diff -r f27ba31b7e96 -r 6f7b9039715f no_shm_selinux/Flask.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/no_shm_selinux/Flask.thy	Tue Dec 17 13:30:21 2013 +0800
@@ -0,0 +1,1654 @@
+theory Flask
+imports Main Flask_type OS_type_def File_renaming 
+begin
+
+(****** Policy-language-level rule match/auxiliary functions ******)
+
+fun patt_match :: "'a patt \<Rightarrow> 'a \<Rightarrow> bool"
+where
+  "patt_match (Single x) y = (x = y)"
+| "patt_match (Set s)    x = (x \<in> s)"
+| "patt_match (Tilde s)  x = (x \<notin> s)"
+| "patt_match Asterisk   x = True"
+
+(* match a target with a patt, but including the SELF special 
+ * patt_match_self patt target source 
+ *)
+fun patt_match_self :: "(type_t patt) target_with_self_t \<Rightarrow> type_t \<Rightarrow> type_t \<Rightarrow> bool"
+where
+  "patt_match_self Self                  tt st = (tt = st)"
+| "patt_match_self (Not_self (Single p)) tt st = (tt = p)"
+| "patt_match_self (Not_self (Set p))    tt st = (tt \<in> p)"
+| "patt_match_self (Not_self (Tilde p))  tt st = (tt \<notin> p)"
+| "patt_match_self (Not_self Asterisk)   tt st = True"
+
+(* list lookup: find the first one *)
+fun lookup :: "'a list \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'a option"
+where
+  "lookup [] P = None"
+| "lookup (x # xs) P = (if (P x) then Some x else lookup xs P)"
+
+(* list member: if exists an element satisfy predicate P *)
+fun member :: "'a list \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool"
+where
+  "member [] P = False"
+| "member (x # xs) P = (if (P x) then True else member xs P)"
+
+(******* socket related aux functions ********)
+fun is_remote_request :: "t_event \<Rightarrow> bool"    
+where
+  "is_remote_request (Accept p fd addr port fd' i)  = True"
+| "is_remote_request (Connect p fd addr)   = True"
+| "is_remote_request (SendSock p fd) = True"
+| "is_remote_request (RecvSock p fd) = True"
+| "is_remote_request e         = False"
+
+fun ip_of_addr:: "t_sock_addr \<Rightarrow> t_inet_addr"
+where
+  "ip_of_addr (INET_ADDR saddr sport) = saddr"
+
+fun port_of_addr:: "t_sock_addr \<Rightarrow> t_socket_port"
+where
+  "port_of_addr (INET_ADDR saddr sport) = sport"
+
+fun after_bind:: "t_sock_state option \<Rightarrow> bool"
+where
+  "after_bind None = False"
+| "after_bind (Some SS_create) = False"
+| "after_bind _= True" (* after bind , it should have local addr *)
+
+fun after_conacc:: "t_sock_state option \<Rightarrow> bool"
+where
+  "after_conacc None = False"
+| "after_conacc (Some SS_create) = False"
+| "after_conacc (Some SS_bind) = False"
+| "after_conacc (Some SS_listen) = False"
+| "after_conacc _ = True"  (* after connect or accept, it should have remote addr *)
+
+(* inet_addr_compare addr1 addr2 \<equiv> addr1 \<le> addr2; but in \<le>'s def,there is a \<exists>, is not a function*)
+fun subnet_addr :: "t_inet_addr \<Rightarrow> t_inet_addr \<Rightarrow> bool"
+where
+  "subnet_addr []     addr   = True"
+| "subnet_addr (x#xs) []     = False"
+| "subnet_addr (x#xs) (y#ys) = ((x = y) \<and> subnet_addr xs ys)"
+
+definition port_in_range :: "t_socket_port \<Rightarrow> t_socket_port \<Rightarrow> t_socket_port \<Rightarrow> bool"
+where
+  "port_in_range minp maxp p \<equiv> ((p < maxp \<or> p = maxp) \<and> (p > minp \<or> p = minp))"
+
+(* initial value of initiate state constrains aux function *)
+definition bidirect_in_init :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b option) \<Rightarrow> bool"
+where
+  "bidirect_in_init S f \<equiv>  (\<forall> a. a \<in> S \<longrightarrow> (\<exists> b. f a = Some b)) \<and> 
+                           (\<forall> a b. f a = Some b \<longrightarrow> a \<in> S)" 
+
+fun is_file_itag :: "t_inode_tag \<Rightarrow> bool"
+where
+  "is_file_itag Tag_FILE = True"
+| "is_file_itag tag      = False"
+
+fun is_dir_itag :: "t_inode_tag \<Rightarrow> bool"
+where
+  "is_dir_itag Tag_DIR = True"
+| "is_dir_itag tag     = False"
+
+fun is_file_dir_itag :: "t_inode_tag \<Rightarrow> bool"
+where
+  "is_file_dir_itag Tag_FILE = True"
+| "is_file_dir_itag Tag_DIR  = True"
+| "is_file_dir_itag tag      = False"
+
+fun is_tcp_itag :: "t_inode_tag \<Rightarrow> bool"
+where
+  "is_tcp_itag Tag_TCP_SOCK = True"
+| "is_tcp_itag tag      = False"
+
+fun is_udp_itag :: "t_inode_tag \<Rightarrow> bool"
+where
+  "is_udp_itag Tag_UDP_SOCK = True"
+| "is_udp_itag tag     = False"
+
+fun is_sock_itag :: "t_inode_tag \<Rightarrow> bool"
+where
+  "is_sock_itag Tag_TCP_SOCK = True"
+| "is_sock_itag Tag_UDP_SOCK = True"
+| "is_sock_itag tag          = False"
+
+
+locale init = 
+  fixes 
+      init_files :: "t_file set"
+  and init_procs :: "t_process set"
+  and init_nodes :: "t_node set"
+  and init_sockets :: "t_socket set"
+  and init_users :: "user_t set"
+
+  and init_file_of_proc_fd ::   "t_process \<Rightarrow> t_fd \<rightharpoonup> t_file"
+  and init_fds_of_proc ::       "t_process \<Rightarrow> t_fd set"
+  and init_oflags_of_proc_fd::  "t_process \<Rightarrow> t_fd \<rightharpoonup> t_open_flags"
+  and init_inum_of_file ::      "t_file \<rightharpoonup> t_inode_num"
+  and init_inum_of_socket::     "t_socket  \<rightharpoonup> t_inode_num" 
+  and init_itag_of_inum::       "t_inode_num \<rightharpoonup> t_inode_tag"
+  and init_files_hung_by_del :: "t_file set"
+(*
+  and init_file_at_writing_by:: "t_file \<Rightarrow> (t_process \<times> t_fd) set"
+*)
+  and init_socket_state ::      "t_socket \<rightharpoonup> t_sock_state"
+
+  and init_msgqs :: "t_msgq set"
+  and init_msgs_of_queue:: "t_msgq \<Rightarrow> t_msg list"
+(*
+  and init_shms :: "t_shm set"
+  and init_procs_of_shm :: "t_shm \<Rightarrow> (t_process \<times> t_shm_attach_flag) set" 
+  and init_flag_of_proc_shm :: "t_process \<Rightarrow> t_shm \<rightharpoonup> t_shm_attach_flag"
+*)
+  assumes
+      parent_file_in_init: "\<And> f pf. \<lbrakk>parent f = Some pf; f \<in> init_files\<rbrakk> \<Longrightarrow> pf \<in> init_files"
+  and root_is_dir: "\<exists> i. init_inum_of_file [] = Some i \<and> init_itag_of_inum i = Some Tag_DIR"
+  and init_file_has_inum:  "\<And> f. f \<in> init_files \<Longrightarrow> \<exists> im. init_inum_of_file f = Some im"
+  and inof_has_file_tag:   "\<And> f im. init_inum_of_file f = Some im \<Longrightarrow> f \<in> init_files \<and> (\<exists> tag. init_itag_of_inum im = Some tag \<and> is_file_dir_itag tag)"
+  and init_file_no_son:    "\<And> f im. \<lbrakk>init_inum_of_file f = Some im; init_itag_of_inum im = Some Tag_FILE\<rbrakk> \<Longrightarrow> \<not> (\<exists> f' \<in> init_files. parent f' = Some f)"
+  and init_parentf_is_dir: "\<And> f pf im. \<lbrakk>parent f = Some pf; f \<in> init_files; init_inum_of_file pf = Some im\<rbrakk> \<Longrightarrow> init_itag_of_inum im = Some Tag_DIR"
+
+  and init_files_hung_valid:   "\<And> f. f \<in> init_files_hung_by_del \<Longrightarrow> f \<in> init_files \<and> (\<exists> p fd. init_file_of_proc_fd p fd = Some f)"
+  and init_files_hung_valid':  "\<And> f im. \<lbrakk>f \<in> init_files_hung_by_del; init_inum_of_file f = Some im\<rbrakk> \<Longrightarrow> 
+                                         init_itag_of_inum im = Some Tag_FILE \<or> (init_itag_of_inum im = Some Tag_DIR \<and> \<not> (\<exists> f'. f' \<in> init_files \<and> f \<prec> f'))"
+
+  and init_procfds_valid:      "\<And> p fd. fd \<in> init_fds_of_proc p \<Longrightarrow> p \<in> init_procs \<and> ((\<exists> f \<in> init_files. init_file_of_proc_fd p fd = Some f) \<or> (p, fd) \<in> init_sockets)"
+  and init_filefd_valid:       "\<And> p fd f. init_file_of_proc_fd p fd = Some f \<Longrightarrow> fd \<in> init_fds_of_proc p \<and> (\<exists> im. init_inum_of_file f = Some im \<and> init_itag_of_inum im = Some Tag_FILE) \<and> p \<in> init_procs \<and> (\<exists> flags. init_oflags_of_proc_fd p fd = Some flags) \<and> (p, fd) \<notin> init_sockets"
+  and init_fileflag_valid:     "\<And> p fd flags. init_oflags_of_proc_fd p fd = Some flags \<Longrightarrow> \<exists> f. init_file_of_proc_fd p fd = Some f"
+
+(*
+  and init_procs_has_shm:      "\<And> p h flag. (p,flag) \<in> init_procs_of_shm h \<Longrightarrow> p \<in> init_procs \<and> h \<in> init_shms \<and> init_flag_of_proc_shm p h = Some flag"
+  and init_shmflag_has_proc:   "\<And> p h flag. init_flag_of_proc_shm p h = Some flag \<Longrightarrow> (p, flag) \<in> init_procs_of_shm h"
+*)
+  and init_msgs_distinct:      "\<And> q. distinct (init_msgs_of_queue q)"
+  and init_msgs_valid:         "\<And> m q. m \<in> set (init_msgs_of_queue q) \<Longrightarrow> q \<in> init_msgqs"
+
+  and init_socket_has_inode:   "\<And> p fd. (p, fd) \<in> init_sockets \<Longrightarrow> \<exists> im. init_inum_of_socket (p, fd) = Some im \<and> p \<in> init_procs \<and> fd \<in> init_fds_of_proc p \<and> init_file_of_proc_fd p fd = None"
+  and inos_has_sock_tag:  "\<And> s im. init_inum_of_socket s = Some im \<Longrightarrow> s \<in> init_sockets \<and> (\<exists> tag. init_itag_of_inum im = Some tag \<and> is_sock_itag tag)"
+(*
+  and init_netobj_has_state:   "bidirect_in_init init_netobjs init_netobj_state"
+  and init_netobj_has_socktype:"bidirect_in_init init_netobjs init_netobj_socktype"
+  and init_netobj_has_laddr:   "\<And> s. after_bind (init_netobj_state s) \<Longrightarrow> init_netobj_localaddr s \<noteq> None"
+  and init_netobj_has_raddr:   "\<And> s. after_conacc (init_netobj_state s) \<Longrightarrow> init_netobj_remoteaddr s \<noteq> None"
+*)
+
+  and init_finite_sets:   "finite init_files \<and> finite init_procs \<and> (\<forall> p. finite (init_fds_of_proc p)) \<and> finite init_shms \<and> finite init_msgqs \<and> finite init_users" 
+
+begin 
+
+definition is_init_file:: "t_file \<Rightarrow> bool"
+where
+  "is_init_file f \<equiv> (case (init_inum_of_file f) of
+                       Some i \<Rightarrow> (case (init_itag_of_inum i) of
+                                    Some Tag_FILE \<Rightarrow> True
+                                  | _             \<Rightarrow> False)
+                     | _      \<Rightarrow> False)"
+
+definition is_init_dir:: "t_file \<Rightarrow> bool"
+where
+  "is_init_dir f \<equiv> (case (init_inum_of_file f) of
+                      Some i \<Rightarrow> (case (init_itag_of_inum i) of
+                                   Some Tag_DIR \<Rightarrow> True
+                                 | _            \<Rightarrow> False)
+                    | _      \<Rightarrow> False)"
+
+definition is_init_tcp_sock:: "t_socket \<Rightarrow> bool"
+where
+  "is_init_tcp_sock s \<equiv> (case (init_inum_of_socket s) of
+                           Some i \<Rightarrow> (case (init_itag_of_inum i) of
+                                        Some Tag_TCP_SOCK \<Rightarrow> True
+                                      | _                 \<Rightarrow> False)
+                         | _      \<Rightarrow> False)"
+
+definition is_init_udp_sock:: "t_socket \<Rightarrow> bool"
+where
+  "is_init_udp_sock s \<equiv> (case (init_inum_of_socket s) of
+                           Some i \<Rightarrow> (case (init_itag_of_inum i) of
+                                        Some Tag_UDP_SOCK \<Rightarrow> True
+                                      | _                 \<Rightarrow> False)
+                         | _      \<Rightarrow> False)"
+
+definition init_same_inode_files :: "t_file \<Rightarrow> t_file set"
+where
+  "init_same_inode_files f \<equiv> if is_init_file f then {f'. init_inum_of_file f = init_inum_of_file f'} else {}"
+
+fun init_alive :: "t_object \<Rightarrow> bool"
+where
+  "init_alive (O_proc p)     = (p \<in> init_procs)"
+| "init_alive (O_file f)     = (is_init_file f)"
+| "init_alive (O_dir  f)     = (is_init_dir  f)"
+| "init_alive (O_fd p fd)    = (fd \<in> init_fds_of_proc p)"
+| "init_alive (O_node n)     = (n \<in> init_nodes)"
+| "init_alive (O_tcp_sock k) = (is_init_tcp_sock k)"
+| "init_alive (O_udp_sock k) = (is_init_udp_sock k)"
+(*
+| "init_alive (O_shm  h)     = (h \<in> init_shms)"
