1(*
2 * Copyright 2020, Data61, CSIRO (ABN 41 687 119 230)
3 *
4 * SPDX-License-Identifier: BSD-2-Clause
5 *)
6
7(* Author: Andrew Boyton, 2012
8   Maintainers: Gerwin Klein <kleing at cse.unsw.edu.au>
9                Rafal Kolanski <rafal.kolanski at nicta.com.au>
10*)
11
12chapter "Defining some separation logic maps-to predicates on top of the instantiation."
13
14theory Separation_D
15imports Abstract_Separation_D
16begin
17
18type_synonym sep_pred = "sep_state \<Rightarrow> bool"
19
20definition
21  state_sep_projection :: "cdl_state \<Rightarrow> sep_state"
22where
23  "state_sep_projection \<equiv> \<lambda>s. SepState (cdl_objects s) (cdl_ghost_state s)"
24
25(* This turns a separation logic predicate into a predicate on the capDL state. *)
26abbreviation
27  lift' :: "(sep_state \<Rightarrow> 'a) \<Rightarrow> cdl_state \<Rightarrow> 'a" ("<_>")
28where
29  "<P> s \<equiv> P (state_sep_projection s)"
30
31(* The generalisation of the maps to operator for separation logic. *)
32definition
33  sep_map_general :: "cdl_object_id \<Rightarrow> cdl_object \<Rightarrow> cdl_components \<Rightarrow> sep_pred"
34where
35  "sep_map_general p obj gs \<equiv> \<lambda>s. sep_heap s = [p \<mapsto> obj] \<and> sep_ghost_state s p = gs"
36
37(* Alternate definition without the [p \<mapsto> obj] notation. *)
38lemma sep_map_general_def2:
39  "sep_map_general p obj gs s =
40   (dom (sep_heap s) = {p} \<and> ko_at obj p (sep_heap s) \<and> sep_ghost_state s p = gs)"
41  apply (clarsimp simp: sep_map_general_def object_at_def)
42  apply (rule)
43   apply clarsimp
44  apply (clarsimp simp: fun_upd_def)
45  apply (rule ext)
46  apply (fastforce simp: dom_def split:if_split)
47  done
48
49(* There is an object there. *)
50definition
51  sep_map_i :: "cdl_object_id \<Rightarrow> cdl_object \<Rightarrow> sep_pred" ("_ \<mapsto>i _" [76,71] 76)
52where
53  "p \<mapsto>i obj \<equiv> sep_map_general p obj UNIV"
54
55(* The fields are there (and there are no caps). *)
56definition
57  sep_map_f :: "cdl_object_id \<Rightarrow> cdl_object \<Rightarrow> sep_pred" ("_ \<mapsto>f _" [76,71] 76)
58where
59  "p \<mapsto>f obj \<equiv> sep_map_general p (update_slots Map.empty obj) {None}"
60
61(* There is that cap there. *)
62definition
63  sep_map_c :: "cdl_cap_ref \<Rightarrow> cdl_cap \<Rightarrow> sep_pred" ("_ \<mapsto>c _" [76,71] 76)
64where
65  "p \<mapsto>c cap \<equiv> \<lambda>s. let (obj_id, slot) = p; heap = sep_heap s in
66  \<exists>obj. sep_map_general obj_id obj {Some slot} s \<and> object_slots obj = [slot \<mapsto> cap]"
67
68definition
69  sep_any :: "('a \<Rightarrow> 'b \<Rightarrow> sep_pred) \<Rightarrow> ('a \<Rightarrow> sep_pred)" where
70  "sep_any m \<equiv> (\<lambda>p s. \<exists>v. (m p v) s)"
71
72abbreviation "sep_any_map_i \<equiv> sep_any sep_map_i"
73notation sep_any_map_i ("_ \<mapsto>i -" 76)
74
75abbreviation "sep_any_map_c \<equiv> sep_any sep_map_c"
76notation sep_any_map_c ("_ \<mapsto>c -" 76)
77
78end
79