theory del_vs_del_s

imports Main rc_theory os_rc obj2sobj_prop all_sobj_prop 

begin

context tainting_s_complete begin

lemma deleted_has_del_event_proc:

  "\<lbrakk>deleted (Proc p) s; valid s\<rbrakk> \<Longrightarrow> \<exists> s' p'. Kill p' p # s' \<preceq> s \<and> \<not> deleted (Proc p) s'"

apply (induct s, simp)

apply (frule valid_cons, frule valid_os)

by (case_tac a, auto simp:no_junior_def)

lemma deleted_has_del_event_ipc:

  "\<lbrakk>deleted (IPC i) s; valid s\<rbrakk> \<Longrightarrow> \<exists> s' p. DeleteIPC p i # s' \<preceq> s \<and> \<not> deleted (IPC i) s'"

apply (induct s, simp)

apply (frule valid_cons, frule valid_os)

by (case_tac a, auto simp:no_junior_def)

lemma deleted_has_del_event_file:

  "\<lbrakk>deleted (File f) s; valid s\<rbrakk> \<Longrightarrow> \<exists> s' p. DeleteFile p f # s' \<preceq> s \<and> \<not> deleted (File f) s'"

apply (induct s, simp)

apply (frule valid_cons, frule valid_os)

by (case_tac a, auto simp:no_junior_def)

(*

lemma noJ_Anc: "x \<prec> y = (x \<preceq> y \<and> x \<noteq> y)"

apply (simp add: no_junior_expand)

by (auto simp:is_ancestor_def)

lemma noJ_Anc': "x \<prec> y \<Longrightarrow> (x \<preceq> y \<and> x \<noteq> y)"

by (simp add:noJ_Anc)

lemma noJ_Anc'': "\<lbrakk>x \<preceq> y; x \<noteq> y\<rbrakk> \<Longrightarrow> x \<prec> y"

by (simp add:noJ_Anc)

lemma deled_imp_no_childfs:

  "\<lbrakk>valid (DeleteFile p f # s); f \<prec> childf\<rbrakk> \<Longrightarrow> childf \<notin> current_files s"

apply (frule valid_cons, drule valid_os, rule notI, clarsimp dest!:noJ_Anc')

apply (drule ancient_has_parent, simp, clarsimp)

apply (drule_tac af = sonf and f = childf in ancient_file_in_current, simp+)

apply (case_tac sonf, simp)

apply (erule_tac x = a in allE, simp)

done

lemma deled_imp_childfs_deleted:

  "\<lbrakk>valid (DeleteFile p f # s); f \<prec> childf; childf \<in> init_files\<rbrakk>

  \<Longrightarrow> deleted (File childf) s"

apply (drule deled_imp_no_childfs, simp+)

apply (erule_tac P = "childf \<in> current_files s" in swap)

by (drule not_deleted_imp_exists, simp+)

lemma initf_deled_imp_childf_deled:

  "\<lbrakk>deleted (File f) s; valid s\<rbrakk> \<Longrightarrow> \<exists> s' p. DeleteFile p f # s' \<preceq> s \<and> \<not> deleted (File f) s' \<and> (\<forall> childf \<in> init_files. f \<prec> childf \<longrightarrow> deleted (File childf) s')"

apply (drule deleted_has_del_event_file, simp, clarify)

apply (rule_tac x = s' in exI, rule_tac x = p in exI, simp)

apply (rule ballI, rule impI, frule vs_history, simp)

by (erule deled_imp_childfs_deleted, simp+)

lemma initf_deled_imp_childf_deled:

  "\<lbrakk>deleted (File f) s; valid s; f \<prec> childf; childf \<in> init_files\<rbrakk> 

  \<Longrightarrow> \<exists> s'. s' \<preceq> s \<and> valid s' \<and> deleted (File childf) s'"

apply (drule deleted_has_del_event_file, simp, clarify)

apply (frule vs_history, simp, frule valid_cons)

apply (drule deled_imp_childfs_deleted, simp, simp)

apply (rule_tac x = s' in exI, auto elim:no_juniorE)

done

lemma deleted_has_del_event_allchildf:

