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_PiCong2 :  ∀ (L : atoms) (G : context) (D : available_props) rho (A1 B1 A2 B2 : tm),
DefEq G D A1 A2 a_Star
→ (∀ x : atom,
x `notin` L
→ DefEq ([(x, Tm A1)] ++ G) D (open_tm_wrt_tm B1 (a_Var_f x))
(open_tm_wrt_tm B2 (a_Var_f x)) a_Star)
→ DefEq G D (a_Pi rho A1 B1) (a_Pi rho A2 B2) a_Star.