(* --------------------------------------------------- *)
Inductive context_DefEq : available_props -> context -> context -> Prop :=
| Nul_Eqcontext: forall D, context_DefEq D nil nil
| Factor_Eqcontext_tm: forall G1 G2 D A A' x,
context_DefEq D G1 G2 ->
DefEq G1 D A A' a_Star ->
DefEq G2 D A A' a_Star ->
context_DefEq D ([(x, Tm A)] ++ G1) ([(x, Tm A')] ++ G2)
| Factor_Eqcontext_co: forall D G1 G2 Phi1 Phi2 c,
context_DefEq D G1 G2 ->
Iso G1 D Phi1 Phi2 ->
Iso G2 D Phi1 Phi2 ->
context_DefEq D ([(c, Co Phi1)] ++ G1) ([(c, Co Phi2)] ++ G2).
Hint Constructors context_DefEq.

Lemma DefEqIso_regularity :
(forall G0 a A, Typing G0 a A -> True ) /\
(forall G0 phi,  PropWff G0 phi -> True ) /\
(forall G D p1 p2, Iso G D p1 p2 ->
PropWff G p1 /\ PropWff G p2) /\
(forall G D A B T,   DefEq G D A B T ->
Typing G A T /\ Typing G B T) /\
(forall G0, Ctx G0 -> True).