eauto.
Qed.
Ltac an_use_binder f x :=
pick fresh x and apply f; eauto;
unfold open_tm_wrt_tm; simpl; simpl_env; eauto;
match goal with
[ |- AnnTyping ?ctx ?a ?A ] =>
assert (AnnCtx ctx); [econstructor; eauto|idtac]
end.

Lemma FixDef_FixTy :
AnnTyping nil FixDef FixTy.