(*<*)

theory Init_prop

imports Main OS_type_def Flask Flask_type Static_type Static

begin

(*>*)

context init begin

lemma init_files_prop1: "init_inum_of_file f = Some im \<Longrightarrow> f \<in> init_files"

by (simp add:inof_has_file_tag)

lemma init_files_prop2: "finite init_files" 

by (simp add:init_finite_sets)

lemma init_files_prop3: "f \<in> init_files \<Longrightarrow> init_inum_of_file f \<noteq> None"

by (auto dest:init_file_has_inum)

lemma init_files_prop4: "(f \<in> init_files) = (f \<in> current_files [])"

apply (simp add:current_files_def, rule iffI)

using init_files_prop1 init_files_prop3 by auto

lemmas init_files_props = init_file_has_inum init_files_prop1 init_files_prop2 init_files_prop3 init_files_prop4

lemma init_inumof_prop1: "init_inum_of_file f = Some im \<Longrightarrow> \<exists> tag. init_itag_of_inum im = Some tag"

by (auto dest:inof_has_file_tag)

lemma init_inumof_prop2:  "init_inum_of_file f = Some im \<Longrightarrow> init_itag_of_inum im \<noteq> None"

by (auto dest:inof_has_file_tag)

lemma init_inumof_prop3: "\<lbrakk>init_inum_of_file f = Some im; init_itag_of_inum im = Some tag\<rbrakk> \<Longrightarrow> is_file_dir_itag tag"

by (auto dest:inof_has_file_tag)

lemmas init_inum_of_file_props = init_files_prop1 init_inumof_prop1 init_inumof_prop2 init_inumof_prop3

lemma init_inumos_prop1: "init_inum_of_socket s = Some im \<Longrightarrow> s \<in> init_sockets"

by (auto dest:inos_has_sock_tag)

lemma init_inumos_prop2: "init_inum_of_socket s = Some im \<Longrightarrow> init_itag_of_inum im = Some Tag_TCP_SOCK \<or> init_itag_of_inum im = Some Tag_UDP_SOCK"

apply (auto dest!:inos_has_sock_tag) 

apply (case_tac tag, simp+)

done

lemma init_inumos_prop3: "init_inum_of_socket s = Some im \<Longrightarrow> init_itag_of_inum im \<noteq> None"

by (auto dest:inos_has_sock_tag)

lemma init_inumos_prop4: "init_inum_of_socket s = Some im \<Longrightarrow> \<exists> tag. init_itag_of_inum im = Some tag \<and> is_sock_itag tag"

by (auto dest!:inos_has_sock_tag) 

lemmas init_inum_of_socket_props = init_inumos_prop1 init_inumos_prop2 init_inumos_prop3 init_inumos_prop4

lemma init_sockets_prop1: "(p, fd) \<in> init_sockets \<Longrightarrow> p \<in> init_procs"

by (auto dest: init_socket_has_inode)

lemma init_sockets_prop2: "(p, fd) \<in> init_sockets \<Longrightarrow> fd \<in> init_fds_of_proc p"

by (auto dest:init_socket_has_inode)

lemma init_sockets_prop3: "s \<in> init_sockets \<Longrightarrow> \<exists> im. init_inum_of_socket s = Some im"

by (case_tac s, auto dest:init_socket_has_inode)

lemma init_sockets_prop4: "s \<in> init_sockets \<Longrightarrow> init_inum_of_socket s \<noteq> None"

by (simp add:init_sockets_prop3)

lemma init_sockets_prop5: "s \<in> init_sockets = (s \<in> current_sockets [])"

apply (simp add:current_sockets_def, rule iffI)

using init_sockets_prop4 inos_has_sock_tag apply auto

apply (case_tac s, auto)

done

lemmas init_sockets_props = init_sockets_prop1 init_sockets_prop2 init_sockets_prop3 init_sockets_prop4 init_sockets_prop5

lemma is_init_file_prop1: "is_init_file f = (f \<in> init_files \<and> is_file [] f)"

by (auto simp add:is_init_file_def is_file_def init_inum_of_file_props split:option.splits)

lemma is_init_file_prop2: "is_init_file f = (init_alive (O_file f))"

