theory Dynamic_static

imports Main Flask Static Init_prop Valid_prop Tainted_prop Delete_prop Co2sobj_prop S2ss_prop S2ss_prop2

 Temp

begin

context tainting_s begin

fun remove_create_flag :: "t_open_flags \<Rightarrow> t_open_flags"

where

  "remove_create_flag (mflag, oflags) = (mflag, oflags - {OF_CREAT})"

fun all_procs :: "t_state \<Rightarrow> t_process set"

where

  "all_procs [] = init_procs"

| "all_procs (Clone p p' fds shms # s) = insert p' (all_procs s)"

| "all_procs (e # s) = all_procs s"

definition brandnew_proc :: "t_state \<Rightarrow> t_process"

where

  "brandnew_proc s \<equiv> next_nat (all_procs s)"

(*

definition brandnew_proc :: "t_state \<Rightarrow> t_process"

where

  "brandnew_proc s \<equiv> next_nat ({p | p s'. p \<in> current_procs s' \<and> s' \<preceq> s})"

 another approach:

  brandnew_proc = next_nat (all_procs s),

    where all_procs is a event-trace listener *)

(*

lemma brandnew_proc_prop1:

  "\<lbrakk>s' \<preceq> s; valid s\<rbrakk> \<Longrightarrow> brandnew_proc s \<notin> current_procs s'"

apply (frule vd_preceq, simp)

apply (simp add:brandnew_proc_def)

apply (auto)

sorry

lemma brandnew_proc_prop2:

  "\<lbrakk>p \<in> current_procs s'; s' \<preceq> s; valid s\<rbrakk> \<Longrightarrow> brandnew_proc s \<noteq> p"

by (auto dest:brandnew_proc_prop1)

lemma brandnew_proc_prop3:

  "\<lbrakk>p \<in> current_procs s; valid (e # s)\<rbrakk> \<Longrightarrow> brandnew_proc (e # s) \<noteq> p"

apply (rule brandnew_proc_prop2, simp)

apply (rule no_juniorI, simp+)

done

*)

(* enrich s target_proc duplicated_pro *)

fun enrich_proc :: "t_state \<Rightarrow> t_process \<Rightarrow> t_process \<Rightarrow> t_state"

where 

  "enrich_proc [] tp dp = []"

| "enrich_proc (Execve p f fds # s) tp dp = (

     if (tp = p) 

     then Execve dp f fds # Execve p f fds # (enrich_proc s tp dp)

     else Execve p f fds # (enrich_proc s tp dp))"

| "enrich_proc (Clone p p' fds shms # s) tp dp = (

     if (tp = p') 

     then Clone p dp fds shms # Clone p p' fds shms # s

     else Clone p dp fds shms # (enrich_proc s tp dp))"

| "enrich_proc (Open p f flags fd opt # s) tp dp = (

     if (tp = p)

     then Open dp f (remove_create_flag flags) fd opt # Open p f flags fd opt # (enrich_proc s tp dp)

     else Open p f flags fd opt # (enrich_proc s tp dp))"

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

     if (tp = p)

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

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

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

     if (tp = p)

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

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

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

     if (tp = p) 

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

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

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

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

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

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

     if (tp = p) then Exit p # s

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

| "enrich_proc (e # s) tp dp = e # (enrich_proc s tp dp)"

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

where

  "is_created_proc s p \<equiv> p \<in> init_procs \<longrightarrow> deleted (O_proc p) s"

lemma enrich_proc_prop:

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

   \<Longrightarrow> valid (enrich_proc s p p') \<and> 

proof (induct s)

  case Nil

  thus ?case by (auto simp:is_created_proc_def)

next

  case (Cons e s)

