theory obj2sobj_prop imports Main rc_theory os_rc deleted_propbegincontext tainting_s_complete begin(** file 2 sfile **)lemma init_son_deleted_D: "\<lbrakk>deleted (File pf) s; f # pf \<in> init_files; valid s\<rbrakk> \<Longrightarrow> deleted (File (f # pf)) s"apply (induct s, simp)by (frule valid_cons, frule valid_os, case_tac a, auto dest:init_notin_curf_deleted)lemma init_parent_undeleted_I: "\<lbrakk>\<not> deleted (File (f # pf)) s; f # pf \<in> init_files; valid s\<rbrakk> \<Longrightarrow> \<not> deleted (File pf) s"by (rule notI, simp add:init_son_deleted_D)lemma source_dir_in_init: "source_dir s f = Some sd \<Longrightarrow> sd \<in> init_files"by (induct f, auto split:if_splits)lemma source_dir_of_init: "\<lbrakk>source_dir [] f = Some sd; f \<in> init_files\<rbrakk> \<Longrightarrow> f = sd"by (induct f, auto)lemma source_dir_of_init': "f \<in> init_files \<Longrightarrow> source_dir [] f = Some f"by (induct f, auto)lemma init_not_curf_imp_deleted: "\<lbrakk>f \<in> init_files; f \<notin> current_files s; valid s\<rbrakk> \<Longrightarrow> deleted (File f) s"apply (induct s, simp)apply (frule valid_cons, frule valid_os, case_tac a, auto)donelemma source_dir_of_init'': "\<lbrakk>f \<in> init_files; \<not> deleted (File f) s; valid s\<rbrakk> \<Longrightarrow> source_dir s f = Some f"by (induct f, auto)lemma source_dir_createf: "valid (CreateFile p (f#pf) # s) \<Longrightarrow> source_dir (CreateFile p (f#pf) # s) = (source_dir s) ((f#pf) := source_dir s pf)"apply (frule valid_os, frule valid_cons)apply (rule ext, induct_tac x)apply (auto dest:init_not_curf_imp_deleted)donelemma source_dir_createf': "valid (CreateFile p f # s) \<Longrightarrow> source_dir (CreateFile p f # s) = (source_dir s) (f := (case (parent f) of Some pf \<Rightarrow> source_dir s pf | _ \<Rightarrow> None))"apply (frule valid_os, case_tac f, simp+)apply (drule source_dir_createf, auto)donelemma source_dir_other: "\<lbrakk>valid (e # s); \<forall> p f. e \<noteq> CreateFile p f; \<forall> p f. e \<noteq> DeleteFile p f\<rbrakk> \<Longrightarrow> source_dir (e#s) = source_dir s"apply (rule ext, induct_tac x, simp)apply (auto dest:not_deleted_cons_D)apply (case_tac [!] e, auto)donelemma source_dir_deletef: "valid (DeleteFile p f # s) \<Longrightarrow> source_dir (DeleteFile p f # s) f' = (if (source_dir s f') = Some f then parent f else (source_dir s f'))"apply (frule valid_os, frule valid_cons)apply (case_tac "f \<in> init_files")apply (induct_tac f', simp)apply (auto dest!:init_parent_undeleted_I intro:parent_file_in_init' intro!: source_dir_of_init'')[1]apply (induct_tac f', auto)donelemma source_dir_deletef': "valid (DeleteFile p f # s) \<Longrightarrow> source_dir (DeleteFile p f # s) = (\<lambda> f'. (if (source_dir s f') = Some f then parent f else (source_dir s f')) )"by (auto dest:source_dir_deletef)lemmas source_dir_simps = source_dir_of_init' source_dir_of_init'' source_dir_createf' source_dir_deletef' source_dir_otherdeclare source_dir.simps [simp del]lemma source_dir_is_ancient: "source_dir s f = Some sd ==> sd \<preceq> f"apply (induct f)by (auto simp:source_dir.simps no_junior_def split:if_splits)lemma no_junior_trans: "\<lbrakk>f \<preceq> f'; f' \<preceq> f''\<rbrakk> \<Longrightarrow> f \<preceq> f''"by (auto elim:no_juniorE)lemma ancient_has_parent: "[| f \<preceq> f'; f \<noteq> f'|] ==> \<exists> sonf. parent sonf = Some f \<and> sonf \<preceq> f' "apply (induct f')apply (simp add:no_junior_def)apply (case_tac "f = f'")apply (rule_tac x = "a # f'" in exI, simp