(* Idem: *)

(* Specifications: *)
Lemma not_exd_A_1_nil:
forall (m:fmap)(k:dim)(x:dart),
inv_qhmap m -> ~ exd m x -> A_1 m k x = nil.
Lemma exd_A_1_exd :
forall (m : fmap) (k : dim) (x : dart),
inv_qhmap m -> exd m (A_1 m k x) -> exd m x.
Lemma exd_cA_1_exd :
forall (m:fmap)(k:dim)(z:dart),
inv_qhmap m ->
exd m (cA_1 m k z) -> exd m z.
Lemma F_1_nil : forall m:fmap,
inv_qhmap m -> F_1 m nil = nil.
Lemma predf_exd_F_1 : forall (m:fmap)(x:dart),
inv_qhmap m -> predf m x -> exd m (F_1 m x).
Lemma succf_F: forall (m:fmap)(x:dart),
inv_qhmap m -> (succf m x <-> F m x <> nil).
Lemma predf_F_1: forall (m:fmap)(y:dart),
inv_qhmap m -> (predf m y <-> F_1 m y <> nil).
Lemma F_1_F : forall (m:fmap)(x:dart),
inv_qhmap m -> exd m x -> exd m (F m x) ->
F_1 m (F m x) = x.
Lemma inj_F_succf :
forall m:fmap, inv_qhmap m ->
inj_dart (succf m) (F m).