theory Enrich
imports Main Flask Static Init_prop Valid_prop Tainted_prop Delete_prop Co2sobj_prop S2ss_prop S2ss_prop2
Temp
begin
(* enriched objects, closely related to static objects, so are only 3 kinds *)
datatype t_enrich_obj =
E_proc "t_process" "t_msgq" "t_msgq"
| E_file_link "t_file"
| E_file "t_file" "nat"
(*
| E_fd "t_process" "t_fd"
| E_inum "nat" *)
| E_msgq "t_msgq"
(*
| E_msg "t_msgq" "t_msg"
*)
(* objects that need dynamic indexing, all nature-numbers *)
datatype t_index_obj =
I_proc "t_process"
| I_file "t_file"
| I_fd "t_process" "t_fd"
| I_inum "nat"
| I_msgq "t_msgq"
| I_msg "t_msgq" "t_msg"
context tainting_s begin
fun no_del_event:: "t_event list \<Rightarrow> bool"
where
"no_del_event [] = True"
| "no_del_event (Kill p p' # \<tau>) = False"
| "no_del_event (Exit p # s) = False"
| "no_del_event (CloseFd p fd # \<tau>) = False"
| "no_del_event (UnLink p f # \<tau>) = False"
| "no_del_event (Rmdir p f # \<tau>) = False"
(*
| "no_del_event (Rename p f f' # \<tau>) = False"
*)
| "no_del_event (RemoveMsgq p q # \<tau>) = False"
(*
| "no_del_event (RecvMsg p q m # \<tau>) = False"
*)
| "no_del_event (_ # \<tau>) = no_del_event \<tau>"
fun all_inums :: "t_state \<Rightarrow> t_inode_num set"
where
"all_inums [] = current_inode_nums []"
| "all_inums (Open p f flags fd opt # s) = (
case opt of
None \<Rightarrow> all_inums s
| Some i \<Rightarrow> all_inums s \<union> {i} )"
| "all_inums (Mkdir p f i # s) = (all_inums s \<union> {i})"
| "all_inums (CreateSock p af st fd i # s) = (all_inums s \<union> {i})"
| "all_inums (Accept p fd addr lport fd' i # s) = (all_inums s \<union> {i})"
| "all_inums (_ # s) = all_inums s"
fun all_fds :: "t_state \<Rightarrow> t_process \<Rightarrow> t_fd set"
where
"all_fds [] = init_fds_of_proc"
| "all_fds (Open p f flags fd ipt # s) = (all_fds s) (p := all_fds s p \<union> {fd})"
| "all_fds (CreateSock p sf st fd i # s) = (all_fds s) (p := all_fds s p \<union> {fd})"
| "all_fds (Accept p fd' raddr port fd i # s) = (all_fds s) (p := all_fds s p \<union> {fd})"
| "all_fds (Clone p p' fds # s) = (all_fds s) (p' := fds)"
| "all_fds (_ # s) = all_fds s"
fun all_msgqs:: "t_state \<Rightarrow> t_msgq set"
where
"all_msgqs [] = {}"
| "all_msgqs (CreateMsgq p q # s) = all_msgqs s \<union> {q}"
| "all_msgqs (e # s) = all_msgqs s"
fun all_msgs:: "t_state \<Rightarrow> t_msgq \<Rightarrow> t_msg set"
where
"all_msgs [] q = {}"
| "all_msgs (CreateMsgq p q # s) q' = (if q' = q then {} else all_msgs s q')"
| "all_msgs (SendMsg p q m # s) q' = (if q' = q then all_msgs s q \<union> {m} else all_msgs s q')"
| "all_msgs (_ # s) q' = all_msgs s q'"
fun all_files:: "t_state \<Rightarrow> t_file set"
where
"all_files [] = init_files "
| "all_files (Open p f flags fd opt # s) = (if opt = None then all_files s else (all_files s \<union> {f}))"
| "all_files (Mkdir p f inum # s) = all_files s \<union> {f}"
| "all_files (LinkHard p f f' # s) = all_files s \<union> {f'}"
| "all_files (e # s) = all_files s"
(*
fun notin_all:: "t_state \<Rightarrow> t_enrich_obj \<Rightarrow> bool"
where
"notin_all s (E_proc p) = (p \<notin> all_procs s)"
| "notin_all s (E_file f) = (f \<notin> all_files s \<and> (\<exists> pf. parent f = Some pf \<and> is_dir s pf))"
| "notin_all s (E_fd p fd) = (fd \<notin> all_fds s p)"
| "notin_all s (E_inum i) = (i \<notin> all_inums s)"
| "notin_all s (E_msgq q) = (q \<notin> all_msgqs s)"
| "notin_all s (E_msg q m) = (m \<notin> all_msgs s q)"
*)
fun nums_of_recvmsg:: "t_state \<Rightarrow> t_process \<Rightarrow> nat"
where
"nums_of_recvmsg [] p' = 0"
| "nums_of_recvmsg (RecvMsg p q m # s) p' =
(if p' = p then Suc (nums_of_recvmsg s p) else nums_of_recvmsg s p')"
| "nums_of_recvmsg (e # s) p' = nums_of_recvmsg s p'"
lemma nums_of_recv_0:
"\<lbrakk>p \<notin> current_procs s; no_del_event s; valid s\<rbrakk> \<Longrightarrow> nums_of_recvmsg s p = 0"
apply (induct s, simp)
apply (frule vt_grant_os, frule vd_cons, case_tac a)
apply (auto)
done
lemma new_msgq_1:
"\<lbrakk>new_msgq s \<le> q; q \<le> new_msgq s - Suc 0\<rbrakk> \<Longrightarrow> False"
apply (subgoal_tac "new_msgq s \<noteq> 0")
apply (simp, simp add:new_msgq_def next_nat_def)
done
fun notin_cur:: "t_state \<Rightarrow> t_enrich_obj \<Rightarrow> bool"
where
"notin_cur s (E_proc p qmin qmax) =
(p \<notin> current_procs s \<and> qmin = new_msgq s \<and> qmax = new_msgq s + (nums_of_recvmsg s p) - 1)"
| "notin_cur s (E_file f inum) =
(f \<notin> current_files s \<and> (\<exists> pf. parent f = Some pf \<and> is_dir s pf) \<and> inum \<notin> current_inode_nums s)"
| "notin_cur s (E_file_link f) =
(f \<notin> current_files s \<and> (\<exists> pf. parent f = Some pf \<and> is_dir s pf))"
| "notin_cur s (E_msgq q) = (q \<notin> current_msgqs s)"
lemma not_all_procs_cons:
"p \<notin> all_procs (e # s) \<Longrightarrow> p \<notin> all_procs s"
by (case_tac e, auto)
lemma not_all_procs_prop:
"\<lbrakk>p' \<notin> all_procs s; p \<in> current_procs s; valid s\<rbrakk> \<Longrightarrow> p' \<noteq> p"
apply (induct s, rule notI, simp)
apply (frule vt_grant_os, frule vd_cons, frule not_all_procs_cons, simp, rule notI)
apply (case_tac a, auto)
done
lemma not_all_procs_prop2:
"p' \<notin> all_procs s \<Longrightarrow> p' \<notin> init_procs"
apply (induct s, simp)
by (case_tac a, auto)
lemma not_all_procs_prop3:
"p' \<notin> all_procs s \<Longrightarrow> p' \<notin> current_procs s"
apply (induct s, simp)
by (case_tac a, auto)
lemma not_all_msgqs_cons:
"p \<notin> all_msgqs (e # s) \<Longrightarrow> p \<notin> all_msgqs s"
by (case_tac e, auto)
lemma not_all_msgqs_prop:
"\<lbrakk>p' \<notin> all_msgqs s; p \<in> current_msgqs s; valid s\<rbrakk> \<Longrightarrow> p' \<noteq> p"
apply (induct s, rule notI, simp)
apply (frule vt_grant_os, frule vd_cons, frule not_all_msgqs_cons, simp, rule notI)
apply (case_tac a, auto)
done
lemma not_all_msgqs_prop3:
"p' \<notin> all_msgqs s \<Longrightarrow> p' \<notin> current_msgqs s"
apply (induct s, simp)
by (case_tac a, auto)
fun enrich_not_alive :: "t_state \<Rightarrow> t_enrich_obj \<Rightarrow> t_index_obj \<Rightarrow> bool"
where
"enrich_not_alive s obj (I_file f) =
(f \<notin> current_files s \<and> (\<forall> inum. obj \<noteq> E_file f inum) \<and> obj \<noteq> E_file_link f)"
| "enrich_not_alive s obj (I_proc p) = (p \<notin> current_procs s \<and> (\<forall> qmin qmax. obj \<noteq> E_proc p qmin qmax))"
| "enrich_not_alive s obj (I_fd p fd) =
