(*<*)

theory Current_prop

imports Main Flask_type Flask My_list_prefix Init_prop Valid_prop Delete_prop

begin

(*>*)

context flask begin

lemma procs_of_shm_prop1: "\<lbrakk> p_flag \<in> procs_of_shm s h; valid s\<rbrakk> \<Longrightarrow> h \<in> current_shms s"

apply (induct s arbitrary:p_flag)

apply (case_tac p_flag, simp, drule init_procs_has_shm, simp)

apply (frule vd_cons, frule vt_grant_os)

apply (case_tac a, auto split:if_splits option.splits)

done

lemma procs_of_shm_prop2: "\<lbrakk>(p, flag) \<in> procs_of_shm s h; valid s\<rbrakk> \<Longrightarrow> p \<in> current_procs s"

apply (induct s arbitrary:p flag)

apply (simp, drule init_procs_has_shm, simp)

apply (frule vd_cons, frule vt_grant_os)

apply (case_tac a, auto split:if_splits option.splits)

done

lemma procs_of_shm_prop3: "\<lbrakk>(p, flag) \<in> procs_of_shm s h; (p, flag') \<in> procs_of_shm s h; valid s\<rbrakk>

  \<Longrightarrow> flag = flag'"

apply (induct s arbitrary:p flag flag')

apply (simp, drule_tac flag = flag in init_procs_has_shm, drule_tac flag = flag' in init_procs_has_shm, simp)

apply (frule vd_cons, frule vt_grant_os)

apply (case_tac a, auto split:if_splits option.splits dest:procs_of_shm_prop2)

done

lemma procs_of_shm_prop4: "\<lbrakk>(p, flag) \<in> procs_of_shm s h; valid s\<rbrakk> \<Longrightarrow> flag_of_proc_shm s p h = Some flag"

apply (induct s arbitrary:p flag)

apply (simp, drule init_procs_has_shm, simp)

apply (frule vd_cons, frule vt_grant_os)

apply (case_tac a, auto split:if_splits option.splits dest:procs_of_shm_prop2)

done

lemma procs_of_shm_prop4':

  "\<lbrakk>flag_of_proc_shm s p h = None; valid s\<rbrakk> \<Longrightarrow> \<forall> flag. (p, flag) \<notin> procs_of_shm s h"

by (auto dest:procs_of_shm_prop4)

lemma not_init_intro_proc:

  "\<lbrakk>p \<notin> current_procs s; valid s\<rbrakk> \<Longrightarrow> deleted (O_proc p) s \<or> p \<notin> init_procs"

using not_deleted_init_proc by auto

lemma not_init_intro_proc':

  "\<lbrakk>p \<notin> current_procs s; valid s\<rbrakk> \<Longrightarrow> \<not> (\<not> deleted (O_proc p) s \<and> p \<in> init_procs)"

using not_deleted_init_proc by auto

lemma info_shm_flow_in_procs:

  "\<lbrakk>info_flow_shm s p p'; valid s\<rbrakk> \<Longrightarrow> p \<in> current_procs s \<and> p' \<in> current_procs s"

(*********** simpset for info_flow_shm **************)

lemma info_flow_shm_prop1: 

  "\<lbrakk>info_flow_shm s p p'; p \<noteq> p'; valid s\<rbrakk> 

   \<Longrightarrow> \<exists> h h' flag. (p, SHM_RDWR) \<in> procs_of_shm s h \<and> (p', flag) \<in> procs_of_shm s h'"

by (induct rule: info_flow_shm.induct, auto)

lemma info_flow_shm_cases:

  "\<lbrakk>info_flow_shm \<tau> pa pb; \<And>p s. \<lbrakk>s = \<tau> ; pa = p; pb = p; p \<in> current_procs s\<rbrakk> \<Longrightarrow> P;

  \<And>s p p' h p'' flag. \<lbrakk>s = \<tau>; pa = p; pb = p''; info_flow_shm s p p'; (p', SHM_RDWR) \<in> procs_of_shm s h;

