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 ann_remove_available_mutual:
(forall G1  a A,   AnnTyping G1 a A -> True) /\
(forall G1  phi,   AnnPropWff G1 phi -> True) /\
(forall G1 D g p1 p2, AnnIso G1 D g p1 p2 ->
AnnIso G1 (AtomSetImpl.inter D (dom G1)) g p1 p2) /\
(forall G1 D g A B,   AnnDefEq G1 D g A B ->
AnnDefEq G1 (AtomSetImpl.inter D (dom G1)) g A B) /\
(forall G1 ,       AnnCtx G1 -> True).