theory Tainted_prop 
imports Main Flask Flask_type Init_prop Current_files_prop Current_sockets_prop Delete_prop Proc_fd_of_file_prop Current_prop Alive_prop
begin

context tainting begin

fun Tainted :: "t_state \<Rightarrow> t_object set"
where
  "Tainted [] = seeds"
| "Tainted (Clone p p' fds shms # s) = 
     (if (O_proc p) \<in> Tainted s then Tainted s \<union> {O_proc p'} else Tainted s)"
| "Tainted (Execve p f fds # s) = 
     (if (O_file f) \<in> Tainted s then Tainted s \<union> {O_proc p} else Tainted s)"
| "Tainted (Kill p p' # s) = Tainted s - {O_proc p'}"
| "Tainted (Ptrace p p' # s) = 
     (if (O_proc p) \<in> Tainted s 
      then Tainted s \<union> {O_proc p'' | p''. info_flow_shm s p' p''}
      else if (O_proc p') \<in> Tainted s 
           then Tainted s \<union> {O_proc p'' | p''. info_flow_shm s p p''}
                else Tainted s)"
| "Tainted (Exit p # s) = Tainted s - {O_proc p}"
| "Tainted (Open p f flags fd opt # s) = 
     (case opt of
        Some inum \<Rightarrow> (if (O_proc p) \<in> Tainted s
                      then Tainted s \<union> {O_file f}
                      else Tainted s)
      | _         \<Rightarrow> Tainted s)" 
| "Tainted (ReadFile p fd # s) = 
     (case (file_of_proc_fd s p fd) of
        Some f \<Rightarrow> if (O_file f) \<in> Tainted s
                  then Tainted s \<union> {O_proc p' | p'. info_flow_shm s p p'}
                  else Tainted s
      | None   \<Rightarrow> Tainted s)"
| "Tainted (WriteFile p fd # s) = 
     (case (file_of_proc_fd s p fd) of 
        Some f \<Rightarrow> if (O_proc p) \<in> Tainted s
                  then Tainted s \<union> {O_file f' | f'. has_same_inode s f f'}
                  else Tainted s
      | None   \<Rightarrow> Tainted s)"
| "Tainted (CloseFd p fd # s) = 
     (case (file_of_proc_fd s p fd) of
        Some f \<Rightarrow> ( if ((proc_fd_of_file s f = {(p,fd)}) \<and> (f \<in> files_hung_by_del s))
                    then Tainted s - {O_file f} else Tainted s )
      | _      \<Rightarrow> Tainted s)"
| "Tainted (UnLink p f # s) = 
     (if (proc_fd_of_file s f = {}) then Tainted s - {O_file f} else Tainted s)"
| "Tainted (LinkHard p f f' # s) = 
     (if (O_file f \<in> Tainted s) then Tainted s \<union> {O_file f'} else Tainted s)"
| "Tainted (Truncate p f len # s) = 
     (if (len > 0 \<and> O_proc p \<in> Tainted s)
      then Tainted s \<union> {O_file f' | f'. has_same_inode s f f'}
      else Tainted s)"
| "Tainted (Attach p h flag # s) = 
     (if (O_proc p \<in> Tainted s \<and> flag = SHM_RDWR) 
      then Tainted s \<union> {O_proc p' | p' flag'. (p', flag') \<in> procs_of_shm s h}
      else if (\<exists> p'. O_proc p' \<in> Tainted s \<and> (p', SHM_RDWR) \<in> procs_of_shm s h)
           then Tainted s \<union> {O_proc p}
           else Tainted s)"
| "Tainted (SendMsg p q m # s) = 
     (if (O_proc p \<in> Tainted s) then Tainted s \<union> {O_msg q m} else Tainted s)"
| "Tainted (RecvMsg p q m # s) = 
     (if (O_msg q m \<in> Tainted s) 
      then (Tainted s \<union> {O_proc p' | p'. info_flow_shm s p p'}) - {O_msg q m}
      else Tainted s)"
| "Tainted (RemoveMsgq p q # s) = Tainted s - {O_msg q m| m. O_msg q m \<in> Tainted s}"
| "Tainted (e # s) = Tainted s"

lemma valid_Tainted_obj:
  "\<lbrakk>obj \<in> Tainted s; valid s\<rbrakk> \<Longrightarrow> (\<forall> f. obj \<noteq> O_dir f) \<and> (\<forall> q. obj \<noteq> O_msgq q) \<and> (\<forall> h. obj \<noteq> O_shm h) \<and> (\<forall> p fd. obj \<noteq> O_fd p fd) \<and> (\<forall> s. obj \<noteq> O_tcp_sock s) \<and> (\<forall> s. obj \<noteq> O_udp_sock s)"
apply (induct s, simp, drule seeds_in_init, case_tac obj, simp+)
apply (frule vd_cons, frule vt_grant_os, case_tac a)
apply (auto split:if_splits option.splits)
done

