Library Coq.Structures.Orders

Require Export Relations Morphisms Setoid Equalities.
Set Implicit Arguments.
Unset Strict Implicit.

Ordered types

First, signatures with only the order relations

Module Type HasLt (Import T:Typ).
Parameter Inline(40) lt : t -> t -> Prop.
End HasLt.

Module Type HasLe (Import T:Typ).
Parameter Inline(40) le : t -> t -> Prop.
End HasLe.

Module Type EqLt := Typ <+ HasEq <+ HasLt.
Module Type EqLe := Typ <+ HasEq <+ HasLe.
Module Type EqLtLe := Typ <+ HasEq <+ HasLt <+ HasLe.

Versions with nice notations

Module Type LtNotation (E:EqLt).
  Infix "<" := E.lt.
  Notation "x > y" := (y<x) (only parsing).
  Notation "x < y < z" := (x<y /\ y<z).
End LtNotation.

Module Type LeNotation (E:EqLe).
  Infix "<=" := E.le.
  Notation "x >= y" := (y<=x) (only parsing).
  Notation "x <= y <= z" := (x<=y /\ y<=z).
End LeNotation.

Module Type LtLeNotation (E:EqLtLe).
  Include LtNotation E <+ LeNotation E.
  Notation "x <= y < z" := (x<=y /\ y<z).
  Notation "x < y <= z" := (x<y /\ y<=z).
End LtLeNotation.

Module Type EqLtNotation (E:EqLt) := EqNotation E <+ LtNotation E.
Module Type EqLeNotation (E:EqLe) := EqNotation E <+ LeNotation E.
Module Type EqLtLeNotation (E:EqLtLe) := EqNotation E <+ LtLeNotation E.

Module Type EqLt' := EqLt <+ EqLtNotation.
Module Type EqLe' := EqLe <+ EqLeNotation.
Module Type EqLtLe' := EqLtLe <+ EqLtLeNotation.

Versions with logical specifications

Module Type IsStrOrder (Import E:EqLt).
  Declare Instance lt_strorder : StrictOrder lt.
  Declare Instance lt_compat : Proper (eq ==>eq ==>iff) lt.
End IsStrOrder.

Module Type LeIsLtEq (Import E:EqLtLe').
  Axiom le_lteq : forall x y, x <=y <-> x <y \/ x ==y.
End LeIsLtEq.

Module Type StrOrder := EqualityType <+ HasLt <+ IsStrOrder.
Module Type StrOrder' := StrOrder <+ EqLtNotation.

Versions with a decidable ternary comparison

Module Type HasCmp (Import T:Typ).
  Parameter Inline compare : t -> t -> comparison.
End HasCmp.

Module Type CmpNotation (T:Typ)(C:HasCmp T).
  Infix "?=" := C.compare (at level 70, no associativity).
End CmpNotation.

Module Type CmpSpec (Import E:EqLt')(Import C:HasCmp E).
  Axiom compare_spec : forall x y, CompareSpec (x ==y) (x <y) (y <x) (compare x y).
End CmpSpec.

Module Type HasCompare (E:EqLt) := HasCmp E <+ CmpSpec E.

Module Type DecStrOrder := StrOrder <+ HasCompare.
Module Type DecStrOrder' := DecStrOrder <+ EqLtNotation <+ CmpNotation.

Module Type OrderedType <: DecidableType := DecStrOrder <+ HasEqDec.
Module Type OrderedType' := OrderedType <+ EqLtNotation <+ CmpNotation.

Module Type OrderedTypeFull := OrderedType <+ HasLe <+ LeIsLtEq.
Module Type OrderedTypeFull' :=
OrderedTypeFull <+ EqLtLeNotation <+ CmpNotation.

NB: in OrderedType, an eq_dec could be deduced from compare. But adding this redundant field allows seeing an OrderedType as a DecidableType.

Versions with eq being the usual Leibniz equality of Coq

Module Type UsualStrOrder := UsualEqualityType <+ HasLt <+ IsStrOrder.
Module Type UsualDecStrOrder := UsualStrOrder <+ HasCompare.
Module Type UsualOrderedType <: UsualDecidableType <: OrderedType
:= UsualDecStrOrder <+ HasEqDec.
Module Type UsualOrderedTypeFull := UsualOrderedType <+ HasLe <+ LeIsLtEq.

NB: in UsualOrderedType, the field lt_compat is useless since eq is Leibniz, but it should be there for subtyping.

