(*<*)
theory Current_prop
imports Main Flask_type Flask My_list_prefix Init_prop Valid_prop Delete_prop
begin
(*>*)

ML {*quick_and_dirty := true*}

context flask begin

lemma procs_of_shm_prop1: "\<lbrakk> p_flag \<in> procs_of_shm s h; valid s\<rbrakk> \<Longrightarrow> h \<in> current_shms s"
apply (induct s arbitrary:p_flag)
apply (case_tac p_flag, simp, drule init_procs_has_shm, simp)
apply (frule vd_cons, frule vt_grant_os)
apply (case_tac a, auto split:if_splits option.splits)
done

lemma procs_of_shm_prop2: "\<lbrakk>(p, flag) \<in> procs_of_shm s h; valid s\<rbrakk> \<Longrightarrow> p \<in> current_procs s"
apply (induct s arbitrary:p flag)
apply (simp, drule init_procs_has_shm, simp)
apply (frule vd_cons, frule vt_grant_os)
apply (case_tac a, auto split:if_splits option.splits)
done

lemma procs_of_shm_prop2':
  "\<lbrakk>p \<notin> current_procs s; valid s\<rbrakk> \<Longrightarrow> \<forall> flag h. (p, flag) \<notin> procs_of_shm s h"
by (auto dest:procs_of_shm_prop2)

lemma procs_of_shm_prop3: "\<lbrakk>(p, flag) \<in> procs_of_shm s h; (p, flag') \<in> procs_of_shm s h; valid s\<rbrakk>
  \<Longrightarrow> flag = flag'"
apply (induct s arbitrary:p flag flag')
apply (simp, drule_tac flag = flag in init_procs_has_shm, drule_tac flag = flag' in init_procs_has_shm, simp)
apply (frule vd_cons, frule vt_grant_os)
apply (case_tac a, auto split:if_splits option.splits dest:procs_of_shm_prop2)
done

lemma procs_of_shm_prop4: "\<lbrakk>(p, flag) \<in> procs_of_shm s h; valid s\<rbrakk> \<Longrightarrow> flag_of_proc_shm s p h = Some flag"
apply (induct s arbitrary:p flag)
apply (simp, drule init_procs_has_shm, simp)
apply (frule vd_cons, frule vt_grant_os)
apply (case_tac a, auto split:if_splits option.splits dest:procs_of_shm_prop2)
done

lemma procs_of_shm_prop4':
  "\<lbrakk>flag_of_proc_shm s p h = None; valid s\<rbrakk> \<Longrightarrow> \<forall> flag. (p, flag) \<notin> procs_of_shm s h"
by (auto dest:procs_of_shm_prop4)

lemma not_init_intro_proc:
  "\<lbrakk>p \<notin> current_procs s; valid s\<rbrakk> \<Longrightarrow> deleted (O_proc p) s \<or> p \<notin> init_procs"
using not_deleted_init_proc by auto

lemma not_init_intro_proc':
  "\<lbrakk>p \<notin> current_procs s; valid s\<rbrakk> \<Longrightarrow> \<not> (\<not> deleted (O_proc p) s \<and> p \<in> init_procs)"
using not_deleted_init_proc by auto

lemma info_shm_flow_in_procs:
  "\<lbrakk>info_flow_shm s p p'; valid s\<rbrakk> \<Longrightarrow> p \<in> current_procs s \<and> p' \<in> current_procs s"
by (auto intro:procs_of_shm_prop2 simp:info_flow_shm_def one_flow_shm_def)

lemma flag_of_proc_shm_prop1:
  "\<lbrakk>flag_of_proc_shm s p h = Some flag; valid s\<rbrakk> \<Longrightarrow> (p, flag) \<in> procs_of_shm s h"
apply (induct s arbitrary:p flag)
apply (simp, drule init_shmflag_has_proc, simp)
apply (frule vd_cons, frule vt_grant_os)
apply (case_tac a, auto split:if_splits option.splits dest:procs_of_shm_prop2)
done

(*********** simpset for one_flow_shm **************)

lemma one_flow_not_self:
  "one_flow_shm s h p p \<Longrightarrow> False"
by (simp add:one_flow_shm_def)

lemma one_flow_shm_attach:
  "valid (Attach p h flag # s) \<Longrightarrow> one_flow_shm (Attach p h flag # s) = (\<lambda> h' pa pb. 
     if (h' = h) 
     then (pa = p \<and> pb \<noteq> p \<and> flag = SHM_RDWR \<and> (\<exists> flagb. (pb, flagb) \<in> procs_of_shm s h)) \<or>
          (pb = p \<and> pa \<noteq> p \<and> (pa, SHM_RDWR) \<in> procs_of_shm s h) \<or>
          (one_flow_shm s h pa pb)               
     else one_flow_shm s h' pa pb        )"
apply (rule ext, rule ext, rule ext, frule vd_cons, frule vt_grant_os)
by (auto simp add: one_flow_shm_def)

lemma one_flow_shm_detach:
  "valid (Detach p h # s) \<Longrightarrow> one_flow_shm (Detach p h # s) = (\<lambda> h' pa pb.
     if (h' = h) 
     then (pa \<noteq> p \<and> pb \<noteq> p \<and> one_flow_shm s h' pa pb)
     else one_flow_shm s h' pa pb)"
apply (rule ext, rule ext, rule ext, frule vt_grant_os)
by (auto simp:one_flow_shm_def)

lemma one_flow_shm_deleteshm:
  "valid (DeleteShM p h # s) \<Longrightarrow> one_flow_shm (DeleteShM p h # s) = (\<lambda> h' pa pb. 
     if (h' = h) 
     then False
     else one_flow_shm s h' pa pb)"
apply (rule ext, rule ext, rule ext, frule vt_grant_os)
by (auto simp: one_flow_shm_def)

lemma one_flow_shm_clone:
  "valid (Clone p p' fds shms # s) \<Longrightarrow> one_flow_shm (Clone p p' fds shms # s) = (\<lambda> h pa pb. 
     if (pa = p' \<and> pb \<noteq> p' \<and> h \<in> shms)
     then (if (pb = p) then (flag_of_proc_shm s p h = Some SHM_RDWR) else one_flow_shm s h p pb)
     else if (pb = p' \<and> pa \<noteq> p' \<and> h \<in> shms)
          then (if (pa = p) then (flag_of_proc_shm s p h = Some SHM_RDWR) else one_flow_shm s h pa p)
          else one_flow_shm s h pa pb)"
apply (rule ext, rule ext, rule ext, frule vt_grant_os, frule vd_cons, clarsimp)
apply (frule_tac p = p' in procs_of_shm_prop2', simp)
apply (auto simp:one_flow_shm_def intro:procs_of_shm_prop4 flag_of_proc_shm_prop1)
done

lemma one_flow_shm_execve:
  "valid (Execve p f fds # s) \<Longrightarrow> one_flow_shm (Execve p f fds # s) = (\<lambda> h pa pb. 
     pa \<noteq> p \<and> pb \<noteq> p \<and> one_flow_shm s h pa pb    )"
apply (rule ext, rule ext, rule ext, frule vt_grant_os, frule vd_cons)
by (auto simp:one_flow_shm_def)

lemma one_flow_shm_kill:
  "valid (Kill p p' # s) \<Longrightarrow> one_flow_shm (Kill p p' # s) = (\<lambda> h pa pb. 
     pa \<noteq> p' \<and> pb \<noteq> p' \<and> one_flow_shm s h pa pb                 )"
apply (rule ext, rule ext, rule ext, frule vt_grant_os, frule vd_cons)
by (auto simp:one_flow_shm_def)

