Library Stdlib.Structures.DecidableType

Require Export SetoidList.
Require Equalities.

Set Implicit Arguments.

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').

  #[local]
  Hint Unfold eqk eqke : core.
  #[local]
  Hint Extern 2 (eqke ?a ?b) => split : core.

   Lemma eqke_eqk : forall x x', eqke x x' -> eqk x x'.

  Lemma eqk_refl : forall e, eqk e e.

  Lemma eqke_refl : forall e, eqke e e.

  Lemma eqk_sym : forall e e', eqk e e' -> eqk e' e.

  Lemma eqke_sym : forall e e', eqke e e' -> eqke e' e.

  Lemma eqk_trans : forall e e' e'', eqk e e' -> eqk e' e'' -> eqk e e''.

  Lemma eqke_trans : forall e e' e'', eqke e e' -> eqke e' e'' -> eqke e e''.

  #[local]
  Hint Resolve eqk_trans eqke_trans eqk_refl eqke_refl : core.
  #[local]
  Hint Immediate eqk_sym eqke_sym : core.

  Global Instance eqk_equiv : Equivalence eqk.

  Global Instance eqke_equiv : Equivalence eqke.

  Lemma InA_eqke_eqk :
     forall x m, InA eqke x m -> InA eqk x m.
  #[local]
  Hint Resolve InA_eqke_eqk : core.

  Lemma InA_eqk : forall p q m, eqk p q -> InA eqk p m -> InA eqk q m.

  Definition MapsTo (k:key)(e:elt):= InA eqke (k ,e ).
  Definition In k m := exists e:elt , MapsTo k e m.

  #[local]
  Hint Unfold MapsTo In : core.

  Lemma In_alt : forall k l, In k l <-> exists e , InA eqk (k ,e ) l.

  Lemma MapsTo_eq : forall l x y e, eq x y -> MapsTo x e l -> MapsTo y e l.

  Lemma In_eq : forall l x y, eq x y -> In x l -> In y l.

  Lemma In_inv : forall k k' e l, In k ((k',e ) :: l) -> eq k k' \/ In k 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.

  Lemma In_inv_3 : forall x x' l,
      InA eqke x (x' :: l) -> ~ eqk x x' -> InA eqke x l.

End Elt.

#[global]
Hint Unfold eqk eqke : core.
#[global]
Hint Extern 2 (eqke ?a ?b) => split : core.
#[global]
Hint Resolve eqk_trans eqke_trans eqk_refl eqke_refl : core.
#[global]
Hint Immediate eqk_sym eqke_sym : core.
#[global]
Hint Resolve InA_eqke_eqk : core.
#[global]
Hint Unfold MapsTo In : core.
#[global]
Hint Resolve In_inv_2 In_inv_3 : core.

End KeyDecidableType.