+*)
+| "init_alive (O_msgq q)     = (q \<in> init_msgqs)"
+| "init_alive (O_msg q m)    = (m \<in> set (init_msgs_of_queue q) \<and> q \<in> init_msgqs)"
+
+
+(************ system listeners, event-related ***********)
+
+fun file_of_proc_fd ::  "t_state \<Rightarrow> t_process \<Rightarrow> t_fd \<Rightarrow> t_file option"
+where
+  "file_of_proc_fd [] p' fd' = (init_file_of_proc_fd p' fd')"
+| "file_of_proc_fd (Open p f flags fd iopt # \<tau>) p' fd' = 
+     (if (p' = p \<and> fd = fd') then Some f else file_of_proc_fd \<tau> p' fd')"
+| "file_of_proc_fd (CloseFd p fd # \<tau>) p' fd' = 
+     (if (p = p' \<and> fd = fd') then None else file_of_proc_fd \<tau> p' fd')"
+(*deleted: if (f\<^isub>3 \<preceq> f\<^isub>1) then None else *)
+(*
+| "file_of_proc_fd (Rename p f\<^isub>2 f\<^isub>3 # \<tau>) p' fd = 
+     (case (file_of_proc_fd \<tau> p' fd) of
+        Some f\<^isub>1 \<Rightarrow> (if (f\<^isub>2 \<preceq> f\<^isub>1) then Some (file_after_rename f\<^isub>2 f\<^isub>3 f\<^isub>1) 
+                    else Some f\<^isub>1)
+      | None    \<Rightarrow> None )"
+*)
+| "file_of_proc_fd (Execve p f fds # \<tau>) p' fd = 
+     (if (p' = p \<and> fd \<in> fds) then file_of_proc_fd \<tau> p fd 
+      else if (p' = p) then None
+      else file_of_proc_fd \<tau> p' fd) "
+| "file_of_proc_fd (Clone p p' fds # \<tau>) p'' fd = 
+     (if (p'' = p' \<and> fd \<in> fds) then file_of_proc_fd \<tau> p fd
+      else if (p'' = p') then None
+      else file_of_proc_fd \<tau> p'' fd)"
+| "file_of_proc_fd (Kill p\<^isub> p\<^isub>' # \<tau>) p'' fd = 
+     (if (p'' = p\<^isub>') then None else file_of_proc_fd \<tau> p'' fd)"
+| "file_of_proc_fd (Exit p # \<tau>) p' fd = 
+     (if (p = p') then None else file_of_proc_fd \<tau> p' fd)"
+| "file_of_proc_fd (_ # \<tau>) p fd = file_of_proc_fd \<tau> p fd"
+
+definition proc_fd_of_file :: "t_state \<Rightarrow> t_file \<Rightarrow> (t_process \<times> t_fd) set"
+where
+  "proc_fd_of_file \<tau> f = {(p, fd) | p fd. file_of_proc_fd \<tau> p fd = Some f}"
+
+definition proc_file_fds :: "t_state \<Rightarrow> t_process \<Rightarrow> t_fd set"
+where
+  "proc_file_fds s p \<equiv> {fd. \<exists> f. file_of_proc_fd s p fd = Some f}"
+
+definition init_proc_file_fds:: "t_process \<Rightarrow> t_fd set"
+where
+  "init_proc_file_fds p \<equiv> {fd. \<exists> f. init_file_of_proc_fd p fd = Some f}"
+
+(****** files and directories ******)
+
+fun files_hung_by_del :: "t_state \<Rightarrow> t_file set"
+where
+  "files_hung_by_del [] = init_files_hung_by_del"
+| "files_hung_by_del (UnLink p f # \<tau>) = (
+     if (proc_fd_of_file \<tau> f = {}) then files_hung_by_del \<tau> 
+     else insert f (files_hung_by_del \<tau>))"
+(* | files_hung_by_del (Rmdir p f) is not need, for open can only apply to file not dir *)
+| "files_hung_by_del (CloseFd p fd # \<tau>) = (
+     case (file_of_proc_fd \<tau> p fd) of
+       Some f \<Rightarrow> (if (proc_fd_of_file \<tau> f = {(p, fd)}) 
+                  then files_hung_by_del \<tau> - {f}
+                  else files_hung_by_del \<tau>)
+     | None   \<Rightarrow> files_hung_by_del \<tau>            )"
+(*
+| "files_hung_by_del (Rename p f\<^isub>2 f\<^isub>3 # \<tau>) = {file_after_rename f\<^isub>2 f\<^isub>3 f\<^isub>1| f\<^isub>1. f\<^isub>1 \<in> files_hung_by_del \<tau>}" (* for rename(2) does not infect on fd of the file, see man -S 2 rename *)
+*)
+| "files_hung_by_del (e # \<tau>) = files_hung_by_del \<tau>"
+declare files_hung_by_del.simps [simp del]
+
+fun inum_of_file :: "t_state \<Rightarrow> (t_file \<rightharpoonup> t_inode_num)"
+where
+  "inum_of_file [] = init_inum_of_file"
+| "inum_of_file (Open p f flags fd (Some inum) # \<tau>) = (inum_of_file \<tau>) (f := Some inum)" 
+| "inum_of_file (CloseFd p fd # \<tau>) = (
+     case (file_of_proc_fd \<tau> p fd) of
+       Some f \<Rightarrow> ( if ((proc_fd_of_file \<tau> f = {(p, fd)}) \<and> (f \<in> files_hung_by_del \<tau>)) 
+                   then (inum_of_file \<tau>) (f := None)
+                   else inum_of_file \<tau> )
+     | _      \<Rightarrow> (inum_of_file \<tau>)    )"
+| "inum_of_file (UnLink p f # \<tau>) = (
+     if (proc_fd_of_file \<tau> f = {}) 
+     then (inum_of_file \<tau>) (f := None)
+     else inum_of_file \<tau>            )" 
+| "inum_of_file (Rmdir p f # \<tau>) = (inum_of_file \<tau>) (f := None)"
+| "inum_of_file (Mkdir p f ino # \<tau>) = (inum_of_file \<tau>) (f:=  Some ino)"
+| "inum_of_file (LinkHard p f\<^isub>1 f\<^isub>2 # \<tau>) = (inum_of_file \<tau>) (f\<^isub>2 := inum_of_file \<tau> f\<^isub>1)"
+(*
+| "inum_of_file (Rename p f\<^isub>2 f\<^isub>3 # \<tau>) = (\<lambda> f. 
+     if (f\<^isub>2 \<preceq> f) then None
+     else if (f\<^isub>3 \<preceq> f) then inum_of_file \<tau> (file_before_rename f\<^isub>2 f\<^isub>3 f)
+     else inum_of_file \<tau> f         )"
+*)
+| "inum_of_file (e # \<tau>) = inum_of_file \<tau>" 
+
+definition current_files :: "t_state \<Rightarrow> t_file set"
+where
+  "current_files \<tau> = {f. \<exists> i. inum_of_file \<tau> f = Some i}" 
+
+(* experimentary: for the delete obj's (file or socket) inode num, it is unnecessary for itag_of_inum to be NONE, cause this is the situation blocked by inum_of_file / inum_of_socket !!! *)  
+(* delete a file, just recycle its inode num, but not destroy its inode \<dots>, so it is irrelative to these events, like closeFd, Rmdir, UnLink \<dots>*)
+fun itag_of_inum :: "t_state \<Rightarrow> (t_inode_num \<rightharpoonup> t_inode_tag)"
+where
+  "itag_of_inum [] = init_itag_of_inum"
+| "itag_of_inum (Open p f flags fd (Some ino) # \<tau>) = (itag_of_inum \<tau>) (ino := Some Tag_FILE)"
+| "itag_of_inum (Mkdir p f ino # \<tau>) = (itag_of_inum \<tau>) (ino := Some Tag_DIR)" 
+| "itag_of_inum (CreateSock p af st fd ino # \<tau>) = 
+     (case st of 
+        STREAM \<Rightarrow> (itag_of_inum \<tau>) (ino := Some Tag_TCP_SOCK)
+      | DGRAM  \<Rightarrow> (itag_of_inum \<tau>) (ino := Some Tag_UDP_SOCK) )"
+| "itag_of_inum (Accept p fd addr lport' fd' ino # \<tau>) = 
+     (itag_of_inum \<tau>) (ino := Some Tag_TCP_SOCK)"
+| "itag_of_inum (_ # \<tau>) = itag_of_inum \<tau>"  
+
+definition is_file:: "t_state \<Rightarrow> t_file \<Rightarrow> bool"
+where
+  "is_file \<tau> f \<equiv> (case (inum_of_file \<tau> f) of
+                    Some i \<Rightarrow> (case (itag_of_inum \<tau> i) of
+                                 Some Tag_FILE \<Rightarrow> True
+                               | _             \<Rightarrow> False)
+                  | _      \<Rightarrow> False)"
+
+definition is_dir:: "t_state \<Rightarrow> t_file \<Rightarrow> bool"
+where
+  "is_dir \<tau> f \<equiv> (case (inum_of_file \<tau> f) of
+                    Some i \<Rightarrow> (case (itag_of_inum \<tau> i) of
+                                 Some Tag_DIR \<Rightarrow> True
+                               | _            \<Rightarrow> False)
+                  | _      \<Rightarrow> False)"
+
+definition dir_is_empty :: "t_state \<Rightarrow> t_file \<Rightarrow> bool"
+where
+  "dir_is_empty \<tau> f \<equiv> is_dir \<tau> f \<and> \<not> (\<exists> f'. f' \<in> current_files \<tau> \<and> f \<prec> f')"
+
+definition same_inode_files :: "t_state \<Rightarrow> t_file \<Rightarrow> t_file set"
+where
+  "same_inode_files s f \<equiv> if is_file s f then {f'. inum_of_file s f = inum_of_file s f'} else {}"
+
+definition has_same_inode :: "t_state \<Rightarrow> t_file \<Rightarrow> t_file \<Rightarrow> bool"
+where 
+  "has_same_inode s fa fb \<equiv> fb \<in> same_inode_files s fa"
+
+fun inum_of_socket :: "t_state \<Rightarrow> (t_socket \<rightharpoonup> t_inode_num)"
+where
+  "inum_of_socket [] = init_inum_of_socket"
+| "inum_of_socket (CloseFd p fd # \<tau>) = 
+     (inum_of_socket \<tau>) ((p, fd):= None)"
+| "inum_of_socket (CreateSock p af st fd ino # \<tau>) = 
+     (inum_of_socket \<tau>) ((p, fd) := Some ino)"
+| "inum_of_socket (Accept p fd addr lport' fd' ino # \<tau>) = 
+     (inum_of_socket \<tau>) ((p, fd') := Some ino)"
+| "inum_of_socket (Clone p p' fds # \<tau>) = 
+     (\<lambda> (p'', fd). if (p'' = p' \<and> fd \<in> fds) then inum_of_socket \<tau> (p, fd) 
+                   else if (p'' = p') then None
+                   else inum_of_socket \<tau> (p'',fd))" 
+| "inum_of_socket (Execve p f fds # \<tau>) = 
+     (\<lambda> (p', fd). if (p' = p \<and> fd \<in> fds) then inum_of_socket \<tau> (p, fd)
+                  else if (p' = p) then None
+                  else inum_of_socket \<tau> (p', fd))"
+| "inum_of_socket (Kill p\<^isub>1 p\<^isub>2 # \<tau>) = 
+     (\<lambda> (p, fd). if (p = p\<^isub>2) then None else inum_of_socket \<tau> (p, fd) )"
+| "inum_of_socket (Exit p # \<tau>) = 
+     (\<lambda> (p', fd). if (p' = p) then None else inum_of_socket \<tau> (p', fd))"
+| "inum_of_socket (_ # \<tau>) = inum_of_socket \<tau>" 
+
+definition current_sockets :: "t_state \<Rightarrow> t_socket set"
+where
+  "current_sockets \<tau> = {s. \<exists> i. inum_of_socket \<tau> s = Some i}"
+
+definition is_tcp_sock :: "t_state \<Rightarrow> t_socket \<Rightarrow> bool"
+where
+  "is_tcp_sock \<tau> s \<equiv> (case (inum_of_socket \<tau> s) of
+                        Some i \<Rightarrow> (case (itag_of_inum \<tau> i) of
+                                     Some Tag_TCP_SOCK \<Rightarrow> True
+                                   | _                 \<Rightarrow> False)
+                      | _      \<Rightarrow> False)"
+
+definition is_udp_sock :: "t_state \<Rightarrow> t_socket \<Rightarrow> bool"
+where
+  "is_udp_sock \<tau> s \<equiv> (case (inum_of_socket \<tau> s) of
+                        Some i \<Rightarrow> (case (itag_of_inum \<tau> i) of
+                                     Some Tag_UDP_SOCK \<Rightarrow> True
+                                   | _                 \<Rightarrow> False)
+                      | _      \<Rightarrow> False)"
+
+fun current_procs :: "t_state \<Rightarrow> t_process set"
+where
+  "current_procs [] = init_procs"
+| "current_procs (Clone p p' fds # \<tau>) = (insert p' (current_procs \<tau>))"
+| "current_procs (Exit p # \<tau>) = (current_procs \<tau> - {p})"
+| "current_procs (Kill p p' # \<tau>) = (current_procs \<tau> - {p'})"
+| "current_procs (_ # \<tau>) = current_procs \<tau>" 
+
+fun current_proc_fds :: "t_state \<Rightarrow> t_process \<Rightarrow> t_fd set" 
+where  
+  "current_proc_fds [] = init_fds_of_proc"
+| "current_proc_fds (Open p f flags fd iopt # \<tau>) = 
+     (current_proc_fds \<tau>) (p:= insert fd (current_proc_fds \<tau> p))"
+| "current_proc_fds (CreateSock p sf st newfd newi # \<tau>) =
+     (current_proc_fds \<tau>) (p:= insert newfd (current_proc_fds \<tau> p))" 
+| "current_proc_fds (Accept p fd raddr port fd' ino # \<tau>) = 
+     (current_proc_fds \<tau>) (p:= insert fd' (current_proc_fds \<tau> p))"
+| "current_proc_fds (CloseFd p fd # \<tau>) = 
+     (current_proc_fds \<tau>) (p:= (current_proc_fds \<tau> p) - {fd})"
+| "current_proc_fds (Clone p p' fds # \<tau>) = 
+     (current_proc_fds \<tau>) (p':= fds)"
+| "current_proc_fds (Execve p f fds # \<tau>) = 
+     (current_proc_fds \<tau>) (p:= fds)"
+| "current_proc_fds (Exit p # \<tau>) = (current_proc_fds \<tau>) (p := {})"
+| "current_proc_fds (Kill p p' # \<tau>) = (current_proc_fds \<tau>) (p' := {})"
+| "current_proc_fds (_ # \<tau>) = current_proc_fds \<tau>"
+
+(*
+fun current_shms :: "t_state \<Rightarrow> t_shm set"
+where
+  "current_shms [] = init_shms"
+| "current_shms (CreateShM p newshm # \<tau>) = insert newshm (current_shms \<tau>)"
+| "current_shms (DeleteShM p s # \<tau>) = (current_shms \<tau>) - {s}"
+| "current_shms (e # \<tau>) = current_shms \<tau> "
+
+fun flag_of_proc_shm :: "t_state \<Rightarrow> t_process \<Rightarrow> t_shm \<Rightarrow> t_shm_attach_flag option"
+where
+  "flag_of_proc_shm [] = init_flag_of_proc_shm"
+| "flag_of_proc_shm (Attach p h flag # s) = (\<lambda> p' h'. 
