cfd2sfd, OF_create flag should not be in the sfd, cause all-duplicated sfd could not have this flag
theory Enrich
imports Main Flask Static Init_prop Valid_prop Tainted_prop Delete_prop Co2sobj_prop S2ss_prop S2ss_prop2
Temp
begin
(* objects that need dynamic indexing, all nature-numbers *)
datatype t_enrich_obj =
E_proc "t_process"
| E_file "t_file"
| E_fd "t_process" "t_fd"
| E_inum "nat"
| E_msgq "t_msgq"
| E_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 [] = init_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 = set (init_msgs_of_queue 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)"
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_prop2:
"p' \<notin> all_msgqs s \<Longrightarrow> p' \<notin> init_msgqs"
apply (induct s, simp)
by (case_tac a, auto)
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_enrich_obj \<Rightarrow> bool"
where
"enrich_not_alive s obj (E_file f) = (f \<notin> current_files s \<and> obj \<noteq> E_file f)"
| "enrich_not_alive s obj (E_proc p) = (p \<notin> current_procs s \<and> obj \<noteq> E_proc p)"
| "enrich_not_alive s obj (E_fd p fd) =
((p \<in> current_procs s \<longrightarrow> fd \<notin> current_proc_fds s p) \<and> obj \<noteq> E_fd p fd \<and> obj \<noteq> E_proc p)"
| "enrich_not_alive s obj (E_msgq q) = (q \<notin> current_msgqs s \<and> obj \<noteq> E_msgq q)"
| "enrich_not_alive s obj (E_inum i) = (i \<notin> current_inode_nums s \<and> obj \<noteq> E_inum i)"
| "enrich_not_alive s obj (E_msg q m) =
((q \<in> current_msgqs s \<longrightarrow> m \<notin> set (msgs_of_queue s q)) \<and> obj \<noteq> E_msg q m \<and> obj \<noteq> E_msgq q)"
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
(* enrich s target_proc duplicated_pro *)
fun enrich_proc :: "t_state \<Rightarrow> t_process \<Rightarrow> t_process \<Rightarrow> t_state"
where
"enrich_proc [] tp dp = []"
| "enrich_proc (Execve p f fds # s) tp dp = (
if (tp = p)
then Execve dp f (fds \<inter> proc_file_fds s p) # Execve p f fds # (enrich_proc s tp dp)
else Execve p f fds # (enrich_proc s tp dp))"
| "enrich_proc (Clone p p' fds # s) tp dp = (
if (tp = p')
then Clone p dp (fds \<inter> proc_file_fds s p) # Clone p p' fds # s
else Clone p p' fds # (enrich_proc s tp dp))"
| "enrich_proc (Open p f flags fd opt # s) tp dp = (
if (tp = p)
then Open dp f (remove_create_flag flags) fd None # Open p f flags fd opt # (enrich_proc s tp dp)
else Open p f flags fd opt # (enrich_proc s tp dp))"
| "enrich_proc (ReadFile p fd # s) tp dp = (
if (tp = p)
then ReadFile dp fd # ReadFile p fd # (enrich_proc s tp dp)
else ReadFile p fd # (enrich_proc s tp dp))"
(*
| "enrich_proc (CloseFd p fd # s) tp dp = (
if (tp = p \<and> fd \<in> proc_file_fds s p)
then CloseFd dp fd # CloseFd p fd # (enrich_proc s tp dp)
else CloseFd p fd # (enrich_proc s tp dp))"
*)
(*
| "enrich_proc (Attach p h flag # s) tp dp = (
if (tp = p)
then Attach dp h flag # Attach p h flag # (enrich_proc s tp dp)
else Attach p h flag # (enrich_proc s tp dp))"
| "enrich_proc (Detach p h # s) tp dp = (
if (tp = p)
then Detach dp h # Detach p h # (enrich_proc s tp dp)
else Detach p h # (enrich_proc s tp dp))"
*)
(*
| "enrich_proc (Kill p p' # s) tp dp = (
if (tp = p') then Kill p p' # s
else Kill p p' # (enrich_proc s tp dp))"
| "enrich_proc (Exit p # s) tp dp = (
if (tp = p) then Exit p # s
else Exit p # (enrich_proc s tp dp))"
*)
| "enrich_proc (e # s) tp dp = e # (enrich_proc s tp dp)"
definition is_created_proc:: "t_state \<Rightarrow> t_process \<Rightarrow> bool"
where
"is_created_proc s p \<equiv> p \<in> current_procs s \<and> (p \<in> init_procs \<longrightarrow> died (O_proc p) s)"
