Definition hist_incl (h : hist_part) l := forall t e, h t = Some e -> nth_error l t = Some e.
Definition hist_list (h : hist_part) l := forall t e, h t = Some e <-> nth_error l t = Some e.

Lemma hist_list_snoc : forall h l e, hist_list h l ->
hist_list (map_upd h (length l) e) (l ++ [e]).