(*<*)
theory S2ss_prop
imports Main Flask Flask_type Static Static_type Init_prop Tainted_prop Valid_prop Alive_prop Co2sobj_prop Dalive_prop
begin
(*>*)

context tainting_s begin

(* simpset for s2ss*)

lemma s2ss_execve':
  "valid (Execve p f fds # s) \<Longrightarrow> s2ss (Execve p f fds # s) = (
     if (\<exists> p'. p' \<noteq> p \<and> p' \<in> current_procs s \<and> co2sobj s (D_proc p') = co2sobj s (D_proc p))
     then (case (cp2sproc (Execve p f fds # s) p) of
             Some sp \<Rightarrow> s2ss s \<union> {S_proc sp (O_proc p \<in> tainted s \<or> O_file f \<in> tainted s)}
           | _ \<Rightarrow> {} )
     else (case (cp2sproc (Execve p f fds # s) p, cp2sproc s p) of
             (Some sp, Some sp') \<Rightarrow> s2ss s - {S_proc sp' (O_proc p \<in> tainted s)}
                                    \<union> {S_proc sp (O_proc p \<in> tainted s \<or> O_file f \<in> tainted s)}
           | _ \<Rightarrow> {} ) )"
apply (frule vd_cons, frule vt_grant_os, simp split:if_splits)

apply (rule conjI, rule impI, (erule exE|erule conjE)+)
apply (frule_tac p = p in current_proc_has_sp, simp, erule exE)
apply (frule_tac p = p' in current_proc_has_sp, simp, erule exE, simp)
apply (subgoal_tac "p \<in> current_procs (Execve p f fds # s)")
apply (drule_tac p = p and s = "Execve p f fds # s" in current_proc_has_sp, simp)
apply (erule exE, simp)
apply (simp add:s2ss_def, rule set_eqI, rule iffI)
apply (drule CollectD, (erule exE|erule conjE)+)
apply (case_tac "obj = D_proc p")
apply (simp add:co2sobj_execve split:if_splits)
apply (simp add:co2sobj_execve, rule disjI2)
apply (rule_tac x = obj in exI, case_tac obj, (simp add:dalive_simps)+)[1]
apply (simp, erule disjE)
apply (rule_tac x = "D_proc p" in exI, simp)
apply (erule exE| erule conjE)+
apply (case_tac "x = S_proc sp (O_proc p \<in> tainted s)")
apply (rule_tac x = "D_proc p'" in exI)
apply (simp add:dalive_execve co2sobj_execve cp2sproc_execve)
apply (case_tac "obj = D_proc p", simp, simp add:dalive_execve)
apply (frule_tac obj = obj in co2sobj_execve, simp add:dalive_execve)
apply (rule_tac x = obj in exI, simp, simp)

apply (erule conjE, frule current_proc_has_sp, simp, erule exE, rule impI, simp)
apply (subgoal_tac "p \<in> current_procs (Execve p f fds # s)") 
apply (drule_tac p = p and s = "Execve p f fds # s" in current_proc_has_sp, simp)
apply (erule exE, erule conjE, simp)
apply (simp add:s2ss_def, rule set_eqI, rule iffI)
apply (drule CollectD, (erule exE|erule conjE)+)
apply (case_tac "obj = D_proc p", simp)
apply (rule disjI1, simp split:if_splits)
apply (simp add:co2sobj_execve, rule disjI2)
apply (rule conjI,rule_tac x = obj in exI, simp add:dalive_simps split:t_object.splits)
apply (rule notI, simp, case_tac obj)
apply (erule_tac x = nat in allE, simp, (simp split:option.splits)+)
apply (erule disjE, simp)
apply (rule_tac x = "D_proc p" in exI, simp)
apply (erule exE|erule conjE)+
apply (rule_tac x = obj in exI, simp add:dalive_execve)
apply (frule_tac obj = obj in co2sobj_execve, simp add:dalive_execve, simp)
apply (rule impI, simp, simp)
done

lemma s2ss_clone:
  "valid (Clone p p' fds # s) \<Longrightarrow> s2ss (Clone p p' fds # s) = (
     case (cp2sproc (Clone p p' fds # s) p') of
       Some sp \<Rightarrow> s2ss s \<union> {S_proc sp (O_proc p \<in> tainted s)}
     | _       \<Rightarrow> {})"
apply (frule vd_cons, frule vt_grant_os, split option.splits)
apply (rule conjI, rule impI, drule current_proc_has_sp', simp, simp)
apply (rule allI, rule impI, simp add:s2ss_def)
apply (rule set_eqI, rule iffI, drule CollectD, (erule exE|erule conjE)+)
apply (case_tac "obj = D_proc p'", simp)
apply (case_tac "O_proc p' \<in> tainted s", drule tainted_in_current, simp+)
apply (rule disjI1, simp split:if_splits)
apply (simp, rule disjI2)
apply (frule co2sobj_clone, simp)
apply (rule_tac x = obj in exI)
apply (simp add:dalive_simps split:t_dobject.splits)

apply (simp, erule disjE, simp)
apply (rule_tac x = "D_proc p'" in exI, simp)
apply (rule impI, rule notI, drule tainted_in_current, simp+)
apply (erule exE| erule conjE)+
apply (case_tac "obj = D_proc p'", simp)
apply (rule_tac x = obj in exI)
apply (frule dalive_clone)
apply (case_tac obj)
apply (auto simp:co2sobj_clone split:t_dobject.splits simp del:co2sobj.simps)
done

(*
definition s2ss_shm_no_backup:: "t_state \<Rightarrow> t_process \<Rightarrow> t_static_state"
where
  "s2ss_shm_no_backup s pfrom \<equiv> {S_proc sp False | sp p. info_flow_shm s pfrom p \<and> cp2sproc s p = Some sp \<and>
     (\<not> (\<exists> p'. \<not> info_flow_shm s pfrom p' \<and> p' \<in> current_procs s \<and> co2sobj s (O_proc p') = Some (S_proc sp False)))}"

definition update_s2ss_shm:: "t_state \<Rightarrow> t_process \<Rightarrow> t_static_state" 
where
  "update_s2ss_shm s pfrom \<equiv> s2ss s 
     \<union> {S_proc sp True| sp p. info_flow_shm s pfrom p \<and> cp2sproc s p = Some sp}
     - (s2ss_shm_no_backup s pfrom)"

lemma s2ss_shm_no_bk_elim:
  "\<lbrakk>S_proc sp False \<notin> s2ss_shm_no_backup s pfrom; co2sobj s (O_proc p) = Some (S_proc sp False); 
    valid s; info_flow_shm s pfrom p\<rbrakk>
   \<Longrightarrow> \<exists> p'. \<not> info_flow_shm s pfrom p' \<and> p' \<in> current_procs s \<and> co2sobj s (O_proc p') = Some (S_proc sp False)"
apply (auto simp:s2ss_shm_no_backup_def co2sobj.simps split:option.splits)
apply (erule_tac x = p in allE, auto)
apply (rule_tac x = p' in exI, auto)
done

lemma s2ss_shm_no_bk_intro1:
  "\<lbrakk>co2sobj s' obj = Some x; \<forall> p. obj \<noteq> O_proc p\<rbrakk> \<Longrightarrow> x \<notin> s2ss_shm_no_backup s pfrom"
apply (case_tac obj)
apply (auto simp:co2sobj.simps s2ss_shm_no_backup_def split:option.splits)
done

lemma s2ss_shm_no_bk_intro2:
  "\<lbrakk>co2sobj s' obj = Some x; obj \<in> tainted s'; valid s'\<rbrakk> \<Longrightarrow> x \<notin> s2ss_shm_no_backup s pfrom"
apply (case_tac obj)

apply (auto simp:co2sobj.simps s2ss_shm_no_backup_def split:option.splits)
done

lemma s2ss_shm_no_bk_intro3:
  "\<lbrakk>co2sobj s (O_proc p) = Some x; \<not> info_flow_shm s pfrom p; p \<in> current_procs s
   \<rbrakk> \<Longrightarrow> x \<notin> s2ss_shm_no_backup s pfrom"
apply (auto simp add:s2ss_shm_no_backup_def split:option.splits)
apply (rule_tac x = p in exI, simp)
done

lemma s2ss_shm_no_bk_intro4:
  "\<lbrakk>co2sobj s (O_proc p) = Some x; info_flow_shm s pfrom p; 
    \<not> info_flow_shm s pfrom p'; p' \<in> current_procs s; co2sobj s (O_proc p') = Some x\<rbrakk>
   \<Longrightarrow> x \<notin> s2ss_shm_no_backup s pfrom"
apply (rule notI)
apply (auto simp add:s2ss_shm_no_backup_def co2sobj.simps split:option.splits)
done  
*)

lemma tainted_ptrace':
  "tainted (Ptrace p p' # s) = 
     (if (O_proc p \<in> tainted s \<and> O_proc p' \<notin> tainted s)
      then tainted s \<union> {O_proc p'}
      else if (O_proc p' \<in> tainted s \<and> O_proc p \<notin> tainted s)
           then tainted s \<union> {O_proc p}
                else tainted s)"
by auto

(*
lemma co2sobj_some_caseD:
  "\<lbrakk>co2sobj s obj = Some sobj; \<And> p. \<lbrakk>co2sobj s obj = Some sobj; obj = O_proc p\<rbrakk> \<Longrightarrow> P (O_proc p);
    \<And> f. \<lbrakk>co2sobj s obj = Some sobj; obj = O_file f\<rbrakk> \<Longrightarrow> P (O_file f); 
    \<And> f. \<lbrakk>co2sobj s obj = Some sobj; obj = O_dir f\<rbrakk> \<Longrightarrow> P (O_dir f); 
    \<And> q. \<lbrakk>co2sobj s obj = Some sobj; obj = O_msgq q\<rbrakk> \<Longrightarrow> P (O_msgq q)\<rbrakk>
   \<Longrightarrow> P obj"
by (case_tac obj, auto)
*)

definition update_s2ss_obj :: "t_state \<Rightarrow> t_static_state \<Rightarrow> t_dobject \<Rightarrow> t_sobject \<Rightarrow> t_sobject \<Rightarrow> t_static_state"
where
  "update_s2ss_obj s ss obj sobj sobj' = 
     (if (\<exists> obj'. dalive s obj' \<and> obj' \<noteq> obj \<and> co2sobj s obj' = Some sobj)
      then ss \<union> {sobj'}
      else ss - {sobj} \<union> {sobj'})"

ML {*
fun my_setiff_tac i = 
     (etac @{thm CollectE} i 
      ORELSE (     asm_full_simp_tac (HOL_ss addsimps @{thms Set.insert_iff}) i
              THEN etac @{thm disjE} i)
      ORELSE (     asm_full_simp_tac (HOL_ss addsimps @{thms Set.Diff_iff}) i
              THEN etac @{thm conjE} i 
              THEN (REPEAT (etac @{thm CollectE} i))))
THEN (REPEAT ((        etac @{thm exE} 
               ORELSE' etac @{thm conjE}
               ORELSE' etac @{thm bexE}) i))
THEN (rtac @{thm CollectI} i
      ORELSE (     asm_full_simp_tac (HOL_ss addsimps @{thms Set.insert_iff}) i))

*}

ML {*
fun my_seteq_tac i = 
     (simp_tac (HOL_ss addsimps @{thms s2ss_def}) 1)
THEN (rtac @{thm set_eqI} i)
THEN (rtac @{thm iffI} i)
THEN my_setiff_tac i
*}

ML {*fun my_clarify_tac i = 
REPEAT ((  rtac @{thm impI}
   ORELSE' rtac @{thm allI}
   ORELSE' rtac @{thm ballI}
   ORELSE' rtac @{thm conjI}
   ORELSE' etac @{thm conjE}
   ORELSE' etac @{thm exE}
   ORELSE' etac @{thm bexE}
   ORELSE' etac @{thm disjE}) i)
*}

lemma co2sobj_sproc_imp:
  "co2sobj s obj = Some (S_proc sp tag) \<Longrightarrow> \<exists> p. obj = D_proc p"
by (case_tac obj, auto simp:co2sobj.simps split:option.splits)

lemma co2sobj_sfile_imp:
  "co2sobj s obj = Some (S_file sfs tag) \<Longrightarrow> \<exists> f. obj = D_file f"
by (case_tac obj, auto simp:co2sobj.simps split:option.splits)

lemma co2sobj_sdir_imp:
  "co2sobj s obj = Some (S_dir sf) \<Longrightarrow> \<exists> f. obj = D_dir f"
by (case_tac obj, auto simp:co2sobj.simps split:option.splits)

lemma co2sobj_smsgq_imp:
  "co2sobj s obj = Some (S_msgq sq) \<Longrightarrow> \<exists> q. obj = D_msgq q"
by (case_tac obj, auto simp:co2sobj.simps split:option.splits)

lemma s2ss_execve:
  "valid (Execve p f fds # s) \<Longrightarrow> 
     (case (cp2sproc s p, cp2sproc (Execve p f fds # s) p) of
        (Some sp, Some sp') \<Rightarrow> s2ss (Execve p f fds # s) = 
            update_s2ss_obj s (s2ss s) (D_proc p) (S_proc sp (O_proc p \<in> tainted s))
                          (S_proc sp' (O_proc p \<in> tainted s \<or> O_file f \<in> tainted s))
      | _ \<Rightarrow> s2ss (Execve p f fds # s) = {})"
apply (frule vd_cons, frule vt_grant_os)
apply (clarsimp simp only:os_grant.simps)
apply (case_tac "cp2sproc s p")
apply (drule current_proc_has_sp', simp, simp)
apply (case_tac "cp2sproc (Execve p f fds # s) p")
apply (drule current_proc_has_sp', simp, simp, simp)
apply (simp add:update_s2ss_obj_def)
apply (tactic {*my_clarify_tac 1*})
apply (frule co2sobj_sproc_imp, erule exE, simp split:option.splits)
apply (simp add:s2ss_execve')
apply (rule impI)
apply (erule_tac x = pa in allE, simp)
apply (rule impI)
apply (simp add:s2ss_execve')
apply (rule impI)
apply (tactic {*my_clarify_tac 1*})
apply (simp split:option.splits)
apply (erule_tac x = "D_proc p'" in allE, simp)
done

lemma s2ss_ptrace1_aux: "x \<notin> {x. P x} \<Longrightarrow> \<not> P x" by simp

lemma s2ss_ptrace1:
  "\<lbrakk>valid (Ptrace p p' # s); O_proc p \<in> tainted s; O_proc p' \<notin> tainted s\<rbrakk>
   \<Longrightarrow> (case (cp2sproc s p') of 
          Some sp' \<Rightarrow> s2ss (Ptrace p p' # s) = 
                     update_s2ss_obj s (s2ss s) (D_proc p') (S_proc sp' False) (S_proc sp' True)
        | _        \<Rightarrow> s2ss (Ptrace p p' # s) = {})"
apply (frule vd_cons, frule vt_grant_os)
apply (clarsimp simp only:os_grant.simps)
apply (case_tac "cp2sproc s p'")
apply (drule current_proc_has_sp', simp+)
apply (case_tac "cp2sproc s p")
apply (drule current_proc_has_sp', simp+)

apply (simp add:update_s2ss_obj_def)
apply (tactic {*my_clarify_tac 1*})
apply (simp add:s2ss_def)
apply (tactic {*my_seteq_tac 1*})
apply (case_tac "obj = D_proc p'")
apply (rule disjI1, simp add:co2sobj.simps cp2sproc_other)
apply (rule disjI2, simp, rule_tac x = obj in exI)
apply (simp add:co2sobj_ptrace is_file_simps is_dir_simps dalive_other split:t_dobject.splits if_splits)
apply (tactic {*my_setiff_tac 1*})
apply (rule_tac x = "D_proc p'" in exI, simp add:co2sobj.simps cp2sproc_other)
apply (tactic {*my_setiff_tac 1*})
apply (case_tac "obj = D_proc p'")
apply (rule_tac x = obj' in exI)
apply (simp add:co2sobj_ptrace dalive_other split:t_dobject.splits if_splits)
apply (auto simp:co2sobj.simps)[1]
apply (rule_tac x = obj in exI, simp add:co2sobj_ptrace dalive_other split:t_dobject.splits)
apply (auto simp:co2sobj.simps)[1]

apply (rule impI)
apply (simp add:s2ss_def)
apply (tactic {*my_seteq_tac 1*})
apply (case_tac "obj = D_proc p'")
apply (rule disjI1, simp add:co2sobj.simps cp2sproc_other)
apply (rule disjI2, simp, rule conjI, rule_tac x = obj in exI)
apply (simp add:co2sobj_ptrace is_file_simps is_dir_simps dalive_other split:t_dobject.splits if_splits)
apply (rule notI, simp)
apply (frule_tac obj = obj in co2sobj_sproc_imp, erule exE, simp)
apply (erule_tac x = obj in allE, simp add:co2sobj_ptrace cp2sproc_other split:option.splits)
apply (tactic {*my_setiff_tac 1*})
apply (rule_tac x = "D_proc p'" in exI, simp add:co2sobj.simps cp2sproc_other)
apply (tactic {*my_setiff_tac 1*})
apply (case_tac "obj = D_proc p'")
apply (simp add:co2sobj.simps cp2sproc_other)
apply (rule_tac x = obj in exI, simp add:co2sobj_ptrace dalive_other split:t_dobject.splits)
apply (auto simp:co2sobj.simps)[1]
done

