theory Static

imports Static_type OS_type_def Flask_type Flask

begin

locale tainting_s = tainting 

begin

(*

definition init_sectxt_of_file:: "t_file \<Rightarrow> security_context_t option"

where

  "init_sectxt_of_file f \<equiv> 

     if (is_init_file f) then init_sectxt_of_obj (O_file f)

     else if (is_init_dir f) then init_sectxt_of_obj (O_dir f)

     else None"

*)

definition sroot :: "t_sfile"

where

  "sroot \<equiv> (Init [], sec_of_root, None, {})"

definition init_cf2sfile :: "t_file \<Rightarrow> t_sfile option"

where

  "init_cf2sfile f \<equiv> 

     case (parent f) of 

       None \<Rightarrow> Some sroot

     | Some pf \<Rightarrow> if (is_init_file f) then 

 (case (init_sectxt_of_obj (O_file f), init_sectxt_of_obj (O_dir pf), get_parentfs_ctxts [] pf) of

    (Some sec, Some psec, Some aseclist) \<Rightarrow> Some (Init f, sec, Some psec, set aseclist)

  | _ \<Rightarrow> None)    else 

 (case (init_sectxt_of_obj (O_dir f), init_sectxt_of_obj (O_dir pf), get_parentfs_ctxts [] pf) of

    (Some sec, Some psec, Some aseclist) \<Rightarrow> Some (Init f, sec, Some psec, set aseclist)

  | _ \<Rightarrow> None)"

definition init_cf2sfiles :: "t_file \<Rightarrow> t_sfile set"

where

  "init_cf2sfiles f \<equiv> {sf. \<exists>f' \<in> init_same_inode_files f. init_cf2sfile f' = Some sf}"

definition init_cfd2sfd :: "t_process \<Rightarrow> t_fd \<Rightarrow> (security_context_t \<times> t_open_flags \<times> t_sfile) option"

where

  "init_cfd2sfd p fd = 

     (case (init_file_of_proc_fd p fd, init_oflags_of_proc_fd p fd, init_sectxt_of_obj (O_fd p fd)) of

        (Some f, Some flags, Some sec) \<Rightarrow> (case (init_cf2sfile f) of 

                                             Some sf \<Rightarrow> Some (sec, flags, sf)

                                           | _ \<Rightarrow> None)

      | _ \<Rightarrow> None)"

definition init_cfds2sfds :: "t_process \<Rightarrow> (security_context_t \<times> t_open_flags \<times> t_sfile) set"

where

  "init_cfds2sfds p \<equiv> { sfd. \<exists> fd \<in> init_proc_file_fds p. init_cfd2sfd p fd = Some sfd}"

definition init_ch2sshm :: "t_shm \<Rightarrow> t_sshm option"

where

  "init_ch2sshm h \<equiv> (case (init_sectxt_of_obj (O_shm h)) of

                       Some sec \<Rightarrow> Some (Init h, sec)

                     | _        \<Rightarrow> None)"

definition init_cph2spshs 

  :: "t_process \<Rightarrow> (t_sshm \<times> t_shm_attach_flag) set"

where

  "init_cph2spshs p \<equiv> {(sh, flag)| sh flag h. (p, flag) \<in> init_procs_of_shm h \<and> 

                                             init_ch2sshm h = Some sh}"

definition init_cp2sproc :: "t_process \<Rightarrow> t_sproc option"

where

  "init_cp2sproc p \<equiv> (case (init_sectxt_of_obj (O_proc p)) of 

       Some sec \<Rightarrow> Some (Init p, sec, (init_cfds2sfds p), (init_cph2spshs p))

     | None     \<Rightarrow> None)"

(* acient files of init-file 

definition init_o2s_afs :: "t_file \<Rightarrow> security_context_t set"

where

  "init_o2s_afs f \<equiv> {sec. \<exists> f'. f' \<preceq> f \<and> init_sectxt_of_obj (O_dir f') = Some sec}" *)

definition init_cm2smsg :: "t_msgq \<Rightarrow> t_msg \<Rightarrow> t_smsg option"

where

  "init_cm2smsg q m \<equiv> (case (init_sectxt_of_obj (O_msg q m)) of 

                         Some sec \<Rightarrow> Some (Init m, sec, (O_msg q m) \<in> seeds)

                       | _ \<Rightarrow> None)"

