theory Dynamic2staticimports Main Flask Static Init_prop Valid_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 is_file_has_sfile: "is_file s f \<Longrightarrow> \<exists> sf. cf2sfile s f True = Some sf"sorrylemma is_dir_has_sfile: "is_dir s f \<Longrightarrow> \<exists> sf. cf2sfile s f False = Some sf"sorrylemma is_file_imp_alive: "is_file s f \<Longrightarrow> alive s (O_file f)"sorrylemma d2s_main': "\<lbrakk>alive s obj; co2sobj s obj= Some sobj\<rbrakk> \<Longrightarrow> sobj \<in> (s2ss s)"apply (induct s)apply (simp add:s2ss_def)apply (rule_tac x = obj in exI, simp)sorrylemma tainted_prop1: "obj \<in> tainted s \<Longrightarrow> alive s obj"sorrylemma tainted_prop2: "obj \<in> tainted s \<Longrightarrow> valid s"sorrylemma 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_prop1, frule tainted_prop2)apply (simp add:s2ss_def)apply (case_tac sobj, simp_all)apply (case_tac [!] obj, simp_all split:option.splits)apply (rule_tac x = "O_proc nat" in exI, simp)apply (rule_tac x = "O_file list" in exI, simp)defer defer deferapply (case_tac prod1, simp, case_tac prod2, clarsimp)apply (rule conjI)apply (rule_tac x = "O_msgq nat1" in exI, simp)sorry (* doable, need properties about cm2smsg and cq2smsgq *)lemma delq_imp_delqm: "deleted (O_msgq q) s \<Longrightarrow> deleted (O_msg q m) s"apply (induct s, simp)by (case_tac a, auto)lemma undel_init_file_remains: "\<lbrakk>is_init_file f; \<not> deleted (O_file f) s\<rbrakk> \<Longrightarrow> is_file s f"sorrytheorem 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_prop2) hence static: "s2ss s \<in> static" using d2s_main by auto from tainted have alive: "alive s obj" using tainted_prop1 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) apply (frule undel_init_file_remains, simp, 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) apply (rule_tac x = list in exI, simp) apply (case_tac list, auto split:option.splits simp:is_init_dir_props delq_imp_delqm) done 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 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 (induct s, simp)apply (frule vd_cons)apply (case_tac a, auto)(* need simpset for s2ss *)sorrylemma 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)+apply (erule exE, erule conjE)+sorrylemma tainted_s_imp_tainted: "\<lbrakk>tainted_s ss sobj; ss \<in> static\<rbrakk> \<Longrightarrow> \<exists> obj s. s2ss s = ss \<and> valid s \<and> co2sobj s obj = Some sobj \<and> obj \<in> tainted s"sorrytheorem 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 s2ss: "s2ss s = ss" and co2sobj: "co2sobj s obj' = Some sobj" and tainted: "obj' \<in> tainted s" and vs: "valid s" by blast from co2sobj related have eq:"obj = obj'" apply (case_tac obj', case_tac [!] obj, case_tac [!] sobj) apply auto apply (auto split:option.splits if_splits) apply (case_tac a, simp+) apply (simp add:cp2sproc_def split:option.splits if_splits) apply simp sorry with tainted 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