lemma s2ss_ptrace2:
  "\<lbrakk>valid (Ptrace p p' # s); O_proc p' \<in> tainted s; O_proc p \<notin> tainted s\<rbrakk>
   \<Longrightarrow> (case (cp2sproc s p) of 
          Some sp \<Rightarrow> s2ss (Ptrace p p' # s) = 
                    update_s2ss_obj s (s2ss s) (D_proc p) (S_proc sp False) (S_proc sp True)
        | _       \<Rightarrow> s2ss (Ptrace p p' # s) = {})"
apply (frule vd_cons, frule vt_grant_os)
apply (clarsimp simp only:os_grant.simps)
apply (case_tac "cp2sproc s p'")
apply (drule current_proc_has_sp', simp+)
apply (case_tac "cp2sproc s p")
apply (drule current_proc_has_sp', simp+)

apply (simp add:update_s2ss_obj_def)
apply (tactic {*my_clarify_tac 1*})
apply (simp add:s2ss_def)
apply (tactic {*my_seteq_tac 1*})
apply (case_tac "obj = D_proc p")
apply (rule disjI1, simp add:co2sobj.simps cp2sproc_other)
apply (rule disjI2, simp, rule_tac x = obj in exI)
apply (simp add:co2sobj_ptrace is_file_simps is_dir_simps dalive_other split:t_dobject.splits if_splits)
apply (tactic {*my_setiff_tac 1*})
apply (rule_tac x = "D_proc p" in exI, simp add:co2sobj.simps cp2sproc_other)
apply (tactic {*my_setiff_tac 1*})
apply (case_tac "obj = D_proc p")
apply (rule_tac x = obj' in exI)
apply (simp add:co2sobj_ptrace dalive_other split:t_dobject.splits if_splits)
apply (auto simp:co2sobj.simps)[1]
apply (rule_tac x = obj in exI, simp add:co2sobj_ptrace dalive_other split:t_dobject.splits)
apply (auto simp:co2sobj.simps)[1]

apply (rule impI)
apply (simp add:s2ss_def)
apply (tactic {*my_seteq_tac 1*})
apply (case_tac "obj = D_proc p")
apply (rule disjI1, simp add:co2sobj.simps cp2sproc_other)
apply (rule disjI2, simp, rule conjI, rule_tac x = obj in exI)
apply (simp add:co2sobj_ptrace is_file_simps is_dir_simps dalive_other split:t_dobject.splits if_splits)
apply (rule notI, simp)
apply (frule_tac obj = obj in co2sobj_sproc_imp, erule exE, simp)
apply (erule_tac x = obj in allE, simp add:co2sobj_ptrace cp2sproc_other split:option.splits)
apply (tactic {*my_setiff_tac 1*})
apply (rule_tac x = "D_proc p" in exI, simp add:co2sobj.simps cp2sproc_other)
apply (tactic {*my_setiff_tac 1*})
apply (case_tac "obj = D_proc p")
apply (simp add:co2sobj.simps cp2sproc_other)
apply (rule_tac x = obj in exI, simp add:co2sobj_ptrace dalive_other split:t_dobject.splits)
apply (auto simp:co2sobj.simps)[1]
done

lemma s2ss_ptrace3:
  "\<lbrakk>valid (Ptrace p p' # s); (O_proc p' \<in> tainted s) = (O_proc p \<in> tainted s)\<rbrakk>
   \<Longrightarrow> s2ss (Ptrace p p' # s) = s2ss s"
unfolding s2ss_def
apply (frule vd_cons, frule vt_grant_os, rule set_eqI, rule iffI)
apply (erule CollectE, (erule exE|erule conjE)+, rule CollectI)
apply (rule_tac x = obj in exI)
apply (frule dalive_other, simp+)
apply (frule_tac obj = obj in co2sobj_ptrace, simp)
apply (auto split:t_dobject.splits option.splits if_splits)[1]

apply (tactic {*my_setiff_tac 1*})
apply (rule_tac x = obj in exI)
apply (frule dalive_other, simp+)
apply (frule_tac obj = obj in co2sobj_ptrace, simp)
apply (case_tac "cp2sproc s p'")
apply (drule current_proc_has_sp', simp, simp)
apply (case_tac "cp2sproc s p")
apply (drule current_proc_has_sp', simp, simp)
apply (case_tac "O_proc p' \<in> tainted s")
apply (auto split:t_dobject.splits option.splits if_splits simp:co2sobj.simps)
done

lemma s2ss_ptrace:
  "valid (Ptrace p p' # s) \<Longrightarrow> s2ss (Ptrace p p' # s) = (
     if (O_proc p \<in> tainted s \<and> O_proc p' \<notin> tainted s) 
     then (case (cp2sproc s p') of 
             Some sp \<Rightarrow> update_s2ss_obj s (s2ss s) (D_proc p') (S_proc sp False) (S_proc sp True)
           | _       \<Rightarrow> {})
     else if (O_proc p' \<in> tainted s \<and> O_proc p \<notin> tainted s)
          then (case (cp2sproc s p) of 
                  Some sp \<Rightarrow> update_s2ss_obj s (s2ss s) (D_proc p) (S_proc sp False) (S_proc sp True)
                | _       \<Rightarrow> {})
          else s2ss s                                   )"
apply (case_tac "O_proc p \<in> tainted s \<and> O_proc p' \<notin> tainted s")
apply (drule s2ss_ptrace1, simp, simp, simp split:option.splits)
apply (case_tac "O_proc p' \<in> tainted s \<and> O_proc p \<notin> tainted s")
apply (drule s2ss_ptrace2, simp, simp, simp split:option.splits)
apply (drule s2ss_ptrace3, auto)
done

lemma s2ss_kill':
  "valid (Kill p p' # s) \<Longrightarrow> s2ss (Kill p p' # s) = (
     if (\<exists> p''. p'' \<in> current_procs s \<and> p'' \<noteq> p' \<and> co2sobj s (D_proc p'') = co2sobj s (D_proc p'))
     then s2ss s 
     else (case (co2sobj s (D_proc p')) of
             Some sp \<Rightarrow> s2ss s - {sp}
           | _       \<Rightarrow> {}))"
apply (frule vt_grant_os, frule vd_cons)
unfolding s2ss_def
apply (simp split:if_splits, rule conjI)
apply (rule impI, (erule exE|erule conjE)+)
apply (split option.splits)
apply (drule current_proc_has_sp', simp, simp)
apply (simp split: option.splits, (erule conjE)+)
apply (rule set_eqI, rule iffI, erule CollectE, (erule exE|erule conjE)+, rule CollectI)
apply (rule_tac x = obj in exI)
apply (simp add:co2sobj_kill dalive_kill split:t_dobject.splits if_splits)
apply (erule CollectE, erule exE, erule conjE, rule CollectI)
apply (case_tac obj)
apply (case_tac "nat = p'")
apply (rule_tac x = "D_proc p''" in exI)
apply (simp add:cp2sproc_kill dalive_kill
  split:t_dobject.splits if_splits option.splits)
apply (rule_tac x = "D_proc nat" in exI)
apply (clarsimp simp add:cp2sproc_kill dalive_kill
  split:t_dobject.splits if_splits option.splits)
apply (rule_tac x = obj in exI, frule dalive_kill, simp add:co2sobj_kill del:co2sobj.simps)+

apply (rule impI, erule conjE, frule current_proc_has_sp, simp, erule exE, simp)
apply (rule set_eqI, rule iffI)
apply (erule CollectE, erule exE, erule conjE, rule DiffI)
apply (rule CollectI, rule_tac x = obj in exI)
apply (simp add:co2sobj_kill dalive_kill split:t_dobject.splits if_splits)
apply (rule notI, simp, case_tac obj)
apply (erule_tac x = nat in allE)
apply (simp add:co2sobj_kill cp2sproc_kill split:option.splits)
apply (simp split:option.splits)+
apply (case_tac obj)
apply (case_tac "nat = p'")
apply (rule_tac x = "D_proc p''" in exI)
apply (simp add:cp2sproc_kill dalive_kill
  split:t_dobject.splits if_splits option.splits)
apply (rule_tac x = "D_proc nat" in exI)
apply (clarsimp simp add:cp2sproc_kill dalive_kill
  split:t_dobject.splits if_splits option.splits)
apply (rule_tac x = obj in exI, frule dalive_kill, simp add:co2sobj_kill del:co2sobj.simps)+
done

definition del_s2ss_obj :: "t_state \<Rightarrow> t_static_state \<Rightarrow> t_dobject \<Rightarrow> t_sobject \<Rightarrow> t_static_state"
where
  "del_s2ss_obj s ss obj sobj \<equiv> 
     if (\<exists> obj'. dalive s obj' \<and> obj' \<noteq> obj \<and> co2sobj s obj' = Some sobj)
     then ss
     else ss - {sobj}"

lemma del_update_s2ss_obj:
  "update_s2ss_obj s ss obj sobj sobj' = del_s2ss_obj s ss obj sobj \<union> {sobj'}"
by (auto simp:update_s2ss_obj_def del_s2ss_obj_def split:if_splits)

lemma s2ss_kill:
  "valid (Kill p p' # s) \<Longrightarrow> (
      case (cp2sproc s p') of 
        Some sp \<Rightarrow> s2ss (Kill p p' # s) = del_s2ss_obj s (s2ss s) (D_proc p') (S_proc sp (O_proc p' \<in> tainted s))
      | _       \<Rightarrow> s2ss (Kill p p' # s) = {})"
apply (frule vd_cons, frule vt_grant_os)
apply (clarsimp simp only:os_grant.simps)
apply (split option.splits, rule conjI, rule impI)
apply (drule current_proc_has_sp', simp, simp)
apply (rule allI, rule impI)
apply (simp add:del_s2ss_obj_def split:option.splits)
apply (tactic {*my_clarify_tac 1*})
apply (frule co2sobj_sproc_imp, erule exE)
apply (simp add:s2ss_kill')
apply (rule impI)
apply (erule_tac x = pa in allE, simp)
apply (rule impI)
apply (simp add:s2ss_kill')
apply (rule impI)
apply (tactic {*my_clarify_tac 1*})
apply (simp split:option.splits)
apply (erule_tac x = "D_proc p''" in allE, simp)
done

lemma s2ss_exit':
  "valid (Exit p # s) \<Longrightarrow> s2ss (Exit p # s) = (
     if (\<exists> p'. p' \<in> current_procs s \<and> p' \<noteq> p \<and> co2sobj s (D_proc p') = co2sobj s (D_proc p))
     then s2ss s 
     else (case (co2sobj s (D_proc p)) of
             Some sp \<Rightarrow> s2ss s - {sp}
           | _       \<Rightarrow> {}))"
apply (frule vt_grant_os, frule vd_cons)
unfolding s2ss_def
apply (simp split:if_splits, rule conjI)
apply (rule impI, (erule exE|erule conjE)+)
apply (split option.splits)
apply (drule current_proc_has_sp', simp, simp)
apply (simp split: option.splits, (erule conjE)+)
apply (rule set_eqI, rule iffI, erule CollectE, (erule exE|erule conjE)+, rule CollectI)
apply (rule_tac x = obj in exI)
apply (simp add:co2sobj_exit dalive_exit split:t_dobject.splits if_splits)
apply (erule CollectE, erule exE, erule conjE, rule CollectI)
apply (case_tac obj)
apply (case_tac "nat = p")
apply (rule_tac x = "D_proc p'" in exI)
apply (simp add:cp2sproc_exit dalive_exit
  split:t_dobject.splits if_splits option.splits)
apply (rule_tac x = "D_proc nat" in exI)
apply (clarsimp simp add:cp2sproc_exit dalive_exit
  split:t_dobject.splits if_splits option.splits)
apply (rule_tac x = obj in exI, frule dalive_exit, simp add:co2sobj_exit del:co2sobj.simps)+

apply (rule impI, frule current_proc_has_sp, simp, erule exE, simp)
apply (rule set_eqI, rule iffI)
apply (erule CollectE, erule exE, erule conjE, rule DiffI)
apply (rule CollectI, rule_tac x = obj in exI)
apply (simp add:co2sobj_exit dalive_exit split:t_dobject.splits if_splits)
apply (rule notI, simp, case_tac obj)
apply (erule_tac x = nat in allE)
apply (simp add:co2sobj_exit cp2sproc_exit split:option.splits)
apply (simp split:option.splits)+
apply (case_tac obj)
apply (case_tac "nat = p")
apply (rule_tac x = "D_proc p'" in exI)
apply (simp add:cp2sproc_exit dalive_exit
  split:t_dobject.splits if_splits option.splits)
apply (rule_tac x = "D_proc nat" in exI)
apply (clarsimp simp add:cp2sproc_exit dalive_exit
  split:t_dobject.splits if_splits option.splits)
apply (rule_tac x = obj in exI, frule dalive_exit, simp add:co2sobj_exit del:co2sobj.simps)+
done

lemma s2ss_exit:
  "valid (Exit p # s) \<Longrightarrow> (
      case (cp2sproc s p) of 
        Some sp \<Rightarrow> s2ss (Exit p # s) = del_s2ss_obj s (s2ss s) (D_proc p) (S_proc sp (O_proc p \<in> tainted s))
      | _       \<Rightarrow> s2ss (Exit p # s) = {})"
apply (frule vd_cons, frule vt_grant_os)
apply (clarsimp simp only:os_grant.simps)
apply (split option.splits, rule conjI, rule impI)
apply (drule current_proc_has_sp', simp, simp)
apply (rule allI, rule impI)
apply (simp add:del_s2ss_obj_def split:option.splits)
apply (tactic {*my_clarify_tac 1*})
apply (frule co2sobj_sproc_imp, erule exE)
apply (simp add:s2ss_exit')
apply (rule impI)
apply (erule_tac x = pa in allE, simp)
apply (rule impI)
apply (simp add:s2ss_exit')
apply (rule impI)
apply (tactic {*my_clarify_tac 1*})
apply (simp split:option.splits)
apply (erule_tac x = "D_proc p'" in allE, simp)
done

lemma dalive_has_sobj':
  "\<lbrakk>co2sobj s obj = None; valid s\<rbrakk> \<Longrightarrow> \<not> dalive s obj"
apply (case_tac obj)
apply (auto split:option.splits)
oops

declare co2sobj.simps [simp del]

lemma co2sobj_open_none:
  "\<lbrakk>valid (Open p f flag fd None # s); dalive s obj\<rbrakk> \<Longrightarrow> co2sobj (Open p f flag fd None # s) obj = (
      if (obj = D_proc p) 
      then (case (cp2sproc (Open p f flag fd None # s) p) of
              Some sp \<Rightarrow> Some (S_proc sp (O_proc p \<in> tainted s))
           | _       \<Rightarrow> None)
      else co2sobj s obj)"
apply (frule vt_grant_os, frule vd_cons)
apply (frule_tac obj = obj in co2sobj_open, simp add:dalive_open)
apply (auto split:t_dobject.splits option.splits dest!:current_proc_has_sp')
done

lemma co2sobj_open_some:
  "\<lbrakk>valid (Open p f flag fd (Some i) # s); dalive s obj\<rbrakk> \<Longrightarrow> co2sobj (Open p f flag fd (Some i) # s) obj = (
      if (obj = D_proc p) 
      then (case (cp2sproc (Open p f flag fd (Some i) # s) p) of
              Some sp \<Rightarrow> Some (S_proc sp (O_proc p \<in> tainted s))
           | _       \<Rightarrow> None)
      else if (obj = D_file f) 
           then (case (cf2sfile (Open p f flag fd (Some i) # s) f) of
                   Some sf \<Rightarrow> Some (S_file {sf} (O_proc p \<in> tainted s))
                 | _       \<Rightarrow> None)
           else co2sobj s obj)"
apply (frule vt_grant_os, frule vd_cons)
apply (frule_tac obj = obj in co2sobj_open, simp add:dalive_open)
apply (auto split:t_dobject.splits option.splits dest!:current_proc_has_sp')
done

lemma co2sobj_proc_obj:
  "\<lbrakk>co2sobj s obj = Some x; co2sobj s (D_proc p) = Some x\<rbrakk>
   \<Longrightarrow> \<exists> p'. obj = D_proc p'"
by (case_tac obj, auto simp:co2sobj.simps split:option.splits)

lemma s2ss_open_none:
  "valid (Open p f flag fd None # s) \<Longrightarrow> s2ss (Open p f flag fd None # s) = (
      case (co2sobj s (D_proc p), co2sobj (Open p f flag fd None # s) (D_proc p)) of
        (Some sp, Some sp') \<Rightarrow> 
           if (\<exists> p'. p' \<in> current_procs s \<and> p' \<noteq> p \<and> co2sobj s (D_proc p') = Some sp)
           then s2ss s \<union> {sp'}
           else s2ss s - {sp} \<union> {sp'} 
      | _                   \<Rightarrow> {} )"
unfolding s2ss_def
apply (frule vt_grant_os, frule vd_cons)
apply (case_tac "co2sobj s (D_proc p)", simp add:co2sobj.simps split:option.splits)
apply (drule current_proc_has_sp', simp, simp)
apply (case_tac "co2sobj (Open p f flag fd None # s) (D_proc p)")
apply (simp add:co2sobj.simps split:option.splits)
apply (drule current_proc_has_sp', simp, simp)
apply (rule set_eqI, rule iffI, erule CollectE, erule exE, erule conjE, simp)
apply (simp add:dalive_open)
apply (rule conjI, rule impI, erule exE, (erule conjE)+)
apply (rule Meson.disj_comm, rule disjCI, case_tac "obj = D_proc p", simp)
apply (rule_tac x = obj in exI, simp add:co2sobj_open_none dalive_open split:t_dobject.splits)
apply (rule impI)
apply (case_tac "obj = D_proc p", simp)
apply (rule Meson.disj_comm, rule disjCI, rule conjI)
apply (rule_tac x = obj in exI, simp add:co2sobj_open_none split:t_dobject.splits)
apply (rule notI)
apply (simp add:co2sobj_open_none split:option.splits)
apply (frule_tac co2sobj_proc_obj, simp, erule exE)
apply (erule_tac x = p' in allE, simp split:t_dobject.splits)

