Proof using.
move => n.
elim => //.
move => n' l IH H_in.
case: H_in => H_in.
rewrite H_in.
by left.
right.
exact: IH.
Qed.

Lemma remove_tot_map_eq :
forall h failed,
map tot_map_name (remove name_eq_dec h failed) =
remove name_eq_dec (tot_map_name h) (map tot_map_name failed).