theory Enrich2

imports Main Flask Static Init_prop Valid_prop Tainted_prop Delete_prop Co2sobj_prop S2ss_prop S2ss_prop2

 Temp Enrich Proc_fd_of_file_prop

begin

context tainting_s begin

where

lemma enrich_msgq_s2ss:

  ""

thm cp2sproc_def

(* enrich s target_proc duplicated_pro *)

fun enrich_proc :: "t_state \<Rightarrow> t_process \<Rightarrow> t_process \<Rightarrow> nat \<Rightarrow> t_state"

where 

  "enrich_proc [] tp dp n = []"

| "enrich_proc (Execve p f fds # s) tp dp n = (

     if (tp = p) 

     then Execve dp f (fds \<inter> proc_file_fds s p) # Execve p f fds # (enrich_proc s tp dp n)

     else Execve p f fds # (enrich_proc s tp dp n))"

| "enrich_proc (Clone p p' fds # s) tp dp n = (

     if (tp = p') 

     then Clone p dp (fds \<inter> proc_file_fds s p) # Clone p p' fds # s

     else Clone p p' fds # (enrich_proc s tp dp n))"

| "enrich_proc (Open p f flags fd opt # s) tp dp n= (

     if (tp = p)

     then Open dp f (remove_create_flag flags) fd None # Open p f flags fd opt # (enrich_proc s tp dp n)

     else Open p f flags fd opt # (enrich_proc s tp dp n))"

| "enrich_proc (ReadFile p fd # s) tp dp n = (

     if (tp = p) 

     then ReadFile dp fd # ReadFile p fd # (enrich_proc s tp dp n)

     else ReadFile p fd # (enrich_proc s tp dp n))"

| "enrich_proc (RecvMsg p q m # s) tp dp n = (

     if (tp = p) 

     then RecvMsg dp n m # RecvMsg p q m # (enrich_msgq (enrich_proc s tp dp (n+1)) q n)

     else RecvMsg p q m # (enrich_proc s tp dp n))"

(*

| "enrich_proc (CloseFd p fd # s) tp dp = (

     if (tp = p \<and> fd \<in> proc_file_fds s p)

     then CloseFd dp fd # CloseFd p fd # (enrich_proc s tp dp)

     else CloseFd p fd # (enrich_proc s tp dp))"

*)

(*

| "enrich_proc (Attach p h flag # s) tp dp = (

     if (tp = p)

     then Attach dp h flag # Attach p h flag # (enrich_proc s tp dp)

     else Attach p h flag # (enrich_proc s tp dp))"

| "enrich_proc (Detach p h # s) tp dp = (

     if (tp = p) 

     then Detach dp h # Detach p h # (enrich_proc s tp dp)

     else Detach p h # (enrich_proc s tp dp))"

*)

(*

| "enrich_proc (Kill p p' # s) tp dp = (

     if (tp = p') then Kill p p' # s

     else Kill p p' # (enrich_proc s tp dp))"

| "enrich_proc (Exit p # s) tp dp = (

     if (tp = p) then Exit p # s

     else Exit p # (enrich_proc s tp dp))"

*)

| "enrich_proc (e # s) tp dp n = e # (enrich_proc s tp dp n)"

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

where

  "is_created_proc s p \<equiv> p \<in> current_procs s \<and> (p \<in> init_procs \<longrightarrow> died (O_proc p) s)"

definition is_created_proc':: "t_state \<Rightarrow> t_process \<Rightarrow> bool"

where

  "is_created_proc' s p \<equiv> p \<in> current_procs s \<and> p \<notin> init_procs"

lemma created_proc_clone:

  "valid (Clone p p' fds # s) \<Longrightarrow> 

   is_created_proc (Clone p p' fds # s) tp = (if (tp = p') then True else is_created_proc s tp)"

apply (drule vt_grant_os)

apply (auto simp:is_created_proc_def dest:not_all_procs_prop2)

using not_died_init_proc

by auto

lemma created_proc_exit: 

  "is_created_proc (Exit p # s) tp = (if (tp = p) then False else is_created_proc s tp)"

by (simp add:is_created_proc_def)

lemma created_proc_kill:

  "is_created_proc (Kill p p' # s) tp = (if (tp = p') then False else is_created_proc s tp)"

by (simp add:is_created_proc_def)

lemma created_proc_other:

  "\<lbrakk>\<And> p p' fds. e \<noteq> Clone p p' fds;

    \<And> p. e \<noteq> Exit p;

    \<And> p p'. e \<noteq> Kill p p'\<rbrakk> \<Longrightarrow> is_created_proc (e # s) tp = is_created_proc s tp"

by (case_tac e, auto simp:is_created_proc_def)

lemmas is_created_proc_simps = created_proc_clone created_proc_exit created_proc_kill created_proc_other

lemma no_del_died:

  "\<lbrakk>no_del_event s; died obj s\<rbrakk> \<Longrightarrow> (\<exists> p fd. obj = O_fd p fd \<or> obj = O_tcp_sock (p, fd) \<or> obj = O_udp_sock (p, fd))

  \<or> (\<exists> q m. obj = O_msg q m) "

apply (induct s)

apply simp

apply (case_tac a)

apply (auto split:option.splits)

done

lemma no_del_created_eq:

  "no_del_event s \<Longrightarrow> is_created_proc s p = is_created_proc' s p"

apply (induct s)

apply (simp add:is_created_proc_def is_created_proc'_def)

apply (case_tac a)

apply (auto simp add:is_created_proc_def is_created_proc'_def dest:no_del_died)

done

lemma enrich_proc_dup_in:

  "\<lbrakk>is_created_proc s p; p' \<notin> all_procs s; no_del_event s; valid s\<rbrakk>

   \<Longrightarrow> p' \<in> current_procs (enrich_proc s p p' i)"

apply (induct s, simp add:is_created_proc_def)

apply (frule vt_grant_os, frule vd_cons)

apply (case_tac a)

apply ( auto simp:is_created_proc_def Let_def enrich_msgq_cur_procs

  dest:not_all_procs_prop3)

sorry

lemma enrich_proc_dup_ffd:

  "\<lbrakk>file_of_proc_fd s p fd = Some f; is_created_proc s p; p' \<notin> all_procs s; valid s\<rbrakk>

   \<Longrightarrow> file_of_proc_fd (enrich_proc s p p' i) p' fd = Some f"

apply (induct s, simp add:is_created_proc_def)

apply (frule vt_grant_os, frule vd_cons)

apply (case_tac a, auto simp:is_created_proc_def proc_file_fds_def Let_def

  dest:not_all_procs_prop3 split:if_splits option.splits)

sorry

lemma enrich_proc_dup_ffd':

  "\<lbrakk>file_of_proc_fd (enrich_proc s p p' i) p' fd = Some f; is_created_proc s p; p' \<notin> all_procs s;

    no_del_event s; valid s\<rbrakk>

   \<Longrightarrow> file_of_proc_fd s p fd = Some f"

apply (induct s, simp add:is_created_proc_def)

apply (frule vt_grant_os, frule vd_cons)

apply (case_tac a, auto simp:is_created_proc_def proc_file_fds_def Let_def

  dest:not_all_procs_prop3 split:if_splits option.splits)

sorry

lemma enrich_proc_dup_ffd_eq:

  "\<lbrakk>is_created_proc s p; p' \<notin> all_procs s; no_del_event s; valid s\<rbrakk>

  \<Longrightarrow> file_of_proc_fd (enrich_proc s p p' i) p' fd = file_of_proc_fd s p fd"

apply (case_tac "file_of_proc_fd s p fd")

apply (case_tac[!] "file_of_proc_fd (enrich_proc s p p' i) p' fd")

apply (auto dest:enrich_proc_dup_ffd enrich_proc_dup_ffd')

apply (drule_tac i = i in enrich_proc_dup_ffd, simp+)

done

lemma enrich_proc_cur_msgqs:

  "\<lbrakk>valid s\<rbrakk> \<Longrightarrow> current_msgqs (enrich_proc s p p' i) = current_msgqs s \<union> {q'. q' \<ge> new_msgq s \<and> q' \<le> new_msgq s + (nums_of_recvmsg s p) - 1}"

apply (induct s, simp)

apply (auto)[1]

apply (drule new_msgq_1, simp, simp)

apply (frule vt_grant_os, frule vd_cons)

sorry

lemma enrich_proc_not_alive:

  "\<lbrakk>enrich_not_alive s (E_proc p' (new_msgq s) (new_msgq s + (nums_of_recvmsg s p) - 1)) obj; 

    is_created_proc s p; p' \<notin> all_procs s; no_del_event s; valid s\<rbrakk>

  \<Longrightarrow> enrich_not_alive (enrich_proc s p p' (new_msgq s)) (E_proc p' (new_msgq s) (new_msgq s + (nums_of_recvmsg s p) - 1)) obj"

apply (case_tac obj, simp_all)

prefer 5

apply (simp add:enrich_proc_cur_msgqs)

apply (rule impI, rule notI)

apply simp

apply (auto)[1]

defer

apply simp

apply (rule impI, rule notI)

defer

apply (subgoal_tac "new_msgq s \<noteq> 0")

apply simp

apply arith

apply (simp_all add:enrich_proc_cur_procs enrich_proc_cur_files enrich_proc_cur_inums 

  enrich_proc_cur_msgqs enrich_proc_cur_msgs enrich_proc_cur_fds)

defer

apply (rule impI, rule notI)

sorry

lemma enrich_proc_dup_fflags:

  "\<lbrakk>flags_of_proc_fd s p fd = Some flag; is_created_proc s p; p' \<notin> all_procs s; valid s\<rbrakk>

   \<Longrightarrow> flags_of_proc_fd (enrich_proc s p p') p' fd = Some (remove_create_flag flag) \<or>

       flags_of_proc_fd (enrich_proc s p p') p' fd = Some flag"

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

apply (frule vt_grant_os, frule vd_cons)

apply (case_tac a, auto simp:is_created_proc_def proc_file_fds_def is_creat_flag_def Let_def

  dest:not_all_procs_prop3 split:if_splits option.splits)

sorry

lemma enrich_proc_dup_ffds:

  "\<lbrakk>is_created_proc s p; p' \<notin> all_procs s; no_del_event s; valid s\<rbrakk>

   \<Longrightarrow> proc_file_fds (enrich_proc s p p') p' = proc_file_fds s p"

apply (auto simp:proc_file_fds_def)

apply (rule_tac x = f in exI) 

apply (erule enrich_proc_dup_ffd', simp+)

apply (rule_tac x = f in exI)

apply (erule enrich_proc_dup_ffd, simp+)

done

lemma enrich_proc_dup_ffds_eq_fds:

  "\<lbrakk>is_created_proc s p; p' \<notin> all_procs s; no_del_event s; valid s\<rbrakk>

   \<Longrightarrow> current_proc_fds (enrich_proc s p p') p' = proc_file_fds s p"

apply (induct s arbitrary:p)

apply (simp add: is_created_proc_def)

apply (frule not_all_procs_prop3)

apply (frule vd_cons, frule vt_grant_os, case_tac a)

apply (auto split:if_splits option.splits dest:proc_fd_in_fds set_mp not_all_procs_prop3 

  simp:proc_file_fds_def is_created_proc_def Let_def)

sorry

lemma enrich_proc_dup_ffds':

  "\<lbrakk>fd \<notin> current_proc_fds (enrich_proc s p p') p'; is_created_proc s p; p' \<notin> all_procs s; no_del_event s; valid s\<rbrakk>

   \<Longrightarrow> fd \<notin> proc_file_fds s p \<and> file_of_proc_fd s p fd = None"

apply (auto simp:enrich_proc_dup_ffds_eq_fds)

apply (simp add:proc_file_fds_def)

done

lemma enrich_proc_cur_inof:

  "\<lbrakk>valid s; no_del_event s\<rbrakk> \<Longrightarrow> inum_of_file (enrich_proc s p p') f = inum_of_file s f"

apply (induct s arbitrary:f)

apply simp

apply (frule vd_cons, frule vt_grant_os, frule vt_grant)

apply (case_tac a, auto)

apply (auto split:option.splits simp del:grant.simps simp add:Let_def)

sorry

lemma not_all_procs_sock:

  "\<lbrakk>p' \<notin> all_procs s; valid s\<rbrakk> \<Longrightarrow> inum_of_socket s (p', fd) = None"

apply (frule not_all_procs_prop3)

apply (case_tac "inum_of_socket s (p', fd)", simp_all)

apply (drule cn_in_curp', simp+)

done

lemma enrich_proc_cur_inos:

  "\<lbrakk>valid s; no_del_event s; p' \<notin> all_procs s\<rbrakk> 

   \<Longrightarrow> inum_of_socket (enrich_proc s p p') (tp, fd) = inum_of_socket s (tp, fd)"

apply (induct s arbitrary:tp)

apply simp

apply (frule vd_cons, frule vt_grant_os)

apply (case_tac a, auto split:option.splits simp:not_all_procs_sock Let_def)

apply (simp add:proc_file_fds_def, erule exE)

apply (case_tac "inum_of_socket s (nat1, fd)", simp_all)

apply (drule filefd_socket_conflict, simp_all add:current_sockets_def)

sorry

lemma enrich_proc_cur_inums:

  "\<lbrakk>p' \<notin> all_procs s; no_del_event s; valid s\<rbrakk>

   \<Longrightarrow> current_inode_nums (enrich_proc s p p') = current_inode_nums s"

apply (auto simp:current_inode_nums_def current_file_inums_def 

  current_sock_inums_def enrich_proc_cur_inof enrich_proc_cur_inos)

done

lemma enrich_proc_cur_itag:

  "\<lbrakk>valid s; no_del_event s; p' \<notin> all_procs s\<rbrakk> 

   \<Longrightarrow> itag_of_inum (enrich_proc s p p') i = itag_of_inum s i"

apply (induct s)

apply simp

apply (frule vd_cons, frule vt_grant_os)

apply (case_tac a, auto split:option.splits t_socket_type.splits simp:Let_def)

sorry

lemma enrich_proc_cur_tcps:

  "\<lbrakk>valid s; no_del_event s; p' \<notin> all_procs s\<rbrakk> 

   \<Longrightarrow> is_tcp_sock (enrich_proc s p p') = is_tcp_sock s"

apply (rule ext, case_tac x)

apply (auto simp add:is_tcp_sock_def enrich_proc_cur_itag enrich_proc_cur_inos

  split:option.splits t_inode_tag.splits)

done

lemma enrich_proc_cur_udps:

  "\<lbrakk>valid s; no_del_event s; p' \<notin> all_procs s\<rbrakk> 

   \<Longrightarrow> is_udp_sock (enrich_proc s p p') = is_udp_sock s"

apply (rule ext, case_tac x)

apply (auto simp add:is_udp_sock_def enrich_proc_cur_itag enrich_proc_cur_inos 

  split:option.splits t_inode_tag.splits)

done

lemma enrich_proc_cur_msgqs:

  "\<lbrakk>q \<in> current_msgqs s; valid s\<rbrakk> \<Longrightarrow> q \<in> current_msgqs (enrich_proc s p p')"

apply (induct s, simp)

apply (frule vt_grant_os, frule vd_cons)

apply (case_tac a, auto simp:Let_def)

sorry

lemma enrich_proc_cur_msgs:

  "\<lbrakk>q \<in> current_msgqs s; valid s\<rbrakk> \<Longrightarrow> msgs_of_queue (enrich_proc s p p') q = msgs_of_queue s q"

apply (induct s, simp)

apply (frule_tac p = p and p' = p' in enrich_proc_cur_msgqs, simp)

apply (frule vt_grant_os, frule vd_cons)

apply (case_tac a, auto simp:Let_def)

sorry

lemma enrich_proc_cur_procs:

  "\<lbrakk>p' \<notin> all_procs s; no_del_event s; is_created_proc s p; valid s\<rbrakk> 

   \<Longrightarrow> current_procs (enrich_proc s p p') = current_procs s \<union> {p'}"

apply (induct s, simp add:is_created_proc_def)

apply (frule vt_grant_os, frule vd_cons)

apply (case_tac a, auto simp:is_created_proc_simps Let_def)

sorry

lemma enrich_proc_cur_files:

  "\<lbrakk>valid s; no_del_event s\<rbrakk> \<Longrightarrow> current_files (enrich_proc s p p') = current_files s"

apply (auto simp:current_files_def)

apply (simp add: enrich_proc_cur_inof)+

done

lemma enrich_proc_cur_fds1:

  "\<lbrakk>p' \<notin> all_procs s; no_del_event s; is_created_proc s p; valid s; tp \<in> current_procs s\<rbrakk>

   \<Longrightarrow> current_proc_fds (enrich_proc s p p') tp = current_proc_fds s tp"

apply (induct s, simp add:is_created_proc_def)

apply (frule vt_grant_os, frule vd_cons)

apply (frule not_all_procs_prop3)

apply (case_tac a)

sorry

(*

apply (auto simp:is_created_proc_simps)

done

*)

lemma enrich_proc_cur_fds1':

  "\<lbrakk>p' \<notin> all_procs s; no_del_event s; is_created_proc s p; valid s; tp \<noteq> p'\<rbrakk>

   \<Longrightarrow> current_proc_fds (enrich_proc s p p') tp = current_proc_fds s tp"

apply (induct s, simp add:is_created_proc_def)

apply (frule vt_grant_os, frule vd_cons)

apply (frule not_all_procs_prop3) sorry  (*

apply (case_tac a)

apply (auto simp:is_created_proc_simps)

done

*)

lemma enrich_proc_cur_fds:

  "\<lbrakk>p' \<notin> all_procs s; no_del_event s; is_created_proc s p; valid s\<rbrakk>

   \<Longrightarrow> current_proc_fds (enrich_proc s p p') tp = (if (tp = p') then proc_file_fds s p else current_proc_fds s tp)"

apply (simp add:enrich_proc_cur_fds1' enrich_proc_dup_ffds_eq_fds split:if_splits)

done

lemma enrich_proc_not_alive:

  "\<lbrakk>enrich_not_alive s (E_proc p') obj; is_created_proc s p; p' \<notin> all_procs s; no_del_event s; valid s\<rbrakk>

  \<Longrightarrow> enrich_not_alive (enrich_proc s p p') (E_proc p') obj"

apply (case_tac obj)

apply (simp_all add:enrich_proc_cur_procs enrich_proc_cur_files enrich_proc_cur_inums 

  enrich_proc_cur_msgqs enrich_proc_cur_msgs enrich_proc_cur_fds)

defer

apply (rule impI, rule notI)

sorry

lemma enrich_proc_filefd:

  "\<lbrakk>file_of_proc_fd s tp fd = Some f; is_created_proc s p; p' \<notin> all_procs s; no_del_event s; valid s\<rbrakk>

  \<Longrightarrow> file_of_proc_fd (enrich_proc s p p') tp fd = Some f"

apply (induct s arbitrary:tp, simp add:is_created_proc_def)

apply (frule vt_grant_os, frule vd_cons)

apply (frule not_all_procs_prop3)

apply (case_tac a)

apply (auto simp:is_created_proc_simps dest:proc_fd_in_procs split:if_splits)

sorry

lemma enrich_proc_flagfd:

  "\<lbrakk>flags_of_proc_fd s tp fd = Some f; is_created_proc s p; p' \<notin> all_procs s; no_del_event s; valid s\<rbrakk>

  \<Longrightarrow> flags_of_proc_fd (enrich_proc s p p') tp fd = Some f"

apply (induct s arbitrary:tp, simp add:is_created_proc_def)

apply (frule vt_grant_os, frule vd_cons)

apply (frule not_all_procs_prop3)

apply (case_tac a)

apply (auto simp:is_created_proc_simps dest:proc_fd_in_procs current_fflag_has_ffd split:if_splits option.splits)

sorry

lemma enrich_proc_hungs:

  "\<lbrakk>valid s; no_del_event s\<rbrakk> \<Longrightarrow> files_hung_by_del (enrich_proc s p p') = files_hung_by_del s"

apply (induct s, simp)

apply (frule vt_grant_os, frule vd_cons)

apply (case_tac a, auto simp:files_hung_by_del.simps)

sorry

lemma enrich_proc_is_file:

  "\<lbrakk>valid s; no_del_event s; p' \<notin> all_procs s\<rbrakk> 

   \<Longrightarrow> is_file (enrich_proc s p p') = is_file s"

apply (rule ext, case_tac x)

apply (auto simp add:is_file_def enrich_proc_cur_itag enrich_proc_cur_inof

  split:option.splits t_inode_tag.splits)

done

lemma enrich_proc_is_dir:

  "\<lbrakk>valid s; no_del_event s; p' \<notin> all_procs s\<rbrakk> 

   \<Longrightarrow> is_dir (enrich_proc s p p') = is_dir s"

apply (rule ext, case_tac x)

apply (auto simp add:is_dir_def enrich_proc_cur_itag enrich_proc_cur_inof

  split:option.splits t_inode_tag.splits)

done

lemma enrich_proc_alive:

  "\<lbrakk>alive s obj; valid s; is_created_proc s p; p' \<notin> all_procs s; no_del_event s\<rbrakk>

   \<Longrightarrow> alive (enrich_proc s p p') obj"

apply (case_tac obj)

apply (simp_all add:enrich_proc_is_file enrich_proc_is_dir enrich_proc_cur_msgqs 

  enrich_proc_cur_msgs enrich_proc_cur_procs enrich_proc_cur_fds

  enrich_proc_cur_tcps enrich_proc_cur_udps)

apply (rule impI, simp)

apply (drule current_proc_fds_in_curp, simp, simp add:not_all_procs_prop3)

done

lemma enrich_proc_prop:

  "\<lbrakk>valid s; is_created_proc s p; p' \<notin> all_procs s; no_del_event s\<rbrakk>

   \<Longrightarrow> valid (enrich_proc s p p') \<and> 

       (\<forall> tp. tp \<in> current_procs s \<longrightarrow> cp2sproc (enrich_proc s p p') tp = cp2sproc s tp) \<and>

       (\<forall> f. f \<in> current_files s \<longrightarrow> cf2sfile (enrich_proc s p p') f = cf2sfile s f) \<and> 

       (\<forall> q. q \<in> current_msgqs s \<longrightarrow> cq2smsgq (enrich_proc s p p') q = cq2smsgq s q) \<and> 

       (\<forall> tp fd. fd \<in> proc_file_fds s tp \<longrightarrow> cfd2sfd (enrich_proc s p p') tp fd = cfd2sfd s tp fd) \<and>

       (cp2sproc (enrich_proc s p p') p' = cp2sproc s p) \<and>

       (\<forall> fd. fd \<in> proc_file_fds s p \<longrightarrow> cfd2sfd (enrich_proc s p p') p' fd = cfd2sfd s p fd)"

proof (induct s)

  case Nil

  thus ?case by (auto simp:is_created_proc_def)

next

  case (Cons e s)

  hence vd_cons': "valid (e # s)" and created_cons: "is_created_proc (e # s) p"

    and all_procs_cons: "p' \<notin> all_procs (e # s)" and vd: "valid s" 

    and os: "os_grant s e" and grant: "grant s e" 

    and nodel_cons: "no_del_event (e # s)"

    by (auto dest:vd_cons' vt_grant_os vt_grant)

  from all_procs_cons have all_procs: "p' \<notin> all_procs s" by (case_tac e, auto)

  from nodel_cons have nodel: "no_del_event s" by (case_tac e, auto)

  from Cons have pre: "is_created_proc s p \<Longrightarrow> valid (enrich_proc s p p') \<and>

     (\<forall>tp. tp \<in> current_procs s \<longrightarrow> cp2sproc (enrich_proc s p p') tp = cp2sproc s tp) \<and>

     (\<forall>f. f \<in> current_files s \<longrightarrow> cf2sfile (enrich_proc s p p') f = cf2sfile s f) \<and>

     (\<forall>q. q \<in> current_msgqs s \<longrightarrow> cq2smsgq (enrich_proc s p p') q = cq2smsgq s q) \<and>

     (\<forall>tp fd. fd \<in> proc_file_fds s tp \<longrightarrow> cfd2sfd (enrich_proc s p p') tp fd = cfd2sfd s tp fd) \<and>     

