diff options
Diffstat (limited to 'theories/Lists/List.v')
| -rw-r--r-- | theories/Lists/List.v | 42 |
1 files changed, 26 insertions, 16 deletions
diff --git a/theories/Lists/List.v b/theories/Lists/List.v index 30f1dec2..ca5f154e 100644 --- a/theories/Lists/List.v +++ b/theories/Lists/List.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Setoid. @@ -27,7 +29,6 @@ Module ListNotations. Notation "[ ]" := nil (format "[ ]") : list_scope. Notation "[ x ]" := (cons x nil) : list_scope. Notation "[ x ; y ; .. ; z ]" := (cons x (cons y .. (cons z nil) ..)) : list_scope. -Notation "[ x ; .. ; y ]" := (cons x .. (cons y nil) ..) (compat "8.4") : list_scope. End ListNotations. Import ListNotations. @@ -419,7 +420,7 @@ Section Elts. Proof. unfold lt; induction n as [| n hn]; simpl. - destruct l; simpl; [ inversion 2 | auto ]. - - destruct l as [| a l hl]; simpl. + - destruct l; simpl. * inversion 2. * intros d ie; right; apply hn; auto with arith. Qed. @@ -1280,7 +1281,7 @@ End Fold_Right_Recursor. partition l = ([], []) <-> l = []. Proof. split. - - destruct l as [|a l' _]. + - destruct l as [|a l']. * intuition. * simpl. destruct (f a), (partition l'); now intros [= -> ->]. - now intros ->. @@ -2110,13 +2111,13 @@ Section Exists_Forall. {Exists l} + {~ Exists l}. Proof. intro Pdec. induction l as [|a l' Hrec]. - - right. now rewrite Exists_nil. + - right. abstract now rewrite Exists_nil. - destruct Hrec as [Hl'|Hl']. * left. now apply Exists_cons_tl. * destruct (Pdec a) as [Ha|Ha]. + left. now apply Exists_cons_hd. - + right. now inversion_clear 1. - Qed. + + right. abstract now inversion 1. + Defined. Inductive Forall : list A -> Prop := | Forall_nil : Forall nil @@ -2152,9 +2153,9 @@ Section Exists_Forall. - destruct Hrec as [Hl'|Hl']. + destruct (Pdec a) as [Ha|Ha]. * left. now apply Forall_cons. - * right. now inversion_clear 1. - + right. now inversion_clear 1. - Qed. + * right. abstract now inversion 1. + + right. abstract now inversion 1. + Defined. End One_predicate. @@ -2179,6 +2180,16 @@ Section Exists_Forall. * now apply Exists_cons_hd. Qed. + Lemma neg_Forall_Exists_neg (P:A->Prop) (l:list A) : + (forall x:A, {P x} + { ~ P x }) -> + ~ Forall P l -> + Exists (fun x => ~ P x) l. + Proof. + intro Dec. + apply Exists_Forall_neg; intros. + destruct (Dec x); auto. + Qed. + Lemma Forall_Exists_dec (P:A->Prop) : (forall x:A, {P x} + { ~ P x }) -> forall l:list A, @@ -2186,9 +2197,8 @@ Section Exists_Forall. Proof. intros Pdec l. destruct (Forall_dec P Pdec l); [left|right]; trivial. - apply Exists_Forall_neg; trivial. - intro x. destruct (Pdec x); [now left|now right]. - Qed. + now apply neg_Forall_Exists_neg. + Defined. Lemma Forall_impl : forall (P Q : A -> Prop), (forall a, P a -> Q a) -> forall l, Forall P l -> Forall Q l. |