lemma Tainted_in_current:
  "\<lbrakk>obj \<in> Tainted s; valid s\<rbrakk> \<Longrightarrow> alive s obj"
apply (induct s, simp)
apply (drule seeds_in_init, case_tac obj, simp_all add:is_file_nil)
apply (frule vd_cons, frule valid_Tainted_obj, simp, frule vt_grant_os, case_tac a)
apply (auto simp:alive_simps split:if_splits option.splits t_object.splits
           intro:has_same_inode_prop2 has_same_inode_prop1 procs_of_shm_prop2 
            dest:info_shm_flow_in_procs)
done 

lemma Tainted_proc_in_current:
  "\<lbrakk>O_proc p \<in> Tainted s; valid s\<rbrakk> \<Longrightarrow> p \<in> current_procs s"
by (drule Tainted_in_current, simp+)


lemma info_flow_shm_Tainted:
  "\<lbrakk>O_proc p \<in> Tainted s; info_flow_shm s p p'; valid s\<rbrakk> \<Longrightarrow> O_proc p' \<in> Tainted s"
proof (induct s arbitrary:p p')
  case Nil
  thus ?case by (simp add:flow_shm_in_seeds)
next
  case (Cons e s)
  hence p1: "O_proc p \<in> Tainted (e # s)" and p2: "info_flow_shm (e # s) p p'" and p3: "valid (e # s)"  
    and p4: "\<And> p p'. \<lbrakk>O_proc p \<in> Tainted s; info_flow_shm s p p'\<rbrakk> \<Longrightarrow> O_proc p' \<in> Tainted s" 
    and p5: "valid s" and p6: "os_grant s e"
    by (auto dest:vd_cons intro:vd_cons vt_grant_os)
  have p4': 
    "\<And> p p' h flag. \<lbrakk>O_proc p \<in> Tainted s; (p, SHM_RDWR) \<in> procs_of_shm s h; (p', flag) \<in> procs_of_shm s h\<rbrakk> 
                \<Longrightarrow> O_proc p' \<in> Tainted s"
    by (rule p4, auto simp:info_flow_shm_def one_flow_shm_def procs_of_shm_prop2 p5)    
  from p2 p3 have p7: "p \<in> current_procs (e # s)" and p8: "p' \<in> current_procs (e # s)" 
    by (auto dest:info_shm_flow_in_procs) 
  show ?case
  proof (cases "self_shm s p p'")
    case True with p1 show ?thesis by simp
  next
    case False
    with p1 p2 p5 p6 p7 p8 p3 show ?thesis
    apply (case_tac e)(*
    prefer 7
    apply (simp add:info_flow_shm_simps split:if_splits option.splits)
    apply (rule allI|rule impI|rule conjI)+
    apply simp
    apply (case_tac "O_proc p \<in> Tainted s", drule_tac p'=p' in p4, simp+)
    apply simp




    apply (auto simp:info_flow_shm_simps one_flow_shm_def dest:Tainted_in_current 
  intro:p4 p4' split:if_splits option.splits)
    apply (auto simp:info_flow_shm_def one_flow_shm_def)



    apply (auto simp:one_flow_shm_def intro:p4 p4' split:if_splits option.splits)



    prefer 7
    apply (simp split:if_splits option.splits)
    apply (rule allI|rule impI|rule conjI)+


    apply (auto dest:p4'   procs_of_shm_prop2 Tainted_in_current split:if_splits option.splits)[1]

    apply (erule disjE, drule_tac p = p and p' = p' in p4', simp+)
    apply (erule disjE, rule disjI2, rule disjI2, rule_tac x = h in exI, simp, rule_tac x= toflag in exI, simp)
    apply ((erule exE|erule conjE)+)
    

    apply (auto simp:info_flow_shm_def dest:p4'
           procs_of_shm_prop2 Tainted_in_current split:if_splits option.splits)[1]
    apply (drule_tac p = p and p' = p' in p4')
    apply (erule_tac x = ha in allE, simp)
    apply (drule_tac p = "nat1" and p' = p' in p4')
    apply (auto dest:p4'[where p = nat1 and p' = p'])
    
apply (induct s) 
apply simp defer
apply (frule vd_cons, frule vt_grant_os, case_tac a)
apply (auto simp:info_flow_shm_def elim!:disjE)
sorry *)
  sorry
qed
qed

lemma tainted_imp_Tainted:
  "obj \<in> tainted s \<Longrightarrow> obj \<in> Tainted s"
apply (induct rule:tainted.induct)  (*
apply (simp_all add:vd_cons) 
apply (rule impI)

