(*<*)

theory Co2sobj_prop

imports Main Flask Flask_type Static Static_type Sectxt_prop Init_prop Current_files_prop Current_sockets_prop

begin

(*<*)

context tainting_s begin

(************ simpset for cf2sfile ***************)

declare get_parentfs_ctxts.simps [simp del]

  "valid (Open p f flag fd None # s) \<Longrightarrow> cf2sfile (Open p f flag fd None # s) f b = cf2sfile s f b"

apply (frule vd_cons, frule vt_grant_os)

          simp:cf2sfile_def is_file_simps is_dir_simps current_files_simps sectxt_of_obj_simps 

               get_parentfs_ctxts_simps)

lemma sroot_only:

  "cf2sfile s [] = (\<lambda> b. if b then None else Some sroot)"

by (rule ext, simp add:cf2sfile_def)

lemma cf2sfile_open_some1:

  "\<lbrakk>valid (Open p f flag fd (Some inum) # s); f' \<in> current_files s\<rbrakk>

   \<Longrightarrow> cf2sfile (Open p f flag fd (Some inum) # s) f' = cf2sfile s f'"

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

apply (frule parentf_in_current', simp+)

               get_parentfs_ctxts_simps)

done

lemma cf2sfile_open_some2:

  "\<lbrakk>valid (Open p f flag fd (Some inum) # s); is_file s f'\<rbrakk>

   \<Longrightarrow> cf2sfile (Open p f flag fd (Some inum) # s) f' True = cf2sfile s f' True"

apply (frule vd_cons, drule is_file_in_current)

by (simp add:cf2sfile_open_some1)

lemma cf2sfile_open_some3:

  "\<lbrakk>valid (Open p f flag fd (Some inum) # s); is_dir s f'\<rbrakk>

   \<Longrightarrow> cf2sfile (Open p f flag fd (Some inum) # s) f' False = cf2sfile s f' False"

apply (frule vd_cons, drule is_dir_in_current)

by (simp add:cf2sfile_open_some1)

lemma cf2sfile_open_some:

  "valid (Open p f flag fd (Some inum) # s) \<Longrightarrow> cf2sfile (Open p f flag fd (Some inum) # s) f True = (

     case (parent f) of

       Some pf \<Rightarrow> (case (sectxt_of_obj (Open p f flag fd (Some inum) # s) (O_file f), sectxt_of_obj s (O_dir pf), 

                         get_parentfs_ctxts s pf) of

                    (Some sec, Some psec, Some asecs) \<Rightarrow> Some (Created, sec, Some psec, set asecs)

                  | _ \<Rightarrow> None)

     | None \<Rightarrow> None)"

apply (frule vd_cons, frule vt_grant_os, frule noroot_events)

apply (auto split:if_splits option.splits    dest!:current_has_sec'

          simp:cf2sfile_def is_file_simps is_dir_simps current_files_simps sectxt_of_obj_simps 

               get_parentfs_ctxts_simps)

lemma cf2sfile_open:

  "valid (Open p f flag fd opt # s) \<Longrightarrow> cf2sfile (Open p f flag fd opt # s) = (\<lambda> f' b. 

     if (opt = None) then cf2sfile s f' b

     else if (f' = f \<and> b = True) 

     then (case (parent f) of

             Some pf \<Rightarrow> (case (sectxt_of_obj s (O_file f), sectxt_of_obj s (O_dir pf), 

                 get_parentfs_ctxts s pf) of

                          (Some sec, Some psec, Some asecs) \<Rightarrow>

                              Some (if (\<not> deleted (O_file f) s \<and> f \<in> init_files) 

                                    then Init f else Created, sec, Some psec, set asecs)

                        | _ \<Rightarrow> None)

           | None \<Rightarrow> None)

     else cf2sfile s f' b)"