apply (simp split:if_splits)
apply (erule disjE, rule_tac x = "D_proc p" in exI, simp)
apply (erule exE, erule conjE, case_tac "obj = D_proc p")
apply (rule_tac x = "D_proc p'" in exI, simp add:co2sobj_open_none)
apply (rule_tac x = obj in exI, simp add:co2sobj_open_none dalive_open)
apply (erule disjE, rule_tac x = "D_proc p" in exI, simp)
apply (erule conjE, erule exE, erule conjE, case_tac "obj = D_proc p")
apply (rule_tac x = "D_proc p'" in exI, simp add:co2sobj_open_none)
apply (rule_tac x = obj in exI, simp add:co2sobj_open_none dalive_open)
done

lemma s2ss_open_some:
  "valid (Open p f flag fd (Some i) # s) \<Longrightarrow> s2ss (Open p f flag fd (Some i) # s) = (
      case (co2sobj s (D_proc p), co2sobj (Open p f flag fd (Some i) # s) (D_proc p),
            co2sobj (Open p f flag fd (Some i) # s) (D_file f)) of
        (Some sp, Some sp', Some sf) \<Rightarrow> 
           if (\<exists> p'. p' \<in> current_procs s \<and> p' \<noteq> p \<and> co2sobj s (D_proc p') = Some sp)
           then s2ss s \<union> {sp', sf} 
           else s2ss s - {sp} \<union> {sp', sf} 
      | _                   \<Rightarrow> {} )"
unfolding s2ss_def
apply (frule vt_grant_os, frule vd_cons)
apply (case_tac "co2sobj s (D_proc p)", simp add:co2sobj.simps split:option.splits)
apply (drule current_proc_has_sp', simp, simp)
apply (case_tac "co2sobj (Open p f flag fd (Some i) # s) (D_proc p)")
apply (simp add:co2sobj.simps split:option.splits)
apply (drule current_proc_has_sp', simp, simp)
apply (case_tac "co2sobj (Open p f flag fd (Some i) # s) (D_file f)")
apply (simp add:co2sobj.simps split:option.splits)
apply (clarsimp split del:if_splits)

apply (rule set_eqI, rule iffI, erule CollectE, erule exE, erule conjE)
apply (split if_splits, rule conjI, rule impI, erule exE, erule conjE, erule conjE)
apply (case_tac "obj = D_proc p", simp, case_tac "obj = D_file f", simp)
apply (rule UnI1, rule CollectI, rule_tac x = obj in exI)
apply (simp add:co2sobj_open dalive_open split:t_dobject.splits option.splits)
apply (rule impI, case_tac "obj = D_proc p", simp, case_tac "obj = D_file f", simp)
apply (rule UnI1, rule DiffI, rule CollectI, rule_tac x = obj in exI)
apply (simp add:co2sobj_open dalive_open split:t_dobject.splits)
apply (frule_tac obj = obj in co2sobj_open_some, simp+)
apply (simp add:dalive_open)
apply (rule notI, simp)
apply (frule_tac obj = obj and p = p in co2sobj_proc_obj, simp+, erule exE)
apply (erule_tac x = p' in allE, simp)

apply (simp split:if_splits, erule disjE)
apply (rule_tac x = "D_proc p" in exI, simp)
apply (erule disjE, rule_tac x = "D_file f" in exI, simp add:is_file_simps)
apply (erule exE, erule conjE)
apply (case_tac "obj = D_proc p", simp)
apply (rule_tac x = "D_proc p'" in exI, simp add:co2sobj_open_some)
apply (case_tac "obj = D_file f", simp add:is_file_in_current)
apply (rule_tac x = obj in exI, simp add:co2sobj_open_some dalive_open)
apply (erule disjE, rule_tac x = "D_proc p" in exI, simp)
apply (erule disjE, rule_tac x = "D_file f" in exI, simp add:is_file_simps)
apply (erule conjE, erule exE, erule conjE)
apply (case_tac "obj = D_proc p", simp)
apply (case_tac "obj = D_file f", simp add:is_file_in_current)
apply (rule_tac x = obj in exI, simp add:co2sobj_open_some dalive_open)
done

lemma s2ss_open':
  "valid (Open p f flag fd opt # s) \<Longrightarrow> s2ss (Open p f flag fd opt # s) = (
     if opt = None
     then (case (co2sobj s (D_proc p), co2sobj (Open p f flag fd opt # s) (D_proc p)) of
        (Some sp, Some sp') \<Rightarrow> 
           if (\<exists> p'. p' \<in> current_procs s \<and> p' \<noteq> p \<and> co2sobj s (D_proc p') = Some sp)
           then s2ss s \<union> {sp'}
           else s2ss s - {sp} \<union> {sp'} 
      | _                   \<Rightarrow> {} )
     else (case (co2sobj s (D_proc p), co2sobj (Open p f flag fd opt # s) (D_proc p),
            co2sobj (Open p f flag fd opt # s) (D_file f)) of
        (Some sp, Some sp', Some sf) \<Rightarrow> 
           if (\<exists> p'. p' \<in> current_procs s \<and> p' \<noteq> p \<and> co2sobj s (D_proc p') = Some sp)
           then s2ss s \<union> {sp', sf} 
           else s2ss s - {sp} \<union> {sp', sf} 
      | _                   \<Rightarrow> {} ) )"
apply (case_tac opt)
apply (simp add:s2ss_open_some s2ss_open_none)+
done

lemma co2sobj_proc_eq_some:
  "\<lbrakk>co2sobj s (D_proc p) = Some sp; co2sobj s obj = Some sp\<rbrakk>
   \<Longrightarrow> \<exists> p'. obj = D_proc p'"
apply (case_tac obj, case_tac[!] sp)
by (auto simp:co2sobj.simps split:option.splits)

lemma s2ss_open:
  "valid (Open p f flag fd opt # s) \<Longrightarrow> 
    (case (co2sobj s (D_proc p), co2sobj (Open p f flag fd opt # s) (D_proc p), 
           co2sobj (Open p f flag fd opt # s) (D_file f)) of
       (Some sp, Some sp', Some sf) \<Rightarrow> s2ss (Open p f flag fd opt # s) = (
       if opt = None
       then update_s2ss_obj s (s2ss s) (D_proc p) sp sp'
       else update_s2ss_obj s (s2ss s) (D_proc p) sp sp' \<union> {sf})
     | _ \<Rightarrow> s2ss (Open p f flag fd opt # s) = {})"
apply (frule vt_grant_os, frule vd_cons, clarsimp simp only:os_grant.simps)
apply (case_tac "co2sobj s (D_proc p)")
apply (simp add:co2sobj.simps split:option.splits)
apply (drule current_proc_has_sp', simp, simp)
apply (drule current_proc_has_sp', simp, simp)
apply (case_tac "co2sobj (Open p f flag fd opt # s) (D_proc p)")
apply (simp add:co2sobj.simps split:option.splits)
apply (drule current_proc_has_sp', simp, simp)
apply (drule current_proc_has_sp', simp, simp)
apply (case_tac "co2sobj (Open p f flag fd opt # s) (D_file f)")
apply (simp add:co2sobj.simps split:option.splits)
apply (simp split:option.splits add:s2ss_open' update_s2ss_obj_def)
apply (auto)
apply (erule_tac x = "D_proc p'" in allE, simp)
apply (frule_tac obj = obj' in co2sobj_proc_eq_some, simp, erule exE, simp)
apply (erule_tac x = "p'" in allE, simp)
apply (erule_tac x = "D_proc p'" in allE, simp)
apply (frule_tac obj = obj' in co2sobj_proc_eq_some, simp, erule exE, simp)
apply (erule_tac x = "p'" in allE, simp)
apply (erule_tac x = "D_proc p'" in allE, simp)
apply (frule_tac obj = obj' in co2sobj_proc_eq_some, simp, erule exE, simp)
apply (erule_tac x = "p'" in allE, simp)
done

lemma s2ss_readfile:
  "valid (ReadFile p fd # s) \<Longrightarrow> s2ss (ReadFile p fd # s) = (
     case (file_of_proc_fd s p fd) of 
       Some f \<Rightarrow> if (O_file f \<in> tainted s \<and> O_proc p \<notin> tainted s)
                 then (case (cp2sproc s p) of 
                         Some sp \<Rightarrow> update_s2ss_obj s (s2ss s) (D_proc p) (S_proc sp False) (S_proc sp True)
                       | _       \<Rightarrow> {})
                 else s2ss s
     | _      \<Rightarrow> {})"
apply (frule vt_grant_os, frule vd_cons, clarsimp simp only:os_grant.simps)
apply (case_tac "cp2sproc s p")
apply (drule current_proc_has_sp', simp+)
apply (rule conjI, rule impI, erule conjE)

apply (simp add:update_s2ss_obj_def)
apply (tactic {*my_clarify_tac 1*})
apply (frule co2sobj_sproc_imp, erule exE, simp split:option.splits)
apply (simp add:s2ss_def)
apply (tactic {*my_seteq_tac 1*})
apply (case_tac "obj = D_proc p")
apply (rule disjI1, simp add:co2sobj_readfile)
apply (rule disjI2, simp)
apply (rule_tac x = obj in exI)
apply (simp add:dalive_other co2sobj_readfile split:t_dobject.splits option.splits)
apply (tactic {*my_setiff_tac 1*})
apply (rule_tac x= "D_proc p" in exI, simp add:dalive_other co2sobj_readfile)
apply (tactic {*my_setiff_tac 1*})
apply (case_tac "obj = D_proc p")
apply (rule_tac x = "D_proc pa" in exI, simp add:dalive_other co2sobj_readfile)
apply (simp add:co2sobj.simps)
apply (rule_tac x = obj in exI)
apply (auto simp add:dalive_other co2sobj_readfile split:t_dobject.splits option.splits)[1]
apply (tactic {*my_clarify_tac 1*})
apply (simp add:s2ss_def)
apply (tactic {*my_seteq_tac 1*})
apply (case_tac "obj = D_proc p")
apply (rule disjI1, simp add:co2sobj_readfile)
apply (rule disjI2, rule DiffI)
apply (simp, rule_tac x = obj in exI)
apply (simp add:dalive_other co2sobj_readfile split:t_dobject.splits option.splits)
apply (rule notI, erule_tac x = obj in allE)
apply (auto simp add:dalive_other co2sobj_readfile split:t_dobject.splits option.splits)[1]
apply (tactic {*my_setiff_tac 1*})
apply (rule_tac x = "D_proc p" in exI, simp add:dalive_other co2sobj_readfile)
apply (tactic {*my_setiff_tac 1*})
apply (rule_tac x = obj in exI)
apply (auto simp add:dalive_other co2sobj_readfile split:t_dobject.splits option.splits)[1]
apply (simp add:co2sobj.simps)

apply (simp add:s2ss_def, rule impI)
apply (tactic {*my_seteq_tac 1*})
apply (rule_tac x = obj in exI)
apply (simp add:dalive_other co2sobj_readfile split:t_dobject.splits option.splits if_splits)
apply (simp add:co2sobj.simps)
apply (tactic {*my_setiff_tac 1*})
apply (rule_tac x = obj in exI)
apply (auto simp add:dalive_other co2sobj_readfile split:t_dobject.splits option.splits if_splits)
apply (simp add:co2sobj.simps)
done

lemma same_inode_files_prop9:
  "is_file s f \<Longrightarrow> f \<in> same_inode_files s f"
by (simp add:same_inode_files_def)

lemma cf2sfiles_prop:
  "\<lbrakk>f \<in> same_inode_files s f'; valid s\<rbrakk> \<Longrightarrow> cf2sfiles s f = cf2sfiles s f'"
apply (auto simp:cf2sfiles_def)
apply (rule_tac x = f'a in bexI, simp)
apply (erule same_inode_files_prop4, simp)
apply (rule_tac x = f'a in bexI, simp)
apply (drule same_inode_files_prop5)
apply (erule same_inode_files_prop4, simp)
done

lemma co2sobj_writefile_unchange:
  "\<lbrakk>valid (WriteFile p fd # s); dalive s obj; file_of_proc_fd s p fd = Some f;
    O_proc p \<in> tainted s \<longrightarrow> O_file f \<in> tainted s\<rbrakk> 
   \<Longrightarrow> co2sobj (WriteFile p fd # s) obj = co2sobj s obj"
apply (frule vd_cons, frule co2sobj_writefile, simp, simp split:t_dobject.splits if_splits)
apply (simp add:co2sobj.simps)
apply (case_tac "O_proc p \<in> tainted s")
apply (simp add:same_inodes_tainted)+
done

lemma s2ss_writefile':
  "valid (WriteFile p fd # s) \<Longrightarrow> s2ss (WriteFile p fd # s) = (
     case (file_of_proc_fd s p fd) of
       Some f \<Rightarrow> if (O_proc p \<in> tainted s \<and> O_file f \<notin> tainted s)
                 then (if (\<exists> f'. f' \<notin> same_inode_files s f \<and> is_file s f' \<and>
                                 co2sobj s (D_file f') = co2sobj s (D_file f))
                       then s2ss s \<union> {S_file (cf2sfiles s f) True}
                       else s2ss s - {S_file (cf2sfiles s f) False} 
                                   \<union> {S_file (cf2sfiles s f) True})
                 else s2ss s
     | _      \<Rightarrow> {})"
apply (frule vd_cons, frule vt_grant_os)
apply (clarsimp split:option.splits)
unfolding s2ss_def
apply (rule conjI|rule impI|erule exE|erule conjE)+
apply (rule set_eqI, rule iffI, erule CollectE, erule exE, erule conjE)
apply (frule_tac obj =  obj in co2sobj_writefile, simp add:dalive_other)
apply (simp split:t_dobject.splits if_splits)
apply (rule disjI2, rule_tac x= "D_proc nat" in exI, simp)
apply (rule disjI1, simp add:cf2sfiles_prop)
apply (rule disjI2, rule_tac x = obj in exI, simp add:is_file_simps)
apply (rule disjI2, rule_tac x = obj in exI, simp add:is_dir_simps)
apply (rule disjI2, rule_tac x = obj in exI, simp)
apply (simp add:co2sobj.simps)

apply (erule disjE)
apply (rule_tac x = "D_file aa" in exI, simp add:is_file_simps file_of_pfd_is_file)
apply (frule_tac obj = "D_file aa" in co2sobj_writefile, simp add:file_of_pfd_is_file)
apply (simp split:if_splits add:same_inode_files_def file_of_pfd_is_file)
apply (erule exE, erule conjE, erule conjE)
apply (case_tac obj)
apply (rule_tac x = obj in exI, simp add:co2sobj_writefile)
apply (case_tac "list \<in> same_inode_files s aa")
apply (frule_tac f = list and f' = aa in cf2sfiles_prop, simp)
apply (rule_tac x = "D_file f'" in exI, simp add:co2sobj_writefile is_file_simps)
apply (rule conjI, rule impI, simp add:same_inode_files_prop5)
apply (rule impI, simp add:co2sobj.simps same_inodes_tainted)
apply (rule_tac x = "D_file list" in exI, simp add:co2sobj_writefile is_file_simps)
apply (rule impI, simp add:same_inode_files_prop5)
apply (rule_tac x = obj in exI, simp add:co2sobj_writefile is_dir_simps)
apply (rule_tac x = obj in exI, simp add:co2sobj_writefile)

apply (rule impI, rule impI, simp, rule set_eqI, rule iffI, erule CollectE, (erule conjE|erule exE)+)
apply (rule CollectI, rule_tac x = obj in exI, simp add:dalive_simps)
apply (simp add:co2sobj_writefile split:t_dobject.splits if_splits)
apply (simp add:co2sobj.simps same_inodes_tainted)
apply (case_tac "O_proc p \<in> tainted s", simp, simp)
apply (erule CollectE, (erule conjE|erule exE)+, rule CollectI)
apply (rule_tac x = obj in exI)
apply (simp add:co2sobj_writefile_unchange dalive_simps)

apply (rule impI| rule conjI|erule conjE)+
apply (rule set_eqI, rule iffI, erule CollectE, erule exE, erule conjE)
apply (simp add:dalive_simps co2sobj_writefile split:t_dobject.splits)
apply (rule disjI2, rule conjI, rule_tac x = obj in exI, simp,
  rule notI, simp add:co2sobj.simps split:option.splits)
apply (simp split:if_splits)
apply (rule disjI1, simp add:cf2sfiles_prop)
apply (rule disjI2, rule conjI, rule_tac x = obj in exI, simp)
apply (rule notI, simp add:co2sobj.simps split:option.splits)
apply (erule_tac x = list in allE, simp add:same_inode_files_prop5)
apply (simp add:co2sobj.simps)
apply (rule disjI2, rule conjI, rule_tac x = obj in exI, simp split:option.splits add:co2sobj.simps)
apply (rule notI, simp add:co2sobj.simps split:option.splits)
apply (rule disjI2, rule conjI, rule_tac x = obj in exI, simp,
  rule notI, simp add:co2sobj.simps split:option.splits)
apply (simp add:co2sobj.simps)
apply (erule disjE)
apply (rule_tac x= "D_file aa" in exI)
apply ( simp add:co2sobj_writefile dalive_simps file_of_pfd_is_file)
apply (rule impI, simp add:same_inode_files_def file_of_pfd_is_file)
apply (erule exE|erule conjE)+
apply (case_tac obj)
apply (rule_tac x = obj in exI, simp add:co2sobj_writefile)
apply (case_tac "list \<in> same_inode_files s aa")
apply (frule cf2sfiles_prop, simp, simp add:co2sobj.simps same_inodes_tainted)
apply (rule_tac x = obj in exI, simp add:co2sobj_writefile is_file_simps)
apply (rule impI, simp add:same_inode_files_prop5)
apply (rule_tac x = obj in exI, simp add:co2sobj_writefile is_dir_simps)
apply (rule_tac x = obj in exI, simp add:co2sobj_writefile)

