Definition AnnIso_respects_atoms_eq   := third  ann_respects_atoms_eq_mutual.
Definition AnnDefEq_respects_atoms_eq := fourth ann_respects_atoms_eq_mutual.
Instance AnnIso_atoms_eq_mor : Morphisms.Proper
(eq ==> AtomSetImpl.Equal ==> eq ==> eq ==> eq ==> iff)
AnnIso.
Instance AnnDefEq_atoms_eq_mor : Morphisms.Proper
(eq ==> AtomSetImpl.Equal ==> eq ==> eq ==> eq ==> iff)
AnnDefEq.

Lemma AnnDefEq_strengthen_available_tm :
forall G D g A B, AnnDefEq G D g A B ->  forall x A', binds x (Tm A') G ->
forall D', D' [=] remove x D ->
AnnDefEq G D' g A B.