theory Enrich

imports Main Flask Static Init_prop Valid_prop Tainted_prop Delete_prop Co2sobj_prop S2ss_prop S2ss_prop2

 Temp Enrich

begin

context tainting_s begin

lemma get_parentfs_ctxts_prop:

  "\<lbrakk>get_parentfs_ctxts s (a # f) = Some ctxts; sectxt_of_obj s (O_dir f) = Some ctxt; valid s\<rbrakk>

   \<Longrightarrow> ctxt \<in> set (ctxts)"

apply (induct f)

apply (auto split:option.splits)

done

lemma search_check_allp_intro:

  "\<lbrakk>search_check s sp pf; get_parentfs_ctxts s pf = Some ctxts; valid s; is_dir s pf\<rbrakk>

   \<Longrightarrow> search_check_allp sp (set ctxts)"

apply (case_tac pf)

apply (simp split:option.splits if_splits add:search_check_allp_def)

apply (rule ballI)

apply (auto simp:search_check_ctxt_def search_check_dir_def split:if_splits option.splits)

apply (auto simp:search_check_allp_def search_check_file_def)

apply (frule is_dir_not_file, simp)

done

lemma search_check_leveling_f:

  "\<lbrakk>search_check s sp pf; parent f = Some pf; is_file s f; valid s;

    sectxt_of_obj s (O_file f) = Some fctxt; search_check_file sp fctxt\<rbrakk>

   \<Longrightarrow> search_check s sp f"

apply (case_tac f, simp+)

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

apply (frule parentf_is_dir_prop2, simp)

apply (erule get_pfs_secs_prop, simp)

apply (erule_tac search_check_allp_intro, simp_all)

apply (simp add:parentf_is_dir_prop2)

done

lemma enrich_proc_prop:

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

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

       (\<forall> obj. alive s obj \<longrightarrow> alive (enrich_proc s p p') obj) \<and> 

       (\<forall> obj. enrich_not_alive s obj \<longrightarrow> enrich_not_alive (enrich_proc s p p') obj) \<and>

       (files_hung_by_del (enrich_proc s p p') = files_hung_by_del s) \<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 f. file_of_proc_fd s tp fd = Some f \<longrightarrow> file_of_proc_fd (enrich_proc s p p') tp fd = Some f) \<and>

       (\<forall> tp fd flags. flags_of_proc_fd s tp fd = Some flags \<longrightarrow> 

                      flags_of_proc_fd (enrich_proc s p p') tp fd = Some flags) \<and>

       (\<forall> q. msgs_of_queue (enrich_proc s p p') q = msgs_of_queue 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"

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

     (\<forall>obj. alive s obj \<longrightarrow> alive (enrich_proc s p p') obj) \<and>

     (\<forall>obj. enrich_not_alive s obj \<longrightarrow> enrich_not_alive (enrich_proc s p p') obj) \<and>

     files_hung_by_del (enrich_proc s p p') = files_hung_by_del s \<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 f. file_of_proc_fd s tp fd = Some f \<longrightarrow> file_of_proc_fd (enrich_proc s p p') tp fd = Some f) \<and>

     (\<forall>tp fd flags.

         flags_of_proc_fd s tp fd = Some flags \<longrightarrow> flags_of_proc_fd (enrich_proc s p p') tp fd = Some flags) \<and>

     (\<forall>q. msgs_of_queue (enrich_proc s p p') q = msgs_of_queue 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 by auto

  have alive_pre: "is_created_proc s p \<Longrightarrow> (\<forall>obj. alive s obj \<longrightarrow> alive (enrich_proc s p p') obj)"

    using pre by simp

  hence curf_pre: "is_created_proc s p \<Longrightarrow> (\<forall>f. f \<in> current_files s \<longrightarrow> f \<in> current_files (enrich_proc s p p'))"

    using vd apply auto

    apply (drule is_file_or_dir, simp)

    apply (erule disjE)

    apply (erule_tac x = "O_file f" in allE, simp add:is_file_in_current)

    apply (erule_tac x = "O_dir f" in allE, simp add:is_dir_in_current)

    done

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

  proof-

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

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

      apply (frule pre')

      apply (erule_tac s = s in enrich_valid_intro_cons)

      apply (simp_all add:os grant vd pre)

      done  

    moreover 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 ffd_remain: "\<forall>tp fd f. file_of_proc_fd s tp fd = Some f \<longrightarrow> 

