(* --------------------------------------------------- *)
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 context_tm_binding_defeq: forall D (G1 G2: context) A x,
Ctx G1 -> Ctx G2 -> context_DefEq D G1 G2 ->
binds x (Tm A) G1 -> exists A', (binds x (Tm A') G2) /\ DefEq G2 D A A' a_Star.