+     if (p' = p \<and> h' = h) then Some flag else flag_of_proc_shm s p' h')"
+| "flag_of_proc_shm (Detach p h # s) = (\<lambda> p' h'. 
+     if (p' = p \<and> h' = h) then None else flag_of_proc_shm s p' h')"
+| "flag_of_proc_shm (CreateShM p h # s) = (\<lambda> p' h'. 
+     if (h' = h) then None else flag_of_proc_shm s p' h')"
+| "flag_of_proc_shm (DeleteShM p h # s) = (\<lambda> p' h'. 
+     if (h' = h) then None else flag_of_proc_shm s p' h')"
+| "flag_of_proc_shm (Clone p p' fds shms # s) = (\<lambda> p'' h.
+     if (p'' = p' \<and> h \<in> shms) then flag_of_proc_shm s p h
+     else if (p'' = p') then None
+     else flag_of_proc_shm s p'' h)"
+| "flag_of_proc_shm (Execve p f fds # s) = (\<lambda> p' h. 
+     if (p' = p) then None else flag_of_proc_shm s p' h)"
+| "flag_of_proc_shm (Kill p p' # s) = (\<lambda> p'' h. 
+     if (p'' = p') then None else flag_of_proc_shm s p'' h)"
+| "flag_of_proc_shm (Exit p # s) = (\<lambda> p' h. 
+     if (p' = p) then None else flag_of_proc_shm s p' h)"
+| "flag_of_proc_shm (e # s) = flag_of_proc_shm s"
+
+fun procs_of_shm :: "t_state \<Rightarrow> t_shm \<Rightarrow> (t_process \<times> t_shm_attach_flag) set"
+where
+  "procs_of_shm [] = init_procs_of_shm"
+| "procs_of_shm (CreateShM p h # \<tau>) = 
+    (procs_of_shm \<tau>) (h := {})"  (* creator may not be the sharer of the shm *)
+| "procs_of_shm (Attach p h flag # \<tau>) = 
+    (procs_of_shm \<tau>) (h := insert (p, flag) (procs_of_shm \<tau> h))"
+| "procs_of_shm (Detach p h # \<tau>) =
+    (procs_of_shm \<tau>) (h := (procs_of_shm \<tau> h) - {(p,flag) | flag. (p, flag) \<in> procs_of_shm \<tau> h})"
+| "procs_of_shm (DeleteShM p h # \<tau>) = (procs_of_shm \<tau>) (h := {})"
+| "procs_of_shm (Clone p p' fds shms # \<tau>) = 
+    (\<lambda> h. if (h \<in> shms) 
+          then (procs_of_shm \<tau> h) \<union> {(p', flag) | flag. (p, flag) \<in> procs_of_shm \<tau> h}
+          else procs_of_shm \<tau> h)"
+| "procs_of_shm (Execve p f fds # \<tau>) = 
+    (\<lambda> h. procs_of_shm \<tau> h - {(p, flag) | flag. (p, flag) \<in> procs_of_shm \<tau> h})"
+| "procs_of_shm (Kill p p' # s) = 
+    (\<lambda> h. (procs_of_shm s h) - {(p', flag) | flag. (p', flag) \<in> procs_of_shm s h})"
+| "procs_of_shm (Exit p # s) = 
+    (\<lambda> h. (procs_of_shm s h) - {(p, flag) | flag. (p, flag) \<in> procs_of_shm s h})"
+| "procs_of_shm (e # \<tau>) = procs_of_shm \<tau>"
+
+(*
+inductive info_flow_shm :: "t_state \<Rightarrow> t_process \<Rightarrow> t_process \<Rightarrow> bool"
+where
+  "p \<in> current_procs s \<Longrightarrow> info_flow_shm s p p"
+| "\<lbrakk>info_flow_shm s p p'; (p', SHM_RDWR) \<in> procs_of_shm s h; (p'', flag) \<in> procs_of_shm s h; p' \<noteq> p''\<rbrakk>
+   \<Longrightarrow> info_flow_shm s p p''"
+*)
+definition one_flow_shm :: "t_state \<Rightarrow> t_shm \<Rightarrow> t_process \<Rightarrow> t_process \<Rightarrow> bool"
+where
+  "one_flow_shm s h from to \<equiv> from \<noteq> to \<and> (from, SHM_RDWR) \<in> procs_of_shm s h \<and> 
+                              (\<exists> flag. (to, flag) \<in> procs_of_shm s h)"
+
+fun self_shm :: "t_state \<Rightarrow> t_process \<Rightarrow> t_process \<Rightarrow> bool"
+where
+  "self_shm s p p' = (p = p' \<and> p' \<in> current_procs s)"
+
+definition info_flow_shm :: "t_state \<Rightarrow> t_process \<Rightarrow> t_process \<Rightarrow> bool"
+where
+  "info_flow_shm s from to \<equiv> (self_shm s from to) \<or> (\<exists> h. one_flow_shm s h from to)" 
+*)
+
+fun current_msgqs :: "t_state \<Rightarrow> t_msg set"
+where
+  "current_msgqs [] = init_msgqs"
+| "current_msgqs (CreateMsgq p q # \<tau>) = insert q (current_msgqs \<tau>)"
+| "current_msgqs (RemoveMsgq p q # \<tau>) = (current_msgqs \<tau>) - {q}"
+| "current_msgqs (_ # \<tau>) = current_msgqs \<tau>"
+
+fun msgs_of_queue :: "t_state \<Rightarrow> t_msgq \<Rightarrow> t_msg list"
+where
+  "msgs_of_queue [] = init_msgs_of_queue"
+| "msgs_of_queue (CreateMsgq p q # \<tau>) = (msgs_of_queue \<tau>)(q := [])"
+| "msgs_of_queue (SendMsg p q m  # \<tau>) = (msgs_of_queue \<tau>)(q := msgs_of_queue \<tau> q @ [m])"
+| "msgs_of_queue (RecvMsg p q m  # \<tau>) = (msgs_of_queue \<tau>)(q := tl (msgs_of_queue \<tau> q))"
+| "msgs_of_queue (RemoveMsgq p q # \<tau>) = (msgs_of_queue \<tau>)(q := [])" 
+| "msgs_of_queue (_ # \<tau>) = msgs_of_queue \<tau>"         
+
+definition current_file_inums :: "t_state \<Rightarrow> t_inode_num set"
+where
+  "current_file_inums \<tau> \<equiv> {im. \<exists> f. inum_of_file \<tau> f = Some im}"
+
+definition current_sock_inums :: "t_state \<Rightarrow> t_inode_num set"
+where
+  "current_sock_inums \<tau> = {im. \<exists> s. inum_of_socket \<tau> s = Some im}"
+
+definition current_inode_nums :: "t_state \<Rightarrow> nat set"
+where
+  "current_inode_nums \<tau> = current_file_inums \<tau> \<union> current_sock_inums \<tau>"
+
+fun flags_of_proc_fd :: "t_state \<Rightarrow> t_process \<Rightarrow> t_fd \<Rightarrow> t_open_flags option"
+where
+  "flags_of_proc_fd [] p fd = init_oflags_of_proc_fd p fd" 
+| "flags_of_proc_fd (Open p f flags fd iopt # \<tau>) p' fd' = 
+     (if (p = p' \<and> fd = fd') then Some flags else flags_of_proc_fd \<tau> p' fd')"
+| "flags_of_proc_fd (CloseFd p fd # \<tau>) p' fd' = 
+     (if (p = p' \<and> fd = fd') then None else flags_of_proc_fd \<tau> p' fd')"
+| "flags_of_proc_fd (CreateSock p af st fd ino # \<tau>) p' fd' = 
+     (if (p = p' \<and> fd = fd') then None else flags_of_proc_fd \<tau> p' fd')"
+| "flags_of_proc_fd (Accept p fd addr lport' fd' ino # \<tau>) p' fd'' = 
+     (if (p = p' \<and> fd' = fd'') then None else flags_of_proc_fd \<tau> p' fd'')"
+| "flags_of_proc_fd (Clone p p' fds # \<tau>) p'' fd = 
+     (if (p' = p'' \<and> fd \<in> fds) then flags_of_proc_fd \<tau> p fd else flags_of_proc_fd \<tau> p'' fd)"  
+| "flags_of_proc_fd (Execve p f fds # \<tau>) p' fd = 
+     (if (p' = p \<and> fd \<in> fds) then flags_of_proc_fd \<tau> p fd else flags_of_proc_fd \<tau> p' fd)"
+| "flags_of_proc_fd (Kill p\<^isub>1 p\<^isub>2 # \<tau>) p fd = 
+     (if (p = p\<^isub>2) then None else flags_of_proc_fd \<tau> p fd)"
+| "flags_of_proc_fd (Exit p # \<tau>) p' fd' = 
+     (if (p = p') then None else flags_of_proc_fd \<tau> p' fd')"
+| "flags_of_proc_fd (e # \<tau>) p fd = flags_of_proc_fd \<tau> p fd" 
+
+(* 
+fun file_at_writing_by:: "t_state \<Rightarrow> t_file \<Rightarrow> (t_process \<times> t_fd) set"
+where
+  "file_at_writing_by [] f = init_file_at_writing_by f"  
+| "file_at_writing_by (Open p f flags fd (Some inum) # \<tau>) f' = (
+     if (f' = f) 
+     then ( if (is_write_flag flags) 
+            then {(p, fd)}
+            else {} )
+     else file_at_writing_by \<tau> f'                              )"
+| "file_at_writing_by (Open p f flags fd None # \<tau>) f' = (
+     if (f' = f \<and> is_write_flag flags) 
+     then insert (p, fd) (file_at_writing_by \<tau> f) 
+     else file_at_writing_by \<tau> f'                       )"
+| "file_at_writing_by (Mkdir p f inum # \<tau>) f' = 
+     (if (f' = f) then {} else file_at_writing_by \<tau> f')"
+| "file_at_writing_by (LinkHard p f f' # \<tau>) f'' = 
+     (if (f'' = f') then file_at_writing_by \<tau> f else file_at_writing_by \<tau> f'')"
+| "file_at_writing_by (Rename p f\<^isub>2 f\<^isub>3 # \<tau>) f = (
+     if (f\<^isub>3 \<preceq> f) then file_at_writing_by \<tau> (file_before_rename f\<^isub>2 f\<^isub>3 f)
+     else file_at_writing_by \<tau> f               )"
+| "file_at_writing_by (CloseFd p fd # \<tau>) f = 
+     (file_at_writing_by \<tau> f - {(p, fd)})"
+| "file_at_writing_by (Clone p p' fds shms # \<tau>) f = 
+     (file_at_writing_by \<tau> f) \<union> {(p',fd)| fd. fd \<in> fds \<and> (p, fd) \<in> file_at_writing_by \<tau> f}"
+| "file_at_writing_by (Execve p f fds # \<tau>) f' = 
+     (if (f' = f) then {} else file_at_writing_by \<tau> f')"
+| "file_at_writing_by (Exit p # \<tau>) f = 
+     (file_at_writing_by \<tau> f - {(p, fd)| fd. (p, fd) \<in> file_at_writing_by \<tau> f})"
+| "file_at_writing_by (Kill p p' # \<tau>) f = 
+     (file_at_writing_by \<tau> f - {(p', fd)| fd. (p', fd) \<in> file_at_writing_by \<tau> f})"
+| "file_at_writing_by (e # \<tau>) f            =  file_at_writing_by \<tau> f"
+*)
+
+definition current_users :: "t_state \<Rightarrow> user_t set"
+where
+  "current_users \<tau> \<equiv> init_users"  
+
+fun update_with_shuthow :: "t_sock_state option \<Rightarrow> t_shutdown_how \<Rightarrow> t_sock_state option"
+where
+  "update_with_shuthow (Some (SS_trans Trans_Recv)) SHUT_RD = Some (SS_trans Trans_No)"
+| "update_with_shuthow (Some (SS_trans Trans_RS))   SHUT_RD = Some (SS_trans Trans_Send)"
+| "update_with_shuthow (Some (SS_trans Trans_Send)) SHUT_RD = Some (SS_trans Trans_Send)"
+| "update_with_shuthow (Some (SS_trans Trans_No))   SHUT_RD = Some (SS_trans Trans_No)"
+| "update_with_shuthow (Some (SS_trans Trans_Recv)) SHUT_WR = Some (SS_trans Trans_Recv)"
+| "update_with_shuthow (Some (SS_trans Trans_RS))   SHUT_WR = Some (SS_trans Trans_Recv)"
+| "update_with_shuthow (Some (SS_trans Trans_Send)) SHUT_WR = Some (SS_trans Trans_No)"
+| "update_with_shuthow (Some (SS_trans Trans_No))   SHUT_WR = Some (SS_trans Trans_No)"
+| "update_with_shuthow (Some (SS_trans Trans_Recv)) SHUT_RDWR = Some (SS_trans Trans_No)"
+| "update_with_shuthow (Some (SS_trans Trans_RS))   SHUT_RDWR = Some (SS_trans Trans_No)"
+| "update_with_shuthow (Some (SS_trans Trans_Send)) SHUT_RDWR = Some (SS_trans Trans_No)"
+| "update_with_shuthow (Some (SS_trans Trans_No))   SHUT_RDWR = Some (SS_trans Trans_No)"
+| "update_with_shuthow opt1                         how       = None" 
+
+fun socket_state :: "t_state \<Rightarrow> t_socket \<Rightarrow> t_sock_state option"
+where
+  "socket_state [] = init_socket_state"
+| "socket_state (CloseFd p fd # \<tau>) = 
+     (socket_state \<tau>) ((p, fd):= None)"
+| "socket_state (CreateSock p af st fd ino # \<tau>) = 
+     (socket_state \<tau>) ((p, fd) := Some SS_create)"
+| "socket_state (Bind p fd addr # \<tau>) = 
+     (socket_state \<tau>) ((p, fd) := Some SS_bind)"
+| "socket_state (Connect p fd addr # \<tau>) = 
+     (socket_state \<tau>) ((p, fd) := Some (SS_trans Trans_RS))"
+| "socket_state (Listen p fd # \<tau>) = 
+     (socket_state \<tau>) ((p, fd) := Some SS_listen)"
+| "socket_state (Accept p fd addr lport' fd' ino # \<tau>) = 
+     (socket_state \<tau>) ((p, fd') := Some (SS_trans Trans_RS))"
+| "socket_state (Shutdown p fd sh # \<tau>) = 
+     (socket_state \<tau>) ((p, fd) := update_with_shuthow (socket_state \<tau> (p, fd)) sh)"
+| "socket_state (Clone p p' fds # \<tau>) = 
+     (\<lambda> (p'', fd). if (p'' = p' \<and> fd \<in> fds) then socket_state \<tau> (p, fd) 
+                   else if (p'' = p') then None
+                   else socket_state \<tau> (p'', fd))" 
+| "socket_state (Execve p f fds # \<tau>) = 
+     (\<lambda> (p', fd). if (p' = p \<and> fd \<in> fds) then socket_state \<tau> (p, fd) 
+                  else if (p' = p) then None 
+                  else socket_state \<tau> (p', fd))" 
+| "socket_state (Kill p\<^isub>1 p\<^isub>2 # \<tau>) = 
+     (\<lambda> (p, fd). if (p = p\<^isub>2) then None else socket_state \<tau> (p, fd))"
+| "socket_state (Exit p # \<tau>) = 
+     (\<lambda> (p', fd'). if (p = p') then None else socket_state \<tau> (p', fd'))"
