theory Enrich

imports Main Flask Static Init_prop Valid_prop Tainted_prop Delete_prop Co2sobj_prop S2ss_prop S2ss_prop2

 Temp

begin

datatype t_enrich_obj = 

  E_proc "t_process"

| E_file "t_file"

| E_fd   "t_process" "t_fd"

| E_inum "nat"

| E_msgq "t_msgq"

| E_msg  "t_msgq"    "t_msg"

context tainting_s begin

(* enrich s target_proc duplicated_pro *)

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

where 

  "enrich_proc [] tp dp = []"

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

     if (tp = p) 

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

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

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

     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))"

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

     if (tp = p)

     then Open dp f (remove_create_flag flags) fd opt # Open p f flags fd opt # (enrich_proc s tp dp)

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

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

     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 = e # (enrich_proc s tp dp)"

lemma enrich_search_check:

  assumes grant: "search_check s (up, rp, tp) f"

  and cf2sf: "\<forall> f. f \<in> current_files s \<longrightarrow> cf2sfile s' f = cf2sfile s f"

  and vd: "valid s" and f_in: "is_file s f"  and f_in': "is_file s' f"

  and sec: "sectxt_of_obj s' (O_file f) = sectxt_of_obj s (O_file f)"

  shows "search_check s' (up, rp, tp) f"

proof (cases f)

  case Nil

  with f_in vd have "False" 

    by (auto dest:root_is_dir') 

  thus ?thesis by simp

next

  case (Cons n pf)

  from vd f_in obtain sf where sf: "cf2sfile s f = Some sf"

    apply (drule_tac is_file_in_current, drule_tac current_file_has_sfile, simp)

    apply (erule exE, simp)

    done

  then obtain psfs where psfs: "get_parentfs_ctxts s pf = Some psfs" using Cons

    by (auto simp:cf2sfile_def split:option.splits if_splits)

  from sf cf2sf f_in have sf': "cf2sfile s' f = Some sf" by (auto dest:is_file_in_current)

  then obtain psfs' where psfs': "get_parentfs_ctxts s' pf = Some psfs'"using Cons

    by (auto simp:cf2sfile_def split:option.splits if_splits)

  with sf sf' psfs have psfs_eq: "set psfs' = set psfs" using Cons f_in f_in'

    apply (simp add:cf2sfile_def split:option.splits)

    apply (case_tac sf, simp)

    done

  show ?thesis using grant f_in f_in' psfs psfs' psfs_eq sec

    apply (simp add:Cons split:option.splits)

    by (case_tac a, simp)

qed

lemma enrich_search_check':

  assumes grant: "search_check s (up, rp, tp) f"

  and cf2sf: "\<forall> f. f \<in> current_files s \<longrightarrow> cf2sfile s' f = cf2sfile s f"

  and vd: "valid s" and vd': "valid s'" and f_in: "is_dir s f"  and f_in': "is_dir s' f"

  and sec: "sectxt_of_obj s' (O_dir f) = sectxt_of_obj s (O_dir f)"

  shows "search_check s' (up, rp, tp) f"

proof (cases f)

  case Nil

  have "sectxt_of_obj s' (O_dir []) = sectxt_of_obj s (O_dir [])"

    using cf2sf 

    apply (erule_tac x = "[]" in allE)

    by (auto simp:cf2sfile_def root_sec_remains vd vd')

  thus ?thesis using grant Nil 

    by auto

next

  case (Cons n pf)

  from vd f_in obtain sf where sf: "cf2sfile s f = Some sf"

    apply (drule_tac is_dir_in_current, drule_tac current_file_has_sfile, simp)

    apply (erule exE, simp)

    done

  then obtain psfs where psfs: "get_parentfs_ctxts s pf = Some psfs" using Cons

    by (auto simp:cf2sfile_def split:option.splits if_splits)

  from sf cf2sf f_in have sf': "cf2sfile s' f = Some sf" by (auto dest:is_dir_in_current)

