(*<*)theory Co2sobj_propimports Main Flask Flask_type Static Static_type Sectxt_prop Init_prop Current_files_prop Current_sockets_prop Delete_prop Proc_fd_of_file_prop Tainted_prop Valid_prop Init_prop Alive_propbegin(*<*)ML {*fun print_ss ss =letval {simps, congs, procs, ...} = Raw_Simplifier.dest_ss ssinsimpsend*}context tainting_s begin(********************* cm2smsg simpset ***********************)lemma cm2smsg_other: "\<lbrakk>valid (e # s); \<forall> p q m. e \<noteq> SendMsg p q m; \<forall> p q m. e \<noteq> RecvMsg p q m; \<forall> p q. e \<noteq> RemoveMsgq p q\<rbrakk> \<Longrightarrow> cm2smsg (e # s) = cm2smsg s"apply (frule vt_grant_os, frule vd_cons, rule ext, rule ext)unfolding cm2smsg_defapply (case_tac e)apply (auto simp:sectxt_of_obj_simps tainted_eq_Tainted split:t_object.splits option.splits if_splits dest:not_deleted_init_msg dest!:current_has_sec')donelemma cm2smsg_sendmsg: "valid (SendMsg p q m # s) \<Longrightarrow> cm2smsg (SendMsg p q m # s) = (\<lambda> q' m'. if (q' = q \<and> m' = m) then (case (sectxt_of_obj (SendMsg p q m # s) (O_msg q m)) of Some sec \<Rightarrow> Some (Created, sec, O_msg q m \<in> tainted (SendMsg p q m # s)) | _ \<Rightarrow> None) else cm2smsg s q' m') "apply (frule vd_cons, frule vt_grant_os, rule ext, rule ext)apply (auto simp:cm2smsg_def tainted_eq_Tainted sectxt_of_obj_simps split:if_splits option.splits dest:not_deleted_init_msg dest!:current_has_sec')donelemma cm2smsg_recvmsg1: "\<lbrakk>q' \<noteq> q; valid (RecvMsg p q m # s)\<rbrakk> \<Longrightarrow> cm2smsg (RecvMsg p q m # s) q' = cm2smsg s q'"apply (frule vd_cons, frule vt_grant_os, rule ext)apply (auto simp:cm2smsg_def tainted_eq_Tainted sectxt_of_obj_simps split:if_splits option.splits)donelemma cm2smsg_recvmsg2: "\<lbrakk>m' \<noteq> m; valid (RecvMsg p q m # s)\<rbrakk> \<Longrightarrow> cm2smsg (RecvMsg p q m # s) q m' = cm2smsg s q m'"apply (frule vd_cons, frule vt_grant_os)apply (auto simp:cm2smsg_def tainted_eq_Tainted sectxt_of_obj_simps split:if_splits option.splits)donelemma cm2smsg_recvmsg1': "\<lbrakk>valid (RecvMsg p q m # s); q' \<noteq> q\<rbrakk> \<Longrightarrow> cm2smsg (RecvMsg p q m # s) q' = cm2smsg s q'"apply (frule vd_cons, frule vt_grant_os, rule ext)apply (auto simp:cm2smsg_def tainted_eq_Tainted sectxt_of_obj_simps split:if_splits option.splits)donelemma cm2smsg_recvmsg2': "\<lbrakk>valid (RecvMsg p q m # s); m' \<noteq> m\<rbrakk> \<Longrightarrow> cm2smsg (RecvMsg p q m # s) q m' = cm2smsg s q m'"apply (frule vd_cons, frule vt_grant_os)apply (auto simp:cm2smsg_def tainted_eq_Tainted sectxt_of_obj_simps split:if_splits option.splits)donelemma cm2smsg_removemsgq: "\<lbrakk>q' \<noteq> q; valid (RemoveMsgq p q # s)\<rbrakk> \<Longrightarrow> cm2smsg (RemoveMsgq p q # s) q' = cm2smsg s q'"apply (frule vd_cons, frule vt_grant_os, rule ext)apply (auto simp:cm2smsg_def tainted_eq_Tainted sectxt_of_obj_simps split:if_splits option.splits)donelemmas cm2smsg_simps = cm2smsg_nil_prop