add:no_junior_def)apply (frule no_junior_noteq, simp)apply clarsimpapply (rule_tac x = sonf in exI, simp add:no_junior_trans)donelemma source_dir_prop: "[|\<forall>fn. fn # f' \<notin> current_files s; source_dir s f = Some f'; f \<in> current_files s; valid s|] ==> f = f'" apply (drule source_dir_is_ancient) apply (case_tac "f = f'", simp) apply (drule ancient_has_parent, simp, clarsimp) apply (drule_tac ancient_file_in_current, simp+) apply (case_tac sonf, auto) donelemma current_file_has_sd: "\<lbrakk>f \<in> current_files s; valid s\<rbrakk> \<Longrightarrow> \<exists> sd. source_dir s f = Some sd"apply (induct s arbitrary:f, simp add:source_dir_of_init')apply (frule valid_cons, frule valid_os, case_tac a, auto simp:source_dir_simps)apply (case_tac list, simp)apply (rule_tac f = f in cannot_del_root, simp+)donelemma current_file_has_sd': "\<lbrakk>source_dir s f = None; valid s\<rbrakk> \<Longrightarrow> f \<notin> current_files s"by (rule notI, auto dest:current_file_has_sd)lemma current_file_has_sfile: "\<lbrakk>f \<in> current_files s; valid s\<rbrakk> \<Longrightarrow> \<exists> et sd. cf2sfile s f = Some (et, sd)"apply (frule current_file_has_sd, simp+)apply (frule current_file_has_etype, auto)donelemma current_file_has_sfile': "\<lbrakk>f \<in> current_files s; valid s\<rbrakk> \<Longrightarrow> \<exists> sf. cf2sfile s f = Some sf"by (auto dest:current_file_has_sfile)(*lemma not_deleted_sf_remains: "\<lbrakk>f \<in> current_files s; \<not> deleted (File f) s; valid s\<rbrakk> \<Longrightarrow> "*)lemma current_proc_has_sproc: "\<lbrakk>p \<in> current_procs s; valid s\<rbrakk> \<Longrightarrow> \<exists> r fr pt u. cp2sproc s p = Some (r, fr, pt, u)"apply (frule current_proc_has_role, simp+)apply (frule current_proc_has_type, simp)apply (frule current_proc_has_forcedrole, simp)apply (frule current_proc_has_owner, auto)donelemma current_proc_has_sproc': "\<lbrakk>p \<in> current_procs s; valid s\<rbrakk> \<Longrightarrow> \<exists> sp. cp2sproc s p = Some sp"by (auto dest!:current_proc_has_sproc)lemma current_ipc_has_sipc: "\<lbrakk>i \<in> current_ipcs s; valid s\<rbrakk> \<Longrightarrow> \<exists> t. ci2sipc s i = Some t"by (drule current_ipc_has_type, auto)lemma init_file_has_sobj: "f \<in> init_files \<Longrightarrow> \<exists> t sd. init_obj2sobj (File f) = SFile (t, sd) (Some f)"by (frule init_file_has_etype, clarsimp)lemma init_proc_has_sobj: assumes pinit:"p \<in> init_processes" shows "\<exists> r fr pt u. init_obj2sobj (Proc p) = SProc (r, fr, pt, u) (Some p)"proof - from pinit obtain r where "init_currentrole p = Some r" using init_proc_has_role by (auto simp:bidirect_in_init_def) moreover from pinit obtain fr where "init_proc_forcedrole p = Some fr" using init_proc_has_frole by (auto simp:bidirect_in_init_def) moreover from pinit obtain pt where "init_process_type p = Some pt" using init_proc_has_type by (auto simp:bidirect_in_init_def) moreover from pinit obtain u where "init_owner p = Some u" using init_proc_has_owner by (auto simp:bidirect_in_init_def) ultimately show ?thesis by auto qedlemma init_ipc_has_sobj: "i \<in> init_ipcs \<Longrightarrow> \<exists> t. init_obj2sobj (IPC i) = SIPC t (Some i)"using init_ipc_has_typeby (auto simp:bidirect_in_init_def)lemma init_obj_has_sobj: "exists [] obj \<Longrightarrow> init_obj2sobj obj \<noteq> Unknown"apply (case_tac obj)apply (simp_all only:exists.simps current_procs.simps current_ipcs.simps current_files.simps) apply (auto dest!:init_proc_has_sobj init_file_has_sobj init_ipc_has_sobj)donelemma