Definition Iso_respects_atoms_eq   := third  respects_atoms_eq_mutual.Definition DefEq_respects_atoms_eq := fourth respects_atoms_eq_mutual.

Lemma strengthen_available_noncovar: (forall G1 a A, Typing G1 a A -> True) /\ (forall G1 phi, PropWff G1 phi -> True) /\ (forall G1 D p1 p2, Iso G1 D p1 p2 -> forall x, not (exists phi, binds x (Co phi) G1) -> Iso G1 (remove x D) p1 p2) /\ (forall G1 D A B A1,DefEq G1 D A B A1 -> forall x, not (exists phi, binds x (Co phi) G1) -> DefEq G1 (remove x D) A B A1) /\ (forall G1 , Ctx G1 -> True).