apply (rule impI, rule impI)
apply (rule set_eqI, rule iffI, erule CollectE,erule exE,erule conjE,rule CollectI)
apply (rule_tac x = obj in exI)
apply (simp add:co2sobj_writefile_unchange dalive_simps)
apply (erule CollectE, erule exE, erule conjE)
apply (rule CollectI, rule_tac x = obj in exI)
apply (simp add:co2sobj_writefile_unchange dalive_simps)
done

definition update_s2ss_sfile_tainted:: "t_state \<Rightarrow> t_static_state \<Rightarrow> t_file \<Rightarrow> bool \<Rightarrow> t_static_state"
where
 "update_s2ss_sfile_tainted s ss f tag \<equiv>
    if (\<exists> f'. is_file s f' \<and> f' \<notin> same_inode_files s f \<and> 
              co2sobj s (D_file f') = Some (S_file (cf2sfiles s f) False))
       then ss \<union> {S_file (cf2sfiles s f) True}
       else ss - {S_file (cf2sfiles s f) False} 
               \<union> {S_file (cf2sfiles s f) True}"

lemma s2ss_writefile:
  "valid (WriteFile p fd # s) \<Longrightarrow> s2ss (WriteFile p fd # s) = (
     case (file_of_proc_fd s p fd) of
       Some f \<Rightarrow> if (O_proc p \<in> tainted s \<and> O_file f \<notin> tainted s)
                 then update_s2ss_sfile_tainted s (s2ss s) f True
                 else s2ss s
     | _      \<Rightarrow> {})"
apply (drule s2ss_writefile')
apply (simp)
apply (case_tac "file_of_proc_fd s p fd", simp)
apply (simp add:update_s2ss_sfile_tainted_def)
apply auto
apply (erule_tac x = f' in allE, simp add:co2sobj.simps)+
done

definition update_s2ss_sfile_del :: "t_state \<Rightarrow> t_static_state \<Rightarrow> t_file \<Rightarrow> t_sfile \<Rightarrow> t_static_state"
where 
  "update_s2ss_sfile_del s ss f sf \<equiv> 
     if (same_inode_files s f = {f})
     then ss
     else ss \<union> {S_file (cf2sfiles s f - {sf}) (O_file f \<in> tainted s)}"

definition del_s2ss_file:: "t_state \<Rightarrow> t_static_state \<Rightarrow> t_file \<Rightarrow> t_sfile \<Rightarrow> t_static_state"
where
  "del_s2ss_file s ss f sf = 
     (if (\<exists> f' \<in> same_inode_files s f. f' \<noteq> f \<and> cf2sfile s f' = Some sf)
      then ss
      else if (\<exists> f'. is_file s f' \<and> f' \<notin> same_inode_files s f \<and> co2sobj s (D_file f') = co2sobj s (D_file f))
           then update_s2ss_sfile_del s ss f sf
           else update_s2ss_sfile_del s (ss - {S_file (cf2sfiles s f) (O_file f \<in> tainted s)}) f sf)"


lemma dalive_co2sobj_closefd1:
  "\<lbrakk>dalive s obj; valid (CloseFd p fd # s); 
    file_of_proc_fd s p fd = Some f; \<not> (f \<in> files_hung_by_del s \<and> proc_fd_of_file s f = {(p, fd)})\<rbrakk>
   \<Longrightarrow> dalive (CloseFd p fd # s) obj"
apply (case_tac obj)
by (auto simp:dalive_simps is_file_simps is_dir_simps split:option.splits)

lemma dalive_co2sobj_closefd3:
  "\<lbrakk>dalive s obj; valid (CloseFd p fd # s); obj \<noteq> D_file f;
    file_of_proc_fd s p fd = Some f; f \<in> files_hung_by_del s; proc_fd_of_file s f = {(p, fd)}\<rbrakk>
   \<Longrightarrow> dalive (CloseFd p fd # s) obj"
apply (case_tac obj)
by (auto simp:dalive_simps is_file_simps is_dir_simps split:option.splits)

lemma dalive_co2sobj_closefd2:
  "\<lbrakk>dalive s obj; valid (CloseFd p fd # s); file_of_proc_fd s p fd = None\<rbrakk>
   \<Longrightarrow> dalive (CloseFd p fd # s) obj"
apply (case_tac obj)
by (auto simp:dalive_simps is_file_simps is_dir_simps split:option.splits)

lemma dalive_co2sobj_closefd':
  "\<lbrakk>co2sobj (CloseFd p fd # s) obj = Some sobj; dalive (CloseFd p fd # s) obj; 
    valid (CloseFd p fd # s)\<rbrakk> \<Longrightarrow> dalive s obj"
apply (case_tac obj)
by (auto simp:dalive_simps is_file_simps is_dir_simps split:option.splits if_splits)

lemma same_inode_files_prop10:
  "\<lbrakk>same_inode_files s f \<noteq> {f}; is_file s f\<rbrakk> \<Longrightarrow> \<exists> f'. f' \<in> same_inode_files s f \<and> f' \<noteq> f"
by (auto simp:same_inode_files_def split:if_splits)

lemma same_inode_files_prop11:
  "f \<in> same_inode_files s f' \<Longrightarrow> is_file s f"
by (auto simp:same_inode_files_def is_file_def split:if_splits)

lemma same_inode_files_prop11':
  "f \<in> same_inode_files s f' \<Longrightarrow> is_file s f'"
by (auto simp:same_inode_files_def is_file_def split:if_splits)

lemma s2ss_closefd:
  "valid (CloseFd p fd # s) \<Longrightarrow> s2ss (CloseFd p fd # s) = (
     case (file_of_proc_fd s p fd) of
       Some f \<Rightarrow> if (f \<in> files_hung_by_del s \<and> proc_fd_of_file s f = {(p, fd)})
                 then (case (cf2sfile s f, cp2sproc s p, cp2sproc (CloseFd p fd # s) p) of
                         (Some sf, Some sp, Some sp') \<Rightarrow> 
                            (del_s2ss_file s (
                               update_s2ss_obj s (s2ss s) (D_proc p) 
                                 (S_proc sp (O_proc p \<in> tainted s))
                                 (S_proc sp' (O_proc p \<in> tainted s))) f sf)
                      | _ \<Rightarrow> {})
                 else (case (cp2sproc s p, cp2sproc (CloseFd p fd # s) p) of 
                         (Some sp, Some sp') \<Rightarrow> 
                            (update_s2ss_obj s (s2ss s) (D_proc p)
                               (S_proc sp (O_proc p \<in> tainted s))
                               (S_proc sp' (O_proc p \<in> tainted s)))
                      | _ \<Rightarrow> {})
      | _     \<Rightarrow> s2ss s)"
apply (frule vd_cons, frule vt_grant_os)
apply (clarsimp simp only:os_grant.simps)
apply (frule current_proc_has_sp, simp, erule exE)
apply (case_tac "file_of_proc_fd s p fd")

apply (simp add:s2ss_def)
apply (rule set_eqI, rule iffI, erule CollectE, erule exE, erule conjE, rule CollectI)
apply (rule_tac x = obj in exI, simp add:dalive_co2sobj_closefd')
apply (frule co2sobj_closefd, simp)
apply (frule cp2sproc_closefd, simp)
apply (simp add:proc_file_fds_def split:t_dobject.splits)
apply (simp split:if_splits add:co2sobj.simps)
apply (erule CollectE, erule exE, erule conjE, rule CollectI)
apply (rule_tac x = obj in exI, simp add:dalive_co2sobj_closefd2)
apply (frule_tac obj = obj in co2sobj_closefd, simp add:dalive_co2sobj_closefd2)
apply (frule cp2sproc_closefd, simp)
apply (auto simp add:proc_file_fds_def co2sobj.simps 
            split:t_dobject.splits option.splits if_splits)[1]

apply (case_tac "cp2sproc (CloseFd p fd # s) p")
apply (drule current_proc_has_sp', simp, simp)
apply (case_tac "cf2sfile s a")
apply (drule current_file_has_sfile', simp, simp add:file_of_pfd_in_current)
apply (simp)

apply (rule conjI, rule impI, erule conjE)
apply (simp add:del_s2ss_file_def)
apply (rule conjI|rule impI|erule exE|erule conjE|erule bexE)+

apply (simp add:update_s2ss_obj_def)
apply (rule conjI|rule impI|erule exE|erule conjE|erule bexE)+
apply (tactic {*my_seteq_tac 1*})
apply simp
apply (case_tac "obj = D_proc p")
apply (simp add:co2sobj.simps)
apply (rule disjI2, rule_tac x = obj in exI)
apply (frule_tac obj = obj in co2sobj_closefd, simp add:dalive_simps)
apply (simp add:dalive_simps co2sobj.simps split:t_dobject.splits if_splits)
apply (tactic {*my_setiff_tac 1*})
apply (rule_tac x = "D_proc p" in exI)
apply (simp add:co2sobj.simps)
apply (tactic {*my_setiff_tac 1*})
apply (case_tac "obj = D_proc p")
apply (rule_tac x = obj' in exI)
apply (frule co2sobj_sproc_imp, erule exE, simp add:co2sobj_closefd)
apply (simp add:co2sobj.simps)
apply (case_tac "obj = D_file a")
apply (rule_tac x = "D_file f'" in exI)
apply (case_tac "f' = a", simp add:same_inode_files_prop9 file_of_pfd_is_file)
apply (frule_tac obj = "D_file f'" in co2sobj_closefd, simp add:dalive_simps)
apply (simp add:dalive_simps co2sobj.simps split:t_dobject.splits if_splits)
apply (rule_tac x = obj in exI)
apply (frule_tac obj = obj in dalive_co2sobj_closefd3, simp+)
apply (frule_tac obj = obj in co2sobj_closefd, simp)
apply (auto simp add:dalive_simps co2sobj.simps split:t_dobject.splits if_splits)[1]

apply (rule impI)+
apply (tactic {*my_seteq_tac 1*})
apply (case_tac "obj = D_proc p", rule disjI1, simp add:co2sobj.simps)
apply (rule disjI2)
apply (case_tac "obj = D_file a", simp add:dalive_simps)
apply (rule DiffI, simp)
apply (rule_tac x = obj in exI)
apply (frule_tac obj = obj in dalive_co2sobj_closefd', simp+)
apply (frule_tac obj = obj in co2sobj_closefd, simp)
apply (simp add:dalive_simps co2sobj.simps split:t_dobject.splits if_splits)
apply (simp, rule notI, simp, frule co2sobj_sproc_imp, erule exE, simp add:co2sobj_closefd)
apply (erule_tac x = "D_proc pa" in allE, simp)
apply (tactic {*my_setiff_tac 1*})
apply (rule_tac x = "D_proc p" in exI, simp add:co2sobj.simps)
apply (tactic {*my_setiff_tac 1*})
apply (case_tac "obj = D_proc p", simp add:co2sobj.simps)
apply (case_tac "obj = D_file a", rule_tac x = "D_file f'" in exI)
apply (case_tac "f' = a", simp add:same_inode_files_prop9 file_of_pfd_is_file)
apply (frule_tac obj = "D_file f'" in co2sobj_closefd, simp add:dalive_simps)
apply (simp add:dalive_simps co2sobj.simps split:t_dobject.splits if_splits)
apply (rule_tac x = obj in exI)
apply (frule_tac obj = obj in dalive_co2sobj_closefd3, simp+)
apply (frule_tac obj = obj in co2sobj_closefd, simp)
apply (auto simp add:co2sobj.simps split:t_dobject.splits if_splits)[1]

apply (rule impI, tactic {*my_seteq_tac 1*})
apply (simp add:update_s2ss_obj_def update_s2ss_sfile_del_def)
apply (rule conjI| rule impI|erule exE|erule conjE)+
apply (case_tac obj)
apply (case_tac "nat = p", simp add:co2sobj.simps)
apply (rule disjI2, rule_tac x = obj in exI)
apply (frule_tac obj = obj in co2sobj_closefd, simp, simp add:dalive_simps)
apply (case_tac "list = a", simp add:dalive_simps)
apply (rule disjI2, rule_tac x = obj in exI)
apply (frule_tac obj = obj in co2sobj_closefd, simp, simp add:dalive_simps)
apply (rule disjI2, rule_tac x = obj in exI)
apply (frule_tac obj = obj in co2sobj_closefd, simp, simp add:dalive_simps)
apply (rule disjI2, rule_tac x = obj in exI)
apply (frule_tac obj = obj in co2sobj_closefd, simp, simp add:dalive_simps)
apply (rule conjI| rule impI|erule exE|erule conjE)+
apply (case_tac obj)
apply (case_tac "nat = p", simp add:co2sobj.simps)
apply (rule disjI2, rule disjI2, rule_tac x = obj in exI)
apply (frule_tac obj = obj in co2sobj_closefd, simp)
apply (auto simp add:co2sobj.simps split:t_dobject.splits if_splits)[1]
apply (case_tac "list = a", simp add:dalive_simps)
apply (case_tac "list \<in> same_inode_files s a", rule disjI1)
apply (simp add:co2sobj_simps split:if_splits option.splits t_sobject.splits)
apply (simp add:co2sobj.simps same_inodes_tainted cf2sfiles_prop)
apply (erule bexE, erule conjE)
apply (erule_tac x = f'' in ballE, simp, simp)
apply (rule disjI2, rule disjI2, rule_tac x = obj in exI)
apply (frule_tac obj = obj in co2sobj_closefd, simp, simp add:dalive_simps)
apply (rule disjI2, rule disjI2, rule_tac x = obj in exI)
apply (frule_tac obj = obj in co2sobj_closefd, simp, simp add:dalive_simps)
apply (rule disjI2, rule disjI2, rule_tac x = obj in exI)
apply (frule_tac obj = obj in co2sobj_closefd, simp, simp add:dalive_simps)
apply (rule impI, rule conjI, rule impI)
apply (case_tac obj)
apply (case_tac "nat = p", simp add:co2sobj.simps)
apply (rule disjI2, rule conjI, rule_tac x = obj in exI)
apply (frule_tac obj = obj in co2sobj_closefd, simp, simp add:dalive_simps)
apply (rule notI, simp add:co2sobj_closefd, erule_tac x = obj in allE, simp add:is_file_simps)
apply (case_tac "list = a", simp add:dalive_simps)
apply (rule disjI2, rule conjI, rule_tac x = obj in exI)
apply (frule_tac obj = obj in co2sobj_closefd, simp, simp add:dalive_simps)
apply (rule notI, simp add:co2sobj_closefd, erule_tac x = obj in allE, simp add:is_file_simps)
apply (rule disjI2, rule conjI, rule_tac x = obj in exI)
apply (frule_tac obj = obj in co2sobj_closefd, simp, simp add:dalive_simps)
apply (rule notI, simp add:co2sobj_closefd, erule_tac x = obj in allE, simp add:is_dir_simps)
apply (rule disjI2, rule conjI, rule_tac x = obj in exI)
apply (frule_tac obj = obj in co2sobj_closefd, simp, simp add:dalive_simps)
apply (rule notI, simp add:co2sobj_closefd, erule_tac x = obj in allE, simp)
apply (rule impI)
apply (case_tac obj)
apply (case_tac "nat = p", simp add:co2sobj.simps)
apply (rule disjI2, rule disjI2, rule conjI, rule_tac x = obj in exI)
apply (frule_tac obj = obj in co2sobj_closefd, simp)
apply (auto simp add:co2sobj.simps split:t_dobject.splits if_splits)[1]
apply (rule notI, simp add:co2sobj_closefd)
apply (erule_tac x = obj in allE, simp)
apply (case_tac "list = a", simp add:dalive_simps)
apply (case_tac "list \<in> same_inode_files s a", rule disjI1)
apply (simp add:co2sobj_simps split:if_splits option.splits t_sobject.splits)
apply (simp add:co2sobj.simps same_inodes_tainted cf2sfiles_prop)
apply (erule bexE, erule conjE)
apply (erule_tac x = f'' in ballE, simp, simp)
apply (rule disjI2, rule disjI2, rule conjI, rule_tac x = obj in exI)
apply (frule_tac obj = obj in co2sobj_closefd, simp, simp add:dalive_simps)
apply (rule notI, simp add:co2sobj_closefd, erule_tac x = obj in allE, simp add:is_file_simps)
apply (rule disjI2, rule disjI2, rule conjI, rule_tac x = obj in exI)
apply (frule_tac obj = obj in co2sobj_closefd, simp, simp add:dalive_simps)
apply (rule notI, simp add:co2sobj_closefd, erule_tac x = obj in allE, simp add:is_dir_simps)
apply (rule disjI2, rule disjI2, rule conjI, rule_tac x = obj in exI)
apply (frule_tac obj = obj in co2sobj_closefd, simp, simp add:dalive_simps)
apply (rule notI, simp add:co2sobj_closefd, erule_tac x = obj in allE, simp)

