theory Enrich2

imports Main Flask Static Init_prop Valid_prop Tainted_prop Delete_prop Co2sobj_prop S2ss_prop S2ss_prop2

 Temp Enrich Proc_fd_of_file_prop

begin

context tainting_s begin

fun enrich_msgq :: "t_state \<Rightarrow> t_msgq \<Rightarrow> t_msgq \<Rightarrow> t_state"

where

  "enrich_msgq [] tq dq = []"

| "enrich_msgq (CreateMsgq p q # s) tq dq = 

    (if (tq = q) 

     then (CreateMsgq p dq # CreateMsgq p q # s)

     else CreateMsgq p q # (enrich_msgq s tq dq))"

| "enrich_msgq (SendMsg p q m # s) tq dq = 

    (if (tq = q) 

     then (SendMsg p dq m # SendMsg p q m # (enrich_msgq s tq dq))

     else SendMsg p q m # (enrich_msgq s tq dq))"

| "enrich_msgq (RecvMsg p q m # s) tq dq = 

    (if (tq = q) 

     then (RecvMsg p dq m # RecvMsg p q m # (enrich_msgq s tq dq))

     else RecvMsg p q m # (enrich_msgq s tq dq))"

| "enrich_msgq (e # s) tq dq = e # (enrich_msgq s tq dq)"

lemma enrich_msgq_duq_in:

  "\<lbrakk>q' \<notin> current_msgqs s; q \<in> current_msgqs s; no_del_event s; valid s\<rbrakk>

   \<Longrightarrow> q' \<in> current_msgqs (enrich_msgq s q q')"

apply (induct s, simp)

apply (frule vt_grant_os, frule vd_cons, case_tac a)

apply auto

done

lemma enrich_msgq_duq_sms:

  "\<lbrakk>q' \<notin> current_msgqs s; q \<in> current_msgqs s; no_del_event s; valid s\<rbrakk>

   \<Longrightarrow> msgs_of_queue (enrich_msgq s q q') q' = msgs_of_queue s q"

apply (induct s, simp)

apply (frule vt_grant_os, frule vd_cons, case_tac a)

apply auto

done

lemma enrich_msgq_cur_inof:

   \<Longrightarrow> inum_of_file (enrich_msgq s q q') f = inum_of_file s f"

apply (induct s arbitrary:f, simp)

apply (frule vt_grant_os, frule vd_cons, case_tac a)

apply (auto split:option.splits)

done

lemma enrich_msgq_cur_inos:

   \<Longrightarrow> inum_of_socket (enrich_msgq s q q') = inum_of_socket s"

apply (rule ext)

apply (induct s, simp)

apply (frule vt_grant_os, frule vd_cons, case_tac a)

apply (auto split:option.splits)

done

lemma enrich_msgq_cur_inos':

   \<Longrightarrow> inum_of_socket (enrich_msgq s q q') sock = inum_of_socket s sock"

apply (simp add:enrich_msgq_cur_inos)

done

lemma enrich_msgq_cur_inums:

   \<Longrightarrow> current_inode_nums (enrich_msgq s q q') = current_inode_nums s"

apply (auto simp:current_inode_nums_def current_file_inums_def 

  current_sock_inums_def enrich_msgq_cur_inof enrich_msgq_cur_inos)

done

lemma enrich_msgq_cur_itag:

   \<Longrightarrow> itag_of_inum (enrich_msgq s q q') = itag_of_inum s"

apply (rule ext)

apply (induct s, simp)

apply (frule vt_grant_os, frule vd_cons, case_tac a)

apply (auto split:option.splits t_socket_type.splits)

done

lemma enrich_msgq_cur_tcps:

   \<Longrightarrow> is_tcp_sock (enrich_msgq s q q') = is_tcp_sock s"

apply (rule ext)

apply (auto simp:is_tcp_sock_def enrich_msgq_cur_itag enrich_msgq_cur_inos

  split:option.splits t_inode_tag.splits)

done

lemma enrich_msgq_cur_udps:

   \<Longrightarrow> is_udp_sock (enrich_msgq s q q') = is_udp_sock s"

apply (rule ext)

apply (auto simp:is_udp_sock_def enrich_msgq_cur_itag enrich_msgq_cur_inos

  split:option.splits t_inode_tag.splits)

done

lemma enrich_msgq_cur_msgqs:

   \<Longrightarrow> current_msgqs (enrich_msgq s q q') = current_msgqs s \<union> {q'}"

apply (induct s, simp)

apply (frule vt_grant_os, frule vd_cons)

apply (case_tac a, auto)

done

lemma enrich_msgq_cur_msgs:

  "\<lbrakk>q' \<notin> current_msgqs s; q \<in> current_msgqs s; no_del_event s; valid s\<rbrakk>

apply (rule ext, simp, rule conjI, rule impI)

apply (simp add:enrich_msgq_duq_sms)

apply (induct s, simp)

apply (frule vt_grant_os, frule vd_cons)

apply (case_tac a, auto)

done

lemma enrich_msgq_cur_procs:

   \<Longrightarrow> current_procs (enrich_msgq s q q') = current_procs s"

apply (induct s, simp)

apply (frule vt_grant_os, frule vd_cons)

apply (case_tac a, auto)

done

lemma enrich_msgq_cur_files:

   \<Longrightarrow> current_files (enrich_msgq s q q') = current_files s"

apply (auto simp:current_files_def)

apply (simp add:enrich_msgq_cur_inof)+

done

lemma enrich_msgq_cur_fds:

   \<Longrightarrow> current_proc_fds (enrich_msgq s q q') = current_proc_fds s"

apply (induct s, simp)

apply (frule vt_grant_os, frule vd_cons, case_tac a)

apply auto

done

lemma enrich_msgq_filefd:

   \<Longrightarrow> file_of_proc_fd (enrich_msgq s q q') = file_of_proc_fd s"

apply (rule ext, rule ext)

apply (induct s, simp)

apply (frule vt_grant_os, frule vd_cons, case_tac a)

apply auto

done

lemma enrich_msgq_flagfd:

   \<Longrightarrow> flags_of_proc_fd (enrich_msgq s q q') = flags_of_proc_fd s"

apply (rule ext, rule ext)

apply (induct s, simp)

apply (frule vt_grant_os, frule vd_cons, case_tac a)

apply auto

done

lemma enrich_msgq_proc_fds:

   \<Longrightarrow> proc_file_fds (enrich_msgq s q q') = proc_file_fds s"

apply (auto simp:proc_file_fds_def enrich_msgq_filefd)

done

lemma enrich_msgq_hungs:

  "\<lbrakk>no_del_event s; valid s\<rbrakk>

   \<Longrightarrow> files_hung_by_del (enrich_msgq s q q') = files_hung_by_del s"

apply (induct s, simp)

apply (frule vt_grant_os, frule vd_cons, case_tac a)

apply (auto simp:files_hung_by_del.simps)

done

lemma enrich_msgq_is_file:

   \<Longrightarrow> is_file (enrich_msgq s q q') = is_file s"

apply (rule ext)

apply (auto simp add:is_file_def enrich_msgq_cur_itag enrich_msgq_cur_inof

  split:option.splits t_inode_tag.splits)

done

lemma enrich_msgq_is_dir:

   \<Longrightarrow> is_dir (enrich_msgq s q q') = is_dir s"

apply (rule ext)

apply (auto simp add:is_dir_def enrich_msgq_cur_itag enrich_msgq_cur_inof

  split:option.splits t_inode_tag.splits)

done

lemma enrich_msgq_alive:

  "\<lbrakk>alive s obj; valid s; q \<in> current_msgqs s; q' \<notin> current_msgqs s; no_del_event s\<rbrakk>

   \<Longrightarrow> alive (enrich_msgq s q q') obj"

apply (case_tac obj)

apply (simp_all add:enrich_msgq_is_file enrich_msgq_is_dir enrich_msgq_cur_msgqs 

  enrich_msgq_cur_msgs enrich_msgq_cur_procs enrich_msgq_cur_fds

  enrich_msgq_cur_tcps enrich_msgq_cur_udps)

apply (rule impI, simp)

done

lemma enrich_msgq_alive':

  "\<lbrakk>alive (enrich_msgq s q q') obj; valid s; q \<in> current_msgqs s; q' \<notin> current_msgqs s; no_del_event s\<rbrakk>

   \<Longrightarrow> alive s obj \<or> obj = O_msgq q' \<or> (\<exists> m. obj = O_msg q' m \<and> alive s (O_msg q m))"

apply (case_tac obj)

apply (simp_all add:enrich_msgq_is_file enrich_msgq_is_dir enrich_msgq_cur_msgqs 

  enrich_msgq_cur_msgs enrich_msgq_cur_procs enrich_msgq_cur_fds

  enrich_msgq_cur_tcps enrich_msgq_cur_udps)

apply (auto split:if_splits)

done

(* enrich s target_proc duplicated_pro *)

where 

     if (tp = p) 

     if (tp = p') 

     then Clone p dp (fds \<inter> proc_file_fds s p) # Clone p p' fds # s

     if (tp = p)

     if (tp = p) 

     if (tp = p) 

(*

| "enrich_proc (CloseFd p fd # s) tp dp = (

     if (tp = p \<and> fd \<in> proc_file_fds s p)

     then CloseFd dp fd # CloseFd p fd # (enrich_proc s tp dp)

     else CloseFd p fd # (enrich_proc s tp dp))"

*)