((p \<in> current_procs s \<longrightarrow> fd \<notin> current_proc_fds s p) \<and> (\<forall> qmin qmax. obj \<noteq> E_proc p qmin qmax))"
| "enrich_not_alive s obj (I_msgq q) = (q \<notin> current_msgqs s \<and> obj \<noteq> E_msgq q \<and>
(case obj of
E_proc p qmin qmax \<Rightarrow> \<not> (q \<ge> qmin \<and> q \<le> qmax)
| _ \<Rightarrow> True) )"
| "enrich_not_alive s obj (I_inum i) = (i \<notin> current_inode_nums s \<and> (\<forall> f. obj \<noteq> E_file f i))"
| "enrich_not_alive s obj (I_msg q m) =
((q \<in> current_msgqs s \<longrightarrow> m \<notin> set (msgs_of_queue s q)) \<and> obj \<noteq> E_msgq q \<and>
(case obj of
E_proc p qmin qmax \<Rightarrow> \<not> (q \<ge> qmin \<and> q \<le> qmax)
| _ \<Rightarrow> True) )"
lemma file_has_parent: "\<lbrakk>is_file s f; valid s\<rbrakk> \<Longrightarrow> \<exists> pf. is_dir s pf \<and> parent f = Some pf"
apply (case_tac f)
apply (simp, drule root_is_dir', simp+)
apply (simp add:parentf_is_dir_prop2)
done
lemma enrich_search_check:
assumes grant: "search_check s (up, rp, tp) f"
and cf2sf: "\<forall> f. f \<in> current_files s \<longrightarrow> cf2sfile s' f = cf2sfile s f"
and vd: "valid s" and f_in: "is_file s f" and f_in': "is_file s' f"
and sec: "sectxt_of_obj s' (O_file f) = sectxt_of_obj s (O_file f)"
shows "search_check s' (up, rp, tp) f"
proof (cases f)
case Nil
with f_in vd have "False"
by (auto dest:root_is_dir')
thus ?thesis by simp
next
case (Cons n pf)
from vd f_in obtain sf where sf: "cf2sfile s f = Some sf"
apply (drule_tac is_file_in_current, drule_tac current_file_has_sfile, simp)
apply (erule exE, simp)
done
then obtain psfs where psfs: "get_parentfs_ctxts s pf = Some psfs" using Cons
by (auto simp:cf2sfile_def split:option.splits if_splits)
from sf cf2sf f_in have sf': "cf2sfile s' f = Some sf" by (auto dest:is_file_in_current)
then obtain psfs' where psfs': "get_parentfs_ctxts s' pf = Some psfs'"using Cons
by (auto simp:cf2sfile_def split:option.splits if_splits)
with sf sf' psfs have psfs_eq: "set psfs' = set psfs" using Cons f_in f_in'
apply (simp add:cf2sfile_def split:option.splits)
apply (case_tac sf, simp)
done
show ?thesis using grant f_in f_in' psfs psfs' psfs_eq sec
apply (simp add:Cons split:option.splits)
by (case_tac a, simp)
qed
lemma enrich_search_check':
assumes grant: "search_check s (up, rp, tp) f"
and cf2sf: "\<forall> f. f \<in> current_files s \<longrightarrow> cf2sfile s' f = cf2sfile s f"
and vd: "valid s" and vd': "valid s'" and f_in: "is_dir s f" and f_in': "is_dir s' f"
and sec: "sectxt_of_obj s' (O_dir f) = sectxt_of_obj s (O_dir f)"
shows "search_check s' (up, rp, tp) f"
proof (cases f)
case Nil
have "sectxt_of_obj s' (O_dir []) = sectxt_of_obj s (O_dir [])"
using cf2sf
apply (erule_tac x = "[]" in allE)
by (auto simp:cf2sfile_def root_sec_remains vd vd')
thus ?thesis using grant Nil
by auto
next
case (Cons n pf)
from vd f_in obtain sf where sf: "cf2sfile s f = Some sf"
apply (drule_tac is_dir_in_current, drule_tac current_file_has_sfile, simp)
apply (erule exE, simp)
done
then obtain psfs where psfs: "get_parentfs_ctxts s pf = Some psfs" using Cons
by (auto simp:cf2sfile_def split:option.splits if_splits)
from sf cf2sf f_in have sf': "cf2sfile s' f = Some sf" by (auto dest:is_dir_in_current)
then obtain psfs' where psfs': "get_parentfs_ctxts s' pf = Some psfs'"using Cons
by (auto simp:cf2sfile_def split:option.splits if_splits)
with sf sf' psfs have psfs_eq: "set psfs' = set psfs" using Cons f_in f_in'
apply (drule_tac is_dir_not_file)
apply (drule is_dir_not_file)
apply (simp add:cf2sfile_def split:option.splits)
apply (case_tac sf, simp)
done
show ?thesis using grant f_in f_in' psfs psfs' psfs_eq sec
apply (drule_tac is_dir_not_file)
apply (drule_tac is_dir_not_file)
apply (simp add:Cons split:option.splits)
by (case_tac a, simp)
qed
lemma proc_filefd_has_sfd: "\<lbrakk>fd \<in> proc_file_fds s p; valid s\<rbrakk> \<Longrightarrow> \<exists> sfd. cfd2sfd s p fd = Some sfd"
apply (simp add:proc_file_fds_def)
apply (auto dest: current_filefd_has_sfd)
done
lemma enrich_inherit_fds_check:
assumes grant: "inherit_fds_check s (up, nr, nt) p fds" and vd: "valid s"
and cfd2sfd: "\<forall> p fd. fd \<in> proc_file_fds s p\<longrightarrow> cfd2sfd s' p fd = cfd2sfd s p fd"
and fd_in: "fds \<subseteq> proc_file_fds s p" and fd_in': "fds \<subseteq> proc_file_fds s' p"
shows "inherit_fds_check s' (up, nr, nt) p fds"
proof-
have "\<And> fd. fd \<in> fds \<Longrightarrow> sectxt_of_obj s' (O_fd p fd) = sectxt_of_obj s (O_fd p fd)"
proof-
fix fd
assume fd_in_fds: "fd \<in> fds"
hence fd_in_cfds: "fd \<in> proc_file_fds s p"
and fd_in_cfds': "fd \<in> proc_file_fds s' p"
using fd_in fd_in' by auto
with cfd2sfd
have cfd_eq: "cfd2sfd s' p fd = cfd2sfd s p fd" by auto
from fd_in_cfds obtain f where ffd: "file_of_proc_fd s p fd = Some f"
by (auto simp:proc_file_fds_def)
moreover have "flags_of_proc_fd s p fd \<noteq> None"
using ffd vd by (auto dest:current_filefd_has_flags)
moreover have "sectxt_of_obj s (O_fd p fd) \<noteq> None"
using fd_in_cfds vd
apply (rule_tac notI)
by (auto dest!:current_has_sec' file_fds_subset_pfds[where p = p] intro:vd)
moreover have "cf2sfile s f \<noteq> None"
apply (rule notI)
apply (drule current_file_has_sfile')
using ffd
by (auto simp:vd is_file_in_current dest:file_of_pfd_is_file)
ultimately show "sectxt_of_obj s' (O_fd p fd) = sectxt_of_obj s (O_fd p fd)"
using cfd_eq
by (auto simp:cfd2sfd_def split:option.splits)
qed
hence "sectxts_of_fds s' p fds = sectxts_of_fds s p fds"
by (simp add:sectxts_of_fds_def)
thus ?thesis using grant
by (simp add:inherit_fds_check_def)
qed
lemma enrich_inherit_fds_check_dup:
assumes grant: "inherit_fds_check s (up, nr, nt) p fds" and vd: "valid s"
and cfd2sfd: "\<forall> fd. fd \<in> proc_file_fds s p \<longrightarrow> cfd2sfd s' p' fd = cfd2sfd s p fd"
and fd_in: "fds' \<subseteq> fds \<inter> proc_file_fds s p"
shows "inherit_fds_check s' (up, nr, nt) p' fds'"
proof-
have "sectxts_of_fds s' p' fds' \<subseteq> sectxts_of_fds s p fds"
proof-
have "\<And> fd sfd. \<lbrakk>fd \<in> fds'; sectxt_of_obj s' (O_fd p' fd) = Some sfd\<rbrakk>
\<Longrightarrow> \<exists> fd \<in> fds. sectxt_of_obj s (O_fd p fd) = Some sfd"
proof-
fix fd sfd
assume fd_in_fds': "fd \<in> fds'"
and sec: "sectxt_of_obj s' (O_fd p' fd) = Some sfd"
from fd_in_fds' fd_in
have fd_in_fds: "fd \<in> fds" and fd_in_cfds: "fd \<in> proc_file_fds s p"
by auto
from fd_in_cfds obtain f where ffd: "file_of_proc_fd s p fd = Some f"