  \<Longrightarrow> valid (enrich_proc s p p') \<and>

     (alive s obj \<longrightarrow> alive (enrich_proc s p p') obj \<and> co2sobj (enrich_proc s p p') obj = co2sobj s obj)"

    by auto

    by (auto dest:vd_cons vt_grant vt_grant_os)

  have c1: "valid (enrich_proc (e # s) p p')"

    apply (case_tac e)

    sorry

  moreover have c2: "p' \<in> current_procs (enrich_proc (e # s) p p')"

    sorry

  moreover have c3: "co2sobj (enrich_proc (e # s) p p') (O_proc p') = co2sobj (enrich_proc (e # s) p p') (O_proc p)"

    sorry

  moreover have c4: "alive (e # s) obj \<longrightarrow>

     alive (enrich_proc (e # s) p p') obj \<and> co2sobj (enrich_proc (e # s) p p') obj = co2sobj (e # s) obj"

    sorry

  ultimately show ?case by auto

qed

lemma "alive s obj \<Longrightarrow> alive (enrich_proc s p p') obj"

apply (induct s, simp)

apply (case_tac a, case_tac[!] obj)

apply (auto simp:is_file_def is_dir_def split:option.splits t_inode_tag.splits)

thm is_file_other

lemma enrich_proc_valid:

  "\<lbrakk>p \<in> current_procs s; valid s; p \<in> init_procs \<longrightarrow> deleted (O_proc p) s; p' \<notin> current_procs s\<rbrakk> \<Longrightarrow> valid (enrich_proc s p p')"

apply (induct s, simp)

apply (frule vd_cons, frule vt_grant, frule vt_grant_os, case_tac a)

apply (auto intro!:valid.intros(2))

prefer 28

end

(* for any created obj, we can enrich trace with events that create new objs with the same static-properties *)

definition enriched:: "t_state \<Rightarrow> t_object set \<Rightarrow> t_state \<Rightarrow> bool"

where

  "enriched s objs s' \<equiv> \<forall> obj \<in> objs. \<exists> obj'. \<not> alive s obj' \<and> obj' \<notin> objs \<and>

                                                alive s' obj' \<and> co2sobj s' obj' = co2sobj s' obj"

definition reserved:: "t_state \<Rightarrow> t_object set \<Rightarrow> t_state \<Rightarrow> bool"

where

  "reserved s objs s' \<equiv> \<forall> obj. alive s obj \<longrightarrow> alive s' obj \<and> co2sobj s' obj = co2sobj s obj"

definition enrichable :: "t_state \<Rightarrow> t_object set \<Rightarrow> bool"

where

  "enrichable s objs \<equiv> \<exists> s'. valid s' \<and> s2ss s' = s2ss s \<and> enriched s objs s' \<and> reserved s objs s'"

fun is_created :: "t_state \<Rightarrow> t_object \<Rightarrow> bool"

where

  "is_created s (O_file f) = (\<forall> f' \<in> same_inode_files s f. init_alive (O_file f') \<longrightarrow> deleted (O_file f') s)"

| "is_created s obj        = (init_alive obj \<longrightarrow> deleted obj s)"

definition is_inited :: "t_state \<Rightarrow> t_object \<Rightarrow> bool"

where

  "is_inited s obj \<equiv> init_alive obj \<and> \<not> deleted obj s"

(*

lemma is_inited_eq_not_created:

  "is_inited s obj = (\<not> is_created s obj)"

by (auto simp:is_created_def is_inited_def)

*)

lemma many_sq_imp_sms:

  "\<lbrakk>S_msgq (Create, sec, sms) \<in> ss; ss \<in> static\<rbrakk> \<Longrightarrow> \<forall> sm \<in> (set sms). is_many_smsg sm"

sorry

(* recorded in our static world *)

fun recorded :: "t_object \<Rightarrow> bool"

where

  "recorded (O_proc p)     = True"

| "recorded (O_file f)     = True"

| "recorded (O_dir  f)     = True"

| "recorded (O_node n)     = False" (* cause socket is temperary not considered *)

| "recorded (O_shm  h)     = True"

| "recorded (O_msgq q)     = True"

| "recorded _              = False"

lemma cf2sfile_fi_init_file:

  "\<lbrakk>cf2sfile s f = Some (Init f', sec, psec, asecs); is_file s f; valid s\<rbrakk> 

   \<Longrightarrow> is_init_file f \<and> \<not> deleted (O_file f) s"

apply (simp add:cf2sfile_def sroot_def split:option.splits if_splits)

apply (case_tac f, simp, drule root_is_dir', simp+)

done

lemma root_not_deleted:

  "valid s \<Longrightarrow> \<not> deleted (O_dir []) s"

apply (induct s, simp)

apply (frule vd_cons, frule vt_grant_os, case_tac a)

by auto

lemma cf2sfile_fi_init_dir:

  "\<lbrakk>cf2sfile s f = Some (Init f', sec, psec, asecs); is_dir s f; valid s\<rbrakk> 

   \<Longrightarrow> is_init_dir f \<and> \<not> deleted (O_dir f) s"

apply (simp add:cf2sfile_def sroot_def split:option.splits if_splits)

apply (case_tac f, simp add:root_is_init_dir root_not_deleted, simp)

apply (drule file_dir_conflict, simp+)

done

lemma is_created_imp_many: 

  "\<lbrakk>is_created s obj; co2sobj s obj = Some sobj; alive s obj; valid s\<rbrakk> \<Longrightarrow> is_many sobj"

apply (case_tac obj, auto simp:co2sobj.simps split:option.splits)

apply (case_tac [!] a)

apply (auto simp:cp2sproc_def ch2sshm_def cq2smsgq_def cf2sfiles_def same_inode_files_def

  split:option.splits if_splits)

apply (frule cf2sfile_fi_init_file, simp add:is_file_def, simp)

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

apply (frule cf2sfile_fi_init_dir, simp+)+

done

lemma anotherp_imp_manysp: 

  "\<lbrakk>cp2sproc s p = Some sp; co2sobj s (O_proc p') = co2sobj s (O_proc p); p' \<noteq> p;

    p' \<in> current_procs s; p \<in> current_procs s\<rbrakk> 

   \<Longrightarrow> is_many_sproc sp"

by (case_tac sp, auto simp:cp2sproc_def co2sobj.simps split:option.splits if_splits)

lemma is_file_has_sfs:

  "\<lbrakk>is_file s f; valid s; cf2sfile s f = Some sf\<rbrakk> 

   \<Longrightarrow> \<exists> sfs. co2sobj s (O_file f) = Some (S_file sfs (O_file f \<in> Tainted s)) \<and> sf \<in> sfs"

apply (rule_tac x = "{sf' | f' sf'. cf2sfile s f' = Some sf' \<and> f' \<in> same_inode_files s f}" in exI)

apply (auto simp:co2sobj.simps cf2sfiles_def tainted_eq_Tainted)

apply (rule_tac x = f in exI, simp add:same_inode_files_prop9)

done

declare Product_Type.split_paired_Ex Product_Type.split_paired_All [simp del]

lemma current_proc_in_s2ss:

  "\<lbrakk>cp2sproc s p = Some sp; p \<in> current_procs s; valid s\<rbrakk>

   \<Longrightarrow> S_proc sp (O_proc p \<in> Tainted s) \<in> s2ss s"

apply (simp add:s2ss_def, rule_tac x = "O_proc p" in exI)

apply (auto simp:co2sobj.simps tainted_eq_Tainted)

done

lemma current_file_in_s2ss:

  "\<lbrakk>co2sobj s (O_file f) = Some (S_file sfs tagf); is_file s f; valid s\<rbrakk>

   \<Longrightarrow> S_file sfs tagf \<in> s2ss s"

by (simp add:s2ss_def, rule_tac x = "O_file f" in exI, simp)

declare npctxt_execve.simps grant_execve.simps search_check.simps [simp del]

lemma npctxt_execve_eq_sec:

  "\<lbrakk>sectxt_of_obj (Execve p f fds # s) (O_proc p) = Some sec'; sectxt_of_obj s (O_proc p) = Some sec;

    sectxt_of_obj s (O_file f) = Some fsec; valid (Execve p f fds # s)\<rbrakk>

   \<Longrightarrow> npctxt_execve sec fsec = Some sec'"

by (case_tac sec, case_tac fsec, auto simp:npctxt_execve.simps sectxt_of_obj_simps split:option.splits)

lemma npctxt_execve_eq_cp2sproc:

  "\<lbrakk>cp2sproc (Execve p f fds # s) p = Some (pi', sec', sfds', shms'); valid (Execve p f fds # s);

    cp2sproc s p = Some (pi, sec, sfds, shms); cf2sfile s f = Some (fi, fsec, psec, asecs)\<rbrakk>