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

apply (auto split:if_splits option.splits 

             simp:sectxt_of_obj_simps)

lemma cf2sfile_keep_path: "\<lbrakk>f \<preceq> f'; \<tau> \<in> vt rc_cs\<rbrakk> \<Longrightarrow> cf2sfile \<tau> f \<preceq> cf2sfile \<tau> f'"

apply (induct f', simp add:no_junior_def)

apply (case_tac "f = a # f'", simp add:cf2sfile.simps)

apply (drule no_junior_noteq, simp, simp add:cf2sfile.simps)

done

lemma ckp'_aux: "\<not> f \<preceq> a # f' \<Longrightarrow> \<not> f \<preceq> f'"

by (auto simp:no_junior_def)

lemma cf2sfile_keep_path': "\<lbrakk>\<not> f \<preceq> f'; \<tau> \<in> vt rc_cs; f \<in> current_files \<tau>; f' \<in> current_files \<tau>\<rbrakk> \<Longrightarrow> \<not> cf2sfile \<tau> f \<preceq> cf2sfile \<tau> f'"

apply (induct f', simp add:no_junior_def cf2sfile.simps, rule notI, drule cf2sfile_root_file, simp)

apply (case_tac "f = a # f'", simp add:cf2sfile.simps)

apply (rule notI)

apply (frule ckp'_aux, simp, frule parentf_in_current', simp+)

apply (case_tac "cf2sfile \<tau> f = cf2sfile \<tau> (a # f')", drule cf2sfile_inj, simp+)

apply (simp add:cf2sfile.simps)

by (drule no_junior_noteq, simp+)

lemma cf2sfile_keep_path'': "\<lbrakk>cf2sfile \<tau> f \<preceq> cf2sfile \<tau> f'; \<tau> \<in> vt rc_cs; f \<in> current_files \<tau>; f' \<in> current_files \<tau>\<rbrakk> \<Longrightarrow> f \<preceq> f'"

using cf2sfile_keep_path'

by (auto)

lemma cf2sfile_open_some': "\<lbrakk>f \<in> current_files \<tau>; Open p f' flags fd (Some inum) # \<tau> \<in> vt rc_cs\<rbrakk> \<Longrightarrow> cf2sfile (Open p f' flags fd (Some inum) # \<tau>) f = cf2sfile \<tau> f"

apply (frule vt_cons')

apply (drule vt_grant_os)

apply (induct f)

apply (simp add:cf2sfile.simps)+

apply (frule parentf_in_current', simp)

apply (auto simp:os_grant.simps index_of_file.simps split:option.splits)

done

lemma cf2sfile_open_some: "\<lbrakk>Open p f flags fd (Some inum) # \<tau> \<in> vt rc_cs; parent f = Some pf\<rbrakk> 

                           \<Longrightarrow> cf2sfile (Open p f flags fd (Some inum) # \<tau>) f = (SCrea (Suc (length \<tau>)) # (cf2sfile \<tau> pf))"

apply (case_tac f, simp)

apply (frule vt_grant_os)

apply (simp add:cf2sfile.simps os_grant.simps current_files_simps index_of_file.simps cf2sfile_open_some')

done

lemma cf2sfile_open_none: "cf2sfile (Open p f flags fd None # \<tau>) f' = cf2sfile \<tau> f'"

apply (induct f')

by (simp add:cf2sfile.simps index_of_file.simps split:option.splits nat.splits)+

lemma cf2sfile_open: "\<lbrakk>Open p f flags fd opt # \<tau> \<in> vt rc_cs; f' \<in> current_files (Open p f flags fd opt # \<tau>)\<rbrakk> \<Longrightarrow> 

                       cf2sfile (Open p f flags fd opt # \<tau>) f' = (

                         if (opt = None) then cf2sfile \<tau> f'

                         else if (f' = f) then (case (parent f) of

                                                  Some pf \<Rightarrow> (SCrea (Suc (length \<tau>)) # (cf2sfile \<tau> pf))

                                                | _       \<Rightarrow> [])

                              else cf2sfile \<tau> f'                                                          )"

apply (frule vt_grant_os)

by (auto simp:os_grant.simps current_files_simps intro:cf2sfile_open_none cf2sfile_open_some cf2sfile_open_some' split:if_splits option.splits)

lemma cf2sfile_mkdir_some': "\<lbrakk>Mkdir p f' inum # \<tau> \<in> vt rc_cs; f \<in> current_files \<tau>\<rbrakk> \<Longrightarrow> cf2sfile (Mkdir p f' inum # \<tau>) f = cf2sfile \<tau> f"

apply (frule vt_cons', drule vt_grant_os)

apply (induct f, (simp add:cf2sfile.simps)+)

apply (frule parentf_in_current', simp)

apply (auto simp:os_grant.simps index_of_file.simps split:option.splits)

done

lemma cf2sfile_mkdir_some: "\<lbrakk>Mkdir p f inum # \<tau> \<in> vt rc_cs; parent f = Some pf\<rbrakk>