                       (p'', flag) \<in> procs_of_shm s h\<rbrakk>\<Longrightarrow> P\<rbrakk>

  \<Longrightarrow> P"

by (erule info_flow_shm.cases, auto)

definition one_flow_shm :: "t_state \<Rightarrow> t_process \<Rightarrow> t_process \<Rightarrow> bool"

where

  "one_flow_shm s p p' \<equiv> p \<noteq> p' \<and> (\<exists> h flag. (p, SHM_RDWR) \<in> procs_of_shm s h \<and> (p', flag) \<in> procs_of_shm s h)"

inductive flows_shm :: "t_state \<Rightarrow> t_process \<Rightarrow> t_process \<Rightarrow> bool"

where

  "p \<in> current_procs s \<Longrightarrow> flows_shm s p p"

| "\<lbrakk>flows_shm s p p'; one_flow_shm s p' p''\<rbrakk> \<Longrightarrow> flows_shm s p p''"

definition attached_procs :: "t_state \<Rightarrow> t_shm \<Rightarrow> t_process set"

where

  "attached_procs s h \<equiv> {p. \<exists> flag. (p, flag) \<in> procs_of_shm s h}"

definition flowed_procs:: "t_state \<Rightarrow> t_shm \<Rightarrow> t_process set"

where

  "flowed_procs s h \<equiv> {p'. \<exists> p \<in> attached_procs s h. flows_shm s p p'}"

fun Info_flow_shm :: "t_state \<Rightarrow> t_process \<Rightarrow> t_process set"

where

  "Info_flow_shm [] = (\<lambda> p. {p'. flows_shm [] p p'})"

lemma info_flow_shm_attach:

  "valid (Attach p h flag # s) \<Longrightarrow> info_flow_shm (Attach p h flag # s) = (\<lambda> pa pb. (info_flow_shm s pa pb) \<or> 

     (if (pa = p) 

      then (if (flag = SHM_RDWR) 

            then (\<exists> flag. (pb, flag) \<in> procs_of_shm s h)

            else (pb = p)) 

      else (if (pb = p) 

            then (pa, SHM_RDWR) \<in> procs_of_shm s h

            else info_flow_shm s pa pb)) )"

apply (frule vd_cons, frule vt_grant_os, rule ext, rule ext)

apply (case_tac "info_flow_shm s pa pb", simp)

thm info_flow_shm.cases

apply (auto split:if_splits intro:info_flow_shm.intros elim:info_flow_shm_cases)

apply (erule info_flow_shm_cases, simp, simp split:if_splits)

apply (rule_tac p = pa and p' = p' in info_flow_shm.intros(2), simp+)

apply (rule notI, erule info_flow_shm.cases, simp+)

pr 5

lemma has_same_inode_in_current:

  "\<lbrakk>has_same_inode s f f'; valid s\<rbrakk> \<Longrightarrow> f \<in> current_files s \<and> f' \<in> current_files s"

by (auto simp add:has_same_inode_def current_files_def)

lemma has_same_inode_prop1:

  "\<lbrakk>has_same_inode s f f'; is_file s f; valid s\<rbrakk> \<Longrightarrow> is_file s f'"

by (auto simp:has_same_inode_def is_file_def)

lemma has_same_inode_prop1':

  "\<lbrakk>has_same_inode s f f'; is_file s f'; valid s\<rbrakk> \<Longrightarrow> is_file s f"

by (auto simp:has_same_inode_def is_file_def)

lemma has_same_inode_prop2:

  "\<lbrakk>has_same_inode s f f'; file_of_proc_fd s p fd = Some f; valid s\<rbrakk> \<Longrightarrow> is_file s f'"

apply (drule has_same_inode_prop1)

apply (simp add:file_of_pfd_is_file, simp+)

done

lemma has_same_inode_prop2':

  "\<lbrakk>has_same_inode s f f'; file_of_proc_fd s p fd = Some f'; valid s\<rbrakk> \<Longrightarrow> is_file s f"

apply (drule has_same_inode_prop1')

apply (simp add:file_of_pfd_is_file, simp+)

done

lemma tobj_in_init_alive:

  "tobj_in_init obj \<Longrightarrow> init_alive obj"

by (case_tac obj, auto)

lemma tobj_in_alive:

  "tobj_in_init obj \<Longrightarrow> alive [] obj"

by (case_tac obj, auto simp:is_file_nil)

end

end