lemma one_flow_shm_exit:
  "valid (Exit p # s) \<Longrightarrow> one_flow_shm (Exit p # s) = (\<lambda> h pa pb. 
     pa \<noteq> p \<and> pb \<noteq> p \<and> one_flow_shm s h pa pb                          )"
apply (rule ext, rule ext, rule ext, frule vt_grant_os, frule vd_cons)
by (auto simp:one_flow_shm_def)

lemma one_flow_shm_other:
  "\<lbrakk>valid (e # s); 
    \<forall> p h flag. e \<noteq> Attach p h flag;
    \<forall> p h. e \<noteq> Detach p h;
    \<forall> p h. e \<noteq> DeleteShM p h;
    \<forall> p p' fds shms. e \<noteq> Clone p p' fds shms;
    \<forall> p f fds. e \<noteq> Execve p f fds;
    \<forall> p p'. e \<noteq> Kill p p';
    \<forall> p. e \<noteq> Exit p
   \<rbrakk> \<Longrightarrow> one_flow_shm (e # s) = one_flow_shm s"
apply (rule ext, rule ext, rule ext, frule vt_grant_os, frule vd_cons)
apply (case_tac e, auto simp:one_flow_shm_def dest:procs_of_shm_prop2)
apply (drule procs_of_shm_prop1, auto)
done

lemmas one_flow_shm_simps = one_flow_shm_other one_flow_shm_attach one_flow_shm_detach one_flow_shm_deleteshm
  one_flow_shm_clone one_flow_shm_execve one_flow_shm_kill one_flow_shm_exit

type_synonym t_edge_shm = "t_process \<times> t_shm \<times> t_process"
fun Fst:: "t_edge_shm \<Rightarrow> t_process" where "Fst (a, b, c) = a"
fun Snd:: "t_edge_shm \<Rightarrow> t_shm" where "Snd (a, b, c) = b"
fun Trd:: "t_edge_shm \<Rightarrow> t_process" where "Trd (a, b, c) = c"

fun edge_related:: "t_edge_shm list \<Rightarrow> t_process \<Rightarrow> t_shm \<Rightarrow> bool"
where
  "edge_related [] p h = False"
| "edge_related ((from, shm, to) # path) p h = 
     (if (((p = from) \<or> (p = to)) \<and> (h = shm)) then True 
      else edge_related path p h)"
         
inductive path_by_shm :: "t_state \<Rightarrow> t_process \<Rightarrow> t_edge_shm list \<Rightarrow> t_process \<Rightarrow> bool"
where
  pbs1: "p \<in> current_procs s \<Longrightarrow> path_by_shm s p [] p"
| pbs2: "\<lbrakk>path_by_shm s p path p'; one_flow_shm s h p' p''; p'' \<notin> set (map Fst path)\<rbrakk> 
         \<Longrightarrow> path_by_shm s p ((p', h, p'')# path) p''"


lemma one_step_path: "\<lbrakk>one_flow_shm s h p p'; valid s\<rbrakk> \<Longrightarrow> path_by_shm s p [(p, h, p')] p'"
apply (rule_tac path = "[]" and p = p in path_by_shm.intros(2))
apply (rule path_by_shm.intros(1))
apply (auto intro:procs_of_shm_prop2 simp:one_flow_shm_def)
done

lemma pbs_prop1:
  "path_by_shm s p path p' \<Longrightarrow> ((path = []) = (p = p')) \<and> (path \<noteq> [] \<longrightarrow> p \<in> set (map Fst path))"
apply (erule path_by_shm.induct, simp)
apply (auto simp:one_flow_shm_def)
done

lemma pbs_prop2:
  "path_by_shm s p path p' \<Longrightarrow> (path = []) = (p = p')"
by (simp add:pbs_prop1)

lemma pbs_prop2':
  "path_by_shm s p path p \<Longrightarrow> path = []"
by (simp add:pbs_prop2)

lemma pbs_prop3:
  "\<lbrakk>path_by_shm s p path p'; path \<noteq> []\<rbrakk> \<Longrightarrow> p \<in> set (map Fst path)"
by (drule pbs_prop1, auto)

lemma pbs_prop4[rule_format]:
  "path_by_shm s p path p'\<Longrightarrow> path \<noteq> [] \<longrightarrow> p' \<in> set (map Trd path)"
by (erule path_by_shm.induct, auto)

lemma pbs_prop5[rule_format]:
  "path_by_shm s p path p' \<Longrightarrow> path \<noteq> [] \<longrightarrow> p' \<notin> set (map Fst path)"
by (erule path_by_shm.induct, auto simp:one_flow_shm_def)

lemma pbs_prop6_aux:
  "path_by_shm s pa pathac pc \<Longrightarrow> valid s \<longrightarrow> (\<forall> pb \<in> set (map Fst pathac). \<exists> pathab pathbc. path_by_shm s pa pathab pb \<and> path_by_shm s pb pathbc pc \<and> pathac = pathbc @ pathab)"
apply (erule path_by_shm.induct)
apply simp
apply clarify
apply (case_tac "pb = p'", simp)
apply (rule_tac x = path in exI, simp)
apply (erule one_step_path, simp)
apply (erule_tac x = pb in ballE, simp_all, clarsimp)
apply (rule_tac x = pathab in exI, simp)
apply (erule pbs2, auto)
done

lemma pbs_prop6:
  "\<lbrakk>path_by_shm s pa pathac pc; pb \<in> set (map Fst pathac); valid s\<rbrakk>
   \<Longrightarrow> \<exists> pathab pathbc. path_by_shm s pa pathab pb \<and> path_by_shm s pb pathbc pc \<and> pathac = pathbc @ pathab"
by (drule pbs_prop6_aux, auto)

lemma pbs_prop7_aux:
  "path_by_shm s pa pathac pc \<Longrightarrow> valid s \<longrightarrow> (\<forall> pb \<in> set (map Trd pathac). \<exists> pathab pathbc. path_by_shm s pa pathab pb \<and> path_by_shm s pb pathbc pc \<and> pathac = pathbc @ pathab)"
apply (erule path_by_shm.induct)
apply simp
apply clarify
apply (case_tac "pb = p''", simp)
apply (rule_tac x = "(p',h,p'') # path" in exI, simp)
apply (rule conjI, erule pbs2, simp+)
apply (rule pbs1, clarsimp simp:one_flow_shm_def procs_of_shm_prop2)
apply (erule_tac x = pb in ballE, simp_all, clarsimp)
apply (rule_tac x = pathab in exI, simp)
apply (erule pbs2, auto)
done

lemma pbs_prop7:
  "\<lbrakk>path_by_shm s pa pathac pc; pb \<in> set (map Trd pathac); valid s\<rbrakk>
   \<Longrightarrow> \<exists> pathab pathbc. path_by_shm s pa pathab pb \<and> path_by_shm s pb pathbc pc \<and> pathac = pathbc @ pathab"
by (drule pbs_prop7_aux, drule mp, simp, erule_tac x = pb in ballE, auto)