  then obtain psfs' where psfs': "get_parentfs_ctxts s' pf = Some psfs'"using Cons

    by (auto simp:cf2sfile_def split:option.splits if_splits)

  with sf sf' psfs have psfs_eq: "set psfs' = set psfs" using Cons f_in f_in'

    apply (drule_tac is_dir_not_file)

    apply (drule is_dir_not_file)    

    apply (simp add:cf2sfile_def split:option.splits)

    apply (case_tac sf, simp)

    done

  show ?thesis using grant f_in f_in' psfs psfs' psfs_eq sec

    apply (drule_tac is_dir_not_file)

    apply (drule_tac is_dir_not_file)

    apply (simp add:Cons split:option.splits)

    by (case_tac a, simp)

qed

lemma proc_filefd_has_sfd: "\<lbrakk>fd \<in> proc_file_fds s p; valid s\<rbrakk> \<Longrightarrow> \<exists> sfd. cfd2sfd s p fd = Some sfd"

apply (simp add:proc_file_fds_def)

apply (auto dest: current_filefd_has_sfd)

done

lemma enrich_inherit_fds_check:

  assumes grant: "inherit_fds_check s (up, nr, nt) p fds"  and vd: "valid s"

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

  and fd_in: "fds \<subseteq> proc_file_fds s p" and fd_in': "fds \<subseteq> proc_file_fds s' p"

  shows "inherit_fds_check s' (up, nr, nt) p fds"

proof-

  have "\<And> fd. fd \<in> fds \<Longrightarrow> sectxt_of_obj s' (O_fd p fd) = sectxt_of_obj s (O_fd p fd)"

  proof-

    fix fd

    assume fd_in_fds: "fd \<in> fds"

    hence fd_in_cfds: "fd \<in> proc_file_fds s p" 

      and fd_in_cfds': "fd \<in> proc_file_fds s' p" 

      using fd_in fd_in' by auto

    with cfd2sfd

    have cfd_eq: "cfd2sfd s' p fd = cfd2sfd s p fd" by auto

    from fd_in_cfds obtain f where ffd: "file_of_proc_fd s p fd = Some f"

      by (auto simp:proc_file_fds_def)

    moreover have "flags_of_proc_fd s p fd \<noteq> None"

      using ffd vd by (auto dest:current_filefd_has_flags)

    moreover have "sectxt_of_obj s (O_fd p fd) \<noteq> None"

      using fd_in_cfds vd

      apply (rule_tac notI)

      by (auto dest!:current_has_sec' file_fds_subset_pfds[where p = p] intro:vd)

    moreover have "cf2sfile s f \<noteq> None"

      apply (rule notI)

      apply (drule current_file_has_sfile')

      using ffd

      by (auto simp:vd is_file_in_current dest:file_of_pfd_is_file)

    ultimately show "sectxt_of_obj s' (O_fd p fd) = sectxt_of_obj s (O_fd p fd)"

      using cfd_eq

      by (auto simp:cfd2sfd_def split:option.splits)

  qed

  hence "sectxts_of_fds s' p fds = sectxts_of_fds s p fds"

    by (simp add:sectxts_of_fds_def)

  thus ?thesis using grant

    by (simp add:inherit_fds_check_def)

qed

lemma not_all_procs_cons:

  "p \<notin> all_procs (e # s) \<Longrightarrow> p \<notin> all_procs s"

by (case_tac e, auto)

lemma not_all_procs_prop:

  "\<lbrakk>p' \<notin> all_procs s; p \<in> current_procs s; valid s\<rbrakk> \<Longrightarrow> p' \<noteq> p"

apply (induct s, rule notI, simp)

apply (frule vt_grant_os, frule vd_cons, frule not_all_procs_cons, simp, rule notI)

apply (case_tac a, auto)

done

fun enrich_not_alive :: "t_state \<Rightarrow> t_enrich_obj \<Rightarrow> bool"

where

  "enrich_not_alive s (E_file f) = (f \<notin> current_files s)"

| "enrich_not_alive s (E_proc p) = (p \<notin> current_procs s)"

| "enrich_not_alive s (E_fd p fd) = (p \<in> current_procs s \<longrightarrow> fd \<notin> current_proc_fds s p)"

| "enrich_not_alive s (E_msgq q) = (q \<notin> current_msgqs s)"

| "enrich_not_alive s (E_inum i) = (i \<notin> current_inode_nums s)"

| "enrich_not_alive s (E_msg q m) = (q \<in> current_msgqs s \<longrightarrow> m \<notin> set (msgs_of_queue s q))"

lemma file_has_parent: "\<lbrakk>is_file s f; valid s\<rbrakk> \<Longrightarrow> \<exists> pf. is_dir s pf \<and> parent f = Some pf"

apply (case_tac f)

apply (simp, drule root_is_dir', simp+)

apply (simp add:parentf_is_dir_prop2)

done

lemma enrich_valid_intro_cons:

  assumes vs': "valid s'"

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

    and alive: "\<forall> obj. alive s obj \<longrightarrow> alive s' obj"

    and alive': "\<forall> obj. enrich_not_alive s obj \<longrightarrow> enrich_not_alive s' obj"

    and hungs: "files_hung_by_del s' = files_hung_by_del s"

    and cp2sp: "\<forall> p. p \<in> current_procs s \<longrightarrow> cp2sproc s' p = cp2sproc s p"

    and cf2sf: "\<forall> f. f \<in> current_files s \<longrightarrow> cf2sfile s' f = cf2sfile s f"

    and cq2sq: "\<forall> q. q \<in> current_msgqs s \<longrightarrow> cq2smsgq s' q = cq2smsgq s q"

    and ffd_remain: "\<forall> p fd f. file_of_proc_fd s p fd = Some f \<longrightarrow> file_of_proc_fd s' p fd = Some f"

    and fflags_remain: "\<forall> p fd flags. flags_of_proc_fd s p fd = Some flags \<longrightarrow> flags_of_proc_fd s' p fd = Some flags"

    and sms_remain: "\<forall> q. msgs_of_queue s' q = msgs_of_queue s q"

   (* and empty_remain: "\<forall> f. dir_is_empty s f \<longrightarrow> dir_is_empty s' f" *)

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

  shows "valid (e # s')"

proof (cases e)

  case (Execve p f fds)

  have p_in: "p \<in> current_procs s'" using os alive

    apply (erule_tac x = "O_proc p" in allE)

    by (auto simp:Execve)

  have f_in: "is_file s' f" using os alive

    apply (erule_tac x = "O_file f" in allE)

    by (auto simp:Execve)

  have fd_in: "fds \<subseteq> proc_file_fds s' p" using os alive ffd_remain

    by (auto simp:Execve proc_file_fds_def)

  have "os_grant s' e" using p_in f_in fd_in by (simp add:Execve)

  moreover have "grant s' e"

  proof-

    from grant obtain up rp tp uf rf tf 

      where p1: "sectxt_of_obj s (O_proc p) = Some (up, rp, tp)"

      and p2: "sectxt_of_obj s (O_file f) = Some (uf, rf, tf)" 

      by (simp add:Execve split:option.splits, blast)

    with grant obtain pu nr nt where p3: "npctxt_execve (up, rp, tp) (uf, rf, tf) = Some (pu, nr, nt)"

      by (simp add:Execve split:option.splits del:npctxt_execve.simps, blast)

    from p1 have p1': "sectxt_of_obj s' (O_proc p) = Some (up, rp, tp)"

      using os cp2sp

      apply (erule_tac x = p in allE)

      by (auto simp:Execve co2sobj.simps cp2sproc_def split:option.splits)

    from os have f_in': "is_file s f" by (simp add:Execve)

    from vd os have "\<exists> sf. cf2sfile s f = Some sf"

      by (auto dest!:is_file_in_current current_file_has_sfile simp:Execve)