+| "socket_state (e # \<tau>) = socket_state \<tau>"  
+
+definition socket_in_trans :: "t_state \<Rightarrow> t_socket \<Rightarrow> bool"
+where
+  "socket_in_trans \<tau> s \<equiv> (case (socket_state \<tau> s) of
+                            Some st \<Rightarrow> (case st of 
+                                          SS_trans para \<Rightarrow> True
+                                        | _             \<Rightarrow> False)
+                          | _       \<Rightarrow> False)"
+
+definition socket_in_sending :: "t_state \<Rightarrow> t_socket \<Rightarrow> bool"
+where
+  "socket_in_sending \<tau> s \<equiv> (socket_state \<tau> s = Some (SS_trans Trans_Send)) \<or> (socket_state \<tau> s = Some (SS_trans Trans_RS))"
+
+definition socket_in_recving :: "t_state \<Rightarrow> t_socket \<Rightarrow> bool"
+where
+  "socket_in_recving \<tau> s \<equiv> (socket_state \<tau> s = Some (SS_trans Trans_Recv)) \<or> (socket_state \<tau> s = Some (SS_trans Trans_RS))"
+
+
+(******* admissable check by the OS *******)
+fun os_grant :: "t_state \<Rightarrow> t_event \<Rightarrow> bool"
+where
+  "os_grant \<tau> (Open p f flags fd inumopt) = (
+     case inumopt of
+       Some ino \<Rightarrow>
+    (p \<in> current_procs \<tau> \<and> f \<notin> current_files \<tau> \<and> 
+     (\<exists> pf. parent f = Some pf \<and> is_dir \<tau> pf \<and> pf \<notin> files_hung_by_del \<tau>) \<and>
+     is_creat_flag flags \<and> fd \<notin> (current_proc_fds \<tau> p) \<and> ino \<notin> (current_inode_nums \<tau>))
+   |   None     \<Rightarrow>
+    (p \<in> current_procs \<tau> \<and> \<not> is_creat_excl_flag flags \<and> is_file \<tau> f \<and>
+     fd \<notin> (current_proc_fds \<tau> p))           )"
+(*(if (is_dir \<tau> f) then (is_dir_flag flags \<and> \<not> is_write_flag flags) else (\<not> is_dir_flag flags)),
+  here open is not applied to directories, readdir is for that, but it is not security-related *)
+| "os_grant \<tau> (CloseFd p fd)                  = 
+    (p \<in> current_procs \<tau> \<and> fd \<in> current_proc_fds \<tau> p)"
+| "os_grant \<tau> (UnLink p f)                    = 
+    (p \<in> current_procs \<tau> \<and> is_file \<tau> f \<and> f \<notin> files_hung_by_del \<tau>)"
+| "os_grant \<tau> (Rmdir p f)                     = 
+    (p \<in> current_procs \<tau> \<and> dir_is_empty \<tau> f \<and> f \<notin> files_hung_by_del \<tau> \<and> f \<noteq> [])" (* root file cannot be deleted *)
+| "os_grant \<tau> (Mkdir p f ino)                 = 
+    (p \<in> current_procs \<tau> \<and> f \<notin> current_files \<tau> \<and> 
+     (\<exists> pf. parent f = Some pf \<and> is_dir \<tau> pf \<and> pf \<notin> files_hung_by_del \<tau>) \<and>
+     ino \<notin> (current_inode_nums \<tau>))"
+| "os_grant \<tau> (LinkHard p f\<^isub>1 f\<^isub>2)              = 
+    (p \<in> current_procs \<tau> \<and> is_file \<tau> f\<^isub>1 \<and> f\<^isub>2 \<notin> current_files \<tau> \<and> 
+     (\<exists> pf\<^isub>2. parent f\<^isub>2 = Some pf\<^isub>2 \<and> is_dir \<tau> pf\<^isub>2 \<and> pf\<^isub>2 \<notin> files_hung_by_del \<tau>))"
+| "os_grant \<tau> (Truncate p f len)              = 
+    (p \<in> current_procs \<tau> \<and> is_file \<tau> f)"
+(*
+| "os_grant \<tau> (FTruncate p fd len)            = 
+    (p \<in> current_procs \<tau> \<and> fd \<in> current_proc_fds \<tau> p \<and> 
+     (\<exists> f. file_of_proc_fd \<tau> p fd = Some f \<and> f \<in> current_files \<tau> \<and> is_file \<tau> f) \<and> 
+     (\<exists> flags. flags_of_proc_fd \<tau> p fd = Some flags \<and> is_write_flag flags))"
+*)
+| "os_grant \<tau> (ReadFile p fd)                 = 
+    (p \<in> current_procs \<tau> \<and> fd \<in> current_proc_fds \<tau> p \<and> 
+     (\<exists> f. file_of_proc_fd \<tau> p fd = Some f \<and> f \<in> current_files \<tau>) \<and> 
+     (\<exists> flags. flags_of_proc_fd \<tau> p fd = Some flags \<and> is_read_flag flags))"
+| "os_grant \<tau> (WriteFile p fd)                = 
+    (p \<in> current_procs \<tau> \<and> fd \<in> current_proc_fds \<tau> p \<and> 
+     (\<exists> f. file_of_proc_fd \<tau> p fd = Some f \<and> f \<in> current_files \<tau>) \<and> 
+     (\<exists> flags. flags_of_proc_fd \<tau> p fd = Some flags \<and> is_write_flag flags))" 
+| "os_grant \<tau> (Execve p f fds)                = 
+    (p \<in> current_procs \<tau> \<and> is_file \<tau> f \<and> fds \<subseteq> proc_file_fds \<tau> p)" (* file_at_writing_by \<tau> f = {} \<and> fds \<subseteq> current_proc_fds \<tau> p *)
+(*
+| "os_grant \<tau> (Rename p f\<^isub>2 f\<^isub>3)                = 
+    (p \<in> current_procs \<tau> \<and> f\<^isub>2 \<in> current_files \<tau> \<and> \<not>(f\<^isub>2 \<preceq> f\<^isub>3) \<and> f\<^isub>3 \<notin> current_files \<tau> \<and>
+     (\<exists> pf\<^isub>3. parent f\<^isub>3 = Some pf\<^isub>3 \<and> pf\<^isub>3 \<in> current_files \<tau> \<and> is_dir \<tau> pf\<^isub>3 \<and> 
+             pf\<^isub>3 \<notin> files_hung_by_del \<tau>) )"
+*)
+| "os_grant \<tau> (CreateMsgq p q)                = 
+    (p \<in> current_procs \<tau> \<and> q \<notin> (current_msgqs \<tau>))"
+| "os_grant \<tau> (SendMsg p q m)                 = 
+    (p \<in> current_procs \<tau> \<and> q \<in> current_msgqs \<tau> \<and> m \<notin> (set (msgs_of_queue \<tau> q)))"
+| "os_grant \<tau> (RecvMsg p q m)                 = 
+    (p \<in> current_procs \<tau> \<and> q \<in> current_msgqs \<tau> \<and> m = hd (msgs_of_queue \<tau> q) \<and> msgs_of_queue \<tau> q \<noteq> [])"
+| "os_grant \<tau> (RemoveMsgq p q)                = 
+    (p \<in> current_procs \<tau> \<and> q \<in> current_msgqs \<tau>)"
+(*
+| "os_grant \<tau> (CreateShM p h)                 = 
+    (p \<in> current_procs \<tau> \<and> h \<notin> (current_shms \<tau>))"
+| "os_grant \<tau> (Attach p h flag)               = 
+    (p \<in> current_procs \<tau> \<and> h \<in> current_shms \<tau> \<and> \<not> (\<exists> flag. (p, flag) \<in> procs_of_shm \<tau> h))"
+| "os_grant \<tau> (Detach p h)                    = 
+    (p \<in> current_procs \<tau> \<and> h \<in> current_shms \<tau> \<and> (\<exists> flag. (p, flag) \<in> procs_of_shm \<tau> h))"
+| "os_grant \<tau> (DeleteShM p h)                 = 
+    (p \<in> current_procs \<tau> \<and> h \<in> current_shms \<tau>)"
+*)
+| "os_grant \<tau> (CreateSock p af st fd ino)     = 
+    (p \<in> current_procs \<tau> \<and> af = AF_INET \<and> fd \<notin> (current_proc_fds \<tau> p) \<and> 
+     ino \<notin> (current_inode_nums \<tau>))"
+| "os_grant \<tau> (Accept p fd addr port fd' ino) = 
+    (p \<in> current_procs \<tau> \<and> fd \<in> current_proc_fds \<tau> p \<and> (p, fd) \<in> current_sockets \<tau> \<and> 
+     fd' \<notin> (current_proc_fds \<tau> p) \<and> socket_state \<tau> (p, fd) = Some SS_listen \<and> 
+     is_tcp_sock \<tau> (p, fd) \<and> ino \<notin> (current_inode_nums \<tau>))"
+| "os_grant \<tau> (Bind p fd addr)                = 
+    (p \<in> current_procs \<tau> \<and> fd \<in> current_proc_fds \<tau> p \<and> (p, fd) \<in> current_sockets \<tau> \<and> 
+     socket_state \<tau> (p, fd) = Some SS_create)" (* ?? !! (case addr of INET_ADDR ip port \<Rightarrow> ip \<in> local_ipaddrs)) *)
+| "os_grant \<tau> (Connect p fd addr)             = 
+    (p \<in> current_procs \<tau> \<and> fd \<in> current_proc_fds \<tau> p \<and> (p, fd) \<in> current_sockets \<tau> \<and> 
+     socket_state \<tau> (p, fd) = Some SS_bind)"
+| "os_grant \<tau> (Listen p fd)                   = 
+    (p \<in> current_procs \<tau> \<and> fd \<in> current_proc_fds \<tau> p \<and> is_tcp_sock \<tau> (p, fd) \<and>
+     socket_state \<tau> (p, fd) = Some SS_bind)" (* Listen and Accept should only be used for TCP sever side *)
+| "os_grant \<tau> (SendSock p fd)                 = 
+    (p \<in> current_procs \<tau> \<and> fd \<in> current_proc_fds \<tau> p \<and> (p, fd) \<in> current_sockets \<tau> \<and> 
+     socket_in_sending \<tau> (p, fd))"
+| "os_grant \<tau> (RecvSock p fd)                 = 
+    (p \<in> current_procs \<tau> \<and> fd \<in> current_proc_fds \<tau> p \<and> (p, fd) \<in> current_sockets \<tau> \<and> 
+     socket_in_recving \<tau> (p, fd))"
+| "os_grant \<tau> (Shutdown p fd sh)              = 
+    (p \<in> current_procs \<tau> \<and> fd \<in> current_proc_fds \<tau> p \<and> (p, fd) \<in> current_sockets \<tau> \<and> 
+     socket_in_trans \<tau> (p, fd))"
+(*
+| "os_grant \<tau> (ChangeOwner p u)               = (p \<in> current_procs \<tau> \<and> u \<in> current_users \<tau>)"
+*)
+| "os_grant \<tau> (Clone p p' fds)           = 
+    (p \<in> current_procs \<tau> \<and> p' \<notin> (current_procs \<tau>) \<and> fds \<subseteq> proc_file_fds \<tau> p)" (*\<and> 
+     (\<forall> h \<in> shms. \<exists> flag. (p, flag) \<in> procs_of_shm \<tau> h) current_proc_fds \<rightarrow> proc_file_fds *)
+| "os_grant \<tau> (Kill p\<^isub>1 p\<^isub>2)                    = 
+    (p\<^isub>1 \<in> current_procs \<tau> \<and> p\<^isub>2 \<in> current_procs \<tau>)" (* a process can kill itself right? *)
+| "os_grant \<tau> (Exit p)                        = 
+    (p \<in> current_procs \<tau>)"
+| "os_grant \<tau> (Ptrace p\<^isub>1 p\<^isub>2)                  = 
+    (p\<^isub>1 \<in> current_procs \<tau> \<and> p\<^isub>2 \<in> current_procs \<tau>)" (* a process can trace itself right? *)
+
+fun alive :: "t_state \<Rightarrow> t_object \<Rightarrow> bool"
+where
+  "alive s (O_proc p)     = (p \<in> current_procs s)"
+| "alive s (O_file f)     = (is_file s f)"
+| "alive s (O_dir  f)     = (is_dir s f)"
+| "alive s (O_fd p fd)    = (fd \<in> current_proc_fds s p)"
+| "alive s (O_node n)     = (n \<in> init_nodes)"
+| "alive s (O_tcp_sock k) = (is_tcp_sock s k)"
+| "alive s (O_udp_sock k) = (is_udp_sock s k)"
+(*
+| "alive s (O_shm  h)     = (h \<in> current_shms s)"
+*)
+| "alive s (O_msgq q)     = (q \<in> current_msgqs s)"
+| "alive s (O_msg q m)    = (m \<in> set (msgs_of_queue s q) \<and> q \<in> current_msgqs s)"
+
+(* deleted objects should be "taintable" objects, objects like fd should not be in consideration *)
+(* actually, deleted should be renamed as "vanished", cause "exit/closefd" *)
+fun deleted :: "t_object \<Rightarrow> t_event list \<Rightarrow> bool"
+where
+  "deleted obj [] = False"
+| "deleted obj (Kill p p' # s) = (obj = O_proc p' \<or> deleted obj s)"
+(*
+| "deleted obj (Kill p p' # s)  = ((obj = O_proc p') \<or> (\<exists> fd \<in> current_proc_fds s p'. obj = O_fd p' fd) \<or> 
+                                  (\<exists> fd. obj = O_tcp_sock (p', fd) \<and> is_tcp_sock s (p',fd)) \<or>
+                                  (\<exists> fd. obj = O_udp_sock (p', fd) \<and> is_udp_sock s (p',fd)) \<or>
+                                  deleted obj s)"
+| "deleted obj (CloseFd p fd # s) = ((obj = O_fd p fd) \<or> (obj = O_tcp_sock (p,fd) \<and> is_tcp_sock s (p,fd)) \<or>
+                                     (obj = O_udp_sock (p,fd) \<and> is_udp_sock s (p, fd)) \<or> 
+                                     (\<exists> f. obj = O_file f \<and> proc_fd_of_file s f = {(p, fd)} \<and> 
+                                           f \<in> files_hung_by_del s) \<or> deleted obj s)"
+| "deleted obj (Execve p f fds # s) = ((\<exists> fd \<in> current_proc_fds s p. obj = O_fd p fd \<and> fd \<notin> fds) \<or> 