lemma pbs_prop8:
  "path_by_shm s p path p' \<Longrightarrow> (set (map Fst path) - {p}) = (set (map Trd path) - {p'})"
proof (induct rule:path_by_shm.induct)
  case (pbs1 p s)
  thus ?case by simp
next
  case (pbs2 s p path p' h p'')
  assume p1:"path_by_shm s p path p'" and p2: "set (map Fst path) - {p} = set (map Trd path) - {p'}"
    and p3: "one_flow_shm s h p' p''" and p4: "p'' \<notin> set (map Fst path)" 
  show "set (map Fst ((p', h, p'') # path)) - {p} = set (map Trd ((p', h, p'') # path)) - {p''}"
    (is "?left = ?right")
  proof (cases "path = []")
    case True
    with p1 have "p = p'" by (drule_tac pbs_prop2, simp)
    thus ?thesis using True
      using p2 by (simp)
  next
    case False
    with p1 have a1: "p \<noteq> p'" by (drule_tac pbs_prop2, simp)
    from p3 have a2: "p' \<noteq> p''" by (simp add:one_flow_shm_def)
    from p1 False have a3: "p' \<in> set (map Trd path)" by (drule_tac pbs_prop4, simp+)
    from p4 p1 False have a4: "p \<noteq> p''" by (drule_tac pbs_prop3, auto)
    with p2 a2 p4 have a5: "p'' \<notin> set (map Trd path)" by auto
    
    have "?left = (set (map Fst path) - {p}) \<union> {p'}" using a1 by auto
    moreover have "... = (set (map Trd path) - {p'}) \<union> {p'}"  
      using p2 by auto
    moreover have "... = set (map Trd path)" using a3 by auto
    moreover have "... = set (map Trd path) - {p''}" using a5 by simp
    moreover have "... = ?right" by simp
    ultimately show ?thesis by simp
  qed
qed

lemma pbs_prop9_aux[rule_format]:
  "path_by_shm s p path p' \<Longrightarrow> h \<in> set (map Snd path) \<and> valid s \<longrightarrow> (\<exists> pa pb patha pathb. path_by_shm s p patha pa \<and> path_by_shm s pb pathb p' \<and> one_flow_shm s h pa pb \<and> path = pathb @ [(pa, h, pb)] @ patha \<and> h \<notin> set (map Snd patha))"
apply (erule path_by_shm.induct, simp)
apply (rule impI, case_tac "h \<in> set (map Snd path)", simp_all)
apply (erule exE|erule conjE)+
apply (rule_tac x = pa in exI, rule_tac x = pb in exI, rule_tac x = patha in exI, simp)
apply (rule pbs2, auto)
apply (rule_tac x = p' in exI, rule_tac x = p'' in exI, rule_tac x = path in exI, simp)
apply (rule pbs1, clarsimp simp:one_flow_shm_def procs_of_shm_prop2)
done

lemma pbs_prop9:
  "\<lbrakk>h \<in> set (map Snd path); path_by_shm s p path p'; valid s\<rbrakk>
   \<Longrightarrow> \<exists> pa pb patha pathb. path_by_shm s p patha pa \<and> path_by_shm s pb pathb p' \<and> 
        one_flow_shm s h pa pb \<and> path = pathb @ [(pa, h, pb)] @ patha \<and> h \<notin> set (map Snd patha)"
by (rule pbs_prop9_aux, auto)

lemma path_by_shm_trans_aux[rule_format]:
  "path_by_shm s p' path' p'' \<Longrightarrow> valid s \<longrightarrow> (\<forall> p path. path_by_shm s p path p' \<longrightarrow> (\<exists> path''. path_by_shm s p path'' p''))"
proof (induct rule:path_by_shm.induct)
  case (pbs1 p s)
  thus ?case
    by (clarify, rule_tac x = path in exI, simp)
next
  case (pbs2 s p path p' h p'')
  hence p1: "path_by_shm s p path p'" and p2: "one_flow_shm s h p' p''" 
    and p3: "valid s \<longrightarrow> (\<forall>pa path. path_by_shm s pa path p \<longrightarrow> (\<exists>path''. path_by_shm s pa path'' p'))"
    and p4: "p'' \<notin> set (map Fst path)" by auto
  show ?case
  proof clarify
    fix pa path'
    assume p5: "path_by_shm s pa path' p" and p6: "valid s"
    with p3 obtain path'' where a1: "path_by_shm s pa path'' p'" by auto
    have p3': "\<forall>pa path. path_by_shm s pa path p \<longrightarrow> (\<exists>path''. path_by_shm s pa path'' p')" 
      using p3 p6 by simp
    show "\<exists>path''. path_by_shm s pa path'' p''"
    proof (cases "p'' \<in> set (map Fst path'')")
      case True
      then obtain res where "path_by_shm s pa res p''" using a1 pbs_prop6 p6 by blast
      thus ?thesis by auto
    next
      case False
      with p2 a1 show ?thesis 
        apply (rule_tac x = "(p', h, p'') # path''" in exI)
        apply (rule path_by_shm.intros(2), auto)
        done
    qed
  qed
qed

lemma path_by_shm_trans:
  "\<lbrakk>path_by_shm s p path p'; path_by_shm s p' path' p''; valid s\<rbrakk> \<Longrightarrow> \<exists> path''. path_by_shm s p path'' p''"
by (drule_tac p' = p' and p'' = p'' in path_by_shm_trans_aux, auto)

lemma path_by_shm_intro1_prop:
  "\<not> path_by_shm s p [] p \<Longrightarrow> p \<notin> current_procs s"
by (auto dest:path_by_shm.intros(1))

lemma path_by_shm_intro3:
  "\<lbrakk>path_by_shm s p path from; (from, SHM_RDWR) \<in> procs_of_shm s h; (to, flag) \<in> procs_of_shm s h; 
    to \<notin> set (map Fst path); from \<noteq> to\<rbrakk>
   \<Longrightarrow> path_by_shm s p ((from, h, to)#path) to"
apply (rule path_by_shm.intros(2), simp_all)
by (auto simp:one_flow_shm_def)

lemma path_by_shm_intro4:
  "\<lbrakk>(p, flag) \<in> procs_of_shm s h; valid s\<rbrakk> \<Longrightarrow> path_by_shm s p [] p"
by (drule procs_of_shm_prop2, simp, simp add:path_by_shm.intros(1))

lemma path_by_shm_intro5:
  "\<lbrakk>(from, SHM_RDWR) \<in> procs_of_shm s h; (to,flag) \<in> procs_of_shm s h; valid s; from \<noteq> to\<rbrakk>
   \<Longrightarrow> path_by_shm s from [(from, h, to)] to"
apply (rule_tac p' = "from" and h = h in path_by_shm.intros(2))
apply (rule path_by_shm.intros(1), simp add:procs_of_shm_prop2)
apply (simp add:one_flow_shm_def, rule_tac x = flag in exI, auto)
done

