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-(************************************************************************)
-(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
-(* \VV/ **************************************************************)
-(* // * This file is distributed under the terms of the *)
-(* * GNU Lesser General Public License Version 2.1 *)
-(************************************************************************)
-
-(*i $Id: ListSet.v,v 1.1.2.1 2004/07/16 19:31:28 herbelin Exp $ i*)
-
-(** A Library for finite sets, implemented as lists
- A Library with similar interface will soon be available under
- the name TreeSet in the theories/Trees directory *)
-
-(** PolyList is loaded, but not exported.
- This allow to "hide" the definitions, functions and theorems of PolyList
- and to see only the ones of ListSet *)
-
-Require PolyList.
-
-Set Implicit Arguments.
-V7only [Implicits nil [1].].
-
-Section first_definitions.
-
- Variable A : Set.
- Hypothesis Aeq_dec : (x,y:A){x=y}+{~x=y}.
-
- Definition set := (list A).
-
- Definition empty_set := (!nil ?) : set.
-
- Fixpoint set_add [a:A; x:set] : set :=
- Cases x of
- | nil => (cons a nil)
- | (cons a1 x1) => Cases (Aeq_dec a a1) of
- | (left _) => (cons a1 x1)
- | (right _) => (cons a1 (set_add a x1))
- end
- end.
-
-
- Fixpoint set_mem [a:A; x:set] : bool :=
- Cases x of
- | nil => false
- | (cons a1 x1) => Cases (Aeq_dec a a1) of
- | (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 :=
- Cases x of
- | nil => empty_set
- | (cons a1 x1) => Cases (Aeq_dec a a1) of
- | (left _) => x1
- | (right _) => (cons a1 (set_remove a x1))
- end
- end.
-
- Fixpoint set_inter [x:set] : set -> set :=
- Cases x of
- | nil => [y]nil
- | (cons a1 x1) => [y]if (set_mem a1 y)
- then (cons a1 (set_inter x1 y))
- else (set_inter x1 y)
- end.
-
- Fixpoint set_union [x,y:set] : set :=
- Cases y of
- | nil => x
- | (cons 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:set] : set -> set :=
- [y]Cases x of
- | nil => nil
- | (cons 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 1!A).
-
- Lemma set_In_dec : (a:A; x:set){(set_In a x)}+{~(set_In a x)}.
-
- Proof.
- Unfold set_In.
- (*** Realizer set_mem. Program_all. ***)
- 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 :
- (B:Set)(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.
- Induction x; Simpl; Intros.
- Assumption.
- Elim (Aeq_dec a a0); Auto with datatypes.
- Qed.
-
- Lemma set_mem_ind2 :
- (B:Set)(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.
- 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 :
- (a:A)(x:set)(set_mem a x)=true -> (set_In a x).
- Proof.
- Induction x; Simpl.
- Discriminate.
- Intros a0 l; Elim (Aeq_dec a a0); Auto with datatypes.
- Qed.
-
- Lemma set_mem_correct2 :
- (a:A)(x:set)(set_In a x) -> (set_mem a x)=true.
- Proof.
- 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 :
- (a:A)(x:set)(set_mem a x)=false -> ~(set_In a x).
- Proof.
- 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 :
- (a:A)(x:set)~(set_In a x) -> (set_mem a x)=false.
- Proof.
- Induction x; Simpl.
- Tauto.
- Intros a0 l; Elim (Aeq_dec a a0).
- Intros; Elim H0; Auto with datatypes.
- Tauto.
- Qed.
-
- Lemma set_add_intro1 : (a,b:A)(x:set)
- (set_In a x) -> (set_In a (set_add b x)).
-
- Proof.
- Unfold set_In; 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 : (a,b:A)(x:set)
- a=b -> (set_In a (set_add b x)).
-
- Proof.
- Unfold set_In; 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.
-
- Hints Resolve set_add_intro1 set_add_intro2.
-
- Lemma set_add_intro : (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 : (a,b:A)(x:set)
- (set_In a (set_add b x)) -> a=b\/(set_In a x).
-
- Proof.
- Unfold set_In.
- 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 : (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.
-
- Hints Resolve set_add_intro set_add_elim set_add_elim2.
-
- Lemma set_add_not_empty : (a:A)(x:set)~(set_add a x)=empty_set.
- Proof.
- Induction x; Simpl.
- Discriminate.
- Intros; Elim (Aeq_dec a a0); Intros; Discriminate.
- Qed.
-
-
- Lemma set_union_intro1 : (a:A)(x,y:set)
- (set_In a x) -> (set_In a (set_union x y)).
- Proof.
- Induction y; Simpl; Auto with datatypes.
- Qed.
-
- Lemma set_union_intro2 : (a:A)(x,y:set)
- (set_In a y) -> (set_In a (set_union x y)).
- Proof.
- Induction y; Simpl.
- Tauto.
- Intros; Elim H0; Auto with datatypes.
- Qed.
-
- Hints Resolve set_union_intro2 set_union_intro1.
-
- Lemma set_union_intro : (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 : (a:A)(x,y:set)
- (set_In a (set_union x y)) -> (set_In a x)\/(set_In a y).
- Proof.
- Induction y; Simpl.
- Auto with datatypes.
- Intros.
- Generalize (set_add_elim H0).
- Intros [H1 | H1].
- Auto with datatypes.
- Tauto.
- Qed.
-
- Lemma set_union_emptyL : (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 Orelse Contradiction.
- Qed.
-
-
- Lemma set_union_emptyR : (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 Orelse Contradiction.
- Qed.
-
-
- Lemma set_inter_intro : (a:A)(x,y:set)
- (set_In a x) -> (set_In a y) -> (set_In a (set_inter x y)).
- Proof.
- 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 : (a:A)(x,y:set)
- (set_In a (set_inter x y)) -> (set_In a x).
- Proof.
- 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 : (a:A)(x,y:set)
- (set_In a (set_inter x y)) -> (set_In a y).
- Proof.
- 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.
-
- Hints Resolve set_inter_elim1 set_inter_elim2.
-
- Lemma set_inter_elim : (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 : (a:A)(x,y:set)
- (set_In a x) -> ~(set_In a y) -> (set_In a (set_diff x y)).
- Proof.
- 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 : (a:A)(x,y:set)
- (set_In a (set_diff x y)) -> (set_In a x).
- Proof.
- 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 : (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 : (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.
-
-Hints Resolve set_diff_intro set_diff_trivial.
-
-
-End first_definitions.
-
-Section other_definitions.
-
- Variables A,B : Set.
-
- Definition set_prod : (set A) -> (set B) -> (set A*B) := (list_prod 1!A 2!B).
-
- (** [B^A], set of applications from [A] to [B] *)
- Definition set_power : (set A) -> (set B) -> (set (set A*B)) :=
- (list_power 1!A 2!B).
-
- Definition set_map : (A->B) -> (set A) -> (set B) := (map 1!A 2!B).
-
- Definition set_fold_left : (B -> A -> B) -> (set A) -> B -> B :=
- (fold_left 1!B 2!A).
-
- Definition set_fold_right : (A -> B -> B) -> (set A) -> B -> B :=
- [f][x][b](fold_right f b x).
-
-
-End other_definitions.
-
-V7only [Implicits nil [].].
-Unset Implicit Arguments.
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