path: root/theories/Structures/OrderedType.v

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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 Export SetoidList Morphisms OrdersTac.
Set Implicit Arguments.
Unset Strict Implicit.

(** NB: This file is here only for compatibility with earlier version of
     [FSets] and [FMap]. Please use [Structures/Orders.v] directly now. *)

(** * Ordered types *)

Inductive Compare (X : Type) (lt eq : X -> X -> Prop) (x y : X) : Type :=
  | LT : lt x y -> Compare lt eq x y
  | EQ : eq x y -> Compare lt eq x y
  | GT : lt y x -> Compare lt eq x y.

Arguments LT [X lt eq x y] _.
Arguments EQ [X lt eq x y] _.
Arguments GT [X lt eq x y] _.

Module Type MiniOrderedType.

  Parameter Inline t : Type.

  Parameter Inline eq : t -> t -> Prop.
  Parameter Inline lt : 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 lt_trans : forall x y z : t, lt x y -> lt y z -> lt x z.
  Axiom lt_not_eq : forall x y : t, lt x y -> ~ eq x y.

  Parameter compare : forall x y : t, Compare lt eq x y.

  Hint Immediate eq_sym.
  Hint Resolve eq_refl eq_trans lt_not_eq lt_trans.

End MiniOrderedType.

Module Type OrderedType.
  Include MiniOrderedType.

  (** A [eq_dec] can be deduced from [compare] below. But adding this
     redundant field allows seeing an OrderedType as a DecidableType. *)
  Parameter eq_dec : forall x y, { eq x y } + { ~ eq x y }.

End OrderedType.

Module MOT_to_OT (Import O : MiniOrderedType) <: OrderedType.
  Include O.

  Definition eq_dec : forall x y : t, {eq x y} + {~ eq x y}.
  Proof.
   intros; elim (compare x y); intro H; [ right | left | right ]; auto.
   assert (~ eq y x); auto.
  Defined.

End MOT_to_OT.

(** * Ordered types properties *)

(** Additional properties that can be derived from signature
    [OrderedType]. *)

Module OrderedTypeFacts (Import O: OrderedType).

  Instance eq_equiv : Equivalence eq.
  Proof. split; [ exact eq_refl | exact eq_sym | exact eq_trans ]. Qed.

  Lemma lt_antirefl : forall x, ~ lt x x.
  Proof.
   intros; intro; absurd (eq x x); auto.
  Qed.

  Instance lt_strorder : StrictOrder lt.
  Proof. split; [ exact lt_antirefl | exact lt_trans]. Qed.

  Lemma lt_eq : forall x y z, lt x y -> eq y z -> lt x z.
  Proof.
   intros; destruct (compare x z) as [Hlt|Heq|Hlt]; auto.
   elim (lt_not_eq H); apply eq_trans with z; auto.
   elim (lt_not_eq (lt_trans Hlt H)); auto.
  Qed.

  Lemma eq_lt : forall x y z, eq x y -> lt y z -> lt x z.
  Proof.
   intros; destruct (compare x z) as [Hlt|Heq|Hlt]; auto.
   elim (lt_not_eq H0); apply eq_trans with x; auto.
   elim (lt_not_eq (lt_trans H0 Hlt)); auto.
  Qed.

  Instance lt_compat : Proper (eq==>eq==>iff) lt.
  Proof.
  apply proper_sym_impl_iff_2; auto with *.
  intros x x' Hx y y' Hy H.
  apply eq_lt with x; auto.
  apply lt_eq with y; auto.
  Qed.

  Lemma lt_total : forall x y, lt x y \/ eq x y \/ lt y x.
  Proof. intros; destruct (compare x y); auto. Qed.

  Module TO.
   Definition t := t.
   Definition eq := eq.
   Definition lt := lt.
   Definition le x y := lt x y \/ eq x y.
  End TO.
  Module IsTO.
   Definition eq_equiv := eq_equiv.
   Definition lt_strorder := lt_strorder.
   Definition lt_compat := lt_compat.
   Definition lt_total := lt_total.
   Lemma le_lteq x y : TO.le x y <-> lt x y \/ eq x y.
   Proof. reflexivity. Qed.
  End IsTO.
  Module OrderTac := !MakeOrderTac TO IsTO.
  Ltac order := OrderTac.order.

  Lemma le_eq x y z : ~lt x y -> eq y z -> ~lt x z. Proof. order. Qed.
  Lemma eq_le x y z : eq x y -> ~lt y z -> ~lt x z. Proof. order. Qed.
  Lemma neq_eq x y z : ~eq x y -> eq y z -> ~eq x z. Proof. order. Qed.
  Lemma eq_neq x y z : eq x y -> ~eq y z -> ~eq x z. Proof. order. Qed.
  Lemma le_lt_trans x y z : ~lt y x -> lt y z -> lt x z. Proof. order. Qed.
  Lemma lt_le_trans x y z : lt x y -> ~lt z y -> lt x z. Proof. order. Qed.
  Lemma le_neq x y : ~lt x y -> ~eq x y -> lt y x. Proof. order. Qed.
  Lemma le_trans x y z : ~lt y x -> ~lt z y -> ~lt z x. Proof. order. Qed.
  Lemma le_antisym x y : ~lt y x -> ~lt x y -> eq x y. Proof. order. Qed.
  Lemma neq_sym x y : ~eq x y -> ~eq y x. Proof. order. Qed.
  Lemma lt_le x y : lt x y -> ~lt y x. Proof. order. Qed.
  Lemma gt_not_eq x y : lt y x -> ~ eq x y. Proof. order. Qed.
  Lemma eq_not_lt x y : eq x y -> ~ lt x y. Proof. order. Qed.
  Lemma eq_not_gt x y : eq x y -> ~ lt y x. Proof. order. Qed.
  Lemma lt_not_gt x y : lt x y -> ~ lt y x. Proof. order. Qed.