by (auto simp add:is_init_file_def is_file_def init_inum_of_file_props split:option.splits)

lemmas is_init_file_props = is_init_file_prop1 is_init_file_prop2

lemma is_init_dir_prop1: "is_init_dir f = (f \<in> init_files \<and> is_dir [] f)"

by (auto simp add:is_init_dir_def is_dir_def init_inum_of_file_props split:option.splits)

lemma is_init_dir_prop2: "is_init_dir f = (init_alive (O_dir f))"

by (auto simp add:is_init_dir_def is_dir_def init_inum_of_file_props split:option.splits)

lemmas is_init_dir_props = is_init_dir_prop1 is_init_dir_prop2

lemma is_init_udp_sock_prop1: "is_init_udp_sock s = (s \<in> init_sockets \<and> is_udp_sock [] s)"

apply (auto simp add:is_init_udp_sock_def is_udp_sock_def init_inum_of_socket_props 

                dest:init_socket_has_inode split:option.splits)       

done

lemma is_init_udp_sock_prop2: "is_init_udp_sock s = (init_alive (O_udp_sock s))"

apply (auto simp add:is_init_udp_sock_def is_udp_sock_def init_inum_of_socket_props 

                dest:init_socket_has_inode split:option.splits)       

done

lemmas is_init_udp_sock_props = is_init_udp_sock_prop1 is_init_udp_sock_prop2

lemma is_init_tcp_sock_prop1: "is_init_tcp_sock s = (s \<in> init_sockets \<and> is_tcp_sock [] s)"

apply (auto simp add:is_init_tcp_sock_def is_tcp_sock_def init_inum_of_socket_props 

                dest:init_socket_has_inode split:option.splits)       

done

lemma is_init_tcp_sock_prop2: "is_init_tcp_sock s = (init_alive (O_tcp_sock s))"

apply (auto simp add:is_init_tcp_sock_def is_tcp_sock_def init_inum_of_socket_props 

                dest:init_socket_has_inode split:option.splits)       

done

lemmas is_init_tcp_sock_props = is_init_tcp_sock_prop1 is_init_tcp_sock_prop2

lemma init_parent_file_prop1: 

  "\<lbrakk>parent f = Some pf; f \<in> init_files\<rbrakk> \<Longrightarrow> is_init_dir pf"

apply (frule parent_file_in_init, simp, frule_tac f = pf in init_files_prop3)

apply (clarsimp, drule_tac im = y in init_parentf_is_dir, simp+)

by (simp add:is_init_dir_def)

lemma init_parent_file_prop1': 

  "a # f \<in> init_files \<Longrightarrow> is_init_dir f"

by (rule_tac pf = f in init_parent_file_prop1, auto)

lemma init_parent_file_prop2:

  "\<lbrakk>parent f = Some pf; is_init_file f\<rbrakk> \<Longrightarrow> is_init_dir pf"

by (rule init_parent_file_prop1, simp, simp add: is_init_file_props)

lemma init_parent_file_prop2':

  "is_init_file (f#pf) \<Longrightarrow> is_init_dir pf"

apply (rule init_parent_file_prop2)

by auto

lemma init_parent_file_prop3:

  "\<lbrakk>parent f = Some pf; is_init_dir f\<rbrakk> \<Longrightarrow> is_init_dir pf"

by (rule init_parent_file_prop1, simp, simp add: is_init_dir_props)

lemma init_parent_file_prop3':

  "is_init_dir (f#pf) \<Longrightarrow> is_init_dir pf"

apply (rule init_parent_file_prop3)

by auto

lemma parent_file_in_init': "a # f \<in> init_files \<Longrightarrow> f \<in> init_files"

by (subgoal_tac "parent (a # f) = Some f", drule parent_file_in_init, auto)

lemmas init_parent_file_props = parent_file_in_init init_parent_file_prop1 parent_file_in_init' init_parent_file_prop1' init_parent_file_prop2 init_parent_file_prop2' init_parent_file_prop3 init_parent_file_prop3'

lemma root_in_filesystem:  "[] \<in> init_files" 

using init_files_prop1 root_is_dir by auto

lemma root_is_init_dir: "is_init_dir []"

using root_is_dir

by (auto simp add:is_init_dir_def split:option.splits)

