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|
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(** A Library for finite sets, implemented as lists *)
(** List is loaded, but not exported.
This allow to "hide" the definitions, functions and theorems of List
and to see only the ones of ListSet *)
Require Import List.
Set Implicit Arguments.
Section first_definitions.
Variable A : Type.
Hypothesis Aeq_dec : forall x y:A, {x = y} + {x <> y}.
Definition set := list A.
Definition empty_set : set := nil.
Fixpoint set_add (a:A) (x:set) : set :=
match x with
| nil => a :: nil
| a1 :: x1 =>
match Aeq_dec a a1 with
| left _ => a1 :: x1
| right _ => a1 :: set_add a x1
end
end.
Fixpoint set_mem (a:A) (x:set) : bool :=
match x with
| nil => false
| a1 :: x1 =>
match Aeq_dec a a1 with
| left _ => true
| right _ => set_mem a x1
end
end.
(** If [a] belongs to [x], removes [a] from [x]. If not, does nothing *)
Fixpoint set_remove (a:A) (x:set) : set :=
match x with
| nil => empty_set
| a1 :: x1 =>
match Aeq_dec a a1 with
| left _ => x1
| right _ => a1 :: set_remove a x1
end
end.
Fixpoint set_inter (x:set) : set -> set :=
match x with
| nil => fun y => nil
| a1 :: x1 =>
fun y =>
if set_mem a1 y then a1 :: set_inter x1 y else set_inter x1 y
end.
Fixpoint set_union (x y:set) : set :=
match y with
| nil => x
| a1 :: y1 => set_add a1 (set_union x y1)
end.
(** returns the set of all els of [x] that does not belong to [y] *)
Fixpoint set_diff (x y:set) : set :=
match x with
| nil => nil
| a1 :: x1 =>
if set_mem a1 y then set_diff x1 y else set_add a1 (set_diff x1 y)
end.
Definition set_In : A -> set -> Prop := In (A:=A).
Lemma set_In_dec : forall (a:A) (x:set), {set_In a x} + {~ set_In a x}.
Proof.
unfold set_In.
(*** Realizer set_mem. Program_all. ***)
simple induction x.
auto.
intros a0 x0 Ha0. case (Aeq_dec a a0); intro eq.
rewrite eq; simpl; auto with datatypes.
elim Ha0.
auto with datatypes.
right; simpl; unfold not; intros [Hc1| Hc2];
auto with datatypes.
Qed.
Lemma set_mem_ind :
forall (B:Type) (P:B -> Prop) (y z:B) (a:A) (x:set),
(set_In a x -> P y) -> P z -> P (if set_mem a x then y else z).
Proof.
simple induction x; simpl; intros.
assumption.
elim (Aeq_dec a a0); auto with datatypes.
Qed.
Lemma set_mem_ind2 :
forall (B:Type) (P:B -> Prop) (y z:B) (a:A) (x:set),
(set_In a x -> P y) ->
(~ set_In a x -> P z) -> P (if set_mem a x then y else z).
Proof.
simple induction x; simpl; intros.
apply H0; red; trivial.
case (Aeq_dec a a0); auto with datatypes.
intro; apply H; intros; auto.
apply H1; red; intro.
case H3; auto.
Qed.
Lemma set_mem_correct1 :
forall (a:A) (x:set), set_mem a x = true -> set_In a x.
Proof.
simple induction x; simpl.
discriminate.
intros a0 l; elim (Aeq_dec a a0); auto with datatypes.
Qed.
Lemma set_mem_correct2 :
forall (a:A) (x:set), set_In a x -> set_mem a x = true.
Proof.
simple induction x; simpl.
intro Ha; elim Ha.
intros a0 l; elim (Aeq_dec a a0); auto with datatypes.
intros H1 H2 [H3| H4].
absurd (a0 = a); auto with datatypes.
auto with datatypes.
Qed.
Lemma set_mem_complete1 :
forall (a:A) (x:set), set_mem a x = false -> ~ set_In a x.
