theory Tainted_prop 

begin

context tainting begin

fun Tainted :: "t_state \<Rightarrow> t_object set"

where

  "Tainted [] = seeds"

| "Tainted (Clone p p' fds shms # s) = 

     (if (O_proc p) \<in> Tainted s then Tainted s \<union> {O_proc p'} else Tainted s)"

| "Tainted (Execve p f fds # s) = 

     (if (O_file f) \<in> Tainted s then Tainted s \<union> {O_proc p} else Tainted s)"

| "Tainted (Kill p p' # s) = Tainted s - {O_proc p'}"

| "Tainted (Ptrace p p' # s) = 

     (if (O_proc p) \<in> Tainted s 

      then Tainted s \<union> {O_proc p'' | p''. info_flow_shm s p' p''}

      else if (O_proc p') \<in> Tainted s 

           then Tainted s \<union> {O_proc p'' | p''. info_flow_shm s p p''}

                else Tainted s)"

| "Tainted (Exit p # s) = Tainted s - {O_proc p}"

| "Tainted (Open p f flags fd opt # s) = 

     (case opt of

        Some inum \<Rightarrow> (if (O_proc p) \<in> Tainted s

                      then Tainted s \<union> {O_file f}

                      else Tainted s)

      | _         \<Rightarrow> Tainted s)" 

| "Tainted (ReadFile p fd # s) = 

     (case (file_of_proc_fd s p fd) of

        Some f \<Rightarrow> if (O_file f) \<in> Tainted s

                  then Tainted s \<union> {O_proc p' | p'. info_flow_shm s p p'}

                  else Tainted s

      | None   \<Rightarrow> Tainted s)"

| "Tainted (WriteFile p fd # s) = 

     (case (file_of_proc_fd s p fd) of 

        Some f \<Rightarrow> if (O_proc p) \<in> Tainted s

                  then Tainted s \<union> {O_file f' | f'. has_same_inode s f f'}

                  else Tainted s

      | None   \<Rightarrow> Tainted s)"

| "Tainted (CloseFd p fd # s) = 

     (case (file_of_proc_fd s p fd) of

        Some f \<Rightarrow> ( if ((proc_fd_of_file s f = {(p,fd)}) \<and> (f \<in> files_hung_by_del s))

                    then Tainted s - {O_file f} else Tainted s )

      | _      \<Rightarrow> Tainted s)"

| "Tainted (UnLink p f # s) = 

     (if (proc_fd_of_file s f = {}) then Tainted s - {O_file f} else Tainted s)"

| "Tainted (LinkHard p f f' # s) = 

     (if (O_file f \<in> Tainted s) then Tainted s \<union> {O_file f'} else Tainted s)"

| "Tainted (Truncate p f len # s) = 

     (if (len > 0 \<and> O_proc p \<in> Tainted s)

      then Tainted s \<union> {O_file f' | f'. has_same_inode s f f'}

      else Tainted s)"

| "Tainted (Attach p h flag # s) = 

     (if (O_proc p \<in> Tainted s \<and> flag = SHM_RDWR) 

      then Tainted s \<union> {O_proc p' | p' flag'. (p', flag') \<in> procs_of_shm s h}

      else if (\<exists> p'. O_proc p' \<in> Tainted s \<and> (p', SHM_RDWR) \<in> procs_of_shm s h)

           then Tainted s \<union> {O_proc p}

           else Tainted s)"

| "Tainted (SendMsg p q m # s) = 

     (if (O_proc p \<in> Tainted s) then Tainted s \<union> {O_msg q m} else Tainted s)"

| "Tainted (RecvMsg p q m # s) = 

     (if (O_msg q m \<in> Tainted s) 

      then (Tainted s \<union> {O_proc p' | p'. info_flow_shm s p p'}) - {O_msg q m}

      else Tainted s)"

| "Tainted (RemoveMsgq p q # s) = Tainted s - {O_msg q m| m. O_msg q m \<in> Tainted s}"

| "Tainted (e # s) = Tainted s"

lemma valid_Tainted_obj:

  "\<lbrakk>obj \<in> Tainted s; valid s\<rbrakk> \<Longrightarrow> (\<forall> f. obj \<noteq> O_dir f) \<and> (\<forall> q. obj \<noteq> O_msgq q) \<and> (\<forall> h. obj \<noteq> O_shm h) \<and> (\<forall> p fd. obj \<noteq> O_fd p fd) \<and> (\<forall> s. obj \<noteq> O_tcp_sock s) \<and> (\<forall> s. obj \<noteq> O_udp_sock s)"

apply (induct s, simp, drule seeds_in_init, case_tac obj, simp+)

apply (frule vd_cons, frule vt_grant_os, case_tac a)

apply (auto split:if_splits option.splits)

done

lemma Tainted_in_current:

  "\<lbrakk>obj \<in> Tainted s; valid s\<rbrakk> \<Longrightarrow> alive s obj"

apply (induct s, simp)

apply (drule seeds_in_init, case_tac obj, simp_all add:is_file_nil)

apply (frule vd_cons, frule valid_Tainted_obj, simp, frule vt_grant_os, case_tac a)

apply (auto simp:alive_simps split:if_splits option.splits t_object.splits

           intro:has_same_inode_prop2 has_same_inode_prop1 procs_of_shm_prop2 

            dest:info_shm_flow_in_procs)

done 

lemma Tainted_proc_in_current:

  "\<lbrakk>O_proc p \<in> Tainted s; valid s\<rbrakk> \<Longrightarrow> p \<in> current_procs s"

by (drule Tainted_in_current, simp+)

lemma info_flow_shm_Tainted:

  "\<lbrakk>O_proc p \<in> Tainted s; info_flow_shm s p p'; valid s\<rbrakk> \<Longrightarrow> O_proc p' \<in> Tainted s"

proof (induct s arbitrary:p p')

  case Nil

  thus ?case by (simp add:flow_shm_in_seeds)

next

  case (Cons e s)

  hence p1: "O_proc p \<in> Tainted (e # s)" and p2: "info_flow_shm (e # s) p p'" and p3: "valid (e # s)"  

    and p4: "\<And> p p'. \<lbrakk>O_proc p \<in> Tainted s; info_flow_shm s p p'\<rbrakk> \<Longrightarrow> O_proc p' \<in> Tainted s" 

    and p5: "valid s" and p6: "os_grant s e"

    by (auto dest:vd_cons intro:vd_cons vt_grant_os)

  have p4': 

    "\<And> p p' h flag. \<lbrakk>O_proc p \<in> Tainted s; (p, SHM_RDWR) \<in> procs_of_shm s h; (p', flag) \<in> procs_of_shm s h\<rbrakk> 

                \<Longrightarrow> O_proc p' \<in> Tainted s"

  show ?case

  proof (cases "self_shm s p p'")

    case True with p1 show ?thesis by simp

  next

    case False

    with p1 p2 p5 p6 p7 p8 p3 show ?thesis

    prefer 7

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

    apply (rule allI|rule impI|rule conjI)+

    apply simp

    apply (case_tac "O_proc p \<in> Tainted s", drule_tac p'=p' in p4, simp+)

    apply simp

    apply (auto simp:info_flow_shm_simps one_flow_shm_def dest:Tainted_in_current 

  intro:p4 p4' split:if_splits option.splits)

    apply (auto simp:info_flow_shm_def one_flow_shm_def)

    apply (auto simp:one_flow_shm_def intro:p4 p4' split:if_splits option.splits)

    prefer 7

    apply (simp split:if_splits option.splits)

    apply (rule allI|rule impI|rule conjI)+

    apply (auto dest:p4'   procs_of_shm_prop2 Tainted_in_current split:if_splits option.splits)[1]

    apply (erule disjE, drule_tac p = p and p' = p' in p4', simp+)

    apply (erule disjE, rule disjI2, rule disjI2, rule_tac x = h in exI, simp, rule_tac x= toflag in exI, simp)

    apply ((erule exE|erule conjE)+)

    apply (auto simp:info_flow_shm_def dest:p4'

           procs_of_shm_prop2 Tainted_in_current split:if_splits option.splits)[1]

    apply (drule_tac p = p and p' = p' in p4')

    apply (erule_tac x = ha in allE, simp)

    apply (drule_tac p = "nat1" and p' = p' in p4')

    apply (auto dest:p4'[where p = nat1 and p' = p'])

apply simp defer

apply (frule vd_cons, frule vt_grant_os, case_tac a)

apply (auto simp:info_flow_shm_def elim!:disjE)

sorry *)

  sorry

qed

lemma tainted_imp_Tainted:

  "obj \<in> tainted s \<Longrightarrow> obj \<in> Tainted s"

apply (auto intro:info_flow_shm_Tainted dest:vd_cons)

apply (case_tac e, auto split:option.splits if_splits simp:alive_simps)

done

lemma Tainted_imp_tainted:

  "\<lbrakk>obj \<in> Tainted s; valid s\<rbrakk> \<Longrightarrow> obj \<in> tainted s"

lemma tainted_in_current:

  "obj \<in> tainted s \<Longrightarrow> alive s obj"

apply (drule seeds_in_init, simp add:tobj_in_alive)

apply (erule has_same_inode_prop2, simp, simp add:vd_cons)

apply (frule vt_grant_os, simp)

apply (erule has_same_inode_prop1, simp, simp add:vd_cons)

done

lemma tainted_is_valid:

  "obj \<in> tainted s \<Longrightarrow> valid s"

by (erule tainted.induct, auto intro:valid.intros)

lemma t_remain_app:

  "\<lbrakk>obj \<in> tainted s; \<not> deleted obj (s' @ s); valid (s' @ s)\<rbrakk> 

  \<Longrightarrow> obj \<in> tainted (s' @ s)"

apply (induct s', simp)

apply (simp (no_asm) only:cons_app_simp_aux, rule t_remain)

apply (simp_all add:not_deleted_cons_D vd_cons)

apply (drule tainted_in_current, simp add:not_deleted_imp_alive_cons)

done

lemma valid_tainted_obj:

  "obj \<in> tainted s \<Longrightarrow> (\<forall> f. obj \<noteq> O_dir f) \<and> (\<forall> q. obj \<noteq> O_msgq q) \<and> (\<forall> h. obj \<noteq> O_shm h) \<and> (\<forall> p fd. obj \<noteq> O_fd p fd) \<and> (\<forall> s. obj \<noteq> O_tcp_sock s) \<and> (\<forall> s. obj \<noteq> O_udp_sock s)"

apply (erule tainted.induct)

apply (drule seeds_in_init)

by auto

lemma dir_not_tainted: "O_dir f \<in> tainted s \<Longrightarrow> False"

by (auto dest:valid_tainted_obj)

lemma msgq_not_tainted: "O_msgq q \<in> tainted s \<Longrightarrow> False"

by (auto dest:valid_tainted_obj)

lemma shm_not_tainted: "O_shm h \<in> tainted s \<Longrightarrow> False"

by (auto dest:valid_tainted_obj)

end

end