cm2smsg_sendmsg cm2smsg_recvmsg1' cm2smsg_recvmsg2' cm2smsg_removemsgq cm2smsg_otherlemma init_cm2smsg_has_smsg: "\<lbrakk>m \<in> set (init_msgs_of_queue q); q \<in> init_msgqs\<rbrakk> \<Longrightarrow> \<exists> sm. init_cm2smsg q m = Some sm"by (auto simp:init_cm2smsg_def split:option.splits dest:init_msg_has_ctxt)lemma hd_set_prop1: "hd l \<notin> set l \<Longrightarrow> l = []"by (case_tac l, auto)lemma tl_set_prop1: "a \<in> set (tl l) \<Longrightarrow> a \<in> set l"by (case_tac l, auto)lemma current_has_smsg: "\<lbrakk>m \<in> set (msgs_of_queue s q); q \<in> current_msgqs s; valid s\<rbrakk> \<Longrightarrow> \<exists> sm. cm2smsg s q m = Some sm"apply (induct s)apply (simp only:cm2smsg_nil_prop msgs_of_queue.simps current_msgqs.simps init_cm2smsg_has_smsg)apply (frule vt_grant_os, frule vd_cons, case_tac a)apply (auto simp:cm2smsg_def sectxt_of_obj_simps split:if_splits option.splits dest!:current_has_sec' hd_set_prop1 dest:not_deleted_init_msg tl_set_prop1)done lemma current_has_smsg': "\<lbrakk>cm2smsg s q m = None; q \<in> current_msgqs s; valid s\<rbrakk> \<Longrightarrow> m \<notin> set (msgs_of_queue s q)"by (auto dest:current_has_smsg)lemma cqm2sms_has_sms_aux: "\<forall> m \<in> set ms. sectxt_of_obj s (O_msg q m) \<noteq> None \<Longrightarrow> (\<exists> sms. cqm2sms s q ms = Some sms)"by (induct ms, auto split:option.splits simp:cm2smsg_def)lemma current_has_sms: "\<lbrakk>q \<in> current_msgqs s; valid s\<rbrakk> \<Longrightarrow> \<exists> sms. cqm2sms s q (msgs_of_queue s q) = Some sms"apply (rule cqm2sms_has_sms_aux)apply (auto dest:current_msg_has_sec)donelemma current_has_sms': "\<lbrakk>cqm2sms s q (msgs_of_queue s q) = None; valid s\<rbrakk> \<Longrightarrow> q \<notin> current_msgqs s"by (auto dest:current_has_sms)(********************* cqm2sms simpset ***********************) lemma cqm2sms_other: "\<lbrakk>valid (e # s); \<forall> p m. e \<noteq> CreateMsgq p q; \<forall> p q m. e \<noteq> SendMsg p q m; \<forall> p q m. e \<noteq> RecvMsg p q m; \<forall> p q. e \<noteq> RemoveMsgq p q\<rbrakk> \<Longrightarrow> cqm2sms (e # s) = cqm2sms s"apply (rule ext, rule ext)apply (induct_tac xa, simp)apply (frule vt_grant_os, frule vd_cons, case_tac e)apply (auto split:option.splits dest:cm2smsg_other) donelemma cqm2sms_createmsgq: "valid (CreateMsgq p q # s) \<Longrightarrow> cqm2sms (CreateMsgq p q # s) = (\<lambda> q' ms'. if (q' = q \<and> ms' = []) then Some [] else cqm2sms s q' ms')"apply (rule ext, rule ext)apply (frule vt_grant_os, frule vd_cons, induct_tac ms')apply (auto split:if_splits option.splits dest:cm2smsg_other)donelemma cqm2sms_2: "cqm2sms s q (ms @ [m]) = (case (cqm2sms s q ms, cm2smsg s q m) of (Some sms, Some sm) \<Rightarrow> Some (sms @ [sm]) | _ \<Rightarrow> None)"apply (induct ms, simp split:option.splits)by (auto split:option.splits)lemmas cqm2sms_simps2 = cqm2sms.simps(1) cqm2sms_2declare cqm2sms.simps [simp del]lemma cqm2sms_prop1: "cqm2sms s q ms = None \<Longrightarrow> \<exists> m \<in> set ms. cm2smsg s q m = None"by (induct ms, auto simp:cqm2sms.simps split:option.splits)lemma cqm2sms_sendmsg1: "\<lbrakk>valid (SendMsg p q m # s); m \<notin> set ms\<rbrakk> \<Longrightarrow> cqm2sms (SendMsg p q m # s) q' ms = cqm2sms s q' ms"apply (frule vt_grant_os, frule vd_cons)apply (frule cm2smsg_sendmsg)apply (induct ms rule:rev_induct)apply (auto split:if_splits option.splits simp:cqm2sms_simps2)donelemma cqm2sms_sendmsg2: "\<lbrakk>valid (SendMsg p q m # s); q' \<noteq> q\<rbrakk> \<Longrightarrow> cqm2sms (SendMsg p q m # s) q' ms = cqm2sms s q' ms"apply (frule vt_grant_os, frule vd_cons)apply (frule cm2smsg_sendmsg)apply (induct ms rule:rev_induct)apply (auto split:if_splits option.splits simp:cqm2sms_simps2)donelemma cqm2sms_sendmsg3: "\<lbrakk>valid (SendMsg p q m # s); ms' = msgs_of_queue (SendMsg p q m # s) q\<rbrakk> \<Longrightarrow> cqm2sms (SendMsg p q m # s) q ms' = (case (cqm2sms s q (msgs_of_queue s q), sectxt_of_obj (SendMsg p q m # s) (O_msg q m)) of (Some sms, Some sec) \<Rightarrow> Some (sms @ [(Created, sec, O_msg q m \<in> tainted (SendMsg p q m # s))]) | _ \<Rightarrow> None)"apply (frule vt_grant_os, frule vd_cons)apply (frule cm2smsg_sendmsg)apply (auto split:if_splits option.splits simp:cqm2sms_simps2 cqm2sms_sendmsg1)donelemma cqm2sms_sendmsg: "\<lbrakk>valid (SendMsg p q m # s); ms' = msgs_of_queue (SendMsg p q m # s) q'\<rbrakk> \<Longrightarrow> cqm2sms (SendMsg p q m # s) q' ms' = ( if (q' = q) then (case (cqm2sms s q (msgs_of_queue s q), sectxt_of_obj (SendMsg p q m # s) (O_msg q m)) of (Some sms, Some sec) \<Rightarrow> Some (sms @ [(Created, sec, O_msg q m \<in> tainted (SendMsg p q m # s))]) | _ \<Rightarrow> None) else cqm2sms s q' ms' )"apply (simp split:if_splits add:cqm2sms_sendmsg2 cqm2sms_sendmsg3)donelemma cqm2sms_recvmsg1: "\<lbrakk>valid (RecvMsg p q m # s); m \<notin> set ms\<rbrakk> \<Longrightarrow> cqm2sms (RecvMsg p q m # s) q ms = cqm2sms s q ms"apply (frule vt_grant_os, frule vd_cons)apply (induct ms rule:rev_induct)apply (auto split:if_splits option.splits simp:cqm2sms_simps2 cm2smsg_recvmsg2')donelemma cqm2sms_recvmsg2: "\<lbrakk>valid (RecvMsg p q m # s); q' \<noteq> q\<rbrakk> \<Longrightarrow> cqm2sms (RecvMsg p q m # s) q' ms = cqm2sms s q' ms"apply (frule vt_grant_os, frule vd_cons)apply (induct ms rule:rev_induct)apply (auto split:if_splits option.splits simp:cqm2sms_simps2 cm2smsg_recvmsg1')donelemma cqm2sms_recvmsg: "\<lbrakk>valid (RecvMsg p q m # s); ms = msgs_of_queue (RecvMsg p q m # s) q'\<rbrakk> \<Longrightarrow> cqm2sms (RecvMsg p q m # s) q' ms = ( if (q' = q) then (case (cqm2sms s q (msgs_of_queue s q)) of Some sms \<Rightarrow> Some (tl sms) | _ \<Rightarrow> None) else cqm2sms s q' ms)"apply (frule vt_grant_os, frule vd_cons)apply (auto split:if_splits option.splits simp:cqm2sms_recvmsg1 cqm2sms_recvmsg2 dest!:current_has_sms')apply (case_tac "msgs_of_queue s q", simp)apply (frule_tac ms = "tl (msgs_of_queue s q)" in cqm2sms_recvmsg1)apply (drule_tac q = q in distinct_queue_msgs, simp+)apply (case_tac a, auto simp:cqm2sms.simps split:option.splits if_splits)donelemma cqm2sms_removemsgq: "\<lbrakk>valid (RemoveMsgq p q # s); q' \<noteq> q; q' \<in> current_msgqs s\<rbrakk> \<Longrightarrow> cqm2sms (RemoveMsgq p q # s) q' ms = cqm2sms s q' ms"apply (frule vt_grant_os, frule vd_cons)apply (induct ms rule:rev_induct)apply (auto simp:cqm2sms_simps2 cm2smsg_removemsgq split:option.splits if_splits)donelemmas cqm2sms_simps = cqm2sms_other cqm2sms_createmsgq cqm2sms_sendmsg cqm2sms_recvmsg cqm2sms_removemsgq(********************* cq2smsgq simpset ***********************) lemma cq2smsgq_other: "\<lbrakk>valid (e # s); \<forall> p q. e \<noteq> CreateMsgq p q; \<forall> p q m. e \<noteq> SendMsg p q m; \<forall> p q m. e \<noteq> RecvMsg p q m; \<forall> p q. e \<noteq> RemoveMsgq p q\<rbrakk> \<Longrightarrow> cq2smsgq (e # s) = cq2smsgq s" apply (frule cqm2sms_other, simp+)apply (frule vd_cons, frule vt_grant_os, rule ext, case_tac e) apply (auto simp:cq2smsgq_def sectxt_of_obj_simps split:t_object.splits option.splits if_splits dest:not_deleted_init_msg) donelemma cq2smsg_createmsgq: "valid (CreateMsgq p q # s) \<Longrightarrow> cq2smsgq (CreateMsgq p q # s) = (cq2smsgq s) (q := case (sectxt_of_obj s (O_proc p)) of Some psec \<Rightarrow> Some (Created, (fst psec, R_object, (snd o snd) psec), []) | None \<Rightarrow> None)"apply (frule vd_cons, frule vt_grant_os)apply (frule cqm2sms_createmsgq)apply (rule ext, auto simp:cq2smsgq_def sec_createmsgq split:option.splits if_splits dest:not_deleted_init_msgq)donelemma cq2smsg_sendmsg: "valid (SendMsg p q m # s) \<Longrightarrow> cq2smsgq (SendMsg p q m # s) = (cq2smsgq s) (q := case (cq2smsgq s q, sectxt_of_obj (SendMsg p q m # s) (O_msg q m)) of (Some (qi, sec, sms), Some msec) \<Rightarrow> Some (qi, sec, sms @ [(Created, msec, O_msg q m \<in> tainted (SendMsg p q m # s))]) | _ \<Rightarrow> None)"apply (frule vd_cons, frule vt_grant_os)apply (rule ext)apply (frule_tac q' = x in cqm2sms_sendmsg, simp)apply (auto simp:cq2smsgq_def sectxt_of_obj_simps split:option.splits if_splits dest!:current_has_sms' current_has_sec')donelemma current_has_smsgq: "\<lbrakk>q \<in> current_msgqs s; valid s\<rbrakk> \<Longrightarrow> \<exists> sq. cq2smsgq s q = Some sq"by (auto simp:cq2smsgq_def split:option.splits dest!:current_has_sec' current_has_sms')lemma current_has_smsgq': "\<lbrakk>cq2smsgq s q = None; valid s\<rbrakk> \<Longrightarrow> q \<notin> current_msgqs s"by (auto dest:current_has_smsgq)lemma cq2smsg_recvmsg: "valid (RecvMsg p q m # s) \<Longrightarrow> cq2smsgq (RecvMsg p q m # s) = (cq2smsgq s) (q := case (cq2smsgq s q) of Some (qi, sec, sms) \<Rightarrow> Some (qi, sec, tl sms) | _ \<Rightarrow> None)"apply (frule vd_cons, frule vt_grant_os)apply (rule ext, frule_tac q' = x in cqm2sms_recvmsg, simp)apply (auto simp add:cq2smsgq_def sectxt_of_obj_simps split:option.splits