lemma root_is_init_dir': "is_init_file [] \<Longrightarrow> False"

using root_is_dir

by (auto simp:is_init_file_def split:option.splits)

lemma init_files_hung_prop1: "f \<in> init_files_hung_by_del \<Longrightarrow> f \<in> init_files"

by (auto dest:init_files_hung_valid)

lemma init_files_hung_prop2: "f \<in> init_files_hung_by_del \<Longrightarrow> \<exists> p fd. init_file_of_proc_fd p fd = Some f"

by (auto dest:init_files_hung_valid)

lemmas init_files_hung_by_del_props = init_files_hung_prop1 init_files_hung_prop2 init_files_hung_valid'

lemma init_fds_of_proc_prop1: "fd \<in> init_fds_of_proc p \<Longrightarrow> p \<in> init_procs"

by (auto dest!:init_procfds_valid)

lemma init_fds_of_proc_prop2: "fd \<in> init_fds_of_proc p \<Longrightarrow> (\<exists> f \<in> init_files. init_file_of_proc_fd p fd = Some f) \<or> (p, fd) \<in> init_sockets"

by (auto dest:init_procfds_valid)

lemmas init_fds_of_proc_props = init_fds_of_proc_prop1 init_fds_of_proc_prop2

lemma init_filefd_prop1: "init_file_of_proc_fd p fd = Some f \<Longrightarrow> f \<in> init_files"

by (auto dest!:init_filefd_valid intro:init_files_prop1)

lemma init_filefd_prop2: "init_file_of_proc_fd p fd = Some f \<Longrightarrow> p \<in> init_procs"

by (auto dest:init_filefd_valid)

lemma init_filefd_prop3: "init_file_of_proc_fd p fd = Some f \<Longrightarrow> fd \<in> init_fds_of_proc p"

by (auto dest:init_filefd_valid)

lemma init_filefd_prop4: "init_file_of_proc_fd p fd = Some f \<Longrightarrow> \<exists> flags. init_oflags_of_proc_fd p fd = Some flags"

by (auto dest:init_filefd_valid)

lemma init_filefd_prop5: "init_file_of_proc_fd p fd = Some f \<Longrightarrow> is_init_file f"

by (auto dest:init_filefd_valid simp:is_init_file_def)

lemmas init_file_of_proc_fd_props = init_filefd_prop1 init_filefd_prop2 init_filefd_prop3 init_filefd_prop4 init_filefd_prop5

lemma init_oflags_prop1: "init_oflags_of_proc_fd p fd = Some flags \<Longrightarrow> p \<in> init_procs"

by (auto dest:init_fileflag_valid init_file_of_proc_fd_props)

lemma init_oflags_prop2: "init_oflags_of_proc_fd p fd = Some flags \<Longrightarrow> fd \<in> init_fds_of_proc p"

by (auto dest:init_fileflag_valid init_file_of_proc_fd_props)

lemmas init_oflags_of_proc_fd_props = init_oflags_prop1 init_oflags_prop2 init_fileflag_valid

(*

lemma init_socketstate_prop1: "s \<in> init_sockets \<Longrightarrow> init_socket_state s \<noteq> None"

using init_socket_has_state

by (case_tac s, simp add:bidirect_in_init_def)

lemma init_socketstate_prop2: "s \<in> init_sockets \<Longrightarrow> \<exists> t. init_socket_state s = Some t"

using init_socket_has_state

by (case_tac s, simp add:bidirect_in_init_def)

lemma init_socketstate_prop3: "init_socket_state s = Some t \<Longrightarrow> s \<in> init_sockets"

using init_socket_has_state

by (case_tac s, simp add:bidirect_in_init_def)

lemmas init_socket_state_props = init_socketstate_prop1 init_socketstate_prop2 init_socketstate_prop3