+                                       (\<exists> fd. obj = O_tcp_sock (p, fd) \<and> is_tcp_sock s (p,fd) \<and> fd \<notin> fds) \<or>
+                                       (\<exists> fd. obj = O_udp_sock (p, fd) \<and> is_udp_sock s (p,fd) \<and> fd \<notin> fds) \<or>
+                                       deleted obj s)"
+| "deleted obj (Clone p p' fds # s) = ((\<exists> fd \<in> current_proc_fds s p. obj = O_fd p' fd \<and> fd \<notin> fds) \<or> 
+                                            deleted obj s)"
+*)
+| "deleted obj (CloseFd p fd # s) = ((\<exists> f. obj = O_file f \<and> proc_fd_of_file s f = {(p, fd)} \<and> 
+                                      f \<in> files_hung_by_del s) \<or> deleted obj s)"
+| "deleted obj (UnLink p f # s) = ((proc_fd_of_file s f = {} \<and> obj = O_file f) \<or> deleted obj s)"
+| "deleted obj (Rmdir p f # s)  = ((obj = O_dir  f) \<or> deleted obj s)"
+(*
+| "deleted obj (Exit p # s) = ((obj = O_proc p) \<or> (\<exists> fd \<in> current_proc_fds s p. obj = O_fd p fd) \<or> 
+                               (\<exists> fd. obj = O_tcp_sock (p, fd) \<and> is_tcp_sock s (p,fd)) \<or>
+                               (\<exists> fd. obj = O_udp_sock (p, fd) \<and> is_udp_sock s (p,fd)) \<or> deleted obj s)"
+*)
+(*
+| "deleted obj (Rename p f f' # s)  = 
+     (case obj of
+        O_file f'' \<Rightarrow> f \<preceq> f'' \<or> deleted obj s
+      | O_dir  f'' \<Rightarrow> f \<preceq> f'' \<or> deleted obj s
+      | _ \<Rightarrow> deleted obj s)"
+*)
+| "deleted obj (RemoveMsgq p q # s) = (obj = O_msgq q \<or> deleted obj s)"
+(*
+| "deleted obj (DeleteShM p h # s) = (obj = O_shm h \<or> deleted obj s)"
+
+| "deleted obj (RecvMsg p q m # s)  = (obj = O_msg q m \<or> deleted obj s)"
+*)
+| "deleted obj (_ # s) = deleted obj s"
+
+(* actually, died should be renamed as "vanished", cause "exit/closefd" *)
+fun died :: "t_object \<Rightarrow> t_event list \<Rightarrow> bool"
+where
+  "died obj [] = False"
+| "died obj (Kill p p' # s)  = ((obj = O_proc p') \<or> (\<exists> fd \<in> current_proc_fds s p'. obj = O_fd p' fd) \<or> 
+                                  (\<exists> fd. obj = O_tcp_sock (p', fd) \<and> is_tcp_sock s (p',fd)) \<or>
+                                  (\<exists> fd. obj = O_udp_sock (p', fd) \<and> is_udp_sock s (p',fd)) \<or>
+                                  died obj s)"
+| "died obj (CloseFd p fd # s) = ((obj = O_fd p fd) \<or> (obj = O_tcp_sock (p,fd) \<and> is_tcp_sock s (p,fd)) \<or>
+                                     (obj = O_udp_sock (p,fd) \<and> is_udp_sock s (p, fd)) \<or> 
+                                     (\<exists> f. obj = O_file f \<and> proc_fd_of_file s f = {(p, fd)} \<and> 
+                                           f \<in> files_hung_by_del s) \<or> died obj s)"
+| "died obj (Execve p f fds # s) = ((\<exists> fd \<in> current_proc_fds s p. obj = O_fd p fd \<and> fd \<notin> fds) \<or> 
+                                       (\<exists> fd. obj = O_tcp_sock (p, fd) \<and> is_tcp_sock s (p,fd) \<and> fd \<notin> fds) \<or>
+                                       (\<exists> fd. obj = O_udp_sock (p, fd) \<and> is_udp_sock s (p,fd) \<and> fd \<notin> fds) \<or>
+                                       died obj s)"
+| "died obj (Clone p p' fds # s) = ((\<exists> fd \<in> current_proc_fds s p. obj = O_fd p' fd \<and> fd \<notin> fds) \<or> 
+                                            died obj s)"
+| "died obj (UnLink p f # s) = ((proc_fd_of_file s f = {} \<and> obj = O_file f) \<or> died obj s)"
+| "died obj (Rmdir p f # s)  = ((obj = O_dir f) \<or> died obj s)"
+| "died obj (Exit p # s) = ((obj = O_proc p) \<or> (\<exists> fd \<in> current_proc_fds s p. obj = O_fd p fd) \<or> 
+                               (\<exists> fd. obj = O_tcp_sock (p, fd) \<and> is_tcp_sock s (p,fd)) \<or>
+                               (\<exists> fd. obj = O_udp_sock (p, fd) \<and> is_udp_sock s (p,fd)) \<or> died obj s)"
+(*
+| "died obj (Rename p f f' # s)  = 
+     (case obj of
+        O_file f'' \<Rightarrow> f \<preceq> f'' \<or> died obj s
+      | O_dir  f'' \<Rightarrow> f \<preceq> f'' \<or> died obj s
+      | _ \<Rightarrow> died obj s)"
+*)
+| "died obj (RemoveMsgq p q # s) = (obj = O_msgq q \<or> (\<exists> m. obj = O_msg q m ) \<or> died obj s)"
+(*
+| "died obj (DeleteShM p h # s) = (obj = O_shm h \<or> died obj s)"
+*)
+| "died obj (RecvMsg p q m # s)  = (obj = O_msg q m \<or> died obj s)"
+| "died obj (_ # s) = died obj s"
+
+end
+
+
+
+locale flask = init + 
+ fixes
+   (***************** Policy-specific Parameters *************)
+      type_transition_rules :: "type_transition_rule_list_t"
+  and allowed_rules :: "allowed_rule_list_t"
+  and role_allowed_rules :: "role_allow_rule_list_t"
+  and role_declaration_rules :: "role_declarations_list_t"
+  and role_transition_rules:: "role_transition_rule_list_t"
+  and constrains_rules :: "constrains_list_t"
+
+  and init_type_of_obj :: "t_object \<rightharpoonup> type_t"
+  and init_user_of_obj :: "t_object \<rightharpoonup> user_t"
+  and init_role_of_proc :: "t_process \<rightharpoonup> role_t"
+
+  assumes
+   
+      init_obj_has_type: "\<And> obj. init_alive obj \<Longrightarrow> \<exists> t. init_type_of_obj obj = Some t"
+  and init_type_has_obj: "\<And> obj t. init_type_of_obj obj = Some t \<Longrightarrow> init_alive obj"
+  and init_obj_has_user: "\<And> obj. init_alive obj \<Longrightarrow> \<exists> u. init_user_of_obj obj = Some u"
+  and init_user_has_obj: "\<And> obj u. init_user_of_obj obj = Some u \<Longrightarrow> init_alive obj \<and> u \<in> init_users"
+  and init_proc_has_role: "bidirect_in_init init_procs init_role_of_proc"
+
+begin
+
+(*********** policy-specified computations, not-event-related ************)
+
+(* security_transition_sid : type_transition_rules
+ * It's a system-call offered by security-server? So access-control-checked too?
+ * according to selinux, it deals only with execve and file-creation
+ *  is_execve: if is the execve case
+ *  is_file: if is the creation of file/dir
+ *  change from is_execve to is_file, because the new message by default use
+ *  the type of the sending process too. 
+ *)
+
+definition type_transition :: "type_t \<Rightarrow> type_t \<Rightarrow> security_class_t \<Rightarrow> bool \<Rightarrow> type_t"
+where
+  "type_transition st rt tc IS_file \<equiv> (
+     case (lookup type_transition_rules 
+             (\<lambda> (stp, rtp, tc', dt). patt_match stp st \<and> patt_match rtp rt \<and> tc = tc')) of
+       None     \<Rightarrow> (if IS_file then rt else st)
+     | Some (stp,rtp,tc',dt) \<Rightarrow> dt)"
+
+(* allowed_rule_list *)
+definition TE_allow :: "type_t \<Rightarrow> type_t \<Rightarrow> security_class_t \<Rightarrow> permission_t \<Rightarrow> bool"
+where
+  "TE_allow st tt tc perm \<equiv> member allowed_rules
+      (\<lambda> (stp, ttp, tc', pp). tc = tc' \<and> patt_match stp st \<and> patt_match_self ttp tt st \<and>
+                              patt_match pp perm)"
+
+(* role_allow_rule_list_t *)
+definition role_transition :: "role_t \<Rightarrow> type_t \<Rightarrow> role_t option"
+where
+  "role_transition r t \<equiv> 
+     (case (lookup role_transition_rules (\<lambda> (pr, ft, nr). pr = r \<and> t = ft)) of
+        None \<Rightarrow> None
+      | Some (pr,ft,nr) \<Rightarrow> Some nr)"
+
+definition role_decl_check :: "role_t \<Rightarrow> type_t \<Rightarrow> bool"
+where
+  "role_decl_check r t \<equiv> member role_declaration_rules (\<lambda> (role, types). r = role \<and> t \<in> types)"
+
+definition role_allow :: "role_t \<Rightarrow> role_t \<Rightarrow> bool"
+where
+  "role_allow r nr \<equiv> member role_allowed_rules (\<lambda> (from, to). r \<in> from \<and> nr \<in> to)"
+
+(* here don't use lookup, because constrains should all be satisfied, 
+ * while transitions always choose the "first" one 
+ *)
+definition constrains_satisfied :: 
+  "security_context_t \<Rightarrow> security_context_t \<Rightarrow> security_class_t \<Rightarrow> permission_t \<Rightarrow> bool"
+where
+  "constrains_satisfied sctxt tctxt c p \<equiv> 
+    (fold (\<lambda> (cset, pset, P) res. res \<and> (if (c \<in> cset \<and> p \<in> pset) 
+                                            then P sctxt tctxt else True))
+          constrains_rules True)"
+
+(* main interface for TE_allow check and constrains check *)
+fun permission_check:: 
+  "security_context_t \<Rightarrow> security_context_t \<Rightarrow> security_class_t \<Rightarrow> permission_t \<Rightarrow> bool"
+where
+  "permission_check (su,sr,st) (tu,tr,tt) c p =
+    ((TE_allow st tt c p) \<and> (constrains_satisfied (su,sr,st) (tu,tr,tt) c p))"
+declare permission_check.simps [simp del]
+
+(* no changeowner ??? : by adding "relabel obj sectxt" event or "login" event? 
+ * or this explanation: the running of the sever will not grant any login but running 
+ * of the server-programs! 
+ *)
+
+fun user_of_obj :: "t_state \<Rightarrow> t_object \<Rightarrow> user_t option"
+where 
+  "user_of_obj [] = init_user_of_obj"
+| "user_of_obj (Clone p p' fds # s) = 
+    (\<lambda> obj. case obj of 
+              O_proc p'' \<Rightarrow> if (p'' = p') then user_of_obj s (O_proc p) else user_of_obj s obj
+            | O_fd p'' fd  \<Rightarrow> if (p'' = p' \<and> fd \<in> fds) then user_of_obj s (O_fd p fd)
+                              else if (p'' = p') then None
+                                   else user_of_obj s obj
+            | O_tcp_sock (p'',fd) \<Rightarrow> if (p'' = p' \<and> fd \<in> fds) then user_of_obj s (O_tcp_sock (p, fd))
+                                     else if (p'' = p') then None
+                                          else user_of_obj s obj
+            | O_udp_sock (p'',fd) \<Rightarrow> if (p'' = p' \<and> fd \<in> fds) then user_of_obj s (O_udp_sock (p, fd))
+                                     else if (p'' = p') then None
+                                          else user_of_obj s obj 
+            | _ \<Rightarrow> user_of_obj s obj)"
+| "user_of_obj (Open p f flags fd iopt # s) = 
+    (case iopt of
+       None \<Rightarrow> (user_of_obj s) ((O_fd p fd) := user_of_obj s (O_proc p))
+     | _    \<Rightarrow> ((user_of_obj s) ((O_fd p fd) := user_of_obj s (O_proc p)))
+               ((O_file f) := user_of_obj s (O_proc p)))"
+| "user_of_obj (Mkdir p f inum # s) =
+    (user_of_obj s) ((O_dir f) := user_of_obj s (O_proc p))"
+| "user_of_obj (LinkHard p f f' # s) = 
+    (user_of_obj s) ((O_file f') := user_of_obj s (O_proc p))"
+(*
+| "user_of_obj (Rename p f2 f3 # s) = 
+    (\<lambda> obj. case obj of 
+              O_file f \<Rightarrow> if (f3 \<preceq> f) then user_of_obj s (O_file (file_before_rename f2 f3 f))
+                          else user_of_obj s obj
+            | O_dir  f \<Rightarrow> if (f3 \<preceq> f) then user_of_obj s (O_dir (file_before_rename f2 f3 f))
+                          else user_of_obj s obj
+            | _        \<Rightarrow> user_of_obj s obj)"
+*)
+| "user_of_obj (CreateMsgq p q # s) = 
+    (user_of_obj s) ((O_msgq q) := user_of_obj s (O_proc p))" 
+| "user_of_obj (SendMsg p q m # s) = 
+    (user_of_obj s) ((O_msg q m) := user_of_obj s (O_proc p))"
+(*
+| "user_of_obj (CreateShM p h # s) = 
+    (user_of_obj s) ((O_shm h) := user_of_obj s (O_proc p))"
+*)
+| "user_of_obj (CreateSock p af st fd inum # s) =  (\<lambda> obj. 
