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Diffstat (limited to 'theories/FSets/OrderedTypeEx.v')
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diff --git a/theories/FSets/OrderedTypeEx.v b/theories/FSets/OrderedTypeEx.v new file mode 100644 index 00000000..1c5a4054 --- /dev/null +++ b/theories/FSets/OrderedTypeEx.v @@ -0,0 +1,248 @@ +(***********************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) +(* \VV/ *************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(***********************************************************************) + +(* Finite sets library. + * Authors: Pierre Letouzey and Jean-Christophe Filliâtre + * Institution: LRI, CNRS UMR 8623 - Université Paris Sud + * 91405 Orsay, France *) + +(* $Id: OrderedTypeEx.v 8836 2006-05-20 21:34:27Z letouzey $ *) + +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 t : Set. + Definition eq := @eq t. + Parameter 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. +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. + Proof. unfold lt in |- *; intros; apply lt_trans with y; auto. Qed. + + Lemma lt_not_eq : forall x y : t, lt x y -> ~ eq x y. + Proof. unfold lt, eq in |- *; intros; omega. Qed. + + Definition compare : forall x y : t, Compare lt eq x y. + Proof. + intros; case (lt_eq_lt_dec x y). + simple destruct 1; intro. + constructor 1; auto. + constructor 2; auto. + intro; constructor 3; auto. + Qed. + +End Nat_as_OT. + + +(** [Z] is an ordered type with respect to the usual order on integers. *) + +Open 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. + Proof. intros; omega. Qed. + + Lemma lt_not_eq : forall x y, x<y -> ~ x=y. + Proof. intros; omega. Qed. + + Definition compare : forall x y, Compare lt eq x y. + Proof. + intros x y; case_eq (x ?= y); intros. + apply EQ; unfold eq; apply Zcompare_Eq_eq; auto. + apply LT; unfold lt, Zlt; auto. + apply GT; unfold lt, Zlt; rewrite <- Zcompare_Gt_Lt_antisym; auto. + Defined. + +End Z_as_OT. + +(** [positive] is an ordered type with respect to the usual order on natural numbers. *) + +Open 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. + Proof. + unfold lt; intros x y z. + change ((Zpos x < Zpos y)%Z -> (Zpos y < Zpos z)%Z -> (Zpos x < Zpos z)%Z). + omega. + Qed. + + Lemma lt_not_eq : forall x y : t, lt x y -> ~ eq x y. + Proof. + intros; intro. + rewrite H0 in H. + unfold lt in H. + rewrite Pcompare_refl in H; discriminate. + Qed. + + Definition compare : forall x y : t, Compare lt eq x y. + Proof. + intros x y. + case_eq ((x ?= y) Eq); intros. + apply EQ; apply Pcompare_Eq_eq; auto. + apply LT; unfold lt; auto. + apply GT; unfold lt. + replace Eq with (CompOpp Eq); auto. + rewrite <- Pcompare_antisym; rewrite H; auto. + Qed. + +End Positive_as_OT. + + +(** [N] is an ordered type with respect to the usual order on natural numbers. *) + +Open 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 p q:= Nle q p = false. + + Definition lt_trans := Nlt_trans. + + Lemma lt_not_eq : forall x y : t, lt x y -> ~ eq x y. + Proof. + intros; intro. + rewrite H0 in H. + unfold lt in H. + rewrite Nle_refl in H; discriminate. + Qed. + + Definition compare : forall x y : t, Compare lt eq x y. + Proof. + intros x y. + case_eq ((x ?= y)%N); intros. + apply EQ; apply Ncompare_Eq_eq; auto. + apply LT; unfold lt; auto. + generalize (Nle_Ncompare y x). + destruct (Nle y x); auto. + rewrite <- Ncompare_antisym. + destruct (x ?= y)%N; simpl; try discriminate. + intros (H0,_); elim H0; auto. + apply GT; unfold lt. + generalize (Nle_Ncompare x y). + destruct (Nle x y); auto. + destruct (x ?= y)%N; simpl; try discriminate. + intros (H0,_); elim H0; auto. + Qed. + +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. + Proof. + intros (x1,x2); red; simpl; auto. + Qed. + + Lemma eq_sym : forall x y : t, eq x y -> eq y x. + Proof. + intros (x1,x2) (y1,y2); unfold eq; simpl; intuition. + Qed. + + Lemma eq_trans : forall x y z : t, eq x y -> eq y z -> eq x z. + Proof. + intros (x1,x2) (y1,y2) (z1,z2); unfold eq; simpl; intuition eauto. + Qed. + + Lemma lt_trans : forall x y z : t, lt x y -> lt y z -> lt x z. + Proof. + intros (x1,x2) (y1,y2) (z1,z2); unfold eq, lt; simpl; intuition. + left; eauto. + left; eapply MO1.lt_eq; eauto. + left; eapply MO1.eq_lt; eauto. + right; split; eauto. + Qed. + + Lemma lt_not_eq : forall x y : t, lt x y -> ~ eq x y. + Proof. + intros (x1,x2) (y1,y2); unfold eq, lt; simpl; intuition. + apply (O1.lt_not_eq H0 H1). + apply (O2.lt_not_eq H3 H2). + Qed. + + Definition compare : forall x y : t, Compare lt eq x y. + intros (x1,x2) (y1,y2). + destruct (O1.compare x1 y1). + apply LT; unfold lt; auto. + destruct (O2.compare x2 y2). + apply LT; unfold lt; auto. + apply EQ; unfold eq; auto. + apply GT; unfold lt; auto. + apply GT; unfold lt; auto. + Qed. + +End PairOrderedType. + |