apply (simp add:update_s2ss_sfile_del_def update_s2ss_obj_def split:if_splits)
apply (erule disjE, rule_tac x = "D_proc p" in exI, simp add:co2sobj.simps)
apply (erule exE, erule conjE)
apply (case_tac "obj = D_proc p")
apply (rule_tac x = obj' in exI)
apply (frule co2sobj_sproc_imp, erule exE, simp add:co2sobj_closefd)
apply (simp add:co2sobj.simps)
apply (case_tac "obj = D_file a")
apply (rule_tac x = "D_file f'" in exI)
apply (case_tac "f' = a", simp add:same_inode_files_prop9 file_of_pfd_is_file)
apply (frule_tac obj = "D_file f'" in co2sobj_closefd, simp add:dalive_simps)
apply (simp add:dalive_simps co2sobj.simps split:t_dobject.splits if_splits)
apply (rule_tac x = obj in exI)
apply (frule_tac obj = obj in dalive_co2sobj_closefd3, simp+)
apply (frule_tac obj = obj in co2sobj_closefd, simp)
apply (auto simp add:dalive_simps co2sobj.simps split:t_dobject.splits if_splits)[1]
apply (erule disjE, rule_tac x = "D_proc p" in exI, simp add:co2sobj.simps)
apply (erule conjE, erule exE, erule conjE)
apply (case_tac "obj = D_proc p")
apply (simp add:co2sobj.simps)
apply (case_tac "obj = D_file a")
apply (rule_tac x = "D_file f'" in exI)
apply (case_tac "f' = a", simp add:same_inode_files_prop9 file_of_pfd_is_file)
apply (frule_tac obj = "D_file f'" in co2sobj_closefd, simp add:dalive_simps)
apply (simp add:dalive_simps co2sobj.simps split:t_dobject.splits if_splits)
apply (rule_tac x = obj in exI)
apply (frule_tac obj = obj in dalive_co2sobj_closefd3, simp+)
apply (frule_tac obj = obj in co2sobj_closefd, simp)
apply (auto simp add:dalive_simps co2sobj.simps split:t_dobject.splits if_splits)[1]
apply (erule disjE)
apply (drule same_inode_files_prop10, simp add:file_of_pfd_is_file, erule exE, erule conjE)
apply (rule_tac x = "D_file f'a" in exI)
apply (frule same_inode_files_prop11)
apply (frule_tac obj = "D_file f'a" in co2sobj_closefd)
apply (simp add:dalive_simps)+
apply (frule_tac f = "f'a" in is_file_has_sfile', simp, erule exE)
apply (simp add:co2sobj.simps cf2sfiles_prop same_inodes_tainted split:if_splits)
apply (rule impI, erule bexE, erule conjE, erule_tac x = f'' in ballE, simp, simp)
apply (erule bexE, erule conjE, erule_tac x = f'' in ballE, simp, simp)
apply (erule disjE)
apply (rule_tac x = "D_proc p" in exI, simp add:co2sobj.simps)
apply (erule exE, erule conjE)
apply (case_tac "obj = D_proc p")
apply (rule_tac x = obj' in exI)
apply (frule co2sobj_sproc_imp, erule exE, simp add:co2sobj_closefd)
apply (simp add:co2sobj.simps)
apply (case_tac obj)
apply (rule_tac x = obj in exI)
apply (simp add:co2sobj_closefd)
apply (case_tac "list \<in> same_inode_files s a")
apply (rule_tac x = "D_file f'" in exI)
apply (simp add:co2sobj_simps is_file_simps split:if_splits option.splits t_sobject.splits)
apply (rule conjI, rule notI, simp add:same_inode_files_prop9)
apply (rule impI, simp add:co2sobj.simps cf2sfiles_prop same_inodes_tainted)
apply (rule_tac x = obj in exI)
apply (simp add:co2sobj_closefd is_file_simps)
apply (rule notI, simp add:same_inode_files_prop9)
apply (rule_tac x = obj in exI, simp add:co2sobj_closefd is_dir_simps)
apply (rule_tac x = obj in exI, simp add:co2sobj_closefd)

apply (erule disjE)
apply (drule same_inode_files_prop10, simp add:file_of_pfd_is_file, erule exE, erule conjE)
apply (rule_tac x = "D_file f'a" in exI)
apply (frule same_inode_files_prop11)
apply (frule_tac obj = "D_file f'a" in co2sobj_closefd)
apply (simp add:dalive_simps)+
apply (frule_tac f = "f'a" in is_file_has_sfile', simp, erule exE)
apply (simp add:co2sobj.simps cf2sfiles_prop same_inodes_tainted split:if_splits)
apply (rule impI, erule bexE, erule conjE, erule_tac x = f'' in ballE, simp, simp)
apply (erule bexE, erule conjE, erule_tac x = f'' in ballE, simp, simp)
apply (erule disjE)
apply (rule_tac x = "D_proc p" in exI)
apply (simp add:co2sobj.simps)
apply (erule conjE, erule exE, erule conjE)
apply (case_tac "obj = D_proc p")
apply (simp add:co2sobj.simps)
apply (case_tac obj)
apply (rule_tac x = obj in exI)
apply (simp add:co2sobj_closefd)
apply (case_tac "list \<in> same_inode_files s a")
apply (rule_tac x = "D_file f'" in exI)
apply (simp add:co2sobj_simps is_file_simps split:if_splits option.splits t_sobject.splits)
apply (rule conjI, rule notI, simp add:same_inode_files_prop9)
apply (rule impI, simp add:co2sobj.simps cf2sfiles_prop same_inodes_tainted)
apply (rule_tac x = obj in exI)
apply (simp add:co2sobj_closefd is_file_simps)
apply (rule notI, simp add:same_inode_files_prop9)
apply (rule_tac x = obj in exI, simp add:co2sobj_closefd is_dir_simps)
apply (rule_tac x = obj in exI, simp add:co2sobj_closefd)

apply (rule impI, rule conjI, rule impI)
apply (tactic {*my_seteq_tac 1*})
apply (simp add:update_s2ss_obj_def update_s2ss_sfile_del_def)
apply (rule conjI| rule impI|erule exE|erule conjE)+
apply (case_tac obj)
apply (case_tac "nat = p", simp add:co2sobj.simps)
apply (rule disjI2, rule_tac x = obj in exI)
apply (frule_tac obj = obj in co2sobj_closefd, simp, simp add:dalive_simps)
apply (case_tac "list = a", simp add:dalive_simps)
apply (rule disjI2, rule_tac x = obj in exI)
apply (frule_tac obj = obj in co2sobj_closefd, simp, clarsimp simp:dalive_simps split:if_splits)
apply (rule disjI2, rule_tac x = obj in exI)
apply (frule_tac obj = obj in co2sobj_closefd, simp, simp add:dalive_simps)
apply (rule disjI2, rule_tac x = obj in exI)
apply (frule_tac obj = obj in co2sobj_closefd, simp, simp add:dalive_simps)
apply (rule conjI| rule impI|erule exE|erule conjE)+
apply (case_tac obj)
apply (case_tac "nat = p", simp add:co2sobj.simps)
apply (rule disjI2, rule conjI, rule_tac x = obj in exI)
apply (frule_tac obj = obj in co2sobj_closefd, simp)
apply (auto simp add:co2sobj.simps split:t_dobject.splits if_splits)[1]
apply (rule notI, simp, erule_tac x = "D_proc nat" in allE, simp add:co2sobj_closefd)
apply (case_tac "list = a", simp add:dalive_simps)
apply (case_tac "list \<in> same_inode_files s a", rule disjI2)
apply (simp add:co2sobj_simps split:if_splits option.splits t_sobject.splits)
apply (simp add:co2sobj.simps same_inodes_tainted cf2sfiles_prop)
apply (erule bexE, erule conjE)
apply (rule conjI, rule_tac x = "D_file f''" in exI)
apply (simp add:same_inode_files_prop11 co2sobj.simps cf2sfiles_prop same_inodes_tainted)
apply (rule notI, simp)
apply (rule disjI2, rule conjI, rule_tac x = obj in exI)
apply (frule_tac obj = obj in co2sobj_closefd, simp, simp add:dalive_simps)
apply (rule notI, simp add:co2sobj.simps)
apply (rule disjI2, rule conjI, rule_tac x = obj in exI)
apply (frule_tac obj = obj in co2sobj_closefd, simp, simp add:dalive_simps)
apply (rule notI, simp add:co2sobj.simps split:option.splits)
apply (rule disjI2, rule conjI, rule_tac x = obj in exI)
apply (frule_tac obj = obj in co2sobj_closefd, simp, simp add:dalive_simps)
apply (rule notI, simp add:co2sobj.simps split:option.splits)

apply (erule bexE, erule conjE)
apply (simp add:update_s2ss_obj_def split:if_splits)
apply (erule disjE, rule_tac x = "D_proc p" in exI, simp add:co2sobj.simps)
apply (erule exE, erule conjE)
apply (case_tac "obj = D_proc p")
apply (rule_tac x = obj' in exI)
apply (frule co2sobj_sproc_imp, erule exE, simp add:co2sobj_closefd)
apply (simp add:co2sobj.simps)
apply (case_tac "obj = D_file a")
apply (rule_tac x = "D_file f'" in exI)
apply (frule_tac obj = "D_file f'" in co2sobj_closefd, simp add:dalive_simps same_inode_files_prop11)
apply (simp add:dalive_simps co2sobj.simps split:t_dobject.splits if_splits)
apply (rule conjI)
apply (rule impI)
apply (rule_tac x = f' in ballE, simp, simp, simp)
apply (simp add:same_inode_files_prop11 co2sobj.simps cf2sfiles_prop same_inodes_tainted)
apply (rule_tac x = obj in exI)
apply (frule_tac obj = obj in dalive_co2sobj_closefd3, simp+)
apply (frule_tac obj = obj in co2sobj_closefd, simp)
apply (auto simp add:dalive_simps co2sobj.simps split:t_dobject.splits if_splits)[1]
apply (erule disjE, rule_tac x = "D_proc p" in exI, simp add:co2sobj.simps)
apply (erule conjE, erule exE, erule conjE)
apply (case_tac "obj = D_proc p")
apply (simp add:co2sobj.simps)
apply (case_tac "obj = D_file a")
apply (rule_tac x = "D_file f'" in exI)
apply (frule_tac obj = "D_file f'" in co2sobj_closefd, simp add:dalive_simps same_inode_files_prop11)
apply (simp add:dalive_simps co2sobj.simps split:t_dobject.splits if_splits)
apply (rule conjI)
apply (rule impI)
apply (rule_tac x = f' in ballE, simp, simp, simp)
apply (simp add:same_inode_files_prop11 co2sobj.simps cf2sfiles_prop same_inodes_tainted)
apply (rule_tac x = obj in exI)
apply (frule_tac obj = obj in dalive_co2sobj_closefd3, simp+)
apply (frule_tac obj = obj in co2sobj_closefd, simp)
apply (auto simp add:dalive_simps co2sobj.simps split:t_dobject.splits if_splits)[1]

apply (rule impI)
apply (tactic {*my_seteq_tac 1*})
apply (simp add:update_s2ss_obj_def update_s2ss_sfile_del_def)
apply (rule conjI| rule impI|erule exE|erule conjE)+
apply (case_tac obj)
apply (case_tac "nat = p", simp add:co2sobj.simps)
apply (rule disjI2, rule_tac x = obj in exI)
apply (frule_tac obj = obj in co2sobj_closefd, simp, simp add:dalive_simps)
apply (case_tac "list = a", simp add:dalive_simps)
apply (rule disjI2, rule_tac x = obj in exI)
apply (frule_tac obj = obj in co2sobj_closefd, simp, clarsimp simp:dalive_simps split:if_splits)
apply (rule disjI2, rule_tac x = obj in exI)
apply (frule_tac obj = obj in co2sobj_closefd, simp, simp add:dalive_simps)
apply (rule disjI2, rule_tac x = obj in exI)
apply (frule_tac obj = obj in co2sobj_closefd, simp, simp add:dalive_simps)
apply (frule_tac obj = obj in co2sobj_closefd, simp, rule notI, simp)
apply (frule_tac obj = obj in co2sobj_sfile_imp, erule exE, simp add:is_file_simps split:if_splits)
apply (erule_tac x= f in allE, simp add:co2sobj.simps)
apply (rule conjI| rule impI|erule exE|erule conjE)+
apply (case_tac obj)
apply (case_tac "nat = p", simp add:co2sobj.simps)
apply (rule disjI2, rule notI, simp)
apply (rule disjI2, rule conjI, rule disjI2, rule_tac x = obj in exI)
apply (frule_tac obj = obj in co2sobj_closefd, simp)
apply (auto simp add:co2sobj.simps split:t_dobject.splits if_splits)[1]
apply (rule notI, simp add:co2sobj.simps split:option.splits)
apply (case_tac "list = a", simp add:dalive_simps)
apply (case_tac "list \<in> same_inode_files s a", rule disjI1)
apply (simp add:co2sobj_simps split:if_splits option.splits t_sobject.splits)
apply (simp add:co2sobj.simps same_inodes_tainted cf2sfiles_prop)
apply (erule bexE, erule conjE)
apply (erule_tac x = f'' in ballE, simp, simp)
apply (rule disjI2, rule conjI, rule disjI2, rule_tac x = obj in exI)
apply (simp add:is_file_simps co2sobj_closefd)
apply (rule notI, simp add:co2sobj_closefd)
apply (erule_tac x = list in allE, simp add:is_file_simps co2sobj.simps) 
apply (rule disjI2, rule conjI, rule disjI2, rule_tac x = obj in exI)
apply (frule_tac obj = obj in co2sobj_closefd, simp, simp add:dalive_simps)
apply (rule notI, simp add:co2sobj.simps split:option.splits)
apply (rule disjI2, rule conjI, rule disjI2, rule_tac x = obj in exI)
apply (frule_tac obj = obj in co2sobj_closefd, simp, simp add:dalive_simps)
apply (rule notI, simp add:co2sobj.simps split:option.splits)
apply (rule impI, rule conjI, rule impI)

apply (case_tac obj)
apply (case_tac "nat = p", simp add:co2sobj.simps)
apply (rule notI, simp)
apply (rule conjI, rule disjI2, rule conjI, rule_tac x = obj in exI)
apply (frule_tac obj = obj in co2sobj_closefd, simp)
apply (auto simp add:co2sobj.simps split:t_dobject.splits if_splits)[1]
apply (erule_tac x = obj in allE, simp add:co2sobj_closefd)
apply (rule notI, simp add:co2sobj.simps split:option.splits)
apply (case_tac "list = a", simp add:dalive_simps)
apply (case_tac "list \<in> same_inode_files s a", rule conjI, rule disjI2, rule conjI)
apply (simp add:co2sobj_simps split:if_splits option.splits t_sobject.splits)
apply (simp add:co2sobj.simps)
apply (rule notI, simp add:co2sobj.simps)
apply (rule conjI, rule disjI2, rule conjI, rule_tac x = obj in exI)
apply (simp add:is_file_simps co2sobj_closefd)
apply (rule notI, simp add:co2sobj.simps)
apply (rule notI, simp add:co2sobj_closefd)
apply (erule_tac x = list in allE, simp add:is_file_simps co2sobj.simps)
apply (rule conjI, rule disjI2, rule conjI, rule_tac x = obj in exI)
apply (frule_tac obj = obj in co2sobj_closefd, simp, simp add:dalive_simps)
apply (rule notI, simp add:co2sobj.simps split:option.splits)+
apply (rule conjI, rule disjI2, rule conjI, rule_tac x = obj in exI)
apply (frule_tac obj = obj in co2sobj_closefd, simp, simp add:dalive_simps)
apply (rule notI, simp add:co2sobj.simps split:option.splits)+
apply (rule impI)
apply (case_tac obj)
apply (case_tac "nat = p", simp add:co2sobj.simps)
apply (rule disjI2, rule notI, simp)
apply (rule disjI2, rule conjI, rule disjI2, rule conjI, rule_tac x = obj in exI)
apply (frule_tac obj = obj in co2sobj_closefd, simp)
apply (auto simp add:co2sobj.simps split:t_dobject.splits if_splits)[1]
apply (erule_tac x = obj in allE, simp add:co2sobj_closefd)
apply (rule notI, simp add:co2sobj.simps split:option.splits)
apply (case_tac "list = a", simp add:dalive_simps)
apply (case_tac "list \<in> same_inode_files s a", rule disjI1)
apply (simp add:co2sobj_closefd split:if_splits option.splits t_sobject.splits)
apply (simp add:co2sobj.simps cf2sfiles_prop same_inodes_tainted)
apply (erule bexE, erule conjE, erule_tac x = "f''" in ballE, simp, simp)
apply (rule disjI2, rule conjI, rule disjI2, rule conjI, rule_tac x = obj in exI)
apply (simp add:is_file_simps co2sobj_closefd)
apply (rule notI, simp add:co2sobj.simps)
apply (rule notI, simp add:co2sobj_closefd)
apply (erule_tac x = list in allE, simp add:is_file_simps co2sobj.simps)
apply (rule disjI2, rule conjI, rule disjI2, rule conjI, rule_tac x = obj in exI)
apply (frule_tac obj = obj in co2sobj_closefd, simp, simp add:dalive_simps)
apply (rule notI, simp add:co2sobj.simps split:option.splits)+
apply (rule disjI2, rule conjI, rule disjI2, rule conjI, rule_tac x = obj in exI)
apply (frule_tac obj = obj in co2sobj_closefd, simp, simp add:dalive_simps)
apply (rule notI, simp add:co2sobj.simps split:option.splits)+