   \<Longrightarrow> npctxt_execve sec fsec = Some sec'"

apply (frule vt_grant_os, frule vd_cons)

apply (rule npctxt_execve_eq_sec, auto simp:cp2sproc_def cf2sfile_def split:option.splits)

apply (case_tac f, auto dest:root_is_dir')

done

lemma seach_check_eq_static:

  "\<lbrakk>cf2sfile s f = Some sf; valid s; is_dir s f \<or> is_file s f\<rbrakk>

   \<Longrightarrow> search_check_s sec sf (is_file s f) = search_check s sec f"

apply (case_tac sf)

apply (induct f)

apply (auto simp:search_check_s_def search_check.simps cf2sfile_def sroot_def 

                 root_sec_remains init_sectxt_prop sec_of_root_valid

           dest!:root_is_dir' current_has_sec' split:option.splits)

apply (simp add:search_check_allp_def)

done

lemma grant_execve_intro_execve:

  "\<lbrakk>cp2sproc (Execve p f fds # s) p = Some (pi', sec', sfds', shms'); valid (Execve p f fds # s);

     cp2sproc s p = Some (pi, sec, sfds, shms); cf2sfile s f = Some (fi, fsec, psec, asecs)\<rbrakk>

    \<Longrightarrow> grant_execve sec fsec sec'"

apply (frule vt_grant_os, frule vd_cons, frule vt_grant)

apply (auto split:option.splits dest!:current_has_sec' simp del:grant_execve.simps simp add:cp2sproc_execve)

apply (erule_tac x = aba in allE, erule_tac x = aca in allE, erule_tac x = bb in allE)

apply (auto simp del:grant_execve.simps simp add:cp2sproc_def cf2sfile_def split:option.splits)

apply (case_tac f, simp, drule root_is_dir', simp, simp, simp)

apply (simp add:sectxt_of_obj_simps)

done

lemma search_check_intro_execve:

  "\<lbrakk>cp2sproc s p = Some (pi, sec, sfds, shms); valid (Execve p f fds # s)\<rbrakk>

   \<Longrightarrow> search_check s sec f"

apply (frule vt_grant_os, frule vd_cons, frule vt_grant)

apply (auto split:option.splits dest!:current_has_sec' simp del:grant_execve.simps simp add:cp2sproc_execve)

apply (erule_tac x = aaa in allE, erule_tac x = ab in allE, erule_tac x = ba in allE)

apply (auto simp add:cp2sproc_def cf2sfile_def split:option.splits)

done

lemma inherit_fds_check_intro_execve:

  "\<lbrakk>cp2sproc (Execve p f fds # s) p = Some (pi', sec', sfds', shms'); valid (Execve p f fds # s)\<rbrakk>

    \<Longrightarrow> inherit_fds_check s sec' p fds"

apply (frule vt_grant_os, frule vd_cons, frule vt_grant)

apply (auto split:option.splits dest!:current_has_sec' simp add:cp2sproc_execve)

apply (erule_tac x = aba in allE, erule_tac x = aca in allE, erule_tac x = bb in allE)

apply (auto simp add:cp2sproc_def cf2sfile_def split:option.splits)

apply (simp add:sectxt_of_obj_simps)

done

lemma execve_sfds_subset: 

  "\<lbrakk>cp2sproc (Execve p f fds # s) p = Some (pi', sec', sfds', shms'); valid (Execve p f fds # s);

    cp2sproc s p = Some (pi, sec, sfds, shms)\<rbrakk>

   \<Longrightarrow> sfds' \<subseteq> sfds"

apply (frule vt_grant_os)

apply (auto simp:cp2sproc_def cpfd2sfds_execve split:option.splits dest!:current_has_sec')

apply (simp add:cpfd2sfds_def)

apply (rule_tac x = fd in bexI, auto simp:proc_file_fds_def)

done

lemma inherit_fds_check_imp_static:

  "\<lbrakk>cp2sproc (Execve p f fds # s) p = Some (pi', sec', sfds', shms');

    inherit_fds_check s sec' p fds; valid (Execve p f fds # s)\<rbrakk>

   \<Longrightarrow> inherit_fds_check_s sec' sfds'"

apply (frule vt_grant_os, frule vd_cons, frule vt_grant)

apply (auto simp:cp2sproc_def cpfd2sfds_execve inherit_fds_check_def inherit_fds_check_s_def split:option.splits)

apply (erule_tac x = "(ad, ae, bc)" in ballE, auto simp:sectxts_of_sfds_def sectxts_of_fds_def)

apply (erule_tac x = fd in ballE, auto simp:cfd2sfd_def split:option.splits)

done

lemma d2s_main_execve_grant_aux:

  "\<lbrakk>cp2sproc (Execve p f fds # s) p = Some (pi', sec', sfds', shms'); valid (Execve p f fds # s);

    cp2sproc s p = Some (pi, sec, sfds, shms); cf2sfile s f = Some (fi, fsec, psec, asecs)\<rbrakk>

   \<Longrightarrow> (npctxt_execve sec fsec = Some sec') \<and> grant_execve sec fsec sec' \<and> 

       search_check_s sec (fi, fsec, psec, asecs) (is_file s f) \<and> 

       inherit_fds_check_s sec' sfds' \<and> sfds' \<subseteq> sfds"

apply (rule conjI, erule_tac pi = pi and sec = sec and sfds = sfds and 

   shms = shms and fi = fi and fsec = fsec and psec = psec and 

   asecs = asecs in npctxt_execve_eq_cp2sproc, simp, simp, simp)

apply (rule conjI, erule_tac pi = pi and sec = sec and sfds = sfds and 

   shms = shms and fi = fi and fsec = fsec and psec = psec and 

   asecs = asecs in grant_execve_intro_execve, simp, simp, simp)

apply (rule conjI, drule_tac sec = sec in search_check_intro_execve, simp)

apply (frule vd_cons, frule vt_grant_os)

apply (drule_tac sec = sec in seach_check_eq_static, simp, simp, simp)

apply (rule conjI, rule inherit_fds_check_imp_static, simp)

apply (erule inherit_fds_check_intro_execve, simp, simp)

apply (erule_tac pi = pi and sfds = sfds and shms = shms in execve_sfds_subset, simp+)

done

lemma d2s_main_execve:

  "\<lbrakk>valid (Execve p f fds # s); s2ss s \<propto> static\<rbrakk> \<Longrightarrow> s2ss (Execve p f fds # s) \<propto> static"

apply (frule vd_cons, frule vt_grant_os, clarsimp)

apply (frule is_file_has_sfile', simp, erule exE, frule is_file_has_sfs, simp+, erule exE, erule conjE)

apply (auto simp:s2ss_execve split:if_splits option.splits dest:current_proc_has_sp')

apply (clarsimp simp add:init_ss_in_def init_ss_eq_def)

apply (rule_tac x = "update_ss ss' (S_proc (ah, (ai, aj, bd), ak, be) (O_proc p \<in> Tainted s))

  (S_proc (ad, (ae, af, bb), ag, bc) (O_proc p \<in> Tainted s \<or> O_file f \<in> Tainted s))" in bexI)

apply (auto simp:update_ss_def elim:Set.subset_insertI2 simp:anotherp_imp_manysp)[1]

apply (case_tac "ah = ad", case_tac "bc = {}", simp)

apply (erule_tac sfs = sfs and fi = a and fsec = "(aa, ab,b)" and pfsec = ac and asecs = ba in s_execve,

  auto intro:current_proc_in_s2ss current_file_in_s2ss split:option.splits dest:d2s_main_execve_grant_aux)[1]

apply (simp add:cp2sproc_execve split:option.splits)

apply (simp add:cp2sproc_def split:option.splits if_splits)

apply (clarsimp simp add:init_ss_in_def init_ss_eq_def)

apply (rule_tac x = "update_ss ss' (S_proc (ah, (ai, aj, bd), ak, be) (O_proc p \<in> Tainted s))

