(*<*)

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_prop2':

  "\<lbrakk>p \<notin> current_procs s; valid s\<rbrakk> \<Longrightarrow> \<forall> flag h. (p, flag) \<notin> procs_of_shm s h"

by (auto dest:procs_of_shm_prop2)

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"

by (auto intro:procs_of_shm_prop2 simp:info_flow_shm_def one_flow_shm_def)

lemma flag_of_proc_shm_prop1:

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

apply (induct s arbitrary:p flag)

apply (simp, drule init_shmflag_has_proc, 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 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 same_inode_files_def 

  is_file_def split:if_splits option.splits)

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 same_inode_files_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 same_inode_files_def 

 split:if_splits option.splits)

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