exists_obj_has_sobj: "\<lbrakk>exists s obj; valid s\<rbrakk> \<Longrightarrow> obj2sobj s obj \<noteq> Unknown"apply (case_tac obj)apply (auto dest!:current_ipc_has_sipc current_proc_has_sproc' current_file_has_sfile' split:option.splits)donelemma current_proc_has_srp: "\<lbrakk>p \<in> current_procs s; valid s\<rbrakk> \<Longrightarrow> \<exists> srp. source_proc s p = Some srp"apply (induct s arbitrary:p, simp)by (frule valid_cons, frule valid_os, case_tac a, auto)lemma current_proc_has_sobj: "\<lbrakk>p \<in> current_procs s; valid s\<rbrakk> \<Longrightarrow> \<exists> r fr t u srp. obj2sobj s (Proc p) = SProc (r,fr,t,u) (Some srp)"apply (frule current_proc_has_sproc')apply (auto dest:current_proc_has_srp)donelemma current_file_has_sobj: "\<lbrakk>f \<in> current_files s; valid s\<rbrakk> \<Longrightarrow> \<exists> t sd srf. obj2sobj s (File f) = SFile (t, sd) srf"by (auto dest:current_file_has_sfile)lemma current_ipc_has_sobj: "\<lbrakk>i \<in> current_ipcs s; valid s\<rbrakk> \<Longrightarrow> \<exists> t sri. obj2sobj s (IPC i) = SIPC t sri"by (auto dest:current_ipc_has_sipc)lemma sobj_has_proc_role: "obj2sobj s (Proc p) = SProc (r, fr, t, u) srp \<Longrightarrow> currentrole s p = Some r"by (auto split:option.splits)lemma chown_role_aux_valid: "\<lbrakk>currentrole s p = Some r; proc_forcedrole s p = Some fr\<rbrakk> \<Longrightarrow> chown_role_aux r fr u = currentrole (ChangeOwner p u # s) p"by (auto split:t_role.splits simp:chown_role_aux_def dest:proc_forcedrole_valid)lemma chown_role_aux_valid': "cp2sproc s p = Some (r, fr, t, u') \<Longrightarrow> chown_role_aux r fr u = currentrole (ChangeOwner p u # s) p"by (rule chown_role_aux_valid, auto split:option.splits)lemma chown_type_aux_valid: "\<lbrakk>currentrole s p = Some r; currentrole (ChangeOwner p u # s) p = Some nr; type_of_process s p = Some t\<rbrakk> \<Longrightarrow> type_of_process (ChangeOwner p u # s) p = Some (chown_type_aux r nr t)" apply (auto split:option.splits t_rc_proc_type.splits dest:default_process_create_type_valid simp:chown_type_aux_def pot_def pct_def)donelemma chown_type_aux_valid': "\<lbrakk>cp2sproc s p = Some (r, fr, t, u'); currentrole (ChangeOwner p u # s) p = Some nr\<rbrakk> \<Longrightarrow> type_of_process (ChangeOwner p u # s) p = Some (chown_type_aux r nr t)"by (rule chown_type_aux_valid, auto split:option.splits)lemma exec_type_aux_valid: "\<lbrakk>currentrole s p = Some r; type_of_process s p = Some t\<rbrakk> \<Longrightarrow> type_of_process (Execute p f # s) p = Some (exec_type_aux r t)" apply (auto split:option.splits t_rc_proc_type.splits dest:default_process_execute_type_valid simp:exec_type_aux_def pet_def)donelemma exec_type_aux_valid': "cp2sproc s p = Some (r, fr, t, u') \<Longrightarrow> type_of_process (Execute p f # s) p = Some (exec_type_aux r t)" by (rule exec_type_aux_valid, auto split:option.splits)lemma non_initf_frole_inherit: "\<lbrakk>f \<notin> init_files; f \<noteq> []\<rbrakk> \<Longrightarrow> forcedrole s f = Some InheritParentRole"apply (induct s) deferapply (case_tac a, auto) apply (induct f, auto split:option.splits dest:init_frole_has_file)donelemma non_initf_irole_inherit: "\<lbrakk>f \<notin> init_files; f \<noteq> []\<rbrakk> \<Longrightarrow> initialrole s f = Some InheritParentRole"apply (induct s) deferapply (case_tac a, auto) apply (induct f, auto split:option.splits dest:init_irole_has_file)donelemma deleted_file_frole_inherit: "\<lbrakk>deleted (File f) s; f \<in> current_files s\<rbrakk> \<Longrightarrow> forcedrole s f = Some