool_6 := eq_true_not_negb (only parsing). (* Compatibility *) @@ -589,17 +611,17 @@ Hint Unfold Is_true: bool. Lemma Is_true_eq_true : forall x:bool, Is_true x -> x = true. Proof. - destruct x; simpl in |- *; tauto. + destr_bool; tauto. Qed. Lemma Is_true_eq_left : forall x:bool, x = true -> Is_true x. Proof. - intros; rewrite H; auto with bool. + intros; subst; auto with bool. Qed. Lemma Is_true_eq_right : forall x:bool, true = x -> Is_true x. Proof. - intros; rewrite <- H; auto with bool. + intros; subst; auto with bool. Qed. Notation Is_true_eq_true2 := Is_true_eq_right (only parsing). @@ -608,34 +630,34 @@ Hint Immediate Is_true_eq_right Is_true_eq_left: bool. Lemma eqb_refl : forall x:bool, Is_true (eqb x x). Proof. - destruct x; simpl; auto with bool. + destr_bool. Qed. Lemma eqb_eq : forall x y:bool, Is_true (eqb x y) -> x = y. Proof. - destruct x; destruct y; simpl; tauto. + destr_bool; tauto. Qed. (** [Is_true] and connectives *) -Lemma orb_prop_elim : +Lemma orb_prop_elim : forall a b:bool, Is_true (a || b) -> Is_true a \/ Is_true b. Proof. - destruct a; destruct b; simpl; tauto. + destr_bool; tauto. Qed. Notation orb_prop2 := orb_prop_elim (only parsing). -Lemma orb_prop_intro : +Lemma orb_prop_intro : forall a b:bool, Is_true a \/ Is_true b -> Is_true (a || b). Proof. - destruct a; destruct b; simpl; tauto. + destr_bool; tauto. Qed. Lemma andb_prop_intro : forall b1 b2:bool, Is_true b1 /\ Is_true b2 -> Is_true (b1 && b2). Proof. - destruct b1; destruct b2; simpl in |- *; tauto. + destr_bool; tauto. Qed. Hint Resolve andb_prop_intro: bool v62. @@ -646,66 +668,65 @@ Notation andb_true_intro2 := Lemma andb_prop_elim : forall a b:bool, Is_true (a && b) -> Is_true a /\ Is_true b. Proof. - destruct a; destruct b; simpl in |- *; try (intro H; discriminate H); - auto with bool. + destr_bool; auto. Qed. Hint Resolve andb_prop_elim: bool v62. Notation andb_prop2 := andb_prop_elim (only parsing). -Lemma eq_bool_prop_intro : - forall b1 b2, (Is_true b1 <-> Is_true b2) -> b1 = b2. -Proof. - destruct b1; destruct b2; simpl in *; intuition. +Lemma eq_bool_prop_intro : + forall b1 b2, (Is_true b1 <-> Is_true b2) -> b1 = b2. +Proof. + destr_bool; tauto. Qed. Lemma eq_bool_prop_elim : forall b1 b2, b1 = b2 -> (Is_true b1 <-> Is_true b2). -Proof. - intros b1 b2; case b1; case b2; intuition. -Qed. +Proof. + destr_bool; tauto. +Qed. Lemma negb_prop_elim : forall b, Is_true (negb b) -> ~ Is_true b. Proof. - destruct b; intuition. + destr_bool; tauto. Qed. Lemma negb_prop_intro : forall b, ~ Is_true b -> Is_true (negb b). Proof. - destruct b; simpl in *; intuition. + destr_bool; tauto. Qed. Lemma negb_prop_classical : forall b, ~ Is_true (negb b) -> Is_true b. Proof. - destruct b; intuition. + destr_bool; tauto. Qed. Lemma negb_prop_involutive : forall b, Is_true b -> ~ Is_true (negb b). Proof. - destruct b; intuition. + destr_bool; tauto. Qed. (** Rewrite rules about andb, orb and if (used in romega) *) -Lemma andb_if : forall (A:Type)(a a':A)(b b' : bool), - (if b && b' then a else a') = +Lemma andb_if : forall (A:Type)(a a':A)(b b' : bool), + (if b && b' then a else a') = (if b then if b' then a else a' else a'). Proof. - destruct b; destruct b'; auto. + destr_bool. Qed. -Lemma negb_if : forall (A:Type)(a a':A)(b:bool), - (if negb b then a else a') = +Lemma negb_if : forall (A:Type)(a a':A)(b:bool), + (if negb b then a else a') = (if b then a' else a). Proof. - destruct b; auto. + destr_bool. Qed. (*****************************************) -(** * Alternative versions of [andb] and [orb] +(** * Alternative versions of [andb] and [orb] with lazy behavior (for vm_compute) *) (*****************************************) -Notation "a &&& b" := (if a then b else false) +Notation "a &&& b" := (if a then b else false) (at level 40, left associativity) : lazy_bool_scope. Notation "a ||| b" := (if a then true else b) (at level 50, left associativity) : lazy_bool_scope. @@ -714,12 +735,51 @@ Open Local Scope lazy_bool_scope. Lemma andb_lazy_alt : forall a b : bool, a && b = a &&& b. Proof. - unfold andb; auto. + reflexivity. Qed. Lemma orb_lazy_alt : forall a b : bool, a || b = a ||| b. Proof. - unfold orb; auto. + reflexivity. +Qed. + +(*****************************************) +(** * Reflect: a specialized inductive type for + relating propositions and booleans, + as popularized by the Ssreflect library. *) +(*****************************************) + +Inductive reflect (P : Prop) : bool -> Set := + | ReflectT : P -> reflect P true + | ReflectF : ~ P -> reflect P false. +Hint Constructors reflect : bool. + +(** Interest: a case on a reflect lemma or hyp performs clever + unification, and leave the goal in a convenient shape + (a bit like case_eq). *) + +(** Relation with iff : *) + +Lemma reflect_iff : forall P b, reflect P b -> (P<->b=true). +Proof. + destruct 1; intuition; discriminate. +Qed. + +Lemma iff_reflect : forall P b, (P<->b=true) -> reflect P b. +Proof. + destr_bool; intuition. Qed. +(** It would be nice to join [reflect_iff] and [iff_reflect] + in a unique [iff] statement, but this isn't allowed since + [iff] is in Prop. *) + +(** Reflect implies decidability of the proposition *) + +Lemma reflect_dec : forall P b, reflect P b -> {P}+{~P}. +Proof. + destruct 1; auto. +Qed. +(** Reciprocally, from a decidability, we could state a + [reflect] as soon as we have a [bool_of_sumbool]. *) |