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Proof_SI.thy
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Proof_SI.thy
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(*
* Copyright 2020, Data61, CSIRO (ABN 41 687 119 230)
*
* SPDX-License-Identifier: GPL-2.0-only
*)
theory Proof_SI
imports
CreateObjects_SI
CreateIRQCaps_SI
DuplicateCaps_SI
InitVSpace_SI
InitTCB_SI
InitCSpace_SI
InitIRQ_SI
StartThreads_SI
begin
lemma parse_bootinfo_sep:
"\<lbrace>\<guillemotleft>((\<And>* (cptr, cap) \<in> set (zip [ustart .e. uend - 1] untyped_caps). (si_cnode_id, unat cptr) \<mapsto>c cap) \<and>*
(\<And>* cptr \<in> set [fstart .e. fend - 1]. (si_cnode_id, unat cptr) \<mapsto>c NullCap) \<and>*
(\<And>* obj_id\<in>(\<Union>cap\<in>set untyped_caps. cap_free_ids cap). obj_id \<mapsto>o Untyped) \<and>* R)
and K (bi_untypes bootinfo = (ustart, uend) \<and>
bi_free_slots bootinfo = (fstart, fend) \<and>
unat ustart < 2 ^ si_cnode_size \<and>
unat (uend - 1) < 2 ^ si_cnode_size \<and>
unat fstart < 2 ^ si_cnode_size \<and>
unat (fend - 1) < 2 ^ si_cnode_size \<and>
uend \<noteq> 0 \<and>
fend \<noteq> 0 \<and>
list_all is_full_untyped_cap untyped_caps \<and>
length untyped_caps = unat uend - unat ustart) \<guillemotright>\<rbrace>
parse_bootinfo bootinfo
\<lbrace>\<lambda>rv.
\<guillemotleft>((\<And>* (cptr, cap) \<in> set (zip (fst rv) untyped_caps). (si_cnode_id, unat cptr) \<mapsto>c cap) \<and>*
(\<And>* cptr \<in> set (snd rv). (si_cnode_id, unat cptr) \<mapsto>c NullCap) \<and>*
(\<And>* obj_id\<in>(\<Union>cap\<in>set untyped_caps. cap_free_ids cap). obj_id \<mapsto>o Untyped) \<and>* R) and
K (rv = ([fst (bi_untypes bootinfo) .e. snd (bi_untypes bootinfo) - 1],
[fst (bi_free_slots bootinfo) .e. snd (bi_free_slots bootinfo) - 1]))\<guillemotright> \<rbrace>"
apply (clarsimp simp: parse_bootinfo_def)
apply (cases bootinfo, clarsimp)
apply wp
apply (clarsimp simp: zip_map1 comp_def split_beta')
done
(* This isn't actually all the combinations, but enough of them for what I needed. *)
lemma object_types_distinct:
"tcb_at x s \<Longrightarrow> \<not> cnode_at x s"
"table_at x s \<Longrightarrow> \<not> cnode_at x s"
"table_at x s \<Longrightarrow> \<not> tcb_at x s"
"capless_at x s \<Longrightarrow> \<not> cnode_at x s"
"capless_at x s \<Longrightarrow> \<not> tcb_at x s"
"capless_at x s \<Longrightarrow> \<not> table_at x s"
"capless_at x s \<Longrightarrow> \<not> pt_at x s"
"capless_at x s \<Longrightarrow> \<not> pd_at x s"
"capless_at x s \<Longrightarrow> \<not> asidpool_at x s"
by (clarsimp simp: object_at_def is_tcb_def is_cnode_def is_pd_def is_pt_def
is_ep_def is_ntfn_def is_asidpool_def is_frame_def
is_untyped_def | rule conjI |
clarsimp split: cdl_object.splits)+
lemma real_objects_some_type:
"well_formed spec \<Longrightarrow>
{obj_id. real_object_at obj_id spec \<and>
\<not> cnode_at obj_id spec \<and>
\<not> tcb_at obj_id spec \<and>
\<not> pt_at obj_id spec \<and>
\<not> pd_at obj_id spec \<and>
\<not> untyped_at obj_id spec \<and>
\<not> ep_at obj_id spec \<and>
\<not> ntfn_at obj_id spec \<and>
\<not> frame_at obj_id spec} = {}"
apply (clarsimp simp: object_at_def is_tcb_def is_cnode_def is_pd_def is_pt_def