                                   file_of_proc_fd (enrich_proc s p p') tp fd = Some f"

        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" 

        proof-

          from pre a2

          have a4: "\<forall> obj. alive s obj \<longrightarrow> alive (enrich_proc s p p') obj"

            by (auto)

          show ?thesis using a3 a4

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

            by (auto dest:vt_grant_os)

        qed

        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))"

        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 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 split:option.splits del:npctxt_execve.simps, blast)

          have p1': "sectxt_of_obj (Execve p f fds # enrich_proc s p p') (O_proc p') = Some (up, rp, tp)"

            using p1 dup_sp a1 a0'

            apply (simp add:sectxt_of_obj_simps)

            by (simp add:cp2sproc_def split:option.splits)

          from os have f_in': "is_file s f"  by simp 

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

            by (auto dest!:is_file_in_current current_file_has_sfile)

          hence p2': "sectxt_of_obj (Execve p f fds # enrich_proc s p p') (O_file f) = Some (uf, rf, tf)" 

            using f_in p2 cf2sf os a1

            apply (erule_tac x = f in allE)

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

            apply (case_tac f, simp)

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

            done 

          from dup_sfd a5 have "\<forall>fd. fd \<in> proc_file_fds s p \<longrightarrow> 

            cfd2sfd (Execve p f fds # enrich_proc s p p') p' fd = cfd2sfd s p fd"

            apply (rule_tac allI)

            apply (erule_tac x = fd in allE, clarsimp)

            apply (drule set_mp, simp)

            apply (auto simp:cfd2sfd_execve proc_file_fds_def a1)

            done

          hence "inherit_fds_check (Execve p f fds # enrich_proc s p p') (up, nr, nt) p' (fds \<inter> proc_file_fds s p)"

            using grant os p1 p2 p3 vd

            apply (clarsimp)

            apply (rule_tac s = s and p = p and fds = fds in enrich_inherit_fds_check_dup)

            apply simp_all

            done

          moreover have "search_check (Execve p f fds # enrich_proc s p p') (up, rp, tp) f"

            using p1 p2 p2' vd cf2sf f_in f_in' grant p3 f_in a1

            apply (rule_tac s = s in enrich_search_check)

            apply (simp_all add:is_file_simps)

            apply (rule allI, rule impI, erule_tac x = fa in allE, simp)

            apply (drule_tac ff = fa in cf2sfile_other')

            by (auto simp:a2 curf_pre)

          ultimately show ?thesis 

            using p1' p2' p3

            apply (simp split:option.splits)

            using grant p1 p2

            apply simp

            done

        qed

        ultimately show ?thesis using a1

          by (erule_tac valid.intros(2), auto)

      qed

    qed

    moreover have "\<And>tp fds. \<lbrakk>valid (Clone tp p fds # s); p' \<noteq> p; p' \<notin> all_procs s\<rbrakk> \<Longrightarrow> 

      valid (Clone tp p' (fds \<inter> proc_file_fds s tp) # Clone tp p fds # s)"

      apply (frule vt_grant_os, frule vt_grant, drule not_all_procs_prop3)

      apply (rule valid.intros(2))

      apply (simp_all split:option.splits add:sectxt_of_obj_simps)

      apply (auto simp add:proc_file_fds_def)[1]

      apply (auto simp:inherit_fds_check_def sectxt_of_obj_simps sectxts_of_fds_def inherit_fds_check_ctxt_def)

      done

    moreover have "\<And>f flags fd opt. \<lbrakk>valid (Open p f flags fd opt # enrich_proc s p p'); 

      is_created_proc s p; valid (Open p f flags fd opt # s); p' \<notin> all_procs s\<rbrakk>

       \<Longrightarrow> valid (Open p' f (remove_create_flag flags) fd None # Open p f flags fd opt # enrich_proc s p p')"

    proof-

      fix f flags fd opt

      assume a1: "valid (Open p f flags fd opt # enrich_proc s p p')" and a2: "is_created_proc s p"

        and a3: "valid (Open p f flags fd opt # s)" and a4: "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 ffd_remain: "\<forall>tp fd f. file_of_proc_fd s tp fd = Some f \<longrightarrow> 

