1 (*<*)

     2 theory Current_prop

     3 imports Main Flask_type Flask My_list_prefix Init_prop Valid_prop Delete_prop

     4 begin

     5 (*>*)

     6 

     7 context flask begin

     8 

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

    10 apply (induct s arbitrary:p_flag)

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

    12 apply (frule vd_cons, frule vt_grant_os)

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

    14 done

    15 

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

    17 apply (induct s arbitrary:p flag)

    18 apply (simp, drule init_procs_has_shm, simp)

    19 apply (frule vd_cons, frule vt_grant_os)

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

    21 done

    22 

    23 lemma procs_of_shm_prop2':

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

    25 by (auto dest:procs_of_shm_prop2)

    26 

    27 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>

    28   \<Longrightarrow> flag = flag'"

    29 apply (induct s arbitrary:p flag flag')

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

    31 apply (frule vd_cons, frule vt_grant_os)

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

    33 done

    34 

    35 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"

    36 apply (induct s arbitrary:p flag)

    37 apply (simp, drule init_procs_has_shm, simp)

    38 apply (frule vd_cons, frule vt_grant_os)

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

    40 done

    41 

    42 lemma procs_of_shm_prop4':

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

    44 by (auto dest:procs_of_shm_prop4)

    45 

    46 lemma not_init_intro_proc:

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

    48 using not_deleted_init_proc by auto

    49 

    50 lemma not_init_intro_proc':

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

    52 using not_deleted_init_proc by auto

    53 

    54 lemma info_shm_flow_in_procs:

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

    56 by (auto intro:procs_of_shm_prop2 simp:info_flow_shm_def one_flow_shm_def)

    57 

    58 lemma flag_of_proc_shm_prop1:

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

    60 apply (induct s arbitrary:p flag)

    61 apply (simp, drule init_shmflag_has_proc, simp)

    62 apply (frule vd_cons, frule vt_grant_os)

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

    64 done

    65 

    66 

    67 

    68 lemma has_same_inode_in_current:

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

    70 by (auto simp add:has_same_inode_def current_files_def same_inode_files_def 

    71   is_file_def split:if_splits option.splits)

    72 

    73 

    74 lemma has_same_inode_prop1:

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

    76 by (auto simp:has_same_inode_def same_inode_files_def is_file_def)

    77 

    78 lemma has_same_inode_prop1':

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

    80 by (auto simp:has_same_inode_def is_file_def same_inode_files_def 

    81  split:if_splits option.splits)

    82 

    83 lemma has_same_inode_prop2:

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

    85 apply (drule has_same_inode_prop1)

    86 apply (simp add:file_of_pfd_is_file, simp+)

    87 done

    88 

    89 lemma has_same_inode_prop2':

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

    91 apply (drule has_same_inode_prop1')

    92 apply (simp add:file_of_pfd_is_file, simp+)

    93 done

    94 

    95 lemma tobj_in_init_alive:

    96   "tobj_in_init obj \<Longrightarrow> init_alive obj"

    97 by (case_tac obj, auto)

    98 

    99 lemma tobj_in_alive:

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

   101 by (case_tac obj, auto simp:is_file_nil)

   102 

   103 end

   104 

   105 end