is_ep_def is_ntfn_def is_asidpool_def is_frame_def is_untyped_def)
apply (clarsimp split: cdl_object.splits)
apply (drule_tac obj_id=x in well_formed_asidpool_at)
apply (clarsimp simp: real_object_at_def object_at_def is_asidpool_def irq_nodes_def is_irq_node_def
split: cdl_object.splits)
by metis
lemma capdl_objects_by_parts:
"well_formed spec \<Longrightarrow>
(sep_map_set_conj P {obj_id. real_object_at obj_id spec}) =
(sep_map_set_conj P {obj_id. cnode_at obj_id spec} \<and>*
sep_map_set_conj P {obj_id. tcb_at obj_id spec} \<and>*
sep_map_set_conj P {obj_id. table_at obj_id spec} \<and>*
sep_map_set_conj P {obj_id. capless_at obj_id spec})"
apply (rule sym)
apply (subst (5) sep_map_set_conj_restrict [where t = "(\<lambda>obj. cnode_at obj spec)"], simp)
apply (subst (6) sep_map_set_conj_restrict [where t = "(\<lambda>obj. tcb_at obj spec)"], simp)
apply (subst (7) sep_map_set_conj_restrict [where t = "(\<lambda>obj. table_at obj spec)"], simp)
apply (subst (8) sep_map_set_conj_restrict [where t = "(\<lambda>obj. capless_at obj spec)"], simp)
apply (clarsimp simp: object_types_distinct real_object_not_irq_node real_objects_some_type
cong: rev_conj_cong)
done
lemma objects_empty_by_parts:
"well_formed spec \<Longrightarrow>
(objects_empty spec t {obj_id. real_object_at obj_id spec}) =
(objects_empty spec t {obj_id. cnode_at obj_id spec} \<and>*
objects_empty spec t {obj_id. tcb_at obj_id spec} \<and>*
objects_empty spec t {obj_id. table_at obj_id spec} \<and>*
objects_empty spec t {obj_id. capless_at obj_id spec})"
by (clarsimp simp: objects_empty_def capdl_objects_by_parts)
lemma objects_initialised_by_parts:
"well_formed spec \<Longrightarrow>
(objects_initialised spec t {obj_id. real_object_at obj_id spec}) =
(objects_initialised spec t {obj_id. cnode_at obj_id spec} \<and>*
objects_initialised spec t {obj_id. tcb_at obj_id spec} \<and>*
objects_initialised spec t {obj_id. table_at obj_id spec} \<and>*
objects_initialised spec t {obj_id. capless_at obj_id spec})"
by (clarsimp simp: objects_initialised_def capdl_objects_by_parts)
lemma object_empty_object_initialised_capless:
"capless_at obj_id spec \<Longrightarrow>
object_empty spec t obj_id = object_initialised spec t obj_id"
apply (rule ext)
apply (clarsimp simp: object_empty_def object_initialised_def)
apply (clarsimp simp: object_initialised_general_def object_default_state_def2)
apply (fastforce simp: object_at_def update_slots_def
object_default_state_def2 spec2s_def
is_ep_def is_ntfn_def is_asidpool_def
is_frame_def is_untyped_def cdl_frame.splits
split: cdl_object.splits)
done
lemma objects_empty_objects_initialised_capless:
"objects_empty spec t {obj_id. capless_at obj_id spec} =
objects_initialised spec t {obj_id. capless_at obj_id spec}"
apply (clarsimp simp: objects_empty_def objects_initialised_def)
apply (rule sep.prod.cong, simp)
apply (clarsimp simp: object_empty_object_initialised_capless)
done
lemma valid_case_prod':
"(\<And>x y. \<lbrace>P x y\<rbrace> f x y \<lbrace>Q\<rbrace>) \<Longrightarrow> \<lbrace>P (fst v) (snd v)\<rbrace> case v of (x, y) \<Rightarrow> f x y \<lbrace>Q\<rbrace>"
by (clarsimp split: prod.splits)
