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(************************************************************************)
(* * The Coq Proof Assistant / The Coq Development Team *)
(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
Require Export SetoidList.
Require Equalities.
Set Implicit Arguments.
Unset Strict Implicit.
(** NB: This file is here only for compatibility with earlier version of
[FSets] and [FMap]. Please use [Structures/Equalities.v] directly now. *)
(** * Types with Equalities, and nothing more (for subtyping purpose) *)
Module Type EqualityType := Equalities.EqualityTypeOrig.
(** * Types with decidable Equalities (but no ordering) *)
Module Type DecidableType := Equalities.DecidableTypeOrig.
(** * Additional notions about keys and datas used in FMap *)
Module KeyDecidableType(D:DecidableType).
Import D.
Section Elt.
Variable elt : Type.
Notation key:=t.
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.
Hint Extern 2 (eqke ?a ?b) => split.
(* eqke is stricter than eqk *)
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.
Proof. auto. Qed.
Lemma eqke_refl : forall e, eqke e e.
Proof. auto. Qed.
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.
Proof. unfold eqke; intuition. Qed.
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''.
Proof.
unfold eqke; intuition; [ eauto | congruence ].
Qed.
Hint Resolve eqk_trans eqke_trans eqk_refl eqke_refl.
Hint Immediate eqk_sym eqke_sym.
Global Instance eqk_equiv : Equivalence eqk.
Proof. split; eauto. Qed.
Global Instance eqke_equiv : Equivalence eqke.
Proof. split; eauto. Qed.
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.
Proof.
intros; apply InA_eqA with p; auto using eqk_equiv.
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.
exists e; auto.
destruct IHInA as [e H0].
exists e; auto.
Qed.
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); auto using eqke_equiv.
Qed.
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.
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.
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.
Proof.
inversion_clear 1; compute in H0; intuition.
Qed.
End Elt.
Hint Unfold eqk eqke.
Hint Extern 2 (eqke ?a ?b) => split.
Hint Resolve eqk_trans eqke_trans eqk_refl eqke_refl.
Hint Immediate eqk_sym eqke_sym.
Hint Resolve InA_eqke_eqk.
Hint Unfold MapsTo In.
Hint Resolve In_inv_2 In_inv_3.
End KeyDecidableType.
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