  "\<lbrakk>deleted (File f) s; valid s; f \<preceq> childf; childf \<in> init_files\<rbrakk>

  \<Longrightarrow> \<exists> s' p. DeleteFile p childf # s' \<preceq> s \<and> \<not> deleted (File childf) s'"

apply (case_tac "f = childf")

apply (drule deleted_has_del_event_file, simp, simp)

apply (drule noJ_Anc'', simp)

apply (drule initf_deled_imp_childf_deled, simp+, clarify)

apply (drule deleted_has_del_event_file, simp+, clarify)

apply (rule_tac x = s'a in exI, rule conjI, rule_tac x = p in exI)

apply (erule no_junior_trans, simp+)

done

*)

lemma del_imp_del_s_file:

  assumes initf: "f \<in> init_files"

  and deled: "deleted (File f) s"

  and vs: "valid s"

  shows "file_deletable_s f"

proof -

  from deled vs obtain s' p' where

    his: "DeleteFile p' f # s' \<preceq> s" and fstdel: "\<not> deleted (File f) s'"

    by (drule_tac deleted_has_del_event_file, auto)

  from his vs have "valid (DeleteFile p' f # s')" by (simp add:vs_history)

  hence exp': "p' \<in> current_procs s'" and exf: "f \<in> current_files s'"

    and rc: "rc_grant s' (DeleteFile p' f)" and vs': "valid s'"

    by (auto dest:valid_os valid_rc valid_cons)

  from initf obtain t where etype: "etype_of_file [] f = Some t"

    by (drule_tac init_file_has_etype, simp add:etype_of_file_def, blast)

  from initf have sd: "source_dir [] f = Some f"

    by (simp add:source_dir_of_init') 

  hence "obj2sobj [] (File f) = SFile (t, f) (Some f)"

    using etype initf by (auto simp:obj2sobj.simps)

  with fstdel vs' initf exf etype

  have SF: "obj2sobj s' (File f) = SFile (t, f) (Some f)"

    using obj2sobj_file_remains'''[where s = "[]" and s' = s']

    by (auto simp:obj2sobj.simps)

  from exp' vs' obtain r' fr' pt' u' srp' where

    SP': "obj2sobj s' (Proc p') = SProc (r',fr',pt',u') srp'"

    by (frule_tac current_proc_has_sobj, simp, blast)

  with exp' vs' all_sobjs_I[where s = s' and obj = "Proc p'"]

  have SP'_in: "SProc (r',fr',pt',u') srp' \<in> all_sobjs" by simp

  show ?thesis unfolding file_deletable_s_def

    apply (rule_tac x = t in exI, 

           rule_tac x = "(r',fr',pt',u')" in exI,

           rule_tac x = srp' in exI)

    apply (simp add:SP'_in) 

    using rc SP' SF etype

    by (auto simp:obj2sobj.simps cp2sproc.simps split:option.splits if_splits)

qed

lemma del_imp_del_s_proc:

  assumes initp: "p \<in> init_processes"

  and deled: "deleted (Proc p) s"

  and vs: "valid s"

  shows "proc_deletable_s p"

proof-

  from deled vs obtain s' p' where

    his: "Kill p' p # s' \<preceq> s" and fstdel: "\<not> deleted (Proc p) s'"

    by (drule_tac deleted_has_del_event_proc, auto)

  from his vs have "valid (Kill p' p # s')" by (simp add:vs_history)

  hence exp': "p' \<in> current_procs s'" and exp: "p \<in> current_procs s'"

    and rc: "rc_grant s' (Kill p' p)" and vs': "valid s'"

    by (auto dest:valid_os valid_rc valid_cons)

  from initp fstdel vs' have "source_proc s' p = Some p"

    apply (induct s', simp)