                             \<Longrightarrow> cf2sfile (Mkdir p f inum # \<tau>) f = (SCrea (Suc (length \<tau>)) # (cf2sfile \<tau> pf))"

apply (case_tac f, simp)

apply (frule vt_grant_os)

apply (simp add:cf2sfile.simps os_grant.simps current_files_simps index_of_file.simps cf2sfile_mkdir_some')

done

lemma cf2sfile_mkdir: "\<lbrakk>Mkdir p f inum # \<tau> \<in> vt rc_cs; f' \<in> current_files (Mkdir p f inum # \<tau>)\<rbrakk> \<Longrightarrow>

                        cf2sfile (Mkdir p f inum # \<tau>) f' = (

                          if (f' = f) then (case (parent f) of

                                              Some pf \<Rightarrow> (SCrea (Suc (length \<tau>)) # (cf2sfile \<tau> pf))

                                            | _       \<Rightarrow> [])

                          else cf2sfile \<tau> f'               ) "

apply (frule vt_grant_os)

by (auto simp:os_grant.simps current_files_simps intro:cf2sfile_mkdir_some cf2sfile_mkdir_some' split:if_splits option.splits)

lemma cf2sfile_linkhard_none: "\<lbrakk>LinkHard p f\<^isub>1 f\<^isub>2 # \<tau> \<in> vt rc_cs; f \<in> current_files \<tau>\<rbrakk> \<Longrightarrow> cf2sfile (LinkHard p f\<^isub>1 f\<^isub>2 # \<tau>) f = cf2sfile \<tau> f"

apply (frule vt_cons')

apply (drule vt_grant_os)

apply (induct f)

apply (simp add:cf2sfile.simps)+

apply (frule parentf_in_current', simp)

apply (auto simp:os_grant.simps index_of_file.simps split:option.splits)

done

lemma cf2sfile_linkhard_some: 

  "\<lbrakk>LinkHard p f\<^isub>1 f\<^isub>2 # \<tau> \<in> vt rc_cs; parent f\<^isub>2 = Some pf\<^isub>2\<rbrakk> \<Longrightarrow> cf2sfile (LinkHard p f\<^isub>1 f\<^isub>2 # \<tau>) f\<^isub>2 = (SCrea (Suc (length \<tau>)) # (cf2sfile \<tau> pf\<^isub>2))"

apply (case_tac f\<^isub>2, simp)

apply (frule vt_grant_os)

apply (simp add:cf2sfile.simps os_grant.simps current_files_simps index_of_file.simps cf2sfile_linkhard_none)

done

lemma cf2sfile_linkhard: "\<lbrakk>LinkHard p f\<^isub>1 f\<^isub>2 # \<tau> \<in> vt rc_cs; f \<in> current_files (LinkHard p f\<^isub>1 f\<^isub>2 # \<tau>)\<rbrakk> \<Longrightarrow>

                           cf2sfile (LinkHard p f\<^isub>1 f\<^isub>2 # \<tau>) f = (

                             if (f = f\<^isub>2) then (case (parent f\<^isub>2) of

                                                 Some pf\<^isub>2 \<Rightarrow> SCrea (Suc (length \<tau>)) # (cf2sfile \<tau> pf\<^isub>2)

                                               | _        \<Rightarrow> [])

