theory Delete_prop

imports Main Flask Flask_type Init_prop Alive_prop Current_files_prop

begin

context flask begin

lemma died_cons_I: "died obj s \<Longrightarrow> died obj (e # s)"

by (case_tac e, auto)

lemma not_died_cons_D: "\<not> died obj (e # s) \<Longrightarrow> \<not> died obj s" 

by (auto dest:died_cons_I)

lemma cons_app_simp_aux:

  "(a # b) @ c = a # (b @ c)" by auto

lemma not_died_app_I:

  "died obj s \<Longrightarrow> died obj (s' @ s)"

apply (induct s', simp)

by (simp add:cons_app_simp_aux died_cons_I)

lemma not_died_app_D:

  "\<not> died obj (s' @ s) \<Longrightarrow> \<not> died obj s"

apply (rule notI)

by (simp add:not_died_app_I)

lemma not_died_init_file:

  "\<lbrakk>\<not> died (O_file f) s; valid s; is_init_file f\<rbrakk> \<Longrightarrow> is_file s f"

apply (induct s, simp add:is_file_nil)

apply (frule vd_cons, frule vt_grant_os)

apply (case_tac a) prefer 6 apply (case_tac option)

apply (auto simp:is_file_simps split:option.splits)

done

lemma not_died_init_dir:

  "\<lbrakk>\<not> died (O_dir f) s; valid s; is_init_dir f\<rbrakk> \<Longrightarrow> is_dir s f"

apply (induct s, simp add:is_dir_nil)

apply (frule vd_cons, frule vt_grant_os)

apply (case_tac a) prefer 6 apply (case_tac option)

apply (auto simp:is_dir_simps split:option.splits)

done

lemma not_died_init_proc:

  "\<lbrakk>\<not> died (O_proc p) s; p \<in> init_procs\<rbrakk> \<Longrightarrow> p \<in> current_procs s"

apply (induct s, simp)

by (case_tac a, auto)

lemma current_fd_imp_current_proc:

  "\<lbrakk>fd \<in> current_proc_fds s p; valid s\<rbrakk> \<Longrightarrow> p \<in> current_procs s"

apply (induct s)

apply (simp add:init_fds_of_proc_prop1)

apply (frule vd_cons, drule vt_grant_os, case_tac a)

by (auto split:if_splits option.splits)

lemma not_died_init_fd_aux:

  "\<lbrakk>\<not> died (O_fd p fd) s; valid s; fd \<in> init_fds_of_proc p\<rbrakk> 

   \<Longrightarrow> fd \<in> current_proc_fds s p \<and> \<not> died (O_proc p) s"

apply (induct s arbitrary: p, simp)

apply (frule vd_cons, drule vt_grant_os)

apply (case_tac a, auto dest:current_fd_imp_current_proc)

done

lemma not_died_init_fd2:

  "\<lbrakk>\<not> died (O_fd p fd) s; fd \<in> init_fds_of_proc p; valid s\<rbrakk> \<Longrightarrow> \<not> died (O_proc p) s"

by (auto dest:not_died_init_fd_aux)

lemma not_died_init_fd1:

  "\<lbrakk>\<not> died (O_fd p fd) s; valid s; fd \<in> init_fds_of_proc p\<rbrakk> \<Longrightarrow> fd \<in> current_proc_fds s p"

by (auto dest:not_died_init_fd_aux)

lemma not_died_init_tcp_aux:

  "\<lbrakk>\<not> died (O_tcp_sock (p,fd)) s; valid s; is_init_tcp_sock (p,fd)\<rbrakk> 

   \<Longrightarrow> is_tcp_sock s (p,fd) \<and> \<not> died (O_proc p) s"

apply (induct s arbitrary:p, simp add:is_tcp_sock_nil)

apply (frule vd_cons, frule vt_grant_os)

apply (case_tac a) prefer 6 apply (case_tac option)

by (auto simp:is_tcp_sock_simps split:option.splits t_socket_type.splits 

         dest:is_tcp_sock_imp_curernt_proc)

lemma not_died_init_tcp1:

  "\<lbrakk>\<not> died (O_tcp_sock (p,fd)) s; valid s; is_init_tcp_sock (p,fd)\<rbrakk> 

   \<Longrightarrow> is_tcp_sock s (p,fd)"

by (auto dest:not_died_init_tcp_aux)

lemma not_died_init_tcp2:

  "\<lbrakk>\<not> died (O_tcp_sock (p,fd)) s; valid s; is_init_tcp_sock (p,fd)\<rbrakk> 

   \<Longrightarrow> \<not> died (O_proc p) s"

by (auto dest:not_died_init_tcp_aux)

lemma not_died_init_udp_aux:

  "\<lbrakk>\<not> died (O_udp_sock (p,fd)) s; valid s; is_init_udp_sock (p,fd)\<rbrakk> 