fun init_cqm2sms :: "t_msgq \<Rightarrow> t_msg list \<Rightarrow> (t_smsg list) option"

where

  "init_cqm2sms q []     = Some []"

| "init_cqm2sms q (m#ms) = 

     (case (init_cqm2sms q ms, init_cm2smsg q m) of 

        (Some sms, Some sm) \<Rightarrow> Some (sm # sms)

      | _        \<Rightarrow> None)"

definition init_cq2smsgq :: "t_msgq \<Rightarrow> t_smsgq option"

where

  "init_cq2smsgq q \<equiv> (case (init_sectxt_of_obj (O_msgq q), init_cqm2sms q (init_msgs_of_queue q)) of 

       (Some sec, Some sms) \<Rightarrow> Some (Init q, sec, sms)

     | _ \<Rightarrow> None)"

fun init_obj2sobj :: "t_object \<Rightarrow> t_sobject option"

where

  "init_obj2sobj (O_proc p) = 

     (case (init_cp2sproc p) of 

       Some sp \<Rightarrow> Some (S_proc sp (O_proc p \<in> seeds))

     | _ \<Rightarrow> None)"

| "init_obj2sobj (O_file f) = 

     Some (S_file (init_cf2sfiles f) 

                  (\<exists> f'. f' \<in> (init_same_inode_files f) \<and> O_file f \<in> seeds))"

| "init_obj2sobj (O_dir f) = 

     (case (init_cf2sfile f) of

           Some sf \<Rightarrow> Some (S_dir sf) 

         | _ \<Rightarrow> None)"

| "init_obj2sobj (O_msgq q) = 

     (case (init_cq2smsgq q) of

       Some sq \<Rightarrow> Some (S_msgq sq)

     | _ \<Rightarrow> None)"

| "init_obj2sobj (O_shm h) = 

     (case (init_ch2sshm h) of 

       Some sh \<Rightarrow> Some (S_shm sh)

     | _       \<Rightarrow> None)"  (*

| "init_obj2sobj (O_msg q m) = 

     (case (init_cq2smsgq q, init_cm2smsg q m) of 

        (Some sq, Some sm) \<Rightarrow> Some (S_msg sq sm)

      | _                  \<Rightarrow> None)" *)

| "init_obj2sobj _ = None"

definition  

  "init_static_state \<equiv> {sobj. \<exists> obj. init_alive obj \<and> init_obj2sobj obj = Some sobj}"

(**************** translation from dynamic to static *******************)

definition cf2sfile :: "t_state \<Rightarrow> t_file \<Rightarrow> t_sfile option"

where

  "cf2sfile s f \<equiv>

     case (parent f) of 

       None \<Rightarrow> Some sroot

     | Some pf \<Rightarrow> if (is_file s f) 

     then (case (sectxt_of_obj s (O_file f), sectxt_of_obj s (O_dir pf), get_parentfs_ctxts s pf) of

            (Some sec, Some psec, Some asecs) \<Rightarrow>

 Some (if (\<not> deleted (O_file f) s \<and> is_init_file f) then Init f else Created, sec, Some psec, set asecs)

          | _ \<Rightarrow> None) 

     else (case (sectxt_of_obj s (O_dir f), sectxt_of_obj s (O_dir pf), get_parentfs_ctxts s pf) of

            (Some sec, Some psec, Some asecs) \<Rightarrow>

 Some (if (\<not> deleted (O_dir f) s \<and> is_init_dir f) then Init f else Created, sec, Some psec, set asecs)

          | _ \<Rightarrow> None)"

definition cf2sfiles :: "t_state \<Rightarrow> t_file \<Rightarrow> t_sfile set"

where

  "cf2sfiles s f \<equiv> {sf. \<exists> f' \<in> (same_inode_files s f). cf2sfile s f' = Some sf}"

(* here cf2sfile is passed with True, because, process' fds are only for files not dirs *)

definition cfd2sfd :: "t_state \<Rightarrow> t_process \<Rightarrow> t_fd \<Rightarrow> t_sfd option" 

where

  "cfd2sfd s p fd \<equiv> 

    (case (file_of_proc_fd s p fd, flags_of_proc_fd s p fd, sectxt_of_obj s (O_fd p fd)) of

      (Some f, Some flags, Some sec) \<Rightarrow> (case (cf2sfile s f) of 

                                          Some sf \<Rightarrow> Some (sec, flags, sf)