    hence p2': "sectxt_of_obj s' (O_file f) = Some (uf, rf, tf)" using f_in f_in' p2 cf2sf

      apply (erule_tac x = f in allE)

      apply (auto dest:is_file_in_current simp:cf2sfile_def split:option.splits)

      apply (case_tac f, simp)

      apply (drule_tac s = s in root_is_dir', simp add:vd, simp+)

      done

    have "inherit_fds_check s' (pu, nr, nt) p fds"

    proof-

      have "fds \<subseteq> proc_file_fds s' p" using os ffd_remain Execve

        by (auto simp:proc_file_fds_def)

      thus ?thesis using Execve grant vd cfd2sfd p1 p2 p3 os

        apply (rule_tac s = s in enrich_inherit_fds_check)

        by (simp_all split:option.splits)

    qed

    moreover have "search_check s' (pu, rp, tp) f"

      using p1 p2 p2' vd cf2sf f_in' grant Execve p3 f_in

      apply (rule_tac s = s in enrich_search_check)

      by (simp_all split:option.splits)

    ultimately show ?thesis using p1' p2' p3

      apply (simp add:Execve split:option.splits) 

      using grant Execve p1 p2

      by (simp add:Execve grant p1 p2)

  qed

  ultimately show ?thesis using vs'

    by (erule_tac valid.intros(2), simp+)

next

  case (Clone p p' fds)

  have p_in: "p \<in> current_procs s'" using os alive

    apply (erule_tac x = "O_proc p" in allE)

    by (auto simp:Clone)

  have p'_not_in: "p' \<notin> current_procs s'" using os alive'

    apply (erule_tac x = "E_proc p'" in allE)

    by (auto simp:Clone)

  have fd_in: "fds \<subseteq> proc_file_fds s' p" using os alive ffd_remain

    by (auto simp:Clone proc_file_fds_def)

  have "os_grant s' e" using p_in p'_not_in fd_in by (simp add:Clone)

  moreover have "grant s' e" 

  proof-

    from grant obtain up rp tp 

      where p1: "sectxt_of_obj s (O_proc p) = Some (up, rp, tp)"

      apply (simp add:Clone split:option.splits)

      by (case_tac a, auto)

    from p1 have p1': "sectxt_of_obj s' (O_proc p) = Some (up, rp, tp)"

      using os cp2sp

      apply (erule_tac x = p in allE)

      by (auto simp:Clone co2sobj.simps cp2sproc_def split:option.splits)

    have p2: "inherit_fds_check s' (up, rp, tp) p fds"

    proof-

      have "fds \<subseteq> proc_file_fds s' p" using os ffd_remain Clone

        by (auto simp:proc_file_fds_def)

      thus ?thesis using Clone grant vd cfd2sfd p1 os

        apply (rule_tac s = s in enrich_inherit_fds_check)

        by (simp_all split:option.splits)

    qed

    show ?thesis using p1 p2 p1' grant

      by (simp add:Clone)

  qed

  ultimately show ?thesis using vs'

    by (erule_tac valid.intros(2), simp+)

next

  case (Kill p p')

  have p_in: "p \<in> current_procs s'" using os alive

    apply (erule_tac x = "O_proc p" in allE)

    by (auto simp:Kill)

  have p'_in: "p' \<in> current_procs s'" using os alive

    apply (erule_tac x = "O_proc p'" in allE)

    by (auto simp:Kill)

  have "os_grant s' e" using p_in p'_in by (simp add:Kill)

  moreover have "grant s' e" 

  proof-

    from grant obtain up rp tp up' rp' tp'

      where p1: "sectxt_of_obj s (O_proc p) = Some (up, rp, tp)"

      and p'1: "sectxt_of_obj s (O_proc p') = Some (up', rp', tp')"

      apply (simp add:Kill split:option.splits)

      by (case_tac a, case_tac aa, blast)

    from p1 have p1': "sectxt_of_obj s' (O_proc p) = Some (up, rp, tp)"

      using os cp2sp

      apply (erule_tac x = p in allE)