by (auto simp:proc_file_fds_def)
moreover have "flags_of_proc_fd s p fd \<noteq> None"
using ffd vd by (auto dest:current_filefd_has_flags)
moreover have "cf2sfile s f \<noteq> None"
apply (rule notI)
apply (drule current_file_has_sfile')
using ffd
by (auto simp:vd is_file_in_current dest:file_of_pfd_is_file)
moreover have "sectxt_of_obj s (O_fd p fd) \<noteq> None"
using fd_in_cfds vd
apply (rule_tac notI)
by (auto dest!:current_has_sec' file_fds_subset_pfds[where p = p] intro:vd)
ultimately
have "sectxt_of_obj s (O_fd p fd) = Some sfd"
using fd_in_cfds cfd2sfd sec
apply (erule_tac x = fd in allE)
apply (auto simp:cfd2sfd_def split:option.splits)
done
thus "\<exists> fd \<in> fds. sectxt_of_obj s (O_fd p fd) = Some sfd"
using fd_in_fds
by (rule_tac x = fd in bexI, auto)
qed
thus ?thesis by (auto simp:sectxts_of_fds_def)
qed
thus ?thesis using grant
by (auto simp:inherit_fds_check_def inherit_fds_check_ctxt_def)
qed
lemma enrich_valid_intro_cons:
assumes vs': "valid s'" and vd': "valid (e # s)"
and alive: "\<forall> obj. alive s obj \<longrightarrow> alive s' obj"
and alive': "\<forall> obj. enrich_not_alive s obj' obj \<longrightarrow> enrich_not_alive s' obj' obj"
and hungs: "files_hung_by_del s' = files_hung_by_del s"
and cp2sp: "\<forall> p. p \<in> current_procs s \<longrightarrow> cp2sproc s' p = cp2sproc s p"
and cf2sf: "\<forall> f. f \<in> current_files s \<longrightarrow> cf2sfile s' f = cf2sfile s f"
and cq2sq: "\<forall> q. q \<in> current_msgqs s \<longrightarrow> cq2smsgq s' q = cq2smsgq s q"
and ffd_remain: "\<forall> p fd f. file_of_proc_fd s p fd = Some f \<longrightarrow> file_of_proc_fd s' p fd = Some f"
and fflags_remain: "\<forall> p fd flags. flags_of_proc_fd s p fd = Some flags \<longrightarrow> flags_of_proc_fd s' p fd = Some flags"
and sms_remain: "\<forall> q. q \<in> current_msgqs s \<longrightarrow> msgs_of_queue s' q = msgs_of_queue s q"
(* and empty_remain: "\<forall> f. dir_is_empty s f \<longrightarrow> dir_is_empty s' f" *)
and cfd2sfd: "\<forall> p fd. fd \<in> proc_file_fds s p \<longrightarrow> cfd2sfd s' p fd = cfd2sfd s p fd"
and nodel: "no_del_event (e # s)"
and notin_cur: "notin_cur (e # s) obj'"
shows "valid (e # s')"
proof-
from vd' have os: "os_grant s e" and grant: "grant s e" and vd: "valid s"
by (auto dest:vt_grant_os vt_grant vd_cons)
show ?thesis
proof (cases e)
case (Execve p f fds)
have p_in: "p \<in> current_procs s'" using os alive
apply (erule_tac x = "O_proc p" in allE)
by (auto simp:Execve)
have f_in: "is_file s' f" using os alive
apply (erule_tac x = "O_file f" in allE)
by (auto simp:Execve)
have fd_in: "fds \<subseteq> proc_file_fds s' p" using os alive ffd_remain
by (auto simp:Execve proc_file_fds_def)
have "os_grant s' e" using p_in f_in fd_in by (simp add:Execve)
moreover have "grant s' e"
proof-
from grant obtain up rp tp uf rf tf
where p1: "sectxt_of_obj s (O_proc p) = Some (up, rp, tp)"
and p2: "sectxt_of_obj s (O_file f) = Some (uf, rf, tf)"
by (simp add:Execve split:option.splits, blast)
with grant obtain pu nr nt where p3: "npctxt_execve (up, rp, tp) (uf, rf, tf) = Some (pu, nr, nt)"
by (simp add:Execve split:option.splits del:npctxt_execve.simps, blast)
from p1 have p1': "sectxt_of_obj s' (O_proc p) = Some (up, rp, tp)"
using os cp2sp
apply (erule_tac x = p in allE)
by (auto simp:Execve co2sobj.simps cp2sproc_def split:option.splits)
from os have f_in': "is_file s f" by (simp add:Execve)
from vd os have "\<exists> sf. cf2sfile s f = Some sf"
by (auto dest!:is_file_in_current current_file_has_sfile simp:Execve)
hence p2': "sectxt_of_obj s' (O_file f) = Some (uf, rf, tf)" using f_in f_in' p2 cf2sf
apply (erule_tac x = f in allE)
apply (auto dest:is_file_in_current simp:cf2sfile_def split:option.splits)
apply (case_tac f, simp)
apply (drule_tac s = s in root_is_dir', simp add:vd, simp+)
done
have "inherit_fds_check s' (pu, nr, nt) p fds"
proof-
have "fds \<subseteq> proc_file_fds s' p" using os ffd_remain Execve
by (auto simp:proc_file_fds_def)
thus ?thesis using Execve grant vd cfd2sfd p1 p2 p3 os
apply (rule_tac s = s in enrich_inherit_fds_check)
by (simp_all split:option.splits)
qed
moreover have "search_check s' (pu, rp, tp) f"
using p1 p2 p2' vd cf2sf f_in' grant Execve p3 f_in
apply (rule_tac s = s in enrich_search_check)
by (simp_all split:option.splits)
ultimately show ?thesis using p1' p2' p3
apply (simp add:Execve split:option.splits)
using grant Execve p1 p2
by (simp add:Execve grant p1 p2)
qed
ultimately show ?thesis using vs'
by (erule_tac valid.intros(2), simp+)
next
case (Clone p p' fds)
have p_in: "p \<in> current_procs s'" using os alive
apply (erule_tac x = "O_proc p" in allE)
by (auto simp:Clone)
have p'_not_in: "p' \<notin> current_procs s'" using alive' notin_cur os Clone
apply (erule_tac x = "I_proc p'" in allE)
apply (auto dest:not_all_procs_prop3 simp del:nums_of_recvmsg.simps)
done
have fd_in: "fds \<subseteq> proc_file_fds s' p" using os alive ffd_remain
by (auto simp:Clone proc_file_fds_def)
have "os_grant s' e" using p_in p'_not_in fd_in by (simp add:Clone)
moreover have "grant s' e"
proof-
from grant obtain up rp tp
where p1: "sectxt_of_obj s (O_proc p) = Some (up, rp, tp)"
apply (simp add:Clone split:option.splits)
by (case_tac a, auto)
from p1 have p1': "sectxt_of_obj s' (O_proc p) = Some (up, rp, tp)"
using os cp2sp
apply (erule_tac x = p in allE)
by (auto simp:Clone co2sobj.simps cp2sproc_def split:option.splits)
have p2: "inherit_fds_check s' (up, rp, tp) p fds"
proof-
have "fds \<subseteq> proc_file_fds s' p" using os ffd_remain Clone
by (auto simp:proc_file_fds_def)
thus ?thesis using Clone grant vd cfd2sfd p1 os
apply (rule_tac s = s in enrich_inherit_fds_check)
by (simp_all split:option.splits)
qed
show ?thesis using p1 p2 p1' grant
by (simp add:Clone)
qed
ultimately show ?thesis using vs'
by (erule_tac valid.intros(2), simp+)
next
case (Kill p p')
have p_in: "p \<in> current_procs s'" using os alive
apply (erule_tac x = "O_proc p" in allE)
by (auto simp:Kill)
have p'_in: "p' \<in> current_procs s'" using os alive
apply (erule_tac x = "O_proc p'" in allE)
by (auto simp:Kill)
have "os_grant s' e" using p_in p'_in by (simp add:Kill)
moreover have "grant s' e"
proof-
from grant obtain up rp tp up' rp' tp'
where p1: "sectxt_of_obj s (O_proc p) = Some (up, rp, tp)"
and p'1: "sectxt_of_obj s (O_proc p') = Some (up', rp', tp')"
apply (simp add:Kill split:option.splits)
by (case_tac a, case_tac aa, blast)
from p1 have p1': "sectxt_of_obj s' (O_proc p) = Some (up, rp, tp)"