apply (simp add:update_s2ss_sfile_del_def update_s2ss_obj_def split:if_splits)
apply (erule conjE, erule disjE)
apply (rule_tac x = "D_proc p" in exI, simp add:co2sobj.simps)
apply (erule exE, erule conjE)
apply (case_tac "obj = D_file a", simp add:co2sobj.simps)
apply (case_tac "obj = D_proc p")
apply (rule_tac x = obj' in exI, frule_tac obj = obj' in co2sobj_sproc_imp, erule exE)
apply (frule_tac obj = obj' in dalive_co2sobj_closefd3, simp+)
apply (simp add:co2sobj_closefd)
apply (simp add:co2sobj.simps)
apply (frule_tac obj = obj in dalive_co2sobj_closefd3, simp+)
apply (rule_tac x = obj in exI, simp add:co2sobj_closefd)
apply (auto simp add:co2sobj.simps split:t_dobject.splits if_splits)[1]
apply (erule conjE|erule exE|erule disjE)+
apply (rule_tac x = "D_proc p" in exI, simp add:co2sobj.simps)
apply (erule conjE, erule exE, erule conjE)
apply (case_tac "obj = D_file a", simp add:co2sobj.simps)
apply (case_tac "obj = D_proc p")
apply (simp add:co2sobj.simps)
apply (frule_tac obj = obj in dalive_co2sobj_closefd3, simp+)
apply (rule_tac x = obj in exI, simp add:co2sobj_closefd)
apply (auto simp add:co2sobj.simps split:t_dobject.splits if_splits)[1]
apply (erule conjE|erule exE|erule disjE)+
apply (drule same_inode_files_prop10, simp add:file_of_pfd_is_file, erule exE, erule conjE)
apply (rule_tac x = "D_file f'" in exI)
apply (frule same_inode_files_prop11)
apply (frule_tac obj = "D_file f'" in co2sobj_closefd)
apply (simp add:dalive_simps)+
apply (frule_tac f = "f'" in is_file_has_sfile', simp, erule exE)
apply (simp add:co2sobj.simps cf2sfiles_prop same_inodes_tainted split:if_splits)
apply (rule impI, erule bexE, erule conjE, erule_tac x = f'' in ballE, simp, simp)
apply (erule bexE, erule conjE, erule_tac x = f'' in ballE, simp, simp)
apply (erule conjE, erule disjE)
apply (rule_tac x = "D_proc p" in exI)
apply (simp add:co2sobj.simps)
apply (erule exE, erule conjE)
apply (case_tac "obj = D_proc p")
apply (rule_tac x = "obj'" in exI, simp, frule_tac obj = obj' in co2sobj_sproc_imp, erule exE)
apply (frule_tac obj = obj' in dalive_co2sobj_closefd3, simp+)
apply (simp add:co2sobj_closefd)
apply (simp add:co2sobj.simps)
apply (case_tac "obj = D_file a", simp add:co2sobj.simps)
apply (frule_tac obj = obj in dalive_co2sobj_closefd3, simp+)
apply (case_tac obj)
apply (rule_tac x = obj in exI)
apply (simp add:co2sobj_closefd)
apply (case_tac "list = a", simp add:dalive_simps)
apply (case_tac "list \<in> same_inode_files s a")
apply (simp add:co2sobj.simps cf2sfiles_prop same_inodes_tainted)
apply (rule_tac x = obj in exI, simp add:co2sobj_closefd)
apply (rule_tac x = obj in exI, simp add:co2sobj_closefd)
apply (rule_tac x = obj in exI, simp add:co2sobj_closefd)
apply (erule disjE)
apply (drule same_inode_files_prop10, simp add:file_of_pfd_is_file, erule exE, erule conjE)
apply (rule_tac x = "D_file f'" in exI)
apply (frule same_inode_files_prop11)
apply (frule_tac obj = "D_file f'" in co2sobj_closefd)
apply (simp add:dalive_simps)+
apply (frule_tac f = "f'" in is_file_has_sfile', simp, erule exE)
apply (simp add:co2sobj.simps cf2sfiles_prop same_inodes_tainted split:if_splits)
apply (rule impI, erule bexE, erule conjE, erule_tac x = f'' in ballE, simp, simp)
apply (erule bexE, erule conjE, erule_tac x = f'' in ballE, simp, simp)
apply (erule conjE, erule disjE)
apply (rule_tac x = "D_proc p" in exI)
apply (simp add:co2sobj.simps)
apply (erule conjE, erule exE, erule conjE)
apply (case_tac "obj = D_proc p", simp add:co2sobj.simps)
apply (case_tac "obj = D_file a", simp add:co2sobj.simps)
apply (case_tac obj)
apply (rule_tac x = obj in exI)
apply (simp add:co2sobj_closefd)
apply (case_tac "list \<in> same_inode_files s a")
apply (simp add:co2sobj.simps cf2sfiles_prop same_inodes_tainted)
apply (rule_tac x = obj in exI, simp add:co2sobj_closefd is_file_simps)
apply (rule_tac x = obj in exI, simp add:co2sobj_closefd is_dir_simps)
apply (rule_tac x = obj in exI, simp add:co2sobj_closefd)

apply (rule impI)
apply (simp add:update_s2ss_obj_def)
apply (rule conjI, rule impI, erule exE, erule conjE)
apply (simp add:s2ss_def)
apply (rule set_eqI, rule iffI, erule CollectE, erule exE, erule conjE)
apply (simp)
apply (case_tac "obj = D_proc p")
apply (simp add:co2sobj.simps split:if_splits)
apply (rule disjI2, rule_tac x = obj in exI, erule conjE)
apply (simp add:dalive_co2sobj_closefd')
apply (frule_tac obj = obj in co2sobj_closefd, simp, simp split:t_dobject.splits if_splits)
apply (simp, erule disjE)
apply (rule_tac x = "D_proc p" in exI, simp add:co2sobj.simps)
apply (erule exE, erule conjE)
apply (case_tac "obj = D_proc p")
apply (rule_tac x = obj' in exI, simp add:dalive_co2sobj_closefd1)
apply (frule_tac obj = obj' in co2sobj_closefd, simp add:dalive_co2sobj_closefd1)
apply (clarsimp split:t_dobject.splits if_splits option.splits simp:co2sobj.simps)
apply (rule_tac x = obj in exI, simp add:dalive_co2sobj_closefd1)
apply (frule_tac obj = obj in co2sobj_closefd, simp add:dalive_co2sobj_closefd1)
apply (clarsimp split:t_dobject.splits if_splits option.splits simp: co2sobj.simps)
apply (rule impI)
apply (simp add:s2ss_def)
apply (rule set_eqI, rule iffI, erule CollectE, erule exE, erule conjE)
apply (simp)
apply (case_tac "obj = D_proc p")
apply (rule disjI1, simp add:co2sobj.simps split:if_splits)
apply (rule disjI2, rule conjI, rule_tac x = obj in exI, simp add:dalive_co2sobj_closefd')
apply (frule_tac obj = obj in co2sobj_closefd, simp add:dalive_co2sobj_closefd1)
apply (clarsimp split:t_dobject.splits if_splits option.splits simp: co2sobj.simps)
apply (rule notI, erule_tac x = obj in allE, simp add:dalive_co2sobj_closefd')
apply (frule_tac obj = obj in co2sobj_closefd, simp add:dalive_co2sobj_closefd1)
apply (clarsimp split:t_dobject.splits if_splits option.splits)
apply (simp)
apply (erule disjE)
apply (rule_tac x = "D_proc p" in exI, simp add:co2sobj.simps)
apply (erule exE|erule conjE)+
apply (rule_tac x = obj in exI)
apply (frule_tac obj = obj in co2sobj_closefd, simp add:dalive_co2sobj_closefd1)
apply (clarsimp split:t_dobject.splits if_splits option.splits 
                simp: co2sobj.simps dalive_co2sobj_closefd1)
done

lemma dalive_co2sobj_unlink:
  "\<lbrakk>dalive s obj; valid (UnLink p f # s); obj \<noteq> D_file f\<rbrakk>
   \<Longrightarrow> dalive (UnLink p f # s) obj"
by (auto simp add:dalive_simps split:t_dobject.splits)

lemma s2ss_unlink:
  "valid (UnLink p f # s) \<Longrightarrow> s2ss (UnLink p f # s) = (
     if (proc_fd_of_file s f = {}) 
     then (case (cf2sfile s f) of 
             Some sf \<Rightarrow> del_s2ss_file s (s2ss s) f sf
           | _       \<Rightarrow> {})
     else s2ss s)"
apply (frule vd_cons, frule vt_grant_os, clarsimp split:if_splits)
apply (frule is_file_has_sfile', simp, erule exE, simp)
apply (rule conjI, rule impI)
apply (simp add:update_s2ss_sfile_del_def del_s2ss_file_def)
apply (rule impI|erule conjE|erule exE|rule conjI|erule bexE)+ defer
apply (rule impI|erule conjE|erule exE|rule conjI|erule bexE)+ 

apply (simp add:s2ss_def)
apply (tactic {*my_seteq_tac 1*})
apply (case_tac "obj = D_file f", simp add:is_file_simps)
apply simp
apply (rule conjI)
apply (rule_tac x = obj in exI,simp add:co2sobj_unlink is_file_simps is_dir_simps split:t_dobject.splits)
apply (rule notI, simp, frule_tac obj = obj in co2sobj_sfile_imp, erule exE, simp)
apply (frule_tac obj = obj in co2sobj_unlink, simp)
apply (erule_tac x = fa in allE, simp add:is_file_simps)
apply (simp add:co2sobj.simps)
apply (tactic {*my_setiff_tac 1*})
apply (case_tac "obj = D_file f", simp add:co2sobj.simps)
apply (frule_tac dalive_co2sobj_unlink, simp, simp)
apply (frule_tac obj = obj in co2sobj_unlink, simp)
apply (rule_tac x = obj in exI)
apply (simp add:co2sobj.simps split:t_dobject.splits if_splits)

apply (rule impI|erule conjE|erule exE|rule conjI|erule bexE)+  defer

apply (rule impI|erule conjE|erule exE|rule conjI|erule bexE)+ 
apply (simp add:s2ss_def)
apply (tactic {*my_seteq_tac 1*})
apply (case_tac "obj = D_file f", simp add:dalive_simps)
apply (case_tac obj)
apply (rule disjI2, simp, rule_tac x = obj in exI)
apply (simp add:co2sobj_unlink)
apply (case_tac "list \<in> same_inode_files s f")
apply (rule disjI1)
apply (simp add:co2sobj_unlink)
apply (simp add:co2sobj.simps same_inodes_tainted cf2sfiles_prop split:if_splits)
apply (erule bexE, erule_tac x = f'' in ballE, simp, simp)
apply (rule disjI2, simp, rule_tac x = obj in exI)
apply (simp add:co2sobj_unlink is_file_simps)
apply (rule disjI2, simp, rule_tac x = obj in exI, simp add:co2sobj_unlink is_dir_simps)
apply (rule disjI2, simp, rule_tac x = obj in exI, simp add:co2sobj_unlink)
apply (tactic {*my_setiff_tac 1*})
apply (drule same_inode_files_prop10, simp, erule exE, erule conjE)
apply (rule_tac x = "D_file f'a" in exI, simp add:is_file_simps)
apply (frule_tac obj = "D_file f'a" in co2sobj_unlink, simp add:same_inode_files_prop11 is_file_simps)
apply (simp add:co2sobj.simps same_inodes_tainted cf2sfiles_prop 
  is_file_simps same_inode_files_prop11 split:if_splits)
apply (rule impI, erule bexE, erule_tac x = f'' in ballE, simp, simp)
apply (erule bexE, erule_tac x = f'' in ballE, simp, simp)

apply (tactic {*my_setiff_tac 1*})
apply (case_tac "f' = f", simp add:same_inode_files_prop9) 
apply (case_tac "obj= D_file f")
apply (rule_tac x = "D_file f'" in exI, simp add:is_file_simps)
apply (frule_tac f' = f' in cf2sfiles_unlink, simp add:current_files_simps is_file_in_current)
apply (simp add:co2sobj.simps)
apply (case_tac obj)
apply (rule_tac x = obj in exI, simp add:co2sobj_unlink)
apply (case_tac "list \<in> same_inode_files s f")
apply (rule_tac x = "D_file f'" in exI)
apply (frule_tac f' = f' in cf2sfiles_unlink, simp add:current_files_simps is_file_in_current)
apply (simp add:co2sobj.simps is_file_simps cf2sfiles_prop same_inodes_tainted)
apply (rule_tac x = obj in exI, simp add:co2sobj_unlink is_file_simps)
apply (rule_tac x = obj in exI, simp add:co2sobj_unlink is_dir_simps)
apply (rule_tac x = obj in exI, simp add:co2sobj_unlink)

apply (rule impI|erule conjE|erule exE|rule conjI|erule bexE)+  defer

apply (rule impI|erule conjE|erule exE|rule conjI|erule bexE)+ 
apply (simp add:s2ss_def)
apply (tactic {*my_seteq_tac 1*})
apply (case_tac "obj = D_file f", simp add:dalive_simps, simp)
apply (case_tac obj)
apply (rule disjI2, rule conjI, simp, rule_tac x = obj in exI)
apply (simp add:co2sobj_unlink)
apply (rule notI, simp add:co2sobj.simps split:option.splits)
apply (case_tac "list \<in> same_inode_files s f")
apply (rule disjI1)
apply (simp add:co2sobj_unlink)
apply (simp add:co2sobj.simps same_inodes_tainted cf2sfiles_prop split:if_splits)
apply (erule bexE, erule_tac x = f'' in ballE, simp, simp)
apply (rule disjI2, rule conjI, simp, rule_tac x = obj in exI)
apply (simp add:co2sobj_unlink is_file_simps)
apply (rule notI, simp add:co2sobj_unlink)
apply (erule_tac x = list in allE, simp add:co2sobj.simps is_file_simps)
apply (rule disjI2, rule conjI, simp, rule_tac x = obj in exI, simp add:co2sobj_unlink is_dir_simps)
apply (rule notI, simp add:co2sobj.simps split:option.splits)
apply (rule disjI2, rule conjI, simp, rule_tac x = obj in exI, simp add:co2sobj_unlink)
apply (rule notI, simp add:co2sobj.simps split:option.splits)
apply (tactic {*my_setiff_tac 1*})
apply (drule same_inode_files_prop10, simp, erule exE, erule conjE)
apply (rule_tac x = "D_file f'" in exI, simp add:is_file_simps)
apply (frule_tac obj = "D_file f'" in co2sobj_unlink, simp add:same_inode_files_prop11 is_file_simps)
apply (simp add:co2sobj.simps same_inodes_tainted cf2sfiles_prop 
  is_file_simps same_inode_files_prop11 split:if_splits)
apply (rule impI, erule bexE, erule_tac x = f'' in ballE, simp, simp)
apply (erule bexE, erule_tac x = f'' in ballE, simp, simp)
apply (tactic {*my_setiff_tac 1*}, simp)
apply (case_tac "obj = D_file f", simp add:co2sobj.simps)
apply (case_tac obj)
apply (rule_tac x = obj in exI, simp add:co2sobj_unlink)
apply (case_tac "list \<in> same_inode_files s f")
apply (simp add:co2sobj.simps cf2sfiles_prop same_inodes_tainted)
apply (rule_tac x = obj in exI, simp add:co2sobj_unlink is_file_simps)
apply (rule_tac x = obj in exI, simp add:co2sobj_unlink is_dir_simps)
apply (rule_tac x = obj in exI, simp add:co2sobj_unlink)

apply (rule impI)
apply (simp add:s2ss_def)
apply (tactic {*my_seteq_tac 1*})
apply (rule_tac x = obj in exI)
apply (simp add:co2sobj_unlink is_file_simps is_dir_simps split:t_dobject.splits)
apply (tactic {*my_setiff_tac 1*})
apply (rule_tac x = obj in exI)
apply (subgoal_tac "dalive (UnLink p f # s) obj")
apply (auto simp add:co2sobj_unlink is_file_simps is_dir_simps split:t_dobject.splits)[1]
apply (auto simp add:co2sobj_unlink dalive_simps split:t_dobject.splits)[1]

apply (simp add:s2ss_def)
apply (tactic {*my_seteq_tac 1*})
apply (case_tac "obj = D_file f", simp add:dalive_simps)
apply (rule_tac x = obj in exI, simp add:co2sobj_unlink is_file_simps is_dir_simps split:t_dobject.splits)
apply (tactic {*my_setiff_tac 1*})
apply (case_tac "obj = D_file f")
apply (rule_tac x = "D_file f'" in exI)
apply (auto simp add:co2sobj_unlink is_file_simps is_dir_simps split:t_dobject.splits)[1]
apply (rule_tac x =obj in exI)
apply (subgoal_tac "dalive (UnLink p f # s) obj")
apply (auto simp add:co2sobj_unlink is_file_simps is_dir_simps split:t_dobject.splits)[1]
apply (auto simp add:co2sobj_unlink dalive_simps split:t_dobject.splits)[1]

apply (simp add:s2ss_def)
apply (tactic {*my_seteq_tac 1*})
apply (case_tac "obj = D_file f", simp add:dalive_simps)
apply (rule_tac x = obj in exI, simp add:co2sobj_unlink is_file_simps is_dir_simps split:t_dobject.splits if_splits)
apply (tactic {*my_setiff_tac 1*})
apply (case_tac "obj = D_file f")
apply (rule_tac x = "D_file f'" in exI)
apply (auto simp add:co2sobj_unlink is_file_simps is_dir_simps same_inode_files_prop9 split:t_dobject.splits)[1]
apply (case_tac obj)
apply (rule_tac x = obj in exI, simp add:co2sobj_unlink)
apply (case_tac "list \<in> same_inode_files s f")
apply (rule_tac x = "D_file f'" in exI)
apply (simp add:dalive_simps co2sobj.simps)
apply (rule conjI, rule notI, simp add:same_inode_files_prop9)
apply (rule impI, frule_tac f' = f' in cf2sfiles_unlink)
apply (simp add:current_files_simps is_file_simps is_file_in_current)
apply (simp add:same_inodes_tainted cf2sfiles_prop)
apply (rule_tac x = obj in exI, simp add:co2sobj_unlink is_file_simps)
apply (rule_tac x = obj in exI, simp add:co2sobj_unlink is_dir_simps)
apply (rule_tac x = obj in exI, simp add:co2sobj_unlink)