InheritParentRole"apply (induct s, simp)apply (case_tac a, auto) donelemma deleted_file_irole_inherit: "\<lbrakk>deleted (File f) s; f \<in> current_files s\<rbrakk> \<Longrightarrow> initialrole s f = Some InheritParentRole"apply (induct s, simp)apply (case_tac a, auto) donelemma sd_deter_efrole: "\<lbrakk>source_dir s f = Some sd; valid s; f \<in> current_files s\<rbrakk> \<Longrightarrow> effforcedrole s f = effforcedrole s sd"apply (induct f)apply (drule source_dir_is_ancient, simp add:no_junior_def)apply (simp add:source_dir.simps split:if_splits)apply (frule parent_file_in_current', simp)apply (case_tac "a # f \<in> init_files", simp)apply (drule_tac deleted_file_frole_inherit, simp, simp add:effforcedrole_def)apply (drule_tac s = s in non_initf_frole_inherit, simp, simp add:effforcedrole_def)donelemma sd_deter_eirole: "\<lbrakk>source_dir s f = Some sd; valid s; f \<in> current_files s\<rbrakk> \<Longrightarrow> effinitialrole s f = effinitialrole s sd"apply (induct f)apply (drule source_dir_is_ancient, simp add:no_junior_def)apply (simp add:source_dir.simps split:if_splits)apply (frule parent_file_in_current', simp)apply (case_tac "a # f \<in> init_files", simp)apply (drule_tac deleted_file_irole_inherit, simp, simp add:effinitialrole_def)apply (drule_tac s = s in non_initf_irole_inherit, simp, simp add:effinitialrole_def)donelemma undel_initf_keeps_frole: "\<lbrakk>f \<in> init_files; \<not> deleted (File f) s; valid s\<rbrakk> \<Longrightarrow> forcedrole s f = init_file_forcedrole f"apply (induct s, simp)apply (frule valid_cons, frule valid_os, case_tac a)apply (auto dest:init_notin_curf_deleted)donelemma undel_initf_keeps_efrole: "\<lbrakk>f \<in> init_files; \<not> deleted (File f) s; valid s\<rbrakk> \<Longrightarrow> effforcedrole s f = erole_functor init_file_forcedrole InheritUpMixed f"apply (induct f)apply (drule undel_initf_keeps_frole, simp, simp)apply (simp add:effforcedrole_def)apply (frule parent_file_in_init', frule init_parent_undeleted_I, simp+)apply (drule undel_initf_keeps_frole, simp, simp)apply (simp add:effforcedrole_def)donelemma undel_initf_keeps_irole: "\<lbrakk>f \<in> init_files; \<not> deleted (File f) s; valid s\<rbrakk> \<Longrightarrow> initialrole s f = init_file_initialrole f"apply (induct s, simp)apply (frule valid_cons, frule valid_os, case_tac a)apply (auto dest:init_notin_curf_deleted)donelemma undel_initf_keeps_eirole: "\<lbrakk>f \<in> init_files; \<not> deleted (File f) s; valid s\<rbrakk> \<Longrightarrow> effinitialrole s f = erole_functor init_file_initialrole UseForcedRole f"apply (induct f)apply (drule undel_initf_keeps_irole, simp, simp)apply (simp add:effinitialrole_def)apply (frule parent_file_in_init', frule init_parent_undeleted_I, simp+)apply (drule undel_initf_keeps_irole, simp, simp)apply (simp add:effinitialrole_def)donelemma source_dir_not_deleted: "source_dir s f = Some sd \<Longrightarrow> \<not> deleted (File sd) s"by (induct f, auto simp:source_dir.simps split:if_splits)lemma exec_role_aux_valid: "\<lbrakk>currentrole s p = Some r; source_dir s f = Some sd; owner s p = Some u; f \<in> current_files s; valid s\<rbrakk> \<Longrightarrow> exec_role_aux r sd u = currentrole (Execute p f # s) p"apply (frule sd_deter_eirole, simp+, frule sd_deter_efrole, simp+)apply (frule source_dir_in_init, drule source_dir_not_deleted)apply (simp add:undel_initf_keeps_eirole undel_initf_keeps_efrole)apply (frule file_has_effinitialrole, simp, frule file_has_effforcedrole, simp)apply (auto split:option.splits t_role.splits