diff options
Diffstat (limited to 'theories/Lists/List.v')
| -rw-r--r-- | theories/Lists/List.v | 71 |
1 files changed, 28 insertions, 43 deletions
diff --git a/theories/Lists/List.v b/theories/Lists/List.v index 4c14008c..69475a6f 100644 --- a/theories/Lists/List.v +++ b/theories/Lists/List.v @@ -1,14 +1,12 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011 *) +(* <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 *) (************************************************************************) -(*i $Id: List.v 14641 2011-11-06 11:59:10Z herbelin $ i*) - -Require Import Le Gt Minus Min Bool. +Require Import Le Gt Minus Bool Setoid. Set Implicit Arguments. @@ -55,9 +53,16 @@ Section Lists. End Lists. -(* Keep these notations local to prevent conflicting notations *) -Local Notation "[ ]" := nil : list_scope. -Local Notation "[ a ; .. ; b ]" := (a :: .. (b :: []) ..) : list_scope. + +(** Standard notations for lists. +In a special module to avoid conflict. *) +Module ListNotations. +Notation " [ ] " := nil : list_scope. +Notation " [ x ] " := (cons x nil) : list_scope. +Notation " [ x ; .. ; y ] " := (cons x .. (cons y nil) ..) : list_scope. +End ListNotations. + +Import ListNotations. (** ** Facts about lists *) @@ -119,7 +124,7 @@ Section Facts. unfold not; intros a H; inversion_clear H. Qed. - Theorem in_split : forall x (l:list A), In x l -> exists l1, exists l2, l = l1++x::l2. + Theorem in_split : forall x (l:list A), In x l -> exists l1 l2, l = l1++x::l2. Proof. induction l; simpl; destruct 1. subst a; auto. @@ -254,7 +259,7 @@ Section Facts. Qed. - (** Compatibility wtih other operations *) + (** Compatibility with other operations *) Lemma app_length : forall l l' : list A, length (l++l') = length l + length l'. Proof. @@ -541,30 +546,21 @@ Section Elts. end. (** Compatibility of count_occ with operations on list *) - Theorem count_occ_In : forall (l : list A) (x : A), In x l <-> count_occ l x > 0. + Theorem count_occ_In (l : list A) (x : A) : In x l <-> count_occ l x > 0. Proof. - induction l as [|y l]. - simpl; intros; split; [destruct 1 | apply gt_irrefl]. - simpl. intro x; destruct (eq_dec y x) as [Heq|Hneq]. - rewrite Heq; intuition. - pose (IHl x). intuition. + induction l as [|y l]; simpl. + - split; [destruct 1 | apply gt_irrefl]. + - destruct eq_dec as [->|Hneq]; rewrite IHl; intuition. Qed. - Theorem count_occ_inv_nil : forall (l : list A), (forall x:A, count_occ l x = 0) <-> l = []. + Theorem count_occ_inv_nil (l : list A) : + (forall x:A, count_occ l x = 0) <-> l = []. Proof. split. - (* Case -> *) - induction l as [|x l]. - trivial. - intro H. - elim (O_S (count_occ l x)). - apply sym_eq. - generalize (H x). - simpl. destruct (eq_dec x x) as [|HF]. - trivial. - elim HF; reflexivity. - (* Case <- *) - intro H; rewrite H; simpl; reflexivity. + - induction l as [|x l]; trivial. + intros H. specialize (H x). simpl in H. + destruct eq_dec as [_|NEQ]; [discriminate|now elim NEQ]. + - now intros ->. Qed. Lemma count_occ_nil : forall (x : A), count_occ [] x = 0. @@ -749,22 +745,11 @@ Section ListOps. Hypothesis eq_dec : forall (x y : A), {x = y}+{x <> y}. - Lemma list_eq_dec : - forall l l':list A, {l = l'} + {l <> l'}. - Proof. - induction l as [| x l IHl]; destruct l' as [| y l']. - left; trivial. - right; apply nil_cons. - right; unfold not; intro HF; apply (nil_cons (sym_eq HF)). - destruct (eq_dec x y) as [xeqy|xneqy]; destruct (IHl l') as [leql'|lneql']; - try (right; unfold not; intro HF; injection HF; intros; contradiction). - rewrite xeqy; rewrite leql'; left; trivial. - Qed. - + Lemma list_eq_dec : forall l l':list A, {l = l'} + {l <> l'}. + Proof. decide equality. Defined. End ListOps. - (***************************************************) (** * Applying functions to the elements of a list *) (***************************************************) @@ -1643,7 +1628,7 @@ Proof. exact Forall2_nil. Qed. Theorem Forall2_app_inv_l : forall A B (R:A->B->Prop) l1 l2 l', Forall2 R (l1 ++ l2) l' -> - exists l1', exists l2', Forall2 R l1 l1' /\ Forall2 R l2 l2' /\ l' = l1' ++ l2'. + exists l1' l2', Forall2 R l1 l1' /\ Forall2 R l2 l2' /\ l' = l1' ++ l2'. Proof. induction l1; intros. exists [], l'; auto. @@ -1654,7 +1639,7 @@ Qed. Theorem Forall2_app_inv_r : forall A B (R:A->B->Prop) l1' l2' l, Forall2 R l (l1' ++ l2') -> - exists l1, exists l2, Forall2 R l1 l1' /\ Forall2 R l2 l2' /\ l = l1 ++ l2. + exists l1 l2, Forall2 R l1 l1' /\ Forall2 R l2 l2' /\ l = l1 ++ l2. Proof. induction l1'; intros. exists [], l; auto. |