definition is_created_proc':: "t_state \<Rightarrow> t_process \<Rightarrow> bool"
where
"is_created_proc' s p \<equiv> p \<in> current_procs s \<and> p \<notin> init_procs"
lemma no_del_died:
"\<lbrakk>no_del_event s; died obj s\<rbrakk> \<Longrightarrow> (\<exists> p fd. obj = O_fd p fd \<or> obj = O_tcp_sock (p, fd) \<or> obj = O_udp_sock (p, fd))
\<or> (\<exists> q m. obj = O_msg q m) "
apply (induct s)
apply simp
apply (case_tac a)
apply (auto split:option.splits)
done
lemma no_del_created_eq:
"no_del_event s \<Longrightarrow> is_created_proc s p = is_created_proc' s p"
apply (induct s)
apply (simp add:is_created_proc_def is_created_proc'_def)
apply (case_tac a)
apply (auto simp add:is_created_proc_def is_created_proc'_def dest:no_del_died)
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. 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_all: "notin_all (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_all os Clone
apply (erule_tac x = "E_proc p'" in allE)
apply (auto dest:not_all_procs_prop3)
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_all Open None
apply (erule_tac x = "E_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_all
apply (erule_tac x = "E_file f" in allE)
apply (case_tac obj')
by auto
have fd_not_in: "fd \<notin> current_proc_fds s' p"
using os alive' p_in Open Some notin_all
apply (erule_tac x = "E_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_all
apply (erule_tac x = "E_inum inum" in allE)
by (case_tac obj', auto)
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_all
apply (clarsimp simp add:dir_is_empty_def f_in)
using alive'
apply (erule_tac x = "E_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)
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_all
apply (erule_tac x = "E_file f" in allE)
by (auto)
have inum_not_in: "inum \<notin> current_inode_nums s'"
using os alive' Mkdir notin_all
apply (erule_tac x = "E_inum inum" in allE)
by (auto)
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_all
apply (erule_tac x = "E_file f'" in allE)
by (auto simp:LinkHard)
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_all
apply (erule_tac x = "E_msgq q" in allE)
by auto
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_all SendMsg q_in
apply (erule_tac x = "E_msg q m" in allE)
apply (case_tac obj', auto dest:not_all_msgqs_prop3)
done
have "os_grant s' e" using p_in q_in m_not_in
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]
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 created_proc_clone:
"valid (Clone p p' fds # s) \<Longrightarrow>
is_created_proc (Clone p p' fds # s) tp = (if (tp = p') then True else is_created_proc s tp)"
apply (drule vt_grant_os)
apply (auto simp:is_created_proc_def dest:not_all_procs_prop2)
using not_died_init_proc
by auto
lemma created_proc_exit:
"is_created_proc (Exit p # s) tp = (if (tp = p) then False else is_created_proc s tp)"
by (simp add:is_created_proc_def)
lemma created_proc_kill:
"is_created_proc (Kill p p' # s) tp = (if (tp = p') then False else is_created_proc s tp)"
by (simp add:is_created_proc_def)
lemma created_proc_other:
"\<lbrakk>\<And> p p' fds. e \<noteq> Clone p p' fds;
\<And> p. e \<noteq> Exit p;
\<And> p p'. e \<noteq> Kill p p'\<rbrakk> \<Longrightarrow> is_created_proc (e # s) tp = is_created_proc s tp"
by (case_tac e, auto simp:is_created_proc_def)
lemmas is_created_proc_simps = created_proc_clone created_proc_exit created_proc_kill created_proc_other
(*