                                        | _       \<Rightarrow> None)

    | _ \<Rightarrow> None)"

definition cpfd2sfds :: "t_state \<Rightarrow> t_process \<Rightarrow> t_sfd set"

where

  "cpfd2sfds s p \<equiv> {sfd. \<exists> fd \<in> proc_file_fds s p. cfd2sfd s p fd = Some sfd}"

definition ch2sshm :: "t_state \<Rightarrow> t_shm \<Rightarrow> t_sshm option"

where

  "ch2sshm s h \<equiv> (case (sectxt_of_obj s (O_shm h)) of

                    Some sec \<Rightarrow> 

 Some (if (\<not> deleted (O_shm h) s \<and> h \<in> init_shms) then Init h else Created, sec)

                  | _ \<Rightarrow> None)"

definition cph2spshs :: "t_state \<Rightarrow> t_process \<Rightarrow> t_sproc_sshm set"

where

  "cph2spshs s p \<equiv> {(sh, flag)| sh flag h. (p, flag) \<in> procs_of_shm s h \<and> ch2sshm s h = Some sh}"

definition cp2sproc :: "t_state \<Rightarrow> t_process \<Rightarrow> t_sproc option"

where

  "cp2sproc s p \<equiv> (case (sectxt_of_obj s (O_proc p)) of 

                     Some sec \<Rightarrow> 

 Some (if (\<not> deleted (O_proc p) s \<and> p \<in> init_procs) then Init p else Created, sec, 

       cpfd2sfds s p, cph2spshs s p)

                   | _ \<Rightarrow> None)"

definition cm2smsg :: "t_state \<Rightarrow> t_msgq \<Rightarrow> t_msg \<Rightarrow> t_smsg option"

where

  "cm2smsg s q m \<equiv> (case (sectxt_of_obj s (O_msg q m)) of 

                      Some sec \<Rightarrow>

 Some (if (\<not> deleted (O_msg q m) s \<and> m \<in> set (init_msgs_of_queue q)) then Init m else Created,

       sec, O_msg q m \<in> tainted s)

                    | _ \<Rightarrow> None)"

fun cqm2sms:: "t_state \<Rightarrow> t_msgq \<Rightarrow> t_msg list \<Rightarrow> (t_smsg list) option"

where 

  "cqm2sms s q [] = Some []"

| "cqm2sms s q (m#ms) = 

     (case (cqm2sms s q ms, cm2smsg s q m) of 

       (Some sms, Some sm) \<Rightarrow> Some (sm#sms) 

     | _ \<Rightarrow> None)"

definition cq2smsgq :: "t_state \<Rightarrow> t_msgq \<Rightarrow> t_smsgq option"

where

  "cq2smsgq s q \<equiv> (case (sectxt_of_obj s (O_msgq q), cqm2sms s q (msgs_of_queue s q)) of 

                     (Some sec, Some sms) \<Rightarrow> 

 Some (if (\<not> deleted (O_msgq q) s \<and> q \<in> init_msgqs) then Init q else Created,

       sec, sms)

                   | _ \<Rightarrow> None)"

fun co2sobj :: "t_state \<Rightarrow> t_object \<Rightarrow> t_sobject option"

where

  "co2sobj s (O_proc p) = 

     (case (cp2sproc s p) of 

        Some sp \<Rightarrow> Some (S_proc sp (O_proc p \<in> tainted s))

      | _       \<Rightarrow> None)"

| "co2sobj s (O_file f) = 

     Some (S_file (cf2sfiles s f) (O_file f \<in> tainted s))"

| "co2sobj s (O_dir f) = 

     (case (cf2sfile s f) of

        Some sf  \<Rightarrow> Some (S_dir sf)

      | _ \<Rightarrow> None)"

| "co2sobj s (O_msgq q) = 

     (case (cq2smsgq s q) of

        Some sq \<Rightarrow> Some (S_msgq sq)

      | _ \<Rightarrow> None)"

| "co2sobj s (O_shm h) = 

     (case (ch2sshm s h) of 

        Some sh \<Rightarrow> Some (S_shm sh)

      | _ \<Rightarrow> None)"

(*

| "co2sobj s (O_msg q m) = 

     (case (cq2smsgq s q, cm2smsg s q m) of 

       (Some sq, Some sm) \<Rightarrow> Some (S_msg sq sm)

     | _ \<Rightarrow> None)"