(* p'' \<notin> set (map Fst path): not duplicated target process;
 * p1 - ha \<rightarrow> p2; p2 - hb \<rightarrow> p3; p3 - ha \<rightarrow> p4; so path_by_shm p1 [(p3,ha,p4), (p2,hb,p3),(p1,ha,p2)] p4,
 * but this could be also path_by_shm p1 [(p1,ha,p4)] p4, so the former one is redundant!  *)

inductive path_by_shm':: "t_state \<Rightarrow> t_process \<Rightarrow> t_edge_shm list \<Rightarrow> t_process \<Rightarrow> bool"
where
  pbs1': "p \<in> current_procs s \<Longrightarrow> path_by_shm' s p [] p"
| pbs2': "\<lbrakk>path_by_shm s p path p'; one_flow_shm s h p' p''; p'' \<notin> set (map Fst path); 
           h \<notin> set (map Snd path)\<rbrakk> 
          \<Longrightarrow> path_by_shm' s p ((p', h, p'')# path) p''"

lemma pbs_imp_pbs'[rule_format]:
  "path_by_shm s p path p' \<Longrightarrow> valid s \<longrightarrow> (\<exists> path'. path_by_shm' s p path' p')"
apply (erule path_by_shm.induct)
apply (rule impI, rule_tac x = "[]" in exI, erule pbs1')
apply (rule impI, simp,  erule exE, case_tac "h \<in> set (map Snd path)")
apply (drule_tac s = s and p = p and p' = p' in pbs_prop9, simp+) defer
apply (rule_tac x = "(p', h, p'') # path" in exI, erule pbs2', simp+) 
apply ((erule exE|erule conjE)+)
apply (rule_tac x = "(pa, h, p'') # patha" in exI)
apply (erule pbs2', auto simp:one_flow_shm_def)
done

lemma pbs'_imp_pbs[rule_format]:
  "path_by_shm' s p path p' \<Longrightarrow> valid s \<longrightarrow> (\<exists> path'. path_by_shm s p path' p')"
apply (erule path_by_shm'.induct)
apply (rule impI, rule_tac x = "[]" in exI, simp add:pbs1)
apply (rule impI, rule_tac x = "(p', h, p'') # path" in exI, simp add:pbs2)
done

definition flow_by_shm :: "t_state \<Rightarrow> t_process \<Rightarrow> t_process \<Rightarrow> bool"
where
  "flow_by_shm s p p' \<equiv> \<exists> path. path_by_shm s p path p'"

lemma flow_by_shm_intro':
  "valid s \<Longrightarrow> flow_by_shm s p p' = (\<exists> path. path_by_shm' s p path p')"
by (auto simp:flow_by_shm_def intro: pbs_imp_pbs' pbs'_imp_pbs)

lemma one_step_flows: "\<lbrakk>one_flow_shm s h p p'; valid s\<rbrakk> \<Longrightarrow> flow_by_shm s p p'"
by (drule one_step_path, auto simp:flow_by_shm_def)

lemma flow_by_shm_trans:
  "\<lbrakk>flow_by_shm s p p'; flow_by_shm s p' p''; valid s\<rbrakk> \<Longrightarrow> flow_by_shm s p p''"
by (auto simp:flow_by_shm_def intro!:path_by_shm_trans)

lemma flow_by_shm_intro1_prop:
  "\<not> flow_by_shm s p p \<Longrightarrow> p \<notin> current_procs s"
by (auto dest:path_by_shm.intros(1) simp:flow_by_shm_def)

lemma flow_by_shm_intro1:
  "p \<in> current_procs s \<Longrightarrow> flow_by_shm s p p"
by (auto dest:path_by_shm.intros(1) simp:flow_by_shm_def)

lemma flow_by_shm_intro2:
  "\<lbrakk>flow_by_shm s p p'; one_flow_shm s h p' p''; valid s\<rbrakk> \<Longrightarrow> flow_by_shm s p p''"
by (auto intro:flow_by_shm_trans dest:one_step_flows)

lemma flow_by_shm_intro3:
  "\<lbrakk>flow_by_shm s p from; (from, SHM_RDWR) \<in> procs_of_shm s h; (to, flag) \<in> procs_of_shm s h; from \<noteq> to; valid s\<rbrakk>
   \<Longrightarrow> flow_by_shm s p to"
apply (rule flow_by_shm_intro2)
by (auto simp:one_flow_shm_def)

lemma flow_by_shm_intro4:
  "\<lbrakk>(p, flag) \<in> procs_of_shm s h; valid s\<rbrakk> \<Longrightarrow> flow_by_shm s p p"
by (drule procs_of_shm_prop2, simp, simp add:flow_by_shm_intro1)

lemma flow_by_shm_intro5:
  "\<lbrakk>(from, SHM_RDWR) \<in> procs_of_shm s h; (to,flag) \<in> procs_of_shm s h; valid s; from \<noteq> to\<rbrakk>
   \<Longrightarrow> flow_by_shm s from  to"
apply (rule_tac p' = "from" and h = h in flow_by_shm_intro2)
apply (rule flow_by_shm_intro1, simp add:procs_of_shm_prop2)
apply (simp add:one_flow_shm_def, rule_tac x = flag in exI, auto)
done

lemma flow_by_shm_intro6:
  "path_by_shm s p path p' \<Longrightarrow> flow_by_shm s p p'"
by (auto simp:flow_by_shm_def)
(********* simpset for inductive Info_flow_shm **********)
term edge_related
lemma path_by_shm_detach1_aux:
  "path_by_shm s' pa path pb \<Longrightarrow> valid (Detach p h # s) \<and> (s' = Detach p h # s) 
     \<longrightarrow> \<not> edge_related path p h \<and> path_by_shm s pa path pb"
apply (erule path_by_shm.induct, simp)
apply (rule impI, rule path_by_shm.intros(1), simp+)
by (auto simp:one_flow_shm_def split:if_splits intro:path_by_shm_intro3)

lemma path_by_shm_detach1:
  "\<lbrakk>path_by_shm (Detach p h # s) pa path pb; valid (Detach p h # s)\<rbrakk> 
   \<Longrightarrow> \<not> edge_related path p h \<and> path_by_shm s pa path pb"
by (auto dest:path_by_shm_detach1_aux)

lemma path_by_shm_detach2_aux[rule_format]:
  "path_by_shm s pa path pb \<Longrightarrow> valid (Detach p h # s) \<and> \<not> edge_related path p h 
   \<longrightarrow> path_by_shm (Detach p h # s) pa path pb"
apply (induct rule:path_by_shm.induct)
apply (rule impI, rule path_by_shm.intros(1), simp)
apply (rule impI, erule conjE, simp split:if_splits)
apply (rule path_by_shm.intros(2), simp)
apply (simp add:one_flow_shm_detach)
apply (rule impI, simp+)
done

lemma path_by_shm_detach2:
  "\<lbrakk>valid (Detach p h # s); \<not> edge_related path p h; path_by_shm s pa path pb\<rbrakk> 
   \<Longrightarrow> path_by_shm (Detach p h # s) pa path pb"
by (auto intro!:path_by_shm_detach2_aux)

lemma path_by_shm_detach:
  "valid (Detach p h # s) \<Longrightarrow>
   path_by_shm (Detach p h # s) pa path pb = (\<not> edge_related path p h  \<and> path_by_shm s pa path pb)"
by (auto dest:path_by_shm_detach1 path_by_shm_detach2)

lemma flow_by_shm_detach:
  "valid (Detach p h # s) \<Longrightarrow> 
   flow_by_shm (Detach p h # s) pa pb = (\<exists> path. \<not> edge_related path p h \<and> path_by_shm s pa path pb)"
by (auto dest:path_by_shm_detach simp:flow_by_shm_def)