Proof.
simple induction x; simpl.
tauto.
intros a0 l; elim (Aeq_dec a a0).
intros; discriminate H0.
unfold not; intros; elim H1; auto with datatypes.
Qed.
Lemma set_mem_complete2 :
forall (a:A) (x:set), ~ set_In a x -> set_mem a x = false.
Proof.
simple induction x; simpl.
tauto.
intros a0 l; elim (Aeq_dec a a0).
intros; elim H0; auto with datatypes.
tauto.
Qed.
Lemma set_add_intro1 :
forall (a b:A) (x:set), set_In a x -> set_In a (set_add b x).
Proof.
unfold set_In; simple induction x; simpl.
auto with datatypes.
intros a0 l H [Ha0a| Hal].
elim (Aeq_dec b a0); left; assumption.
elim (Aeq_dec b a0); right; [ assumption | auto with datatypes ].
Qed.
Lemma set_add_intro2 :
forall (a b:A) (x:set), a = b -> set_In a (set_add b x).
Proof.
unfold set_In; simple induction x; simpl.
auto with datatypes.
intros a0 l H Hab.
elim (Aeq_dec b a0);
[ rewrite Hab; intro Hba0; rewrite Hba0; simpl;
auto with datatypes
| auto with datatypes ].
Qed.
Hint Resolve set_add_intro1 set_add_intro2.
Lemma set_add_intro :
forall (a b:A) (x:set), a = b \/ set_In a x -> set_In a (set_add b x).
Proof.
intros a b x [H1| H2]; auto with datatypes.
Qed.
Lemma set_add_elim :
forall (a b:A) (x:set), set_In a (set_add b x) -> a = b \/ set_In a x.
Proof.
unfold set_In.
simple induction x.
simpl; intros [H1| H2]; auto with datatypes.
simpl; do 3 intro.
elim (Aeq_dec b a0).
simpl; tauto.
simpl; intros; elim H0.
trivial with datatypes.
tauto.
tauto.
Qed.
Lemma set_add_elim2 :
forall (a b:A) (x:set), set_In a (set_add b x) -> a <> b -> set_In a x.
intros a b x H; case (set_add_elim _ _ _ H); intros; trivial.
case H1; trivial.
Qed.
Hint Resolve set_add_intro set_add_elim set_add_elim2.
Lemma set_add_not_empty : forall (a:A) (x:set), set_add a x <> empty_set.
Proof.
simple induction x; simpl.
discriminate.
intros; elim (Aeq_dec a a0); intros; discriminate.
Qed.
Lemma set_union_intro1 :
forall (a:A) (x y:set), set_In a x -> set_In a (set_union x y).
Proof.
simple induction y; simpl; auto with datatypes.
Qed.
Lemma set_union_intro2 :
forall (a:A) (x y:set), set_In a y -> set_In a (set_union x y).
Proof.
simple induction y; simpl.
tauto.
intros; elim H0; auto with datatypes.
Qed.
Hint Resolve set_union_intro2 set_union_intro1.
Lemma set_union_intro :
forall (a:A) (x y:set),
set_In a x \/ set_In a y -> set_In a (set_union x y).
Proof.
intros; elim H; auto with datatypes.
Qed.
Lemma set_union_elim :
forall (a:A) (x y:set),
set_In a (set_union x y) -> set_In a x \/ set_In a y.
Proof.
simple induction y; simpl.
auto with datatypes.
intros.
generalize (set_add_elim _ _ _ H0).
intros [H1| H1].
auto with datatypes.
tauto.
Qed.
Lemma set_union_emptyL :
forall (a:A) (x:set), set_In a (set_union empty_set x) -> set_In a x.
intros a x H; case (set_union_elim _ _ _ H); auto || contradiction.
Qed.
Lemma set_union_emptyR :
forall (a:A) (x:set), set_In a (set_union x empty_set) -> set_In a x.
intros a x H; case (set_union_elim _ _ _ H); auto || contradiction.
Qed.
Lemma set_inter_intro :
forall (a:A) (x y:set),
set_In a x -> set_In a y -> set_In a (set_inter x y).
