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theory Static
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imports Static_type OS_type_def Flask_type Flask
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begin
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locale tainting_s = tainting
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begin
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(*
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definition init_sectxt_of_file:: "t_file \<Rightarrow> security_context_t option"
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where
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"init_sectxt_of_file f \<equiv>
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if (is_init_file f) then init_sectxt_of_obj (O_file f)
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else if (is_init_dir f) then init_sectxt_of_obj (O_dir f)
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else None"
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*)
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definition sroot :: "t_sfile"
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where
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"sroot \<equiv> (Init [], sec_of_root, None, {})"
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definition init_cf2sfile :: "t_file \<Rightarrow> t_sfile option"
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where
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"init_cf2sfile f \<equiv>
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case (parent f) of
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None \<Rightarrow> Some sroot
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| Some pf \<Rightarrow> if (is_init_file f) then
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(case (init_sectxt_of_obj (O_file f), init_sectxt_of_obj (O_dir pf), get_parentfs_ctxts [] pf) of
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(Some sec, Some psec, Some aseclist) \<Rightarrow> Some (Init f, sec, Some psec, set aseclist)
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| _ \<Rightarrow> None) else
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(case (init_sectxt_of_obj (O_dir f), init_sectxt_of_obj (O_dir pf), get_parentfs_ctxts [] pf) of
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(Some sec, Some psec, Some aseclist) \<Rightarrow> Some (Init f, sec, Some psec, set aseclist)
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| _ \<Rightarrow> None)"
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definition init_cf2sfiles :: "t_file \<Rightarrow> t_sfile set"
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where
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"init_cf2sfiles f \<equiv> {sf. \<exists>f' \<in> init_same_inode_files f. init_cf2sfile f' = Some sf}"
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fun remove_create_flag :: "t_open_flags \<Rightarrow> t_open_flags"
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where
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"remove_create_flag (mflag, oflags) = (mflag, oflags - {OF_CREAT})"
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definition init_cfd2sfd :: "t_process \<Rightarrow> t_fd \<Rightarrow> (security_context_t \<times> t_open_flags \<times> t_sfile) option"
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where
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"init_cfd2sfd p fd =
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(case (init_file_of_proc_fd p fd, init_oflags_of_proc_fd p fd, init_sectxt_of_obj (O_fd p fd)) of
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(Some f, Some flags, Some sec) \<Rightarrow> (case (init_cf2sfile f) of
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Some sf \<Rightarrow> Some (sec, remove_create_flag flags, sf)
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| _ \<Rightarrow> None)
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| _ \<Rightarrow> None)"
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definition init_cfds2sfds :: "t_process \<Rightarrow> (security_context_t \<times> t_open_flags \<times> t_sfile) set"
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where
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"init_cfds2sfds p \<equiv> { sfd. \<exists> fd \<in> init_proc_file_fds p. init_cfd2sfd p fd = Some sfd}"
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(*
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definition init_ch2sshm :: "t_shm \<Rightarrow> t_sshm option"
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where
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"init_ch2sshm h \<equiv> (case (init_sectxt_of_obj (O_shm h)) of
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Some sec \<Rightarrow> Some (Init h, sec)
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| _ \<Rightarrow> None)"
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definition init_cph2spshs
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:: "t_process \<Rightarrow> (t_sshm \<times> t_shm_attach_flag) set"
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where
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"init_cph2spshs p \<equiv> {(sh, flag)| sh flag h. (p, flag) \<in> init_procs_of_shm h \<and>
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init_ch2sshm h = Some sh}"
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*)
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definition init_cp2sproc :: "t_process \<Rightarrow> t_sproc option"
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where
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"init_cp2sproc p \<equiv> (case (init_sectxt_of_obj (O_proc p)) of
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Some sec \<Rightarrow> Some (Init p, sec, (init_cfds2sfds p))
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| None \<Rightarrow> None)"
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(* acient files of init-file
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definition init_o2s_afs :: "t_file \<Rightarrow> security_context_t set"
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where
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"init_o2s_afs f \<equiv> {sec. \<exists> f'. f' \<preceq> f \<and> init_sectxt_of_obj (O_dir f') = Some sec}" *)
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definition init_cm2smsg :: "t_msgq \<Rightarrow> t_msg \<Rightarrow> t_smsg option"
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where
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"init_cm2smsg q m \<equiv> (case (init_sectxt_of_obj (O_msg q m)) of
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Some sec \<Rightarrow> Some (Init m, sec, (O_msg q m) \<in> seeds)
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| _ \<Rightarrow> None)"
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fun init_cqm2sms :: "t_msgq \<Rightarrow> t_msg list \<Rightarrow> (t_smsg list) option"
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where
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"init_cqm2sms q [] = Some []"
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| "init_cqm2sms q (m#ms) =
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(case (init_cqm2sms q ms, init_cm2smsg q m) of
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(Some sms, Some sm) \<Rightarrow> Some (sm # sms)
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| _ \<Rightarrow> None)"
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definition init_cq2smsgq :: "t_msgq \<Rightarrow> t_smsgq option"
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where
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"init_cq2smsgq q \<equiv> (case (init_sectxt_of_obj (O_msgq q), init_cqm2sms q (init_msgs_of_queue q)) of
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(Some sec, Some sms) \<Rightarrow> Some (Init q, sec, sms)
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| _ \<Rightarrow> None)"
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fun init_obj2sobj :: "t_object \<Rightarrow> t_sobject option"
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where
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"init_obj2sobj (O_proc p) =
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(case (init_cp2sproc p) of
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Some sp \<Rightarrow> Some (S_proc sp (O_proc p \<in> seeds))