lemma le_list_all:
"\<lbrakk>unat start < 2 ^ si_cnode_size; unat (end - 1) < 2 ^ si_cnode_size\<rbrakk>
\<Longrightarrow> list_all (\<lambda>n. (n::32 word) < 2 ^ si_cnode_size) [start .e. end - 1]"
apply (clarsimp simp: list_all_iff)
apply (subst word_arith_power_alt)
apply simp
by (metis (no_types) dual_order.strict_trans2 unat_less_2_si_cnode_size word_of_int_numeral word_of_int_power_hom)
lemma list_all_drop:
"list_all P xs \<Longrightarrow> list_all P (drop n xs)"
by (fastforce simp: list_all_iff dest: in_set_dropD)
lemma dom_map_of_zip':
"length xs \<le> length ys \<Longrightarrow> dom (map_of (zip xs ys)) = set xs"
apply (subst zip_take_length [symmetric])
apply (subst dom_map_of_zip, simp+)
done
(* FIXME: MOVE (Lib) *)
lemma in_zip_map: "p \<in> set xs \<Longrightarrow> length xs \<le> length ys \<Longrightarrow> map_of (zip xs ys) p \<noteq> None"
using dom_map_of_zip' by blast
lemma map_of_list_allE:
"map_of (zip ys xs) p = Some v \<Longrightarrow> distinct ys \<Longrightarrow> list_all P xs \<Longrightarrow> P v"
apply (induct ys arbitrary: xs; clarsimp)
by (meson in_set_zipE list_all_spec map_of_SomeD)
lemma card_eq_lengthI:
"set xs = ys \<Longrightarrow> distinct xs \<Longrightarrow> length xs = card ys"
by (induct xs arbitrary: ys; fastforce)
lemma length_filter_card:
"\<lbrakk>s_list = sorted_list_of_set s; finite s\<rbrakk>
\<Longrightarrow> length (filter P s_list) = card {x \<in> s. P x}"
by (fastforce intro: card_eq_lengthI)
lemma sys_init_explicit:
"\<lbrakk>well_formed spec;
set obj_ids = dom (cdl_objects spec); distinct obj_ids;
real_ids = [obj_id \<leftarrow> obj_ids. real_object_at obj_id spec];
length obj_ids + length [obj\<leftarrow>obj_ids. cnode_or_tcb_at obj spec] +
card (\<Union>(set ` get_frame_caps spec ` {obj. pd_at obj spec})) \<le> unat fend - unat fstart;
length untyped_caps = unat uend - unat ustart;
distinct_sets (map cap_free_ids untyped_caps);
list_all is_full_untyped_cap untyped_caps;
list_all well_formed_untyped_cap untyped_caps;
list_all (\<lambda>c. \<not> is_device_cap c) untyped_caps;
bi_untypes bootinfo = (ustart, uend);
bi_free_slots bootinfo = (fstart, fend);
unat ustart < 2 ^ si_cnode_size;
unat (uend - 1) < 2 ^ si_cnode_size;
unat fstart < 2 ^ si_cnode_size;
unat (fend - 1) < 2 ^ si_cnode_size;
uend \<noteq> 0; fend \<noteq> 0;
[ustart .e. uend - 1] = untyped_cptrs;
[fstart .e. fend - 1] = free_cptrs;
(map_of (zip [obj\<leftarrow>obj_ids . cnode_or_tcb_at obj spec] (drop (length obj_ids) [fstart .e. fend - 1]))) = dup_caps
\<rbrakk> \<Longrightarrow>
\<lbrace>\<guillemotleft>(\<And>* (cptr, cap) \<in> set (zip untyped_cptrs untyped_caps). (si_cnode_id, unat cptr) \<mapsto>c cap) \<and>*
(\<And>* cptr \<in> set free_cptrs. (si_cnode_id, unat cptr) \<mapsto>c NullCap) \<and>*
(\<And>* obj_id\<in>(\<Union>cap\<in>set untyped_caps. cap_free_ids cap). obj_id \<mapsto>o Untyped) \<and>*
si_objects \<and>*
si_irq_nodes spec \<and>*
(SETSEPCONJ pd_id | pd_at pd_id spec.
frame_duplicates_empty (make_frame_cap_map obj_ids (drop (length obj_ids) free_cptrs) spec)
pd_id spec) \<and>*
R\<guillemotright>\<rbrace>
init_system spec bootinfo obj_ids
\<lbrace>\<lambda>_ s. \<exists>t.