simp:map_comp_def exec_role_aux_def dest:effforcedrole_valid effinitialrole_valid)donelemma exec_role_aux_valid': "\<lbrakk>cp2sproc s p = Some (r, fr, t, u); source_dir s f = Some sd; f \<in> current_files s; valid s\<rbrakk> \<Longrightarrow> exec_role_aux r sd u = currentrole (Execute p f # s) p"by (rule exec_role_aux_valid, auto split:option.splits)lemma cp2sproc_nil_init: "init_obj2sobj (Proc p) = (case (cp2sproc [] p) of Some sp \<Rightarrow> SProc sp (Some p) | _ \<Rightarrow> Unknown)"by (auto split:option.splits)lemma cf2sfile_nil_init: "init_obj2sobj (File f) = (case (cf2sfile [] f) of Some sf \<Rightarrow> SFile sf (Some f) | _ \<Rightarrow> Unknown)"apply (auto split:option.splits simp:etype_of_file_def)apply (case_tac "f \<in> init_files", simp add:source_dir_of_init')apply (induct f, simp+)apply (case_tac "f \<in> init_files", simp add:source_dir_of_init')apply (induct f, simp+)donelemma ci2sipc_nil_init: "init_obj2sobj (IPC i) = (case (ci2sipc [] i) of Some si \<Rightarrow> SIPC si (Some i) | _ \<Rightarrow> Unknown)"by simplemma obj2sobj_nil_init: "exists [] obj \<Longrightarrow> obj2sobj [] obj = init_obj2sobj obj" apply (case_tac obj) apply (auto simp:cf2sfile_nil_init cp2sproc_nil_init ci2sipc_nil_init source_dir_of_init' etype_of_file_def split:if_splits option.splits)done(**** cp2sproc simpset ****)lemma current_proc_has_role': "\<lbrakk>currentrole s p = None; valid s\<rbrakk> \<Longrightarrow> p \<notin> current_procs s"by (rule notI, auto dest:current_proc_has_role)lemma cp2sproc_chown: assumes vs: "valid (ChangeOwner p u # s)" shows "cp2sproc (ChangeOwner p u # s) = (cp2sproc s) (p := (case (cp2sproc s p) of Some (r,fr,pt,u') \<Rightarrow> (case (chown_role_aux r fr u) of Some nr \<Rightarrow> Some (nr,fr,chown_type_aux r nr pt,u) | _ \<Rightarrow> None) | _ \<Rightarrow> None) )" (is "?lhs = ?rhs")proof- have os: "os_grant s (ChangeOwner p u)" and vs': "valid s" using vs by (auto dest:valid_cons valid_os) have "\<And> x. x \<noteq> p \<Longrightarrow> ?lhs x = ?rhs x" by (auto simp:type_of_process.simps split:option.splits t_role.splits) moreover have "?lhs p = ?rhs p" proof- from os have p_in: "p \<in> current_procs s" by (simp+) then obtain r fr t u' where csp: "cp2sproc s p = Some (r, fr, t, u')" using vs' by (drule_tac current_proc_has_sproc, auto) from os have "u \<in> init_users" by simp hence "defrole u \<noteq> None" using init_user_has_role by (auto simp:bidirect_in_init_def) then obtain nr where nrole:"chown_role_aux r fr u = Some nr" by (case_tac fr, auto simp:chown_role_aux_def) have nr_eq: "currentrole (ChangeOwner p u # s) p = chown_role_aux r fr u" using csp by (auto simp:chown_role_aux_valid'[where u = u]) moreover have "type_of_process (ChangeOwner p u # s) p = Some (chown_type_aux r nr t)" using csp nrole nr_eq by (rule_tac fr = fr and u' = u' in chown_type_aux_valid', simp+) moreover have "proc_forcedrole (ChangeOwner p u # s) p = Some fr" using csp by (auto split:option.splits) moreover have "owner (ChangeOwner p u # s) p = Some u" by simp ultimately have "cp2sproc (ChangeOwner p u # s) p = Some (nr, fr, chown_type_aux r nr t, u)" using nrole by (simp) thus ?thesis using csp nrole by simp qed ultimately show ?thesis by (rule_tac ext, auto)qedlemma cp2sproc_crole: "valid (ChangeRole p r # s) \<Longrightarrow> cp2sproc (ChangeRole p r # s) = (cp2sproc s) (p := (case (cp2sproc s p) of Some (r',fr,pt,u) \<Rightarrow> Some (r,fr,pt,u) | _ \<Rightarrow> None) )"apply (frule valid_cons, frule valid_os, simp)apply (frule