                                   file_of_proc_fd (enrich_proc s p p') tp fd = Some f"

        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

      from a4 a3 have a0: "p' \<noteq> p" by (auto dest!:vt_grant_os not_all_procs_prop3 split:option.splits)

      have a5: "p' \<in> current_procs (enrich_proc s p p')" 

        using a2 a3 vd

        apply (erule_tac enrich_proc_dup_in)

        by (simp_all add:vd a4)

      have a6: "is_file (Open p f flags fd opt # enrich_proc s p p') f"

        using a1 a3

        by (auto simp:is_file_open dest:vt_grant_os)

      have a7: "fd \<notin> current_proc_fds (enrich_proc s p p') p'"

        using a2 a4 vd

        apply (simp add:enrich_proc_dup_ffds_eq_fds)

        apply (rule notI)

        apply (drule_tac p = p in file_fds_subset_pfds)

        apply (drule set_mp, simp)

        using a3

        apply (drule_tac vt_grant_os)

        apply (auto split:option.splits)

        done

      from a1 have a8: "valid (enrich_proc s p p')" by (erule_tac valid.cases, auto)

      from a3 have grant: "grant s (Open p f flags fd opt)" and os: "os_grant s (Open p f flags fd opt)"

        by (auto dest:vt_grant_os vt_grant)

      show "valid (Open p' f (remove_create_flag flags) fd None # Open p f flags fd opt # enrich_proc s p p')"

      proof (cases opt)

        case None

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

        proof-

          from pre a2

          have a4: "\<forall> obj. alive s obj \<longrightarrow> alive (enrich_proc s p p') obj"

            by (auto)

          show ?thesis using a3 a4 None

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

            by (auto dest:vt_grant_os)

        qed

        from grant None 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 split:option.splits)

          by (case_tac a, case_tac aa, blast)

        have p1': "sectxt_of_obj (Open p f flags fd opt # enrich_proc s p p') (O_proc p') = Some (up, rp, tp)"

          using p1 dup_sp a1

          apply (simp add:sectxt_of_obj_simps)

          by (simp add:cp2sproc_def split:option.splits)

        from os None have f_in': "is_file s f" by simp 

        from vd os None have "\<exists> sf. cf2sfile s f = Some sf"

          by (auto dest!:is_file_in_current current_file_has_sfile)

        hence p2': "sectxt_of_obj (Open p f flags fd opt # enrich_proc s p p') (O_file f) = Some (uf, rf, tf)" 

          using p2 cf2sf os a1 None f_in' vd f_in

          apply (erule_tac x = f in allE)

          apply (auto dest:is_file_in_current simp:cf2sfile_def sectxt_of_obj_simps 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 (Open p f flags fd opt # enrich_proc s p p') (up, rp, tp) f"

          using p1 p2 p2' vd cf2sf f_in f_in' grant f_in a1 None

          apply (rule_tac s = s in enrich_search_check)

          apply (simp_all add:is_file_simps)

          apply (rule allI, rule impI, erule_tac x = fa in allE, simp)

          apply (simp add:cf2sfile_open_none)

          done

        thus ?thesis using a0 a5 a6 a7 None a1

          apply (rule_tac valid.intros(2))

          apply (simp_all add:a1)

          apply (case_tac flags, simp add:is_creat_excl_flag_def)

          using p1' p2'

          apply simp

          using grant p1 p2

          apply (simp add:oflags_check_remove_create)

          done

      next

        case (Some inum)

        with os obtain pf where parent: "parent f = Some pf" by auto

        with grant Some obtain pu rp pt pfu pfr pft where

          p1: "sectxt_of_obj s (O_proc p) = Some (pu, rp, pt)" 

          and p2: "sectxt_of_obj s (O_dir pf) = Some (pfu, pfr, pft)"

          apply (simp split:option.splits)

          apply (case_tac a, case_tac aa, blast)

          done

        from p1 have p1': "sectxt_of_obj (enrich_proc s p p') (O_proc p) = Some (pu, rp, pt)"

          using cp2sp os

          apply (erule_tac x = p in allE)

          apply (auto split:option.splits simp:cp2sproc_def)

          done

        from os parent Some

        have pf_in: "is_dir s pf" by auto

        hence "\<exists> sf. cf2sfile s pf = Some sf" using vd

          by (auto dest!:is_dir_in_current current_file_has_sfile)