  Hint Resolve gt_not_eq eq_not_lt.
  Hint Immediate eq_lt lt_eq le_eq eq_le neq_eq eq_neq.
  Hint Resolve eq_not_gt lt_antirefl lt_not_gt.

  Lemma elim_compare_eq :
   forall x y : t,
   eq x y -> exists H : eq x y, compare x y = EQ H.
  Proof.
   intros; case (compare x y); intros H'; try (exfalso; order).
   exists H'; auto.
  Qed.

  Lemma elim_compare_lt :
   forall x y : t,
   lt x y -> exists H : lt x y, compare x y = LT H.
  Proof.
   intros; case (compare x y); intros H'; try (exfalso; order).
   exists H'; auto.
  Qed.

  Lemma elim_compare_gt :
   forall x y : t,
   lt y x -> exists H : lt y x, compare x y = GT H.
  Proof.
   intros; case (compare x y); intros H'; try (exfalso; order).
   exists H'; auto.
  Qed.

  Ltac elim_comp :=
    match goal with
      | |- ?e => match e with
           | context ctx [ compare ?a ?b ] =>
                let H := fresh in
                (destruct (compare a b) as [H|H|H]; try order)
         end
    end.

  Ltac elim_comp_eq x y :=
    elim (elim_compare_eq (x:=x) (y:=y));
     [ intros _1 _2; rewrite _2; clear _1 _2 | auto ].

  Ltac elim_comp_lt x y :=
    elim (elim_compare_lt (x:=x) (y:=y));
     [ intros _1 _2; rewrite _2; clear _1 _2 | auto ].

  Ltac elim_comp_gt x y :=
    elim (elim_compare_gt (x:=x) (y:=y));
     [ intros _1 _2; rewrite _2; clear _1 _2 | auto ].

  (** For compatibility reasons *)
  Definition eq_dec := eq_dec.

  Lemma lt_dec : forall x y : t, {lt x y} + {~ lt x y}.
  Proof.
   intros; elim (compare x y); [ left | right | right ]; auto.
  Defined.

  Definition eqb x y : bool := if eq_dec x y then true else false.

  Lemma eqb_alt :
    forall x y, eqb x y = match compare x y with EQ _ => true | _ => false end.
  Proof.
  unfold eqb; intros; destruct (eq_dec x y); elim_comp; auto.
  Qed.

(* Specialization of resuts about lists modulo. *)

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.
Proof. exact (InA_eqA eq_equiv). Qed.

Lemma ListIn_In : forall l x, List.In x l -> In x l.
Proof. exact (In_InA eq_equiv). Qed.

Lemma Inf_lt : forall l x y, lt x y -> Inf y l -> Inf x l.
Proof. exact (InfA_ltA lt_strorder). Qed.

Lemma Inf_eq : forall l x y, eq x y -> Inf y l -> Inf x l.
Proof. exact (InfA_eqA eq_equiv lt_compat). Qed.

Lemma Sort_Inf_In : forall l x a, Sort l -> Inf a l -> In x l -> lt a x.
Proof. exact (SortA_InfA_InA eq_equiv lt_strorder lt_compat). Qed.

Lemma ListIn_Inf : forall l x, (forall y, List.In y l -> lt x y) -> Inf x l.
Proof. exact (@In_InfA t lt). Qed.

Lemma In_Inf : forall l x, (forall y, In y l -> lt x y) -> Inf x l.
Proof. exact (InA_InfA eq_equiv (ltA:=lt)). Qed.

Lemma Inf_alt :
 forall l x, Sort l -> (Inf x l <-> (forall y, In y l -> lt x y)).
Proof. exact (InfA_alt eq_equiv lt_strorder lt_compat). Qed.

Lemma Sort_NoDup : forall l, Sort l -> NoDup l.
Proof. exact (SortA_NoDupA eq_equiv lt_strorder lt_compat). Qed.

End ForNotations.

Hint Resolve ListIn_In Sort_NoDup Inf_lt.
Hint Immediate In_eq Inf_lt.

End OrderedTypeFacts.

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

 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').
  Definition ltk (p p':key*elt) := lt (fst p) (fst p').

  Hint Unfold eqk eqke ltk.
  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.

  (* ltk ignore the second components *)

   Lemma ltk_right_r : forall x k e e', ltk x (k,e) -> ltk x (k,e').
   Proof. auto. Qed.

