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_sub_Tsh : forall h h', hist_sub Tsh h h' <-> (h = h').