diff options
Diffstat (limited to 'theories/Lists')
| -rw-r--r-- | theories/Lists/List.v | 76 | ||||
| -rw-r--r-- | theories/Lists/ListSet.v | 109 |
2 files changed, 181 insertions, 4 deletions
diff --git a/theories/Lists/List.v b/theories/Lists/List.v index fe18686e2..45306caf0 100644 --- a/theories/Lists/List.v +++ b/theories/Lists/List.v @@ -7,7 +7,7 @@ (************************************************************************) Require Setoid. -Require Import PeanoNat Le Gt Minus Bool. +Require Import PeanoNat Le Gt Minus Bool Lt. Set Implicit Arguments. (* Set Universe Polymorphism. *) @@ -1633,6 +1633,80 @@ Section Cutting. end end. + Lemma firstn_nil n: firstn n [] = []. + Proof. induction n; now simpl. Qed. + + Lemma firstn_cons n a l: firstn (S n) (a::l) = a :: (firstn n l). + Proof. now simpl. Qed. + + Lemma firstn_all l: firstn (length l) l = l. + Proof. induction l as [| ? ? H]; simpl; [reflexivity | now rewrite H]. Qed. + + Lemma firstn_all2 n: forall (l:list A), (length l) <= n -> firstn n l = l. + Proof. induction n as [|k iHk]. + - intro. inversion 1 as [H1|?]. + rewrite (length_zero_iff_nil l) in H1. subst. now simpl. + - destruct l as [|x xs]; simpl. + * now reflexivity. + * simpl. intro H. apply Peano.le_S_n in H. f_equal. apply iHk, H. + Qed. + + Lemma firstn_O l: firstn 0 l = []. + Proof. now simpl. Qed. + + Lemma firstn_le_length n: forall l:list A, length (firstn n l) <= n. + Proof. + induction n as [|k iHk]; simpl; [auto | destruct l as [|x xs]; simpl]. + - auto with arith. + - apply Peano.le_n_S, iHk. + Qed. + + Lemma firstn_length_le: forall l:list A, forall n:nat, + n <= length l -> length (firstn n l) = n. + Proof. induction l as [|x xs Hrec]. + - simpl. intros n H. apply le_n_0_eq in H. rewrite <- H. now simpl. + - destruct n. + * now simpl. + * simpl. intro H. apply le_S_n in H. now rewrite (Hrec n H). + Qed. + + Lemma firstn_app n: + forall l1 l2, + firstn n (l1 ++ l2) = (firstn n l1) ++ (firstn (n - length l1) l2). + Proof. induction n as [|k iHk]; intros l1 l2. + - now simpl. + - destruct l1 as [|x xs]. + * unfold firstn at 2, length. now rewrite 2!app_nil_l, <- minus_n_O. + * rewrite <- app_comm_cons. simpl. f_equal. apply iHk. + Qed. + + Lemma firstn_app_2 n: + forall l1 l2, + firstn ((length l1) + n) (l1 ++ l2) = l1 ++ firstn n l2. + Proof. induction n as [| k iHk];intros l1 l2. + - unfold firstn at 2. rewrite <- plus_n_O, app_nil_r. + rewrite firstn_app. rewrite <- minus_diag_reverse. + unfold firstn at 2. rewrite app_nil_r. apply firstn_all. + - destruct l2 as [|x xs]. + * simpl. rewrite app_nil_r. apply firstn_all2. auto with arith. + * rewrite firstn_app. assert (H0 : (length l1 + S k - length l1) = S k). + auto with arith. + rewrite H0, firstn_all2; [reflexivity | auto with arith]. + Qed. + + Lemma firstn_firstn: + forall l:list A, + forall i j : nat, + firstn i (firstn j l) = firstn (min i j) l. + Proof. induction l as [|x xs Hl]. + - intros. simpl. now rewrite ?firstn_nil. + - destruct i. + * intro. now simpl. + * destruct j. + + now simpl. + + simpl. f_equal. apply Hl. + Qed. + Fixpoint skipn (n:nat)(l:list A) : list A := match n with | 0 => l diff --git a/theories/Lists/ListSet.v b/theories/Lists/ListSet.v index 0a0bf0dea..c8ed95cd4 100644 --- a/theories/Lists/ListSet.v +++ b/theories/Lists/ListSet.v @@ -48,7 +48,11 @@ Section first_definitions. end end. - (** If [a] belongs to [x], removes [a] from [x]. If not, does nothing *) + (** If [a] belongs to [x], removes [a] from [x]. If not, does nothing. + Invariant: any element should occur at most once in [x], see