lemma path_by_shm_attach1_aux:
  "path_by_shm s' pa path pb \<Longrightarrow> valid s' \<and> (s' = Attach p h flag # s) \<longrightarrow>
     (path_by_shm s pa path pb) \<or>
     (\<exists> path1 path2 p'. path_by_shm s pa path1 p' \<and> path_by_shm s p path2 pb \<and> 
         (p', SHM_RDWR) \<in> procs_of_shm s h \<and> path = path2 @ [(p', h, p)] @ path1 ) \<or>
     (\<exists> path1 path2 p' flag'. path_by_shm s pa path1 p \<and> path_by_shm s p' path2 pb \<and> 
         (p', flag') \<in> procs_of_shm s h \<and> path = path2 @ [(p, h, p')] @ path1 \<and> flag = SHM_RDWR)"
apply (erule path_by_shm.induct)
apply (simp, rule impI, rule pbs1, simp)
apply (rule impI, erule impE, clarsimp)
apply (erule disjE)
apply (clarsimp simp:one_flow_shm_attach split:if_splits)
apply (erule disjE, clarsimp)
apply (erule_tac x = path in allE, clarsimp)
apply (erule impE, rule pbs1, erule procs_of_shm_prop2, erule vd_cons, simp)
apply (erule disjE, clarsimp)
apply (rule_tac x = path in exI, rule_tac x = "[]" in exI, rule_tac x = p' in exI, simp)
apply (rule pbs1, drule vt_grant_os, clarsimp)
apply (drule_tac p = pa and p' = p' and p'' = p'' in pbs2, simp+)
apply (drule_tac p = pa and p' = p' and p'' = p'' in pbs2, simp+)

apply (erule disjE)
apply ((erule exE|erule conjE)+, simp split:if_splits add:one_flow_shm_attach)
apply (clarsimp simp:one_flow_shm_attach split:if_splits)
apply (erule disjE, clarsimp)
apply (clarsimp)


apply (erule conjE)+



apply (erule conjE, clarsimp simp only:one_flow_shm_attach split:if_splits)
apply simp



lemma path_by_shm_attach1_aux:
  "path_by_shm s' pa path pb \<Longrightarrow> valid s' \<and> (s' = Attach p h flag # s) \<longrightarrow> 
     path_by_shm s pa path pb \<or>
      (if (pa = p \<and> flag = SHM_RDWR)
       then \<exists> p' flagb path'. (p', flagb) \<in> procs_of_shm s h \<and> 
               path_by_shm s p' path' pb \<and> path = path' @ [(p, h, p')]
       else if (pb = p)
            then \<exists> p' path'. path_by_shm s pa path' p' \<and> path = (p', h, p) # path' \<and> 
                   (p', SHM_RDWR) \<in> procs_of_shm s h
            else (\<exists> p' flag' patha pathb. path_by_shm s pa patha p \<and> flag = SHM_RDWR \<and> 
                   (p', flag') \<in> procs_of_shm s h \<and> path_by_shm s p' pathb pb \<and> 
                   path = pathb @ [(p, h, p')] @ patha) \<or>
                 (\<exists> p' patha pathb. path_by_shm s pa patha p' \<and> (p', SHM_RDWR) \<in> procs_of_shm s h \<and> 
                   path_by_shm s p pathb pb \<and> path = pathb @ [(p', h, p)] @ patha))"
proof (induct rule:path_by_shm.induct)
  case (pbs1 proc \<tau>)
  show ?case
  proof (rule impI)
    assume pre: "valid \<tau> \<and> \<tau> = Attach p h flag # s"
    from pbs1 pre have "proc \<in> current_procs s" by simp 
    thus "path_by_shm s proc [] proc \<or>
         (if proc = p \<and> flag = SHM_RDWR
          then \<exists>p' flagb path'.
                  (p', flagb) \<in> procs_of_shm s h \<and> path_by_shm s p' path' proc \<and> [] = path' @ [(p, h, p')]
          else if proc = p
               then \<exists>p' path'.
                       path_by_shm s proc path' p' \<and> [] = (p', h, p) # path' \<and> (p', SHM_RDWR) \<in> procs_of_shm s h
               else (\<exists>p' flag' patha pathb.
                        path_by_shm s proc patha p \<and>
                        flag = SHM_RDWR \<and>
                        (p', flag') \<in> procs_of_shm s h \<and>
                        path_by_shm s p' pathb proc \<and> [] = pathb @ [(p, h, p')] @ patha) \<or>
                    (\<exists>p' patha pathb.
                        path_by_shm s proc patha p' \<and>
                        (p', SHM_RDWR) \<in> procs_of_shm s h \<and>
                        path_by_shm s p pathb proc \<and> [] = pathb @ [(p', h, p)] @ patha))"
      by (auto intro:path_by_shm.intros)
  qed
next
  case (pbs2 \<tau> pa path pb h' pc)
  thus ?case
  proof (rule_tac impI)
    assume p1:"path_by_shm \<tau> pa path pb" and p2: "valid \<tau> \<and> \<tau> = Attach p h flag # s \<longrightarrow>
     path_by_shm s pa path pb \<or>
     (if pa = p \<and> flag = SHM_RDWR
      then \<exists>p' flagb path'. (p', flagb) \<in> procs_of_shm s h \<and> path_by_shm s p' path' pb \<and> path = path' @ [(p, h, p')]
      else if pb = p
           then \<exists>p' path'. path_by_shm s pa path' p' \<and> path = (p', h, p) # path' \<and> (p', SHM_RDWR) \<in> procs_of_shm s h
           else (\<exists>p' flag' pathaa pathb. path_by_shm s pa pathaa p \<and> flag = SHM_RDWR \<and> 
                    (p', flag') \<in> procs_of_shm s h \<and> path_by_shm s p' pathb pb \<and> 
                    path = pathb @ [(p, h, p')] @ pathaa) \<or>
                (\<exists>p' pathaa pathb. path_by_shm s pa pathaa p' \<and> (p', SHM_RDWR) \<in> procs_of_shm s h \<and>
                    path_by_shm s p pathb pb \<and> path = pathb @ [(p', h, p)] @ pathaa))"
      and p3: "one_flow_shm \<tau> h' pb pc" and p4: "valid \<tau> \<and> \<tau> = Attach p h flag # s"
    
    from p2 and p4 have p2': "
      path_by_shm s pa path pb \<or>
     (if pa = p \<and> flag = SHM_RDWR
      then \<exists>p' flagb path'. (p', flagb) \<in> procs_of_shm s h \<and> path_by_shm s p' path' pb \<and> path = path' @ [(p, h, p')]
      else if pb = p
           then \<exists>p' path'. path_by_shm s pa path' p' \<and> path = (p', h, p) # path' \<and> (p', SHM_RDWR) \<in> procs_of_shm s h
           else (\<exists>p' flag' pathaa pathb. path_by_shm s pa pathaa p \<and> flag = SHM_RDWR \<and> 
                    (p', flag') \<in> procs_of_shm s h \<and> path_by_shm s p' pathb pb \<and> 
                    path = pathb @ [(p, h, p')] @ pathaa) \<or>
                (\<exists>p' pathaa pathb. path_by_shm s pa pathaa p' \<and> (p', SHM_RDWR) \<in> procs_of_shm s h \<and>
                    path_by_shm s p pathb pb \<and> path = pathb @ [(p', h, p)] @ pathaa))"
      by (erule_tac impE, simp)
    from p4 have p5: "valid s" and p6: "os_grant s (Attach p h flag)" by (auto intro:vd_cons dest:vt_grant_os)
    from p6 have "p \<in> current_procs s" by simp hence p7:"path_by_shm s p [] p" by (erule_tac path_by_shm.intros)
    from p3 p4 have p8: "if (h' = h) 
     then (pb = p \<and> pc \<noteq> p \<and> flag = SHM_RDWR \<and> (\<exists> flagb. (pc, flagb) \<in> procs_of_shm s h)) \<or>
          (pc = p \<and> pb \<noteq> p \<and> (pb, SHM_RDWR) \<in> procs_of_shm s h) \<or>
          (one_flow_shm s h pb pc)               
     else one_flow_shm s h' pb pc" by (auto simp add:one_flow_shm_attach) 
    