   \<Longrightarrow> is_udp_sock s (p,fd) \<and> \<not> died (O_proc p) s"

apply (induct s arbitrary:p, simp add:is_udp_sock_nil)

apply (frule vd_cons, frule vt_grant_os)

apply (case_tac a) prefer 6 apply (case_tac option)

by (auto simp:is_udp_sock_simps split:option.splits t_socket_type.splits 

         dest:is_udp_sock_imp_curernt_proc)

lemma not_died_init_udp1:

  "\<lbrakk>\<not> died (O_udp_sock (p,fd)) s; valid s; is_init_udp_sock (p,fd)\<rbrakk> 

   \<Longrightarrow> is_udp_sock s (p,fd)"

by (auto dest:not_died_init_udp_aux)

lemma not_died_init_udp2:

  "\<lbrakk>\<not> died (O_udp_sock (p,fd)) s; valid s; is_init_udp_sock (p,fd)\<rbrakk> 

   \<Longrightarrow> \<not> died (O_proc p) s"

by (auto dest:not_died_init_udp_aux)

(*

lemma not_died_init_shm:

  "\<lbrakk>\<not> died (O_shm h) s; h \<in> init_shms\<rbrakk> \<Longrightarrow> h \<in> current_shms s"

apply (induct s, simp)

by (case_tac a, auto)

*)

lemma not_died_init_msgq:

  "\<lbrakk>\<not> died (O_msgq q) s; q \<in> init_msgqs\<rbrakk> \<Longrightarrow> q \<in> current_msgqs s"

apply (induct s, simp)

by (case_tac a, auto)

lemma current_msg_imp_current_msgq:

  "\<lbrakk>m \<in> set (msgs_of_queue s q); valid s\<rbrakk> \<Longrightarrow> q \<in> current_msgqs s"

apply (induct s)

apply (simp add:init_msgs_valid)

apply (frule vd_cons, drule vt_grant_os)

apply (case_tac a, auto split:if_splits)

done

lemma not_died_init_msg:

  "\<lbrakk>\<not> died (O_msg q m) s; valid s; m \<in> set (init_msgs_of_queue q)\<rbrakk> \<Longrightarrow> m \<in> set (msgs_of_queue s q)"

apply (induct s, simp)

apply (frule vd_cons, frule vt_grant_os)

apply (case_tac a, auto dest:current_msg_imp_current_msgq)

apply (case_tac "msgs_of_queue s q", simp+)

done

lemma not_died_imp_alive: (* init_alive obj;  *)

  "\<lbrakk>\<not> died obj s;  valid s; init_alive obj\<rbrakk> \<Longrightarrow> alive s obj"

apply (case_tac obj)

apply (auto dest!: not_died_init_file not_died_init_dir not_died_init_proc

  not_died_init_msg not_died_init_fd1 not_died_init_tcp1 not_died_init_udp1 (* not_died_init_shm *)

  not_died_init_msgq 

           intro:is_file_in_current is_dir_in_current is_tcp_in_current is_udp_in_current

  current_msg_imp_current_msgq)

done

lemma not_died_cur_file_app:

  "\<lbrakk>\<not> died (O_file f) (s' @ s); valid (s' @ s); is_file s f\<rbrakk> \<Longrightarrow> is_file (s' @ s) f"

apply (induct s', simp, simp add:cons_app_simp_aux)

apply (frule vd_cons, frule vt_grant_os, simp)

apply (case_tac a) prefer 6 apply (case_tac option)

apply (auto simp:is_file_simps split:option.splits)

done

lemma not_died_cur_dir_app:

  "\<lbrakk>\<not> died (O_dir f) (s' @ s); valid (s' @ s); is_dir s f\<rbrakk> \<Longrightarrow> is_dir (s' @ s) f"

apply (induct s', simp, simp add:cons_app_simp_aux)

apply (frule vd_cons, frule vt_grant_os, simp)

apply (case_tac a) prefer 6 apply (case_tac option)

apply (auto simp:is_dir_simps split:option.splits)

done

lemma not_died_cur_proc_app:

  "\<lbrakk>\<not> died (O_proc p) (s' @ s); p \<in> current_procs s\<rbrakk> \<Longrightarrow> p \<in> current_procs (s' @ s)"

apply (induct s', simp, simp add:cons_app_simp_aux)

by (case_tac a, auto)

lemma not_died_cur_fd_app:

  "\<lbrakk>\<not> died (O_fd p fd) (s' @ s); valid (s' @ s); fd \<in> current_proc_fds s p\<rbrakk> 

   \<Longrightarrow> fd \<in> current_proc_fds (s' @ s) p"

apply (induct s' arbitrary: p, simp, simp add:cons_app_simp_aux)

apply (frule vd_cons, drule vt_grant_os)

apply (case_tac a, auto dest:current_fd_imp_current_proc)

done

lemma not_died_cur_tcp_app:

  "\<lbrakk>\<not> died (O_tcp_sock (p,fd)) (s' @ s); valid (s' @ s); is_tcp_sock s (p,fd)\<rbrakk> 

   \<Longrightarrow> is_tcp_sock (s' @ s) (p,fd)"

apply (induct s' arbitrary:p, simp, simp add:cons_app_simp_aux)

apply (frule vd_cons, frule vt_grant_os)

apply (case_tac a) prefer 6 apply (case_tac option)

by (auto simp:is_tcp_sock_simps split:option.splits t_socket_type.splits 

         dest:is_tcp_sock_imp_curernt_proc)

lemma not_died_cur_udp_app:

  "\<lbrakk>\<not> died (O_udp_sock (p,fd)) (s' @ s); valid (s' @ s); is_udp_sock s (p,fd)\<rbrakk> 

   \<Longrightarrow> is_udp_sock (s' @ s) (p,fd)"

apply (induct s' arbitrary:p, simp, simp add:cons_app_simp_aux)

apply (frule vd_cons, frule vt_grant_os)

apply (case_tac a) prefer 6 apply (case_tac option)

by (auto simp:is_udp_sock_simps split:option.splits t_socket_type.splits 

         dest:is_udp_sock_imp_curernt_proc)

(*

lemma not_died_cur_shm_app:

  "\<lbrakk>\<not> died (O_shm h) (s' @ s); h \<in> current_shms s\<rbrakk> \<Longrightarrow> h \<in> current_shms (s' @ s)"

apply (induct s', simp, simp add:cons_app_simp_aux)

by (case_tac a, auto)

*)

lemma not_died_cur_msgq_app:

  "\<lbrakk>\<not> died (O_msgq q) (s' @ s); q \<in> current_msgqs s\<rbrakk> \<Longrightarrow> q \<in> current_msgqs (s' @ s)"

apply (induct s', simp, simp add:cons_app_simp_aux)

by (case_tac a, auto)

lemma not_died_cur_msg_app:

  "\<lbrakk>\<not> died (O_msg q m) (s' @ s); valid (s' @ s); m \<in> set (msgs_of_queue s q)\<rbrakk> 

   \<Longrightarrow> m \<in> set (msgs_of_queue (s' @ s) q)"

apply (induct s', simp, simp add:cons_app_simp_aux)

apply (frule vd_cons, frule vt_grant_os)

apply (case_tac a, auto dest:current_msg_imp_current_msgq)

apply (case_tac "msgs_of_queue (s' @ s) q", simp+)

done

lemma not_died_imp_alive_app:

  "\<lbrakk>\<not> died obj (s' @ s); valid (s' @ s); alive s obj\<rbrakk> \<Longrightarrow> alive (s' @ s) obj"

apply (case_tac obj)

apply (auto dest!: not_died_cur_file_app not_died_cur_dir_app

  not_died_cur_proc_app not_died_cur_fd_app not_died_cur_tcp_app not_died_cur_msg_app

  not_died_cur_udp_app  (* not_died_cur_shm_app *) not_died_cur_msgq_app

           intro:is_file_in_current is_dir_in_current is_tcp_in_current is_udp_in_current 

  current_msg_imp_current_msgq)

done

lemma not_died_imp_alive_cons:

  "\<lbrakk>\<not> died obj (e # s); valid (e # s); alive s obj\<rbrakk> \<Longrightarrow> alive (e # s) obj"

using not_died_imp_alive_app[where s = s and s' = "[e]" and obj = obj] 

by auto

end

context tainting begin

lemma deleted_died:

  "appropriate obj \<Longrightarrow> died obj s = (deleted obj s \<or> exited obj s)"

apply (induct s)

apply (simp, case_tac obj, simp+)

apply (case_tac a, case_tac [!] obj, auto)

done

end

(*

lemma nodel_imp_un_died:

  "no_del_event s \<Longrightarrow> \<not> died obj s"

by (induct s, simp, case_tac a,auto)

lemma nodel_exists_remains:

  "\<lbrakk>no_del_event (s'@s); exists s obj\<rbrakk> \<Longrightarrow> exists (s'@s) obj"

apply (drule_tac obj = obj in nodel_imp_un_died)

by (simp add:not_died_imp_exists')

lemma nodel_imp_exists:

  "\<lbrakk>no_del_event s; exists [] obj\<rbrakk> \<Longrightarrow> exists s obj"

apply (drule_tac obj = obj in nodel_imp_un_died)

by (simp add:not_died_imp_exists)

lemma no_del_event_cons_D:

  "no_del_event (e # s) \<Longrightarrow> no_del_event s"

by (case_tac e, auto)

*)

end