     (cp2sproc (enrich_proc s p p') p' = cp2sproc s p) \<and>

     (\<forall> fd. fd \<in> proc_file_fds s p \<longrightarrow> cfd2sfd (enrich_proc s p p') p' fd = cfd2sfd s p fd)"

    using vd all_procs nodel by auto

  from pre have pre_vd: "is_created_proc s p \<Longrightarrow> valid (enrich_proc s p p')" by simp

  have vd_enrich:"is_created_proc s p \<Longrightarrow> valid (e # enrich_proc s p p')" 

    apply (frule pre_vd)

    apply (erule_tac s = s and obj' = "E_proc p'" in enrich_valid_intro_cons)

    apply (simp_all add: pre nodel_cons all_procs_cons vd_cons')

    apply (clarsimp simp:enrich_proc_alive nodel all_procs vd)

    apply (rule allI, rule impI, erule enrich_proc_not_alive)

    apply (simp_all add:nodel all_procs vd enrich_proc_hungs enrich_proc_cur_msgs)

    apply ((rule allI| rule impI)+, erule enrich_proc_filefd)

    apply (simp_all add:nodel all_procs vd)

    apply ((rule allI| rule impI)+, erule enrich_proc_flagfd)

    apply (simp_all add:nodel all_procs vd)

    done  

  have vd_enrich_cons: "valid (enrich_proc (e # s) p p')"

  proof-

    have "\<And>f fds. \<lbrakk>valid (Execve p f fds # enrich_proc s p p'); is_created_proc s p; 

      valid (Execve p f fds # s); p' \<notin> all_procs s\<rbrakk>

      \<Longrightarrow> valid (Execve p' f (fds \<inter> proc_file_fds s p) # Execve p f fds # enrich_proc s p p')"

    proof-

      fix f fds

      assume a1: "valid (Execve p f fds # enrich_proc s p p')" and a2: "is_created_proc s p"

        and a3: "valid (Execve p f fds # s)" and a0: "p' \<notin> all_procs s"

      have cp2sp: "\<forall> tp. tp \<in> current_procs s \<longrightarrow> cp2sproc (enrich_proc s p p') tp = cp2sproc s tp"

        and cf2sf: "\<forall> tf. tf \<in> current_files s \<longrightarrow> cf2sfile (enrich_proc s p p') tf = cf2sfile s tf"

        and cfd2sfd: "\<forall> tp tfd. tfd \<in> proc_file_fds s tp \<longrightarrow> cfd2sfd (enrich_proc s p p') tp tfd = cfd2sfd s tp tfd"

        and dup_sp: "cp2sproc (enrich_proc s p p') p' = cp2sproc s p"

        and dup_sfd: "\<forall> fd. fd \<in> proc_file_fds s p \<longrightarrow> cfd2sfd (enrich_proc s p p') p' fd = cfd2sfd s p fd"

        using pre a2

        by auto

      show "valid (Execve p' f (fds \<inter> proc_file_fds s p) # Execve p f fds # enrich_proc s p p')"

      proof-

        from a0 a3 have a0': "p' \<noteq> p" by (auto dest!:vt_grant_os not_all_procs_prop3)

        from a3 have grant: "grant s (Execve p f fds)" and os: "os_grant s (Execve p f fds)"

          by (auto dest:vt_grant_os vt_grant simp del:os_grant.simps)

        have f_in: "is_file (enrich_proc s p p') f"

          using vd nodel os all_procs

          by (auto dest:vt_grant_os simp:enrich_proc_is_file)

        moreover have a5: "proc_file_fds s p \<subseteq> proc_file_fds (Execve p f fds # enrich_proc s p p') p'" 

          using a3 a0'

          apply (frule_tac vt_grant_os)

          apply (auto simp:proc_file_fds_def)

          apply (rule_tac x = fa in exI)

          apply (erule enrich_proc_dup_ffd)

          apply (simp_all add:vd all_procs a2)

          done

        ultimately have "os_grant (Execve p f fds # enrich_proc s p p') (Execve p' f (fds \<inter> proc_file_fds s p))"

          apply (auto simp:is_file_simps enrich_proc_dup_in a2 vd all_procs a1 enrich_proc_dup_ffds)

          done

        moreover have "grant (Execve p f fds # enrich_proc s p p') (Execve p' f (fds \<inter> proc_file_fds s p))"