Module Type UsualStrOrder' := UsualStrOrder <+ LtNotation.
Module Type UsualDecStrOrder' := UsualDecStrOrder <+ LtNotation.
Module Type UsualOrderedType' := UsualOrderedType <+ LtNotation.
Module Type UsualOrderedTypeFull' := UsualOrderedTypeFull <+ LtLeNotation.

Purely logical versions

Module Type LtIsTotal (Import E:EqLt').
Axiom lt_total : forall x y, x <y \/ x ==y \/ y <x.
End LtIsTotal.

Module Type TotalOrder := StrOrder <+ HasLe <+ LeIsLtEq <+ LtIsTotal.
Module Type UsualTotalOrder <: TotalOrder
:= UsualStrOrder <+ HasLe <+ LeIsLtEq <+ LtIsTotal.

Module Type TotalOrder' := TotalOrder <+ EqLtLeNotation.
Module Type UsualTotalOrder' := UsualTotalOrder <+ LtLeNotation.

Conversions

From compare to eqb, and then eq_dec

Module Compare2EqBool (Import O:DecStrOrder') <: HasEqBool O.

Definition eqb x y :=
match compare x y with Eq => true | _ => false end.

Lemma eqb_eq : forall x y, eqb x y = true <-> x ==y.
Proof.
unfold eqb. intros x y.
destruct (compare_spec x y) as [H|H|H]; split; auto; try discriminate.
intros EQ; rewrite EQ in H; elim (StrictOrder_Irreflexive _ H).
intros EQ; rewrite EQ in H; elim (StrictOrder_Irreflexive _ H).
Qed.

End Compare2EqBool.

Module DSO_to_OT (O:DecStrOrder) <: OrderedType :=
O <+ Compare2EqBool <+ HasEqBool2Dec.

From OrderedType To OrderedTypeFull (adding <=)

Module OT_to_Full (O:OrderedType') <: OrderedTypeFull.
Include O.
Definition le x y := x <y \/ x ==y.
Lemma le_lteq : forall x y, le x y <-> x <y \/ x ==y.
Proof. unfold le; split; auto. Qed.
End OT_to_Full.

From computational to logical versions

Module OTF_LtIsTotal (Import O:OrderedTypeFull') <: LtIsTotal O.
Lemma lt_total : forall x y, x <y \/ x ==y \/ y <x.
Proof. intros; destruct (compare_spec x y); auto. Qed.
End OTF_LtIsTotal.

Module OTF_to_TotalOrder (O:OrderedTypeFull) <: TotalOrder
:= O <+ OTF_LtIsTotal.

Versions with boolean comparisons

This style is used in Mergesort

For stating properties like transitivity of leb, we coerce bool into Prop.

Local Coercion is_true : bool >-> Sortclass.
Hint Unfold is_true.

Module Type HasLeb (Import T:Typ).
Parameter Inline leb : t -> t -> bool.
End HasLeb.

Module Type HasLtb (Import T:Typ).
Parameter Inline ltb : t -> t -> bool.
End HasLtb.

Module Type LebNotation (T:Typ)(E:HasLeb T).
Infix "<=?" := E.leb (at level 35).
End LebNotation.

Module Type LtbNotation (T:Typ)(E:HasLtb T).
Infix "<?" := E.ltb (at level 35).
End LtbNotation.

Module Type LebSpec (T:Typ)(X:HasLe T)(Y:HasLeb T).
Parameter leb_le : forall x y, Y.leb x y = true <-> X.le x y.
End LebSpec.

Module Type LtbSpec (T:Typ)(X:HasLt T)(Y:HasLtb T).
Parameter ltb_lt : forall x y, Y.ltb x y = true <-> X.lt x y.
End LtbSpec.

Module Type LeBool := Typ <+ HasLeb.
Module Type LtBool := Typ <+ HasLtb.
Module Type LeBool' := LeBool <+ LebNotation.
Module Type LtBool' := LtBool <+ LtbNotation.

Module Type LebIsTotal (Import X:LeBool').
Axiom leb_total : forall x y, (x <=? y ) = true \/ (y <=? x ) = true.
End LebIsTotal.

Module Type TotalLeBool := LeBool <+ LebIsTotal.
Module Type TotalLeBool' := LeBool' <+ LebIsTotal.

Module Type LebIsTransitive (Import X:LeBool').
Axiom leb_trans : Transitive X.leb.
End LebIsTransitive.