using os cp2sp
apply (erule_tac x = p in allE)
by (auto simp:Kill co2sobj.simps cp2sproc_def split:option.splits)
from p'1 have p'1': "sectxt_of_obj s' (O_proc p') = Some (up', rp', tp')"
using os cp2sp
apply (erule_tac x = p' in allE)
by (auto simp:Kill co2sobj.simps cp2sproc_def split:option.splits)
show ?thesis using p1 p'1 p1' p'1' grant
by (simp add:Kill)
qed
ultimately show ?thesis using vs'
by (erule_tac valid.intros(2), simp+)
next
case (Ptrace p p')
have p_in: "p \<in> current_procs s'" using os alive
apply (erule_tac x = "O_proc p" in allE)
by (auto simp:Ptrace)
have p'_in: "p' \<in> current_procs s'" using os alive
apply (erule_tac x = "O_proc p'" in allE)
by (auto simp:Ptrace)
have "os_grant s' e" using p_in p'_in by (simp add:Ptrace)
moreover have "grant s' e"
proof-
from grant obtain up rp tp up' rp' tp'
where p1: "sectxt_of_obj s (O_proc p) = Some (up, rp, tp)"
and p'1: "sectxt_of_obj s (O_proc p') = Some (up', rp', tp')"
apply (simp add:Ptrace split:option.splits)
by (case_tac a, case_tac aa, blast)
from p1 have p1': "sectxt_of_obj s' (O_proc p) = Some (up, rp, tp)"
using os cp2sp
apply (erule_tac x = p in allE)
by (auto simp:Ptrace co2sobj.simps cp2sproc_def split:option.splits)
from p'1 have p'1': "sectxt_of_obj s' (O_proc p') = Some (up', rp', tp')"
using os cp2sp
apply (erule_tac x = p' in allE)
by (auto simp:Ptrace co2sobj.simps cp2sproc_def split:option.splits)
show ?thesis using p1 p'1 p1' p'1' grant
by (simp add:Ptrace)
qed
ultimately show ?thesis using vs'
by (erule_tac valid.intros(2), simp+)
next
case (Exit p)
have p_in: "p \<in> current_procs s'" using os alive
apply (erule_tac x = "O_proc p" in allE)
by (auto simp:Exit)
have "os_grant s' e" using p_in by (simp add:Exit)
moreover have "grant s' e"
by (simp add:Exit)
ultimately show ?thesis using vs'
by (erule_tac valid.intros(2), simp+)
next
case (Open p f flags fd opt)
show ?thesis
proof (cases opt)
case None
have p_in: "p \<in> current_procs s'" using os alive
apply (erule_tac x = "O_proc p" in allE)
by (auto simp:Open None)
have f_in: "is_file s' f" using os alive
apply (erule_tac x = "O_file f" in allE)
by (auto simp:Open None)
have fd_not_in: "fd \<notin> current_proc_fds s' p"
using os alive' p_in notin_cur Open None
apply (erule_tac x = "I_fd p fd" in allE)
apply (case_tac obj')
apply (auto dest:not_all_procs_prop3)
done
have "os_grant s' e" using p_in f_in fd_not_in os
by (simp add:Open None)
moreover have "grant s' e"
proof-
from grant obtain up rp tp uf rf tf
where p1: "sectxt_of_obj s (O_proc p) = Some (up, rp, tp)"
and p2: "sectxt_of_obj s (O_file f) = Some (uf, rf, tf)"
apply (simp add:Open None split:option.splits)
by (case_tac a, case_tac aa, blast)
from p1 have p1': "sectxt_of_obj s' (O_proc p) = Some (up, rp, tp)"
using os cp2sp
apply (erule_tac x = p in allE)
by (auto simp:Open None co2sobj.simps cp2sproc_def split:option.splits)
from os have f_in': "is_file s f" by (simp add:Open None)
from vd os have "\<exists> sf. cf2sfile s f = Some sf"
by (auto dest!:is_file_in_current current_file_has_sfile simp:Open None)
hence p2': "sectxt_of_obj s' (O_file f) = Some (uf, rf, tf)" using f_in f_in' p2 cf2sf
apply (erule_tac x = f in allE)
apply (auto dest:is_file_in_current simp:cf2sfile_def split:option.splits)
apply (case_tac f, simp)
apply (drule_tac s = s in root_is_dir', simp add:vd, simp+)
done
have "search_check s' (up, rp, tp) f"
using p1 p2 p2' vd cf2sf f_in' grant Open None f_in
apply (rule_tac s = s in enrich_search_check)
by (simp_all split:option.splits)
thus ?thesis using p1' p2'
apply (simp add:Open None split:option.splits)
using grant Open None p1 p2
by simp
qed
ultimately show ?thesis using vs'
by (erule_tac valid.intros(2), simp+)
next
case (Some inum)
from os obtain pf where pf_in_s: "is_dir s pf" and parent: "parent f = Some pf"
by (auto simp:Open Some)
have pf_in: "is_dir s' pf" using pf_in_s alive
apply (erule_tac x = "O_dir pf" in allE)
by simp
have p_in: "p \<in> current_procs s'" using os alive
apply (erule_tac x = "O_proc p" in allE)
by (auto simp:Open Some)
have f_not_in: "f \<notin> current_files s'" using os alive' Open Some notin_cur nodel
apply (erule_tac x = "I_file f" in allE)
by (case_tac obj', auto simp:current_files_simps)
have fd_not_in: "fd \<notin> current_proc_fds s' p"
using os alive' p_in Open Some notin_cur
apply (erule_tac x = "I_fd p fd" in allE)
apply (case_tac obj', auto dest:not_all_procs_prop3)
done
have inum_not_in: "inum \<notin> current_inode_nums s'"
using os alive' Open Some notin_cur nodel
apply (erule_tac x = "I_inum inum" in allE)
apply (case_tac obj', auto)
apply (auto simp add:current_inode_nums_def current_file_inums_def split:if_splits)
done
have "os_grant s' e" using p_in pf_in parent f_not_in fd_not_in inum_not_in os
by (simp add:Open Some hungs)
moreover have "grant s' e"
proof-
from grant parent obtain up rp tp uf rf tf
where p1: "sectxt_of_obj s (O_proc p) = Some (up, rp, tp)"
and p2: "sectxt_of_obj s (O_dir pf) = Some (uf, rf, tf)"
apply (simp add:Open Some split:option.splits)
by (case_tac a, case_tac aa, blast)
from p1 have p1': "sectxt_of_obj s' (O_proc p) = Some (up, rp, tp)"
using os cp2sp
apply (erule_tac x = p in allE)
by (auto simp:Open Some co2sobj.simps cp2sproc_def split:option.splits)
from vd os pf_in_s have "\<exists> sf. cf2sfile s pf = Some sf"
by (auto dest!:is_dir_in_current current_file_has_sfile simp:Open Some)
hence p2': "sectxt_of_obj s' (O_dir pf) = Some (uf, rf, tf)" using p2 cf2sf pf_in pf_in_s
apply (erule_tac x = pf in allE)
apply (erule exE, frule_tac s = s in is_dir_in_current, simp)
apply (drule is_dir_not_file, drule is_dir_not_file)
apply (auto simp:cf2sfile_def split:option.splits)
apply (case_tac pf, simp_all)
by (simp add:sroot_def root_sec_remains vd vs')
have "search_check s' (up, rp, tp) pf"
using p1 p2 p2' vd cf2sf pf_in grant Open Some pf_in_s parent vs'
apply (rule_tac s = s in enrich_search_check')
by (simp_all split:option.splits)
thus ?thesis using p1' p2' parent
apply (simp add:Open Some split:option.splits)
using grant Open Some p1 p2
by simp
qed
ultimately show ?thesis using vs'
by (erule_tac valid.intros(2), simp+)
qed
next
case (ReadFile p fd)
have p_in: "p \<in> current_procs s'" using os alive
apply (erule_tac x = "O_proc p" in allE)
by (auto simp:ReadFile)
have fd_in: "fd \<in> current_proc_fds s' p" using os alive
apply (erule_tac x = "O_fd p fd" in allE)
by (auto simp:ReadFile)