*)

| "co2sobj s _ = None"

definition s2ss :: "t_state \<Rightarrow> t_static_state"

where

  "s2ss s \<equiv> {sobj. \<exists> obj. alive s obj \<and> co2sobj s obj = Some sobj}"

(* ******************************************************** *)

fun is_init_sfile :: "t_sfile \<Rightarrow> bool"

where

  "is_init_sfile (Init _, sec, psec,asec) = True"

| "is_init_sfile _ = False"

fun is_many_sfile :: "t_sfile \<Rightarrow> bool"

where

  "is_many_sfile (Created, sec, psec, asec) = True"

| "is_many_sfile _ = False"

fun is_init_sproc :: "t_sproc \<Rightarrow> bool"

where

  "is_init_sproc (Init p, sec, fds, shms) = True"

| "is_init_sproc _                        = False"

fun is_many_sproc :: "t_sproc \<Rightarrow> bool"

where

  "is_many_sproc (Created, sec,fds,shms) = True"

| "is_many_sproc _                       = False"

fun is_many_smsg :: "t_smsg \<Rightarrow> bool"

where

  "is_many_smsg (Created,sec,tag) = True"

| "is_many_smsg _                 = False"

(* wrong def 

fun is_many_smsgq :: "t_smsgq \<Rightarrow> bool"

where

  "is_many_smsgq (Created,sec,sms) = (True \<and> (\<forall> sm \<in> (set sms). is_many_smsg sm))"

| "is_many_smsgq _                 = False"

*)

fun is_many_smsgq :: "t_smsgq \<Rightarrow> bool"

where

  "is_many_smsgq (Created,sec,sms) = True"

| "is_many_smsgq _                 = False"

fun is_many_sshm :: "t_sshm \<Rightarrow> bool"

where

  "is_many_sshm (Created, sec) = True"

| "is_many_sshm _              = False"

fun is_many :: "t_sobject \<Rightarrow> bool"

where

  "is_many (S_proc sp   tag) = is_many_sproc sp"

| "is_many (S_file sfs  tag) = (\<forall> sf \<in> sfs. is_many_sfile sf)"

| "is_many (S_dir  sf      ) = is_many_sfile sf"

| "is_many (S_msgq sq      ) = is_many_smsgq sq"

| "is_many (S_shm  sh      ) = is_many_sshm  sh"

fun is_init_smsgq :: "t_smsgq \<Rightarrow> bool"

where

  "is_init_smsgq (Init q,sec,sms) = True"

| "is_init_smsgq _                = False"

fun is_init_sshm :: "t_sshm \<Rightarrow> bool"

where

  "is_init_sshm (Init h,sec) = True"

| "is_init_sshm _            = False"

fun is_init_sobj :: "t_sobject \<Rightarrow> bool"

where

  "is_init_sobj (S_proc sp tag ) = is_init_sproc sp"

| "is_init_sobj (S_file sfs tag) = (\<exists> sf \<in> sfs. is_init_sfile sf)"

| "is_init_sobj (S_dir  sf     ) = is_init_sfile sf"

| "is_init_sobj (S_msgq sq     ) = is_init_smsgq sq"

| "is_init_sobj (S_shm  sh     ) = is_init_sshm sh"

(*

fun update_ss_sp:: "t_static_state \<Rightarrow> t_sobject \<Rightarrow> t_sobject \<Rightarrow> t_static_state"

where

  "update_ss_sp ss (S_proc sp tag) (S_proc sp' tag') = 

     (if (is_many_sproc sp) then ss \<union> {S_proc sp' tag'} 

      else (ss - {S_proc sp tag}) \<union> {S_proc sp' tag'})"

fun update_ss_sd:: "t_static_state \<Rightarrow> t_sobject \<Rightarrow> t_sobject \<Rightarrow> t_static_state"

where

  "update_ss_sd ss (S_dir sf tag) (S_dir sf' tag') = 

     (if (is_many_sfile sf) then ss \<union> {S_dir sf' tag'} 

      else (ss - {S_dir sf tag}) \<union> {S_dir sf' tag'})"

*)