    
(*
    have "\<And> flagb. (pc, flagb) \<in> procs_of_shm s h 
      \<Longrightarrow> \<exists> p' flagb. (p', flagb) \<in> procs_of_shm s h \<and> path_by_shm s p' [] pc"
      apply (rule_tac x= pc in exI, rule_tac x = flagb in exI, frule procs_of_shm_prop2)
      by (simp add:p5, simp add:path_by_shm.intros(1))
    hence p10: "\<not> path_by_shm s p path pc \<Longrightarrow> (\<exists>p' flagb. (p', flagb) \<in> procs_of_shm s h \<and> path_by_shm s p'  pc) \<or>
      path_by_shm s pa pc"
      using p2' p7 p8 p5
      by (auto split:if_splits dest:path_by_shm.intros(2))      
  (*     apply (rule_tac x = pb in exI, simp add:one_flow_flows, rule_tac x = flagb in exI, simp)+  *) *)

    from p1 have a0: "(path = []) = (pa = pb)" using pbs_prop2 by simp
    have a1:"\<lbrakk>pa = p; flag = SHM_RDWR; \<not> path_by_shm s pa path pb\<rbrakk> \<Longrightarrow> 
      \<exists>p' flagb path'. (p', flagb) \<in> procs_of_shm s h \<and> path_by_shm s p' path' pb \<and> path = path' @ [(p, h, p')]"
      using p2' by auto
    have b1: "\<lbrakk>pa = p; flag = SHM_RDWR; \<not> path_by_shm s pa path pc\<rbrakk> \<Longrightarrow> 
      \<exists>p' flagb path'. (p', flagb) \<in> procs_of_shm s h \<and> path_by_shm s p' path' pc \<and>
        (pb, h', pc) # path = path' @ [(p, h, p')]"
      
      
      using p8 a1 p7 p5 a0 
      apply (auto split:if_splits elim:path_by_shm_intro4)
      apply (rule_tac x = pb in exI, rule conjI, rule_tac x = SHM_RDWR in exI, simp)
      apply (rule_tac x = pc in exI, rule conjI, rule_tac x = flagb in exI, simp)
      apply (rule_tac x = "[]" in exI, rule conjI)
apply (erule path_by_shm_intro4, simp)

      apply (case_tac "path_by_shm s pa path pb", simp) defer
      apply (drule a1, simp+, clarsimp)
      apply (rule conjI, rule_tac x = flagb in exI, simp)
      apply (rule path_by_shm_
      using p2' p8 p5
      apply (auto split:if_splits dest!:pbs_prop2' simp:path_by_shm_intro4)
      apply (drule pbs_prop2', simp)
      apply (erule_tac x = pc in allE, simp add:path_by_shm_intro4)
     
      apply (drule_tac x = "pc" in allE)
      
      apply simp

      sorry
    moreover have "pc = p \<Longrightarrow> (\<exists>p' path'. path_by_shm s pa path' p' \<and>
             (pb, h', pc) # path = path' @ [(p', h, p)] \<and> (p', SHM_RDWR) \<in> procs_of_shm s h) \<or>
      (path_by_shm s pa path pc \<and> \<not> edge_related path p h)"
      using p2' p7 p8 p5
      sorry (*
      apply (auto split:if_splits intro:path_by_shm_intro3 simp:one_flow_shm_def) *)
    moreover have "\<lbrakk>pc \<noteq> p; pa \<noteq> p \<or> flag \<noteq> SHM_RDWR\<rbrakk> \<Longrightarrow> 
      (\<exists>p' flag' pathaa pathb. path_by_shm s pa pathaa p \<and> flag = SHM_RDWR \<and> (p', flag') \<in> procs_of_shm s h \<and>
           path_by_shm s p' pathb pc \<and> (pb, h', pc) # path = pathaa @ [(p, h, p')] @ pathb) \<or>
      (\<exists>p' pathaa pathb. path_by_shm s pa pathaa p' \<and> (p', SHM_RDWR) \<in> procs_of_shm s h \<and>
           path_by_shm s p pathb pc \<and> (pb, h', pc) # path = pathaa @ [(p', h, p)] @ pathb) \<or>
      (path_by_shm s pa path pc \<and> \<not> edge_related path p h)"
      using p2' p7 p8 p5 (*
      apply (auto split:if_splits intro:path_by_shm_intro3 simp:one_flow_shm_def)
      apply (rule_tac x = pc in exI, simp add:path_by_shm_intro4)
      apply (rule_tac x = flagb in exI, simp)      
      done *)
      sorry
    ultimately  
    show "if (pb, h', pc) # path = [] then pa = pc \<and> pa \<in> current_procs s
       else path_by_shm s pa ((pb, h', pc) # path) pc \<and> \<not> edge_related ((pb, h', pc) # path) p h \<or>
       (if pa = p \<and> flag = SHM_RDWR
        then \<exists>p' flagb path'. (p', flagb) \<in> procs_of_shm s h \<and>
                path_by_shm s p' path' pc \<and> (pb, h', pc) # path = path' @ [(p, h, p')]
        else if pc = p
             then \<exists>p' path'. path_by_shm s pa path' p' \<and>
                     (pb, h', pc) # path = (p', h, p) # path' \<and> (p', SHM_RDWR) \<in> procs_of_shm s h
             else (\<exists>p' flag' pathaa pathb. path_by_shm s pa pathaa p \<and> flag = SHM_RDWR \<and>
                      (p', flag') \<in> procs_of_shm s h \<and>
                      path_by_shm s p' pathb pc \<and> (pb, h', pc) # path = pathb @ [(p, h, p')] @ pathaa) \<or>
                  (\<exists>p' pathaa pathb. path_by_shm s pa pathaa p' \<and> (p', SHM_RDWR) \<in> procs_of_shm s h \<and>
                      path_by_shm s p pathb pc \<and> (pb, h', pc) # path = pathb @ [(p', h, p)] @ pathaa))"
        apply (auto split:if_splits)
        using p7 by auto
  qed
qed

lemma path_by_shm_attach1:
  "\<lbrakk>valid (Attach p h flag # s); path_by_shm (Attach p h flag # s) pa pb\<rbrakk>
   \<Longrightarrow> (if path_by_shm s pa pb then True else
     (if (pa = p \<and> flag = SHM_RDWR) 
      then (\<exists> p' flagb. (p', flagb) \<in> procs_of_shm s h \<and> path_by_shm s p' pb)
      else if (pb = p) 
           then (\<exists> p'. (p', SHM_RDWR) \<in> procs_of_shm s h \<and> path_by_shm s pa p')
           else (\<exists> p' flag'. path_by_shm s pa p \<and> flag = SHM_RDWR \<and> (p', flag') \<in> procs_of_shm s h \<and> 
                             path_by_shm s p' pb) \<or>
                (\<exists> p'. path_by_shm s pa p' \<and> (p', SHM_RDWR) \<in> procs_of_shm s h \<and> path_by_shm s p pb)
     )  )"
apply (drule_tac p = p and h = h and flag = flag in path_by_shm_attach1_aux)
by auto

