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+Require Import Coq.Lists.List.
+Require Import Coq.Setoids.Setoid.
+Require Import Coq.Lists.SetoidList.
+Require Import Crypto.Util.Option.
+
+Import ListNotations.
+
+Lemma list_elementwise_eqlistA : forall {T R} (l1 l2 : list T),
+  (forall i, option_eq R (nth_error l1 i) (nth_error l2 i)) -> eqlistA R l1 l2.
+Proof.
+  induction l1, l2; intros H; try solve [ constructor ];
+    pose proof (H 0%nat) as Hfirst; cbn in Hfirst; try solve [ inversion Hfirst ].
+  constructor; [ assumption | ].
+  apply IHl1.
+  intros i; specialize (H (S i)).
+  assumption.
+Qed.