(*

fun sparent :: "t_sfile \<Rightarrow> t_sfile option"

where

  "sparent (Sroot si sec) = None"

| "sparent (Sfile si sec spf) = Some spf"

inductive is_ancesf :: "t_sfile \<Rightarrow> t_sfile \<Rightarrow> bool" 

where

  "is_ancesf sf sf"

| "sparent sf = Some spf \<Longrightarrow> is_ancesf spf sf"

| "\<lbrakk>sparent sf = Some spf; is_ancesf saf spf\<rbrakk> \<Longrightarrow> is_ancesf saf sf"

definition sfile_reparent :: "t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_sfile"

where

  "sfile_reparent (Sroot)"

*)

(*

(* sfds rename aux definitions *)

definition sfds_rename_notrelated 

  :: "t_sfd set \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_sfd set \<Rightarrow> bool"

where

  "sfds_rename_notrelated sfds from to sfds' \<equiv> 

     (\<forall> sec flag sf. (sec,flag,sf) \<in> sfds \<and> (\<not> from \<preceq> sf) \<longrightarrow> (sec,flag,sf) \<in> sfds')"

definition sfds_rename_renamed

  :: "t_sfd set \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_sfd set \<Rightarrow> bool"

where

  "sfds_rename_renamed sfds sf from to sfds' \<equiv> 

     (\<forall> sec flag sf'. (sec,flag,sf) \<in> sfds \<and> (sf \<preceq> sf') \<longrightarrow>

         (sec, flag, file_after_rename sf' from to) \<in> sfds' \<and> (sec,flag,sf) \<notin> sfds')"

definition sfds_rename_remain

  :: "t_sfd set \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_sfd set \<Rightarrow> bool"

where

  "sfds_rename_remain sfds sf from to sfds' \<equiv> 

     (\<forall> sec flag sf'. (sec,flag,sf') \<in> sfds \<and> (sf \<preceq> sf') \<longrightarrow> 

         (sec, flag, sf') \<in> sfds' \<and> (sec,flag, file_after_rename sf' from to) \<notin> sfds')"

  (* for not many, choose on renamed or not *)

definition sfds_rename_choices

  :: "t_sfd set \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_sfd set \<Rightarrow> bool"

where

  "sfds_rename_choices sfds sf from to sfds' \<equiv> 

     sfds_rename_remain sfds sf from to sfds' \<or> sfds_rename_renamed sfds sf from to sfds'"

(* for many, merge renamed with not renamed *)

definition sfds_rename_both

  :: "t_sfd set \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_sfd set \<Rightarrow> bool"

where

  "sfds_rename_both sfds sf from to sfds' \<equiv> 

     (\<forall> sec flag sf'. (sec,flag,sf') \<in> sfds \<and> (sf \<preceq> sf') \<longrightarrow> 

         (sec, flag, sf') \<in> sfds' \<or> (sec,flag, file_after_rename sf' from to) \<in> sfds')"

(* added to the new sfds, are those only under the new sfile *)

definition sfds_rename_added

  :: "t_sfd set \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_sfd set \<Rightarrow> bool"

where

  "sfds_rename_added sfds from to sfds' \<equiv> 

     (\<forall> sec' flag' sf' sec flag. (sec',flag',sf') \<in> sfds' \<and> (sec,flag,sf') \<notin> sfds \<longrightarrow> 

        (\<exists> sf. (sec,flag,sf) \<in> sfds \<and> sf' = file_after_rename from to sf))"

definition sproc_sfds_renamed

  :: "t_static_state \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_static_state \<Rightarrow> bool"

where

  "sproc_sfds_renamed ss sf from to ss' \<equiv>

     (\<forall> pi sec sfds sshms tagp. S_proc (pi,sec,sfds,sshms) tagp \<in> ss \<longrightarrow> 

       (\<exists> sfds'. S_proc (pi,sec,sfds',sshms) tagp \<in> ss \<and> sfds_rename_renamed sfds sf from to sfds'))"

definition sproc_sfds_remain

  :: "t_static_state \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_static_state \<Rightarrow> bool"

where

  "sproc_sfds_remain ss sf from to ss' \<equiv>

     (\<forall> pi sec sfds sshms tagp. S_proc (pi,sec,sfds,sshms) tagp \<in> ss \<longrightarrow> 

       (\<exists> sfds'. S_proc (pi,sec,sfds',sshms) tagp \<in> ss \<and> sfds_rename_remain sfds sf from to sfds'))"