lemma path_by_shm_attach_aux[rule_format]:
  "path_by_shm s pa pb \<Longrightarrow> valid (Attach p h flag # s) \<longrightarrow> path_by_shm (Attach p h flag # s) pa pb"
apply (erule path_by_shm.induct)
apply (rule impI, rule path_by_shm.intros(1), simp)
apply (rule impI, simp, rule_tac h = ha in path_by_shm.intros(2), simp)
apply (auto simp add:one_flow_shm_simps)
done

lemma path_by_shm_attach2:
  "\<lbrakk>valid (Attach p h flag # s); if path_by_shm s pa pb then True else
     (if (pa = p \<and> flag = SHM_RDWR) 
      then (\<exists> p' flagb. (p', flagb) \<in> procs_of_shm s h \<and> path_by_shm s p' pb)
      else if (pb = p) 
           then (\<exists> p'. (p', SHM_RDWR) \<in> procs_of_shm s h \<and> path_by_shm s pa p')
           else (\<exists> p' flag'. path_by_shm s pa p \<and> flag = SHM_RDWR \<and> (p', flag') \<in> procs_of_shm s h \<and> 
                             path_by_shm s p' pb) \<or>
                (\<exists> p'. path_by_shm s pa p' \<and> (p', SHM_RDWR) \<in> procs_of_shm s h \<and> path_by_shm s p pb))\<rbrakk>
   \<Longrightarrow> path_by_shm (Attach p h flag # s) pa pb"
apply (frule vt_grant_os, frule vd_cons)
apply (auto split:if_splits intro:path_by_shm_intro3 simp:one_flow_shm_def intro:path_by_shm_attach_aux)
apply (rule_tac p' = p' in Info_flow_trans)
apply (rule_tac p' = p and h = h in path_by_shm.intros(2))
apply (rule path_by_shm.intros(1), simp)
apply (simp add:one_flow_shm_simps, simp add:one_flow_shm_def)
apply (rule conjI, rule notI, simp, rule_tac x = flagb in exI, simp)
apply (simp add:path_by_shm_attach_aux)

apply (rule_tac p' = p' in Info_flow_trans)
apply (rule_tac p' = p in Info_flow_trans)
apply (simp add:path_by_shm_attach_aux)
apply (rule_tac p' = p and h = h in path_by_shm.intros(2))
apply (rule path_by_shm.intros(1), simp)
apply (simp add:one_flow_shm_simps, simp add:one_flow_shm_def)
apply (rule conjI, rule notI, simp, rule_tac x = flag' in exI, simp)
apply (simp add:path_by_shm_attach_aux)

apply (rule_tac p' = p in Info_flow_trans)
apply (rule_tac p' = p' in Info_flow_trans)
apply (simp add:path_by_shm_attach_aux)
apply (rule_tac p' = p' and h = h in path_by_shm.intros(2))
apply (rule path_by_shm.intros(1), simp add:procs_of_shm_prop2)
apply (simp add:one_flow_shm_simps, simp add:one_flow_shm_def)
apply (rule notI, simp)
apply (simp add:path_by_shm_attach_aux)

apply (rule_tac p' = p in Info_flow_trans)
apply (rule_tac p' = p' in Info_flow_trans)
apply (simp add:path_by_shm_attach_aux)
apply (rule_tac p' = p' and h = h in path_by_shm.intros(2))
apply (rule path_by_shm.intros(1), simp add:procs_of_shm_prop2)
apply (simp add:one_flow_shm_simps, simp add:one_flow_shm_def)
apply (rule notI, simp)
apply (simp add:path_by_shm_attach_aux)
done

lemma path_by_shm_attach:
  "valid (Attach p h flag # s) \<Longrightarrow> path_by_shm (Attach p h flag # s) = (\<lambda> pa pb. 
     path_by_shm s pa pb \<or>
     (if (pa = p \<and> flag = SHM_RDWR) 
      then (\<exists> p' flagb. (p', flagb) \<in> procs_of_shm s h \<and> path_by_shm s p' pb)
      else if (pb = p) 
           then (\<exists> p'. (p', SHM_RDWR) \<in> procs_of_shm s h \<and> path_by_shm s pa p')
           else (\<exists> p' flag'. path_by_shm s pa p \<and> flag = SHM_RDWR \<and> (p', flag') \<in> procs_of_shm s h \<and> 
                             path_by_shm s p' pb) \<or>
                (\<exists> p'. path_by_shm s pa p' \<and> (p', SHM_RDWR) \<in> procs_of_shm s h \<and> path_by_shm s p pb)
     )  )"
apply (rule ext, rule ext, rule iffI)
apply (drule_tac pa = pa and pb = pb in path_by_shm_attach1, simp)
apply (auto split:if_splits)[1]
apply (drule_tac pa = pa and pb = pb in path_by_shm_attach2)
apply (auto split:if_splits)
done
























lemma info_flow_shm_detach:
  "valid (Detach p h # s) \<Longrightarrow> info_flow_shm (Detach p h # s) = (\<lambda> pa pb. 
     self_shm s pa pb \<or> ((p = pa \<or> p = pb) \<and> (\<exists> h'. h' \<noteq> h \<and> one_flow_shm s h' pa pb)) \<or>
     (pa \<noteq> p \<and> pb \<noteq> p \<and> info_flow_shm s pa pb) )"
apply (rule ext, rule ext, frule vt_grant_os)
by (auto simp:info_flow_shm_def one_flow_shm_def)

lemma info_flow_shm_deleteshm:
  "valid (DeleteShM p h # s) \<Longrightarrow> info_flow_shm (DeleteShM p h # s) = (\<lambda> pa pb. 
     self_shm s pa pb \<or> (\<exists> h'. h' \<noteq> h \<and> one_flow_shm s h' pa pb)     )"
apply (rule ext, rule ext, frule vt_grant_os)
by (auto simp:info_flow_shm_def one_flow_shm_def)

lemma info_flow_shm_clone:
  "valid (Clone p p' fds shms # s) \<Longrightarrow> info_flow_shm (Clone p p' fds shms # s) = (\<lambda> pa pb. 
     (pa = p' \<and> pb = p') \<or> (pa = p' \<and> pb \<noteq> p' \<and> (\<exists> h \<in> shms. one_flow_shm s h p pb)) \<or> 
     (pb = p' \<and> pa \<noteq> p' \<and> (\<exists> h \<in> shms. one_flow_shm s h pa p)) \<or> 
     (pa \<noteq> p' \<and> pb \<noteq> p' \<and> info_flow_shm s pa pb))"
apply (rule ext, rule ext, frule vt_grant_os, frule vd_cons, clarsimp)
apply (frule_tac p = p' in procs_of_shm_prop2', simp)
sorry (*
apply (auto simp:info_flow_shm_def one_flow_shm_def)
done *)

lemma info_flow_shm_execve:
  "valid (Execve p f fds # s) \<Longrightarrow> info_flow_shm (Execve p f fds # s) = (\<lambda> pa pb. 
     (pa = p \<and> pb = p) \<or> (pa \<noteq> p \<and> pb \<noteq> p \<and> info_flow_shm s pa pb)    )"
apply (rule ext, rule ext, frule vt_grant_os, frule vd_cons)
by (auto simp:info_flow_shm_def one_flow_shm_def)

