theory Dynamic2static
imports Main Flask Static Init_prop Valid_prop Tainted_prop Delete_prop Co2sobj_prop
begin

context tainting_s begin

lemma 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
*)
thm s2ss_def


sorry

lemma 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 tainted_has_sobj:
  "\<lbrakk>obj \<in> tainted s; valid s\<rbrakk> \<Longrightarrow> \<exists> sobj. co2sobj s obj = Some sobj"
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 (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)
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)

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 obtain sobj where sobj: "co2sobj s obj = Some sobj"
    using vs tainted_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', 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 (rule_tac x = "s2ss s" in bexI, simp)
    apply (rule_tac x = "sobj" in exI, auto)
    done
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)
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)
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 (rule_tac x = "s2ss s" in bexI)
apply (clarify, simp add:init_deled_imp_deled_s)
apply (erule d2s_main)
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 \<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)
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 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'"
sorry

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 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.simps
oops

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)+)

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 (rule conjI)
apply simp
apply (simp add
apply (simp add:co2sobj.simps)
apply (simp add:cm2smsg_def split:option.splits if_splits)
done

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 *)
sorry

lemma 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)
qed



lemma 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