(* for not many, choose on renamed or not *)

definition sproc_sfds_choices

  :: "t_static_state \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_static_state \<Rightarrow> bool"

where

  "sproc_sfds_choices ss sf from to ss' \<equiv>

     (\<forall> pi sec sfds sshms tagp. S_proc (pi,sec,sfds,sshms) tagp \<in> ss \<longrightarrow> 

       (\<exists> sfds'. S_proc (pi,sec,sfds',sshms) tagp \<in> ss \<and> sfds_rename_choices sfds sf from to sfds'))"

(* for many, merge renamed with not renamed *)

definition sproc_sfds_both

  :: "t_static_state \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_static_state \<Rightarrow> bool"

where

  "sproc_sfds_both ss sf from to ss' \<equiv>

     (\<forall> pi sec sfds sshms tagp. S_proc (pi,sec,sfds,sshms) tagp \<in> ss \<longrightarrow> 

       (\<exists> sfds'. S_proc (pi,sec,sfds',sshms) tagp \<in> ss \<and> sfds_rename_both sfds sf from to sfds'))"

(* remove (\<forall> sp tag. S_proc sp tag \<in> ss \<longrightarrow> S_proc sp tag \<in> ss'),

 * cause sfds contains sfs informations *)

definition ss_rename_notrelated 

  :: "t_static_state \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_static_state \<Rightarrow> bool"

where

  "ss_rename_notrelated ss sf sf' ss' \<equiv> 

     (\<forall> sq. S_msgq sq \<in> ss \<longrightarrow> S_msgq sq \<in> ss') \<and> 

     (\<forall> pi sec sfds sshms tagp. S_proc (pi,sec,sfds,sshms) tagp \<in> ss \<longrightarrow> (\<exists> sfds'.  

         S_proc (pi,sec,sfds',sshms) tagp \<in> ss'\<and> sfds_rename_notrelated sfds sf sf' sfds'))  \<and>

     (\<forall> sfs sf'' tag. S_file sfs tag \<in> ss \<and> sf'' \<in> sfs \<and> (\<not> sf \<preceq> sf'') \<longrightarrow> (\<exists> sfs'. 

         S_file sfs tag \<in> ss' \<and> sf'' \<in> sfs')) \<and>

     (\<forall> sf'' tag. S_dir sf'' tag \<in> ss \<and> (\<not> sf \<preceq> sf'') \<longrightarrow> S_dir sf'' tag \<in> ss')" 

(* rename from to, from should definited renamed if not many *)

definition all_descendant_sf_renamed 

  :: "t_static_state \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_static_state \<Rightarrow> bool"

where

  "all_descendant_sf_renamed ss sf from to ss' \<equiv>

     (\<forall> sfs sf' tagf. sf \<preceq> sf' \<and> S_file sfs tagf \<in> ss \<and> sf' \<in> sfs \<longrightarrow> (\<exists> sfs'. 

       S_file sfs' tagf \<in> ss' \<and> file_after_rename from to sf' \<in> sfs' \<and> sf' \<notin> sfs')) \<and> 

     (\<forall> sf' tagf. sf \<preceq> sf' \<and> S_dir sf' tagf \<in> ss \<longrightarrow> S_dir (file_after_rename from to sf') tagf \<in> ss' \<and> S_dir sf' tagf \<notin> ss') \<and>

     sproc_sfds_renamed ss sf from to ss'"

(* not renamed *)

definition all_descendant_sf_remain

  :: "t_static_state \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_static_state \<Rightarrow> bool"

where

  "all_descendant_sf_remain ss sf from to ss' \<equiv>

     (\<forall> sfs sf' tag'. sf \<preceq> sf' \<and> S_file sfs tag' \<in> ss \<and> sf' \<in> sfs \<longrightarrow> (\<exists> sfs'. 

       S_file sfs' tag' \<in> ss \<and> file_after_rename from to sf' \<notin> sfs' \<and> sf' \<in> sfs')) \<and> 

     (\<forall> sf' tag'. sf \<preceq> sf' \<and> S_dir sf' tag' \<in> ss \<longrightarrow> S_dir (file_after_rename from to sf') tag' \<notin> ss' \<and> S_dir sf' tag' \<in> ss') \<and>

     sproc_sfds_remain ss sf from to ss'"

definition all_descendant_sf_choices

  :: "t_static_state \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_static_state \<Rightarrow> bool"

where

  "all_descendant_sf_choices ss sf from to ss' \<equiv>

     all_descendant_sf_renamed ss sf from to ss' \<or> all_descendant_sf_remain ss sf from to ss'"

definition all_descendant_sf_both

  :: "t_static_state \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_static_state \<Rightarrow> bool"

where

  "all_descendant_sf_both ss sf from to ss' \<equiv>

     (\<forall> sfs sf' tag'. sf \<preceq> sf' \<and> S_file sfs tag' \<in> ss \<and> sf' \<in> sfs \<longrightarrow> (\<exists> sfs'. 