        from a1 parent Some have pf_in': "is_dir (enrich_proc s p p') pf" 

          apply (frule_tac vt_grant_os)

          by (clarsimp)

        have p2': "sectxt_of_obj (enrich_proc s p p') (O_dir pf) = Some (pfu, pfr, pft)"

        proof-

          have "cf2sfile (enrich_proc s p p') pf = cf2sfile s pf"

            using cf2sf pf_in

            apply (drule_tac is_dir_in_current)

            apply (erule_tac x = pf in allE)

            by simp

          with pf_in pf_in' p2 vd Some a8

          show ?thesis            

            apply (frule_tac is_dir_not_file)

            apply (frule_tac s = "enrich_proc s p p'" in is_dir_not_file)

            apply (simp add:cf2sfile_def)

            apply (case_tac pf, simp) 

            apply (simp add:sroot_def root_sec_remains)            

            by (auto split:option.splits dest!:current_has_sec' get_pfs_secs_prop' dest:parentf_is_dir_prop1)        

        qed

        from p1 have dup: "sectxt_of_obj (Open p f flags fd (Some inum) # enrich_proc s p p') (O_proc p') 

          = Some (pu, rp, pt)"

          using a1 Some

          apply (simp add:sec_open)

          using dup_sp

          apply (auto split:option.splits if_splits simp:cp2sproc_def)

          done        

        have nsf: "sectxt_of_obj (Open p f flags fd (Some inum) # enrich_proc s p p') (O_file f) 

          = Some (pu, R_object, type_transition pt pft C_file True)"

          using a1 Some p1 p2 parent p2' p1'

          by (simp add:sec_open)

        have "search_check (Open p f flags fd (Some inum) # enrich_proc s p p') (pu, rp, pt) f"

        proof-

          have "search_check (Open p f flags fd (Some inum) # enrich_proc s p p') (pu, rp, pt) pf"

            using grant Some parent p1 p2 vd a1 pf_in pf_in' p2'

            apply simp 

            apply (rule_tac s = s in enrich_search_check') 

            apply (simp_all add:is_dir_simps sectxt_of_obj_simps)

            apply (rule allI, rule impI)

            apply (case_tac "fa = f")

            using os Some

            apply simp

            apply (drule_tac f' = fa in cf2sfile_open)

            apply (simp add:current_files_simps)

            using curf_pre a2

            apply simp

            apply simp

            using cf2sf

            apply simp

            done

          moreover have "is_file (Open p f flags fd (Some inum) # enrich_proc s p p') f"

            using a1 Some

            by (simp add:is_file_open)

          ultimately 

          show ?thesis

            using parent a1 Some nsf

            apply (erule_tac search_check_leveling_f)

            apply (simp_all)

            apply (simp add:search_check_file_def)

            apply (simp add:permission_check.simps)

            sorry

        qed

        thus ?thesis using a0 a5 a6 a7 a1 Some

          apply (rule_tac valid.intros(2))

          apply (simp add:a1)

          apply simp

          apply (case_tac flags, simp add:is_creat_excl_flag_def)

          using grant p1 p2 parent dup nsf

          apply (simp add:oflags_check_remove_create)

          done

      qed

    qed

    moreover have "\<And>fd. \<lbrakk>valid (CloseFd p fd # enrich_proc s p p'); is_created_proc s p; 

      valid (CloseFd p fd # s); p' \<notin> all_procs s; fd \<in> proc_file_fds s p\<rbrakk>

      \<Longrightarrow> valid (CloseFd p' fd # CloseFd p fd # enrich_proc s p p')"

    proof-

      fix fd

      assume a1: "valid (CloseFd p fd # enrich_proc s p p')" and a2: "is_created_proc s p" 

        and a3: "p' \<notin> all_procs s" and a4: "valid (CloseFd p fd # s)"

        and a5: "fd \<in> proc_file_fds s p"

      from a4 a3 have a0: "p' \<noteq> p" by (auto dest!:vt_grant_os not_all_procs_prop3)