obtain f where ffd: "file_of_proc_fd s p fd = Some f"
using os ReadFile by auto
hence f_in_s: "is_file s f" using vd
by (auto intro:file_of_pfd_is_file)
obtain flags where fflag: "flags_of_proc_fd s p fd = Some flags"
using os ReadFile by auto
have ffd_in: "file_of_proc_fd s' p fd = Some f"
using ffd_remain ffd by auto
hence f_in: "is_file s' f" using vs'
by (auto intro:file_of_pfd_is_file)
have flags_in: "flags_of_proc_fd s' p fd = Some flags"
using fflags_remain fflag by auto
have "os_grant s' e" using p_in fd_in ffd_in flags_in fflag os f_in
by (auto simp add:ReadFile is_file_in_current)
moreover have "grant s' e"
proof-
from grant ffd obtain up rp tp uf rf tf ufd rfd tfd
where p1: "sectxt_of_obj s (O_proc p) = Some (up, rp, tp)"
and p2: "sectxt_of_obj s (O_file f) = Some (uf, rf, tf)"
and p3: "sectxt_of_obj s (O_fd p fd) = Some (ufd, rfd, tfd)"
apply (simp add:ReadFile split:option.splits)
by (case_tac a, case_tac aa, case_tac ab, blast)
from p1 have p1': "sectxt_of_obj s' (O_proc p) = Some (up, rp, tp)"
using os cp2sp
apply (erule_tac x = p in allE)
by (auto simp:ReadFile co2sobj.simps cp2sproc_def split:option.splits)
from vd f_in_s have "\<exists> sf. cf2sfile s f = Some sf"
by (auto dest!:is_file_in_current current_file_has_sfile)
hence p2': "sectxt_of_obj s' (O_file f) = Some (uf, rf, tf)" using f_in f_in_s p2 cf2sf
apply (erule_tac x = f in allE)
apply (auto dest:is_file_in_current simp:cf2sfile_def split:option.splits)
apply (case_tac f, simp)
apply (drule_tac s = s in root_is_dir', simp add:vd, simp+)
done
have p3': "sectxt_of_obj s' (O_fd p fd) = Some (ufd, rfd, tfd)"
using cfd2sfd ffd_in ffd p3 f_in f_in_s vd
apply (erule_tac x = p in allE)
apply (erule_tac x = fd in allE)
apply (simp add:proc_file_fds_def)
apply (auto simp:cfd2sfd_def fflag flags_in p3 split:option.splits
dest!:current_file_has_sfile' simp:is_file_in_current)
done
show ?thesis using p1' p2' p3' ffd_in ffd
apply (simp add:ReadFile split:option.splits)
using grant p1 p2 p3 ReadFile
by simp
qed
ultimately show ?thesis using vs'
by (erule_tac valid.intros(2), simp+)
next
case (WriteFile p fd)
have p_in: "p \<in> current_procs s'" using os alive
apply (erule_tac x = "O_proc p" in allE)
by (auto simp:WriteFile)
have fd_in: "fd \<in> current_proc_fds s' p" using os alive
apply (erule_tac x = "O_fd p fd" in allE)
by (auto simp:WriteFile)
obtain f where ffd: "file_of_proc_fd s p fd = Some f"
using os WriteFile by auto
hence f_in_s: "is_file s f" using vd
by (auto intro:file_of_pfd_is_file)
obtain flags where fflag: "flags_of_proc_fd s p fd = Some flags"
using os WriteFile by auto
have ffd_in: "file_of_proc_fd s' p fd = Some f"
using ffd_remain ffd by auto
hence f_in: "is_file s' f" using vs'
by (auto intro:file_of_pfd_is_file)
have flags_in: "flags_of_proc_fd s' p fd = Some flags"
using fflags_remain fflag by auto
have "os_grant s' e" using p_in fd_in ffd_in flags_in fflag os f_in
by (auto simp add:WriteFile is_file_in_current)
moreover have "grant s' e"
proof-
from grant ffd obtain up rp tp uf rf tf ufd rfd tfd
where p1: "sectxt_of_obj s (O_proc p) = Some (up, rp, tp)"
and p2: "sectxt_of_obj s (O_file f) = Some (uf, rf, tf)"
and p3: "sectxt_of_obj s (O_fd p fd) = Some (ufd, rfd, tfd)"
apply (simp add:WriteFile split:option.splits)
by (case_tac a, case_tac aa, case_tac ab, blast)
from p1 have p1': "sectxt_of_obj s' (O_proc p) = Some (up, rp, tp)"
using os cp2sp
apply (erule_tac x = p in allE)
by (auto simp:WriteFile co2sobj.simps cp2sproc_def split:option.splits)
from vd f_in_s have "\<exists> sf. cf2sfile s f = Some sf"
by (auto dest!:is_file_in_current current_file_has_sfile)
hence p2': "sectxt_of_obj s' (O_file f) = Some (uf, rf, tf)" using f_in f_in_s p2 cf2sf
apply (erule_tac x = f in allE)
apply (auto dest:is_file_in_current simp:cf2sfile_def split:option.splits)
apply (case_tac f, simp)
apply (drule_tac s = s in root_is_dir', simp add:vd, simp+)
done
have p3': "sectxt_of_obj s' (O_fd p fd) = Some (ufd, rfd, tfd)"
using cfd2sfd ffd_in ffd p3 f_in f_in_s vd
apply (erule_tac x = p in allE)
apply (erule_tac x = fd in allE)
apply (simp add:proc_file_fds_def)
apply (auto simp:cfd2sfd_def fflag flags_in p3 split:option.splits
dest!:current_file_has_sfile' simp:is_file_in_current)
done
show ?thesis using p1' p2' p3' ffd_in ffd
apply (simp add:WriteFile split:option.splits)
using grant p1 p2 p3 WriteFile
by simp
qed
ultimately show ?thesis using vs'
by (erule_tac valid.intros(2), simp+)
next
case (CloseFd p fd)
have p_in: "p \<in> current_procs s'" using os alive
apply (erule_tac x = "O_proc p" in allE)
by (auto simp:CloseFd)
have fd_in: "fd \<in> current_proc_fds s' p" using os alive
apply (erule_tac x = "O_fd p fd" in allE)
by (auto simp:CloseFd)
have "os_grant s' e" using p_in fd_in
by (auto simp add:CloseFd)
moreover have "grant s' e"
by(simp add:CloseFd)
ultimately show ?thesis using vs'
by (erule_tac valid.intros(2), simp+)
next
case (UnLink p f)
have p_in: "p \<in> current_procs s'" using os alive
apply (erule_tac x = "O_proc p" in allE)
by (auto simp:UnLink)
have f_in: "is_file s' f" using os alive
apply (erule_tac x = "O_file f" in allE)
by (auto simp:UnLink)
from os vd obtain pf where pf_in_s: "is_dir s pf"
and parent: "parent f = Some pf"
by (auto simp:UnLink dest!:file_has_parent)
from pf_in_s alive have pf_in: "is_dir s' pf"
apply (erule_tac x = "O_dir pf" in allE)
by (auto simp:UnLink)
have "os_grant s' e" using p_in f_in os
by (simp add:UnLink hungs)
moreover have "grant s' e"
proof-
from grant parent obtain up rp tp uf rf tf upf rpf tpf
where p1: "sectxt_of_obj s (O_proc p) = Some (up, rp, tp)"
and p2: "sectxt_of_obj s (O_file f) = Some (uf, rf, tf)"
and p3: "sectxt_of_obj s (O_dir pf) = Some (upf, rpf, tpf)"
apply (simp add:UnLink split:option.splits)
by (case_tac a, case_tac aa, case_tac ab, blast)
from p1 have p1': "sectxt_of_obj s' (O_proc p) = Some (up, rp, tp)"
using os cp2sp
apply (erule_tac x = p in allE)
by (auto simp:UnLink co2sobj.simps cp2sproc_def split:option.splits)
from vd os pf_in_s have "\<exists> sf. cf2sfile s f = Some sf"
by (auto dest!:is_file_in_current current_file_has_sfile simp:UnLink)
hence p2': "sectxt_of_obj s' (O_file f) = Some (uf, rf, tf)"
using p2 cf2sf f_in os parent
apply (erule_tac x = f in allE)
apply (erule exE, clarsimp simp:UnLink)
apply (frule_tac s = s in is_file_in_current, simp)
by (auto simp:cf2sfile_def split:option.splits)
from vd os pf_in_s have "\<exists> sf. cf2sfile s pf = Some sf"
