From 669217fe6f82df1b53af5d19ce066720721ea081 Mon Sep 17 00:00:00 2001 From: Robbert Krebbers Date: Wed, 14 Dec 2016 19:17:38 +0100 Subject: [PATCH] Simplify proofs relating nth to lookup. Also make names more consistent. --- theories/prelude/list.v | 23 ++++++----------------- 1 file changed, 6 insertions(+), 17 deletions(-) diff --git a/theories/prelude/list.v b/theories/prelude/list.v index 4348af93..fbe2850c 100644 --- a/theories/prelude/list.v +++ b/theories/prelude/list.v @@ -477,24 +477,13 @@ Lemma list_lookup_middle l1 l2 x n : n = length l1 → (l1 ++ x :: l2) !! n = Some x. Proof. intros ->. by induction l1. Qed. -Lemma nth_lookup_or_length l i d : - {l !! i = Some (nth i l d)} + {(length l ≤ i)%nat}. +Lemma nth_lookup l i d : nth i l d = from_option id d (l !! i). +Proof. revert i. induction l as [|x l IH]; intros [|i]; simpl; auto. Qed. +Lemma nth_lookup_Some l i d x : l !! i = Some x → nth i l d = x. +Proof. rewrite nth_lookup. by intros ->. Qed. +Lemma nth_lookup_or_length l i d : {l !! i = Some (nth i l d)} + {length l ≤ i}. Proof. - revert i; induction l; intros i. - - right. simpl. omega. - - destruct i; simpl. - + left. done. - + destruct (IHl i) as [->|]; [by left|]. - right. omega. -Qed. - -Lemma nth_lookup l i d x : - l !! i = Some x → nth i l d = x. -Proof. - revert i; induction l; intros i; [done|]. - destruct i; simpl. - - congruence. - - apply IHl. + rewrite nth_lookup. destruct (l !! i) eqn:?; eauto using lookup_ge_None_1. Qed. Lemma list_insert_alter l i x : <[i:=x]>l = alter (λ _, x) i l. -- GitLab