                             else cf2sfile \<tau> f                 )"

apply (frule vt_grant_os)

by (auto simp:os_grant.simps current_files_simps intro:cf2sfile_linkhard_none cf2sfile_linkhard_some split:if_splits option.splits)

lemma no_junior_aux: "\<not> f\<^isub>2 \<preceq> a # f \<Longrightarrow> \<not> f\<^isub>2 \<preceq> f"

by (auto simp:no_junior_def)

lemma cf2sfile_rename_aux: "\<lbrakk>Rename p f\<^isub>2 f\<^isub>3 # \<tau> \<in> vt rc_cs; f \<in> current_files \<tau>\<rbrakk> \<Longrightarrow> f \<noteq> f\<^isub>3 \<and> \<not> f\<^isub>3 \<preceq> f"

apply (frule vt_grant_os, simp add:os_grant.simps, (erule exE|erule conjE)+)

apply (rule conjI, rule notI, simp)

apply (rule notI, drule vt_cons', simp add:ancenf_in_current)

done

lemma cf2sfile_rename'1: 

  "\<lbrakk>Rename p f\<^isub>2 f\<^isub>3 # \<tau> \<in> vt rc_cs; f \<in> current_files \<tau>; \<not> (f\<^isub>2 \<preceq> f)\<rbrakk> \<Longrightarrow> cf2sfile (Rename p f\<^isub>2 f\<^isub>3 # \<tau>) f = cf2sfile \<tau> f"

apply (frule vt_cons',frule vt_grant_os,  induct f)

apply (simp add:cf2sfile.simps)

apply (frule parentf_in_current', simp, simp)

apply (frule no_junior_aux, simp)

apply (simp add:os_grant.simps, (erule exE|erule conjE)+)

apply (drule cf2sfile_rename_aux, simp, erule conjE)

apply (auto simp add:cf2sfile.simps index_of_file.simps split:option.splits nat.splits)

done

lemma cf2sfile_rename'2: 

  "\<lbrakk>Rename p f\<^isub>2 f\<^isub>3 # \<tau> \<in> vt rc_cs; \<not> (f\<^isub>3 \<preceq> f)\<rbrakk> \<Longrightarrow> cf2sfile (Rename p f\<^isub>2 f\<^isub>3 # \<tau>) f = cf2sfile \<tau> f"

apply (frule vt_cons', induct f)

by (auto simp add:index_of_file.simps cf2sfile.simps no_junior_def split:option.splits nat.splits)

lemma cf2sfile_rename1:

  "\<lbrakk>Rename p f\<^isub>2 f\<^isub>3 # \<tau> \<in> vt rc_cs; parent f\<^isub>3 = Some pf\<^isub>3\<rbrakk> \<Longrightarrow> cf2sfile (Rename p f\<^isub>2 f\<^isub>3 # \<tau>) f\<^isub>3 = SCrea (Suc (length \<tau>)) # (cf2sfile \<tau> pf\<^isub>3)"

apply (case_tac f\<^isub>3, simp add:cf2sfile.simps)

apply (auto dest!:cf2sfile_rename'2 simp add:no_junior_def cf2sfile.simps index_of_file.simps split:option.splits)

done

lemma index_of_file_rename1: "\<lbrakk>f\<^isub>2 \<preceq> f; f \<noteq> f\<^isub>2\<rbrakk> \<Longrightarrow> index_of_file (Rename p f\<^isub>2 f\<^isub>3 # \<tau>) (file_after_rename f\<^isub>2 f\<^isub>3 f) = index_of_file \<tau> f"

apply (clarsimp simp add:index_of_file.simps split:option.splits)

by (frule_tac f\<^isub>3 = f\<^isub>3 in file_renaming_prop3, simp, erule conjE, simp add:file_renaming_prop5)

lemma cf2sfile_rename2: "\<lbrakk>Rename p f\<^isub>2 f\<^isub>3 # \<tau> \<in> vt rc_cs; (file_before_rename f\<^isub>2 f\<^isub>3 f) \<in> current_files \<tau>; parent f\<^isub>3 = Some pf\<^isub>3; f\<^isub>3 \<preceq> f\<rbrakk> 