+     case obj of
+       O_fd p' fd' \<Rightarrow> if (p' = p \<and> fd' = fd) then user_of_obj s (O_proc p)
+                      else user_of_obj s obj
+     | O_tcp_sock (p', fd') \<Rightarrow> if (p' = p \<and> fd' = fd \<and> st = STREAM) then user_of_obj s (O_proc p)
+                               else user_of_obj s obj
+     | O_udp_sock (p', fd') \<Rightarrow> if (p' = p \<and> fd' = fd \<and> st = DGRAM) then user_of_obj s (O_proc p)
+                               else user_of_obj s obj
+     | _ \<Rightarrow> user_of_obj s obj )"
+| "user_of_obj (Accept p fd addr port fd' inum # s) = (\<lambda> obj. 
+     case obj of
+       O_fd p' fd'' \<Rightarrow> if (p' = p \<and> fd'' = fd') then user_of_obj s (O_proc p)
+                      else user_of_obj s obj
+     | O_tcp_sock (p', fd'') \<Rightarrow> if (p' = p \<and> fd'' = fd') then user_of_obj s (O_proc p)
+                               else user_of_obj s obj
+     | _ \<Rightarrow> user_of_obj s obj )"
+| "user_of_obj (_ # s) = user_of_obj s"
+
+fun type_of_obj :: "t_state \<Rightarrow> (t_object \<Rightarrow> type_t option)"
+where
+  "type_of_obj [] = init_type_of_obj"
+| "type_of_obj (Clone p p' fds # s) = (\<lambda> obj. 
+     case obj of 
+       O_proc p'' \<Rightarrow> if (p'' = p') then type_of_obj s (O_proc p) else type_of_obj s obj
+     | O_fd p'' fd \<Rightarrow> if (p'' = p' \<and> fd \<in> fds) then type_of_obj s (O_fd p fd)
+                      else if (p'' = p') then None
+                           else type_of_obj s obj
+     | O_tcp_sock (p'', fd) \<Rightarrow> if (p'' = p' \<and> fd \<in> fds) then type_of_obj s (O_tcp_sock (p, fd))
+                               else if (p'' = p') then None
+                                    else type_of_obj s obj
+     | O_udp_sock (p'', fd) \<Rightarrow> if (p'' = p' \<and> fd \<in> fds) then type_of_obj s (O_udp_sock (p, fd))
+                               else if (p'' = p') then None
+                                    else type_of_obj s obj
+     | _ \<Rightarrow> type_of_obj s obj )"
+| "type_of_obj (Execve p f fds # s) = (type_of_obj s) ((O_proc p) := 
+    (case (type_of_obj s (O_proc p), type_of_obj s (O_file f)) of 
+       (Some tp, Some tf) \<Rightarrow> Some (type_transition tp tf C_process False)
+     | _                  \<Rightarrow> None)                    )"
+| "type_of_obj (Open p f flags fd (Some inum) # s) = ((type_of_obj s) ((O_file f) := 
+     (case (parent f) of 
+        Some pf \<Rightarrow> (case (type_of_obj s (O_proc p), type_of_obj s (O_dir pf)) of 
+                      (Some tp, Some tpf) \<Rightarrow> Some (type_transition tp tpf C_file True)
+                    | _ \<Rightarrow> None)
+      | _       \<Rightarrow> None)                              )) ((O_fd p fd) := 
+     type_of_obj s (O_proc p))"
+| "type_of_obj (Open p f flags fd None # s) = (type_of_obj s) ((O_fd p fd) := 
+     type_of_obj s (O_proc p))"
+| "type_of_obj (Mkdir p f inum # s) = (type_of_obj s) ((O_dir f) := 
+     (case (parent f) of 
+        Some pf \<Rightarrow> (case (type_of_obj s (O_proc p), type_of_obj s (O_dir pf)) of
+                     (Some tp, Some tpf) \<Rightarrow> Some (type_transition tp tpf C_dir True)
+                   | _ \<Rightarrow> None)
+     | _ \<Rightarrow> None)                                 )"
+| "type_of_obj (LinkHard p f f' # s) = (type_of_obj s) ((O_file f') := 
+    (case (parent f') of 
+       Some pf \<Rightarrow> (case (type_of_obj s (O_proc p), type_of_obj s (O_dir pf)) of
+                    (Some tp, Some tpf) \<Rightarrow> Some (type_transition tp tpf C_file True)
+                  | _ \<Rightarrow> None)
+     | _ \<Rightarrow> None)                                      )"
+(*
+| "type_of_obj (Rename p f f' # s) = (\<lambda> obj. 
+      case obj of
+        O_file f'' \<Rightarrow> (if (f' \<preceq> f'') then type_of_obj s (O_file (file_before_rename f f' f''))
+                       else type_of_obj s (O_file f''))
+      | O_dir f'' \<Rightarrow> (if (f' \<preceq> f'') then type_of_obj s (O_file (file_before_rename f f' f''))
+                      else type_of_obj s (O_file f''))
+      | _ \<Rightarrow> type_of_obj s obj   )"
+*)
+| "type_of_obj (CreateMsgq p q # s) = (type_of_obj s) ((O_msgq q) := 
+     (case (type_of_obj s (O_proc p)) of 
+        Some tp \<Rightarrow> Some tp
+      | _       \<Rightarrow> None)                          )"
+| "type_of_obj (SendMsg p q m # s) = (type_of_obj s) ((O_msg q m) := 
+     (case (type_of_obj s (O_proc p), type_of_obj s (O_msgq q)) of 
+        (Some tp, Some tq) \<Rightarrow> Some (type_transition tp tq C_msg False)
+      | _ \<Rightarrow> None)                               )"
+(*
+| "type_of_obj (CreateShM p h # s) = (type_of_obj s) ((O_shm h) := 
+     (case (type_of_obj s (O_proc p)) of 
+        Some tp \<Rightarrow> Some tp
+      | _       \<Rightarrow> None)                          )"
+*)
+| "type_of_obj (CreateSock p af st fd inum # s) = (\<lambda> obj. 
+     case obj of
+       O_fd p' fd' \<Rightarrow> if (p' = p \<and> fd' = fd) then type_of_obj s (O_proc p)
+                      else type_of_obj s obj
+     | O_tcp_sock (p', fd') \<Rightarrow> if (p' = p \<and> fd' = fd \<and> st = STREAM) then type_of_obj s (O_proc p)
+                               else type_of_obj s obj
+     | O_udp_sock (p', fd') \<Rightarrow> if (p' = p \<and> fd' = fd \<and> st = DGRAM) then type_of_obj s (O_proc p)
+                               else type_of_obj s obj
+     | _ \<Rightarrow> type_of_obj s obj )"
+| "type_of_obj (Accept p fd addr port fd' inum # s) = (\<lambda> obj. 
+     case obj of
+       O_fd p' fd'' \<Rightarrow> if (p' = p \<and> fd'' = fd') then type_of_obj s (O_proc p)
+                      else type_of_obj s obj
+     | O_tcp_sock (p', fd'') \<Rightarrow> if (p' = p \<and> fd'' = fd') then type_of_obj s (O_proc p)
+                               else type_of_obj s obj
+     | _ \<Rightarrow> type_of_obj s obj )"
+| "type_of_obj (_ # s) = type_of_obj s"
+
+fun role_of_proc :: "t_state \<Rightarrow> (t_process \<Rightarrow> role_t option)"
+where
+  "role_of_proc [] = init_role_of_proc"
+| "role_of_proc (Clone p p' fds # s) = 
+    (role_of_proc s) (p' := role_of_proc s p)"
+| "role_of_proc (Execve p f fds # s) = (role_of_proc s) (p := 
+    (case (role_of_proc s p, type_of_obj s (O_file f)) of
+       (Some r, Some ft) \<Rightarrow> role_transition r ft
+     | _ \<Rightarrow> None)                       )"
+| "role_of_proc (_ # s) = role_of_proc s"
+
+fun role_of_obj :: "t_state \<Rightarrow> t_object \<Rightarrow> role_t option"
+where
+  "role_of_obj s (O_proc p) = role_of_proc s p"
+| "role_of_obj s _          = Some R_object"  (* object_r *)
+
+definition sectxt_of_obj :: "t_state \<Rightarrow> t_object \<Rightarrow> security_context_t option"
+where
+  "sectxt_of_obj s obj = (
+     case (user_of_obj s obj, role_of_obj s obj, type_of_obj s obj) of
+       (Some u, Some r, Some t) \<Rightarrow> Some (u, r, t)
+     | _                        \<Rightarrow> None)"
+
+fun init_role_of_obj :: "t_object \<Rightarrow> role_t option"
+where
+  "init_role_of_obj (O_proc p) = init_role_of_proc p"
+| "init_role_of_obj _          = Some R_object"
+
+definition init_sectxt_of_obj :: "t_object \<Rightarrow> security_context_t option"
+where
+  "init_sectxt_of_obj obj \<equiv>
+     (case (init_user_of_obj obj, init_role_of_obj obj, init_type_of_obj obj) of
+        (Some u, Some r, Some t) \<Rightarrow> Some (u,r,t)
+      | _ \<Rightarrow> None)"
+
+definition sec_of_root :: "security_context_t"
+where
+  "sec_of_root \<equiv> (case (init_user_of_obj (O_dir []), init_type_of_obj (O_dir [])) of
+     (Some u, Some t) \<Rightarrow> (u, R_object, t))"
+
+(******* flask granting ********
+ * involves three kinds of rules:
+ *  1. allow rule of types, allowed_rule_list_t, main
+ *  2. allow rule of roles, role_allow_rule_list_t, mainly for execve call
+ *  3. constrains, section 3.4.5, user-specifically defined.