\<guillemotleft>objects_initialised spec t {obj_id. real_object_at obj_id spec} \<and>*
irqs_initialised spec t (used_irqs spec) \<and>*
(\<And>* cptr\<in>set (take (card (dom (cdl_objects spec))) free_cptrs). (si_cnode_id, unat cptr) \<mapsto>c NullCap) \<and>*
si_caps_at t dup_caps spec False {obj_id. cnode_or_tcb_at obj_id spec} \<and>*
si_objects \<and>*
si_objects_extra_caps (dom (cdl_objects spec))
(free_cptrs :: 32 word list)
(untyped_cptrs :: 32 word list) spec \<and>*
(SETSEPCONJ pd_id | pd_at pd_id spec.
frame_duplicates_copied (make_frame_cap_map obj_ids (drop (length obj_ids) free_cptrs) spec)
pd_id spec t) \<and>*
R\<guillemotright> s \<and>
inj_on t (dom (cdl_objects spec)) \<and> dom t = set obj_ids\<rbrace>"
supply [[unify_search_bound = 1000]]
apply clarsimp
apply (frule (1) le_list_all [where start = ustart])
apply (frule (1) le_list_all [where start = fstart])
apply (frule well_formed_objects_card)
apply (insert distinct_card [symmetric, where xs ="[obj\<leftarrow>obj_ids . cnode_or_tcb_at obj spec]"], simp)
apply (frule distinct_card [symmetric])
apply (clarsimp simp: init_system_def, wp valid_case_prod')
apply (rule hoare_ex_wp, rename_tac t, rule_tac t=t in start_threads_sep [sep_wandise], simp)
apply (rule hoare_ex_wp, rename_tac t, rule_tac t=t and
free_cptrs="[fstart .e. fend - 1]" in init_cspace_sep [sep_wandise])
apply (rule hoare_ex_wp, rename_tac t, rule_tac t=t in init_tcbs_sep [sep_wandise])
apply (rule hoare_ex_wp, rename_tac t, rule_tac t=t in init_vspace_sep [sep_wandise])
apply (rule hoare_ex_wp, rename_tac t, rule_tac t=t in init_pd_asids_sep [sep_wandise])
apply (rule hoare_ex_wp, rename_tac t, rule_tac t=t and dev=False in init_irqs_sep [sep_wandise])
apply (rule hoare_ex_wp, rename_tac t, rule_tac t=t and dev=False and
untyped_cptrs = "[ustart .e. uend - 1]" and
free_cptrs_orig = "[fstart .e. fend - 1]" in duplicate_caps_sep [sep_wandise])
apply (rule create_irq_caps_sep [where dev = False,sep_wandise,
where free_cptrs_orig = "[fstart .e. fend - 1]"
and untyped_cptrs = "[ustart .e. uend - 1]"
and orig_caps = "map_of (zip [obj\<leftarrow>obj_ids. real_object_at obj spec]
[fstart .e. fend - 1])"
and spec = spec])
apply (wp sep_wp: create_objects_sep [where untyped_caps = untyped_caps and dev = False])
apply (wp sep_wp: parse_bootinfo_sep [where fstart = fstart
and fend = fend
and ustart = ustart
and uend = uend
and untyped_caps = untyped_caps])
apply (subst objects_initialised_by_parts, assumption)
apply (subst objects_empty_by_parts, assumption)+
apply (subst objects_empty_objects_initialised_capless)+
apply (clarsimp simp: linorder_not_le)
apply (intro conjI allI impI pred_andI | sep_cancel+)+
apply fastforce
apply (clarsimp simp: less_diff_conv)
apply (rule list_all_drop, erule (1) le_list_all)
apply clarsimp
apply (subgoal_tac "map_of (zip (filter (\<lambda>obj. real_object_at obj spec) obj_ids) free_cptrs)
p \<noteq> None")
apply clarsimp
apply (erule map_of_list_allE)
apply (fastforce intro!: List.distinct_filter)
apply (fastforce intro!: le_list_all)
apply (rule in_zip_map)
apply clarsimp
apply (fastforce dest!: real_object_not_irq_node(3))
apply (insert length_filter_le[where xs = obj_ids and P="\<lambda>obj. real_object_at obj spec"],
fastforce)[1]
apply (erule (1) le_list_all)
apply (rule list_all_drop, erule (1) le_list_all)
apply simp
apply (subst dom_map_of_zip')
apply (insert length_filter_le [where xs = obj_ids and P="\<lambda>obj. real_object_at obj spec"],
fastforce)[1]
apply simp
apply (erule (1) le_list_all)
done
(**************************************************
* The top level lemma for system initialisation. *
**************************************************)
(* FIXME, make the acquiring of the object_ids part of sys_init, not a parameter. *)
lemma sys_init:
"\<lbrakk>well_formed spec; obj_ids = sorted_list_of_set (dom (cdl_objects spec))\<rbrakk> \<Longrightarrow>
\<lbrace>\<guillemotleft>valid_boot_info bootinfo spec \<and>* R\<guillemotright>\<rbrace>
init_system spec bootinfo obj_ids
\<lbrace>\<lambda>_ s. \<exists>t.
