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(***********************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *)
(* \VV/ *************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(***********************************************************************)
Require Import Equalities Bool SetoidList RelationPairs.
(** * Keys and datas used in FMap *)
Module KeyDecidableType(Import D:DecidableType).
Section Elt.
Variable elt : Type.
Notation key:=t.
Local Open Scope signature_scope.
Definition eqk : relation (key*elt) := eq @@1.
Definition eqke : relation (key*elt) := eq * Logic.eq.
Hint Unfold eqk eqke.
(* eqke is stricter than eqk *)
Global Instance eqke_eqk : subrelation eqke eqk.
Proof. firstorder. Qed.
(* eqk, eqke are equalities, ltk is a strict order *)
Global Instance eqk_equiv : Equivalence eqk := _.
Global Instance eqke_equiv : Equivalence eqke := _.
(* Additionnal facts *)
Lemma InA_eqke_eqk :
forall x m, InA eqke x m -> InA eqk x m.
Proof.
unfold eqke, RelProd; induction 1; firstorder.
Qed.
Hint Resolve InA_eqke_eqk.
Lemma InA_eqk : forall p q m, eqk p q -> InA eqk p m -> InA eqk q m.
Proof.
intros. rewrite <- H; auto.
Qed.
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.
Proof.
firstorder.
exists x; auto.
induction H.
destruct y; compute in H.
exists e; left; auto.
destruct IHInA as [e H0].
exists e; auto.
Qed.
Lemma In_alt2 : forall k l, In k l <-> Exists (fun p => eq k (fst p)) l.
Proof.
unfold In, MapsTo.
setoid_rewrite Exists_exists; setoid_rewrite InA_alt.
firstorder.
exists (snd x), x; auto.
Qed.
Lemma In_nil : forall k, In k nil <-> False.
Proof.
intros; rewrite In_alt2; apply Exists_nil.
Qed.
Lemma In_cons : forall k p l,
In k (p::l) <-> eq k (fst p) \/ In k l.
Proof.
intros; rewrite !In_alt2, Exists_cons; intuition.
Qed.
Global Instance MapsTo_compat :
Proper (eq==>Logic.eq==>equivlistA eqke==>iff) MapsTo.
Proof.
intros x x' Hx e e' He l l' Hl. unfold MapsTo.
rewrite Hx, He, Hl; intuition.
Qed.
Global Instance In_compat : Proper (eq==>equivlistA eqk==>iff) In.
Proof.
intros x x' Hx l l' Hl. rewrite !In_alt.
setoid_rewrite Hl. setoid_rewrite Hx. intuition.
Qed.
Lemma MapsTo_eq : forall l x y e, eq x y -> MapsTo x e l -> MapsTo y e l.
Proof. intros l x y e EQ. rewrite <- EQ; auto. Qed.
Lemma In_eq : forall l x y, eq x y -> In x l -> In y l.
Proof. intros l x y EQ. rewrite <- EQ; auto. Qed.
Lemma In_inv : forall k k' e l, In k ((k',e) :: l) -> eq k k' \/ In k l.
Proof.
intros; invlist In; invlist MapsTo. compute in * |- ; intuition.
right; exists x; auto.
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.
Proof.
intros; invlist InA; intuition.
Qed.
Lemma In_inv_3 : forall x x' l,
InA eqke x (x' :: l) -> ~ eqk x x' -> InA eqke x l.
Proof.
intros; invlist InA; compute in * |- ; intuition.
Qed.
End Elt.
Hint Unfold eqk eqke.
Hint Extern 2 (eqke ?a ?b) => split.
Hint Resolve InA_eqke_eqk.
Hint Unfold MapsTo In.
Hint Resolve In_inv_2 In_inv_3.
End KeyDecidableType.
(** * PairDecidableType
From two decidable types, we can build a new DecidableType
over their cartesian product. *)
Module PairDecidableType(D1 D2:DecidableType) <: DecidableType.
Definition t := (D1.t * D2.t)%type.
Definition eq := (D1.eq * D2.eq)%signature.
Instance eq_equiv : Equivalence eq := _.
Definition eq_dec : forall x y, { eq x y }+{ ~eq x y }.
Proof.
intros (x1,x2) (y1,y2); unfold eq; simpl.
destruct (D1.eq_dec x1 y1); destruct (D2.eq_dec x2 y2);
compute; intuition.
Defined.
End PairDecidableType.
(** Similarly for pairs of UsualDecidableType *)
Module PairUsualDecidableType(D1 D2:UsualDecidableType) <: UsualDecidableType.
Definition t := (D1.t * D2.t)%type.
Definition eq := @eq t.
Instance eq_equiv : Equivalence eq := _.
Definition eq_dec : forall x y, { eq x y }+{ ~eq x y }.
Proof.
intros (x1,x2) (y1,y2);
destruct (D1.eq_dec x1 y1); destruct (D2.eq_dec x2 y2);
unfold eq, D1.eq, D2.eq in *; simpl;
(left; f_equal; auto; fail) ||
(right; intros [=]; auto).
Defined.
End PairUsualDecidableType.
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