(*

| "enrich_proc (Attach p h flag # s) tp dp = (

     if (tp = p)

     then Attach dp h flag # Attach p h flag # (enrich_proc s tp dp)

     else Attach p h flag # (enrich_proc s tp dp))"

| "enrich_proc (Detach p h # s) tp dp = (

     if (tp = p) 

     then Detach dp h # Detach p h # (enrich_proc s tp dp)

     else Detach p h # (enrich_proc s tp dp))"

*)

(*

| "enrich_proc (Kill p p' # s) tp dp = (

     if (tp = p') then Kill p p' # s

     else Kill p p' # (enrich_proc s tp dp))"

| "enrich_proc (Exit p # s) tp dp = (

     if (tp = p) then Exit p # s

     else Exit p # (enrich_proc s tp dp))"

*)

definition is_created_proc:: "t_state \<Rightarrow> t_process \<Rightarrow> bool"

where

  "is_created_proc s p \<equiv> p \<in> current_procs s \<and> (p \<in> init_procs \<longrightarrow> died (O_proc p) s)"

definition is_created_proc':: "t_state \<Rightarrow> t_process \<Rightarrow> bool"

where

  "is_created_proc' s p \<equiv> p \<in> current_procs s \<and> p \<notin> init_procs"

lemma created_proc_clone:

  "valid (Clone p p' fds # s) \<Longrightarrow> 

   is_created_proc (Clone p p' fds # s) tp = (if (tp = p') then True else is_created_proc s tp)"

apply (drule vt_grant_os)

apply (auto simp:is_created_proc_def dest:not_all_procs_prop2)

using not_died_init_proc

by auto

lemma created_proc_exit: 

  "is_created_proc (Exit p # s) tp = (if (tp = p) then False else is_created_proc s tp)"

by (simp add:is_created_proc_def)

lemma created_proc_kill:

  "is_created_proc (Kill p p' # s) tp = (if (tp = p') then False else is_created_proc s tp)"

by (simp add:is_created_proc_def)

lemma created_proc_other:

  "\<lbrakk>\<And> p p' fds. e \<noteq> Clone p p' fds;

    \<And> p. e \<noteq> Exit p;

    \<And> p p'. e \<noteq> Kill p p'\<rbrakk> \<Longrightarrow> is_created_proc (e # s) tp = is_created_proc s tp"

by (case_tac e, auto simp:is_created_proc_def)

lemmas is_created_proc_simps = created_proc_clone created_proc_exit created_proc_kill created_proc_other

lemma no_del_died:

  "\<lbrakk>no_del_event s; died obj s\<rbrakk> \<Longrightarrow> (\<exists> p fd. obj = O_fd p fd \<or> obj = O_tcp_sock (p, fd) \<or> obj = O_udp_sock (p, fd))

  \<or> (\<exists> q m. obj = O_msg q m) "

apply (induct s)

apply simp

apply (case_tac a)

apply (auto split:option.splits)

done

lemma no_del_created_eq:

  "no_del_event s \<Longrightarrow> is_created_proc s p = is_created_proc' s p"

apply (induct s)

apply (simp add:is_created_proc_def is_created_proc'_def)

apply (case_tac a)

apply (auto simp add:is_created_proc_def is_created_proc'_def dest:no_del_died)

done

lemma enrich_proc_dup_in:

apply (induct s, simp add:is_created_proc_def)

apply (frule vt_grant_os, frule vd_cons)

lemma enrich_proc_dup_ffd:

  "\<lbrakk>file_of_proc_fd s p fd = Some f; is_created_proc s p; p' \<notin> all_procs s; valid s\<rbrakk>

apply (induct s, simp add:is_created_proc_def)

apply (frule vt_grant_os, frule vd_cons)

  dest:not_all_procs_prop3 split:if_splits option.splits)

lemma enrich_proc_dup_ffd':

    no_del_event s; valid s\<rbrakk>

   \<Longrightarrow> file_of_proc_fd s p fd = Some f"

apply (induct s, simp add:is_created_proc_def)

apply (frule vt_grant_os, frule vd_cons)

  dest:not_all_procs_prop3 split:if_splits option.splits)

lemma enrich_proc_dup_ffd_eq:

  "\<lbrakk>is_created_proc s p; p' \<notin> all_procs s; no_del_event s; valid s\<rbrakk>

apply (case_tac "file_of_proc_fd s p fd")

apply (auto dest:enrich_proc_dup_ffd enrich_proc_dup_ffd')

done

lemma enrich_proc_dup_fflags:

  "\<lbrakk>flags_of_proc_fd s p fd = Some flag; is_created_proc s p; p' \<notin> all_procs s; valid s\<rbrakk>