\<guillemotleft>objects_initialised spec t {obj_id. real_object_at obj_id spec} \<and>*
irqs_initialised spec t (used_irqs spec) \<and>*
si_final_objects spec t \<and>*
(EXS map. (SETSEPCONJ pd_id | pd_at pd_id spec. frame_duplicates_copied map pd_id spec t)) \<and>*
R\<guillemotright> s \<and>
inj_on t (set obj_ids) \<and> dom t = set obj_ids\<rbrace>"
apply (insert distinct_card [where xs = "[obj\<leftarrow>obj_ids . cnode_or_tcb_at obj spec]"], simp)
apply (clarsimp simp: valid_boot_info_def si_final_objects_def
sep_conj_exists sep_conj_assoc)
apply (subst ex_conj_increase)+
apply (rule hoare_ex_pre)+
apply (rule hoare_grab_asm)+
apply (rule hoare_chain)
apply (rule sys_init_explicit[where obj_ids="sorted_list_of_set (dom (cdl_objects spec))" and R=R],
(assumption|simp add: unat_less_2_si_cnode_size' length_filter_card)+)
apply sep_solve
apply clarsimp
apply (rule_tac x=t in exI)
apply (clarsimp)
apply (clarsimp simp: si_objects_extra_caps_def si_caps_at_def
sep_conj_exists sep_conj_assoc)
apply (rule_tac x="(map_of (zip [obj \<leftarrow> obj_ids. cnode_or_tcb_at obj spec]
(drop (length obj_ids) [fstart .e. fend - 1])))" in exI)
apply (rule_tac x="[x .e. xa - 1]" in exI)
apply (rule_tac x="[fstart .e. fend - 1]" in exI)
apply (rule_tac x=untyped_capsa in exI)
apply (rule_tac x=all_available_ids in exI)
apply (rule_tac x="make_frame_cap_map obj_ids (drop (card (dom (cdl_objects spec)))
[fstart .e. fend - 1]) spec" in exI)
apply (clarsimp simp: sep_conj_ac)
done
definition injective :: "('a \<Rightarrow> 'b option) \<Rightarrow> bool"
where "injective f \<equiv> inj_on f (dom f)"
lemma sys_init_paper:
"\<lbrakk>well_formed spec; obj_ids = sorted_list_of_set (dom (cdl_objects spec))\<rbrakk> \<Longrightarrow>
\<lbrace>\<guillemotleft>valid_boot_info bootinfo spec \<and>* R\<guillemotright>\<rbrace>
init_system spec bootinfo obj_ids
\<lbrace>\<lambda>_ s. \<exists>\<phi>.
\<guillemotleft>objects_initialised spec \<phi> {obj_id. real_object_at obj_id spec} \<and>*
irqs_initialised spec \<phi> (used_irqs spec) \<and>*
si_final_objects spec \<phi> \<and>*
(EXS map. (SETSEPCONJ pd_id | pd_at pd_id spec. frame_duplicates_copied map pd_id spec \<phi>)) \<and>*
R\<guillemotright> s \<and>
injective \<phi> \<and> dom \<phi> = set obj_ids\<rbrace>"
apply (rule hoare_strengthen_post)
apply (fact sys_init)
apply (fastforce simp: injective_def)
done
end