*)

lemma init_inum_sock_file_noninter: "\<lbrakk>init_inum_of_socket s = Some im; init_inum_of_file f = Some im\<rbrakk> \<Longrightarrow> False"

apply (frule init_inumof_prop1, erule exE, drule init_inumof_prop3, simp)

apply (frule init_inumos_prop2)

apply (case_tac tag, simp+)

done

lemma init_parent_file_has_inum: "\<lbrakk>parent f = Some pf; init_inum_of_file f = Some im\<rbrakk> \<Longrightarrow> \<exists> im. init_inum_of_file pf = Some im"

by (drule init_files_prop1, drule parent_file_in_init, simp, simp add:init_files_props)

lemma init_file_has_no_son': "\<lbrakk>init_itag_of_inum im = Some Tag_FILE; init_inum_of_file f = Some im; parent f' = Some f\<rbrakk> \<Longrightarrow> init_inum_of_file f' = None"

apply (drule init_file_no_son, simp)

by (case_tac "init_inum_of_file f'", auto dest:init_files_prop1)

lemma init_parent_file_is_dir': "\<lbrakk>parent f = Some pf; init_inum_of_file f = Some im; init_inum_of_file pf = Some ipm\<rbrakk> \<Longrightarrow> init_itag_of_inum ipm = Some Tag_DIR"

by (drule init_parentf_is_dir, auto dest:init_files_prop1)

lemma init_file_hung_has_no_son: "\<lbrakk>f \<in> init_files_hung_by_del; parent f' = Some f; init_inum_of_file f' = Some im\<rbrakk> \<Longrightarrow> False"

apply (frule init_files_hung_prop1, drule init_file_has_inum, erule exE)

apply (drule init_files_hung_valid', simp)

apply (frule init_parent_file_is_dir', simp+)

apply (drule init_files_prop1)

apply (erule_tac x = f' in allE, simp)

by (case_tac f', simp_all add:no_junior_def)

end

context flask begin

lemma init_alive_prop: "init_alive obj = alive [] obj"

apply (case_tac obj, simp_all add:is_init_file_props is_init_dir_props is_init_tcp_sock_props

         is_init_udp_sock_props init_files_props init_sockets_props)

done

lemma init_alive_proc: "p \<in> init_procs \<Longrightarrow> init_alive (O_proc p)" by simp

lemma init_alive_file: "is_init_file f \<Longrightarrow> init_alive (O_file f)" by simp

lemma init_alive_dir: "is_init_dir f \<Longrightarrow> init_alive (O_dir f)" by simp

lemma init_alive_fd: "fd \<in> init_fds_of_proc p \<Longrightarrow> init_alive (O_fd p fd)" by simp

lemma init_alive_tcp: "is_init_tcp_sock s \<Longrightarrow> init_alive (O_tcp_sock s)" by simp

lemma init_alive_udp: "is_init_udp_sock s \<Longrightarrow> init_alive (O_udp_sock s)" by simp

lemma init_alive_node: "n \<in> init_nodes \<Longrightarrow> init_alive (O_node n)" by simp

lemma init_alive_shm: "h \<in> init_shms \<Longrightarrow> init_alive (O_shm h)" by simp

lemma init_alive_msgq: "q \<in> init_msgqs \<Longrightarrow> init_alive (O_msgq q)" by simp

lemma init_alive_msg: "\<lbrakk>m \<in> set (init_msgs_of_queue q); q \<in> init_msgqs\<rbrakk>

  \<Longrightarrow> init_alive (O_msg q m)" by simp

lemmas init_alive_intros = init_alive_proc init_alive_file init_alive_dir init_alive_fd 

  init_alive_tcp init_alive_udp init_alive_node init_alive_shm init_alive_msgq init_alive_msg

lemma init_file_type_prop1: "is_init_file f \<Longrightarrow> \<exists> t. init_type_of_obj (O_file f) = Some t"

using init_obj_has_type

by (auto simp:is_init_file_def split:option.splits)

lemma init_file_type_prop2: "is_init_file f \<Longrightarrow> init_type_of_obj (O_file f) \<noteq> None"

by (simp add:init_file_type_prop1)

lemma init_file_type_prop3: "init_type_of_obj (O_file f) = Some t \<Longrightarrow> f \<in> init_files"

apply (drule init_type_has_obj) 

by (simp add:is_init_file_def init_inum_of_file_props split:option.splits)

lemma init_file_type_prop4: "init_type_of_obj (O_file f) = Some t \<Longrightarrow> is_init_file f"

apply (drule init_type_has_obj) 

by (simp add:is_init_file_def init_inum_of_file_props split:option.splits)