      by (auto simp:Kill co2sobj.simps cp2sproc_def split:option.splits)

    from p'1 have p'1': "sectxt_of_obj s' (O_proc p') = Some (up', rp', tp')"

      using os cp2sp

      apply (erule_tac x = p' in allE)

      by (auto simp:Kill co2sobj.simps cp2sproc_def split:option.splits)

    show ?thesis using p1 p'1 p1' p'1' grant

      by (simp add:Kill)

  qed

  ultimately show ?thesis using vs'

    by (erule_tac valid.intros(2), simp+)

next

  case (Ptrace p p')

  have p_in: "p \<in> current_procs s'" using os alive

    apply (erule_tac x = "O_proc p" in allE)

    by (auto simp:Ptrace)

  have p'_in: "p' \<in> current_procs s'" using os alive

    apply (erule_tac x = "O_proc p'" in allE)

    by (auto simp:Ptrace)

  have "os_grant s' e" using p_in p'_in by (simp add:Ptrace)

  moreover have "grant s' e" 

  proof-

    from grant obtain up rp tp up' rp' tp'

      where p1: "sectxt_of_obj s (O_proc p) = Some (up, rp, tp)"

      and p'1: "sectxt_of_obj s (O_proc p') = Some (up', rp', tp')"

      apply (simp add:Ptrace split:option.splits)

      by (case_tac a, case_tac aa, blast)

    from p1 have p1': "sectxt_of_obj s' (O_proc p) = Some (up, rp, tp)"

      using os cp2sp

      apply (erule_tac x = p in allE)

      by (auto simp:Ptrace co2sobj.simps cp2sproc_def split:option.splits)

    from p'1 have p'1': "sectxt_of_obj s' (O_proc p') = Some (up', rp', tp')"

      using os cp2sp

      apply (erule_tac x = p' in allE)

      by (auto simp:Ptrace co2sobj.simps cp2sproc_def split:option.splits)

    show ?thesis using p1 p'1 p1' p'1' grant

      by (simp add:Ptrace)

  qed

  ultimately show ?thesis using vs'

    by (erule_tac valid.intros(2), simp+)

next

  case (Exit p)

  have p_in: "p \<in> current_procs s'" using os alive

    apply (erule_tac x = "O_proc p" in allE)

    by (auto simp:Exit)

  have "os_grant s' e" using p_in by (simp add:Exit)

  moreover have "grant s' e" 

    by (simp add:Exit)

  ultimately show ?thesis using vs'

    by (erule_tac valid.intros(2), simp+)

next

  case (Open p f flags fd opt)

  show ?thesis

  proof (cases opt)

    case None

    have p_in: "p \<in> current_procs s'" using os alive

      apply (erule_tac x = "O_proc p" in allE)

      by (auto simp:Open None)

    have f_in: "is_file s' f" using os alive

      apply (erule_tac x = "O_file f" in allE)

      by (auto simp:Open None)

    have fd_not_in: "fd \<notin> current_proc_fds s' p"

      using os alive' p_in

      apply (erule_tac x = "E_fd p fd" in allE)

      by (simp add:Open None)

    have "os_grant s' e" using p_in f_in fd_not_in os

      by (simp add:Open None)

    moreover have "grant s' e" 

    proof-

      from grant obtain up rp tp uf rf tf 

        where p1: "sectxt_of_obj s (O_proc p) = Some (up, rp, tp)"

        and p2: "sectxt_of_obj s (O_file f) = Some (uf, rf, tf)" 

        apply (simp add:Open None split:option.splits)

        by (case_tac a, case_tac aa, blast)

      from p1 have p1': "sectxt_of_obj s' (O_proc p) = Some (up, rp, tp)"

        using os cp2sp

        apply (erule_tac x = p in allE)

        by (auto simp:Open None co2sobj.simps cp2sproc_def split:option.splits)

      from os have f_in': "is_file s f" by (simp add:Open None)

      from vd os have "\<exists> sf. cf2sfile s f = Some sf"

