Library Coq.FSets.OrderedType
Require Export SetoidList.
Set Implicit Arguments.
Unset Strict Implicit.
Inductive Compare (X : Set) (lt eq : X -> X -> Prop) (x y : X) : Set :=
| 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.
Module Type OrderedType.
Parameter t : Set.
Parameter eq : t -> t -> Prop.
Parameter 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 OrderedType.
Additional properties that can be derived from signature
OrderedType.
Module OrderedTypeFacts (O: OrderedType).
Import O.
Lemma lt_antirefl : forall x, ~ lt x x.
Proof.
intros; intro; absurd (eq x x); auto.
Qed.
Lemma lt_eq : forall x y z, lt x y -> eq y z -> lt x z.
Proof.
intros; destruct (compare x z); auto.
elim (lt_not_eq H); apply eq_trans with z; auto.
elim (lt_not_eq (lt_trans l 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); auto.
elim (lt_not_eq H0); apply eq_trans with x; auto.
elim (lt_not_eq (lt_trans H0 l)); auto.
Qed.
Lemma le_eq : forall x y z, ~lt x y -> eq y z -> ~lt x z.
Proof.
intros; intro; destruct H; apply lt_eq with z; auto.
Qed.
Lemma eq_le : forall x y z, eq x y -> ~lt y z -> ~lt x z.
Proof.
intros; intro; destruct H0; apply eq_lt with x; auto.
Qed.
Lemma neq_eq : forall x y z, ~eq x y -> eq y z -> ~eq x z.
Proof.
intros; intro; destruct H; apply eq_trans with z; auto.
Qed.
Lemma eq_neq : forall x y z, eq x y -> ~eq y z -> ~eq x z.
Proof.
intros; intro; destruct H0; apply eq_trans with x; auto.
Qed.
Hint Immediate eq_lt lt_eq le_eq eq_le neq_eq eq_neq.
Lemma le_lt_trans : forall x y z, ~lt y x -> lt y z -> lt x z.
Proof.
intros; destruct (compare y x); auto.
elim (H l).
apply eq_lt with y; auto.
apply lt_trans with y; auto.
Qed.
Lemma lt_le_trans : forall x y z, lt x y -> ~lt z y -> lt x z.
Proof.
intros; destruct (compare z y); auto.
elim (H0 l).
apply lt_eq with y; auto.
apply lt_trans with y; auto.
Qed.
Lemma le_neq : forall x y, ~lt x y -> ~eq x y -> lt y x.
Proof.
intros; destruct (compare x y); intuition.
Qed.
Lemma neq_sym : forall x y, ~eq x y -> ~eq y x.
Proof.
intuition.
Qed.
Ltac abstraction := match goal with
| H : False |- _ => elim H
| H : lt ?x ?x |- _ => elim (lt_antirefl H)
| H : ~eq ?x ?x |- _ => elim (H (eq_refl x))
| H : eq ?x ?x |- _ => clear H; abstraction
| H : ~lt ?x ?x |- _ => clear H; abstraction
| |- eq ?x ?x => exact (eq_refl x)
| |- lt ?x ?x => elimtype False; abstraction
| |- ~ _ => intro; abstraction
| H1: ~lt ?x ?y, H2: ~eq ?x ?y |- _ =>
generalize (le_neq H1 H2); clear H1 H2; intro; abstraction
| H1: ~lt ?x ?y, H2: ~eq ?y ?x |- _ =>
generalize (le_neq H1 (neq_sym H2)); clear H1 H2; intro; abstraction
| H : lt ?x ?y |- _ => revert H; abstraction
| H : ~lt ?x ?y |- _ => revert H; abstraction
| H : ~eq ?x ?y |- _ => revert H; abstraction
| H : eq ?x ?y |- _ => revert H; abstraction
| _ => idtac
end.
Ltac do_eq a b EQ := match goal with
| |- lt ?x ?y -> _ => let H := fresh "H" in
(intro H;
(generalize (eq_lt (eq_sym EQ) H); clear H; intro H) ||
(generalize (lt_eq H EQ); clear H; intro H) ||
idtac);
do_eq a b EQ
| |- ~lt ?x ?y -> _ => let H := fresh "H" in
(intro H;
(generalize (eq_le (eq_sym EQ) H); clear H; intro H) ||
(generalize (le_eq H EQ); clear H; intro H) ||
idtac);
do_eq a b EQ
| |- eq ?x ?y -> _ => let H := fresh "H" in
(intro H;
(generalize (eq_trans (eq_sym EQ) H); clear H; intro H) ||
(generalize (eq_trans H EQ); clear H; intro H) ||
idtac);
do_eq a b EQ
| |- ~eq ?x ?y -> _ => let H := fresh "H" in
(intro H;
(generalize (eq_neq (eq_sym EQ) H); clear H; intro H) ||
(generalize (neq_eq H EQ); clear H; intro H) ||
idtac);
do_eq a b EQ
| |- lt a ?y => apply eq_lt with b; [exact EQ|]
| |- lt ?y a => apply lt_eq with b; [|exact (eq_sym EQ)]
| |- eq a ?y => apply eq_trans with b; [exact EQ|]
| |- eq ?y a => apply eq_trans with b; [|exact (eq_sym EQ)]
| _ => idtac
end.