apply (simp add:s2ss_def)
apply (tactic {*my_seteq_tac 1*})
apply (case_tac "obj = D_file f", simp add:dalive_simps)
apply (rule_tac x = obj in exI, simp add:co2sobj_unlink is_file_simps is_dir_simps split:t_dobject.splits if_splits)
apply (tactic {*my_setiff_tac 1*})
apply (case_tac "obj = D_file f")
apply (rule_tac x = "D_file f'" in exI)
apply (subgoal_tac "dalive (UnLink p f # s) (D_file f')")
apply (frule same_inode_files_prop11, frule_tac f = f' in is_file_has_sfile', simp add:vd_cons, erule exE)
apply (frule_tac obj = "D_file f'" in co2sobj_unlink, simp)
apply (simp split:if_splits option.splits add:is_file_simps)
apply (simp add:co2sobj.simps cf2sfiles_prop same_inodes_tainted)
apply (auto split:t_sobject.splits)[1]
apply (simp add:is_file_simps same_inode_files_prop11)
apply (case_tac obj)
apply (rule_tac x = obj in exI, simp add:co2sobj_unlink)
apply (case_tac "list \<in> same_inode_files s f")
apply (rule_tac x = "D_file f'" in exI)
apply (subgoal_tac "dalive (UnLink p f # s) (D_file f')")
apply (frule same_inode_files_prop11, frule_tac f = f' in is_file_has_sfile', simp add:vd_cons, erule exE)
apply (frule_tac obj = "D_file f'" in co2sobj_unlink, simp)
apply (simp split:if_splits option.splits add:is_file_simps)
apply (simp add:co2sobj.simps cf2sfiles_prop same_inodes_tainted)
apply (auto split:t_sobject.splits)[1]
apply (simp add:is_file_simps same_inode_files_prop11)
apply (rule_tac x = obj in exI, simp add:co2sobj_unlink is_file_simps)
apply (rule_tac x = obj in exI, simp add:co2sobj_unlink is_dir_simps)
apply (rule_tac x = obj in exI, simp add:co2sobj_unlink)
done

lemma s2ss_rmdir: "valid (Rmdir p f # s) \<Longrightarrow> s2ss (Rmdir p f # s) = (
  case (co2sobj s (D_dir f)) of
    Some sdir \<Rightarrow> del_s2ss_obj s (s2ss s) (D_dir f) sdir
  | _         \<Rightarrow> {})"
apply (frule vd_cons, frule vt_grant_os)
apply (clarsimp simp:dir_is_empty_def)
apply (frule is_dir_has_sdir', simp, erule exE)
apply (simp split:option.splits, rule conjI, rule impI, simp add:co2sobj.simps)
apply (rule allI, rule impI)

apply (simp add:del_s2ss_obj_def)
apply (rule conjI|rule impI|erule exE|erule conjE)+
apply (simp add:s2ss_def)
apply (tactic {*my_seteq_tac 1*})
apply (rule_tac x = obj in exI)
apply (simp add:co2sobj_rmdir is_file_simps is_dir_simps dalive_simps split:t_dobject.splits if_splits)
apply (tactic {*my_setiff_tac 1*})
apply (case_tac "obj = D_dir f")
apply (rule_tac x = obj' in exI)
apply (subgoal_tac "dalive (Rmdir p f # s) obj'")
apply (auto simp add:co2sobj_rmdir is_file_simps is_dir_simps split:t_dobject.splits)[1]
apply (simp add:dalive_rmdir)
apply (rule_tac x = obj in exI)
apply (subgoal_tac "dalive (Rmdir p f # s) obj")
apply (auto simp add:co2sobj_rmdir is_file_simps is_dir_simps split:t_dobject.splits)[1]
apply (simp add:dalive_rmdir)

apply (rule conjI|rule impI|erule exE|erule conjE)+
apply (simp add:s2ss_def)
apply (tactic {*my_seteq_tac 1*})
apply simp
apply (case_tac "obj = D_dir f", simp add:dalive_rmdir)
apply (rule conjI)
apply (rule_tac x = obj in exI, simp add:co2sobj_rmdir dalive_rmdir)
apply (simp add:co2sobj_rmdir)
apply (simp add:dalive_rmdir, erule_tac x = obj in allE, simp)
apply (tactic {*my_setiff_tac 1*}, simp)
apply (case_tac "obj = D_dir f", simp)
apply (rule_tac x = obj in exI, simp add:co2sobj_rmdir dalive_rmdir)
done

lemma s2ss_mkdir: "valid (Mkdir p f inum # s) \<Longrightarrow> s2ss (Mkdir p f inum # s) = (
  case (cf2sfile (Mkdir p f inum # s) f) of
    Some sf \<Rightarrow> (s2ss s) \<union> {S_dir sf}
  | _       \<Rightarrow> {})"
apply (frule vt_grant_os, frule vd_cons, clarsimp)
apply (case_tac "cf2sfile (Mkdir p f inum # s) f")
apply (drule current_file_has_sfile', simp, simp add:current_files_simps, simp)

apply (simp add:s2ss_def)
apply (tactic {*my_seteq_tac 1*}, simp)
apply (case_tac "obj = D_dir f")
apply (rule disjI1, simp add:co2sobj.simps)
apply (rule disjI2, rule_tac x = obj in exI, simp add:co2sobj_mkdir dalive_simps)
apply (tactic {*my_setiff_tac 1*}, simp)
apply (rule_tac x = "D_dir f" in exI, simp add:dalive_mkdir co2sobj.simps)
apply (tactic {*my_setiff_tac 1*})
apply (case_tac "obj = D_dir f", simp add:is_dir_in_current)
apply (rule_tac x = obj in exI, simp add:co2sobj_mkdir dalive_mkdir)
done

definition update_s2ss_sfile_add :: "t_state \<Rightarrow> t_static_state \<Rightarrow> t_file \<Rightarrow> t_sfile \<Rightarrow> t_static_state"
where
 "update_s2ss_sfile_add s ss f sf \<equiv>
    if (\<exists> f'. is_file s f' \<and> f' \<notin> same_inode_files s f \<and> co2sobj s (D_file f') = co2sobj s (D_file f))
       then ss \<union> {S_file (cf2sfiles s f \<union> {sf}) (O_file f \<in> tainted s)}
       else ss - {S_file (cf2sfiles s f) (O_file f \<in> tainted s)} 
               \<union> {S_file (cf2sfiles s f \<union> {sf}) (O_file f \<in> tainted s)}"

lemma s2ss_linkhard: "valid (LinkHard p f f' # s) \<Longrightarrow> s2ss (LinkHard p f f' # s) = (
  case (cf2sfile (LinkHard p f f' # s) f') of
    Some sf \<Rightarrow> update_s2ss_sfile_add s (s2ss s) f sf
  | _       \<Rightarrow> {})"
apply (frule vt_grant_os, frule vd_cons, clarsimp)
apply (split option.splits)
apply (rule conjI, rule impI, drule current_file_has_sfile', simp, simp add:current_files_simps)
apply (rule allI, rule impI)

apply (simp add:update_s2ss_sfile_add_def)
apply (rule conjI, rule impI, erule exE, erule conjE, erule conjE)
apply (simp add:s2ss_def)
apply (tactic {*my_seteq_tac 1*})
apply (case_tac "obj = D_file f'")
apply (rule disjI1, simp add:co2sobj.simps cf2sfiles_linkhard
  same_inode_files_linkhard split:if_splits)
apply (case_tac "O_file f' \<in> tainted s")
apply (drule tainted_in_current, simp, simp add:is_file_in_current dalive.simps, simp)
apply (case_tac obj)
apply (rule disjI2, simp, rule_tac x = obj in exI)
apply (simp add:co2sobj_linkhard dalive_linkhard)
apply (case_tac "list \<in> same_inode_files s f")
apply (rule disjI1, simp add:co2sobj.simps cf2sfiles_linkhard 
  same_inodes_tainted split:if_splits)
apply (rule disjI2, simp, rule_tac x = obj in exI, simp add:co2sobj_linkhard is_file_simps)
apply (rule disjI2, simp, rule_tac x = obj in exI, simp add:co2sobj_linkhard is_dir_simps)
apply (rule disjI2, simp, rule_tac x = obj in exI, simp add:co2sobj_linkhard)
apply (tactic {*my_setiff_tac 1*})
apply (rule_tac x = "D_file f" in exI)
apply (frule_tac obj = "D_file f" in co2sobj_linkhard)
apply (simp add:dalive_linkhard)
apply (simp add:dalive_linkhard same_inode_files_prop9 split:t_dobject.splits)
apply (tactic {*my_setiff_tac 1*})
apply (case_tac "obj = D_file f'", simp add:dalive_linkhard is_file_in_current)
apply (case_tac obj)
apply (rule_tac x = obj in exI, simp add:co2sobj_linkhard dalive_linkhard)
apply (case_tac "list \<in> same_inode_files s f")
apply (rule_tac x = "D_file f'a" in exI, simp add:co2sobj_linkhard dalive_linkhard)
apply (rule conjI, rule impI, simp add:is_file_in_current)
apply (rule impI, simp add:co2sobj.simps cf2sfiles_prop same_inodes_tainted)
apply (rule_tac x = obj in exI, simp add:co2sobj_linkhard dalive_linkhard)
apply (rule_tac x = obj in exI, simp add:co2sobj_linkhard dalive_linkhard)
apply (rule_tac x = obj in exI, simp add:co2sobj_linkhard dalive_linkhard)

apply (rule impI)
apply (simp add:s2ss_def)
apply (tactic {*my_seteq_tac 1*})
apply (case_tac "obj = D_file f'", simp)
apply (rule disjI1, simp add:co2sobj.simps cf2sfiles_linkhard
  same_inode_files_linkhard split:if_splits)
apply (case_tac "O_file f' \<in> tainted s")
apply (drule tainted_in_current, simp, simp add:is_file_in_current dalive.simps, simp)
apply (case_tac obj, simp)
apply (rule disjI2, rule conjI, rule_tac x = obj in exI)
apply (simp add:co2sobj_linkhard dalive_linkhard)
apply (rule notI, simp add:co2sobj.simps split:option.splits)
apply (case_tac "list \<in> same_inode_files s f")
apply (rule disjI1, simp add:co2sobj.simps cf2sfiles_linkhard 
  same_inodes_tainted split:if_splits)
apply (simp, rule disjI2, rule conjI, rule_tac x = obj in exI, simp add:co2sobj_linkhard is_file_simps)
apply (erule_tac x = list in allE, rule notI)
apply (simp add:co2sobj_linkhard is_file_simps)
apply (simp add:co2sobj.simps)
apply (rule disjI2, simp, rule conjI, rule_tac x = obj in exI, simp add:co2sobj_linkhard is_dir_simps)
apply (rule notI, simp add:co2sobj.simps split:option.splits)
apply (rule disjI2, simp, rule conjI, rule_tac x = obj in exI, simp add:co2sobj_linkhard)
apply (rule notI, simp add:co2sobj.simps split:option.splits)
apply (tactic {*my_setiff_tac 1*})
apply (rule_tac x = "D_file f" in exI)
apply (frule_tac obj = "D_file f" in co2sobj_linkhard)
apply (simp add:dalive_linkhard)
apply (simp add:dalive_linkhard same_inode_files_prop9 split:t_dobject.splits)
apply (tactic {*my_setiff_tac 1*})
apply (case_tac "obj = D_file f'", simp add:dalive_linkhard is_file_in_current)
apply (case_tac obj)
apply (rule_tac x = obj in exI, simp add:co2sobj_linkhard dalive_linkhard)
apply (case_tac "list \<in> same_inode_files s f")
apply (simp add:co2sobj.simps cf2sfiles_prop same_inodes_tainted)
apply (rule_tac x = obj in exI, simp add:co2sobj_linkhard dalive_linkhard)
apply (rule_tac x = obj in exI, simp add:co2sobj_linkhard dalive_linkhard)
apply (rule_tac x = obj in exI, simp add:co2sobj_linkhard dalive_linkhard)
done

lemma same_inode_files_prop12:
  "is_file s f \<Longrightarrow> f \<in> same_inode_files s f "
by (auto simp:is_file_def  same_inode_files_def split:option.splits)

lemma s2ss_truncate: "valid (Truncate p f len # s) \<Longrightarrow> s2ss (Truncate p f len # s) = (
  if (O_file f \<notin> tainted s \<and> O_proc p \<in> tainted s \<and> len > 0)
  then update_s2ss_sfile_tainted s (s2ss s) f True
  else s2ss s)"
apply (frule vt_grant_os, frule vd_cons, simp split:if_splits)
apply (rule conjI, rule impI, (erule conjE)+)

apply (simp add:update_s2ss_sfile_tainted_def)
apply (rule conjI|rule impI|erule exE|erule conjE)+
apply (simp add:s2ss_def)
apply (tactic {*my_seteq_tac 1*})
apply (case_tac "obj = D_file f")
apply (rule disjI1, simp add:co2sobj.simps same_inode_files_prop12 cf2sfiles_other)
apply (case_tac obj)
apply (rule disjI2, simp, rule_tac x = obj in exI, simp add:co2sobj_truncate dalive_simps)
apply (case_tac "list \<in> same_inode_files s f")
apply (rule disjI1, simp add:co2sobj.simps cf2sfiles_prop cf2sfiles_other)
apply (rule disjI2, simp, rule_tac x = obj in exI)
apply (simp add:co2sobj_truncate is_file_simps)
apply (rule disjI2, simp, rule_tac x = obj in exI)
apply (simp add:co2sobj_truncate is_dir_simps)
apply (rule disjI2, simp, rule_tac x = obj in exI)
apply (simp add:co2sobj_truncate)
apply (tactic {*my_setiff_tac 1*})
apply (rule_tac x = "D_file f" in exI)
apply (simp add:co2sobj.simps is_file_simps cf2sfiles_other same_inode_files_prop12)
apply (tactic {*my_setiff_tac 1*})
apply (case_tac obj)
apply (rule_tac x = obj in exI, simp add:co2sobj_truncate)
apply (case_tac "list \<in> same_inode_files s f")
apply (rule_tac x = "D_file f'" in exI)
apply (auto simp:co2sobj_truncate is_file_simps is_dir_simps split:t_dobject.splits)[1]
apply (simp add:co2sobj.simps same_inodes_tainted cf2sfiles_prop)
apply (rule_tac x = obj in exI, simp add:co2sobj_truncate is_file_simps)
apply (rule_tac x = obj in exI, simp add:co2sobj_truncate is_dir_simps)
apply (rule_tac x = obj in exI, simp add:co2sobj_truncate)

apply (rule conjI|rule impI|erule exE|erule conjE)+
apply (simp add:s2ss_def)
apply (tactic {*my_seteq_tac 1*})
apply (case_tac "obj = D_file f")
apply (rule disjI1, simp add:co2sobj.simps same_inode_files_prop12 cf2sfiles_other)
apply (case_tac obj)
apply (rule disjI2, simp, rule conjI, rule_tac x = obj in exI, simp add:co2sobj_truncate dalive_simps)
apply (rule notI, simp add:co2sobj.simps split:option.splits)
apply (case_tac "list \<in> same_inode_files s f")
apply (rule disjI1, simp add:co2sobj.simps cf2sfiles_prop cf2sfiles_other)
apply (rule disjI2, simp, rule conjI, rule_tac x = obj in exI)
apply (simp add:co2sobj_truncate is_file_simps)
apply (rule notI, simp add:co2sobj_truncate is_file_simps)
apply (erule_tac x = list in allE)
apply (simp add:co2sobj.simps)
apply (rule disjI2, simp, rule conjI, rule_tac x = obj in exI)
apply (simp add:co2sobj_truncate is_dir_simps)
apply (rule notI, simp add:co2sobj.simps split:option.splits)
apply (rule disjI2, simp, rule conjI, rule_tac x = obj in exI)
apply (simp add:co2sobj_truncate)
apply (rule notI, simp add:co2sobj.simps split:option.splits)
apply (tactic {*my_setiff_tac 1*})
apply (rule_tac x = "D_file f" in exI)
apply (simp add:co2sobj.simps is_file_simps cf2sfiles_other same_inode_files_prop12)
apply (tactic {*my_setiff_tac 1*})
apply (case_tac obj)
apply (rule_tac x = obj in exI, simp add:co2sobj_truncate)
apply (case_tac "list \<in> same_inode_files s f")
apply (simp add:co2sobj.simps same_inodes_tainted cf2sfiles_prop)
apply (rule_tac x = obj in exI, simp add:co2sobj_truncate is_file_simps)
apply (rule_tac x = obj in exI, simp add:co2sobj_truncate is_dir_simps)
apply (rule_tac x = obj in exI, simp add:co2sobj_truncate)

apply (rule impI, simp add:s2ss_def)
apply (tactic {*my_seteq_tac 1*})
apply (rule_tac x = obj in exI)
apply (simp add:dalive_simps co2sobj_truncate)
apply (simp split:t_dobject.splits if_splits add:co2sobj.simps)
apply (case_tac "O_proc p \<in> tainted s", simp add:same_inodes_tainted)
apply simp
apply (tactic {*my_setiff_tac 1*})
apply (rule_tac x = obj in exI)
apply (simp add:dalive_simps co2sobj_truncate)
apply (auto split:t_dobject.splits if_splits simp:co2sobj.simps same_inodes_tainted)
done

