Section GHist.
(* Ghost histories in the style of Nanevsky *)
Context {hist_el : Type}.
Notation hist_part := (nat -> option hist_el).
Definition hist_sub sh (h : hist_part) hr := sh <> Share.bot /\ if eq_dec sh Tsh then h = hr
else map_incl h hr.
Definition ghost_hist (sh : share) (h : hist_part) g :=
own(RA := ref_PCM map_disj_PCM) g (Some (sh, h), None) NoneP.
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.
Definition ghost_ref l g := EX hr : hist_part, !!(hist_list hr l) &&
own(RA := ref_PCM map_disj_PCM) g (None, Some hr) NoneP.
Definition ghost_hist_ref sh (h r : hist_part) g :=
own(RA := ref_PCM map_disj_PCM) g (Some (sh, h), Some r) NoneP.
Definition newer (l : hist_part) t := forall t', l t' <> None -> (t' < t)%nat.
(* We want to be able to remove irrelevant operations from a history, leading to a slightly weaker correspondence between history and list of operations. *)
Inductive hist_list' : hist_part -> list hist_el -> Prop :=
| hist_list'_nil : hist_list' empty_map []
| hist_list'_snoc : forall h l t e (Hlast : newer h t) (Hrest : hist_list' h l),
hist_list' (map_upd h t e) (l ++ [e]).
Hint Resolve hist_list'_nil.

Lemma ghost_hist_init : @valid (ref_PCM (@map_disj_PCM nat hist_el)) (Some (Tsh, empty_map), Some empty_map).