by (auto dest!:is_dir_in_current current_file_has_sfile simp:UnLink)
hence p3': "sectxt_of_obj s' (O_dir pf) = Some (upf, rpf, tpf)" using p3 cf2sf pf_in pf_in_s
apply (erule_tac x = pf in allE)
apply (erule exE, frule_tac s = s in is_dir_in_current, simp)
apply (drule is_dir_not_file, drule is_dir_not_file)
apply (auto simp:cf2sfile_def split:option.splits)
apply (case_tac pf, simp_all)
by (simp add:sroot_def root_sec_remains vd vs')
have "search_check s' (up, rp, tp) f"
using p1 p2 p2' vd cf2sf f_in grant UnLink os parent vs'
apply (rule_tac s = s in enrich_search_check)
by (simp_all split:option.splits)
thus ?thesis using p1' p2' p3' parent
apply (simp add:UnLink split:option.splits)
using grant UnLink p1 p2 p3
by simp
qed
ultimately show ?thesis using vs'
by (erule_tac valid.intros(2), simp+)
next
case (Rmdir p f)
have p_in: "p \<in> current_procs s'" using os alive
apply (erule_tac x = "O_proc p" in allE)
by (auto simp:Rmdir)
have f_in: "is_dir s' f" using os alive
apply (erule_tac x = "O_dir f" in allE)
by (auto simp:Rmdir dir_is_empty_def)
have not_root: "f \<noteq> []" using os
by (auto simp:Rmdir)
from os vd obtain pf where pf_in_s: "is_dir s pf"
and parent: "parent f = Some pf"
apply (auto simp:Rmdir dir_is_empty_def)
apply (case_tac f, simp+)
apply (drule parentf_is_dir_prop1, auto)
done
from pf_in_s alive have pf_in: "is_dir s' pf"
apply (erule_tac x = "O_dir pf" in allE)
by (auto simp:Rmdir)
have empty_in: "dir_is_empty s' f" using os Rmdir notin_cur
apply (clarsimp simp add:dir_is_empty_def f_in)
using alive'
apply (erule_tac x = "I_file f'" in allE)
apply simp
apply (erule disjE)
apply (erule_tac x = f' in allE, simp)
apply (case_tac obj', simp_all)
apply (clarsimp)
apply (drule_tac f' = f in parent_ancen)
apply (simp, rule notI, simp add:noJ_Anc)
apply (case_tac "f = pf")
using vd' Rmdir
apply (simp_all add:is_dir_rmdir)
apply (erule_tac x = pf in allE)
apply (drule_tac f = pf in is_dir_in_current)
apply (simp add:noJ_Anc)
apply (clarsimp)
apply (drule_tac f' = f in parent_ancen)
apply (simp, rule notI, simp add:noJ_Anc)
apply (case_tac "f = pf")
using vd' Rmdir
apply (simp_all add:is_dir_rmdir)
apply (erule_tac x = pf in allE)
apply (drule_tac f = pf in is_dir_in_current)
apply (simp add:noJ_Anc)
done
have "os_grant s' e" using p_in f_in os empty_in
by (simp add:Rmdir hungs)
moreover have "grant s' e"
proof-
from grant parent obtain up rp tp uf rf tf upf rpf tpf
where p1: "sectxt_of_obj s (O_proc p) = Some (up, rp, tp)"
and p2: "sectxt_of_obj s (O_dir f) = Some (uf, rf, tf)"
and p3: "sectxt_of_obj s (O_dir pf) = Some (upf, rpf, tpf)"
apply (simp add:Rmdir split:option.splits)
by (case_tac a, case_tac aa, case_tac ab, blast)
from p1 have p1': "sectxt_of_obj s' (O_proc p) = Some (up, rp, tp)"
using os cp2sp
apply (erule_tac x = p in allE)
by (auto simp:Rmdir co2sobj.simps cp2sproc_def split:option.splits)
from vd os pf_in_s have "\<exists> sf. cf2sfile s f = Some sf"
by (auto dest!:is_dir_in_current current_file_has_sfile simp:dir_is_empty_def Rmdir)
hence p2': "sectxt_of_obj s' (O_dir f) = Some (uf, rf, tf)"
using p2 cf2sf f_in os parent
apply (erule_tac x = f in allE)
apply (erule exE, clarsimp simp:Rmdir dir_is_empty_def)
apply (frule_tac s = s in is_dir_in_current, simp)
apply (drule is_dir_not_file, drule is_dir_not_file)
by (auto simp:cf2sfile_def split:option.splits)
from vd os pf_in_s have "\<exists> sf. cf2sfile s pf = Some sf"
by (auto dest!:is_dir_in_current current_file_has_sfile simp:Rmdir)
hence p3': "sectxt_of_obj s' (O_dir pf) = Some (upf, rpf, tpf)" using p3 cf2sf pf_in pf_in_s
apply (erule_tac x = pf in allE)
apply (erule exE, frule_tac s = s in is_dir_in_current, simp)
apply (drule is_dir_not_file, drule is_dir_not_file)
apply (auto simp:cf2sfile_def split:option.splits)
apply (case_tac pf, simp_all)
by (simp add:sroot_def root_sec_remains vd vs')
have "search_check s' (up, rp, tp) f"
using p1 p2 p2' vd cf2sf f_in grant Rmdir os parent vs'
apply (rule_tac s = s in enrich_search_check')
by (simp_all add:dir_is_empty_def split:option.splits)
thus ?thesis using p1' p2' p3' parent
apply (simp add:Rmdir split:option.splits)
using grant Rmdir p1 p2 p3
by simp
qed
ultimately show ?thesis using vs'
by (erule_tac valid.intros(2), simp+)
next
case (Mkdir p f inum)
from os obtain pf where pf_in_s: "is_dir s pf" and parent: "parent f = Some pf"
by (auto simp:Mkdir)
have pf_in: "is_dir s' pf" using pf_in_s alive
apply (erule_tac x = "O_dir pf" in allE)
by simp
have p_in: "p \<in> current_procs s'" using os alive
apply (erule_tac x = "O_proc p" in allE)
by (auto simp:Mkdir)
have f_not_in: "f \<notin> current_files s'"
using os alive' Mkdir notin_cur
apply (erule_tac x = "I_file f" in allE)
by (auto simp:current_files_simps)
have inum_not_in: "inum \<notin> current_inode_nums s'"
using os alive' Mkdir notin_cur
apply (erule_tac x = "I_inum inum" in allE)
apply (auto simp:current_inode_nums_def current_file_inums_def split:if_splits)
done
have "os_grant s' e" using p_in pf_in parent f_not_in os inum_not_in
by (simp add:Mkdir hungs)
moreover have "grant s' e"
proof-
from grant parent obtain up rp tp uf rf tf
where p1: "sectxt_of_obj s (O_proc p) = Some (up, rp, tp)"
and p2: "sectxt_of_obj s (O_dir pf) = Some (uf, rf, tf)"
apply (simp add:Mkdir split:option.splits)
by (case_tac a, case_tac aa, blast)
from p1 have p1': "sectxt_of_obj s' (O_proc p) = Some (up, rp, tp)"
using os cp2sp
apply (erule_tac x = p in allE)
by (auto simp:Mkdir co2sobj.simps cp2sproc_def split:option.splits)
from vd os pf_in_s have "\<exists> sf. cf2sfile s pf = Some sf"
by (auto dest!:is_dir_in_current current_file_has_sfile simp:Mkdir)
hence p2': "sectxt_of_obj s' (O_dir pf) = Some (uf, rf, tf)" using p2 cf2sf pf_in pf_in_s
apply (erule_tac x = pf in allE)
apply (erule exE, frule_tac s = s in is_dir_in_current, simp)
apply (drule is_dir_not_file, drule is_dir_not_file)
apply (auto simp:cf2sfile_def split:option.splits)
apply (case_tac pf, simp_all)
by (simp add:sroot_def root_sec_remains vd vs')
have "search_check s' (up, rp, tp) pf"
using p1 p2 p2' vd cf2sf pf_in grant Mkdir pf_in_s parent vs'
apply (rule_tac s = s in enrich_search_check')
apply (simp_all split:option.splits)
done
thus ?thesis using p1' p2' parent
apply (simp add:Mkdir split:option.splits)
using grant Mkdir p1 p2
apply simp
done
qed
ultimately show ?thesis using vs'
by (erule_tac valid.intros(2), simp+)
next
case (LinkHard p f f')
from os obtain pf where pf_in_s: "is_dir s pf" and parent: "parent f' = Some pf"