   \<Longrightarrow> flags_of_proc_fd (enrich_proc s p p') p' fd = Some (remove_create_flag flag) \<or>

       flags_of_proc_fd (enrich_proc s p p') p' fd = Some flag"

apply (induct s arbitrary:p, simp add:is_created_proc_def)

apply (frule vt_grant_os, frule vd_cons)

  dest:not_all_procs_prop3 split:if_splits option.splits)

lemma enrich_proc_dup_ffds:

  "\<lbrakk>is_created_proc s p; p' \<notin> all_procs s; no_del_event s; valid s\<rbrakk>

   \<Longrightarrow> proc_file_fds (enrich_proc s p p') p' = proc_file_fds s p"

apply (auto simp:proc_file_fds_def)

apply (rule_tac x = f in exI) 

apply (erule enrich_proc_dup_ffd', simp+)

apply (rule_tac x = f in exI)

apply (erule enrich_proc_dup_ffd, simp+)

done

lemma enrich_proc_dup_ffds_eq_fds:

  "\<lbrakk>is_created_proc s p; p' \<notin> all_procs s; no_del_event s; valid s\<rbrakk>

   \<Longrightarrow> current_proc_fds (enrich_proc s p p') p' = proc_file_fds s p"

apply (induct s arbitrary:p)

apply (simp add: is_created_proc_def)

apply (frule not_all_procs_prop3)

apply (frule vd_cons, frule vt_grant_os, case_tac a)

apply (auto split:if_splits option.splits dest:proc_fd_in_fds set_mp not_all_procs_prop3 

done

lemma enrich_proc_cur_inof:

  "\<lbrakk>valid s; no_del_event s\<rbrakk> \<Longrightarrow> inum_of_file (enrich_proc s p p') f = inum_of_file s f"

apply (induct s arbitrary:f)

apply simp

apply (frule vd_cons, frule vt_grant_os, frule vt_grant)

apply (case_tac a, auto)

lemma not_all_procs_sock:

  "\<lbrakk>p' \<notin> all_procs s; valid s\<rbrakk> \<Longrightarrow> inum_of_socket s (p', fd) = None"

apply (frule not_all_procs_prop3)

apply (case_tac "inum_of_socket s (p', fd)", simp_all)

apply (drule cn_in_curp', simp+)

done

lemma enrich_proc_cur_inos:

  "\<lbrakk>valid s; no_del_event s; p' \<notin> all_procs s\<rbrakk> 

   \<Longrightarrow> inum_of_socket (enrich_proc s p p') (tp, fd) = inum_of_socket s (tp, fd)"

apply (induct s arbitrary:tp)

apply simp

apply (frule vd_cons, frule vt_grant_os)

apply (simp add:proc_file_fds_def, erule exE)

apply (case_tac "inum_of_socket s (nat1, fd)", simp_all)

apply (drule filefd_socket_conflict, simp_all add:current_sockets_def)

lemma enrich_proc_cur_inums:

  "\<lbrakk>p' \<notin> all_procs s; no_del_event s; valid s\<rbrakk>

   \<Longrightarrow> current_inode_nums (enrich_proc s p p') = current_inode_nums s"

apply (auto simp:current_inode_nums_def current_file_inums_def 

  current_sock_inums_def enrich_proc_cur_inof enrich_proc_cur_inos)

done

lemma enrich_proc_cur_itag:

  "\<lbrakk>valid s; no_del_event s; p' \<notin> all_procs s\<rbrakk> 

   \<Longrightarrow> itag_of_inum (enrich_proc s p p') i = itag_of_inum s i"

apply (induct s)

apply simp

apply (frule vd_cons, frule vt_grant_os)

lemma enrich_proc_cur_tcps:

  "\<lbrakk>valid s; no_del_event s; p' \<notin> all_procs s\<rbrakk> 

   \<Longrightarrow> is_tcp_sock (enrich_proc s p p') = is_tcp_sock s"

apply (rule ext, case_tac x)

apply (auto simp add:is_tcp_sock_def enrich_proc_cur_itag enrich_proc_cur_inos

  split:option.splits t_inode_tag.splits)

done

lemma enrich_proc_cur_udps:

  "\<lbrakk>valid s; no_del_event s; p' \<notin> all_procs s\<rbrakk> 

   \<Longrightarrow> is_udp_sock (enrich_proc s p p') = is_udp_sock s"

apply (rule ext, case_tac x)

  split:option.splits t_inode_tag.splits)

done

lemma enrich_proc_cur_msgqs:

apply (induct s, simp)

apply (frule vt_grant_os, frule vd_cons)

lemma enrich_proc_cur_msgs:

apply (induct s, simp)

apply (frule vt_grant_os, frule vd_cons)

lemma enrich_proc_cur_procs:

  "\<lbrakk>p' \<notin> all_procs s; no_del_event s; is_created_proc s p; valid s\<rbrakk>