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| _ \<Rightarrow> None)"
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| "init_obj2sobj (O_file f) =
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Some (S_file (init_cf2sfiles f)
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(\<exists> f'. f' \<in> (init_same_inode_files f) \<and> O_file f \<in> seeds))"
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| "init_obj2sobj (O_dir f) =
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(case (init_cf2sfile f) of
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Some sf \<Rightarrow> Some (S_dir sf)
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| _ \<Rightarrow> None)"
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| "init_obj2sobj (O_msgq q) =
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(case (init_cq2smsgq q) of
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Some sq \<Rightarrow> Some (S_msgq sq)
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| _ \<Rightarrow> None)"
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(*
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| "init_obj2sobj (O_shm h) =
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(case (init_ch2sshm h) of
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Some sh \<Rightarrow> Some (S_shm sh)
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| _ \<Rightarrow> None)"
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| "init_obj2sobj (O_msg q m) =
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(case (init_cq2smsgq q, init_cm2smsg q m) of
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(Some sq, Some sm) \<Rightarrow> Some (S_msg sq sm)
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| _ \<Rightarrow> None)" *)
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| "init_obj2sobj _ = None"
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definition
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"init_static_state \<equiv> {sobj. \<exists> obj. init_alive obj \<and> init_obj2sobj obj = Some sobj}"
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(**************** translation from dynamic to static *******************)
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definition cf2sfile :: "t_state \<Rightarrow> t_file \<Rightarrow> t_sfile option"
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where
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"cf2sfile s f \<equiv>
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case (parent f) of
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None \<Rightarrow> Some sroot
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| Some pf \<Rightarrow> if (is_file s f)
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then (case (sectxt_of_obj s (O_file f), sectxt_of_obj s (O_dir pf), get_parentfs_ctxts s pf) of
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(Some sec, Some psec, Some asecs) \<Rightarrow>
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Some (if (\<not> died (O_file f) s \<and> is_init_file f) then Init f else Created, sec, Some psec, set asecs)
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| _ \<Rightarrow> None)
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else (case (sectxt_of_obj s (O_dir f), sectxt_of_obj s (O_dir pf), get_parentfs_ctxts s pf) of
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(Some sec, Some psec, Some asecs) \<Rightarrow>
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Some (if (\<not> died (O_dir f) s \<and> is_init_dir f) then Init f else Created, sec, Some psec, set asecs)
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| _ \<Rightarrow> None)"
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definition cf2sfiles :: "t_state \<Rightarrow> t_file \<Rightarrow> t_sfile set"
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where
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"cf2sfiles s f \<equiv> {sf. \<exists> f' \<in> (same_inode_files s f). cf2sfile s f' = Some sf}"
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(* here cf2sfile is passed with True, because, process' fds are only for files not dirs *)
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definition cfd2sfd :: "t_state \<Rightarrow> t_process \<Rightarrow> t_fd \<Rightarrow> t_sfd option"
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where
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"cfd2sfd s p fd \<equiv>
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(case (file_of_proc_fd s p fd, flags_of_proc_fd s p fd, sectxt_of_obj s (O_fd p fd)) of
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(Some f, Some flags, Some sec) \<Rightarrow> (case (cf2sfile s f) of
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Some sf \<Rightarrow> Some (sec, remove_create_flag flags, sf)
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| _ \<Rightarrow> None)
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| _ \<Rightarrow> None)"
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definition cpfd2sfds :: "t_state \<Rightarrow> t_process \<Rightarrow> t_sfd set"
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where
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"cpfd2sfds s p \<equiv> {sfd. \<exists> fd \<in> proc_file_fds s p. cfd2sfd s p fd = Some sfd}"
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(*
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definition ch2sshm :: "t_state \<Rightarrow> t_shm \<Rightarrow> t_sshm option"
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where
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"ch2sshm s h \<equiv> (case (sectxt_of_obj s (O_shm h)) of
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Some sec \<Rightarrow>
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Some (if (\<not> died (O_shm h) s \<and> h \<in> init_shms) then Init h else Created, sec)
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| _ \<Rightarrow> None)"
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definition cph2spshs :: "t_state \<Rightarrow> t_process \<Rightarrow> t_sproc_sshm set"
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where
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"cph2spshs s p \<equiv> {(sh, flag)| sh flag h. (p, flag) \<in> procs_of_shm s h \<and> ch2sshm s h = Some sh}"
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*)
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definition cp2sproc :: "t_state \<Rightarrow> t_process \<Rightarrow> t_sproc option"
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where
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"cp2sproc s p \<equiv> (case (sectxt_of_obj s (O_proc p)) of
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Some sec \<Rightarrow>
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Some (if (\<not> died (O_proc p) s \<and> p \<in> init_procs) then Init p else Created, sec,
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cpfd2sfds s p)
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| _ \<Rightarrow> None)"
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definition cm2smsg :: "t_state \<Rightarrow> t_msgq \<Rightarrow> t_msg \<Rightarrow> t_smsg option"
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where
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"cm2smsg s q m \<equiv> (case (sectxt_of_obj s (O_msg q m)) of
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Some sec \<Rightarrow>
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Some (if (\<not> died (O_msg q m) s \<and> m \<in> set (init_msgs_of_queue q)) then Init m else Created,
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sec, O_msg q m \<in> tainted s)
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| _ \<Rightarrow> None)"
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fun cqm2sms:: "t_state \<Rightarrow> t_msgq \<Rightarrow> t_msg list \<Rightarrow> (t_smsg list) option"
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where
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"cqm2sms s q [] = Some []"
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| "cqm2sms s q (m#ms) =
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(case (cqm2sms s q ms, cm2smsg s q m) of
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(Some sms, Some sm) \<Rightarrow> Some (sm#sms)
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| _ \<Rightarrow> None)"
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definition cq2smsgq :: "t_state \<Rightarrow> t_msgq \<Rightarrow> t_smsgq option"