   \<Longrightarrow> cf2sfile (Rename p f\<^isub>2 f\<^isub>3 # \<tau>) f = file_after_rename (cf2sfile \<tau> f\<^isub>2) (SCrea (Suc (length \<tau>)) # (cf2sfile \<tau> pf\<^isub>3)) (cf2sfile \<tau> (file_before_rename f\<^isub>2 f\<^isub>3 f))"

apply (induct f, simp add:no_junior_def)

apply (case_tac "a # f = f\<^isub>3", simp)

apply (drule cf2sfile_rename1, simp, simp add:file_renaming_prop0' file_renaming_prop0)

apply (frule no_junior_noteq, simp, simp)

apply (frule_tac file_renaming_prop1')

apply (frule_tac f = f\<^isub>2 and \<tau> = \<tau> in cf2sfile_keep_path, simp add:vt_cons')

apply (frule_tac f\<^isub>2 = f\<^isub>2 and a = a in file_renaming_prop8')

apply (frule_tac f\<^isub>2 = f\<^isub>2 and a = a in file_renaming_prop6')

apply (frule_tac f = "file_before_rename f\<^isub>2 f\<^isub>3 (a # f)" and \<tau> = \<tau> and f\<^isub>2 = f\<^isub>2 and f\<^isub>3 = f\<^isub>3 and p = p in index_of_file_rename1, simp add:file_before_rename_def)

apply (drule_tac f\<^isub>3 = f\<^isub>3 and f\<^isub>1 = f and a = a and f\<^isub>2 = f\<^isub>2 in file_renaming_prop9', simp)

apply (drule parentf_in_current', simp add:vt_cons')

apply (simp add:cf2sfile.simps)

apply (erule file_renaming_prop6[THEN sym])

done

lemma cf2sfile_rename2': "\<lbrakk>Rename p f\<^isub>2 f\<^isub>3 # \<tau> \<in> vt rc_cs; f \<in> current_files \<tau>; parent f\<^isub>3 = Some pf\<^isub>3; f\<^isub>2 \<preceq> f\<rbrakk> 

   \<Longrightarrow> cf2sfile (Rename p f\<^isub>2 f\<^isub>3 # \<tau>) (file_after_rename f\<^isub>2 f\<^isub>3 f) = file_after_rename (cf2sfile \<tau> f\<^isub>2) (SCrea (Suc (length \<tau>)) # (cf2sfile \<tau> pf\<^isub>3)) (cf2sfile \<tau> f)"

apply (frule_tac f\<^isub>3 = f\<^isub>3 in file_renaming_prop5)

apply (frule_tac f\<^isub>3 = f\<^isub>3 in file_renaming_prop1)

apply (drule_tac f = "file_after_rename f\<^isub>2 f\<^isub>3 f" in cf2sfile_rename2, simp+)

done

lemma cf2sfile_rename3: "\<lbrakk>Rename p f\<^isub>2 f\<^isub>3 # \<tau> \<in> vt rc_cs; f \<in> current_files \<tau>; parent f\<^isub>3 = Some pf\<^isub>3\<rbrakk> 

   \<Longrightarrow> cf2sfile (Rename p f\<^isub>2 f\<^isub>3 # \<tau>) (file_after_rename f\<^isub>2 f\<^isub>3 f) = file_after_rename (cf2sfile \<tau> f\<^isub>2) (SCrea (Suc (length \<tau>)) # (cf2sfile \<tau> pf\<^isub>3)) (cf2sfile \<tau> f)"

apply (case_tac "f\<^isub>2 \<preceq> f")

apply (rule cf2sfile_rename2', simp+)

apply (frule vt_grant_os)

apply (frule_tac \<tau> = \<tau> in cf2sfile_keep_path', simp add:vt_cons', simp add:os_grant.simps, simp)

apply (simp add:file_after_rename_def)

by (rule cf2sfile_rename'1, simp+)

lemma cf2sfile_rename: "\<lbrakk>Rename p f\<^isub>2 f\<^isub>3 # \<tau> \<in> vt rc_cs; f \<in> current_files (Rename p f\<^isub>2 f\<^isub>3 # \<tau>)\<rbrakk> \<Longrightarrow>

                         cf2sfile (Rename p f\<^isub>2 f\<^isub>3 # \<tau>) f = (

                           if (f\<^isub>3 \<preceq> f) then (case (parent f\<^isub>3) of 

                                               Some pf\<^isub>3 \<Rightarrow> file_after_rename (cf2sfile \<tau> f\<^isub>2) (SCrea (Suc (length \<tau>)) # (cf2sfile \<tau> pf\<^isub>3)) (cf2sfile \<tau> (file_before_rename f\<^isub>2 f\<^isub>3 f))