+ * Passed all these 3, then grant Yes, else No 
+ *) 
+
+definition search_check_allp :: "security_context_t \<Rightarrow> security_context_t set \<Rightarrow> bool"
+where
+  "search_check_allp pctxt sectxts = (\<forall> sec \<in> sectxts. permission_check pctxt sec C_dir P_search)"
+
+definition search_check_file :: "security_context_t \<Rightarrow> security_context_t \<Rightarrow> bool"
+where
+  "search_check_file sctxt fctxt \<equiv> permission_check sctxt fctxt C_file P_search"
+
+definition search_check_dir :: "security_context_t \<Rightarrow> security_context_t \<Rightarrow> bool"
+where
+  "search_check_dir sctxt fctxt \<equiv> permission_check sctxt fctxt C_dir P_search"
+
+fun get_parentfs_ctxts :: "t_state \<Rightarrow> t_file \<Rightarrow> (security_context_t list) option"
+where
+  "get_parentfs_ctxts s [] = 
+     (case (sectxt_of_obj s (O_dir [])) of 
+        Some ctxt \<Rightarrow> Some [ctxt]
+      | _         \<Rightarrow> None)"
+| "get_parentfs_ctxts s (f#pf) = 
+     (case (get_parentfs_ctxts s pf, sectxt_of_obj s (O_dir (f#pf))) of
+        (Some ctxts, Some ctxt) \<Rightarrow> Some (ctxt#ctxts)
+      | _ \<Rightarrow> None)"
+
+definition search_check_ctxt:: 
+  "security_context_t \<Rightarrow> security_context_t \<Rightarrow> security_context_t set \<Rightarrow> bool \<Rightarrow> bool"
+where
+  "search_check_ctxt pctxt fctxt asecs if_file \<equiv> (
+      if if_file 
+      then search_check_file pctxt fctxt \<and> search_check_allp pctxt asecs
+      else search_check_dir  pctxt fctxt \<and> search_check_allp pctxt asecs )"
+
+fun search_check :: "t_state \<Rightarrow> security_context_t \<Rightarrow> t_file \<Rightarrow> bool"
+where
+  "search_check s pctxt [] = 
+     (case (sectxt_of_obj s (O_dir [])) of
+        Some fctxt \<Rightarrow> search_check_ctxt pctxt fctxt {} False 
+      | _          \<Rightarrow> False)"
+| "search_check s pctxt (f#pf) = 
+     (if (is_file s (f#pf)) 
+      then (case (sectxt_of_obj s (O_file (f#pf)), get_parentfs_ctxts s pf) of
+              (Some fctxt, Some pfctxts) \<Rightarrow> search_check_ctxt pctxt fctxt (set pfctxts) True
+            | _ \<Rightarrow> False)
+      else (case (sectxt_of_obj s (O_dir (f#pf)), get_parentfs_ctxts s pf) of
+              (Some fctxt, Some pfctxts) \<Rightarrow> search_check_ctxt pctxt fctxt (set pfctxts) False
+            | _ \<Rightarrow> False))"
+
+(* this means we should prove every current fd has a security context! *)
+definition sectxts_of_fds :: "t_state \<Rightarrow> t_process \<Rightarrow> t_fd set \<Rightarrow> security_context_t set"
+where
+  "sectxts_of_fds s p fds \<equiv> {ctxt. \<exists> fd \<in> fds. sectxt_of_obj s (O_fd p fd) = Some ctxt}"
+
+definition inherit_fds_check_ctxt:: "security_context_t \<Rightarrow> security_context_t set \<Rightarrow> bool"
+where
+  "inherit_fds_check_ctxt p fds \<equiv> (\<forall> ctxt \<in> fds. permission_check p ctxt C_fd P_inherit)"
+
+definition inherit_fds_check :: "t_state \<Rightarrow> security_context_t \<Rightarrow> t_process \<Rightarrow> t_fd set \<Rightarrow> bool"
+where
+  "inherit_fds_check s npsectxt p fds \<equiv> inherit_fds_check_ctxt npsectxt (sectxts_of_fds s p fds)"
+
+fun npctxt_execve :: "security_context_t \<Rightarrow> security_context_t \<Rightarrow> security_context_t option"
+where
+  "npctxt_execve (pu,pr,pt) (fu,fr,ft) = 
+     (case (role_transition pr ft) of
+        Some nr \<Rightarrow> Some (pu, nr, type_transition pt ft C_process False)
+      | _       \<Rightarrow>  None)"
+
+fun nfctxt_create :: "security_context_t \<Rightarrow> security_context_t \<Rightarrow> security_class_t \<Rightarrow> security_context_t"
+where
+  "nfctxt_create (pu,pr,pt) (fu,fr,ft) cls = (pu, R_object, type_transition pt ft cls True)"
+
+fun grant_execve :: 
+  "security_context_t \<Rightarrow> security_context_t \<Rightarrow> security_context_t \<Rightarrow> bool"
+where
+  "grant_execve (up,rp,tp) (uf,rf,tf) (up',nr,nt) = 
+         (role_decl_check nr nt \<and> role_allow rp nr \<and> 
+          permission_check (up,rp,tp) (uf,rf,tf) C_file P_execute \<and>
+          permission_check (up,nr,nt) (uf,rf,tf) C_file P_entrypoint \<and>
+          permission_check (up,rp,tp) (up,nr,nt) C_process P_transition \<and>
+          permission_check (up,nr,nt) (uf,rf,tf) C_process P_execute)"
+
+(*
+definition sectxts_of_shms :: "t_state \<Rightarrow> t_shm set \<Rightarrow> security_context_t set"
+where
+  "sectxts_of_shms s shms \<equiv> {ctxt. \<exists> h \<in> shms. sectxt_of_obj s (O_shm h) = Some ctxt}"
+
+definition inherit_shms_check_ctxt:: "security_context_t \<Rightarrow> security_context_t set \<Rightarrow> bool"
+where
+  "inherit_shms_check_ctxt p shms \<equiv> (\<forall> ctxt \<in> shms. permission_check p ctxt C_shm P_inherit)"
+
+definition inherit_shms_check :: "t_state \<Rightarrow> security_context_t \<Rightarrow> t_shm set \<Rightarrow> bool"
+where
+  "inherit_shms_check s npsectxt shms \<equiv> inherit_shms_check_ctxt npsectxt (sectxts_of_shms s shms)"
+*)
+
+fun perm_of_mflag :: "t_open_must_flag \<Rightarrow> permission_t set"
+where
+  "perm_of_mflag OF_RDONLY = {P_read}"
+| "perm_of_mflag OF_WRONLY = {P_write}"
+| "perm_of_mflag OF_RDWR   = {P_read, P_write}"
+
+definition perm_of_oflag :: "t_open_option_flag set \<Rightarrow> permission_t set"
+where
+  "perm_of_oflag oflags \<equiv> 
+    (case (OF_APPEND \<in> oflags, OF_CREAT \<in> oflags) of
+       (True, True) \<Rightarrow> {P_append, P_create} 
+     | (True, _   ) \<Rightarrow> {P_append}
+     | (_   , True) \<Rightarrow> {P_create}
+     | _            \<Rightarrow> {})"
+
+definition perms_of_flags :: "t_open_flags \<Rightarrow> permission_t set"
+where
+  "perms_of_flags flags \<equiv> 
+    (case flags of
+       (mflag,oflags) \<Rightarrow> (perm_of_mflag mflag \<union> perm_of_oflag oflags))"
+
+(*
+definition class_of_flags :: "t_open_flags \<Rightarrow> security_class_t"
+where
+  "class_of_flags flags \<equiv> 
+    (case flags of
+      (mflag, oflags) \<Rightarrow> if (OF_DIRECTORY \<in> oflags) then C_dir else C_file)"
+
+definition obj_of_flags :: "t_open_flags \<Rightarrow> t_file \<Rightarrow> t_object"
+where
+  "obj_of_flags flags f \<equiv> 
+    (case flags of
+      (mflag, oflags) \<Rightarrow> if (OF_DIRECTORY \<in> oflags) then O_dir f else O_file f)"
+*)
+
+definition oflags_check :: "t_open_flags \<Rightarrow> security_context_t \<Rightarrow> security_context_t \<Rightarrow> bool"
+where
+  "oflags_check flags pctxt fctxt \<equiv> 
+     \<forall> perm \<in> (perms_of_flags flags). permission_check pctxt fctxt C_file perm"
+
+fun grant :: "t_state \<Rightarrow> t_event \<Rightarrow> bool"
+where
+  "grant s (Execve p f fds) = 
+    (case (sectxt_of_obj s (O_proc p), sectxt_of_obj s (O_file f)) of
+       (Some (up, rp, tp), Some (uf, rf, tf)) \<Rightarrow> 
+         (case (npctxt_execve (up,rp,tp) (uf,rf,tf)) of 
+            Some (pu,nr,nt) \<Rightarrow> (
+              search_check s (up,rp,tp) f \<and> 
+              grant_execve (up,rp,tp) (uf,rf,tf) (up,nr,nt) \<and> 
+              inherit_fds_check s (up,nr,nt) p fds)
+          | _       \<Rightarrow> False)
+     | _ \<Rightarrow> False )"
+| "grant s (Clone p p' fds) = 
+    (case (sectxt_of_obj s (O_proc p)) of 
+       Some (up, rp, tp) \<Rightarrow>
+         (permission_check (up,rp,tp) (up,rp,tp) C_process P_fork \<and>
+          inherit_fds_check s (up,rp,tp) p fds)          
+     | _ \<Rightarrow> False )" (* \<and> inherit_shms_check s (up,rp,tp) shms *)
+| "grant s (Kill p p') = 
+    (case (sectxt_of_obj s (O_proc p), sectxt_of_obj s (O_proc p')) of  
+       (Some (su,sr,st), Some (tu,tr,tt)) \<Rightarrow> 
+         permission_check (su,sr,st) (tu,tr,tt) C_process P_sigkill
+     | _ \<Rightarrow> False)"
+| "grant s (Ptrace p p') = 
+    (case (sectxt_of_obj s (O_proc p), sectxt_of_obj s (O_proc p')) of  
+       (Some (su,sr,st), Some (tu,tr,tt)) \<Rightarrow> 
+         permission_check (su,sr,st) (tu,tr,tt) C_process P_ptrace
+     | _ \<Rightarrow> False)"
+| "grant s (Exit p) = True"
+| "grant s (Open p f flags fd None) =
+    (case (sectxt_of_obj s (O_proc p), sectxt_of_obj s (O_file f)) of
+      (Some (pu,pr,pt), Some (fu,fr,ft)) \<Rightarrow> 
+        search_check s (pu,pr,pt) f \<and>
+        oflags_check flags (pu,pr,pt) (fu,fr,ft) \<and>
+        permission_check (pu,pr,pt) (pu,pr,pt) C_fd P_setattr
+    | _ \<Rightarrow>False)"
+| "grant s (Open p f flags fd (Some inum)) = 
+    (case (parent f) of 
+       Some pf \<Rightarrow> (case (sectxt_of_obj s (O_proc p), sectxt_of_obj s (O_dir pf)) of
+                    (Some (pu,pr,pt), Some (pfu,pfr,pft)) \<Rightarrow> 
+                        (search_check s (pu,pr,pt) pf \<and>
+                         oflags_check flags (pu,pr,pt) (nfctxt_create (pu,pr,pt) (pfu,pfr,pft) C_file) \<and>
+                         permission_check (pu,pr,pt) (pfu,pfr,pft) C_dir P_add_name \<and>
+                         permission_check (pu,pr,pt) (pu,pr,pt) C_fd P_setattr)
+                  | _ \<Rightarrow> False)
+     | _ \<Rightarrow> False)"
+| "grant s (ReadFile p fd) = 
+    (case (file_of_proc_fd s p fd) of
+       Some f \<Rightarrow> 
+        (case (sectxt_of_obj s (O_proc p), sectxt_of_obj s (O_fd p fd), 
+               sectxt_of_obj s (O_file f)) of 
+           (Some (pu,pr,pt), Some (fdu,fdr,fdt), Some (fu,fr,ft)) \<Rightarrow> 
+             (permission_check (pu,pr,pt) (fdu,fdr,fdt) C_fd P_setattr \<and>
+              permission_check (pu,pr,pt) (fu,fr,ft) C_file P_read)
+         | _ \<Rightarrow> False)
+     | _ \<Rightarrow> False)"
+| "grant s (WriteFile p fd) = 
+    (case (file_of_proc_fd s p fd) of
+       Some f \<Rightarrow> 
+        (case (sectxt_of_obj s (O_proc p), sectxt_of_obj s (O_fd p fd), 
+               sectxt_of_obj s (O_file f)) of 
+           (Some (pu,pr,pt), Some (fdu,fdr,fdt), Some (fu,fr,ft)) \<Rightarrow> 
+             (permission_check (pu,pr,pt) (fdu,fdr,fdt) C_fd P_setattr \<and>
+              permission_check (pu,pr,pt) (fu,fr,ft) C_file P_write)
+         | _ \<Rightarrow> False)
+     | _ \<Rightarrow> False)" 
+| "grant s (CloseFd p fd) = True"
+| "grant s (UnLink p f) = 
+    (case (parent f) of 
+       Some pf \<Rightarrow> 
+         (case (sectxt_of_obj s (O_proc p), sectxt_of_obj s (O_dir pf), 
+                sectxt_of_obj s (O_file f)) of 
+           (Some (pu,pr,pt), Some (du,dr,dt), Some (fu,fr,ft)) \<Rightarrow> 
+             (search_check s (pu,pr,pt) f \<and> 
+              permission_check (pu,pr,pt) (fu,fr,ft) C_file P_unlink \<and>
+              permission_check (pu,pr,pt) (du,dr,dt) C_dir P_remove_name)
+         | _ \<Rightarrow> False)
+    | _ \<Rightarrow> False)"
+| "grant s (Rmdir p f) = 
+    (case (parent f) of 
+       Some pf \<Rightarrow> 
+         (case (sectxt_of_obj s (O_proc p), sectxt_of_obj s (O_dir pf), 
+                sectxt_of_obj s (O_dir f)) of 
+           (Some (pu,pr,pt), Some (du,dr,dt), Some (fu,fr,ft)) \<Rightarrow> 
+             (search_check s (pu,pr,pt) f \<and> 
+              permission_check (pu,pr,pt) (fu,fr,ft) C_dir P_rmdir \<and>
+              permission_check (pu,pr,pt) (du,dr,dt) C_dir P_remove_name)
+         | _ \<Rightarrow> False)
+    | _ \<Rightarrow> False)"
+| "grant s (Mkdir p f inum) = 
+    (case (parent f) of 
+       Some pf \<Rightarrow> 
+         (case (sectxt_of_obj s (O_proc p), sectxt_of_obj s (O_dir pf)) of 
+           (Some (pu,pr,pt), Some (du,dr,dt)) \<Rightarrow> 
+             (search_check s (pu,pr,pt) f \<and> 
+              permission_check (pu,pr,pt) (nfctxt_create (pu,pr,pt) (du,dr,dt) C_dir) C_dir P_create \<and>
+              permission_check (pu,pr,pt) (du,dr,dt) C_dir P_add_name)
+         | _ \<Rightarrow> False)
+    | _ \<Rightarrow> False)"
+| "grant s (LinkHard p f f') = 
+    (case (parent f') of 
+       Some pf \<Rightarrow> 
+         (case (sectxt_of_obj s (O_proc p), sectxt_of_obj s (O_dir pf), 
+                sectxt_of_obj s (O_file f)) of 
+           (Some (pu,pr,pt), Some (du,dr,dt), Some (fu,fr,ft)) \<Rightarrow> 
+             (search_check s (pu,pr,pt) f \<and> search_check s (pu,pr,pt) pf \<and>
+              permission_check (pu,pr,pt) (fu,fr,ft) C_file P_link \<and>
+              permission_check (pu,pr,pt) (du,dr,dt) C_dir P_add_name)
+         | _ \<Rightarrow> False)
+    | _ \<Rightarrow> False)"
+| "grant s (Truncate p f len) = 