current_proc_has_sproc, simp)apply (rule ext, auto split:option.splits)donelemma cp2sproc_exec: assumes vs: "valid (Execute p f # s)" shows "cp2sproc (Execute p f # s) = (cp2sproc s) (p := (case (cp2sproc s p, source_dir s f) of (Some (r,fr,pt,u), Some sd) \<Rightarrow> ( case (exec_role_aux r sd u, erole_functor init_file_forcedrole InheritUpMixed sd) of (Some r', Some fr') \<Rightarrow> Some (r', fr', exec_type_aux r pt, u) | _ \<Rightarrow> None ) | _ \<Rightarrow> None))" (is "?lhs = ?rhs")proof- have os: "os_grant s (Execute p f)" and vs': "valid s" using vs by (auto dest:valid_cons valid_os) have "\<And> x. x \<noteq> p \<Longrightarrow> ?lhs x = ?rhs x" by (auto simp:type_of_process.simps split:option.splits t_role.splits) moreover have "?lhs p = ?rhs p" proof- from os have p_in: "p \<in> current_procs s" by (simp+) then obtain r fr t u where csp: "cp2sproc s p = Some (r, fr, t, u)" using vs' by (drule_tac current_proc_has_sproc, auto) from os have f_in: "f \<in> current_files s" by simp then obtain sd where sdir: "source_dir s f = Some sd" using vs' by (drule_tac current_file_has_sd, auto) have "currentrole (Execute p f # s) p \<noteq> None" using vs p_in by (rule_tac notI, drule_tac current_proc_has_role', simp+) then obtain nr where nrole: "currentrole (Execute p f # s) p = Some nr" by auto have "proc_forcedrole (Execute p f # s) p \<noteq> None" using vs p_in by (rule_tac notI, drule_tac current_proc_has_forcedrole', simp+) then obtain nfr where nfrole: "proc_forcedrole (Execute p f # s) p = Some nfr" by auto have nr_eq: "currentrole (Execute p f # s) p = exec_role_aux r sd u" using csp f_in sdir vs' by (simp only:exec_role_aux_valid') moreover have "type_of_process (Execute p f # s) p = Some (exec_type_aux r t)" using csp by (simp only:exec_type_aux_valid') moreover have nfr_eq: "proc_forcedrole (Execute p f # s) p = erole_functor init_file_forcedrole InheritUpMixed sd" using sdir vs' f_in apply (frule_tac source_dir_in_init, drule_tac source_dir_not_deleted) by (simp add:undel_initf_keeps_efrole sd_deter_efrole) moreover have "owner (Execute p f # s) p = Some u" using csp by (auto split:option.splits) ultimately have "cp2sproc (Execute p f # s) p = Some (nr, nfr, exec_type_aux r t, u)" using nrole nfrole by (simp) moreover have "exec_role_aux r sd u = Some nr" using nrole nr_eq by simp moreover have "erole_functor init_file_forcedrole InheritUpMixed sd = Some nfr" using nfrole nfr_eq by simp ultimately show ?thesis using csp sdir by simp qed ultimately show ?thesis by (rule_tac ext, auto)qedlemma cp2sproc_clone: "valid (Clone p p' # s) \<Longrightarrow> cp2sproc (Clone p p' # s) = (cp2sproc s) (p' := (case (cp2sproc s p) of Some (r, fr, pt, u) \<Rightarrow> Some (r, fr, clone_type_aux r pt, u) | _ \<Rightarrow> None))"apply (frule valid_cons, frule valid_os)apply (rule ext, auto split:option.splits t_rc_proc_type.splits simp:pct_def clone_type_aux_def dest:current_proc_has_type default_process_create_type_valid)donelemma cp2sproc_other: "\<lbrakk>valid (e # s); \<forall> p f. e \<noteq> Execute p f; \<forall> p p'. e \<noteq> Clone p p'; \<forall> p r. e \<noteq> ChangeRole p r; \<forall> p u. e \<noteq> ChangeOwner p u\<rbrakk> \<Longrightarrow> cp2sproc (e#s) = cp2sproc s"by (case_tac e, auto)lemmas cp2sproc_simps = cp2sproc_exec cp2sproc_chown cp2sproc_crole cp2sproc_clone cp2sproc_otherlemma obj2sobj_file: "obj2sobj s obj = SFile sf fopt \<Longrightarrow> \<exists> f. obj = File f"by (case_tac obj, case_tac [!] s, auto split:option.splits if_splits)lemma