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where
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"cq2smsgq s q \<equiv> (case (sectxt_of_obj s (O_msgq q), cqm2sms s q (msgs_of_queue s q)) of
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(Some sec, Some sms) \<Rightarrow>
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Some (if (\<not> died (O_msgq q) s \<and> q \<in> init_msgqs) then Init q else Created,
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sec, sms)
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| _ \<Rightarrow> None)"
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fun co2sobj :: "t_state \<Rightarrow> t_object \<Rightarrow> t_sobject option"
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where
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"co2sobj s (O_proc p) =
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(case (cp2sproc s p) of
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Some sp \<Rightarrow> Some (S_proc sp (O_proc p \<in> tainted s))
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| _ \<Rightarrow> None)"
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| "co2sobj s (O_file f) =
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Some (S_file (cf2sfiles s f) (O_file f \<in> tainted s))"
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| "co2sobj s (O_dir f) =
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(case (cf2sfile s f) of
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Some sf \<Rightarrow> Some (S_dir sf)
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| _ \<Rightarrow> None)"
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| "co2sobj s (O_msgq q) =
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(case (cq2smsgq s q) of
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Some sq \<Rightarrow> Some (S_msgq sq)
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| _ \<Rightarrow> None)"
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(*
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| "co2sobj s (O_shm h) =
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(case (ch2sshm s h) of
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Some sh \<Rightarrow> Some (S_shm sh)
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| _ \<Rightarrow> None)"
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| "co2sobj s (O_msg q m) =
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(case (cq2smsgq s q, cm2smsg s q m) of
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(Some sq, Some sm) \<Rightarrow> Some (S_msg sq sm)
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| _ \<Rightarrow> None)"
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*)
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| "co2sobj s _ = None"
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definition s2ss :: "t_state \<Rightarrow> t_static_state"
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where
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"s2ss s \<equiv> {sobj. \<exists> obj. alive s obj \<and> co2sobj s obj = Some sobj}"
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(* ******************************************************** *)
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fun is_init_sfile :: "t_sfile \<Rightarrow> bool"
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where
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"is_init_sfile (Init _, sec, psec,asec) = True"
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| "is_init_sfile _ = False"
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fun is_many_sfile :: "t_sfile \<Rightarrow> bool"
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where
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"is_many_sfile (Created, sec, psec, asec) = True"
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| "is_many_sfile _ = False"
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fun is_init_sproc :: "t_sproc \<Rightarrow> bool"
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where
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"is_init_sproc (Init p, sec, fds) = True"
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| "is_init_sproc _ = False"
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fun is_many_sproc :: "t_sproc \<Rightarrow> bool"
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where
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"is_many_sproc (Created, sec,fds) = True"
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| "is_many_sproc _ = False"
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fun is_many_smsg :: "t_smsg \<Rightarrow> bool"
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where
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"is_many_smsg (Created,sec,tag) = True"
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| "is_many_smsg _ = False"
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(* wrong def
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fun is_many_smsgq :: "t_smsgq \<Rightarrow> bool"
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where
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"is_many_smsgq (Created,sec,sms) = (True \<and> (\<forall> sm \<in> (set sms). is_many_smsg sm))"
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| "is_many_smsgq _ = False"
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*)
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fun is_many_smsgq :: "t_smsgq \<Rightarrow> bool"
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where
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"is_many_smsgq (Created,sec,sms) = True"
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| "is_many_smsgq _ = False"
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(*
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fun is_many_sshm :: "t_sshm \<Rightarrow> bool"
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where
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"is_many_sshm (Created, sec) = True"
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| "is_many_sshm _ = False"
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*)
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fun is_many :: "t_sobject \<Rightarrow> bool"
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|
292 |
where
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|
293 |
"is_many (S_proc sp tag) = is_many_sproc sp"
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294 |
| "is_many (S_file sfs tag) = (\<forall> sf \<in> sfs. is_many_sfile sf)"
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295 |
| "is_many (S_dir sf ) = is_many_sfile sf"
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296 |
| "is_many (S_msgq sq ) = is_many_smsgq sq"
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|
297 |
(*
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298 |
| "is_many (S_shm sh ) = is_many_sshm sh"
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|
299 |
*)
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300 |
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|
301 |
fun is_init_smsgq :: "t_smsgq \<Rightarrow> bool"
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|
302 |
where
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|
303 |
"is_init_smsgq (Init q,sec,sms) = True"
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304 |
| "is_init_smsgq _ = False"
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305 |
|
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|
306 |
(*
|
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|
307 |
fun is_init_sshm :: "t_sshm \<Rightarrow> bool"
|
|
|
308 |
where
|
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|
309 |
"is_init_sshm (Init h,sec) = True"
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|
310 |
| "is_init_sshm _ = False"
|
|
|
311 |
*)
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|
312 |
fun is_init_sobj :: "t_sobject \<Rightarrow> bool"
|
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|
313 |
where
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|
314 |
"is_init_sobj (S_proc sp tag ) = is_init_sproc sp"
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|
315 |
| "is_init_sobj (S_file sfs tag) = (\<exists> sf \<in> sfs. is_init_sfile sf)"
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|
316 |
| "is_init_sobj (S_dir sf ) = is_init_sfile sf"
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|
317 |
| "is_init_sobj (S_msgq sq ) = is_init_smsgq sq"
|
