Module Type fc_subst_sig.
(* Axiom context_fv_mutual : (forall G (a : tm) A (H: AnnTyping G a A), fv_tm_tm_tm a [<=] dom G /\ fv_co_co_tm a [<=] dom G /\ fv_tm_tm_tm A [<=] dom G /\ fv_co_co_tm A [<=] dom G) /\ (forall G phi (H : AnnPropWff G phi), fv_tm_tm_constraint phi [<=] dom G /\ fv_co_co_constraint phi [<=] dom G) /\ (forall G D g p1 p2 (H : AnnIso G D g p1 p2), fv_tm_tm_co         g  [<=] dom G /\ fv_co_co_co         g  [<=] dom G /\ fv_tm_tm_constraint p1 [<=] dom G /\ fv_co_co_constraint p1 [<=] dom G /\ fv_tm_tm_constraint p2 [<=] dom G /\ fv_co_co_constraint p2 [<=] dom G) /\ (forall G D g A B (H : AnnDefEq G D g A B), fv_tm_tm_co g [<=] dom G /\ fv_co_co_co g [<=] dom G /\ fv_tm_tm_tm A [<=] dom G /\ fv_co_co_tm A [<=] dom G /\ fv_tm_tm_tm B [<=] dom G /\ fv_co_co_tm B [<=] dom G) /\ (forall G (H : AnnCtx G), (forall x A, binds x (Tm A)   G -> fv_tm_tm_tm         A   [<=] dom G /\ fv_co_co_tm         A   [<=] dom G) /\ (forall c phi, binds c (Co phi) G -> fv_tm_tm_constraint phi [<=] dom G /\ fv_co_co_constraint phi [<=] dom G)).  Definition AnnTyping_context_fv  := @first  _ _ _ _ _ context_fv_mutual. Definition AnnPropWff_context_fv := @second _ _ _ _ _ context_fv_mutual. Definition AnnIso_context_fv     := @third  _ _ _ _ _ context_fv_mutual. Definition AnnDefEq_context_fv   := @fourth _ _ _ _ _ context_fv_mutual. Definition AnnCtx_context_fv     := @fifth  _ _ _ _ _ context_fv_mutual. *)
(* -----  exists forms of the binding constructors ----------- *)

Axiom An_CPiCong_inversion :  ∀ (G : context) D (g1 g3 : co) (A1 A2 : tm),
AnnDefEq G D (g_CPiCong g1 g3) A1 A2 ->
exists phi1 phi2 B1 B2 B3,
A1 = (a_CPi phi1 B1) /\
A2 = (a_CPi phi2 B3) /\
AnnIso G D g1 phi1 phi2 /\
AnnTyping G (a_CPi phi1 B1) a_Star /\
AnnTyping G (a_CPi phi2 B3) a_Star /\
AnnTyping G (a_CPi phi1 B2) a_Star /\
(forall c, c `notin` dom G (* \u D \u fv_co_co_tm B2 \u fv_co_co_tm B1
\u fv_co_co_co g3 \u
fv_co_co_co g1 \u fv_co_co_tm B3 *) →
(AnnDefEq ([(c, Co phi1)] ++ G) D (open_co_wrt_co g3 (g_Var_f c))
(open_tm_wrt_co B1 (g_Var_f c)) (open_tm_wrt_co B2 (g_Var_f c))) /\
(open_tm_wrt_co B3 (g_Var_f c) = open_tm_wrt_co B2 (g_Cast (g_Var_f c) (g_Sym g1)))).