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

        using a2 a3 vd

        by (auto intro:enrich_proc_dup_in)

      moreover have "fd \<in> current_proc_fds (enrich_proc s p p') p'"      

        using a5 a2 a3 vd pre'

        apply (simp)

        apply (drule_tac s = "enrich_proc s p p'" and p = p' in file_fds_subset_pfds)

        apply (erule set_mp)

        apply (simp add:enrich_proc_dup_ffds)

        done

      ultimately show "valid (CloseFd p' fd # CloseFd p fd # enrich_proc s p p')"

        apply (rule_tac valid.intros(2))

        apply (simp_all add:a1 a0 a2 pre')

        done

    qed

    moreover have "\<And> fd. \<lbrakk>valid (ReadFile p fd # enrich_proc s p p');

            is_created_proc s p; valid (ReadFile p fd # s); p' \<notin> all_procs s\<rbrakk>

           \<Longrightarrow> valid (ReadFile p' fd # ReadFile p fd # enrich_proc s p p')"

    proof-

      fix fd

      assume a1: "valid (ReadFile p fd # enrich_proc s p p')" and a2: "is_created_proc s p"

        and a3: "valid (ReadFile p fd # s)" and a4: "p' \<notin> all_procs s"

      from a3 have os: "os_grant s (ReadFile p fd)" and grant: "grant s (ReadFile p fd)"

        by (auto dest:vt_grant_os vt_grant)

      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 ffd_remain: "\<forall>tp fd f. file_of_proc_fd s tp fd = Some f \<longrightarrow> 

                                   file_of_proc_fd (enrich_proc s p p') tp fd = Some f"

        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

      have vd_enrich: "valid (enrich_proc s p p')" using a1 by (auto dest:valid.cases)

      show "valid (ReadFile p' fd # ReadFile p fd # enrich_proc s p p')"

      proof-

        have "os_grant (ReadFile p fd # enrich_proc s p p') (ReadFile p' fd)"

          using a1 a2 a4 vd os

          apply (clarsimp simp:current_files_simps enrich_proc_dup_in enrich_proc_dup_ffds_eq_fds)

          apply (simp add:proc_file_fds_def)

          apply (rule conjI)

          apply (rule_tac x = f in exI, simp add:enrich_proc_dup_ffd)

          using curf_pre

          apply (simp)

          apply (drule enrich_proc_dup_fflags)

          apply simp_all

          apply (erule disjE)

          apply auto

          apply (simp add:is_read_flag_def)

          done

        moreover have "grant (ReadFile p fd # enrich_proc s p p') (ReadFile p' fd)"

        proof-

          from grant obtain f sp sfd sf where p1: "file_of_proc_fd s p fd = Some f"

            and p2: "sectxt_of_obj s (O_proc p) = Some sp" 

            and p3: "sectxt_of_obj s (O_fd p fd) = Some sfd"

            and p4: "sectxt_of_obj s (O_file f) = Some sf"

            by (auto split:option.splits) 

          from os obtain flag where p0: "flags_of_proc_fd s p fd = Some flag"

            by auto

          from os have f_in_s: "f \<in> current_files s" using p1 by simp

          from p1 vd have isfile_s: "is_file s f" by (erule_tac file_of_pfd_is_file, simp)

          with alive_pre a2 have isfile_s': "is_file (enrich_proc s p p') f"

            apply simp

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

            done          

          from p0 obtain flag' where p0': "flags_of_proc_fd (enrich_proc s p p') p' fd = Some flag'"

            and p0'': "(flag' = flag) \<or> (flag' = remove_create_flag flag)"

            using a2 a4 vd

            apply (drule_tac enrich_proc_dup_fflags)

            apply auto

            apply (case_tac flag, auto)

            apply (case_tac flag, auto)

            done            

          from p1 have p1': "file_of_proc_fd (enrich_proc s p p') p' fd = Some f"

            using a2 a4 vd

            by (simp add:enrich_proc_dup_ffd)

          from p2 have p2': "sectxt_of_obj (enrich_proc s p p') (O_proc p') = Some sp"

            using dup_sp

            by (auto simp:cp2sproc_def split:option.splits)

          from p3 p1 p1' p0 p0' os have p3': "sectxt_of_obj (enrich_proc s p p') (O_fd p' fd) = Some sfd"

            using dup_sfd

            apply (erule_tac x = fd in allE)

            apply (auto simp:proc_file_fds_def cfd2sfd_def split:option.splits)

            apply (drule current_file_has_sfile')

            apply (simp add:vd, simp)