if_splits dest!:current_has_sec' current_has_sms' current_has_smsgq')donelemma cq2smsg_removemsgq: "\<lbrakk>valid (RemoveMsgq p q # s); q' \<noteq> q; q' \<in> current_msgqs s\<rbrakk> \<Longrightarrow> cq2smsgq (RemoveMsgq p q # s) q' = cq2smsgq s q'"apply (frule vd_cons, frule vt_grant_os)apply (auto simp:cq2smsgq_def sectxt_of_obj_simps cqm2sms_removemsgq split:if_splits option.splits dest!:current_has_sec' current_has_sms' current_has_smsgq')donelemmas cq2smsgq_simps = cq2smsgq_other cq2smsg_sendmsg cq2smsg_recvmsg cq2smsg_removemsgqlemma sm_in_sqsms_init_aux: "\<lbrakk>m \<in> set ms; init_cm2smsg q m = Some sm; q \<in> init_msgqs; init_cqm2sms q ms = Some sms\<rbrakk> \<Longrightarrow> sm \<in> set sms"apply (induct ms arbitrary:m sm sms)by (auto split:if_splits option.splits)lemma sm_in_sqsms_init: "\<lbrakk>m \<in> set (init_msgs_of_queue q); init_cm2smsg q m = Some sm; q \<in> init_msgqs; init_cqm2sms q (init_msgs_of_queue q) = Some sms\<rbrakk> \<Longrightarrow> sm \<in> set sms"by (simp add:sm_in_sqsms_init_aux)lemma cqm2sms_prop0: "\<lbrakk>m \<in> set ms; cm2smsg s q m = Some sm; cqm2sms s q ms = Some sms\<rbrakk> \<Longrightarrow> sm \<in> set sms"apply (induct ms arbitrary:m sm sms)apply (auto simp:cqm2sms.simps split:option.splits)donelemma sm_in_sqsms: "\<lbrakk>m \<in> set (msgs_of_queue s q); q \<in> current_msgqs s; valid s; cq2smsgq s q = Some (qi, qsec, sms); cm2smsg s q m = Some sm\<rbrakk> \<Longrightarrow> sm \<in> set sms"proof (induct s arbitrary:m q qi qsec sms sm) case Nil thus ?case by (simp add:cq2smsga_nil_prop cm2smsg_nil_prop init_cq2smsgq_def sm_in_sqsms_init split:option.splits)next case (Cons e s) hence p1:"\<And> m q qi qsec sms sm. \<lbrakk>m \<in> set (msgs_of_queue s q); q \<in> current_msgqs s; valid s; cq2smsgq s q = Some (qi, qsec, sms); cm2smsg s q m = Some sm\<rbrakk> \<Longrightarrow> sm \<in> set sms" and p2: "m \<in> set (msgs_of_queue (e # s) q)" and p3: "q \<in> current_msgqs (e # s)" and p4: "valid (e # s)" and p5: "cq2smsgq (e # s) q = Some (qi, qsec, sms)" and p6: "cm2smsg (e # s) q m = Some sm" by auto from p4 have os: "os_grant s e" and vd: "valid s" by (auto dest:vd_cons vt_grant_os)(* from p1 have p1': "\<And> m q qi qsec sms sm ms. \<lbrakk>m \<in> set ms; set ms \<subseteq> set (msgs_of_queue s q); q \<in> current_msgqs s; valid s; cq2smsgq s q = Some (qi, qsec, sms); cm2smsg s q m = Some sm\<rbrakk> \<Longrightarrow> sm \<in> set "*) show ?case using p2 p3 p4 p5 p6 vd os apply (case_tac e) apply (auto simp:cq2smsgq_simps cm2smsg_simps split:if_splits option.splits intro:p1) apply (rule_tac m = m and q = q and qi = qi and qsec = qsec in p1, simp+) apply (case_tac "q = nat2", simp) apply (drule cq2smsg_createmsgq, simp, simp) apply (drule_tac m = m and q = q and qi = qi and qsec = "(aa,ab,b)" in p1, simp+) apply (simp add:cq2smsgq_def split:option.splits) apply (rule_tac m = m and sm = sm in cqm2sms_prop0, simp+) apply (simp