Module Type TotalTransitiveLeBool := TotalLeBool <+ LebIsTransitive.
Module Type TotalTransitiveLeBool' := TotalLeBool' <+ LebIsTransitive.

Grouping all boolean comparison functions

Module Type HasBoolOrdFuns (T:Typ) := HasEqb T <+ HasLtb T <+ HasLeb T.

Module Type HasBoolOrdFuns' (T:Typ) :=
HasBoolOrdFuns T <+ EqbNotation T <+ LtbNotation T <+ LebNotation T.

Module Type BoolOrdSpecs (O:EqLtLe)(F:HasBoolOrdFuns O) :=
EqbSpec O O F <+ LtbSpec O O F <+ LebSpec O O F.

Module Type OrderFunctions (E:EqLtLe) :=
HasCompare E <+ HasBoolOrdFuns E <+ BoolOrdSpecs E.
Module Type OrderFunctions' (E:EqLtLe) :=
HasCompare E <+ CmpNotation E <+ HasBoolOrdFuns' E <+ BoolOrdSpecs E.

From OrderedTypeFull to TotalTransitiveLeBool

Module OTF_to_TTLB (Import O : OrderedTypeFull') <: TotalTransitiveLeBool.

Definition leb x y :=
match compare x y with Gt => false | _ => true end.

Lemma leb_le : forall x y, leb x y <-> x <= y.
Proof.
intros. unfold leb. rewrite le_lteq.
destruct (compare_spec x y) as [EQ|LT|GT]; split; auto.
discriminate.
intros LE. elim (StrictOrder_Irreflexive x).
destruct LE as [LT|EQ]. now transitivity y. now rewrite <- EQ in GT.
Qed.

Lemma leb_total : forall x y, leb x y \/ leb y x.
Proof.
intros. rewrite 2 leb_le. rewrite 2 le_lteq.
destruct (compare_spec x y); intuition.
Qed.

Lemma leb_trans : Transitive leb.
Proof.
intros x y z. rewrite !leb_le, !le_lteq.
intros [Hxy|Hxy] [Hyz|Hyz].
left; transitivity y; auto.
left; rewrite <- Hyz; auto.
left; rewrite Hxy; auto.
right; transitivity y; auto.
Qed.

Definition t := t.

End OTF_to_TTLB.

From TotalTransitiveLeBool to OrderedTypeFull

le is leb ... = true. eq is le /\ swap le. lt is le /\ ~swap le.

Local Open Scope bool_scope.

Module TTLB_to_OTF (Import O : TotalTransitiveLeBool') <: OrderedTypeFull.

Definition t := t.

Definition le x y : Prop := x <=? y.
Definition eq x y : Prop := le x y /\ le y x.
Definition lt x y : Prop := le x y /\ ~le y x.

Definition compare x y :=
if x <=? y then (if y <=? x then Eq else Lt) else Gt.

Lemma compare_spec : forall x y, CompSpec eq lt x y (compare x y).
Proof.
intros. unfold compare.
case_eq (x <=? y).
case_eq (y <=? x).
constructor. split; auto.
constructor. split; congruence.
constructor. destruct (leb_total x y); split; congruence.
Qed.

Definition eqb x y := (x <=? y ) && (y <=? x ).

Lemma eqb_eq : forall x y, eqb x y <-> eq x y.
Proof.
intros. unfold eq, eqb, le.
case leb; simpl; intuition; discriminate.
Qed.

Include HasEqBool2Dec.

Instance eq_equiv : Equivalence eq.
Proof.
split.
intros x; unfold eq, le. destruct (leb_total x x); auto.
intros x y; unfold eq, le. intuition.
intros x y z; unfold eq, le. intuition; apply leb_trans with y; auto.
Qed.

Instance lt_strorder : StrictOrder lt.
Proof.
split.
intros x. unfold lt; red; intuition.
intros x y z; unfold lt, le. intuition.
apply leb_trans with y; auto.
absurd (z <=? y); auto.
apply leb_trans with x; 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. unfold eq, lt, le in *.
intuition.
apply leb_trans with x; auto.
apply leb_trans with y; auto.
absurd (y <=? x); auto.
apply leb_trans with x'; auto.
apply leb_trans with y'; auto.
Qed.

Definition le_lteq : forall x y, le x y <-> lt x y \/ eq x y.
Proof.
intros.
unfold lt, eq, le.
split; [ | intuition ].
intros LE.
case_eq (y <=? x); [right|left]; intuition; try discriminate.
Qed.

End TTLB_to_OTF.