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|
318 |
(*
|
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|
319 |
| "is_init_sobj (S_shm sh ) = is_init_sshm sh"
|
|
|
320 |
*)
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|
321 |
(*
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|
322 |
fun update_ss_sp:: "t_static_state \<Rightarrow> t_sobject \<Rightarrow> t_sobject \<Rightarrow> t_static_state"
|
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|
323 |
where
|
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|
324 |
"update_ss_sp ss (S_proc sp tag) (S_proc sp' tag') =
|
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|
325 |
(if (is_many_sproc sp) then ss \<union> {S_proc sp' tag'}
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|
326 |
else (ss - {S_proc sp tag}) \<union> {S_proc sp' tag'})"
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|
327 |
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|
328 |
fun update_ss_sd:: "t_static_state \<Rightarrow> t_sobject \<Rightarrow> t_sobject \<Rightarrow> t_static_state"
|
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|
329 |
where
|
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|
330 |
"update_ss_sd ss (S_dir sf tag) (S_dir sf' tag') =
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|
331 |
(if (is_many_sfile sf) then ss \<union> {S_dir sf' tag'}
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|
332 |
else (ss - {S_dir sf tag}) \<union> {S_dir sf' tag'})"
|
|
|
333 |
*)
|
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|
334 |
|
|
|
335 |
(*
|
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|
336 |
fun sparent :: "t_sfile \<Rightarrow> t_sfile option"
|
|
|
337 |
where
|
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|
338 |
"sparent (Sroot si sec) = None"
|
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|
339 |
| "sparent (Sfile si sec spf) = Some spf"
|
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|
340 |
|
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|
341 |
inductive is_ancesf :: "t_sfile \<Rightarrow> t_sfile \<Rightarrow> bool"
|
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|
342 |
where
|
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|
343 |
"is_ancesf sf sf"
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|
344 |
| "sparent sf = Some spf \<Longrightarrow> is_ancesf spf sf"
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|
345 |
| "\<lbrakk>sparent sf = Some spf; is_ancesf saf spf\<rbrakk> \<Longrightarrow> is_ancesf saf sf"
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|
346 |
|
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|
347 |
definition sfile_reparent :: "t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_sfile"
|
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|
348 |
where
|
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|
349 |
"sfile_reparent (Sroot)"
|
|
|
350 |
*)
|
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|
351 |
|
|
|
352 |
(*
|
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|
353 |
(* sfds rename aux definitions *)
|
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|
354 |
definition sfds_rename_notrelated
|
|
|
355 |
:: "t_sfd set \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_sfd set \<Rightarrow> bool"
|
|
|
356 |
where
|
|
|
357 |
"sfds_rename_notrelated sfds from to sfds' \<equiv>
|
|
|
358 |
(\<forall> sec flag sf. (sec,flag,sf) \<in> sfds \<and> (\<not> from \<preceq> sf) \<longrightarrow> (sec,flag,sf) \<in> sfds')"
|
|
|
359 |
|
|
|
360 |
definition sfds_rename_renamed
|
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|
361 |
:: "t_sfd set \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_sfd set \<Rightarrow> bool"
|
|
|
362 |
where
|
|
|
363 |
"sfds_rename_renamed sfds sf from to sfds' \<equiv>
|
|
|
364 |
(\<forall> sec flag sf'. (sec,flag,sf) \<in> sfds \<and> (sf \<preceq> sf') \<longrightarrow>
|
|
|
365 |
(sec, flag, file_after_rename sf' from to) \<in> sfds' \<and> (sec,flag,sf) \<notin> sfds')"
|
|
|
366 |
|
|
|
367 |
definition sfds_rename_remain
|
|
|
368 |
:: "t_sfd set \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_sfd set \<Rightarrow> bool"
|
|
|
369 |
where
|
|
|
370 |
"sfds_rename_remain sfds sf from to sfds' \<equiv>
|
|
|
371 |
(\<forall> sec flag sf'. (sec,flag,sf') \<in> sfds \<and> (sf \<preceq> sf') \<longrightarrow>
|
|
|
372 |
(sec, flag, sf') \<in> sfds' \<and> (sec,flag, file_after_rename sf' from to) \<notin> sfds')"
|
|
|
373 |
|
|
|
374 |
(* for not many, choose on renamed or not *)
|
|
|
375 |
definition sfds_rename_choices
|
|
|
376 |
:: "t_sfd set \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_sfd set \<Rightarrow> bool"
|
|
|
377 |
where
|
|
|
378 |
"sfds_rename_choices sfds sf from to sfds' \<equiv>
|
|
|
379 |
sfds_rename_remain sfds sf from to sfds' \<or> sfds_rename_renamed sfds sf from to sfds'"
|
|
|
380 |
|
|
|
381 |
(* for many, merge renamed with not renamed *)
|
|
|
382 |
definition sfds_rename_both
|
|
|
383 |
:: "t_sfd set \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_sfd set \<Rightarrow> bool"
|
|
|
384 |
where
|
|
|
385 |
"sfds_rename_both sfds sf from to sfds' \<equiv>
|
|
|
386 |
(\<forall> sec flag sf'. (sec,flag,sf') \<in> sfds \<and> (sf \<preceq> sf') \<longrightarrow>
|
|
|
387 |
(sec, flag, sf') \<in> sfds' \<or> (sec,flag, file_after_rename sf' from to) \<in> sfds')"
|
|
|
388 |
|
|
|
389 |
(* added to the new sfds, are those only under the new sfile *)
|
|
|
390 |
definition sfds_rename_added
|
|
|
391 |
:: "t_sfd set \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_sfd set \<Rightarrow> bool"
|
|
|
392 |
where
|
|
|
393 |
"sfds_rename_added sfds from to sfds' \<equiv>
|
|
|
394 |
(\<forall> sec' flag' sf' sec flag. (sec',flag',sf') \<in> sfds' \<and> (sec,flag,sf') \<notin> sfds \<longrightarrow>
|
|
|
395 |
(\<exists> sf. (sec,flag,sf) \<in> sfds \<and> sf' = file_after_rename from to sf))"
|
|
|
396 |
|
|
|
397 |
definition sproc_sfds_renamed
|
|
|
398 |
:: "t_static_state \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_static_state \<Rightarrow> bool"
|
|
|
399 |
where
|
|
|
400 |
"sproc_sfds_renamed ss sf from to ss' \<equiv>
|
|
|
401 |
(\<forall> pi sec sfds sshms tagp. S_proc (pi,sec,sfds,sshms) tagp \<in> ss \<longrightarrow>
|
|
|
402 |
(\<exists> sfds'. S_proc (pi,sec,sfds',sshms) tagp \<in> ss \<and> sfds_rename_renamed sfds sf from to sfds'))"
|
|
|
403 |
|
|
|
404 |
definition sproc_sfds_remain
|
|
|
405 |
:: "t_static_state \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_static_state \<Rightarrow> bool"
|
|
|
406 |
where
|
|
|
407 |
"sproc_sfds_remain ss sf from to ss' \<equiv>
|
|
|
408 |
(\<forall> pi sec sfds sshms tagp. S_proc (pi,sec,sfds,sshms) tagp \<in> ss \<longrightarrow>
|
|
|
409 |
(\<exists> sfds'. S_proc (pi,sec,sfds',sshms) tagp \<in> ss \<and> sfds_rename_remain sfds sf from to sfds'))"
|
|
|
410 |
|
|
|
411 |
(* for not many, choose on renamed or not *)
|
|
|
412 |
definition sproc_sfds_choices
|
|
|
413 |
:: "t_static_state \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_static_state \<Rightarrow> bool"
|
|
|
414 |
where
|
|
|
415 |
"sproc_sfds_choices ss sf from to ss' \<equiv>
|
|
|
416 |
(\<forall> pi sec sfds sshms tagp. S_proc (pi,sec,sfds,sshms) tagp \<in> ss \<longrightarrow>
|
|
|
417 |
(\<exists> sfds'. S_proc (pi,sec,sfds',sshms) tagp \<in> ss \<and> sfds_rename_choices sfds sf from to sfds'))"
|
|
|
418 |
|
|
|
419 |
(* for many, merge renamed with not renamed *)
|
|
|
420 |
definition sproc_sfds_both
|
|
|
421 |
:: "t_static_state \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_static_state \<Rightarrow> bool"
|
|
|
422 |
where
|
|
|
423 |
"sproc_sfds_both ss sf from to ss' \<equiv>
|
|
|
424 |
(\<forall> pi sec sfds sshms tagp. S_proc (pi,sec,sfds,sshms) tagp \<in> ss \<longrightarrow>
|
|
|
425 |
(\<exists> sfds'. S_proc (pi,sec,sfds',sshms) tagp \<in> ss \<and> sfds_rename_both sfds sf from to sfds'))"
|
|
|
426 |
|
|
|
427 |
(* remove (\<forall> sp tag. S_proc sp tag \<in> ss \<longrightarrow> S_proc sp tag \<in> ss'),
|
|
|
428 |
* cause sfds contains sfs informations *)
|
|
|
429 |
definition ss_rename_notrelated
|
|
|
430 |
:: "t_static_state \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_static_state \<Rightarrow> bool"
|
|
|
431 |
where
|
|
|
432 |
"ss_rename_notrelated ss sf sf' ss' \<equiv>
|
|
|
433 |
(\<forall> sq. S_msgq sq \<in> ss \<longrightarrow> S_msgq sq \<in> ss') \<and>
|
|
|
434 |
(\<forall> pi sec sfds sshms tagp. S_proc (pi,sec,sfds,sshms) tagp \<in> ss \<longrightarrow> (\<exists> sfds'.
