The Coq Proof Assistant

# Library Coq.Structures.OrderedType

Require Export SetoidList Morphisms OrdersTac.
Set Implicit Arguments.
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] _.
Create HintDb ordered_type.
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.
#[global]
Hint Immediate eq_sym : ordered_type.
#[global]
Hint Resolve eq_refl eq_trans lt_not_eq lt_trans : ordered_type.
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}.
End MOT_to_OT.

# Ordered types properties

Additional properties that can be derived from signature OrderedType.
Module OrderedTypeFacts (Import O: OrderedType).
#[global]
Instance eq_equiv : Equivalence eq.
Lemma lt_antirefl : forall x, ~ lt x x.
#[global]
Instance lt_strorder : StrictOrder lt.
Lemma lt_eq : forall x y z, lt x y -> eq y z -> lt x z.
Lemma eq_lt : forall x y z, eq x y -> lt y z -> lt x z.
#[global]
Instance lt_compat : Proper (eq==>eq==>iff) lt.
Qed.
Lemma lt_total : forall x y, lt x y \/ eq x y \/ lt y x.
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.
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.
Lemma eq_le x y z : eq x y -> ~lt y z -> ~lt x z.
Lemma neq_eq x y z : ~eq x y -> eq y z -> ~eq x z.
Lemma eq_neq x y z : eq x y -> ~eq y z -> ~eq x z.
Lemma le_lt_trans x y z : ~lt y x -> lt y z -> lt x z.
Lemma lt_le_trans x y z : lt x y -> ~lt z y -> lt x z.
Lemma le_neq x y : ~lt x y -> ~eq x y -> lt y x.
Lemma le_trans x y z : ~lt y x -> ~lt z y -> ~lt z x.
Lemma le_antisym x y : ~lt y x -> ~lt x y -> eq x y.
Lemma neq_sym x y : ~eq x y -> ~eq y x.
Lemma lt_le x y : lt x y -> ~lt y x.
Lemma gt_not_eq x y : lt y x -> ~ eq x y.
Lemma eq_not_lt x y : eq x y -> ~ lt x y.
Lemma eq_not_gt x y : eq x y -> ~ lt y x.
Lemma lt_not_gt x y : lt x y -> ~ lt y x.
#[global]
Hint Resolve gt_not_eq eq_not_lt : ordered_type.
#[global]
Hint Immediate eq_lt lt_eq le_eq eq_le neq_eq eq_neq : ordered_type.
#[global]
Hint Resolve eq_not_gt lt_antirefl lt_not_gt : ordered_type.
Lemma elim_compare_eq :
forall x y : t,
eq x y -> exists H : eq x y, compare x y = EQ H.
Lemma elim_compare_lt :
forall x y : t,
lt x y -> exists H : lt x y, compare x y = LT H.
Lemma elim_compare_gt :
forall x y : t,
lt y x -> exists H : lt y x, compare x y = GT H.
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}.
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.
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.
#[global]
Hint Resolve ListIn_In Sort_NoDup Inf_lt : ordered_type.
#[global]
Hint Immediate In_eq Inf_lt : ordered_type.
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').
#[local]
Hint Unfold eqk eqke ltk : ordered_type.
#[local]
Hint Extern 2 (eqke ?a ?b) => split : ordered_type.
Lemma eqke_eqk : forall x x', eqke x x' -> eqk x x'.
Lemma ltk_right_r : forall x k e e', ltk x (k,e) -> ltk x (k,e').
Lemma ltk_right_l : forall x k e e', ltk (k,e) x -> ltk (k,e') x.
#[local]
Hint Immediate ltk_right_r ltk_right_l : ordered_type.
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''.
Lemma ltk_trans : forall e e' e'', ltk e e' -> ltk e' e'' -> ltk e e''.
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'.
#[local]
Hint Resolve eqk_trans eqke_trans eqk_refl eqke_refl : ordered_type.
#[local]
Hint Resolve ltk_trans ltk_not_eqk ltk_not_eqke : ordered_type.
#[local]
Hint Immediate eqk_sym eqke_sym : ordered_type.
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 ltk_compat' : Proper (eqke==>eqke==>iff) ltk.
Lemma eqk_not_ltk : forall x x', eqk x x' -> ~ltk x x'.
Lemma ltk_eqk : forall e e' e'', ltk e e' -> eqk e' e'' -> ltk e e''.
Lemma eqk_ltk : forall e e' e'', eqk e e' -> ltk e' e'' -> ltk e e''.
#[local]
Hint Resolve eqk_not_ltk : ordered_type.
#[local]
Hint Immediate ltk_eqk eqk_ltk : ordered_type.
Lemma InA_eqke_eqk :
forall x m, InA eqke x m -> InA eqk x m.
#[local]
Hint Resolve InA_eqke_eqk : ordered_type.
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).
#[local]
Hint Unfold MapsTo In : ordered_type.
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 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.
#[local]
Hint Immediate Inf_eq : ordered_type.
#[local]
Hint Resolve Inf_lt : ordered_type.
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.
#[global]
Hint Unfold eqk eqke ltk : ordered_type.
#[global]
Hint Extern 2 (eqke ?a ?b) => split : ordered_type.
#[global]
Hint Resolve eqk_trans eqke_trans eqk_refl eqke_refl : ordered_type.
#[global]
Hint Resolve ltk_trans ltk_not_eqk ltk_not_eqke : ordered_type.
#[global]
Hint Immediate eqk_sym eqke_sym : ordered_type.
#[global]
Hint Resolve eqk_not_ltk : ordered_type.
#[global]
Hint Immediate ltk_eqk eqk_ltk : ordered_type.
#[global]
Hint Resolve InA_eqke_eqk : ordered_type.
#[global]
Hint Unfold MapsTo In : ordered_type.
#[global]
Hint Immediate Inf_eq : ordered_type.
#[global]
Hint Resolve Inf_lt : ordered_type.
#[global]
Hint Resolve Sort_Inf_NotIn : ordered_type.
#[global]
Hint Resolve In_inv_2 In_inv_3 : ordered_type.
End KeyOrderedType.