theory Dynamic2staticimports Main Flask Static Init_prop Valid_prop Tainted_prop Delete_propbegincontext tainting_s beginlemma d2s_main: "valid s \<Longrightarrow> s2ss s \<in> static"apply (induct s, simp add:s2ss_nil_prop s_init)apply (frule vd_cons, simp)apply (case_tac a, simp_all) (*apply induct s, case tac e, every event analysis*)sorrylemma d2s_main': "\<lbrakk>alive s obj; co2sobj s obj= Some sobj\<rbrakk> \<Longrightarrow> sobj \<in> (s2ss s)"by (simp add:s2ss_def, rule_tac x = obj in exI, simp)lemma alive_has_sobj: "\<lbrakk>alive s obj; valid s\<rbrakk> \<Longrightarrow> \<exists> sobj. co2sobj s obj = Some sobj"sorrylemma t2ts: "obj \<in> tainted s \<Longrightarrow> co2sobj s obj = Some sobj \<Longrightarrow> tainted_s (s2ss s) sobj"apply (frule tainted_in_current, frule tainted_is_valid)apply (simp add:s2ss_def)apply (case_tac sobj, simp_all)apply (case_tac [!] obj, simp_all split:option.splits if_splits)apply (rule_tac x = "O_proc nat" in exI, simp)apply (rule_tac x = "O_file list" in exI, simp)apply (drule dir_not_tainted, simp)apply (drule msgq_not_tainted, simp)apply (drule shm_not_tainted, simp)apply (case_tac prod1, simp, case_tac prod2, clarsimp)apply (rule conjI)apply (rule_tac x = "O_msgq nat1" in exI, simp)apply (rule conjI) deferapply (simp add:cm2smsg_def split:option.splits) sorrylemma delq_imp_delqm: "deleted (O_msgq q) s \<Longrightarrow> deleted (O_msg q m) s"apply (induct s, simp)by (case_tac a, auto)theorem static_complete: assumes undel: "undeletable obj" and tbl: "taintable obj" shows "taintable_s obj"proof- from tbl obtain s where tainted: "obj \<in> tainted s" by (auto simp:taintable_def) hence vs: "valid s" by (simp add:tainted_is_valid) hence static: "s2ss s \<in> static" using d2s_main by auto from tainted have alive: "alive s obj" using tainted_in_current by auto then obtain sobj where sobj: "co2sobj s obj = Some sobj" using vs alive_has_sobj by blast from undel vs have "\<not> deleted obj s" and init_alive: "init_alive obj" by (auto simp:undeletable_def) with vs sobj have "init_obj_related sobj obj" apply (case_tac obj, case_tac [!] sobj) apply (auto split:option.splits if_splits simp:cp2sproc_def ch2sshm_def cq2smsgq_def cm2smsg_def delq_imp_delqm) apply (frule not_deleted_init_file, simp+) (*apply (drule is_file_has_sfile, erule exE) apply (rule_tac x = sf in bexI) apply (case_tac list, auto split:option.splits simp:is_init_file_props)[1] apply (simp add:same_inode_files_def cfs2sfiles_def) *) sorry with tainted t2ts init_alive sobj static show ?thesis unfolding taintable_s_def apply (rule_tac x = "s2ss s" in bexI, simp) apply (rule_tac x = "sobj" in exI, auto) doneqedlemma cp2sproc_pi: "\<lbrakk>cp2sproc s p = Some (Init p', sec, fds, shms); valid s\<rbrakk> \<Longrightarrow> p = p' \<and> \<not> deleted (O_proc p) s \<and> p \<in> init_procs"by (simp add:cp2sproc_def split:option.splits if_splits)lemma cq2smsgq_qi: "\<lbrakk>cq2smsgq s q = Some (Init q', sec, sms); valid s\<rbrakk> \<Longrightarrow> q = q' \<and> \<not> deleted (O_msgq q) s \<and> q \<in> init_msgqs"by (simp add:cq2smsgq_def split:option.splits if_splits)lemma cm2smsg_mi: "\<lbrakk>cm2smsg s q m = Some (Init m', sec, ttag); q \<in> init_msgqs; valid s\<rbrakk> \<Longrightarrow> m = m' \<and> \<not> deleted (O_msg q m) s \<and> m \<in> set (init_msgs_of_queue q) \<and> q \<in> init_msgqs"by (clarsimp simp add:cm2smsg_def split:if_splits