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.