by (auto simp:LinkHard)
have pf_in: "is_dir s' pf" using pf_in_s alive
apply (erule_tac x = "O_dir pf" in allE)
by simp
have p_in: "p \<in> current_procs s'" using os alive
apply (erule_tac x = "O_proc p" in allE)
by (auto simp:LinkHard)
have f'_not_in: "f' \<notin> current_files s'"
using os alive' LinkHard notin_cur vd'
apply (erule_tac x = "I_file f'" in allE)
apply (auto simp:LinkHard current_files_simps)
done
have f_in: "is_file s' f" using os alive
apply (erule_tac x = "O_file f" in allE)
by (auto simp:LinkHard)
have "os_grant s' e" using p_in pf_in parent os f_in f'_not_in
by (simp add:LinkHard hungs)
moreover have "grant s' e"
proof-
from grant parent obtain up rp tp uf rf tf upf rpf tpf
where p1: "sectxt_of_obj s (O_proc p) = Some (up, rp, tp)"
and p2: "sectxt_of_obj s (O_file f) = Some (uf, rf, tf)"
and p3: "sectxt_of_obj s (O_dir pf) = Some (upf, rpf, tpf)"
apply (simp add:LinkHard split:option.splits)
by (case_tac a, case_tac aa, case_tac ab, blast)
from p1 have p1': "sectxt_of_obj s' (O_proc p) = Some (up, rp, tp)"
using os cp2sp
apply (erule_tac x = p in allE)
by (auto simp:LinkHard co2sobj.simps cp2sproc_def split:option.splits)
from vd os pf_in_s have "\<exists> sf. cf2sfile s f = Some sf"
by (auto dest!:is_file_in_current current_file_has_sfile simp:LinkHard)
hence p2': "sectxt_of_obj s' (O_file f) = Some (uf, rf, tf)"
using p2 cf2sf f_in os parent
apply (erule_tac x = f in allE)
apply (erule exE, clarsimp simp:LinkHard)
apply (frule_tac s = s in is_file_in_current, simp)
apply (auto simp:cf2sfile_def split:option.splits)
apply (case_tac f, simp)
by (drule_tac s = s in root_is_dir', simp add:vd, simp+)
from vd os pf_in_s have "\<exists> sf. cf2sfile s pf = Some sf"
by (auto dest!:is_dir_in_current current_file_has_sfile simp:LinkHard)
hence p3': "sectxt_of_obj s' (O_dir pf) = Some (upf, rpf, tpf)" using p3 cf2sf pf_in pf_in_s
apply (erule_tac x = pf in allE)
apply (erule exE, frule_tac s = s in is_dir_in_current, simp)
apply (drule is_dir_not_file, drule is_dir_not_file)
apply (auto simp:cf2sfile_def split:option.splits)
apply (case_tac pf, simp_all)
by (simp add:sroot_def root_sec_remains vd vs')
have "search_check s' (up, rp, tp) f"
using p1 p2 p2' vd cf2sf f_in grant LinkHard os parent vs'
apply (rule_tac s = s in enrich_search_check)
by (simp_all split:option.splits)
moreover have "search_check s' (up, rp, tp) pf"
using p1 p3 p3' vd cf2sf pf_in grant LinkHard os parent vs'
apply (rule_tac s = s in enrich_search_check')
apply (simp_all split:option.splits)
done
ultimately show ?thesis using p1' p2' p3' parent
apply (simp add:LinkHard split:option.splits)
using grant LinkHard p1 p2 p3
apply simp
done
qed
ultimately show ?thesis using vs'
by (erule_tac valid.intros(2), simp+)
next
case (Truncate p f len)
have p_in: "p \<in> current_procs s'" using os alive
apply (erule_tac x = "O_proc p" in allE)
by (auto simp:Truncate)
have f_in: "is_file s' f" using os alive
apply (erule_tac x = "O_file f" in allE)
by (auto simp:Truncate)
have "os_grant s' e" using p_in f_in by (simp add:Truncate)
moreover have "grant s' e"
proof-
from grant obtain up rp tp uf rf tf
where p1: "sectxt_of_obj s (O_proc p) = Some (up, rp, tp)"
and p2: "sectxt_of_obj s (O_file f) = Some (uf, rf, tf)"
apply (simp add:Truncate split:option.splits)
by (case_tac a, case_tac aa, blast)
from p1 have p1': "sectxt_of_obj s' (O_proc p) = Some (up, rp, tp)"
using os cp2sp
apply (erule_tac x = p in allE)
by (auto simp:Truncate co2sobj.simps cp2sproc_def split:option.splits)
from os have f_in': "is_file s f" by (simp add:Truncate)
from vd os have "\<exists> sf. cf2sfile s f = Some sf"
by (auto dest!:is_file_in_current current_file_has_sfile simp:Truncate)
hence p2': "sectxt_of_obj s' (O_file f) = Some (uf, rf, tf)" using f_in f_in' p2 cf2sf
apply (erule_tac x = f in allE)
apply (auto dest:is_file_in_current simp:cf2sfile_def split:option.splits)
apply (case_tac f, simp)
apply (drule_tac s = s in root_is_dir', simp add:vd, simp+)
done
have "search_check s' (up, rp, tp) f"
using p1 p2 p2' vd cf2sf f_in' grant Truncate f_in
apply (rule_tac s = s in enrich_search_check)
by (simp_all split:option.splits)
thus ?thesis using p1' p2'
apply (simp add:Truncate split:option.splits)
using grant Truncate p1 p2
by (simp add:Truncate grant p1 p2)
qed
ultimately show ?thesis using vs'
by (erule_tac valid.intros(2), simp+)
next
case (CreateMsgq p q)
have p_in: "p \<in> current_procs s'" using os alive
apply (erule_tac x = "O_proc p" in allE)
by (auto simp:CreateMsgq)
have q_not_in: "q \<notin> current_msgqs s'"
using os alive' CreateMsgq notin_cur nodel vd
apply (erule_tac x = "I_msgq q" in allE)
apply (auto split:t_enrich_obj.splits)
apply (drule nums_of_recv_0, simp+)
apply (drule new_msgq_1, simp+)
done
have "os_grant s' e" using p_in q_not_in by (simp add:CreateMsgq)
moreover have "grant s' e"
proof-
from grant obtain up rp tp
where p1: "sectxt_of_obj s (O_proc p) = Some (up, rp, tp)"
apply (simp add:CreateMsgq split:option.splits)
by (case_tac a, blast)
from p1 have p1': "sectxt_of_obj s' (O_proc p) = Some (up, rp, tp)"
using os cp2sp
apply (erule_tac x = p in allE)
by (auto simp:CreateMsgq co2sobj.simps cp2sproc_def split:option.splits)
show ?thesis using p1'
apply (simp add:CreateMsgq split:option.splits)
using grant CreateMsgq p1
by simp
qed
ultimately show ?thesis using vs'
by (erule_tac valid.intros(2), simp+)
next
case (RemoveMsgq p q)
have p_in: "p \<in> current_procs s'" using os alive
apply (erule_tac x = "O_proc p" in allE)
by (auto simp:RemoveMsgq)
have q_in: "q \<in> current_msgqs s'" using os alive
apply (erule_tac x = "O_msgq q" in allE)
by (simp add:RemoveMsgq)
have "os_grant s' e" using p_in q_in by (simp add:RemoveMsgq)
moreover have "grant s' e"
proof-
from grant obtain up rp tp uq rq tq
where p1: "sectxt_of_obj s (O_proc p) = Some (up, rp, tp)"
and p2: "sectxt_of_obj s (O_msgq q) = Some (uq, rq, tq)"
apply (simp add:RemoveMsgq split:option.splits)
by (case_tac a, case_tac aa, blast)
from p1 have p1': "sectxt_of_obj s' (O_proc p) = Some (up, rp, tp)"
using os cp2sp
apply (erule_tac x = p in allE)
by (auto simp:RemoveMsgq co2sobj.simps cp2sproc_def split:option.splits)
from p2 have p2': "sectxt_of_obj s' (O_msgq q) = Some (uq, rq, tq)"
using os cq2sq vd
apply (erule_tac x = q in allE)
by (auto simp:RemoveMsgq co2sobj.simps cq2smsgq_def dest!:current_has_sms' split:option.splits)
show ?thesis using p1' p2' grant p1 p2