                                             | _        \<Rightarrow> [])

                           else cf2sfile \<tau> f               )"

apply (frule vt_grant_os)

apply (case_tac "f = f\<^isub>3", clarsimp simp:os_grant.simps, drule cf2sfile_rename1, simp+, simp add:file_renaming_prop0' file_renaming_prop0)

apply (auto simp:os_grant.simps current_files_simps intro:cf2sfile_rename'2 cf2sfile_rename1 split:if_splits option.splits) 

apply (rule cf2sfile_rename2, simp, drule rename_renaming_decom, simp+)

apply (drule_tac f\<^isub>2 = f\<^isub>2 and f\<^isub>1 = f\<^isub>1 and f\<^isub>3 = f\<^isub>3 in file_renaming_prop5, simp+)

done

lemma cf2sfile_other: "\<lbrakk>

                        \<forall> p f flags fd opt. e \<noteq> Open p f flags fd opt;

                        \<forall> p f im. e \<noteq> Mkdir p f im;

                        \<forall> p f\<^isub>1 f\<^isub>2. e \<noteq> LinkHard p f\<^isub>1 f\<^isub>2;

                        \<forall> p f\<^isub>2 f\<^isub>3. e \<noteq> Rename p f\<^isub>2 f\<^isub>3\<rbrakk> \<Longrightarrow> cf2sfile (e # \<tau>) f' = cf2sfile \<tau> f'"

apply (induct f', simp add:cf2sfile.simps index_of_file.simps split:option.splits nat.splits)

apply (case_tac e, auto simp add:cf2sfile.simps index_of_file.simps split:option.splits nat.splits)

done

lemmas cf2sfile_nil = init_cf2sfile

lemma cf2sfile_nil': "f \<in> current_files [] \<Longrightarrow> cf2sfile [] f = map SInit f"

by (simp add:cf2sfile_nil current_files_simps)

lemmas cf2sfile_simps = cf2sfile_nil cf2sfile_nil' cf2sfile_open cf2sfile_mkdir cf2sfile_linkhard cf2sfile_rename cf2sfile_other

lemmas cf2sfile_simpss = cf2sfile_nil cf2sfile_nil' cf2sfile_open_some' cf2sfile_open_some cf2sfile_open_none cf2sfile_open cf2sfile_mkdir_some' cf2sfile_mkdir_some

                         cf2sfile_mkdir cf2sfile_linkhard_none cf2sfile_linkhard_some cf2sfile_linkhard cf2sfile_rename'1 cf2sfile_rename'2 cf2sfile_rename1

                         cf2sfile_rename2 cf2sfile_rename2' cf2sfile_rename3 cf2sfile_rename cf2sfile_other

lemma cf2sfile_open_some'_inum: "\<lbrakk>Open p f' flags fd (Some inum) # \<tau> \<in> vt rc_cs; inum_of_file \<tau> f = Some im\<rbrakk> \<Longrightarrow> cf2sfile (Open p f' flags fd (Some inum) # \<tau>) f = cf2sfile \<tau> f"

by (simp add:cf2sfile_open_some' current_files_def)

lemma cf2sfile_mkdir_some'_inum: "\<lbrakk>Mkdir p f' inum # \<tau> \<in> vt rc_cs; inum_of_file \<tau> f = Some im\<rbrakk> \<Longrightarrow> cf2sfile (Mkdir p f' inum # \<tau>) f = cf2sfile \<tau> f"

by (simp add:cf2sfile_mkdir_some' current_files_def)

lemma cf2sfile_linkhard_none_inum: "\<lbrakk>LinkHard p f\<^isub>1 f\<^isub>2 # \<tau> \<in> vt rc_cs; inum_of_file \<tau> f = Some im\<rbrakk> \<Longrightarrow> cf2sfile (LinkHard p f\<^isub>1 f\<^isub>2 # \<tau>) f = cf2sfile \<tau> f"

by (simp add:cf2sfile_linkhard_none current_files_def)

lemma cf2sfile_rename'1_inum: 