option.splits)lemma ch2sshm_hi: "\<lbrakk>ch2sshm s h = Some (Init h', sec); valid s\<rbrakk> \<Longrightarrow> h = h' \<and> \<not> deleted (O_shm h) s \<and> h \<in> init_shms"by (clarsimp simp:ch2sshm_def split:if_splits option.splits)lemma root_not_deleted: "\<lbrakk>deleted (O_dir []) s; valid s\<rbrakk> \<Longrightarrow> False"apply (induct s, simp)apply (frule vd_cons, frule vt_grant_os, case_tac a, auto)donelemma cf2sfile_fi: "\<lbrakk>cf2sfile s f = Some (Init f', sec, psecopt, asecs); valid s\<rbrakk> \<Longrightarrow> f = f' \<and> (if (is_file s f) then \<not> deleted (O_file f) s \<and> is_init_file f else \<not> deleted (O_dir f) s \<and> is_init_dir f)"apply (case_tac f)by (auto simp:sroot_def cf2sfile_def root_is_init_dir dest!:root_is_dir' root_not_deleted split:if_splits option.splits) lemma init_deled_imp_deled_s: "\<lbrakk>deleted obj s; init_alive obj; sobj \<in> (s2ss s); valid s\<rbrakk> \<Longrightarrow> \<not> init_obj_related sobj obj"apply (rule notI)apply (clarsimp simp:s2ss_def)apply (case_tac obj, case_tac [!] obja, case_tac sobj)apply (auto split:option.splits if_splits dest!:cp2sproc_pi cq2smsgq_qi ch2sshm_hi cm2smsg_mi cf2sfile_fi)apply (auto simp:cf2sfiles_def same_inode_files_def has_same_inode_prop1' is_file_def is_dir_def split:option.splits t_inode_tag.splits dest!:cf2sfile_fi)donelemma deleted_imp_deletable_s: "\<lbrakk>deleted obj s; init_alive obj; valid s\<rbrakk> \<Longrightarrow> deletable_s obj"apply (simp add:deletable_s_def)apply (rule_tac x = "s2ss s" in bexI)apply (clarify, simp add:init_deled_imp_deled_s)apply (erule d2s_main)donetheorem undeletable_s_complete: assumes undel_s: "undeletable_s obj" shows "undeletable obj"proof- from undel_s have init_alive: "init_alive obj" and alive_s: "\<forall> ss \<in> static. \<exists> sobj \<in> ss. init_obj_related sobj obj" using undeletable_s_def by auto have "\<not> (\<exists> s. valid s \<and> deleted obj s)" proof assume "\<exists> s. valid s \<and> deleted obj s" then obtain s where vs: "valid s" and del: "deleted obj s" by auto from vs have vss: "s2ss s \<in> static" by (rule d2s_main) with alive_s obtain sobj where in_ss: "sobj \<in> (s2ss s)" and related: "init_obj_related sobj obj" by auto from init_alive del vs have "deletable_s obj" by (auto elim:deleted_imp_deletable_s) with alive_s show False by (auto simp:deletable_s_def) qed with init_alive show ?thesis by (simp add:undeletable_def)qedtheorem final_offer: "\<lbrakk>undeletable_s obj; \<not> taintable_s obj; init_alive obj\<rbrakk> \<Longrightarrow> \<not> taintable obj"apply (erule swap)by (simp add:static_complete undeletable_s_complete)(************** static \<rightarrow> dynamic ***************)lemma created_can_have_many: "\<lbrakk>valid s; alive s obj; \<not> init_alive obj\<rbrakk> \<Longrightarrow> \<exists> s'. valid s' \<and> alive s' obj \<and> alive s' obj' \<and> s2ss s = s2ss s'"sorrylemma s2d_main: "ss \<in> static \<Longrightarrow> \<exists> s. valid s \<and> s2ss s = ss"apply (erule static.induct)apply (rule_tac x = "[]" in exI, simp add:s2ss_nil_prop valid.intros)apply (erule exE|erule conjE)+sorrylemma tainted_s_in_ss: "tainted_s ss sobj \<Longrightarrow> sobj \<in> ss"apply (case_tac sobj, simp_all)apply (case_tac bool, simp+)apply (case_tac bool, simp+)apply (case_tac prod1, case_tac prod2, simp)thm tainted_s.simpsoopslemma set_eq_D: "\<lbrakk>x \<in> S; {x. P