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_PiCong_exists : forall x1 x2 (G:context) D rho
(g1 g2 : co) (A1 B1 A2 B3 B2 : tm),
x1 `notin` (dom G \u fv_tm_tm_tm B1 \u fv_tm_tm_tm B2 \u  fv_tm_tm_co g2)
-> x2 `notin` (dom G \u fv_tm_tm_tm B2 \u fv_tm_tm_tm B3 \u  fv_tm_tm_co g1)
->  AnnDefEq G D g1 A1 A2
→ AnnDefEq ([(x1, Tm A1)] ++ G) D (open_co_wrt_tm g2 (a_Var_f x1))
(open_tm_wrt_tm B1 (a_Var_f x1)) (open_tm_wrt_tm B2 (a_Var_f x1))
→ (open_tm_wrt_tm B3 (a_Var_f x2) =
open_tm_wrt_tm B2 (a_Conv (a_Var_f x2) (g_Sym g1)))
→ AnnTyping G (a_Pi rho A1 B1) a_Star
→ AnnTyping G (a_Pi rho A2 B3) a_Star
→ AnnTyping G (a_Pi rho A1 B2) a_Star
→ AnnDefEq G D (g_PiCong rho g1 g2) (a_Pi rho A1 B1) (a_Pi rho A2 B3).