        S_file sfs' tag' \<in> ss \<and> file_after_rename from to sf' \<in> sfs' \<or> sf' \<in> sfs')) \<and> 

     (\<forall> sf' tag'. sf \<preceq> sf' \<and> S_dir sf' tag' \<in> ss \<longrightarrow> 

        S_dir (file_after_rename from to sf') tag' \<in> ss' \<or> S_dir sf' tag' \<in> ss') \<and>

     sproc_sfds_both ss sf from to ss'"

definition ss_renamed_file 

  :: "t_static_state \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_static_state \<Rightarrow> bool"

where

  "ss_renamed_file ss sf sf' ss' \<equiv> 

     (if (is_many_sfile sf) 

        then all_descendant_sf_choices ss sf sf sf' ss'

        else all_descendant_sf_renamed ss sf sf sf' ss')"

(* added to the new sfs, are those only under the new sfile *)

definition sfs_rename_added

  :: "t_sfile set \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_sfile set \<Rightarrow> bool"

where

  "sfs_rename_added sfs from to sfs' \<equiv> 

     (\<forall> sf'. sf' \<in> sfs' \<and> sf' \<notin> sfs \<longrightarrow> (\<exists> sf. sf \<in> sfs \<and> sf' = file_after_rename from to sf))"

(* added to the new sfile, are those only under the new sfile *)

definition ss_rename_added

  :: "t_static_state \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_static_state \<Rightarrow> bool"

where

  "ss_rename_added ss from to ss' \<equiv> 

     (\<forall> pi sec fds fds' shms tagp. S_proc (pi, sec, fds,shms) tagp \<in> ss \<and> 

        S_proc (pi,sec,fds',shms) tagp \<in> ss' \<longrightarrow> sfds_rename_added fds from to fds') \<and>

     (\<forall> sq. S_msgq sq \<in> ss' \<longrightarrow> S_msgq sq \<in> ss) \<and>

     (\<forall> sfs sfs' tagf. S_file sfs' tagf \<in> ss' \<and> S_file sfs tagf \<in> ss \<longrightarrow> 

        sfs_rename_added sfs from to sfs') \<and>

     (\<forall> sf' tagf. S_dir sf' tagf \<in> ss' \<and> S_dir sf' tagf \<notin> ss \<longrightarrow> 

        (\<exists> sf. S_dir sf tagf \<in> ss \<and> sf' = file_after_rename from to sf))" 

definition sfile_alive :: "t_static_state \<Rightarrow> t_sfile \<Rightarrow> bool"

where

  "sfile_alive ss sf \<equiv> (\<exists> sfs tagf. sf \<in> sfs \<and> S_file sfs tagf \<in> ss)"

definition sf_alive :: "t_static_state \<Rightarrow> t_sfile \<Rightarrow> bool"

where

  "sf_alive ss sf \<equiv> (\<exists> tagd. S_dir sf tagd \<in> ss) \<or> sfile_alive ss sf"

(* constrains that the new static state should satisfy *)

definition ss_rename:: "t_static_state \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_static_state \<Rightarrow> bool"

where

  "ss_rename ss sf sf' ss' \<equiv> 

     ss_rename_notrelated ss sf sf' ss' \<and>

     ss_renamed_file ss sf sf' ss' \<and> ss_rename_added ss sf sf' ss' \<and>

     (\<forall> sf''. sf_alive ss sf'' \<and> sf \<prec> sf'' \<longrightarrow> 

       (if (is_many_sfile sf'') 

        then all_descendant_sf_choices ss sf'' sf sf' ss'

        else all_descendant_sf_both ss sf'' sf sf' ss'))"

(* two sfile, the last fname should not be equal *)

fun sfile_same_fname:: "t_sfile \<Rightarrow> t_sfile \<Rightarrow> bool"

where

  "sfile_same_fname ((Init n, sec)#spf) ((Init n', sec')#spf') = (n = n')"

| "sfile_same_fname _                   _                      = False"