|
|
|
435 |
S_proc (pi,sec,sfds',sshms) tagp \<in> ss'\<and> sfds_rename_notrelated sfds sf sf' sfds')) \<and>
|
|
|
436 |
(\<forall> sfs sf'' tag. S_file sfs tag \<in> ss \<and> sf'' \<in> sfs \<and> (\<not> sf \<preceq> sf'') \<longrightarrow> (\<exists> sfs'.
|
|
|
437 |
S_file sfs tag \<in> ss' \<and> sf'' \<in> sfs')) \<and>
|
|
|
438 |
(\<forall> sf'' tag. S_dir sf'' tag \<in> ss \<and> (\<not> sf \<preceq> sf'') \<longrightarrow> S_dir sf'' tag \<in> ss')"
|
|
|
439 |
|
|
|
440 |
(* rename from to, from should definited renamed if not many *)
|
|
|
441 |
definition all_descendant_sf_renamed
|
|
|
442 |
:: "t_static_state \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_static_state \<Rightarrow> bool"
|
|
|
443 |
where
|
|
|
444 |
"all_descendant_sf_renamed ss sf from to ss' \<equiv>
|
|
|
445 |
(\<forall> sfs sf' tagf. sf \<preceq> sf' \<and> S_file sfs tagf \<in> ss \<and> sf' \<in> sfs \<longrightarrow> (\<exists> sfs'.
|
|
|
446 |
S_file sfs' tagf \<in> ss' \<and> file_after_rename from to sf' \<in> sfs' \<and> sf' \<notin> sfs')) \<and>
|
|
|
447 |
(\<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>
|
|
|
448 |
sproc_sfds_renamed ss sf from to ss'"
|
|
|
449 |
|
|
|
450 |
(* not renamed *)
|
|
|
451 |
definition all_descendant_sf_remain
|
|
|
452 |
:: "t_static_state \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_static_state \<Rightarrow> bool"
|
|
|
453 |
where
|
|
|
454 |
"all_descendant_sf_remain ss sf from to ss' \<equiv>
|
|
|
455 |
(\<forall> sfs sf' tag'. sf \<preceq> sf' \<and> S_file sfs tag' \<in> ss \<and> sf' \<in> sfs \<longrightarrow> (\<exists> sfs'.
|
|
|
456 |
S_file sfs' tag' \<in> ss \<and> file_after_rename from to sf' \<notin> sfs' \<and> sf' \<in> sfs')) \<and>
|
|
|
457 |
(\<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>
|
|
|
458 |
sproc_sfds_remain ss sf from to ss'"
|
|
|
459 |
|
|
|
460 |
definition all_descendant_sf_choices
|
|
|
461 |
:: "t_static_state \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_static_state \<Rightarrow> bool"
|
|
|
462 |
where
|
|
|
463 |
"all_descendant_sf_choices ss sf from to ss' \<equiv>
|
|
|
464 |
all_descendant_sf_renamed ss sf from to ss' \<or> all_descendant_sf_remain ss sf from to ss'"
|
|
|
465 |
|
|
|
466 |
definition all_descendant_sf_both
|
|
|
467 |
:: "t_static_state \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_static_state \<Rightarrow> bool"
|
|
|
468 |
where
|
|
|
469 |
"all_descendant_sf_both ss sf from to ss' \<equiv>
|
|
|
470 |
(\<forall> sfs sf' tag'. sf \<preceq> sf' \<and> S_file sfs tag' \<in> ss \<and> sf' \<in> sfs \<longrightarrow> (\<exists> sfs'.