   Lemma ltk_right_l : forall x k e e', ltk (k,e) x -> ltk (k,e') x.
   Proof. auto. Qed.
   Hint Immediate ltk_right_r ltk_right_l.

  (* eqk, eqke are equalities, ltk is a strict order *)

  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.

  Lemma ltk_trans : forall e e' e'', ltk e e' -> ltk e' e'' -> ltk e e''.
  Proof. eauto. Qed.

  Lemma ltk_not_eqk : forall e e', ltk e e' -> ~ eqk e e'.
  Proof. unfold eqk, ltk; auto. Qed.

  Lemma ltk_not_eqke : forall e e', ltk e e' -> ~eqke e e'.
  Proof.
    unfold eqke, ltk; intuition; simpl in *; subst.
    exact (lt_not_eq H H1).
  Qed.

  Hint Resolve eqk_trans eqke_trans eqk_refl eqke_refl.
  Hint Resolve ltk_trans ltk_not_eqk ltk_not_eqke.
  Hint Immediate eqk_sym eqke_sym.

  Global Instance eqk_equiv : Equivalence eqk.
  Proof. constructor; eauto. Qed.

  Global Instance eqke_equiv : Equivalence eqke.
  Proof. split; eauto. Qed.

  Global Instance ltk_strorder : StrictOrder ltk.
  Proof. constructor; eauto. intros x; apply (irreflexivity (x:=fst x)). Qed.

  Global Instance ltk_compat : Proper (eqk==>eqk==>iff) ltk.
  Proof.
  intros (x,e) (x',e') Hxx' (y,f) (y',f') Hyy'; compute.
   compute in Hxx'; compute in Hyy'. rewrite Hxx', Hyy'; auto.
  Qed.

  Global Instance ltk_compat' : Proper (eqke==>eqke==>iff) ltk.
  Proof.
  intros (x,e) (x',e') (Hxx',_) (y,f) (y',f') (Hyy',_); compute.
   compute in Hxx'; compute in Hyy'. rewrite Hxx', Hyy'; auto.
  Qed.

  (* Additionnal facts *)

  Lemma eqk_not_ltk : forall x x', eqk x x' -> ~ltk x x'.
   Proof.
     unfold eqk, ltk; simpl; auto.
   Qed.

  Lemma ltk_eqk : forall e e' e'', ltk e e' -> eqk e' e'' -> ltk e e''.
  Proof. eauto. Qed.

  Lemma eqk_ltk : forall e e' e'', eqk e e' -> ltk e' e'' -> ltk e e''.
  Proof.
      intros (k,e) (k',e') (k'',e'').
      unfold ltk, eqk; simpl; eauto.
  Qed.
  Hint Resolve eqk_not_ltk.
  Hint Immediate ltk_eqk eqk_ltk.

  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.

  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.

  (* 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); eauto with *.
  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 Inf_eq : forall l x x', eqk x x' -> Inf x' l -> Inf x l.
  Proof. exact (InfA_eqA eqk_equiv ltk_compat). Qed.

  Lemma Inf_lt : forall l x x', ltk x x' -> Inf x' l -> Inf x l.
  Proof. exact (InfA_ltA ltk_strorder). Qed.

  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.
  Proof.
  exact (SortA_InfA_InA eqk_equiv ltk_strorder ltk_compat).
  Qed.

  Lemma Sort_Inf_NotIn :
      forall l k e, Sort l -> Inf (k,e) l ->  ~In k l.
  Proof.
    intros; red; intros.
    destruct H1 as [e' H2].
    elim (@ltk_not_eqk (k,e) (k,e')).
    eapply Sort_Inf_In; eauto.
    red; simpl; auto.
  Qed.

  Lemma Sort_NoDupA: forall l, Sort l -> NoDupA eqk l.
  Proof.
  exact (SortA_NoDupA eqk_equiv ltk_strorder ltk_compat).
  Qed.

  Lemma Sort_In_cons_1 : forall e l e', Sort (e::l) -> InA eqk e' l -> ltk e e'.
  Proof.
   inversion 1; intros; eapply Sort_Inf_In; eauto.
  Qed.

  Lemma Sort_In_cons_2 : forall l e e', Sort (e::l) -> InA eqk e' (e::l) ->
      ltk e e' \/ eqk e e'.
  Proof.
    inversion_clear 2; auto.
    left; apply Sort_In_cons_1 with l; auto.
  Qed.

  Lemma Sort_In_cons_3 :
    forall x l k e, Sort ((k,e)::l) -> In x l -> ~eq x k.
  Proof.
    inversion_clear 1; red; intros.
    destruct (Sort_Inf_NotIn H0 H1 (In_eq H2 H)).
  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 ltk.
 Hint Extern 2 (eqke ?a ?b) => split.
 Hint Resolve eqk_trans eqke_trans eqk_refl eqke_refl.
 Hint Resolve ltk_trans ltk_not_eqk ltk_not_eqke.
 Hint Immediate eqk_sym eqke_sym.
 Hint Resolve eqk_not_ltk.
 Hint Immediate ltk_eqk eqk_ltk.
 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.