lemma info_flow_shm_kill:
  "valid (Kill p p' # s) \<Longrightarrow> info_flow_shm (Kill p p' # s) = (\<lambda> pa pb. 
     pa \<noteq> p' \<and> pb \<noteq> p' \<and> info_flow_shm s pa pb                 )"
apply (rule ext, rule ext, frule vt_grant_os, frule vd_cons)
by (auto simp:info_flow_shm_def one_flow_shm_def)

lemma info_flow_shm_exit:
  "valid (Exit p # s) \<Longrightarrow> info_flow_shm (Exit p # s) = (\<lambda> pa pb. 
     pa \<noteq> p \<and> pb \<noteq> p \<and> info_flow_shm s pa pb                          )"
apply (rule ext, rule ext, frule vt_grant_os, frule vd_cons)
by (auto simp:info_flow_shm_def one_flow_shm_def)

lemma info_flow_shm_other:
  "\<lbrakk>valid (e # s); 
    \<forall> p h flag. e \<noteq> Attach p h flag;
    \<forall> p h. e \<noteq> Detach p h;
    \<forall> p h. e \<noteq> DeleteShM p h;
    \<forall> p p' fds shms. e \<noteq> Clone p p' fds shms;
    \<forall> p f fds. e \<noteq> Execve p f fds;
    \<forall> p p'. e \<noteq> Kill p p';
    \<forall> p. e \<noteq> Exit p
   \<rbrakk> \<Longrightarrow> info_flow_shm (e # s) = info_flow_shm s"
apply (rule ext, rule ext, frule vt_grant_os, frule vd_cons)
apply (case_tac e, auto simp:info_flow_shm_def one_flow_shm_def dest:procs_of_shm_prop2)
apply (erule_tac x = h in allE, simp)
apply (drule procs_of_shm_prop1, auto)
done


(*
lemma info_flow_shm_prop1: 
  "\<lbrakk>info_flow_shm s p p'; p \<noteq> p'; valid s\<rbrakk> 
   \<Longrightarrow> \<exists> h h' flag. (p, SHM_RDWR) \<in> procs_of_shm s h \<and> (p', flag) \<in> procs_of_shm s h'"
by (induct rule: info_flow_shm.induct, auto)

lemma info_flow_shm_cases:
  "\<lbrakk>info_flow_shm \<tau> pa pb; \<And>p s. \<lbrakk>s = \<tau> ; pa = p; pb = p; p \<in> current_procs s\<rbrakk> \<Longrightarrow> P;
  \<And>s p p' h p'' flag. \<lbrakk>s = \<tau>; pa = p; pb = p''; info_flow_shm s p p'; (p', SHM_RDWR) \<in> procs_of_shm s h;
                       (p'', flag) \<in> procs_of_shm s h\<rbrakk>\<Longrightarrow> P\<rbrakk>
  \<Longrightarrow> P"
by (erule info_flow_shm.cases, auto)

definition one_flow_shm :: "t_state \<Rightarrow> t_process \<Rightarrow> t_process \<Rightarrow> bool"
where
  "one_flow_shm s p p' \<equiv> p \<noteq> p' \<and> (\<exists> h flag. (p, SHM_RDWR) \<in> procs_of_shm s h \<and> (p', flag) \<in> procs_of_shm s h)"

inductive flows_shm :: "t_state \<Rightarrow> t_process \<Rightarrow> t_process \<Rightarrow> bool"
where
  "p \<in> current_procs s \<Longrightarrow> flows_shm s p p"
| "\<lbrakk>flows_shm s p p'; one_flow_shm s p' p''\<rbrakk> \<Longrightarrow> flows_shm s p p''"

definition attached_procs :: "t_state \<Rightarrow> t_shm \<Rightarrow> t_process set"
where
  "attached_procs s h \<equiv> {p. \<exists> flag. (p, flag) \<in> procs_of_shm s h}"

definition flowed_procs:: "t_state \<Rightarrow> t_shm \<Rightarrow> t_process set"
where
  "flowed_procs s h \<equiv> {p'. \<exists> p \<in> attached_procs s h. flows_shm s p p'}"

inductive flowed_shm:: "t_state \<Rightarrow> t_process \<Rightarrow> t_shm set"

fun Info_flow_shm :: "t_state \<Rightarrow> t_process \<Rightarrow> t_process set"
where
  "Info_flow_shm [] = (\<lambda> p. {p'. flows_shm [] p p'})"
| "Info_flow_shm (Attach p h flag # s) = (\<lambda> p'. 
     if (p' = p) then flowed_procs s h 
     else if ()
    "


lemma info_flow_shm_attach:
  "valid (Attach p h flag # s) \<Longrightarrow> info_flow_shm (Attach p h flag # s) = (\<lambda> pa pb. (info_flow_shm s pa pb) \<or> 
     (if (pa = p) 
      then (if (flag = SHM_RDWR) 
            then (\<exists> flag. (pb, flag) \<in> procs_of_shm s h)
            else (pb = p)) 
      else (if (pb = p) 
            then (pa, SHM_RDWR) \<in> procs_of_shm s h
            else info_flow_shm s pa pb)) )"
apply (frule vd_cons, frule vt_grant_os, rule ext, rule ext)
apply (case_tac "info_flow_shm s pa pb", simp)

thm info_flow_shm.cases
apply (auto split:if_splits intro:info_flow_shm.intros elim:info_flow_shm_cases)
apply (erule info_flow_shm_cases, simp, simp split:if_splits)
apply (rule_tac p = pa and p' = p' in info_flow_shm.intros(2), simp+)
apply (rule notI, erule info_flow_shm.cases, simp+)
pr 5
*)
lemmas info_flow_shm_simps = info_flow_shm_other (* info_flow_shm_attach *) info_flow_shm_detach info_flow_shm_deleteshm
  info_flow_shm_clone info_flow_shm_execve info_flow_shm_kill info_flow_shm_exit

lemma has_same_inode_in_current:
  "\<lbrakk>has_same_inode s f f'; valid s\<rbrakk> \<Longrightarrow> f \<in> current_files s \<and> f' \<in> current_files s"
by (auto simp add:has_same_inode_def current_files_def)

lemma has_same_inode_prop1:
  "\<lbrakk>has_same_inode s f f'; is_file s f; valid s\<rbrakk> \<Longrightarrow> is_file s f'"
by (auto simp:has_same_inode_def is_file_def)

lemma has_same_inode_prop1':
  "\<lbrakk>has_same_inode s f f'; is_file s f'; valid s\<rbrakk> \<Longrightarrow> is_file s f"
by (auto simp:has_same_inode_def is_file_def)

lemma has_same_inode_prop2:
  "\<lbrakk>has_same_inode s f f'; file_of_proc_fd s p fd = Some f; valid s\<rbrakk> \<Longrightarrow> is_file s f'"
apply (drule has_same_inode_prop1)
apply (simp add:file_of_pfd_is_file, simp+)
done

lemma has_same_inode_prop2':
  "\<lbrakk>has_same_inode s f f'; file_of_proc_fd s p fd = Some f'; valid s\<rbrakk> \<Longrightarrow> is_file s f"
apply (drule has_same_inode_prop1')
apply (simp add:file_of_pfd_is_file, simp+)
done

lemma tobj_in_init_alive:
  "tobj_in_init obj \<Longrightarrow> init_alive obj"
by (case_tac obj, auto)

lemma tobj_in_alive:
  "tobj_in_init obj \<Longrightarrow> alive [] obj"
by (case_tac obj, auto simp:is_file_nil)

end

end