|
|
|
471 |
S_file sfs' tag' \<in> ss \<and> file_after_rename from to sf' \<in> sfs' \<or> sf' \<in> sfs')) \<and>
|
|
|
472 |
(\<forall> sf' tag'. sf \<preceq> sf' \<and> S_dir sf' tag' \<in> ss \<longrightarrow>
|
|
|
473 |
S_dir (file_after_rename from to sf') tag' \<in> ss' \<or> S_dir sf' tag' \<in> ss') \<and>
|
|
|
474 |
sproc_sfds_both ss sf from to ss'"
|
|
|
475 |
|
|
|
476 |
definition ss_renamed_file
|
|
|
477 |
:: "t_static_state \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_static_state \<Rightarrow> bool"
|
|
|
478 |
where
|
|
|
479 |
"ss_renamed_file ss sf sf' ss' \<equiv>
|
|
|
480 |
(if (is_many_sfile sf)
|
|
|
481 |
then all_descendant_sf_choices ss sf sf sf' ss'
|
|
|
482 |
else all_descendant_sf_renamed ss sf sf sf' ss')"
|
|
|
483 |
|
|
|
484 |
(* added to the new sfs, are those only under the new sfile *)
|
|
|
485 |
definition sfs_rename_added
|
|
|
486 |
:: "t_sfile set \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_sfile set \<Rightarrow> bool"
|
|
|
487 |
where
|
|
|
488 |
"sfs_rename_added sfs from to sfs' \<equiv>
|
|
|
489 |
(\<forall> sf'. sf' \<in> sfs' \<and> sf' \<notin> sfs \<longrightarrow> (\<exists> sf. sf \<in> sfs \<and> sf' = file_after_rename from to sf))"
|
|
|
490 |
|
|
|
491 |
(* added to the new sfile, are those only under the new sfile *)
|
|
|
492 |
definition ss_rename_added
|
|
|
493 |
:: "t_static_state \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_static_state \<Rightarrow> bool"
|
|
|
494 |
where
|
|
|
495 |
"ss_rename_added ss from to ss' \<equiv>
|
|
|
496 |
(\<forall> pi sec fds fds' shms tagp. S_proc (pi, sec, fds,shms) tagp \<in> ss \<and>
|
|
|
497 |
S_proc (pi,sec,fds',shms) tagp \<in> ss' \<longrightarrow> sfds_rename_added fds from to fds') \<and>
|
|
|
498 |
(\<forall> sq. S_msgq sq \<in> ss' \<longrightarrow> S_msgq sq \<in> ss) \<and>
|
|
|
499 |
(\<forall> sfs sfs' tagf. S_file sfs' tagf \<in> ss' \<and> S_file sfs tagf \<in> ss \<longrightarrow>
|
|
|
500 |
sfs_rename_added sfs from to sfs') \<and>
|
|
|
501 |
(\<forall> sf' tagf. S_dir sf' tagf \<in> ss' \<and> S_dir sf' tagf \<notin> ss \<longrightarrow>
|
|
|
502 |
(\<exists> sf. S_dir sf tagf \<in> ss \<and> sf' = file_after_rename from to sf))"
|
|
|
503 |
|
|
|
504 |
definition sfile_alive :: "t_static_state \<Rightarrow> t_sfile \<Rightarrow> bool"
|
|
|
505 |
where
|
|
|
506 |
"sfile_alive ss sf \<equiv> (\<exists> sfs tagf. sf \<in> sfs \<and> S_file sfs tagf \<in> ss)"
|
|
|
507 |
|
|
|
508 |
definition sf_alive :: "t_static_state \<Rightarrow> t_sfile \<Rightarrow> bool"
|
|
|
509 |
where
|
|
|
510 |
"sf_alive ss sf \<equiv> (\<exists> tagd. S_dir sf tagd \<in> ss) \<or> sfile_alive ss sf"
|
|
|
511 |
|
|
|
512 |
(* constrains that the new static state should satisfy *)
|
|
|
513 |
definition ss_rename:: "t_static_state \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_static_state \<Rightarrow> bool"
|
|
|
514 |
where
|
|
|
515 |
"ss_rename ss sf sf' ss' \<equiv>
|
|
|
516 |
ss_rename_notrelated ss sf sf' ss' \<and>
|
|
|
517 |
ss_renamed_file ss sf sf' ss' \<and> ss_rename_added ss sf sf' ss' \<and>
|
|
|
518 |
(\<forall> sf''. sf_alive ss sf'' \<and> sf \<prec> sf'' \<longrightarrow>
|
|
|
519 |
(if (is_many_sfile sf'')
|
|
|
520 |
then all_descendant_sf_choices ss sf'' sf sf' ss'
|
|
|
521 |
else all_descendant_sf_both ss sf'' sf sf' ss'))"
|
|
|
522 |
|
|
|
523 |
(* two sfile, the last fname should not be equal *)
|
|
|
524 |
fun sfile_same_fname:: "t_sfile \<Rightarrow> t_sfile \<Rightarrow> bool"
|
|
|
525 |
where
|
|
|
526 |
"sfile_same_fname ((Init n, sec)#spf) ((Init n', sec')#spf') = (n = n')"
|
|
|
527 |
| "sfile_same_fname _ _ = False"
|
|
|
528 |
|
|
|
529 |
(* no same init sfile/only sfile in the target-to spf, should be in static_state addmissble check *)
|
|
|
530 |
definition ss_rename_no_same_fname:: "t_static_state \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> bool"
|
|
|
531 |
where
|
|
|
532 |
"ss_rename_no_same_fname ss from spf \<equiv>
|
|
|
533 |
\<not> (\<exists> to. sfile_same_fname from to \<and> parent to = Some spf \<and> sf_alive ss to)"
|
|
|
534 |
|
|
|
535 |
(* is not a function, is a relation (one 2 many)
|
|
|
536 |
definition update_ss_rename :: "t_static_state \<Rightarrow> t_sfile \<Rightarrow> t_sfile \<Rightarrow> t_static_state"
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|
537 |
where
|
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|
538 |
"update_ss_rename ss sf sf' \<equiv> if (is_many_sfile sf)
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|
539 |
then (ss \<union> {S_file (file_after_rename sf sf' sf'') tag | sf'' tag. sf \<preceq> sf'' \<and> S_file sf'' tag \<in> ss}
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|
540 |
\<union> {S_dir (file_after_rename sf sf' sf'') tag | sf'' tag. sf \<preceq> sf'' \<and> S_dir sf'' tag \<in> ss})
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|
541 |
else (ss - {S_file sf'' tag | sf'' tag. sf \<preceq> sf'' \<and> S_file sf'' tag \<in> ss}
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|
542 |
- {S_dir sf'' tag | sf'' tag. sf \<preceq> sf'' \<and> S_dir sf'' tag \<in> ss})
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|
543 |
\<union> {S_file (file_after_rename sf sf' sf'') tag | sf'' tag. sf \<preceq> sf'' \<and> S_file sf'' tag \<in> ss}