obj2sobj_proc: "obj2sobj s obj = SProc sp popt \<Longrightarrow> \<exists> p. obj = Proc p"by (case_tac obj, case_tac [!] s, auto split:option.splits if_splits)lemma obj2sobj_ipc: "obj2sobj s obj = SIPC si iopt \<Longrightarrow> \<exists> i. obj = IPC i"by (case_tac obj, case_tac [!] s, auto split:option.splits if_splits)lemma obj2sobj_file': "\<lbrakk>obj2sobj s (File f) = sobj; sobj \<noteq> Unknown\<rbrakk> \<Longrightarrow> \<exists> sf srf. sobj = SFile sf srf"by (case_tac sobj, case_tac [!] s, auto split:option.splits if_splits)lemma obj2sobj_proc': "\<lbrakk>obj2sobj s (Proc p) = sobj; sobj \<noteq> Unknown\<rbrakk> \<Longrightarrow> \<exists> sp srp. sobj = SProc sp srp"by (case_tac sobj, case_tac [!] s, auto split:option.splits if_splits)lemma obj2sobj_ipc': "\<lbrakk>obj2sobj s (IPC i) = sobj; sobj \<noteq> Unknown\<rbrakk> \<Longrightarrow> \<exists> si sri. sobj = SIPC si sri"by (case_tac sobj, case_tac [!] s, auto split:option.splits if_splits)lemma obj2sobj_file_remains_cons: assumes vs: "valid (e#s)" and exf: "f \<in> current_files s" and SF: "obj2sobj s (File f) = SFile sf srf" and notdeled: "\<not> deleted (File f) (e#s)" shows "obj2sobj (e#s) (File f) = SFile sf srf"proof- from vs have os:"os_grant s e" and vs': "valid s" by (auto dest:valid_cons valid_os) from notdeled exf have exf': "f \<in> current_files (e#s)" by (case_tac e, auto) have "etype_of_file (e # s) f = etype_of_file s f" using os vs vs' exf exf' apply (case_tac e, auto simp:etype_of_file_def split:option.splits) by (auto dest:ancient_file_in_current intro!:etype_aux_prop) moreover have "source_dir (e # s) f = source_dir s f" using os vs vs' exf exf' by (case_tac e, auto simp:source_dir_simps dest:source_dir_prop) ultimately show ?thesis using vs SF notdeled by (auto split:if_splits option.splits dest:not_deleted_cons_D)qedlemma obj2sobj_file_remains_cons': "\<lbrakk>valid (e#s); f \<in> current_files s; obj2sobj s (File f) = SFile sf srf; no_del_event (e#s)\<rbrakk> \<Longrightarrow> obj2sobj (e#s) (File f) = SFile sf srf"by (auto intro!:obj2sobj_file_remains_cons nodel_imp_un_deleted simp del:obj2sobj.simps)lemma obj2sobj_file_remains': "\<lbrakk>obj2sobj s (File f) = sobj; sobj \<noteq> Unknown; valid (e#s); f \<in> current_files s; no_del_event (e#s)\<rbrakk> \<Longrightarrow> obj2sobj (e#s) (File f) = sobj"apply (frule obj2sobj_file', simp, (erule exE)+)apply (simp del:obj2sobj.simps)apply (erule obj2sobj_file_remains_cons', simp+)donelemma obj2sobj_file_remains_app: "\<lbrakk>obj2sobj s (File f) = SFile sf srf; valid (s' @ s); f \<in> current_files s; \<not> deleted (File f) (s'@s)\<rbrakk> \<Longrightarrow> obj2sobj (s'@s) (File f) = SFile sf srf"apply (induct s', simp)apply (simp only:cons_app_simp_aux)apply (frule valid_cons, frule not_deleted_cons_D)apply (drule_tac s = "s'@s" in obj2sobj_file_remains_cons, auto simp del:obj2sobj.simps)apply (drule_tac obj = "File f" in not_deleted_imp_exists', simp+)donelemma obj2sobj_file_remains_app': "\<lbrakk>obj2sobj s (File f) = SFile sf srf; valid (s' @ s); f \<in> current_files s; no_del_event (s'@s)\<rbrakk> \<Longrightarrow> obj2sobj (s'@s) (File f) = SFile sf srf"by (auto intro!:obj2sobj_file_remains_app nodel_imp_un_deleted simp del:obj2sobj.simps)lemma obj2sobj_file_remains'': "\<lbrakk>obj2sobj s (File f) = sobj; sobj \<noteq> Unknown; valid (s'@s); f \<in> current_files s; no_del_event (s'@s)\<rbrakk> \<Longrightarrow> obj2sobj (s'@s) (File f) = sobj"apply (frule obj2sobj_file', simp, (erule exE)+)apply (simp del:obj2sobj.simps)apply (erule obj2sobj_file_remains_app', simp+)donelemma obj2sobj_file_remains''': "\<lbrakk>obj2sobj s (File f) = sobj; sobj \<noteq> Unknown; valid (s'@s); f \<in> current_files s; \<not>deleted (File f) (s'@s)\<rbrakk> \<Longrightarrow> obj2sobj (s'@s) (File f) = sobj"apply (frule obj2sobj_file', simp, (erule exE)+)apply (simp del:obj2sobj.simps)by (erule obj2sobj_file_remains_app, simp+)lemma obj2sobj_ipc_remains_cons: "\<lbrakk>valid (e#s); i \<in> current_ipcs s; obj2sobj s (IPC i) = SIPC si sri; \<not> deleted (IPC i) (e#s)\<rbrakk> \<Longrightarrow> obj2sobj (e#s) (IPC i) = SIPC si sri"apply (frule valid_cons, frule valid_os, case_tac e)by (auto split:option.splits dest!:current_proc_has_role')lemma obj2sobj_ipc_remains_cons': "\<lbrakk>valid (e#s); i \<in> current_ipcs s; obj2sobj s (IPC i) = SIPC si sri; no_del_event (e#s)\<rbrakk> \<Longrightarrow> obj2sobj (e#s) (IPC i) = SIPC si sri"by (auto intro!:obj2sobj_ipc_remains_cons nodel_imp_un_deleted simp del:obj2sobj.simps)lemma obj2sobj_ipc_remains': "\<lbrakk>obj2sobj s (IPC i) = sobj; sobj \<noteq> Unknown; valid (e#s); i \<in> current_ipcs s; no_del_event (e#s)\<rbrakk> \<Longrightarrow> obj2sobj (e#s) (IPC i) = sobj"apply (frule obj2sobj_ipc', simp, (erule exE)+)apply (simp del:obj2sobj.simps)apply (erule obj2sobj_ipc_remains_cons', simp+)donelemma obj2sobj_ipc_remains_app: "\<lbrakk>obj2sobj s (IPC i) = SIPC si sri; valid (s'@s); i \<in> current_ipcs s; \<not> deleted (IPC i) (s'@s)\<rbrakk> \<Longrightarrow> obj2sobj (s'@s) (IPC i) = SIPC si sri"apply (induct s', simp)apply (simp only:cons_app_simp_aux)apply (frule valid_cons, frule not_deleted_cons_D)apply (drule_tac s = "s'@s" in obj2sobj_ipc_remains_cons, auto simp del:obj2sobj.simps)apply (drule_tac obj = "IPC i" in not_deleted_imp_exists', simp+)donelemma obj2sobj_ipc_remains_app': "\<lbrakk>obj2sobj s (IPC i) = SIPC si sri; valid (s'@s); i \<in> current_ipcs s; no_del_event (s'@s)\<rbrakk> \<Longrightarrow> obj2sobj (s'@s) (IPC i) = SIPC si sri"by (auto intro!:obj2sobj_ipc_remains_app nodel_imp_un_deleted simp del:obj2sobj.simps)lemma obj2sobj_ipc_remains'': "\<lbrakk>obj2sobj s (IPC i) = sobj; sobj \<noteq> Unknown; valid (s'@s); i \<in> current_ipcs s; no_del_event (s'@s)\<rbrakk> \<Longrightarrow> obj2sobj (s'@s) (IPC i) = sobj"apply (frule obj2sobj_ipc', simp, (erule exE)+)apply (simp del:obj2sobj.simps)apply (erule obj2sobj_ipc_remains_app', simp+)donelemma obj2sobj_ipc_remains''': "\<lbrakk>obj2sobj s (IPC i) = sobj; sobj \<noteq> Unknown; valid (s'@s); i \<in> current_ipcs s; \<not> deleted (IPC i) (s'@s)\<rbrakk> \<Longrightarrow> obj2sobj (s'@s) (IPC i) = sobj"apply (frule obj2sobj_ipc', simp, (erule exE)+)apply (simp del:obj2sobj.simps)apply (erule obj2sobj_ipc_remains_app, simp+)doneendcontext tainting_s_sound beginlemma cp2sproc_clone': "valid (Clone p p' # s) \<Longrightarrow> cp2sproc (Clone p p' # s) = (cp2sproc s) (p' := cp2sproc s p)"by (drule cp2sproc_clone, auto split:option.splits simp:clone_type_unchange clone_type_aux_def)lemmas cp2sproc_simps' = cp2sproc_exec cp2sproc_chown cp2sproc_crole cp2sproc_clone' cp2sproc_otherlemma clone_sobj_keeps_same: "valid (Clone p p' # s) \<Longrightarrow> obj2sobj (Clone p p' # s) (Proc p') = obj2sobj s (Proc p)"apply (frule valid_cons, frule valid_os, clarsimp)apply (auto split:option.splits t_rc_proc_type.splits dest:current_proc_has_role current_proc_has_forcedrole current_proc_has_type current_proc_has_owner default_process_create_type_valid simp:pct_def clone_type_unchange)doneendend