diff options
Diffstat (limited to 'theories/FSets/DecidableType.v')
| -rw-r--r-- | theories/FSets/DecidableType.v | 62 |
1 files changed, 31 insertions, 31 deletions
diff --git a/theories/FSets/DecidableType.v b/theories/FSets/DecidableType.v index 4ba2b191bf..56bcb680df 100644 --- a/theories/FSets/DecidableType.v +++ b/theories/FSets/DecidableType.v @@ -6,7 +6,7 @@ (* * GNU Lesser General Public License Version 2.1 *) (***********************************************************************) -(* $Id$ *) +(* $Id: DecidableType.v,v 1.3 2006/03/03 18:48:37 letouzey Exp $ *) Require Export SetoidList. Set Implicit Arguments. @@ -16,15 +16,15 @@ Unset Strict Implicit. Module Type DecidableType. - Parameter t Set. + Parameter t : Set. - Parameter eq t -> t -> Prop. + Parameter eq : t -> t -> Prop. - Axiom eq_refl forall x : t, eq x x. - Axiom eq_sym forall x y : t, eq x y -> eq y x. - Axiom eq_trans forall x y z : t, eq x y -> eq y z -> eq x z. + Axiom eq_refl : forall x : t, eq x x. + Axiom eq_sym : forall x y : t, eq x y -> eq y x. + Axiom eq_trans : forall x y z : t, eq x y -> eq y z -> eq x z. - Parameter eq_dec forall x y : t, { eq x y } + { ~ eq x y }. + Parameter eq_dec : forall x y : t, { eq x y } + { ~ eq x y }. Hint Immediate eq_sym. Hint Resolve eq_refl eq_trans. @@ -32,15 +32,15 @@ Module Type DecidableType. End DecidableType. -Module PairDecidableType(DDecidableType). +Module PairDecidableType(D:DecidableType). Import D. Section Elt. - Variable elt Set. - Notation key=t. + Variable elt : Set. + Notation key:=t. - Definition eqk (p p'key*elt) := eq (fst p) (fst p'). - Definition eqke (p p'key*elt) := + Definition eqk (p p':key*elt) := eq (fst p) (fst p'). + Definition eqke (p p':key*elt) := eq (fst p) (fst p') /\ (snd p) = (snd p'). Hint Unfold eqk eqke. @@ -48,29 +48,29 @@ Module PairDecidableType(DDecidableType). (* eqke is stricter than eqk *) - Lemma eqke_eqk forall x x', eqke x x' -> eqk x x'. + Lemma eqke_eqk : forall x x', eqke x x' -> eqk x x'. Proof. unfold eqk, eqke; intuition. Qed. (* eqk, eqke are equalities *) - Lemma eqk_refl forall e, eqk e e. + Lemma eqk_refl : forall e, eqk e e. Proof. auto. Qed. - Lemma eqke_refl forall e, eqke e e. + Lemma eqke_refl : forall e, eqke e e. Proof. auto. Qed. - Lemma eqk_sym forall e e', eqk e e' -> eqk e' e. + Lemma eqk_sym : forall e e', eqk e e' -> eqk e' e. Proof. auto. Qed. - Lemma eqke_sym forall e e', eqke e e' -> eqke e' e. + Lemma eqke_sym : forall e e', eqke e e' -> eqke e' e. Proof. unfold eqke; intuition. Qed. - Lemma eqk_trans forall e e' e'', eqk e e' -> eqk e' e'' -> eqk e e''. + Lemma eqk_trans : forall e e' e'', eqk e e' -> eqk e' e'' -> eqk e e''. Proof. eauto. Qed. - Lemma eqke_trans forall e e' e'', eqke e e' -> eqke e' e'' -> eqke e e''. + Lemma eqke_trans : forall e e' e'', eqke e e' -> eqke e' e'' -> eqke e e''. Proof. unfold eqke; intuition; [ eauto | congruence ]. Qed. @@ -78,26 +78,26 @@ Module PairDecidableType(DDecidableType). Hint Resolve eqk_trans eqke_trans eqk_refl eqke_refl. Hint Immediate eqk_sym eqke_sym. - Lemma InA_eqke_eqk + Lemma InA_eqke_eqk : forall x m, InA eqke x m -> InA eqk x m. Proof. unfold eqke; induction 1; intuition. Qed. Hint Resolve InA_eqke_eqk. - Lemma InA_eqk forall p q m, eqk p q -> InA eqk p m -> InA eqk q m. + Lemma InA_eqk : forall p q m, eqk p q -> InA eqk p m -> InA eqk q m. Proof. intros; apply InA_eqA with p; auto; apply eqk_trans; auto. Qed. - Definition MapsTo (kkey)(e:elt):= InA eqke (k,e). - Definition In k m = exists e:elt, MapsTo k e m. + Definition MapsTo (k:key)(e:elt):= InA eqke (k,e). + Definition In k m := exists e:elt, MapsTo k e m. Hint Unfold MapsTo In. (* An alternative formulation for [In k l] is [exists e, InA eqk (k,e) l] *) - Lemma In_alt forall k l, In k l <-> exists e, InA eqk (k,e) l. + Lemma In_alt : forall k l, In k l <-> exists e, InA eqk (k,e) l. Proof. firstorder. exists x; auto. @@ -108,31 +108,31 @@ Module PairDecidableType(DDecidableType). exists e; auto. Qed. - Lemma MapsTo_eq forall l x y e, eq x y -> MapsTo x e l -> MapsTo y e l. + Lemma MapsTo_eq : forall l x y e, eq x y -> MapsTo x e l -> MapsTo y e l. Proof. intros; unfold MapsTo in *; apply InA_eqA with (x,e); eauto. Qed. - Lemma In_eq forall l x y, eq x y -> In x l -> In y l. + Lemma In_eq : forall l x y, eq x y -> In x l -> In y l. Proof. destruct 2 as (e,E); exists e; eapply MapsTo_eq; eauto. Qed. - Lemma In_inv forall k k' e l, In k ((k',e) :: l) -> eq k k' \/ In k l. + Lemma In_inv : forall k k' e l, In k ((k',e) :: l) -> eq k k' \/ In k l. Proof. inversion 1. inversion_clear H0; eauto. destruct H1; simpl in *; intuition. Qed. - Lemma In_inv_2 forall k k' e e' l, - InA eqk (k, e) ((k', e') : l) -> ~ eq k k' -> InA eqk (k, e) l. + Lemma In_inv_2 : forall k k' e e' l, + InA eqk (k, e) ((k', e') :: l) -> ~ eq k k' -> InA eqk (k, e) l. Proof. inversion_clear 1; compute in H0; intuition. Qed. - Lemma In_inv_3 forall x x' l, - InA eqke x (x' : l) -> ~ eqk x x' -> InA eqke x l. + Lemma In_inv_3 : forall x x' l, + InA eqke x (x' :: l) -> ~ eqk x x' -> InA eqke x l. Proof. inversion_clear 1; compute in H0; intuition. Qed. |