Proof.
simple induction x.
auto with datatypes.
simpl; intros a0 l Hrec y [Ha0a| Hal] Hy.
simpl; rewrite Ha0a.
generalize (set_mem_correct1 a y).
generalize (set_mem_complete1 a y).
elim (set_mem a y); simpl; intros.
auto with datatypes.
absurd (set_In a y); auto with datatypes.
elim (set_mem a0 y); [ right; auto with datatypes | auto with datatypes ].
Qed.
Lemma set_inter_elim1 :
forall (a:A) (x y:set), set_In a (set_inter x y) -> set_In a x.
Proof.
simple induction x.
auto with datatypes.
simpl; intros a0 l Hrec y.
generalize (set_mem_correct1 a0 y).
elim (set_mem a0 y); simpl; intros.
elim H0; eauto with datatypes.
eauto with datatypes.
Qed.
Lemma set_inter_elim2 :
forall (a:A) (x y:set), set_In a (set_inter x y) -> set_In a y.
Proof.
simple induction x.
simpl; tauto.
simpl; intros a0 l Hrec y.
generalize (set_mem_correct1 a0 y).
elim (set_mem a0 y); simpl; intros.
elim H0;
[ intro Hr; rewrite <- Hr; eauto with datatypes | eauto with datatypes ].
eauto with datatypes.
Qed.
Hint Resolve set_inter_elim1 set_inter_elim2.
Lemma set_inter_elim :
forall (a:A) (x y:set),
set_In a (set_inter x y) -> set_In a x /\ set_In a y.
Proof.
eauto with datatypes.
Qed.
Lemma set_diff_intro :
forall (a:A) (x y:set),
set_In a x -> ~ set_In a y -> set_In a (set_diff x y).
Proof.
simple induction x.
simpl; tauto.
simpl; intros a0 l Hrec y [Ha0a| Hal] Hay.
rewrite Ha0a; generalize (set_mem_complete2 _ _ Hay).
elim (set_mem a y);
[ intro Habs; discriminate Habs | auto with datatypes ].
elim (set_mem a0 y); auto with datatypes.
Qed.
Lemma set_diff_elim1 :
forall (a:A) (x y:set), set_In a (set_diff x y) -> set_In a x.
Proof.
simple induction x.
simpl; tauto.
simpl; intros a0 l Hrec y; elim (set_mem a0 y).
eauto with datatypes.
intro; generalize (set_add_elim _ _ _ H).
intros [H1| H2]; eauto with datatypes.
Qed.
Lemma set_diff_elim2 :
forall (a:A) (x y:set), set_In a (set_diff x y) -> ~ set_In a y.
intros a x y; elim x; simpl.
intros; contradiction.
intros a0 l Hrec.
apply set_mem_ind2; auto.
intros H1 H2; case (set_add_elim _ _ _ H2); intros; auto.
rewrite H; trivial.
Qed.
Lemma set_diff_trivial : forall (a:A) (x:set), ~ set_In a (set_diff x x).
red; intros a x H.
apply (set_diff_elim2 _ _ _ H).
apply (set_diff_elim1 _ _ _ H).
Qed.
Hint Resolve set_diff_intro set_diff_trivial.
End first_definitions.
Section other_definitions.
Definition set_prod : forall {A B:Type}, set A -> set B -> set (A * B) :=
list_prod.
(** [B^A], set of applications from [A] to [B] *)
Definition set_power : forall {A B:Type}, set A -> set B -> set (set (A * B)) :=
list_power.
Definition set_fold_left {A B:Type} : (B -> A -> B) -> set A -> B -> B :=
fold_left (A:=B) (B:=A).
Definition set_fold_right {A B:Type} (f:A -> B -> B) (x:set A)
(b:B) : B := fold_right f b x.
Definition set_map {A B:Type} (Aeq_dec : forall x y:B, {x = y} + {x <> y})
(f : A -> B) (x : set A) : set B :=
set_fold_right (fun a => set_add Aeq_dec (f a)) x (empty_set B).
End other_definitions.
Unset Implicit Arguments.
|
