Proof using name_map_bijective.
move => n m.
elim => //=.
case => /= n' m' l IH H_eq H_in.
move => H_in'.
case: H_in' => H_in'.
inversion H_in'.
have H_nm := H_eq (n', m').
rewrite /= in H_nm.
case: H_in.
left.
apply tot_map_name_injective in H0.
rewrite H0.
rewrite H_nm //.
by left.
contradict H_in'.
apply: IH.
move => nm H_in_nm.
apply: H_eq.
by right.
move => H_in_nm.
case: H_in.
by right.
Qed.

(* Specifications: *)
Lemma msg_in_map :
forall m l n m',
(forall nm, In nm l -> snd nm = m) ->
In (tot_map_name n, tot_map_msg m') (map (fun nm : name * msg => (tot_map_name (fst nm), tot_map_msg (snd nm))) l) ->
tot_map_msg m' = tot_map_msg m.