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) = Tainted s - {O_file f}"

| "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 (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)

apply (auto intro:has_same_inode_prop2 has_same_inode_prop1)

apply 

lemma Tainted_imp_tainted:

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

apply (induct s arbitrary:obj, rule t_init, simp)

apply (frule vd_cons, frule vt_grant_os)

apply (case_tac a)

apply (auto intro!:t_init t_clone t_execve t_cfile t_read t_write t_link t_trunc t_sendmsg t_recvmsg

             split:if_splits option.splits)

pr 25

lemma tainted_imp_Tainted:

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

  "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)"