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 ----------- *)
(* -----  inversion lemmas for some typing judgments (with maximal co-finite quantification) ----------- *)
(* -------------- name swapping ------------------------ *)

Axiom AnnTyping_co_swap : forall c c0 phi G a A,
c `notin` fv_co_co_tm A ->
c `notin` fv_co_co_tm a ->
c0 `notin` dom G \u {{ c }} ->
AnnTyping ([(c, Co phi)] ++ G) (open_tm_wrt_co a (g_Var_f c))
(open_tm_wrt_co A (g_Var_f c)) ->
AnnTyping ([(c0, Co phi)] ++ G) (open_tm_wrt_co a (g_Var_f c0))
(open_tm_wrt_co A (g_Var_f c0)).