Commit 997c3ed8 authored by Ralf Jung's avatar Ralf Jung

list: relate nth and lookup

parent 0b97288f
......@@ -477,6 +477,26 @@ 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}.
revert i; induction l; intros i.
- right. simpl. omega.
- destruct i; simpl.
+ left. done.
+ destruct (IHl i) as [->|]; [by left|].
right. omega.
Lemma nth_lookup l i d x :
l !! i = Some x → nth i l d = x.
revert i; induction l; intros i; [done|].
destruct i; simpl.
- congruence.
- apply IHl.
Lemma list_insert_alter l i x : <[i:=x]>l = alter (λ _, x) i l.
Proof. by revert i; induction l; intros []; intros; f_equal/=. Qed.
Lemma alter_length f l i : length (alter f i l) = length l.
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