Ltac propagate_eq := abstraction; clear; match goal with
| |- eq ?a ?b -> _ =>
let EQ := fresh "EQ" in (intro EQ; do_eq a b EQ; clear EQ);
propagate_eq
| _ => idtac
end.
Ltac do_lt x y LT := match goal with
| |- lt x y -> _ => intros _; do_lt x y LT
| |- lt y ?z -> _ => let H := fresh "H" in
(intro H; generalize (lt_trans LT H); intro); do_lt x y LT
| |- lt ?z x -> _ => let H := fresh "H" in
(intro H; generalize (lt_trans H LT); intro); do_lt x y LT
| |- lt _ _ -> _ => intro; do_lt x y LT
| |- ~lt y x -> _ => intros _; do_lt x y LT
| |- ~lt x ?z -> _ => let H := fresh "H" in
(intro H; generalize (le_lt_trans H LT); intro); do_lt x y LT
| |- ~lt ?z y -> _ => let H := fresh "H" in
(intro H; generalize (lt_le_trans LT H); intro); do_lt x y LT
| |- ~lt _ _ -> _ => intro; do_lt x y LT
| _ => idtac
end.
Definition hide_lt := lt.
Ltac propagate_lt := abstraction; match goal with
| |- lt ?x ?y -> _ =>
let LT := fresh "LT" in (intro LT; do_lt x y LT;
change (hide_lt x y) in LT);
propagate_lt
| _ => unfold hide_lt in *
end.
Ltac order :=
intros;
propagate_eq;
propagate_lt;
auto;
propagate_lt;
eauto.
Ltac false_order := elimtype False; order.
Lemma gt_not_eq : forall x y, lt y x -> ~ eq x y.
Proof.
order.
Qed.
Lemma eq_not_lt : forall x y : t, eq x y -> ~ lt x y.
Proof.
order.
Qed.
Hint Resolve gt_not_eq eq_not_lt.
Lemma eq_not_gt : forall x y : t, eq x y -> ~ lt y x.
Proof.
order.
Qed.
Lemma lt_not_gt : forall x y : t, lt x y -> ~ lt y x.
Proof.
order.
Qed.
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 solve [false_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 solve [false_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 solve [false_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 solve [ intros; false_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 ].
Lemma eq_dec : forall x y : t, {eq x y} + {~ eq x y}.
Proof.
intros; elim (compare x y); [ right | left | right ]; auto.
Qed.
Lemma lt_dec : forall x y : t, {lt x y} + {~ lt x y}.
Proof.
intros; elim (compare x y); [ left | right | right ]; auto.
Qed.
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.
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_sym eq_trans). Qed.
Lemma ListIn_In : forall l x, List.In x l -> In x l.
Proof. exact (In_InA eq_refl). Qed.
Lemma Inf_lt : forall l x y, lt x y -> Inf y l -> Inf x l.
Proof. exact (InfA_ltA lt_trans). Qed.
Lemma Inf_eq : forall l x y, eq x y -> Inf y l -> Inf x l.
Proof. exact (InfA_eqA eq_lt). 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_refl eq_sym lt_trans lt_eq eq_lt). 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_refl (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_refl eq_sym lt_trans lt_eq eq_lt). Qed.
Lemma Sort_NoDup : forall l, Sort l -> NoDup l.
Proof. exact (SortA_NoDupA eq_refl eq_sym lt_trans lt_not_eq lt_eq eq_lt) . 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 : Set.
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.
Lemma eqke_eqk : forall x x', eqke x x' -> eqk x x'.
Proof.
unfold eqk, eqke; intuition.
Qed.
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.
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.
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.
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.
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_ltk). Qed.
Lemma Inf_lt : forall l x x', ltk x x' -> Inf x' l -> Inf x l.
Proof. exact (InfA_ltA ltk_trans). 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_refl eqk_sym ltk_trans ltk_eqk eqk_ltk).
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_refl eqk_sym ltk_trans ltk_not_eqk ltk_eqk eqk_ltk).
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.