  "\<lbrakk>Rename p f\<^isub>2 f\<^isub>3 # \<tau> \<in> vt rc_cs; inum_of_file \<tau> f = Some im ; \<not> (f\<^isub>2 \<preceq> f)\<rbrakk> \<Longrightarrow> cf2sfile (Rename p f\<^isub>2 f\<^isub>3 # \<tau>) f = cf2sfile \<tau> f"

by (simp add:cf2sfile_rename'1 current_files_def)

lemma cf2sfile_rename2_inum: "\<lbrakk>Rename p f\<^isub>2 f\<^isub>3 # \<tau> \<in> vt rc_cs; inum_of_file \<tau> (file_before_rename f\<^isub>2 f\<^isub>3 f) = Some im; parent f\<^isub>3 = Some pf\<^isub>3; f\<^isub>3 \<preceq> f\<rbrakk> 

   \<Longrightarrow> cf2sfile (Rename p f\<^isub>2 f\<^isub>3 # \<tau>) f = file_after_rename (cf2sfile \<tau> f\<^isub>2) (SCrea (Suc (length \<tau>)) # (cf2sfile \<tau> pf\<^isub>3)) (cf2sfile \<tau> (file_before_rename f\<^isub>2 f\<^isub>3 f))"

by (simp add:cf2sfile_rename2 current_files_def)

lemma cf2sfile_rename2'_inum: "\<lbrakk>Rename p f\<^isub>2 f\<^isub>3 # \<tau> \<in> vt rc_cs; inum_of_file \<tau> f = Some im; parent f\<^isub>3 = Some pf\<^isub>3; f\<^isub>2 \<preceq> f\<rbrakk> 

   \<Longrightarrow> cf2sfile (Rename p f\<^isub>2 f\<^isub>3 # \<tau>) (file_after_rename f\<^isub>2 f\<^isub>3 f) = file_after_rename (cf2sfile \<tau> f\<^isub>2) (SCrea (Suc (length \<tau>)) # (cf2sfile \<tau> pf\<^isub>3)) (cf2sfile \<tau> f)"

by (simp add:cf2sfile_rename2' current_files_def)

lemma cf2sfile_rename3_inum: "\<lbrakk>Rename p f\<^isub>2 f\<^isub>3 # \<tau> \<in> vt rc_cs; inum_of_file \<tau> f = Some im; parent f\<^isub>3 = Some pf\<^isub>3\<rbrakk> 

   \<Longrightarrow> cf2sfile (Rename p f\<^isub>2 f\<^isub>3 # \<tau>) (file_after_rename f\<^isub>2 f\<^isub>3 f) = file_after_rename (cf2sfile \<tau> f\<^isub>2) (SCrea (Suc (length \<tau>)) # (cf2sfile \<tau> pf\<^isub>3)) (cf2sfile \<tau> f)"

by (simp add:cf2sfile_rename3 current_files_def)

lemma cf2sfile_nil'_inum: "inum_of_file [] f = Some im \<Longrightarrow> cf2sfile [] f = map SInit f"

by (simp add:cf2sfile_nil' current_files_def)

lemmas cf2sfile_simps_inum = cf2sfile_nil cf2sfile_nil'_inum cf2sfile_open_some'_inum cf2sfile_open_some cf2sfile_open_none cf2sfile_open cf2sfile_mkdir_some'_inum cf2sfile_mkdir_some

                         cf2sfile_mkdir cf2sfile_linkhard_none_inum cf2sfile_linkhard_some cf2sfile_linkhard cf2sfile_rename'1_inum cf2sfile_rename'2 cf2sfile_rename1

                         cf2sfile_rename2_inum cf2sfile_rename2'_inum cf2sfile_rename3_inum cf2sfile_rename cf2sfile_other

(*************** cp2sproc simpset *********************)

lemma cp2sproc_nil: "p \<in> init_processes \<Longrightarrow> cp2sproc [] p = SInit p"

by (simp add:cp2sproc_def index_of_proc.simps)

lemma cp2sproc_nil': "p \<in> current_procs [] \<Longrightarrow> cp2sproc [] p = SInit p"

by (simp add:cp2sproc_nil current_procs.simps)

lemma cp2sproc_clone: "cp2sproc (Clone p p' # \<tau>) p'' = (

                         if (p'' = p') then SCrea (Suc (length \<tau>))