+    (case (sectxt_of_obj s (O_proc p), sectxt_of_obj s (O_file f)) of 
+       (Some (pu,pr,pt), Some (fu,fr,ft)) \<Rightarrow> 
+          (search_check s (pu,pr,pt) f \<and>
+           permission_check (pu,pr,pt) (fu,fr,ft) C_file P_setattr)
+        | _ \<Rightarrow> False)"
+(*
+| "grant s (Rename p f f') = 
+    (case (parent f, parent f') of
+       (Some pf, Some pf') \<Rightarrow> 
+         (case (sectxt_of_obj s (O_proc p), sectxt_of_obj s (O_dir pf), 
+                sectxt_of_obj s (O_file f), sectxt_of_obj s (O_dir pf')) of
+            (Some (pu,pr,pt), Some (pfu,pfr,pft), Some (fu,fr,ft), 
+             Some (pfu',pfr',pft')) \<Rightarrow> 
+               (search_check s (pu,pr,pt) f \<and>
+                permission_check (pu,pr,pt) (pfu,pfr,pft) C_dir P_remove_name \<and>
+                (if (is_file s f) 
+                 then permission_check (pu,pr,pt) (fu,fr,ft) C_file P_rename
+                 else permission_check (pu,pr,pt) (fu,fr,ft) C_dir P_reparent) \<and>
+                search_check s (pu,pr,pt) pf' \<and>
+                permission_check (pu,pr,pt) (pfu',pfr',pft') C_dir P_add_name)
+         | _ \<Rightarrow> False)
+     | _ \<Rightarrow> False)"
+*)
+| "grant s (CreateMsgq p q) = 
+    (case (sectxt_of_obj s (O_proc p)) of 
+       Some (pu,pr,pt) \<Rightarrow> 
+        (permission_check (pu,pr,pt) (pu,pr,pt) C_msgq P_associate \<and>
+         permission_check (pu,pr,pt) (pu,pr,pt) C_msgq P_create)
+     | _ \<Rightarrow> False)"
+| "grant s (SendMsg p q m) = 
+    (case (sectxt_of_obj s (O_proc p), sectxt_of_obj s (O_msgq q)) of 
+      (Some (pu,pr,pt), Some (qu,qr,qt)) \<Rightarrow> 
+        (permission_check (pu,pr,pt) (qu,qr,qt) C_msgq P_enqueue \<and>
+         permission_check (pu,pr,pt) (qu,qr,qt) C_msgq P_write \<and>
+         permission_check (pu,pr,pt) (pu,pr,pt) C_msg  P_send)
+    | _ \<Rightarrow> False)"
+| "grant s (RecvMsg p q m) = 
+    (case (sectxt_of_obj s (O_proc p),sectxt_of_obj s (O_msgq q),sectxt_of_obj s (O_msg q m)) of 
+      (Some (pu,pr,pt), Some (qu,qr,qt), Some (mu,mr,mt)) \<Rightarrow> 
+         permission_check (pu,pr,pt) (qu,qr,qt) C_msgq P_read \<and>
+         permission_check (pu,pr,pt) (mu,mr,mt) C_msg  P_receive
+    | _ \<Rightarrow> False)"
+| "grant s (RemoveMsgq p q) = 
+    (case (sectxt_of_obj s (O_proc p), sectxt_of_obj s (O_msgq q)) of
+      (Some (pu,pr,pt), Some (qu,qr,qt)) \<Rightarrow> 
+        permission_check (pu,pr,pt) (qu,qr,qt) C_msgq P_destroy
+    | _ \<Rightarrow> False)"
+(*
+| "grant s (CreateShM p h) = 
+    (case (sectxt_of_obj s (O_proc p)) of 
+       Some (pu,pr,pt) \<Rightarrow> 
+        (permission_check (pu,pr,pt) (pu,pr,pt) C_shm P_associate \<and>
+         permission_check (pu,pr,pt) (pu,pr,pt) C_shm P_create)
+     | _ \<Rightarrow> False)"
+| "grant s (Attach p h flag) = 
+    (case (sectxt_of_obj s (O_proc p), sectxt_of_obj s (O_shm h)) of
+       (Some (pu,pr,pt), Some (hu,hr,ht)) \<Rightarrow> 
+         if flag = SHM_RDONLY 
+         then permission_check (pu,pr,pt) (hu,hr,ht) C_shm P_read
+         else permission_check (pu,pr,pt) (hu,hr,ht) C_shm P_read \<and>
+              permission_check (pu,pr,pt) (hu,hr,ht) C_shm P_write
+     | _ \<Rightarrow> False)"
+| "grant s (Detach p h) = True" (*?*)
+| "grant s (DeleteShM p h) = 
+    (case (sectxt_of_obj s (O_proc p), sectxt_of_obj s (O_shm h)) of
+       (Some (pu,pr,pt), Some (hu,hr,ht)) \<Rightarrow> 
+          permission_check (pu,pr,pt) (hu,hr,ht) C_shm P_destroy
+     | _ \<Rightarrow> False)"
+*)
+
+| "grant s _ = False" (* socket event is currently not in consideration *)
+
+inductive valid :: "t_state \<Rightarrow> bool"
+where
+  "valid []"
+| "\<lbrakk>valid s; os_grant s e; grant s e\<rbrakk> \<Longrightarrow> valid (e # s)"
+
+(* tobj: object that can be tainted *)
+fun tobj_in_init :: "t_object \<Rightarrow> bool"
+where
+  "tobj_in_init (O_proc p) = (p \<in> init_procs)"
+| "tobj_in_init (O_file f) = (is_init_file f)"
+(* directory is not taintable 
+| "tobj_in_init (O_dir  f) = (f \<in> init_files)"
+
+| "tobj_in_init (O_node n) = (n \<in> init_nodes)"
+| "tobj_in_init (O_msg q m) = (m \<in> set (init_msgs_of_queue q) \<and> q \<in> init_msgqs)"
+*)
+| "tobj_in_init _ = False" (* other kind of obj cannot be tainted *)
+
+(* no use
+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 (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>"
+*)
+
+end
+
+locale tainting = flask + 
+
+fixes seeds :: "t_object set"
+assumes 
+      seeds_appropriate: "\<And> obj. obj \<in> seeds \<Longrightarrow> appropriate obj"
+  and same_inode_in_seeds: "\<And> f f'. \<lbrakk>O_file f \<in> seeds; has_same_inode [] f f'\<rbrakk> \<Longrightarrow> O_file f' \<in> seeds"
+(*
+  and flow_shm_in_seeds: "\<And> p p'. \<lbrakk>O_proc p \<in> seeds; info_flow_shm [] p p'\<rbrakk> \<Longrightarrow> O_proc p' \<in> seeds"
+*)
+begin
+
+(*
+inductive tainted :: "t_object \<Rightarrow> t_state \<Rightarrow> bool" ("_ \<in> tainted _" [100, 100] 100)
+where
+  t_init:   "obj \<in> seeds \<Longrightarrow> obj \<in> tainted []"
+| t_clone:  "\<lbrakk>O_proc p \<in> tainted s; valid (Clone p p' fds shms # s)\<rbrakk> 
+             \<Longrightarrow> O_proc p' \<in> tainted (Clone p p' fds # s)"
+| t_execve: "\<lbrakk>O_file f \<in> tainted s; valid (Execve p f fds # s)\<rbrakk> 
+             \<Longrightarrow> O_proc p \<in> tainted (Execve p f fds # s)"
+| t_ptrace: "\<lbrakk>O_proc p \<in> tainted s; valid (Ptrace p p' # s); info_flow_shm s p' p''\<rbrakk>
+             \<Longrightarrow> O_proc p'' \<in> tainted (Ptrace p p' # s)"
+| t_ptrace':"\<lbrakk>O_proc p' \<in> tainted s; valid (Ptrace p p' # s); info_flow_shm s p p''\<rbrakk>
+             \<Longrightarrow> O_proc p'' \<in> tainted (Ptrace p p' # s)"
+| t_cfile:  "\<lbrakk>O_proc p \<in> tainted s; valid (Open p f flags fd (Some inum) # s)\<rbrakk>
+             \<Longrightarrow> O_file f \<in> tainted (Open p f flags fd (Some inum) # s)"
+| t_read:   "\<lbrakk>O_file f \<in> tainted s; valid (ReadFile p fd # s); 
+              file_of_proc_fd s p fd = Some f; info_flow_shm s p p'\<rbrakk> 
+             \<Longrightarrow> O_proc p' \<in> tainted (ReadFile p fd # s)"
+| t_write:  "\<lbrakk>O_proc p \<in> tainted s; valid (WriteFile p fd # s);
+              file_of_proc_fd s p fd = Some f; has_same_inode s f f'\<rbrakk> 
+             \<Longrightarrow> O_file f' \<in> tainted (WriteFile p fd # s)"
+(* directory is not tainted 
+| t_mkdir:  "\<lbrakk>O_proc p \<in> tainted s; valid (Mkdir p f inum # s)\<rbrakk> 
+             \<Longrightarrow> O_dir f \<in> tainted (Mkdir p f inum # s)"
+ *)
+| t_link:   "\<lbrakk>O_file f \<in> tainted s; valid (LinkHard p f f' # s)\<rbrakk>
+             \<Longrightarrow> O_file f' \<in> tainted (LinkHard p f f' # s)"
+| t_trunc:  "\<lbrakk>O_proc p \<in> tainted s; valid (Truncate p f len # s); len > 0; has_same_inode s f f'\<rbrakk>
+             \<Longrightarrow> O_file f' \<in> tainted (Truncate p f len # s)"
+(*
+| t_rename: "\<lbrakk>O_file f'' \<in> tainted s; valid (Rename p f f' # s);f \<preceq> f''\<rbrakk>
+             \<Longrightarrow> O_file (file_after_rename f f' f'') \<in> tainted (Rename p f f' # s)"
+| t_rename':"\<lbrakk>O_dir f'' \<in> tainted s; valid (Rename p f f' # s);f \<preceq> f''\<rbrakk>
+             \<Longrightarrow> O_dir (file_after_rename f f' f'') \<in> tainted (Rename p f f' # s)"
+*)
+| t_attach1:"\<lbrakk>O_proc p \<in> tainted s; valid (Attach p h SHM_RDWR # s); (p', flag') \<in> procs_of_shm s h; info_flow_shm s p' p''\<rbrakk>
+             \<Longrightarrow> O_proc p'' \<in> tainted (Attach p h SHM_RDWR # s)"
+| t_attach2:"\<lbrakk>O_proc p' \<in> tainted s; valid (Attach p h flag # s); (p', SHM_RDWR) \<in> procs_of_shm s h; info_flow_shm s p p''\<rbrakk>
+             \<Longrightarrow> O_proc p'' \<in> tainted (Attach p h flag # s)"
+| t_sendmsg:"\<lbrakk>O_proc p \<in> tainted s; valid (SendMsg p q m # s)\<rbrakk>
+             \<Longrightarrow> O_msg q m \<in> tainted (SendMsg p q m # s)"
+| t_recvmsg:"\<lbrakk>O_msg q m \<in> tainted s; valid (RecvMsg p q m # s); info_flow_shm s p p'\<rbrakk>
+             \<Longrightarrow> O_proc p' \<in> tainted (RecvMsg p q m # s)"
+| t_remain: "\<lbrakk>obj \<in> tainted s; valid (e # s); alive (e # s) obj\<rbrakk> 
+             \<Longrightarrow> obj \<in> tainted (e # s)"
+*)
+
+fun tainted :: "t_state \<Rightarrow> t_object set"
+where
+  "tainted [] = seeds"
+| "tainted (Clone p p' fds # s) = 
+     (if (O_proc p) \<in> tainted s then tainted s \<union> {O_proc p'} else tainted s)"
+| "tainted (Execve p f fds # s) = 
+     (if (O_file f) \<in> tainted s then tainted s \<union> {O_proc p} else tainted s)"
+| "tainted (Kill p p' # s) = tainted s - {O_proc p'}"
+| "tainted (Ptrace p p' # s) = 
+     (if (O_proc p \<in> tainted s)
+      then tainted s \<union> {O_proc p'}
+      else if (O_proc p' \<in> tainted s)
+           then tainted s \<union> {O_proc p}
+                else tainted s)"
+| "tainted (Exit p # s) = tainted s - {O_proc p}"
+| "tainted (Open p f flags fd opt # s) = 
+     (case opt of
+        Some inum \<Rightarrow> (if (O_proc p) \<in> tainted s
+                      then tainted s \<union> {O_file f}
+                      else tainted s)
+      | _         \<Rightarrow> tainted s)" 
+| "tainted (ReadFile p fd # s) = 
+     (case (file_of_proc_fd s p fd) of
+        Some f \<Rightarrow> if (O_file f) \<in> tainted s
+                  then tainted s \<union> {O_proc p}
+                  else tainted s
+      | None   \<Rightarrow> tainted s)"
+| "tainted (WriteFile p fd # s) = 
+     (case (file_of_proc_fd s p fd) of 
+        Some f \<Rightarrow> if (O_proc p) \<in> tainted s
+                  then tainted s \<union> {O_file f' | f'. f' \<in> same_inode_files s f}
+                  else tainted s
+      | None   \<Rightarrow> tainted s)"
+| "tainted (CloseFd p fd # s) = 
+     (case (file_of_proc_fd s p fd) of
+        Some f \<Rightarrow> ( if ((proc_fd_of_file s f = {(p,fd)}) \<and> (f \<in> files_hung_by_del s))
+                    then tainted s - {O_file f} else tainted s )
+      | _      \<Rightarrow> tainted s)"
+| "tainted (UnLink p f # s) = 
+     (if (proc_fd_of_file s f = {}) then tainted s - {O_file f} else tainted s)"
+| "tainted (LinkHard p f f' # s) = 
+     (if (O_file f \<in> tainted s) then tainted s \<union> {O_file f'} else tainted s)"
+| "tainted (Truncate p f len # s) = 
+     (if (len > 0 \<and> O_proc p \<in> tainted s)
+      then tainted s \<union> {O_file f' | f'. f' \<in> same_inode_files s f}
+      else tainted s)"
+| "tainted (SendMsg p q m # s) = 
+     (if (O_proc p \<in> tainted s) then tainted s \<union> {O_msg q m} else tainted s)"
+| "tainted (RecvMsg p q m # s) = 
+     (if (O_msg q m \<in> tainted s) 
+      then (tainted s \<union> {O_proc p}) - {O_msg q m}
+      else tainted s)"
+| "tainted (RemoveMsgq p q # s) = tainted s - {O_msg q m| m. O_msg q m \<in> tainted s}"
+| "tainted (e # s) = tainted s"
+
+(* reasonable tainted object, fd and sock and msgq and msg is excluded 
+ * this is different from tainted, which already excluded some of them,
+ * tainted not excluded msg, which does not have the meaning of "tainteable", but 
+ * but acting as a media to "pass" the virus. We normally will not say that 
+ * a message is tableable or not.
+ *)
+fun appropriate :: "t_object \<Rightarrow> bool"
+where
+  "appropriate (O_proc p)     = (p \<in> init_procs)"
+| "appropriate (O_file f)     = (is_init_file f)"
+| "appropriate (O_dir  f)     = False"
+| "appropriate (O_fd p fd)    = False"
+| "appropriate (O_node n)     = False" (* cause socket is temperary not considered *)
+| "appropriate (O_tcp_sock k) = False"
+| "appropriate (O_udp_sock k) = False"
+(*
+| "appropriate (O_shm  h)     = False"
+*)
+| "appropriate (O_msgq q)     = False"
+| "appropriate (O_msg q m)    = False"
+
+definition taintable:: "t_object \<Rightarrow> bool"
+where
+  "taintable obj \<equiv> init_alive obj \<and> appropriate obj \<and> (\<exists> s. obj \<in> tainted s)"
+
+definition undeletable :: "t_object \<Rightarrow> bool"
+where
+  "undeletable obj \<equiv> init_alive obj \<and> \<not> (\<exists> s. valid s \<and> deleted obj s)"
+
+fun exited :: "t_object \<Rightarrow> t_state \<Rightarrow> bool"
+where
+  "exited (O_proc p) s = (Exit p \<in> set s)"
+| "exited _          s = False"
+
+end
+
+end
\ No newline at end of file