(p \<in> current_procs s \<longrightarrow> co2sobj (enrich_proc s p p') (O_proc p') = co2sobj (enrich_proc s p p') (O_proc p)) \<and>
(\<forall> obj. alive s obj \<longrightarrow> alive (enrich_proc s p p') obj) \<and>
(\<forall> p'. p' \<in> current_procs s \<longrightarrow> cp2sproc (enrich_proc s p p') p' = cp2sproc s p) \<and>
(\<forall> f. f \<in> current_files s \<longrightarrow> cf2sfile (enrich_proc s p p') f = cf2sfile s f) \<and>
(Tainted (enrich_proc s p p') = (Tainted s \<union> (if (O_proc p \<in> Tainted s) then {O_proc p'} else {})))"*)
lemma enrich_proc_dup_in:
"\<lbrakk>is_created_proc s p; p' \<notin> all_procs s; valid s\<rbrakk>
\<Longrightarrow> p' \<in> current_procs (enrich_proc s p p')"
apply (induct s, simp add:is_created_proc_def)
apply (frule vt_grant_os, frule vd_cons)
apply (case_tac a, auto simp:is_created_proc_def dest:not_all_procs_prop3)
done
lemma enrich_proc_dup_ffd:
"\<lbrakk>file_of_proc_fd s p fd = Some f; is_created_proc s p; p' \<notin> all_procs s; valid s\<rbrakk>
\<Longrightarrow> file_of_proc_fd (enrich_proc s p p') p' fd = Some f"
apply (induct s, simp add:is_created_proc_def)
apply (frule vt_grant_os, frule vd_cons)
apply (case_tac a, auto simp:is_created_proc_def proc_file_fds_def
dest:not_all_procs_prop3 split:if_splits option.splits)
done
lemma enrich_proc_dup_ffd':
"\<lbrakk>file_of_proc_fd (enrich_proc s p p') p' fd = Some f; is_created_proc s p; p' \<notin> all_procs s;
no_del_event s; valid s\<rbrakk>
\<Longrightarrow> file_of_proc_fd s p fd = Some f"
apply (induct s, simp add:is_created_proc_def)
apply (frule vt_grant_os, frule vd_cons)
apply (case_tac a, auto simp:is_created_proc_def proc_file_fds_def
dest:not_all_procs_prop3 split:if_splits option.splits)
done
lemma enrich_proc_dup_ffd_eq:
"\<lbrakk>is_created_proc s p; p' \<notin> all_procs s; no_del_event s; valid s\<rbrakk>
\<Longrightarrow> file_of_proc_fd (enrich_proc s p p') p' fd = file_of_proc_fd s p fd"
apply (case_tac "file_of_proc_fd s p fd")
apply (case_tac[!] "file_of_proc_fd (enrich_proc s p p') p' fd")
apply (auto dest:enrich_proc_dup_ffd enrich_proc_dup_ffd')
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 enrich_proc_dup_fflags:
"\<lbrakk>flags_of_proc_fd s p fd = Some flag; is_created_proc s p; p' \<notin> all_procs s; valid s\<rbrakk>
\<Longrightarrow> flags_of_proc_fd (enrich_proc s p p') p' fd = Some (remove_create_flag flag) \<or>
flags_of_proc_fd (enrich_proc s p p') p' fd = Some flag"
apply (induct s arbitrary:p, simp add:is_created_proc_def)
apply (frule vt_grant_os, frule vd_cons)
apply (case_tac a, auto simp:is_created_proc_def proc_file_fds_def is_creat_flag_def
dest:not_all_procs_prop3 split:if_splits option.splits)
done
lemma enrich_proc_dup_ffds:
"\<lbrakk>is_created_proc s p; p' \<notin> all_procs s; no_del_event s; valid s\<rbrakk>
\<Longrightarrow> proc_file_fds (enrich_proc s p p') p' = proc_file_fds s p"
apply (auto simp:proc_file_fds_def)
apply (rule_tac x = f in exI)
apply (erule enrich_proc_dup_ffd', simp+)
apply (rule_tac x = f in exI)
apply (erule enrich_proc_dup_ffd, simp+)
done
lemma enrich_proc_dup_ffds_eq_fds:
"\<lbrakk>is_created_proc s p; p' \<notin> all_procs s; no_del_event s; valid s\<rbrakk>
\<Longrightarrow> current_proc_fds (enrich_proc s p p') p' = proc_file_fds s p"
apply (induct s arbitrary:p)
apply (simp add: is_created_proc_def)
apply (frule not_all_procs_prop3)
apply (frule vd_cons, frule vt_grant_os, case_tac a)
apply (auto split:if_splits option.splits dest:proc_fd_in_fds set_mp not_all_procs_prop3
simp:proc_file_fds_def is_created_proc_def)
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