                         else cp2sproc \<tau> p''           )"

by (auto simp:cp2sproc_def index_of_proc.simps d2s_aux.simps)

lemma cp2sproc_other: "\<forall> p p'. e \<noteq> Clone p p' \<Longrightarrow> cp2sproc (e # \<tau>) p'' = cp2sproc \<tau> p''"

apply (case_tac e)

by (auto simp:cp2sproc_def index_of_proc.simps d2s_aux.simps)

lemmas cp2sproc_simps = cp2sproc_nil cp2sproc_nil' cp2sproc_clone cp2sproc_other

(******************** ch2sshm simpset **************************)

lemma ch2sshm_nil: "h \<in> init_shms \<Longrightarrow> ch2sshm [] h = SInit h"

by (simp add:ch2sshm_def index_of_shm.simps)

lemma ch2sshm_nil': "h \<in> current_shms [] \<Longrightarrow> ch2sshm [] h = SInit h"

by (simp add:ch2sshm_nil current_shms.simps)

lemma ch2sshm_createshm: "ch2sshm (CreateShM p h # \<tau>) h' = (if (h' = h) then SCrea (Suc (length \<tau>)) else ch2sshm \<tau> h')"

by (simp add:ch2sshm_def index_of_shm.simps d2s_aux.simps) 

lemma ch2sshm_other: "\<forall> p h. e \<noteq> CreateShM p h \<Longrightarrow> ch2sshm (e # \<tau>) h' = ch2sshm \<tau> h'"

apply (case_tac e)

by (auto simp add:ch2sshm_def index_of_shm.simps d2s_aux.simps) 

lemmas ch2sshm_simps = ch2sshm_nil ch2sshm_nil' ch2sshm_createshm ch2sshm_other

(********************* cm2smsg simpset ***********************)

lemma cm2smsg_nil: "m \<in> init_msgs \<Longrightarrow> cm2smsg [] m = SInit m"

by (simp add:cm2smsg_def index_of_msg.simps)

lemma cm2smsg_nil': "m \<in> current_msgs [] \<Longrightarrow> cm2smsg [] m = SInit m"

by (simp add:cm2smsg_nil current_msgs.simps)

lemma cm2smsg_createmsg: "cm2smsg (CreateMsg p m # \<tau>) m' = (if (m' = m) then SCrea (Suc (length \<tau>)) else cm2smsg \<tau> m')"

by (simp add:cm2smsg_def index_of_msg.simps d2s_aux.simps)

lemma cm2smsg_other: "\<forall> p m. e \<noteq> CreateMsg p m \<Longrightarrow> cm2smsg (e # \<tau>) m' = cm2smsg \<tau> m'"

apply (case_tac e)

by (auto simp:cm2smsg_def index_of_msg.simps d2s_aux.simps)

lemmas cm2smsg_simps = cm2smsg_nil cm2smsg_nil' cm2smsg_createmsg cm2smsg_other

(********************** cfd2fd_s simpset ******************************)

lemma cfd2fd_s_nil: "fd \<in> init_fds_of_proc p \<Longrightarrow> cfd2fd_s [] p fd = SInit fd"

by (simp add:cfd2fd_s_def index_of_fd.simps)

lemma cfd2fd_s_nil': "fd \<in> current_proc_fds [] p \<Longrightarrow> cfd2fd_s [] p fd = SInit fd"

by (simp add:cfd2fd_s_nil current_proc_fds.simps)

lemma cfd2fd_s_open: "cfd2fd_s (Open p f flags fd opt # \<tau>) p' fd' = (

                        if (p = p') then (if (fd = fd') then SCrea (Suc (length \<tau>)) 

                                          else cfd2fd_s \<tau> p' fd')

                        else cfd2fd_s \<tau> p' fd'                      )"

by (simp add:cfd2fd_s_def index_of_fd.simps d2s_aux.simps)

lemma cfd2fd_s_createsock: "cfd2fd_s (CreateSock p af st fd im # \<tau>) p' fd' = (

                              if (p = p') then (if (fd = fd') then SCrea (Suc (length \<tau>))