lemma s2ss_createmsgq: "valid (CreateMsgq p q # s) \<Longrightarrow> s2ss (CreateMsgq p q # s) = 
  (case (cq2smsgq (CreateMsgq p q # s) q) of 
     Some sq \<Rightarrow> s2ss s \<union> {S_msgq sq}
   | _       \<Rightarrow> {})"
apply (frule vd_cons, frule vt_grant_os, clarsimp)
apply (case_tac "cq2smsgq (CreateMsgq p q # s) q")
apply (drule current_has_smsgq', simp+)
apply (simp add:s2ss_def)
apply (tactic {*my_seteq_tac 1*})
apply (case_tac "obj = D_msgq q")
apply (rule disjI1, simp add:co2sobj.simps)
apply (rule disjI2, simp, rule_tac x = obj in exI)
apply (simp add:co2sobj_createmsgq is_file_simps is_dir_simps split:t_dobject.splits if_splits)
apply (tactic {*my_setiff_tac 1*})
apply (rule_tac x = "D_msgq q" in exI, simp add:co2sobj.simps)
apply (tactic {*my_setiff_tac 1*})
apply (case_tac "obj = D_msgq q")
apply simp
apply (rule_tac x = obj in exI)
apply (auto simp add:co2sobj_createmsgq dalive_simps split:t_dobject.splits if_splits)
done

lemma s2ss_sendmsg: "valid (SendMsg p q m # s) \<Longrightarrow> s2ss (SendMsg p q m # s) = (
  case (cq2smsgq s q, cq2smsgq (SendMsg p q m # s) q) of
    (Some sq, Some sq') \<Rightarrow> update_s2ss_obj s (s2ss s) (D_msgq q) (S_msgq sq) (S_msgq sq')
  | _  \<Rightarrow> {})"
apply (frule vd_cons, frule vt_grant_os, clarsimp)
apply (case_tac "cq2smsgq s q")
apply (drule current_has_smsgq', simp+)
apply (case_tac "cq2smsgq (SendMsg p q m # s) q")
apply (drule current_has_smsgq', simp+)

apply (simp add:update_s2ss_obj_def)
apply (tactic {*my_clarify_tac 1*})
apply (simp add:s2ss_def)
apply (tactic {*my_seteq_tac 1*})
apply (case_tac "obj = D_msgq q")
apply (rule disjI1, simp add:co2sobj.simps)
apply (rule disjI2, simp, rule_tac x = obj in exI)
apply (simp add:co2sobj_sendmsg is_file_simps is_dir_simps split:t_dobject.splits if_splits)
apply (tactic {*my_setiff_tac 1*})
apply (rule_tac x = "D_msgq q" in exI, simp add:co2sobj.simps)
apply (tactic {*my_setiff_tac 1*})
apply (case_tac "obj = D_msgq q")
apply (rule_tac x = obj' in exI)
apply (simp add:co2sobj_sendmsg dalive_sendmsg split:t_dobject.splits if_splits)
apply (auto simp:co2sobj.simps)[1]
apply (rule_tac x = obj in exI, simp add:co2sobj_sendmsg dalive_sendmsg split:t_dobject.splits)
apply (auto simp:co2sobj.simps)[1]

apply (rule impI)
apply (simp add:s2ss_def)
apply (tactic {*my_seteq_tac 1*})
apply (case_tac "obj = D_msgq q")
apply (rule disjI1, simp add:co2sobj.simps)
apply (rule disjI2, simp, rule conjI, rule_tac x = obj in exI)
apply (simp add:co2sobj_sendmsg is_file_simps is_dir_simps split:t_dobject.splits if_splits)
apply (rule notI, simp)
apply (frule_tac obj = obj in co2sobj_smsgq_imp, erule exE, simp)
apply (erule_tac x = obj in allE, simp add:co2sobj_sendmsg)
apply (tactic {*my_setiff_tac 1*})
apply (rule_tac x = "D_msgq q" in exI, simp add:co2sobj.simps)
apply (tactic {*my_setiff_tac 1*})
apply (case_tac "obj = D_msgq q")
apply (simp add:co2sobj.simps)
apply (rule_tac x = obj in exI, simp add:co2sobj_sendmsg dalive_sendmsg split:t_dobject.splits)
apply (auto simp:co2sobj.simps)[1]
done

lemma dalive_co2sobj_removemsgq:
  "\<lbrakk>dalive s obj; valid (RemoveMsgq p q # s); obj \<noteq> D_msgq q\<rbrakk> 
   \<Longrightarrow> dalive (RemoveMsgq p q # s) obj"
apply (case_tac obj)
apply (auto simp:is_file_simps is_dir_simps)
done

lemma s2ss_removemsgq: "valid (RemoveMsgq p q # s) \<Longrightarrow> s2ss (RemoveMsgq p q # s) = 
  (case (cq2smsgq s q) of
     Some sq \<Rightarrow> del_s2ss_obj s (s2ss s) (D_msgq q) (S_msgq sq)
   | _       \<Rightarrow> {})"
apply (frule vd_cons, frule vt_grant_os, clarsimp)
apply (split option.splits, rule conjI, rule impI)
apply (drule current_has_smsgq', simp, simp)
apply (rule allI, rule impI)

apply (simp add:del_s2ss_obj_def)
apply (tactic {*my_clarify_tac 1*})
apply (simp add:s2ss_def)
apply (tactic {*my_seteq_tac 1*})
apply (case_tac "obj = D_msgq q", simp)
apply (rule_tac x = obj in exI)
apply (simp add:co2sobj_removemsgq dalive_simps split:t_dobject.splits if_splits)
apply (tactic {*my_setiff_tac 1*})
apply (case_tac "obj = D_msgq q", simp)
apply (rule_tac x = obj' in exI)
apply (frule_tac obj = obj' in co2sobj_smsgq_imp, erule exE)
apply (simp add:co2sobj_removemsgq dalive_simps split:t_dobject.splits if_splits)
apply (simp add:co2sobj.simps)
apply (rule_tac x = obj in exI)
apply (simp add:co2sobj_removemsgq dalive_co2sobj_removemsgq)

apply (rule impI)
apply (simp add:s2ss_def)
apply (tactic {*my_seteq_tac 1*})
apply (case_tac "obj = D_msgq q", simp)
apply (simp, rule conjI, rule_tac x = obj in exI)
apply (simp add:co2sobj_removemsgq dalive_simps split:t_dobject.splits if_splits)
apply (rule notI, simp, frule_tac obj = obj in co2sobj_smsgq_imp, erule exE)
apply (erule_tac x = obj in allE, simp add:co2sobj_removemsgq dalive_co2sobj_removemsgq)
apply (tactic {*my_setiff_tac 1*})
apply (case_tac "obj = D_msgq q", simp)
apply (simp add:co2sobj.simps)
apply (rule_tac x = obj in exI)
apply (simp add:co2sobj_removemsgq dalive_co2sobj_removemsgq)
done

declare Product_Type.split_paired_Ex Product_Type.split_paired_All [simp del]

lemma s2ss_recvmsg: "valid (RecvMsg p q m # s) \<Longrightarrow> s2ss (RecvMsg p q m # s) = (
  case (cq2smsgq s q, cq2smsgq (RecvMsg p q m # s) q, cp2sproc s p) of
    (Some sq, Some sq', Some sp) \<Rightarrow> if (O_msg q m \<in> tainted s \<and> O_proc p \<notin> tainted s)
                                    then update_s2ss_obj s (update_s2ss_obj s (s2ss s)
                                                            (D_proc p) (S_proc sp False) (S_proc sp True))
                                         (D_msgq q) (S_msgq sq) (S_msgq sq')
                                    else update_s2ss_obj s (s2ss s) (D_msgq q) (S_msgq sq) (S_msgq sq')
  | _ \<Rightarrow> {})"
apply (frule vt_grant_os, frule vd_cons)
apply (case_tac "cq2smsgq s q")
apply (drule current_has_smsgq', simp, simp)
apply (case_tac "cq2smsgq (RecvMsg p q m # s) q")
apply (drule current_has_smsgq', simp, simp)
apply (case_tac "cp2sproc s p")
apply (drule current_proc_has_sp', simp, simp+)

apply (tactic {*my_clarify_tac 1*})
apply (simp add:update_s2ss_obj_def)

apply (tactic {*my_clarify_tac 1*})
apply (simp add:s2ss_def)
apply (tactic {*my_seteq_tac 1*})
apply (case_tac "obj = D_msgq q")
apply (rule disjI1, simp add:co2sobj.simps)
apply (case_tac "obj = D_proc p")
apply (rule disjI2, rule disjI1, simp add:co2sobj.simps cp2sproc_other)
apply (rule disjI2, rule disjI2, simp)
apply (rule_tac x = obj in exI)
apply (simp add:dalive_recvmsg co2sobj_recvmsg split:t_dobject.splits if_splits)
apply (tactic {*my_setiff_tac 1*})
apply (rule_tac x = "D_msgq q" in exI, simp add:co2sobj.simps)
apply (tactic {*my_setiff_tac 1*})
apply (rule_tac x = "D_proc p" in exI, simp add:co2sobj.simps cp2sproc_other)
apply (tactic {*my_setiff_tac 1*})
apply (case_tac "obj = D_msgq q")
apply (frule co2sobj_smsgq_imp, erule exE)
apply (rule_tac x = "D_msgq qa" in exI, simp add:dalive_recvmsg co2sobj_recvmsg)
apply (simp add:co2sobj.simps)
apply (case_tac "obj = D_proc p")
apply (frule co2sobj_sproc_imp, erule exE)
apply (rule_tac x = "D_proc pa" in exI, simp add:dalive_recvmsg co2sobj_recvmsg)
apply (simp add:co2sobj.simps)
apply (rule_tac x = obj in exI)
apply (auto simp add:dalive_recvmsg co2sobj_recvmsg split:t_dobject.splits if_splits)[1]

apply (tactic {*my_clarify_tac 1*})
apply (simp add:s2ss_def)
apply (tactic {*my_seteq_tac 1*})
apply (case_tac "obj = D_msgq q")
apply (rule disjI1, simp add:co2sobj.simps)
apply (case_tac "obj = D_proc p")
apply (rule disjI2,  simp add:co2sobj.simps cp2sproc_other)
apply (rule notI, simp)
apply (rule disjI2, simp, rule conjI, rule disjI2)
apply (rule_tac x = obj in exI)
apply (simp add:dalive_recvmsg co2sobj_recvmsg split:t_dobject.splits if_splits)
apply (rule notI, simp)
apply (frule co2sobj_smsgq_imp, erule exE)
apply (erule_tac x = "D_msgq qa" in allE, simp add:dalive_recvmsg co2sobj_recvmsg split:if_splits)
apply (tactic {*my_setiff_tac 1*})
apply (rule_tac x = "D_msgq q" in exI, simp add:co2sobj.simps)
apply (tactic {*my_setiff_tac 1*}, simp, erule disjE)
apply (rule_tac x = "D_proc p" in exI, simp add:co2sobj.simps cp2sproc_other)
apply (erule exE, erule conjE)
apply (case_tac "obj = D_msgq q", simp add:co2sobj.simps)
apply (case_tac "obj = D_proc p")
apply (frule_tac co2sobj_sproc_imp, erule exE)
apply (rule_tac x = "D_proc pa" in exI, simp add:dalive_recvmsg co2sobj_recvmsg)
apply (simp add:co2sobj.simps)
apply (rule_tac x = obj in exI)
apply (auto simp add:dalive_recvmsg co2sobj_recvmsg split:t_dobject.splits if_splits)[1]

apply (tactic {*my_clarify_tac 1*})
apply (simp add:s2ss_def)
apply (tactic {*my_seteq_tac 1*})
apply (case_tac "obj = D_msgq q")
apply (rule disjI1, simp add:co2sobj.simps)
apply (case_tac "obj = D_proc p")
apply (rule disjI2, rule disjI1, simp add:co2sobj.simps cp2sproc_other)
apply (rule disjI2, rule disjI2, simp, rule conjI)
apply (rule_tac x = obj in exI)
apply (simp add:dalive_recvmsg co2sobj_recvmsg split:t_dobject.splits if_splits)
apply (rule notI, simp)
apply (frule co2sobj_sproc_imp, erule exE)
apply (erule_tac x = "D_proc pa" in allE, simp add:co2sobj_recvmsg split:t_dobject.splits)
apply (tactic {*my_setiff_tac 1*})
apply (rule_tac x = "D_msgq q" in exI, simp add:co2sobj.simps)
apply (tactic {*my_setiff_tac 1*})
apply (rule_tac x = "D_proc p" in exI, simp add:co2sobj.simps cp2sproc_other)
apply (tactic {*my_setiff_tac 1*})
apply (case_tac "obj = D_msgq q")
apply (frule co2sobj_smsgq_imp, erule exE)
apply (rule_tac x = "D_msgq qa" in exI, simp add:dalive_recvmsg co2sobj_recvmsg)
apply (simp add:co2sobj.simps)
apply (case_tac "obj = D_proc p")
apply (simp add:co2sobj.simps)
apply (rule_tac x = obj in exI)
apply (auto simp add:dalive_recvmsg co2sobj_recvmsg split:t_dobject.splits if_splits)[1]

apply (tactic {*my_clarify_tac 1*})
apply (simp add:s2ss_def)
apply (tactic {*my_seteq_tac 1*})
apply (case_tac "obj = D_msgq q")
apply (rule disjI1, simp add:co2sobj.simps)
apply (case_tac "obj = D_proc p")
apply (rule disjI2, simp, rule conjI)
apply (rule disjI1, simp add:co2sobj.simps cp2sproc_other)
apply (rule notI, simp add:co2sobj.simps cp2sproc_other)
apply (rule disjI2, simp, rule conjI, rule disjI2, rule conjI)
apply (rule_tac x = obj in exI)
apply (simp add:dalive_recvmsg co2sobj_recvmsg split:t_dobject.splits if_splits)
apply (rule notI, simp, frule co2sobj_sproc_imp, erule exE)
apply (erule_tac x = "D_proc pa" in allE, simp add:co2sobj_recvmsg)
apply (rule notI, simp, frule co2sobj_smsgq_imp, erule exE)
apply (rotate_tac 12, erule_tac x = "D_msgq qa" in allE, simp add:co2sobj_recvmsg)
apply (tactic {*my_setiff_tac 1*})
apply (rule_tac x = "D_msgq q" in exI, simp add:co2sobj.simps)
apply (tactic {*my_setiff_tac 1*}, simp)
apply (tactic {*my_clarify_tac 1*})
apply (rule_tac x = "D_proc p" in exI, simp add:co2sobj.simps cp2sproc_other)
apply (tactic {*my_clarify_tac 1*})
apply (case_tac "obj = D_msgq q")
apply (simp add:co2sobj.simps)
apply (case_tac "obj = D_proc p")
apply (simp add:co2sobj.simps)
apply (rule_tac x = obj in exI)
apply (auto simp add:dalive_recvmsg co2sobj_recvmsg split:t_dobject.splits if_splits)[1]

apply (simp add:update_s2ss_obj_def)

apply (tactic {*my_clarify_tac 1*})
apply (simp add:s2ss_def)
apply (tactic {*my_seteq_tac 1*})
apply (case_tac "obj = D_msgq q")
apply (rule disjI1, simp add:co2sobj.simps)
apply (rule disjI2, simp)
apply (rule_tac x = obj in exI)
apply (simp add:dalive_recvmsg co2sobj_recvmsg split:t_dobject.splits if_splits)
apply (simp add:co2sobj.simps)
apply (tactic {*my_setiff_tac 1*})
apply (rule_tac x = "D_msgq q" in exI, simp add:co2sobj.simps)
apply (tactic {*my_setiff_tac 1*})
apply (case_tac "obj = D_msgq q")
apply (frule co2sobj_smsgq_imp, erule exE)
apply (rule_tac x = "D_msgq qa" in exI, simp add:dalive_recvmsg co2sobj_recvmsg)
apply (simp add:co2sobj.simps)
apply (rule_tac x = obj in exI)
apply (auto simp add:dalive_recvmsg co2sobj_recvmsg split:t_dobject.splits if_splits)[1]
apply (simp add:co2sobj.simps)

apply (tactic {*my_clarify_tac 1*})
apply (simp add:s2ss_def)
apply (tactic {*my_seteq_tac 1*})
apply (case_tac "obj = D_msgq q")
apply (rule disjI1, simp add:co2sobj.simps)
apply (rule disjI2, simp, rule conjI)
apply (rule_tac x = obj in exI)
apply (simp add:dalive_recvmsg co2sobj_recvmsg split:t_dobject.splits if_splits)
apply (simp add:co2sobj.simps)
apply (rule notI, simp)
apply (frule co2sobj_smsgq_imp, erule exE, erule_tac x = "D_msgq qa" in allE)
apply (simp add:co2sobj_recvmsg)
apply (tactic {*my_setiff_tac 1*})
apply (rule_tac x = "D_msgq q" in exI, simp add:co2sobj.simps)
apply (tactic {*my_setiff_tac 1*})
apply (case_tac "obj = D_msgq q")
apply (simp add:co2sobj.simps)
apply (rule_tac x = obj in exI)
apply (auto simp add:dalive_recvmsg co2sobj_recvmsg split:t_dobject.splits if_splits)[1]
apply (simp add:co2sobj.simps)
done

end

end