Definition var_env'_to_env'' (L U : nat) (ve : var_env') :=
  match var_env'_to_var_env''_lemma2 (U - L) L U ve (refl_equal _) with
  | exist ve'' _ => ve''
  end.

Lemma var_env'_to_env''_lemma3 : forall (L U : nat) (ve : var_env'), In _ (Evar_env'' L U) (var_env'_to_env'' L U ve) /\ (forall x : ad, var_lu L U x = true -> in_dom _ x (var_env'_to_env'' L U ve) = ve (nat_of_N x)).