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|
544 |
\<union> {S_dir (file_after_rename sf sf' sf'') tag | sf'' tag. sf \<preceq> sf'' \<and> S_dir sf'' tag \<in> ss}"
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|
545 |
*)
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|
546 |
*)
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|
547 |
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|
548 |
fun sectxt_of_sproc :: "t_sproc \<Rightarrow> security_context_t"
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|
549 |
where
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|
550 |
"sectxt_of_sproc (pi,sec,fds) = sec"
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|
551 |
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|
552 |
fun sectxt_of_sfile :: "t_sfile \<Rightarrow> security_context_t"
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|
553 |
where
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|
554 |
"sectxt_of_sfile (fi,sec,psec,asecs) = sec"
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|
555 |
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|
556 |
fun asecs_of_sfile :: "t_sfile \<Rightarrow> security_context_t set"
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|
557 |
where
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|
558 |
"asecs_of_sfile (fi,sec,psec,asecs) = asecs"
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|
559 |
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|
560 |
definition search_check_s :: "security_context_t \<Rightarrow> t_sfile \<Rightarrow> bool \<Rightarrow> bool"
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|
561 |
where
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|
562 |
"search_check_s pctxt sf if_file =
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|
563 |
(if if_file
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|
564 |
then search_check_ctxt pctxt (sectxt_of_sfile sf) (asecs_of_sfile sf) True
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|
565 |
else search_check_ctxt pctxt (sectxt_of_sfile sf) (asecs_of_sfile sf) False)"
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|
566 |
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|
567 |
definition sectxts_of_sfds :: "t_sfd set \<Rightarrow> security_context_t set"
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|
568 |
where
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|
569 |
"sectxts_of_sfds sfds \<equiv> {ctxt. \<exists> flag sf. (ctxt, flag, sf) \<in> sfds}"
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|
570 |
|
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|
571 |
definition inherit_fds_check_s :: "security_context_t \<Rightarrow> t_sfd set \<Rightarrow> bool"
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|
572 |
where
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|
573 |
"inherit_fds_check_s pctxt sfds \<equiv> inherit_fds_check_ctxt pctxt (sectxts_of_sfds sfds)"
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|
574 |
|
|
|
575 |
(*
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|
576 |
definition sectxts_of_sproc_sshms :: "t_sproc_sshm set \<Rightarrow> security_context_t set"
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|
577 |
where
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|
578 |
"sectxts_of_sproc_sshms sshms \<equiv> {ctxt. \<exists> hi flag. ((hi, ctxt),flag) \<in> sshms}"
|
|
|
579 |
|
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|
580 |
definition inherit_shms_check_s :: "security_context_t \<Rightarrow> t_sproc_sshm set \<Rightarrow> bool"
|
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|
581 |
where
|
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|
582 |
"inherit_shms_check_s pctxt sshms \<equiv> inherit_shms_check_ctxt pctxt (sectxts_of_sproc_sshms sshms)"
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|
583 |
|
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|
584 |
fun info_flow_sshm :: "t_sproc \<Rightarrow> t_sproc \<Rightarrow> bool"
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|
585 |
where
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|
586 |
"info_flow_sshm (pi,sec,fds,shms) (pi',sec',fds',shms') =
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|
587 |
(\<exists> sh flag'. (sh, SHM_RDWR) \<in> shms \<and> (sh, flag') \<in> shms')"
|
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|
588 |
|
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|
589 |
definition info_flow_sproc_sshms :: "t_sproc_sshm set \<Rightarrow> t_sproc_sshm set \<Rightarrow> bool"
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|
590 |
where
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|
591 |
"info_flow_sproc_sshms shms shms' \<equiv> (\<exists> sh flag'. (sh, SHM_RDWR) \<in> shms \<and> (sh, flag') \<in> shms')"
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|
592 |
|
|
|
593 |
|
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|
594 |
fun info_flow_sshm :: "t_sproc \<Rightarrow> t_sproc \<Rightarrow> bool"
|
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|
595 |
where
|
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|
596 |
"info_flow_sshm (pi,sec,fds,shms) (pi',sec',fds',shms') = info_flow_sproc_sshms shms shms'"
|
|
|
597 |
|
|
|
598 |
