Require Export RelationPairs SetoidList Orders.

Set Implicit Arguments.

Module OrderedTypeLists (Import O:OrderedType).

Section ForNotations.

Notation In:=(InA eq).
Notation Inf:=(lelistA lt).
Notation Sort:=(sort lt).
Notation NoDup:=(NoDupA eq).

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

Lemma ListIn_In : forall l x, List.In x l -> In x l.

Lemma Inf_lt : forall l x y, lt x y -> Inf y l -> Inf x l.

Lemma Inf_eq : forall l x y, eq x y -> Inf y l -> Inf x l.

Lemma Sort_Inf_In : forall l x a, Sort l -> Inf a l -> In x l -> lt a x.

Lemma ListIn_Inf : forall l x, (forall y, List.In y l -> lt x y) -> Inf x l.

Lemma In_Inf : forall l x, (forall y, In y l -> lt x y) -> Inf x l.

Lemma Inf_alt :
 forall l x, Sort l -> (Inf x l <-> (forall y, In y l -> lt x y)).

Lemma Sort_NoDup : forall l, Sort l -> NoDup l.

End ForNotations.

Hint Resolve ListIn_In Sort_NoDup Inf_lt.
Hint Immediate In_eq Inf_lt.

End OrderedTypeLists.

Module KeyOrderedType(Import O:OrderedType).
 Module Import MO:=OrderedTypeLists(O).

 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.
  Definition ltk : relation (key*elt) := lt @@1.

  Hint Unfold eqk eqke ltk.


  Global Instance eqke_eqk : subrelation eqke eqk.


  Global Instance eqk_equiv : Equivalence eqk := _.

  Global Instance eqke_equiv : Equivalence eqke := _.

  Global Instance ltk_strorder : StrictOrder ltk := _.

  Global Instance ltk_compat : Proper (eqk==>eqk==>iff) ltk.


  Global Instance pair_compat : Proper (eq==>Logic.eq==>eqke) (@pair key elt).

  Lemma ltk_not_eqk : forall e e´, ltk e e´ -> ~ eqk e e´.

  Lemma ltk_not_eqke : forall e e´, ltk e e´ -> ~eqke e e´.

  Lemma InA_eqke_eqk :
     forall x m, InA eqke x m -> InA eqk x m.
  Hint Resolve InA_eqke_eqk.

  Definition MapsTo (k:key)(e:elt):= InA eqke (k,e).
  Definition In k m := exists e:elt, MapsTo k e m.
  Notation Sort := (sort ltk).
  Notation Inf := (lelistA ltk).

  Hint Unfold MapsTo In.


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

  Lemma In_alt2 : forall k l, In k l <-> Exists (fun p => eq k (fst p)) l.

  Lemma In_nil : forall k, In k nil <-> False.

  Lemma In_cons : forall k p l,
   In k (p::l) <-> eq k (fst p) \/ In k l.

  Global Instance MapsTo_compat :
   Proper (eq==>Logic.eq==>equivlistA eqke==>iff) MapsTo.

  Global Instance In_compat : Proper (eq==>equivlistA eqk==>iff) In.

  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 Inf_eq : forall l x x´, eqk x x´ -> Inf x´ l -> Inf x l.

  Lemma Inf_lt : forall l x x´, ltk x x´ -> Inf x´ l -> Inf x l.

  Hint Immediate Inf_eq.
  Hint Resolve Inf_lt.

  Lemma Sort_Inf_In :
      forall l p q, Sort l -> Inf q l -> InA eqk p l -> ltk q p.

  Lemma Sort_Inf_NotIn :
      forall l k e, Sort l -> Inf (k,e) l -> ~In k l.

  Lemma Sort_NoDupA: forall l, Sort l -> NoDupA eqk l.

  Lemma Sort_In_cons_1 : forall e l e´, Sort (e::l) -> InA eqk e´ l -> ltk e e´.

  Lemma Sort_In_cons_2 : forall l e e´, Sort (e::l) -> InA eqk e´ (e::l) ->
      ltk e e´ \/ eqk e e´.

  Lemma Sort_In_cons_3 :
    forall x l k e, Sort ((k,e)::l) -> In x l -> ~eq x k.

  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.

 Hint Unfold eqk eqke ltk.
 Hint Extern 2 (eqke ?a ?b) => split.
 Hint Resolve ltk_not_eqk ltk_not_eqke.
 Hint Resolve InA_eqke_eqk.
 Hint Unfold MapsTo In.
 Hint Immediate Inf_eq.
 Hint Resolve Inf_lt.
 Hint Resolve Sort_Inf_NotIn.
 Hint Resolve In_inv_2 In_inv_3.

End KeyOrderedType.