Definition Typing_weakening  := first  typing_weakening_mutual.
Definition PropWff_weakening := second typing_weakening_mutual.
Definition Iso_weakening     := third  typing_weakening_mutual.
Definition DefEq_weakening   := fourth typing_weakening_mutual.
End ext_weak_sig.
(**************** Ext Substitution ****************)
Module Type ext_subst_sig.
Include ext_weak_sig.

Axiom tm_substitution_mutual :  (forall G0 b B (H : Typing G0 b B),
forall G a A, Typing G a A ->
forall F x, G0 = (F ++ (x ~ Tm A) ++ G) ->
Typing (map (tm_subst_tm_sort a x) F ++ G)
(tm_subst_tm_tm a x b)
(tm_subst_tm_tm a x B)) /\
(forall G0 phi (H : PropWff G0 phi),
forall G a A, Typing G a A ->
forall F x, G0 = (F ++ (x ~ Tm A) ++ G) ->
PropWff (map (tm_subst_tm_sort a x) F ++ G)
(tm_subst_tm_constraint a x phi)) /\
(forall G0 D p1 p2 (H : Iso G0 D p1 p2),
forall G a A, Typing G a A ->
forall F x, G0 = (F ++ (x ~ Tm A) ++ G) ->
Iso (map (tm_subst_tm_sort a x) F ++ G) D
(tm_subst_tm_constraint a x p1)
(tm_subst_tm_constraint a x p2)) /\
(forall G0 D A B T (H : DefEq G0 D A B T),
forall G a A0, Typing G a A0 ->
forall F x, G0 = (F ++ (x ~ Tm A0) ++ G) ->
DefEq (map (tm_subst_tm_sort a x) F ++ G) D
(tm_subst_tm_tm a x A)
(tm_subst_tm_tm a x B) (tm_subst_tm_tm a x T)) /\
(forall G0 (H : Ctx G0),
forall G a A, Typing G a A ->
forall F x, G0 = (F ++ (x ~ Tm A) ++ G) ->
Ctx (map (tm_subst_tm_sort a x) F ++ G)).