x} = S\<rbrakk> \<Longrightarrow> P x"by autolemma cqm2sms_prop1: "\<lbrakk>cqm2sms s q queue = Some sms; sm \<in> set sms\<rbrakk> \<Longrightarrow> \<exists> m. cm2smsg s q m = Some sm"apply (induct queue arbitrary:sms)apply (auto split:option.splits)donelemma sq_sm_prop: "\<lbrakk>sm \<in> set sms; cq2smsgq s q = Some (qi, qsec, sms); valid s\<rbrakk> \<Longrightarrow> \<exists> m. cm2smsg s q m = Some sm"by (auto simp:cq2smsgq_def split: option.splits intro:cqm2sms_prop1)lemma tainted_s_imp_tainted: "\<lbrakk>tainted_s ss sobj; ss \<in> static\<rbrakk> \<Longrightarrow> \<exists> s obj. valid s \<and> co2sobj s obj = Some sobj \<and> obj \<in> tainted s"apply (drule s2d_main)apply (erule exE, erule conjE, simp add:s2ss_def)apply (rule_tac x = s in exI, simp)apply (case_tac sobj, simp_all)apply (erule conjE, drule_tac S = ss in set_eq_D, simp, (erule exE|erule conjE)+) apply (rule_tac x = obj in exI, simp)apply (case_tac obj, (simp split:option.splits if_splits)+)apply (erule conjE, drule_tac S = ss in set_eq_D, simp, (erule exE|erule conjE)+) apply (rule_tac x = obj in exI, simp)apply (case_tac obj, (simp split:option.splits if_splits)+)apply (case_tac prod1, case_tac prod2, simp)apply ((erule conjE)+, drule_tac S = ss in set_eq_D, simp, (erule exE|erule conjE)+) apply (case_tac obj, simp_all split:option.splits if_splits)apply (drule_tac sm = "(aa, ba, True)" in sq_sm_prop, simp+, erule exE)apply (rule_tac x = "O_msg nat m" in exI)apply (simp)apply (case_tac obj, (simp split:option.splits if_splits)+)apply (erule conjE, drule_tac )lemma has_inode_tainted_aux: "O_file f \<in> tainted s \<Longrightarrow> \<forall> f'. has_same_inode s f f' \<longrightarrow> O_file f' \<in> tainted s"apply (erule tainted.induct)apply (auto intro:tainted.intros simp:has_same_inode_def)(*?? need simpset for tainted *)sorrylemma has_same_inode_tainted: "\<lbrakk>has_same_inode s f f'; O_file f' \<in> tainted s\<rbrakk> \<Longrightarrow> O_file f \<in> tainted s"by (drule has_inode_tainted_aux, auto simp:has_same_inode_def)theorem static_sound: assumes tbl_s: "taintable_s obj" shows "taintable obj"proof- from tbl_s obtain ss sobj where static: "ss \<in> static" and sobj: "tainted_s ss sobj" and related: "init_obj_related sobj obj" and init_alive: "init_alive obj" by (auto simp:taintable_s_def) from static sobj tainted_s_imp_tainted obtain s obj' where co2sobj: "co2sobj s obj' = Some sobj" and tainted': "obj' \<in> tainted s" and vs: "valid s" by blast from co2sobj related vs have eq:"obj = obj' \<or> (\<exists> f f'. obj = O_file f \<and> obj' = O_file f' \<and> has_same_inode s f f')" apply (case_tac obj', case_tac [!] obj) apply (auto split:option.splits if_splits dest!:cp2sproc_pi cq2smsgq_qi ch2sshm_hi cm2smsg_mi cf2sfile_fi) apply (auto simp:cf2sfiles_def same_inode_files_def has_same_inode_def is_file_def is_dir_def split:option.splits t_inode_tag.splits dest!:cf2sfile_fi) done with tainted' have tainted: "obj \<in> tainted s" by (auto intro:has_same_inode_tainted) with vs init_alive show ?thesis by (auto simp:taintable_def)qedlemma ts2t: "obj \<in> tainted_s ss \<Longrightarrow> \<exists> s. obj \<in> tainted s" "obj \<in> tainted_s ss \<Longrightarrow> \<exists> so. so True \<in> ss \<Longrightarrow> so True \<in> ss \<Longrightarrow> \<exists> s. valid s \<and> s2ss s = ss \<Longrightarrow> so True \<in> s2ss s \<Longrightarrow> tainted s obj. "end