for + instance [set_add]. We hence remove here only the first occurrence + of [a] in [x]. *) + Fixpoint set_remove (a:A) (x:set) : set := match x with | nil => empty_set @@ -227,6 +231,68 @@ Section first_definitions. intros; elim (Aeq_dec a a0); intros; discriminate. Qed. + Lemma set_add_iff a b l : In a (set_add b l) <-> a = b \/ In a l. + Proof. + split. apply set_add_elim. apply set_add_intro. + Qed. + + Lemma set_add_nodup a l : NoDup l -> NoDup (set_add a l). + Proof. + induction 1 as [|x l H H' IH]; simpl. + - constructor; [ tauto | constructor ]. + - destruct (Aeq_dec a x) as [<-|Hax]; constructor; trivial. + rewrite set_add_iff. intuition. + Qed. + + Lemma set_remove_1 (a b : A) (l : set) : + In a (set_remove b l) -> In a l. + Proof. + induction l as [|x xs Hrec]. + - intros. auto. + - simpl. destruct (Aeq_dec b x). + * tauto. + * intro H. destruct H. + + rewrite H. apply in_eq. + + apply in_cons. apply Hrec. assumption. + Qed. + + Lemma set_remove_2 (a b:A) (l : set) : + NoDup l -> In a (set_remove b l) -> a <> b. + Proof. + induction l as [|x l IH]; intro ND; simpl. + - tauto. + - inversion_clear ND. + destruct (Aeq_dec b x) as [<-|Hbx]. + + congruence. + + destruct 1; subst; auto. + Qed. + + Lemma set_remove_3 (a b : A) (l : set) : + In a l -> a <> b -> In a (set_remove b l). + Proof. + induction l as [|x xs Hrec]. + - now simpl. + - simpl. destruct (Aeq_dec b x) as [<-|Hbx]; simpl; intuition. + congruence. + Qed. + + Lemma set_remove_iff (a b : A) (l : set) : + NoDup l -> (In a (set_remove b l) <-> In a l /\ a <> b). + Proof. + split; try split. + - eapply set_remove_1; eauto. + - eapply set_remove_2; eauto. + - destruct 1; apply set_remove_3; auto. + Qed. + + Lemma set_remove_nodup a l : NoDup l -> NoDup (set_remove a l). + Proof. + induction 1 as [|x l H H' IH]; simpl. + - constructor. + - destruct (Aeq_dec a x) as [<-|Hax]; trivial. + constructor; trivial. + rewrite set_remove_iff; trivial. intuition. + Qed. Lemma set_union_intro1 : forall (a:A) (x y:set), set_In a x -> set_In a (set_union x y). @@ -264,18 +330,26 @@ Section first_definitions. tauto. Qed. + Lemma set_union_iff a l l': In a (set_union l l') <-> In a l \/ In a l'. + Proof. + split. apply set_union_elim. apply set_union_intro. + Qed. + + Lemma set_union_nodup l l' : NoDup l -> NoDup l' -> NoDup (set_union l l'). + Proof. + induction 2 as [|x' l' ? ? IH]; simpl; trivial. now apply set_add_nodup. + 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). @@ -326,6 +400,21 @@ Section first_definitions. eauto with datatypes. Qed. + Lemma set_inter_iff a l l' : In a (set_inter l l') <-> In a l /\ In a l'. + Proof. + split. + - apply set_inter_elim. + - destruct 1. now apply set_inter_intro. + Qed. + + Lemma set_inter_nodup l l' : NoDup l -> NoDup l' -> NoDup (set_inter l l'). + Proof. + induction 1 as [|x l H H' IH]; intro Hl'; simpl. + - constructor. + - destruct (set_mem x l'); auto. + constructor; auto. rewrite set_inter_iff; tauto. + 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). @@ -360,6 +449,20 @@ Section first_definitions. rewrite H; trivial. Qed. + Lemma set_diff_iff a l l' : In a (set_diff l l') <-> In a l /\ ~In a l'. + Proof. + split. + - split; [eapply set_diff_elim1 | eapply set_diff_elim2]; eauto. + - destruct 1. now apply set_diff_intro. + Qed. + + Lemma set_diff_nodup l l' : NoDup l -> NoDup l' -> NoDup (set_diff l l'). + Proof. + induction 1 as [|x l H H' IH]; intro Hl'; simpl. + - constructor. + - destruct (set_mem x l'); auto using set_add_nodup. + 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). |