by (simp add:RemoveMsgq)
qed
ultimately show ?thesis using vs'
by (erule_tac valid.intros(2), simp+)
next
case (SendMsg p q m)
have p_in: "p \<in> current_procs s'" using os alive
apply (erule_tac x = "O_proc p" in allE)
by (auto simp:SendMsg)
have q_in: "q \<in> current_msgqs s'" using os alive
apply (erule_tac x = "O_msgq q" in allE)
by (simp add:SendMsg)
have m_not_in: "m \<notin> set (msgs_of_queue s' q)"
using os alive' notin_cur SendMsg q_in nodel vd
apply (erule_tac x = "I_msg q m" in allE)
apply (case_tac obj', auto dest:not_all_msgqs_prop3)
apply (drule nums_of_recv_0, simp+)
apply (drule new_msgq_1, simp+)
done
have "os_grant s' e" using p_in q_in m_not_in sms_remain os
by (simp add:SendMsg)
moreover have "grant s' e"
proof-
from grant obtain up rp tp uq rq tq
where p1: "sectxt_of_obj s (O_proc p) = Some (up, rp, tp)"
and p2: "sectxt_of_obj s (O_msgq q) = Some (uq, rq, tq)"
apply (simp add:SendMsg split:option.splits)
by (case_tac a, case_tac aa, blast)
from p1 have p1': "sectxt_of_obj s' (O_proc p) = Some (up, rp, tp)"
using os cp2sp
apply (erule_tac x = p in allE)
by (auto simp:SendMsg co2sobj.simps cp2sproc_def split:option.splits)
from p2 have p2': "sectxt_of_obj s' (O_msgq q) = Some (uq, rq, tq)"
using os cq2sq vd
apply (erule_tac x = q in allE)
by (auto simp:SendMsg co2sobj.simps cq2smsgq_def dest!:current_has_sms' split:option.splits)
show ?thesis using p1' p2' grant p1 p2
by (simp add:SendMsg)
qed
ultimately show ?thesis using vs'
by (erule_tac valid.intros(2), simp+)
next
case (RecvMsg p q m)
have p_in: "p \<in> current_procs s'" using os alive
apply (erule_tac x = "O_proc p" in allE)
by (auto simp:RecvMsg)
have q_in: "q \<in> current_msgqs s'" using os alive
apply (erule_tac x = "O_msgq q" in allE)
by (simp add:RecvMsg)
have m_in: "m = hd (msgs_of_queue s' q)"
and sms_not_empty: "msgs_of_queue s' q \<noteq> []"
using os sms_remain
by (auto simp:RecvMsg)
have "os_grant s' e" using p_in q_in m_in sms_not_empty os
by (simp add:RecvMsg)
moreover have "grant s' e"
proof-
from grant obtain up rp tp uq rq tq um rm tm
where p1: "sectxt_of_obj s (O_proc p) = Some (up, rp, tp)"
and p2: "sectxt_of_obj s (O_msgq q) = Some (uq, rq, tq)"
and p3: "sectxt_of_obj s (O_msg q m) = Some (um, rm, tm)"
apply (simp add:RecvMsg split:option.splits)
by (case_tac a, case_tac aa, case_tac ab, blast)
from p1 have p1': "sectxt_of_obj s' (O_proc p) = Some (up, rp, tp)"
using os cp2sp
apply (erule_tac x = p in allE)
by (auto simp:RecvMsg co2sobj.simps cp2sproc_def split:option.splits)
from p2 have p2': "sectxt_of_obj s' (O_msgq q) = Some (uq, rq, tq)"
using os cq2sq vd
apply (erule_tac x = q in allE)
by (auto simp:RecvMsg co2sobj.simps cq2smsgq_def dest!:current_has_sms' split:option.splits)
from p3 have p3': "sectxt_of_obj s' (O_msg q m) = Some (um, rm, tm)"
using sms_remain cq2sq vd os p2 p2' p3
apply (erule_tac x = q in allE)
apply (erule_tac x = q in allE)
apply (clarsimp simp:RecvMsg)
apply (simp add:cq2smsgq_def split:option.splits if_splits)
apply (drule current_has_sms', simp, simp)
apply (case_tac "msgs_of_queue s q", simp)
apply (simp add:cqm2sms.simps split:option.splits)
apply (auto simp add:cm2smsg_def split:option.splits if_splits)[1]
done
show ?thesis using p1' p2' p3' grant p1 p2 p3
by (simp add:RecvMsg)
qed
ultimately show ?thesis using vs'
by (erule_tac valid.intros(2), simp+)
next
case (CreateSock p af st fd inum)
show ?thesis using grant
by (simp add:CreateSock)
next
case (Bind p fd addr)
show ?thesis using grant
by (simp add:Bind)
next
case (Connect p fd addr)
show ?thesis using grant
by (simp add:Connect)
next
case (Listen p fd)
show ?thesis using grant
by (simp add:Listen)
next
case (Accept p fd addr port fd' inum)
show ?thesis using grant
by (simp add:Accept)
next
case (SendSock p fd)
show ?thesis using grant
by (simp add:SendSock)
next
case (RecvSock p fd)
show ?thesis using grant
by (simp add:RecvSock)
next
case (Shutdown p fd how)
show ?thesis using grant
by (simp add:Shutdown)
qed
qed
lemma current_proc_fds_in_curp:
"\<lbrakk>fd \<in> current_proc_fds s p; valid s\<rbrakk> \<Longrightarrow> p \<in> current_procs s"
apply (induct s)
apply (simp add:init_fds_of_proc_prop1)
apply (frule vt_grant_os, frule vd_cons)
apply (case_tac a, auto split:if_splits option.splits)
done
lemma get_parentfs_ctxts_prop:
"\<lbrakk>get_parentfs_ctxts s (a # f) = Some ctxts; sectxt_of_obj s (O_dir f) = Some ctxt; valid s\<rbrakk>
\<Longrightarrow> ctxt \<in> set (ctxts)"
apply (induct f)
apply (auto split:option.splits)
done
lemma search_check_allp_intro:
"\<lbrakk>search_check s sp pf; get_parentfs_ctxts s pf = Some ctxts; valid s; is_dir s pf\<rbrakk>
\<Longrightarrow> search_check_allp sp (set ctxts)"
apply (case_tac pf)
apply (simp split:option.splits if_splits add:search_check_allp_def)
apply (rule ballI)
apply (auto simp:search_check_ctxt_def search_check_dir_def split:if_splits option.splits)
apply (auto simp:search_check_allp_def search_check_file_def)
apply (frule is_dir_not_file, simp)
done
lemma search_check_leveling_f:
"\<lbrakk>search_check s sp pf; parent f = Some pf; is_file s f; valid s;
sectxt_of_obj s (O_file f) = Some fctxt; search_check_file sp fctxt\<rbrakk>
\<Longrightarrow> search_check s sp f"
apply (case_tac f, simp+)
apply (auto split:option.splits simp:search_check_ctxt_def)
apply (frule parentf_is_dir_prop2, simp)
apply (erule get_pfs_secs_prop, simp)
apply (erule_tac search_check_allp_intro, simp_all)
apply (simp add:parentf_is_dir_prop2)
done
lemma current_fflag_in_fds:
"\<lbrakk>flags_of_proc_fd s p fd = Some flag; valid s\<rbrakk> \<Longrightarrow> fd \<in> current_proc_fds s p"
apply (induct s arbitrary:p)
apply (simp add:flags_of_proc_fd.simps file_of_proc_fd.simps init_oflags_prop2)
apply (frule vd_cons, frule vt_grant_os, case_tac a)
apply (auto split:if_splits option.splits dest:proc_fd_in_fds)
done
lemma current_fflag_has_ffd:
"\<lbrakk>flags_of_proc_fd s p fd = Some flag; valid s\<rbrakk> \<Longrightarrow> \<exists> f. file_of_proc_fd s p fd = Some f"
apply (induct s arbitrary:p)
apply (simp add: file_of_proc_fd.simps init_fileflag_valid)
apply (frule vd_cons, frule vt_grant_os, case_tac a)
apply (auto split:if_splits option.splits dest:proc_fd_in_fds)
done
lemma oflags_check_remove_create:
"oflags_check flags sp sf \<Longrightarrow> oflags_check (remove_create_flag flags) sp sf"
apply (case_tac flags)
apply (auto simp:oflags_check_def perms_of_flags_def perm_of_oflag_def split:bool.splits)
done
end
end