theory Dynamic_static

imports Main Flask Static Init_prop Valid_prop Tainted_prop Delete_prop Co2sobj_prop S2ss_prop S2ss_prop2

 Temp Enrich

begin

context tainting_s begin

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

apply (induct s, simp)

apply (case_tac a, case_tac[!] obj) sorry (*

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

*)

sorry

(* 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)

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)

sorry (*

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 vd have "fds \<subseteq> proc_file_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)

      sorry

    ultimately have "os_grant s' e" 

      by (simp add:p4)

    moreover have "grant s' e"

      sorry

    ultimately have "valid (e # s')"

      using a1 by (erule_tac valid.intros(2), simp+)

    thus ?thesis

      apply (simp add:enrichable_def)

      apply (rule_tac x = "e # s'" in exI)

      apply (simp)

    sorry

qed

qed

lemma s2d_main_execve:

  "\<lbrakk>grant_execve pctxt fsec pctxt'; ss \<in> static; S_proc (pi, pctxt, fds, shms) tagp \<in> ss; S_file sfs tagf \<in> ss;

     (fi, fsec, pfsec, asecs) \<in> sfs; npctxt_execve pctxt fsec = Some pctxt'; 

    search_check_s pctxt (fi, fsec, pfsec, asecs) True; inherit_fds_check_s pctxt' fds'; fds' \<subseteq> fds; valid s;

    s2ss s = ss\<rbrakk> \<Longrightarrow> \<exists>s. valid s \<and> 

    s2ss s = update_ss ss (S_proc (pi, pctxt, fds, shms) tagp) (S_proc (pi, pctxt', fds', {}) (tagp \<or> tagf))"

apply (simp add:update_ss_def)

thm update_ss_def

sorry

(*

lemma s2d_main_execve:

  "ss \<in> static \<Longrightarrow> \<exists> s. valid s \<and> s2ss s = ss"

apply (erule static.induct)

apply (rule_tac x = "[]" in exI, simp add:s2ss_nil_prop valid.intros)

apply (erule exE|erule conjE)+

apply (rule s2d_main_execve, simp+)

apply (erule exE|erule conjE)+

sorry

*)

(*********************** uppest-level 3 theorems ***********************)

lemma enrichability: 

  "\<lbrakk>valid s; \<forall> obj \<in> objs. alive s obj \<and> is_created s obj \<and> recorded obj\<rbrakk>

   \<Longrightarrow> enrichable s objs" sorry (* 

proof (induct s arbitrary:objs)

  case Nil

  hence "objs = {}" 

    apply (auto)

    apply (erule_tac x = x in ballE)

    apply (case_tac x)

    apply (auto simp:init_alive_prop)

    sorry (*     done *)

  thus ?case using Nil unfolding enrichable_def enriched_def reserved_def

    by (rule_tac x = "[]" in exI, auto)

next

  case (Cons e s)

  hence 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 os: "os_grant s e" and se: "grant s e" and vd: "valid s"

    by (auto dest:vt_grant_os vd_cons vt_grant simp:enrichable_def)

  show ?case 

  proof (cases e)

    case (Execve p f fds) 

    hence p4: "e = Execve p f fds" by simp

    from p3 have p5: "is_inited s (O_proc p) \<Longrightarrow> (O_proc p) \<notin> objs" 

      by (auto simp:is_created_def is_inited_def p4 elim!:ballE[where x = "O_proc p"])

    show "enrichable (e # s) objs"

    proof (cases "is_inited s (O_proc p)")

      case True

      with p5 have a1: "(O_proc p) \<notin> objs" by simp

      with p3 p4 p2 have a2: "\<forall> obj \<in> objs. alive s obj \<and> is_created s obj" and a2': "\<forall> obj \<in> objs. recorded obj"

        apply (auto simp:is_created_def alive_simps is_inited_def)

        apply (erule_tac x = obj in ballE, auto simp:alive_simps split:t_object.splits)

        done

      then obtain s' where a3: "valid s'" and a4: "s2ss s' = s2ss s"

        and a5: "enriched s objs s'" and a6: "reserved s objs s'"

        using p1 apply (simp add:enrichable_def) sorry

      from a5 p4 p2 a2' have a7: "enriched s objs (e # s')"

        apply (clarsimp simp add:enriched_def co2sobj_execve)

        apply (erule_tac x = obj in ballE, clarsimp)

        apply (rule_tac x = obj' in exI, auto simp:co2sobj_execve alive_simps)

        thm enriched_def

obtain s' where p6:"enriched s objs s'"

        apply (simp add: alive_simps enrichable_def)       

        apply auto apply (rule ballI, rule_tac x = obj in exI)

        have p6:"enriched (e # s) objs (e # s)"

        apply (simp add:enriched_def alive_simps)       

        apply auto apply (rule ballI, rule_tac x = obj in exI)

      have "enrich (e # s) objs (e # s)"

        apply (simp add:enrich_def p4)

        sorry

      moreover have "reserve (e # s) objs (e # s)"

        sorry

      ultimately show ?thesis using p2

        apply (simp add:enrichable_def)

        by (rule_tac x = "e # s" in exI, simp)

    next

thm enrichable_def

      apply (simp add:enrichable_def p4)

    apply auto

    apply (auto simp:enrichable_def)

apply (induct s)

done

qed 

*)

lemma d2s_main:

  "valid s \<Longrightarrow> s2ss s \<propto> static"

apply (induct s, simp add:s2ss_nil_prop init_ss_in_def)

apply (rule_tac x = "init_static_state" in bexI, simp, simp add:s_init)

apply (frule vd_cons, frule vt_grant_os, simp)

apply (case_tac a) 

apply (clarsimp simp add:s2ss_execve)

apply (rule conjI, rule impI)

sorry

lemma s2d_main:

  "ss \<in> static \<Longrightarrow> \<exists> s. valid s \<and> s2ss s = ss"

apply (erule static.induct)

apply (rule_tac x = "[]" in exI, simp add:s2ss_nil_prop valid.intros)

apply (erule exE|erule conjE)+

apply (simp add:update_ss_def)

sorry

lemma deleted_imp_deleted_s:

  "\<lbrakk>deleted obj s; valid s\<rbrakk> \<Longrightarrow> \<exists> ss \<in> static. deleted_s ss obj "

sorry

lemma deleted_s_imp_deleted:

  "\<lbrakk>deleted_s ss obj; ss \<in> static\<rbrakk> \<Longrightarrow> \<exists> s. valid s \<and> deleted obj s"

sorry

lemma updeletable_s_sound_comp:

  "undeletable_s obj = undeletable obj"

apply (simp add:undeletable_def undeletable_s_def)

apply (auto dest:deleted_imp_deleted_s deleted_s_imp_deleted)

done

end

end

(* 

lemma s2d_main':

  "ss \<in> static \<Longrightarrow> \<exists> s. valid s \<and> s2ss s \<doteq> ss"

apply (erule static.induct)

apply (rule_tac x = "[]" in exI, simp add:s2ss_nil_prop valid.intros)

apply (erule exE|erule conjE)+

apply (simp add:update_ss_def)

sorry

*)