theory Dynamic2static

imports Main Flask Static Init_prop Valid_prop Tainted_prop Delete_prop Co2sobj_prop S2ss_prop S2ss_prop2

begin

context tainting_s begin

lemma many_sq_imp_sms:

  "\<lbrakk>S_msgq (Create, sec, sms) \<in> ss; ss \<in> static\<rbrakk> \<Longrightarrow> \<forall> sm \<in> (set sms). is_many_smsg sm"

sorry

definition init_ss_eq:: "t_static_state \<Rightarrow> t_static_state \<Rightarrow> bool" (infix "\<doteq>" 100)

where

  "ss \<doteq> ss' \<equiv> ss \<subseteq> ss' \<and> {sobj. is_init_sobj sobj \<and> sobj \<in> ss'} \<subseteq> ss"

lemma [simp]: "ss \<doteq> ss"

by (auto simp:init_ss_eq_def)

definition init_ss_in:: "t_static_state \<Rightarrow> t_static_state set \<Rightarrow> bool" (infix "\<propto>" 101)

where

  "ss \<propto> sss \<equiv> \<exists> ss' \<in> sss. ss \<doteq> ss'"

lemma s2ss_included_sobj:

  "\<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 init_ss_in_prop:

  "\<lbrakk>s2ss s \<propto> static; co2sobj s obj = Some sobj; alive s obj; init_obj_related sobj obj\<rbrakk>

   \<Longrightarrow> \<exists> ss \<in> static. sobj \<in> ss"

apply (simp add:init_ss_in_def init_ss_eq_def)

apply (erule bexE, erule conjE)

apply (rule_tac x = ss' in bexI, auto dest!:s2ss_included_sobj)

done

lemma d2s_main_execve:

  "valid (Execve p f fds # s) \<Longrightarrow> s2ss (Execve p f fds # s) \<in> static"

apply (frule vd_cons, frule vt_grant_os, clarsimp simp:s2ss_execve)

sorry

lemma d2s_main:

  "valid s \<Longrightarrow> s2ss s \<propto> static"

apply (induct s, simp add:s2ss_nil_prop init_ss_in_def)

apply (rule_tac x = "init_static_state" in bexI, simp, simp add:s_init)

apply (frule vd_cons, frule vt_grant_os, simp)

apply (case_tac a) 

apply (clarsimp simp add:s2ss_execve)

apply (rule conjI, rule impI)

sorry

lemma 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 (case_tac sobj, simp_all)

apply (case_tac [!] obj, simp_all add:co2sobj.simps split:option.splits if_splits)

apply (drule dir_not_tainted, simp)

apply (drule msgq_not_tainted, simp)

apply (drule shm_not_tainted, simp)

done

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 tainted_s_subset_prop:

  "\<lbrakk>tainted_s ss sobj; ss \<subseteq> ss'\<rbrakk> \<Longrightarrow> tainted_s ss' sobj"

apply (case_tac sobj)

apply auto

done

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 \<propto> static" using d2s_main by auto

  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:co2sobj.simps cp2sproc_def ch2sshm_def cq2smsgq_def cm2smsg_def delq_imp_delqm)

    apply (frule not_deleted_init_file, simp+) 

    apply (drule is_file_has_sfile', simp, erule exE)

    apply (rule_tac x = sf in bexI)

    apply (case_tac list, auto split:option.splits simp:is_init_file_props)[1]

    apply (drule root_is_init_dir', simp)

    apply (frule not_deleted_init_file, simp, simp)

    apply (simp add:cf2sfile_def split:option.splits if_splits)

    apply (simp add:cf2sfiles_def)

    apply (rule_tac x = list in bexI, simp, simp add:same_inode_files_def not_deleted_init_file)

    apply (frule not_deleted_init_dir, simp+)

    apply (simp add:cf2sfile_def split:option.splits if_splits)

    apply (case_tac list, simp add:sroot_def, simp)

    apply (drule file_dir_conflict, simp+)

    done

  with tainted t2ts init_alive sobj static

  show ?thesis unfolding taintable_s_def 

    apply (simp add:init_ss_in_def)

    apply (erule bexE)

    apply (simp add:init_ss_eq_def)

    apply (rule_tac x = "ss'" in bexI)

    apply (rule_tac x = "sobj" in exI)

    by (auto intro:tainted_s_subset_prop)

qed

lemma 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)

done

lemma 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 simp:co2sobj.simps)

apply (auto simp:cf2sfiles_def same_inode_files_def has_same_inode_prop1' is_file_def is_dir_def co2sobj.simps

           split:option.splits t_inode_tag.splits dest!:cf2sfile_fi)

done

lemma 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 (frule d2s_main)

apply (simp add:init_ss_in_def)

apply (erule bexE)

apply (rule_tac x = ss' in bexI)

apply (auto simp add: init_ss_eq_def dest!:init_deled_imp_deled_s)

apply (case_tac obj, case_tac [!] sobj)

apply auto

apply (erule set_mp)

apply (simp)

apply auto

apply (rule_tac x = "(Init list, (aa, ab, b), ac, ba)" in bexI)

apply auto

done

theorem 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 \<propto> static" by (rule d2s_main) 

    with alive_s obtain sobj where in_ss: "sobj \<in> (s2ss s)" 

    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)

qed

theorem 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 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)+

sorry

lemma set_eq_D:

  "\<lbrakk>x \<in> S; {x. P x} = S\<rbrakk> \<Longrightarrow> P x"

by auto

lemma 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 simp:cqm2sms.simps split:option.splits)

done

lemma 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)+)

done

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

  show ?thesis by (auto simp:taintable_def)

qed

end