    apply (frule valid_cons, frule valid_os, frule not_deleted_cons_D, simp)

    by (case_tac a, auto dest:not_deleted_imp_exists)

  with exp initp vs' obtain r fr pt u 

    where SP: "obj2sobj s' (Proc p) = SProc (r,fr,pt,u) (Some p)"

    apply (frule_tac current_proc_has_sobj, simp)

    by (simp add:obj2sobj.simps split:option.splits, blast)

  with exp vs' all_sobjs_I[where s = s' and obj = "Proc p"]

  have SP_in: "SProc (r,fr,pt,u) (Some p) \<in> all_sobjs" by simp

  from exp' vs' obtain r' fr' pt' u' srp' where

    SP': "obj2sobj s' (Proc p') = SProc (r',fr',pt',u') srp'"

    by (frule_tac current_proc_has_sobj, simp, blast)

  with exp' vs' all_sobjs_I[where s = s' and obj = "Proc p'"]

  have SP'_in: "SProc (r',fr',pt',u') srp' \<in> all_sobjs" by simp

  show ?thesis unfolding proc_deletable_s_def

    apply (rule_tac x = r in exI, rule_tac x = fr in exI,

           rule_tac x = pt in exI, rule_tac x = u in exI,

           rule_tac x = "(r',fr',pt',u')" in exI,

           rule_tac x = srp' in exI)

    apply (simp add:SP_in SP'_in) 

    using rc SP SP'

    by (auto simp:obj2sobj.simps cp2sproc.simps split:option.splits)

qed

lemma del_imp_del_s_ipc:

  assumes initi: "i \<in> init_ipcs"

  and deled: "deleted (IPC i) s"

  and vs: "valid s"

  shows "ipc_deletable_s i"

proof-

  from deled vs obtain s' p' where

    his: "DeleteIPC p' i # s' \<preceq> s" and fstdel: "\<not> deleted (IPC i) s'"

    by (drule_tac deleted_has_del_event_ipc, auto)

  from his vs have "valid (DeleteIPC p' i # s')" by (simp add:vs_history)

  hence exp': "p' \<in> current_procs s'" and exi: "i \<in> current_ipcs s'"

    and rc: "rc_grant s' (DeleteIPC p' i)" and vs': "valid s'"

    by (auto dest:valid_os valid_rc valid_cons)

  from initi obtain t where type: "init_ipc_type i = Some t" 

    using init_ipc_has_type by (auto simp:bidirect_in_init_def)

  with fstdel vs' initi exi have SI: "obj2sobj s' (IPC i) = SIPC t (Some i)"

    using obj2sobj_ipc_remains'''[where s = "[]" and s' = s']

    by (auto simp:obj2sobj.simps)

  from exp' vs' obtain r' fr' pt' u' srp' where

    SP': "obj2sobj s' (Proc p') = SProc (r',fr',pt',u') srp'"

    by (frule_tac current_proc_has_sobj, simp, blast)

  with exp' vs' all_sobjs_I[where s = s' and obj = "Proc p'"]

  have SP'_in: "SProc (r',fr',pt',u') srp' \<in> all_sobjs" by simp

  show ?thesis unfolding ipc_deletable_s_def

    apply (rule_tac x = t in exI, 

           rule_tac x = "(r',fr',pt',u')" in exI,

           rule_tac x = srp' in exI)

    apply (simp add: SP'_in) 

    using rc SP' SI type

    by (auto simp:obj2sobj.simps cp2sproc.simps split:option.splits if_splits)

qed

lemma deleted_imp_deletable_s:

  "\<lbrakk>deleted obj s; exists [] obj; valid s\<rbrakk> \<Longrightarrow> deletable_s obj"

apply (case_tac obj)  

apply (simp add:del_imp_del_s_proc)  

apply (simp add:del_imp_del_s_file) 

apply (simp add:del_imp_del_s_ipc)

done

(*

lemma all_sobjs_E3:

  assumes prem: "sobj \<in> all_sobjs" 

  shows "\<exists> s obj. valid s \<and> obj2sobj s obj = sobj \<and> sobj_source_eq_obj sobj obj \<and> 

                  initp_intact_but s sobj \<and> exists s obj \<and> no_del_event s"

proof-

  obtain t where "etype_of_file [] [] = Some t" 

    using root_in_filesystem current_file_has_etype[of "[]" "[]"] vs_nil 

    by auto

  with root_in_filesystem

  have "obj2sobj [] (File []) = SFile (t,[]) (Some [])"

    by (auto simp:obj2sobj.simps source_dir_of_init' 

            split:option.splits if_splits)

  moreover have "initp_intact []"

    by (auto simp:initp_intact_def cp2sproc.simps obj2sobj.simps 

            split:option.splits)

  ultimately show ?thesis

    using prem vs_nil root_in_filesystem 

    apply (drule_tac s' = "[]" and obj' = "File []" in all_sobjs_E2)

    apply (simp+, (erule exE|erule conjE)+)

    by (rule_tac x = s in exI, simp, rule_tac x = obj in exI, simp+)

qed

lemma del_s_imp_del_proc:

  assumes initp: "p \<in> init_processes"

  and del_s: "proc_deletable_s p"

  shows "\<exists> s. valid s \<and> deleted (Proc p) s"

proof-

  from del_s obtain r fr pt u sp' srp' 

    where Target: "SProc (r,fr,pt,u) (Some p) \<in> all_sobjs" 

    and Killer: "SProc sp' srp' \<in> all_sobjs"

    and rc: "(role_of_sproc sp', Proc_type pt, DELETE) \<in> compatible"

    using proc_deletable_s_def by auto

  from Target obtain s where vs: "valid s" 

    and "obj2sobj s (Proc p) = SProc (r,fr,pt,u) (Some p)" 

    and "exists s (Proc p)"  and "no_del_event s"

    and "initp_intact_but s (SProc (r,fr,pt,u) (Some p))"

    apply (drule_tac all_sobjs_E3, clarsimp)

    by (frule obj2sobj_proc, clarsimp)

  with Killer obtain s' p' where vs': "valid (s' @ s)" and

    SP : "obj2sobj (s' @ s) (Proc p) = SProc (r,fr,pt,u) (Some p)" and

    exp: "exists (s' @ s) (Proc p)"  and

    SP': "obj2sobj (s' @ s) (Proc p') = SProc sp' srp'" and

    exp': "exists (s' @ s) (Proc p')"

    apply (drule_tac obj' = "Proc p" and s' = s in all_sobjs_E1, auto)

    apply (frule_tac obj = obj in obj2sobj_proc, erule exE)

    apply (auto intro:nodel_exists_remains)

    apply blast

    apply (frule_tac obj = "Proc p" in nodel_exists_remains)

lemma deletable_s_imp_deleted:

  "deletable_s obj \<Longrightarrow> \<exists> s. valid s \<and> deleted obj s"

apply (case_tac obj)

apply (simp add:deletable_s.simps)

lemma valid_kill_imp_proc_del_s:

  "\<lbrakk>valid (Kill p' p # s); p \<in> init_processes; \<not> deleted (Proc p) s\<rbrakk> \<Longrightarrow> proc_deletable_s p"

apply (frule valid_os, frule valid_

lemma build_phase: "f \<in> init_files \<and> file_deletable_s f \<longrightarrow> (\<exists> s. \<forall> childf. valids s \<and> f \<preceq> childf \<and> childf \<in> init_files \<and> etype_of_file [] f = Some t \<longrightarrow> (\<exists> p. p \<in> current_procs s \<and> currentrole s p = Some r \<and> (r, File_type t, DELETE) \<in> compatible))"

 thm all_sobjs_E0

lemma del_phase: "f \<in> init_files \<and> file_deletable_s f \<and> (\<forall> childf. f \<preceq> childf \<and> childf \<notin> current_files s \<and> valid s"

*)

end

end