lemmas init_file_types_props = init_file_type_prop1 init_file_type_prop2 init_file_type_prop3 init_file_type_prop4

lemma init_dir_type_prop1: "is_init_dir f \<Longrightarrow> \<exists> t. init_type_of_obj (O_dir f) = Some t"

using init_obj_has_type

by (auto simp:is_init_dir_def split:option.splits)

lemma init_dir_type_prop2: "is_init_dir f \<Longrightarrow> init_type_of_obj (O_dir f) \<noteq> None"

by (simp add:init_dir_type_prop1)

lemma init_dir_type_prop3: "init_type_of_obj (O_dir f) = Some t \<Longrightarrow> f \<in> init_files"

apply (drule init_type_has_obj) 

by (simp add:is_init_dir_def init_inum_of_file_props split:option.splits)

lemma init_dir_type_prop4: "init_type_of_obj (O_dir f) = Some t \<Longrightarrow> is_init_dir f"

apply (drule init_type_has_obj) 

by (simp add:is_init_dir_def init_inum_of_file_props split:option.splits)

lemmas init_dir_types_props = init_dir_type_prop1 init_dir_type_prop2 init_dir_type_prop3 init_dir_type_prop4

lemma init_procrole_prop1: "init_role_of_proc p = Some r \<Longrightarrow> p \<in> init_procs"

using init_proc_has_role

by (auto simp:bidirect_in_init_def)

lemma init_procrole_prop2: "p \<in> init_procs \<Longrightarrow> \<exists> r. init_role_of_proc p = Some r"

using init_proc_has_role

by (auto simp:bidirect_in_init_def)

lemma init_procrole_prop3: "p \<in> init_procs \<Longrightarrow> init_role_of_proc p \<noteq> None"

using init_proc_has_role

by (auto simp:bidirect_in_init_def)

lemmas init_role_of_proc_props = init_procrole_prop1 init_procrole_prop2 init_procrole_prop3

lemma init_file_user_prop1: "is_init_file f \<Longrightarrow> \<exists> t. init_user_of_obj (O_file f) = Some t"

apply (simp only: is_init_file_prop2)

by (drule init_obj_has_user, auto)

lemma init_file_user_prop2: "is_init_file f \<Longrightarrow> init_user_of_obj (O_file f) \<noteq> None"

by (simp add:init_file_user_prop1)

lemma init_file_user_prop3: "init_user_of_obj (O_file f) = Some t \<Longrightarrow> f \<in> init_files"

apply (drule init_user_has_obj) 

by (simp add:is_init_file_def init_inum_of_file_props split:option.splits)

lemma init_file_user_prop4: "init_user_of_obj (O_file f) = Some t \<Longrightarrow> is_init_file f"

apply (drule init_user_has_obj) 

by (simp add:is_init_file_def init_inum_of_file_props split:option.splits)

lemma init_file_user_prop5: "init_user_of_obj (O_file f) = Some u \<Longrightarrow> u \<in> init_users"

by (simp add:init_user_has_obj)

lemmas init_file_users_props = init_file_user_prop1 init_file_user_prop2 init_file_user_prop3 init_file_user_prop4 init_file_user_prop5

lemma init_dir_user_prop1: "is_init_dir f \<Longrightarrow> \<exists> t. init_user_of_obj (O_dir f) = Some t"

apply (simp only: is_init_dir_prop2)

by (drule init_obj_has_user, auto)

lemma init_dir_user_prop2: "is_init_dir f \<Longrightarrow> init_user_of_obj (O_dir f) \<noteq> None"

by (simp add:init_dir_user_prop1)

lemma init_dir_user_prop3: "init_user_of_obj (O_dir f) = Some t \<Longrightarrow> f \<in> init_files"

apply (drule init_user_has_obj) 

by (simp add:is_init_dir_def init_inum_of_file_props split:option.splits)

lemma init_dir_user_prop4: "init_user_of_obj (O_dir f) = Some t \<Longrightarrow> is_init_dir f"

apply (drule init_user_has_obj) 

by (simp add:is_init_dir_def init_inum_of_file_props split:option.splits)

lemma init_dir_user_prop5: "init_user_of_obj (O_dir f) = Some u \<Longrightarrow> u \<in> init_users"

by (simp add:init_user_has_obj)

lemmas init_dir_users_props = init_dir_user_prop1 init_dir_user_prop2 init_dir_user_prop3 init_dir_user_prop4 init_dir_user_prop5

end

context tainting_s begin

lemma init_file_has_ctxt:

  "is_init_file f \<Longrightarrow> \<exists> sec. init_sectxt_of_obj (O_file f) = Some sec"

apply (simp add:init_sectxt_of_obj_def split:option.splits)

apply (rule conjI, rule init_obj_has_user, simp add:is_init_file_props)

by (simp add:init_file_types_props)

lemma init_file_has_ctxt':

  "init_sectxt_of_obj (O_file f) = None \<Longrightarrow> \<not> is_init_file f"

by (rule notI, drule init_file_has_ctxt, simp)

lemma init_dir_has_ctxt:

  "is_init_dir f \<Longrightarrow> \<exists> sec. init_sectxt_of_obj (O_dir f) = Some sec"

apply (simp add:init_sectxt_of_obj_def split:option.splits)

apply (rule conjI, rule init_obj_has_user, simp add:is_init_dir_props)

by (simp add:init_dir_types_props)

lemma init_dir_has_ctxt':

  "init_sectxt_of_obj (O_dir f) = None \<Longrightarrow> \<not> is_init_dir f"

by (rule notI, drule init_dir_has_ctxt, simp)

lemma init_proc_has_ctxt:

  "p \<in> init_procs \<Longrightarrow> \<exists> sec. init_sectxt_of_obj (O_proc p) = Some sec"

apply (simp add:init_sectxt_of_obj_def split:option.splits)

apply (rule conjI, rule init_obj_has_user, simp)

apply (frule init_alive_proc, drule init_obj_has_type)

by (drule init_procrole_prop2, auto)

lemma init_proc_has_ctxt':

  "init_sectxt_of_obj (O_proc p) = None \<Longrightarrow> p \<notin> init_procs"

by (rule notI, drule init_proc_has_ctxt, simp)

lemma init_fd_has_ctxt:

  "fd \<in> init_fds_of_proc p \<Longrightarrow> \<exists> sec. init_sectxt_of_obj (O_fd p fd) = Some sec"

apply (simp add:init_sectxt_of_obj_def split:option.splits)

apply (rule conjI, rule init_obj_has_user, simp add:is_init_dir_props)

apply (drule init_alive_intros)

apply (drule init_obj_has_type, clarsimp)

done

lemma init_fd_has_ctxt':

  "init_sectxt_of_obj (O_fd p fd) = None \<Longrightarrow> fd \<notin> init_fds_of_proc p"

by (rule notI, drule init_fd_has_ctxt, simp)

lemma init_node_has_ctxt:

  "n \<in> init_nodes \<Longrightarrow> \<exists> sec. init_sectxt_of_obj (O_node n) = Some sec"

apply (simp add:init_sectxt_of_obj_def split:option.splits)

apply (rule conjI, rule init_obj_has_user, simp add:is_init_dir_props)

apply (drule init_alive_intros)

apply (drule init_obj_has_type, clarsimp)

done

lemma init_node_has_ctxt':

  "init_sectxt_of_obj (O_node n) = None \<Longrightarrow> n \<notin> init_nodes"

by (rule notI, drule init_node_has_ctxt, simp)

lemma init_tcp_has_ctxt:

  "is_init_tcp_sock s \<Longrightarrow> \<exists> sec. init_sectxt_of_obj (O_tcp_sock s) = Some sec"

apply (simp add:init_sectxt_of_obj_def split:option.splits)

apply (rule conjI, rule init_obj_has_user, simp add:is_init_dir_props)

apply (drule init_alive_intros)

apply (drule init_obj_has_type, clarsimp)

done

lemma init_tcp_has_ctxt':

  "init_sectxt_of_obj (O_tcp_sock s) = None \<Longrightarrow> \<not> is_init_tcp_sock s"

by (rule notI, drule init_tcp_has_ctxt, simp)

lemma init_udp_has_ctxt:

