The Coq Proof Assistant

# Library Coq.Structures.Orders

Require Export Relations Morphisms Setoid Equalities.
Set Implicit Arguments.

# 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).
#[global]
Declare Instance lt_strorder : StrictOrder lt.
#[global]
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.
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.
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.
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.
#[global]
Hint Unfold is_true : core.
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 70, no associativity).
End LebNotation.
Module Type LtbNotation (T:Typ)(E:HasLtb T).
Infix "<?" := E.ltb (at level 70, no associativity).
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.
Lemma leb_total : forall x y, leb x y \/ leb y x.
Lemma leb_trans : Transitive leb.
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).
Definition eqb x y := (x <=? y) && (y <=? x).
Lemma eqb_eq : forall x y, eqb x y <-> eq x y.
Include HasEqBool2Dec.
#[global]
Instance eq_equiv : Equivalence eq.
#[global]
Instance lt_strorder : StrictOrder lt.
#[global]
Instance lt_compat : Proper (eq ==> eq ==> iff) lt.
Definition le_lteq : forall x y, le x y <-> lt x y \/ eq x y.
End TTLB_to_OTF.