        by (auto dest!:is_file_in_current current_file_has_sfile simp:Open None)

      hence p2': "sectxt_of_obj s' (O_file f) = Some (uf, rf, tf)" using f_in f_in' p2 cf2sf

        apply (erule_tac x = f in allE)

        apply (auto dest:is_file_in_current simp:cf2sfile_def split:option.splits)

        apply (case_tac f, simp)

        apply (drule_tac s = s in root_is_dir', simp add:vd, simp+)

        done

      have "search_check s' (up, rp, tp) f"

        using p1 p2 p2' vd cf2sf f_in' grant Open None f_in

        apply (rule_tac s = s in enrich_search_check)

        by (simp_all split:option.splits)

      thus ?thesis using p1' p2' 

        apply (simp add:Open None split:option.splits) 

        using grant Open None p1 p2 

        by simp

    qed

    ultimately show ?thesis using vs'

      by (erule_tac valid.intros(2), simp+)

  next

    case (Some inum)

    from os obtain pf where pf_in_s: "is_dir s pf" and parent: "parent f = Some pf"

      by (auto simp:Open Some)

    have pf_in: "is_dir s' pf" using pf_in_s alive

      apply (erule_tac x = "O_dir pf" in allE)

      by simp

    have p_in: "p \<in> current_procs s'" using os alive

      apply (erule_tac x = "O_proc p" in allE)

      by (auto simp:Open Some)

    have f_not_in: "f \<notin> current_files s'" using os alive'

      apply (erule_tac x = "E_file f" in allE)

      by (auto simp:Open Some)

    have fd_not_in: "fd \<notin> current_proc_fds s' p"

      using os alive' p_in

      apply (erule_tac x = "E_fd p fd" in allE)

      by (simp add:Open Some)

    have inum_not_in: "inum \<notin> current_inode_nums s'"

      using os alive' 

      apply (erule_tac x = "E_inum inum" in allE)

      by (simp add:Open Some)

    have "os_grant s' e" using p_in pf_in parent f_not_in fd_not_in inum_not_in os

      by (simp add:Open Some hungs)

    moreover have "grant s' e"  

    proof-

      from grant parent obtain up rp tp uf rf tf 

        where p1: "sectxt_of_obj s (O_proc p) = Some (up, rp, tp)"

        and p2: "sectxt_of_obj s (O_dir pf) = Some (uf, rf, tf)" 

        apply (simp add:Open Some split:option.splits)

        by (case_tac a, case_tac aa, blast)

      from p1 have p1': "sectxt_of_obj s' (O_proc p) = Some (up, rp, tp)"

        using os cp2sp

        apply (erule_tac x = p in allE)

        by (auto simp:Open Some co2sobj.simps cp2sproc_def split:option.splits)

      from vd os pf_in_s have "\<exists> sf. cf2sfile s pf = Some sf"

        by (auto dest!:is_dir_in_current current_file_has_sfile simp:Open Some)

      hence p2': "sectxt_of_obj s' (O_dir pf) = Some (uf, rf, tf)" using p2 cf2sf pf_in pf_in_s

        apply (erule_tac x = pf in allE)

        apply (erule exE, frule_tac s = s in is_dir_in_current, simp)

        apply (drule is_dir_not_file, drule is_dir_not_file)

        apply (auto simp:cf2sfile_def split:option.splits)

        apply (case_tac pf, simp_all)

        by (simp add:sroot_def root_sec_remains vd vs')

      have "search_check s' (up, rp, tp) pf"

        using p1 p2 p2' vd cf2sf pf_in grant Open Some pf_in_s parent vs'

        apply (rule_tac s = s in enrich_search_check')

        by (simp_all split:option.splits)

      thus ?thesis using p1' p2' parent 

        apply (simp add:Open Some split:option.splits) 

        using grant Open Some p1 p2 

        by simp

    qed

    ultimately show ?thesis using vs'

      by (erule_tac valid.intros(2), simp+)