            apply (drule current_file_has_sfile')

            apply (simp add:vd, simp)

            done

          from p4 have p4': "sectxt_of_obj (enrich_proc s p p') (O_file f) = Some sf" 

            using f_in_s cf2sf isfile_s isfile_s' a1 vd_enrich vd

            apply (erule_tac x = f in allE)

            apply (simp)

            apply (auto simp:cf2sfile_def split:option.splits 

              dest!:current_has_sec' get_pfs_secs_prop' dest:parentf_is_dir is_file_in_current)

            apply (case_tac f, simp, drule root_is_dir', simp, simp, simp)

            done

          show ?thesis using p1' p2' p3' p4' a1

            apply (simp add:sectxt_of_obj_simps)

            using grant p1 p2 p3 p4 

            apply simp

            done

        qed

        ultimately show ?thesis

          using a1

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

      qed

    qed

    ultimately show ?thesis 

      using created_cons vd_cons all_procs_cons

      apply (case_tac e)

      apply (auto simp:is_created_proc_simps split:if_splits)

      done

  qed

  moreover have "\<forall>obj. alive (e # s) obj \<longrightarrow> alive (enrich_proc (e # s) p p') obj"

    sorry

  moreover have "\<forall>obj. enrich_not_alive (e # s) obj \<longrightarrow> enrich_not_alive (enrich_proc (e # s) p p') obj"

    sorry

  moreover have "files_hung_by_del (enrich_proc (e # s) p p') = files_hung_by_del (e # s)"

    sorry

  moreover have "\<forall>tp. tp \<in> current_procs (e # s) \<longrightarrow> cp2sproc (enrich_proc (e # s) p p') tp = cp2sproc (e # s) tp"

    sorry

  moreover have "\<forall>f. f \<in> current_files (e # s) \<longrightarrow> cf2sfile (enrich_proc (e # s) p p') f = cf2sfile (e # s) f"

    sorry

  moreover have "\<forall>q. q \<in> current_msgqs (e # s) \<longrightarrow> cq2smsgq (enrich_proc (e # s) p p') q = cq2smsgq (e # s) q"

    sorry

  moreover have "\<forall>tp fd f. file_of_proc_fd (e # s) tp fd = Some f \<longrightarrow> 

    file_of_proc_fd (enrich_proc (e # s) p p') tp fd = Some f"

    sorry

  moreover have "\<forall>tp fd flags. flags_of_proc_fd (e # s) tp fd = Some flags \<longrightarrow>

        flags_of_proc_fd (enrich_proc (e # s) p p') tp fd = Some flags"

    sorry

  moreover have "\<forall>q. msgs_of_queue (enrich_proc (e # s) p p') q = msgs_of_queue (e # s) q"

    sorry 

  moreover have "\<forall>tp fd. fd \<in> proc_file_fds (e # s) tp \<longrightarrow> 

    cfd2sfd (enrich_proc (e # s) p p') tp fd = cfd2sfd (e # s) tp fd"

    sorry 

  moreover have "cp2sproc (enrich_proc (e # s) p p') p' = cp2sproc (e # s) p"

  proof-

    from pre have b0: "is_created_proc s p \<Longrightarrow> cp2sproc (enrich_proc s p p') p' = cp2sproc s p" by simp

    have b1: "\<And> tp f fds. \<lbrakk>valid (enrich_proc (Execve tp f fds # s) p p'); valid (Execve tp f fds # s);

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

      \<Longrightarrow> cp2sproc (enrich_proc (Execve tp f fds # s) p p') p' = cp2sproc (Execve tp f fds # s) p"

    proof-

      fix tp f fds

      assume a1: "valid (enrich_proc (Execve tp f fds # s) p p')"

      and a2: "valid (Execve tp f fds # s)" and a3: "is_created_proc (Execve tp f fds # s) p"

      and a4: "p' \<notin> all_procs (Execve tp f fds # s)"

      show "cp2sproc (enrich_proc (Execve tp f fds # s) p p') p' = cp2sproc (Execve tp f fds # s) p"

      proof (cases "tp = p")

        case True

        show ?thesis using True a1 a2 a3 a4

          thm not_all_procs_prop3

          apply (auto simp add:cp2sproc_execve is_created_proc_def split:option.splits dest!:current_has_sec'