Module Type ext_invert_sig.
Include ext_subst_sig.
(* ---------- inversion lemmas ---------------- *)
(* ---------- context conversion -------------- *)
(* Terms still type check even after varying the context *)
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).
(* ---------------- regularity lemmas ------------------- *)
(* ------- smart constructors --------- *)

Axiom E_CPi2 :  ∀ (L : atoms) (G : context) (phi : constraint) (B : tm),
(∀ c : atom, c `notin` L → Typing ([(c, Co phi)] ++ G) (open_tm_wrt_co B (g_Var_f c)) a_Star) ->
Typing G (a_CPi phi B) a_Star.