  "is_init_udp_sock s \<Longrightarrow> \<exists> sec. init_sectxt_of_obj (O_udp_sock s) = Some sec"

apply (simp add:init_sectxt_of_obj_def split:option.splits)

apply (rule conjI, rule init_obj_has_user, simp add:is_init_dir_props)

by (drule init_alive_intros, drule init_obj_has_type, clarsimp)

lemma init_udp_has_ctxt':

  "init_sectxt_of_obj (O_udp_sock s) = None \<Longrightarrow> \<not> is_init_udp_sock s"

by (rule notI, drule init_udp_has_ctxt, simp)

lemma init_shm_has_ctxt:

  "h \<in> init_shms \<Longrightarrow> \<exists> sec. init_sectxt_of_obj (O_shm h) = Some sec"

apply (simp add:init_sectxt_of_obj_def split:option.splits)

apply (rule conjI, rule init_obj_has_user, simp add:is_init_dir_props)

by (drule init_alive_intros, drule init_obj_has_type, clarsimp)

lemma init_shm_has_ctxt':

  "init_sectxt_of_obj (O_shm h) = None \<Longrightarrow> h \<notin> init_shms"

by (rule notI, drule init_shm_has_ctxt, simp)

lemma init_msgq_has_ctxt:

  "q \<in> init_msgqs \<Longrightarrow> \<exists> sec. init_sectxt_of_obj (O_msgq q) = Some sec"

apply (simp add:init_sectxt_of_obj_def split:option.splits)

apply (rule conjI, rule init_obj_has_user, simp add:is_init_dir_props)

by (drule init_alive_intros, drule init_obj_has_type, clarsimp)

lemma init_msgq_has_ctxt':

  "init_sectxt_of_obj (O_msgq q) = None \<Longrightarrow> q \<notin> init_msgqs"

by (rule notI, drule init_msgq_has_ctxt, simp)

lemma init_msg_has_ctxt:

  "\<lbrakk>m \<in> set (init_msgs_of_queue q); q \<in> init_msgqs\<rbrakk> \<Longrightarrow> \<exists> sec. init_sectxt_of_obj (O_msg q m) = Some sec"

apply (simp add:init_sectxt_of_obj_def split:option.splits)

apply (rule conjI, rule init_obj_has_user, simp add:is_init_dir_props)

by (drule init_alive_intros, simp, drule init_obj_has_type, clarsimp)

lemma init_msg_has_ctxt':

  "init_sectxt_of_obj (O_msg q m) = None \<Longrightarrow> m \<notin> set (init_msgs_of_queue q) \<or> q \<notin> init_msgqs"

by (auto dest:init_msg_has_ctxt)

lemma init_rootf_has_ctxt:

  "\<exists> sec. init_sectxt_of_obj (O_dir []) = Some sec"

apply (rule init_dir_has_ctxt, simp add:is_init_dir_def split:option.splits)

using root_is_dir by auto

lemma init_rootf_has_ctxt':

  "init_sectxt_of_obj (O_dir []) = None \<Longrightarrow> False" 

using init_rootf_has_ctxt by auto

lemmas init_has_ctxt = init_file_has_ctxt init_dir_has_ctxt init_proc_has_ctxt init_fd_has_ctxt

  init_node_has_ctxt init_tcp_has_ctxt init_udp_has_ctxt init_shm_has_ctxt init_msgq_has_ctxt

  init_msg_has_ctxt init_rootf_has_ctxt

lemmas init_has_ctxt' = init_file_has_ctxt' init_dir_has_ctxt' init_proc_has_ctxt' init_fd_has_ctxt'

  init_node_has_ctxt' init_tcp_has_ctxt' init_udp_has_ctxt' init_shm_has_ctxt' init_msgq_has_ctxt'

  init_msg_has_ctxt' init_rootf_has_ctxt'

lemma sec_of_root_valid:

  "init_sectxt_of_obj (O_dir []) = Some sec_of_root"

using init_rootf_has_ctxt

by (auto simp:init_sectxt_of_obj_def sec_of_root_def split:option.splits)

lemma sec_of_root_is_tuple:

  "\<exists> u t. sec_of_root = (u, R_object, t)"

using sec_of_root_valid

by (auto simp:sec_of_root_def init_sectxt_of_obj_def split:option.splits)