inductive info_flow_sshm :: "t_sproc \<Rightarrow> t_sproc \<Rightarrow> bool"
|
|
|
599 |
where
|
|
|
600 |
"info_flow_sshm sp sp"
|
|
|
601 |
| "\<lbrakk>info_flow_sshm sp (pi,sec,fds,shms); info_flow_sproc_sshms shms shms'\<rbrakk>
|
|
|
602 |
\<Longrightarrow> info_flow_sshm sp (pi',sec',fds',shms')"
|
|
|
603 |
*)
|
|
|
604 |
|
|
|
605 |
(*
|
|
|
606 |
fun smsg_related :: "t_msg \<Rightarrow> t_smsg list \<Rightarrow> bool"
|
|
|
607 |
where
|
|
|
608 |
"smsg_related m [] = False"
|
|
|
609 |
| "smsg_related m ((mi, sec, tag)#sms) =
|
|
|
610 |
(if (mi = Init m) then True else smsg_related m sms)"
|
|
|
611 |
|
|
|
612 |
fun smsgq_smsg_related :: "t_msgq \<Rightarrow> t_msg \<Rightarrow> t_smsgq \<Rightarrow> bool"
|
|
|
613 |
where
|
|
|
614 |
"smsgq_smsg_related q m (qi, sec, smsgslist) = ((qi = Init q) \<and> (smsg_related m smsgslist))"
|
|
|
615 |
|
|
|
616 |
fun smsg_relatainted :: "t_msg \<Rightarrow> t_smsg list \<Rightarrow> bool"
|
|
|
617 |
where
|
|
|
618 |
"smsg_relatainted m [] = False"
|
|
|
619 |
| "smsg_relatainted m ((mi, sec, tag)#sms) =
|
|
|
620 |
(if (mi = Init m \<and> tag = True) then True else smsg_relatainted m sms)"
|
|
|
621 |
|
|
|
622 |
fun smsgq_smsg_relatainted :: "t_msgq \<Rightarrow> t_msg \<Rightarrow> t_smsgq \<Rightarrow> bool"
|
|
|
623 |
where
|
|
|
624 |
"smsgq_smsg_relatainted q m (qi, sec, smsgslist) =
|
|
|
625 |
((qi = Init q) \<and> (smsg_relatainted m smsgslist))"
|
|
|
626 |
*)
|
|
|
627 |
|
|
|
628 |
fun sfile_related :: "t_sfile \<Rightarrow> t_file \<Rightarrow> bool"
|
|
|
629 |
where
|
|
|
630 |
"sfile_related (fi,sec,psec,asecs) f = (fi = Init f)"
|
|
|
631 |
(*
|
|
|
632 |
fun sproc_related :: "t_process \<Rightarrow> t_sproc \<Rightarrow> bool"
|
|
|
633 |
where
|
|
|
634 |
"sproc_related p (pi, sec, fds, shms) = (pi = Init p)"
|
|
|
635 |
*)
|
|
|
636 |
fun init_obj_related :: "t_sobject \<Rightarrow> t_object \<Rightarrow> bool"
|
|
|
637 |
where
|
|
|
638 |
"init_obj_related (S_proc (pi, sec, fds) tag) (O_proc p') = (pi = Init p')"
|
|
|
639 |
| "init_obj_related (S_file sfs tag) (O_file f) = (\<exists> sf \<in> sfs. sfile_related sf f)"
|
|
|
640 |
| "init_obj_related (S_dir sf) (O_dir f) = (sfile_related sf f)"
|
|
|
641 |
| "init_obj_related (S_msgq (qi, sec, sms)) (O_msgq q') = (qi = Init q')"
|
|
|
642 |
(*
|
|
|
643 |
| "init_obj_related (S_shm (hi, sec)) (O_shm h') = (hi = Init h')"
|
|
|
644 |
|
|
|
645 |
| "init_obj_related (S_msg (qi, sec, sms) (mi, secm, tagm)) (O_msg q' m') = (qi = Init q' \<and> mi = Init m')"
|
|
|
646 |
*)
|
|
|
647 |
| "init_obj_related _ _ = False"
|
|
|
648 |
|
|
|
649 |
|
|
|
650 |
|
|
|
651 |
(***************** for backward proof when Instancing static objects ******************)
|
|
|
652 |
|
|
|
653 |
fun all_procs :: "t_state \<Rightarrow> t_process set"
|
|
|
654 |
where
|
|
|
655 |
"all_procs [] = init_procs"
|
|
|
656 |
| "all_procs (Clone p p' fds # s) = insert p' (all_procs s)"
|
|
|
657 |
| "all_procs (e # s) = all_procs s"
|
|
|
658 |
|
|
|
659 |
definition next_nat :: "nat set \<Rightarrow> nat"
|
|
|
660 |
where
|
|
|
661 |
"next_nat nset = (Max nset) + 1"
|
|
|
662 |
|
|
|
663 |
definition new_proc :: "t_state \<Rightarrow> t_process"
|
|
|
664 |
where
|
|
|
665 |
"new_proc \<tau> = next_nat (current_procs \<tau>)"
|
|
|
666 |
|
|
|
667 |
definition brandnew_proc :: "t_state \<Rightarrow> t_process"
|
|
|
668 |
where
|
|
|
669 |
"brandnew_proc s \<equiv> next_nat (all_procs s)"
|
|
|
670 |
|
|
|
671 |
definition new_inode_num :: "t_state \<Rightarrow> nat"
|
|
|
672 |
where
|
|
|
673 |
"new_inode_num \<tau> = next_nat (current_inode_nums \<tau>)"
|
|
|
674 |
|
|
|
675 |
definition new_msgq :: "t_state \<Rightarrow> t_msgq"
|
|
|
676 |
where
|
|
|
677 |
"new_msgq s = next_nat (current_msgqs s)"
|
|
|
678 |
|
|
|
679 |
definition new_msg :: "t_state \<Rightarrow> t_msgq \<Rightarrow> nat"
|
|
|
680 |
where
|
|
|
681 |
"new_msg s q = next_nat (set (msgs_of_queue s q))"
|
|
|
682 |
|
|
|
683 |
(*
|
|
|
684 |
definition new_shm :: "t_state \<Rightarrow> nat"
|
|
|
685 |
where
|
|
|
686 |
"new_shm \<tau> = next_nat (current_shms \<tau>)"
|
|
|
687 |
*)
|
|
|
688 |
|
|
|
689 |
definition new_proc_fd :: "t_state \<Rightarrow> t_process \<Rightarrow> t_fd"
|
|
|
690 |
where
|
|
|
691 |
"new_proc_fd \<tau> p = next_nat (current_proc_fds \<tau> p)"
|
|
|
692 |
|
|
|
693 |
definition all_fname_under_dir:: "t_file \<Rightarrow> t_state \<Rightarrow> t_fname set"
|
|
|
694 |
where
|
|
|
695 |
"all_fname_under_dir d \<tau> = {fn. \<exists> f. fn # d = f \<and> f \<in> current_files \<tau>}"
|
|
|
696 |
|
|
|
697 |
fun fname_all_a:: "nat \<Rightarrow> t_fname"
|
|
|
698 |
where
|
|
|
699 |
"fname_all_a 0 = []" |
|
|
|
700 |
"fname_all_a (Suc n) = ''a''@(fname_all_a n)"
|
|
|
701 |
|
|
|
702 |
definition fname_length_set :: "t_fname set \<Rightarrow> nat set"
|
|
|
703 |
where
|
|
|
704 |
"fname_length_set fns = length`fns"
|
|
|
705 |
|
|
|
706 |
definition next_fname:: "t_file \<Rightarrow> t_state \<Rightarrow> t_fname"
|
|
|
707 |
where
|
|
|
708 |
"next_fname pf \<tau> = fname_all_a ((Max (fname_length_set (all_fname_under_dir pf \<tau>))) + 1)"
|
|
|
709 |
|
|
|
710 |
definition new_childf:: "t_file \<Rightarrow> t_state \<Rightarrow> t_file"
|
|
|
711 |
where
|
|
|
712 |
"new_childf pf \<tau> = next_fname pf \<tau> # pf"
|
|
|
713 |
|
|
|
714 |
end
|
|
|
715 |
|
|
|
716 |
end |