# Library Coq.Structures.OrderedTypeEx

Require Import OrderedType.
Require Import ZArith.
Require Import Omega.
Require Import NArith Ndec.
Require Import Compare_dec.

# Examples of Ordered Type structures.

First, a particular case of OrderedType where the equality is the usual one of Coq.
Module Type UsualOrderedType.
Parameter Inline t : Type.
Definition eq := @eq t.
Parameter Inline lt : t -> t -> Prop.
Definition eq_refl := @refl_equal t.
Definition eq_sym := @sym_eq t.
Definition eq_trans := @trans_eq t.
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.
Parameter eq_dec : forall x y : t, { eq x y } + { ~ eq x y }.
End UsualOrderedType.
a UsualOrderedType is in particular an OrderedType.
Module UOT_to_OT (U:UsualOrderedType) <: OrderedType := U.
nat is an ordered type with respect to the usual order on natural numbers.
Module Nat_as_OT <: UsualOrderedType.
Definition t := nat.
Definition eq := @eq nat.
Definition eq_refl := @refl_equal t.
Definition eq_sym := @sym_eq t.
Definition eq_trans := @trans_eq t.
Definition lt := lt.
Lemma lt_trans : forall x y z : t, lt x y -> lt y z -> lt x z.
Lemma lt_not_eq : forall x y : t, lt x y -> ~ eq x y.
Definition compare : forall x y : t, Compare lt eq x y.
Definition eq_dec := eq_nat_dec.
End Nat_as_OT.
Z is an ordered type with respect to the usual order on integers.
Open Local Scope Z_scope.
Module Z_as_OT <: UsualOrderedType.
Definition t := Z.
Definition eq := @eq Z.
Definition eq_refl := @refl_equal t.
Definition eq_sym := @sym_eq t.
Definition eq_trans := @trans_eq t.
Definition lt (x y:Z) := (x < y).
Lemma lt_trans : forall x y z, x < y -> y < z -> x < z.
Lemma lt_not_eq : forall x y, x < y -> ~ x = y.
Definition compare : forall x y, Compare lt eq x y.
Definition eq_dec := Z_eq_dec.
End Z_as_OT.
positive is an ordered type with respect to the usual order on natural numbers.
Open Local Scope positive_scope.
Module Positive_as_OT <: UsualOrderedType.
Definition t:=positive.
Definition eq:=@eq positive.
Definition eq_refl := @refl_equal t.
Definition eq_sym := @sym_eq t.
Definition eq_trans := @trans_eq t.
Definition lt p q:= (p ?= q) Eq = Lt.
Lemma lt_trans : forall x y z : t, lt x y -> lt y z -> lt x z.
Lemma lt_not_eq : forall x y : t, lt x y -> ~ eq x y.
Definition compare : forall x y : t, Compare lt eq x y.
Definition eq_dec : forall x y, { eq x y } + { ~ eq x y }.
End Positive_as_OT.
N is an ordered type with respect to the usual order on natural numbers.
Open Local Scope positive_scope.
Module N_as_OT <: UsualOrderedType.
Definition t:=N.
Definition eq:=@eq N.
Definition eq_refl := @refl_equal t.
Definition eq_sym := @sym_eq t.
Definition eq_trans := @trans_eq t.
Definition lt:=Nlt.
Definition lt_trans := Nlt_trans.
Definition lt_not_eq := Nlt_not_eq.
Definition compare : forall x y : t, Compare lt eq x y.
Definition eq_dec : forall x y, { eq x y } + { ~ eq x y }.
End N_as_OT.
From two ordered types, we can build a new OrderedType over their cartesian product, using the lexicographic order.
Module PairOrderedType(O1 O2:OrderedType) <: OrderedType.
Module MO1:=OrderedTypeFacts(O1).
Module MO2:=OrderedTypeFacts(O2).
Definition t := prod O1.t O2.t.
Definition eq x y := O1.eq (fst x) (fst y) /\ O2.eq (snd x) (snd y).
Definition lt x y :=
O1.lt (fst x) (fst y) \/
( O1.eq (fst x) (fst y) /\ O2.lt (snd x) (snd y)).
Lemma eq_refl : forall x : t, eq x x.
Lemma eq_sym : forall x y : t, eq x y -> eq y x.
Lemma eq_trans : forall x y z : t, eq x y -> eq y z -> eq x z.
Lemma lt_trans : forall x y z : t, lt x y -> lt y z -> lt x z.
Lemma lt_not_eq : forall x y : t, lt x y -> ~ eq x y.
Definition compare : forall x y : t, Compare lt eq x y.
Defined.
Definition eq_dec : forall x y : t, { eq x y } + { ~ eq x y }.
End PairOrderedType.
Even if positive can be seen as an ordered type with respect to the usual order (see above), we can also use a lexicographic order over bits (lower bits are considered first). This is more natural when using positive as indexes for sets or maps (see FSetPositive and FMapPositive.
Module PositiveOrderedTypeBits <: UsualOrderedType.
Definition t:=positive.
Definition eq:=@eq positive.
Definition eq_refl := @refl_equal t.
Definition eq_sym := @sym_eq t.
Definition eq_trans := @trans_eq t.
Fixpoint bits_lt (p q:positive) : Prop :=
match p, q with
| xH, xI _ => True
| xH, _ => False
| xO p, xO q => bits_lt p q
| xO _, _ => True
| xI p, xI q => bits_lt p q
| xI _, _ => False
end.
Definition lt:=bits_lt.
Lemma bits_lt_trans :
forall x y z : positive, bits_lt x y -> bits_lt y z -> bits_lt x z.
Lemma lt_trans : forall x y z : t, lt x y -> lt y z -> lt x z.
Lemma bits_lt_antirefl : forall x : positive, ~ bits_lt x x.
Lemma lt_not_eq : forall x y : t, lt x y -> ~ eq x y.
Definition compare : forall x y : t, Compare lt eq x y.
Lemma eq_dec (x y: positive): { x = y } + { x <> y }.
End PositiveOrderedTypeBits.