apply (rule disjI2, rule_tac x = flag' in exI, simp)
apply ((rule impI)+, erule_tac x = p' in allE, simp)
*)
apply (auto intro:info_flow_shm_Tainted dest:vd_cons)
apply (case_tac e, auto split:option.splits if_splits simp:alive_simps)
done

lemma Tainted_imp_tainted_aux_remain:
  "\<lbrakk>obj \<in> Tainted s; valid (e # s); alive (e # s) obj; \<And> obj. obj \<in> Tainted s \<Longrightarrow> obj \<in> tainted s\<rbrakk>
   \<Longrightarrow> obj \<in> tainted (e # s)"
by (rule t_remain, auto)

lemma Tainted_imp_tainted:
  "\<lbrakk>obj \<in> Tainted s; valid s\<rbrakk> \<Longrightarrow> obj \<in> tainted s"
apply (induct s arbitrary:obj, simp add:t_init)
apply (frule Tainted_in_current, simp+)
apply (frule vt_grant_os, frule vd_cons, case_tac a)
apply (auto split:if_splits option.splits elim:Tainted_imp_tainted_aux_remain intro:tainted.intros)
done

lemma tainted_eq_Tainted:
  "valid s \<Longrightarrow> (obj \<in> tainted s) = (obj \<in> Tainted s)"
by (rule iffI, auto intro:Tainted_imp_tainted tainted_imp_Tainted)

lemma info_flow_shm_tainted:
  "\<lbrakk>O_proc p \<in> tainted s; info_flow_shm s p p'; valid s\<rbrakk> \<Longrightarrow> O_proc p' \<in> tainted s"
by (simp only:tainted_eq_Tainted info_flow_shm_Tainted)

lemma has_same_inode_Tainted:
  "\<lbrakk>O_file f \<in> Tainted s; has_same_inode s f f'; valid s\<rbrakk> \<Longrightarrow> O_file f' \<in> Tainted s"
apply (induct s arbitrary:f f', simp add:same_inode_in_seeds)
apply (frule vt_grant_os, frule vd_cons, case_tac a)
apply (auto split:if_splits option.splits simp:has_same_inode_def dest:iof's_im_in_cim)
apply (drule_tac obj = "O_file list2" in Tainted_in_current, simp, simp add:is_file_in_current) 
done

lemma has_same_inode_tainted:
  "\<lbrakk>O_file f \<in> tainted s; has_same_inode s f f'; valid s\<rbrakk> \<Longrightarrow> O_file f' \<in> tainted s"
by (simp add:has_same_inode_Tainted tainted_eq_Tainted)





lemma tainted_in_current:
  "obj \<in> tainted s \<Longrightarrow> alive s obj"
apply (erule tainted.induct)
apply (auto dest:vt_grant_os vd_cons info_shm_flow_in_procs procs_of_shm_prop2 simp:is_file_simps)
apply (drule seeds_in_init, simp add:tobj_in_alive)
apply (erule has_same_inode_prop2, simp, simp add:vd_cons)
apply (frule vt_grant_os, simp)
apply (erule has_same_inode_prop1, simp, simp add:vd_cons)
done

lemma tainted_is_valid:
  "obj \<in> tainted s \<Longrightarrow> valid s"
by (erule tainted.induct, auto intro:valid.intros)

lemma t_remain_app:
  "\<lbrakk>obj \<in> tainted s; \<not> deleted obj (s' @ s); valid (s' @ s)\<rbrakk> 
  \<Longrightarrow> obj \<in> tainted (s' @ s)"
apply (induct s', simp)
apply (simp (no_asm) only:cons_app_simp_aux, rule t_remain)
apply (simp_all add:not_deleted_cons_D vd_cons)
apply (drule tainted_in_current, simp add:not_deleted_imp_alive_cons)
done

lemma valid_tainted_obj:
  "obj \<in> tainted s \<Longrightarrow> (\<forall> f. obj \<noteq> O_dir f) \<and> (\<forall> q. obj \<noteq> O_msgq q) \<and> (\<forall> h. obj \<noteq> O_shm h) \<and> (\<forall> p fd. obj \<noteq> O_fd p fd) \<and> (\<forall> s. obj \<noteq> O_tcp_sock s) \<and> (\<forall> s. obj \<noteq> O_udp_sock s)"
apply (erule tainted.induct)
apply (drule seeds_in_init)
by auto

lemma dir_not_tainted: "O_dir f \<in> tainted s \<Longrightarrow> False"
by (auto dest:valid_tainted_obj)

lemma msgq_not_tainted: "O_msgq q \<in> tainted s \<Longrightarrow> False"
by (auto dest:valid_tainted_obj)

lemma shm_not_tainted: "O_shm h \<in> tainted s \<Longrightarrow> False"
by (auto dest:valid_tainted_obj)

end

end