  (S_proc (ad, (ae, af, bb), ag, bc) (O_proc p \<in> Tainted s \<or> O_file f \<in> Tainted s))" in bexI)

apply (rule conjI, simp add:update_ss_def)

apply (rule conjI, simp add:update_ss_def)

apply (auto)[1]

apply (simp add:update_ss_def)

apply (rule conjI, rule impI)

apply (rule subsetI, clarsimp)

apply (erule impE)

apply (erule set_mp, simp)

apply (case_tac ah, simp+)

apply (rule impI, rule subsetI, clarsimp)

apply (erule set_mp, simp)

apply (case_tac "ah = ad", case_tac "bc = {}", simp)

apply (erule_tac sfs = sfs and fi = a and fsec = "(aa, ab,b)" and pfsec = ac and asecs = ba in s_execve,

  auto intro:current_proc_in_s2ss current_file_in_s2ss split:option.splits dest:d2s_main_execve_grant_aux)[1]

apply (simp add:cp2sproc_execve split:option.splits)

apply (simp add:cp2sproc_def split:option.splits if_splits)

done

lemma co2sobj_eq_alive_proc_imp:

  "\<lbrakk>co2sobj s obj = co2sobj s (O_proc p); alive s (O_proc p); valid s\<rbrakk>

   \<Longrightarrow> \<exists> p'. obj = O_proc p'"

by (auto simp add:co2sobj.simps split:option.splits dest:current_proc_has_sp' intro:co2sobj_sproc_imp)

lemma enrichable_execve:

 assumes p1: "\<And> objs. \<forall> obj \<in> objs. alive s obj \<and> is_created s obj \<and> recorded obj

              \<Longrightarrow> \<exists> s'. valid s' \<and> s2ss s' = s2ss s \<and> enriched s objs s' \<and> reserved s objs s'"

  and p2: "valid (e # s)" and p3: "\<forall>obj\<in>objs. alive (e # s) obj \<and> is_created (e # s) obj \<and> recorded obj"

  and p4: "e = Execve p f fds" 

  shows "enrichable (e # s) objs"

proof-

  from p2 have os: "os_grant s e" and se: "grant s e" and vd: "valid s"

    by (auto dest:vt_grant_os vd_cons vt_grant)

  from p3 have recorded: "\<forall> obj \<in> objs. recorded obj" by auto

  from p3 p4 p2 have p1': "\<forall> obj \<in> objs. alive s obj \<and> is_created s obj"

    apply clarify

    apply (erule_tac x = obj in ballE, simp add:alive_simps)

    apply (case_tac obj, auto simp:same_inode_files_simps)

    done

  then obtain s' where a1: "valid s'" and a2: "s2ss s' = s2ss s" and a3: "enriched s objs s'"

    and a4: "reserved s objs s'"

    using p1 recorded by metis 

  show ?thesis

  proof (cases "O_proc p \<in> objs")

    case True

    hence p_in: "p \<in> current_procs s'" using a4 os p4

      by (auto simp:reserved_def elim:allE[where x = "O_proc p"])

    with a1 a3 True obtain p' where b1: "\<not> alive s (O_proc p')" and b2: "O_proc p' \<notin> objs" 

      and b3: "alive s' (O_proc p')" and b4: "co2sobj s' (O_proc p') = co2sobj s' (O_proc p)"

      apply (simp only:enriched_def)

      apply (erule_tac x = "O_proc p" in ballE)

      apply (erule exE|erule conjE)+

      apply (frule co2sobj_eq_alive_proc_imp, auto)

      done

    have "valid (Execve p' f fds # e # s')"

      sorry

    moreover have "s2ss (Execve p' f fds # e # s') = s2ss (e # s)"

      sorry

    moreover have "enriched (e # s) objs (Execve p' f fds # e # s')"

      sorry

    moreover have "reserved (e # s) objs (Execve p' f fds # e # s')"

      sorry

    ultimately show ?thesis

      apply (simp add:enrichable_def)

      apply (rule_tac x = "Execve p' f fds # e # s'" in exI)

      by auto

  next

    case False

    from a4 os p4 have "p \<in> current_procs s'"

      apply (simp add:reserved_def)

      by (erule_tac x = "O_proc p" in allE, auto)      

    moreover from a4 os p4 have "is_file s' f"

      apply (simp add:reserved_def)

      by (erule_tac x = "O_file f" in allE, auto)

    moreover from a4 os p4 have "fds \<subseteq> current_proc_fds s' p"

      apply (rule_tac subsetI, clarsimp simp:reserved_def current_proc_fds.simps)

      apply (erule_tac x = "O_fd p x" in allE, erule impE)

      apply (simp, erule set_mp, simp+)

      done

    ultimately have "os_grant s' e" 

      by (simp add:p4)

    moreover have "grant s' e"