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

Lemma typing_weakening_mutual: (forall G0 a A, Typing G0 a A -> forall E F G, (G0 = F ++ G) -> Ctx (F ++ E ++ G) -> Typing (F ++ E ++ G) a A) /\ (forall G0 phi, PropWff G0 phi -> forall E F G, (G0 = F ++ G) -> Ctx (F ++ E ++ G) -> PropWff (F ++ E ++ G) phi) /\ (forall G0 D p1 p2, Iso G0 D p1 p2 -> forall E F G, (G0 = F ++ G) -> Ctx (F ++ E ++ G) -> Iso (F ++ E ++ G) D p1 p2) /\ (forall G0 D A B T, DefEq G0 D A B T -> forall E F G, (G0 = F ++ G) -> Ctx (F ++ E ++ G) -> DefEq (F ++ E ++ G) D A B T) /\ (forall G0, Ctx G0 -> forall E F G, (G0 = F ++ G) -> Ctx (F ++ E ++ G) -> Ctx (F ++ E ++ G)).