Commit 669217fe authored by Robbert Krebbers's avatar Robbert Krebbers
Browse files

Simplify proofs relating nth to lookup.

Also make names more consistent.
parent b1fa82f0
...@@ -477,24 +477,13 @@ Lemma list_lookup_middle l1 l2 x n : ...@@ -477,24 +477,13 @@ Lemma list_lookup_middle l1 l2 x n :
n = length l1 (l1 ++ x :: l2) !! n = Some x. n = length l1 (l1 ++ x :: l2) !! n = Some x.
Proof. intros ->. by induction l1. Qed. Proof. intros ->. by induction l1. Qed.
Lemma nth_lookup_or_length l i d : Lemma nth_lookup l i d : nth i l d = from_option id d (l !! i).
{l !! i = Some (nth i l d)} + {(length l i)%nat}. 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. Proof.
revert i; induction l; intros i. rewrite nth_lookup. destruct (l !! i) eqn:?; eauto using lookup_ge_None_1.
- 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.
Qed. Qed.
Lemma list_insert_alter l i x : <[i:=x]>l = alter (λ _, x) i l. Lemma list_insert_alter l i x : <[i:=x]>l = alter (λ _, x) i l.
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