add:cqm2sms_recvmsg) doneqed(****************** cf2sfile path simpset ***************)lemma sroot_only: "cf2sfile s [] = Some sroot"by (simp add:cf2sfile_def)lemma not_file_is_dir: "\<lbrakk>\<not> is_file s f; f \<in> current_files s; valid s\<rbrakk> \<Longrightarrow> is_dir s f"by (auto simp:is_file_def current_files_def is_dir_def dest:finum_has_itag finum_has_ftag' split:t_inode_tag.splits option.splits)lemma not_dir_is_file: "\<lbrakk>\<not> is_dir s f; f \<in> current_files s; valid s\<rbrakk> \<Longrightarrow> is_file s f"by (auto simp:is_file_def current_files_def is_dir_def dest:finum_has_itag finum_has_ftag' split:t_inode_tag.splits option.splits)lemma is_file_or_dir: "\<lbrakk>f \<in> current_files s; valid s\<rbrakk> \<Longrightarrow> is_file s f \<or> is_dir s f"by (auto dest:not_dir_is_file)lemma current_file_has_sfile: "\<lbrakk>f \<in> current_files s; valid s\<rbrakk> \<Longrightarrow> \<exists> sf. cf2sfile s f = Some sf"apply (induct f)apply (rule_tac x = "sroot" in exI, simp add:sroot_only)apply (frule parentf_in_current', simp, clarsimp)apply (frule parentf_is_dir'', simp)apply (frule is_file_or_dir, simp)apply (auto dest!:current_has_sec' simp:cf2sfile_def split:option.splits if_splits dest!:get_pfs_secs_prop')donedefinition sectxt_of_pf :: "t_state \<Rightarrow> t_file \<Rightarrow> security_context_t option"where "sectxt_of_pf s f = (case f of [] \<Rightarrow> None | (a # pf) \<Rightarrow> sectxt_of_obj s (O_dir pf))"definition get_parentfs_ctxts' :: "t_state \<Rightarrow> t_file \<Rightarrow> (security_context_t list) option"where "get_parentfs_ctxts' s f = (case f of [] \<Rightarrow> None | (a # pf) \<Rightarrow> get_parentfs_ctxts s pf)"lemma is_file_has_sfile: "\<lbrakk>is_file s f; valid s\<rbrakk> \<Longrightarrow> \<exists> sec psec asecs. cf2sfile s f = Some (if (\<not> deleted (O_file f) s \<and> is_init_file f) then Init f else Created, sec, Some psec, set asecs) \<and> (sectxt_of_obj s (O_file f) = Some sec) \<and> (sectxt_of_pf s f = Some psec) \<and> (get_parentfs_ctxts' s f = Some asecs)"apply (case_tac f, simp, drule root_is_dir', simp, simp)apply (frule is_file_in_current)apply (drule current_file_has_sfile, simp)apply (auto simp:cf2sfile_def sectxt_of_pf_def get_parentfs_ctxts'_def split:if_splits option.splits)donelemma is_file_has_sfile': "\<lbrakk>is_file s f; valid s\<rbrakk> \<Longrightarrow> \<exists> sf. cf2sfile s f = Some sf"by (drule is_file_has_sfile, auto)lemma is_dir_has_sfile: "\<lbrakk>is_dir s f; valid s\<rbrakk> \<Longrightarrow> (case f of [] \<Rightarrow> cf2sfile s f = Some sroot | a # pf \<Rightarrow> (\<exists> sec psec asecs. cf2sfile s f = Some (if (\<not> deleted (O_dir f) s \<and> is_init_dir f) then Init f else Created, sec, Some psec, set asecs) \<and> (sectxt_of_obj s (O_dir f) = Some sec) \<and> (sectxt_of_obj s (O_dir pf) = Some psec) \<and> (get_parentfs_ctxts s pf = Some asecs)))"apply (case_tac f, simp add:sroot_only)apply (frule is_dir_in_current, frule is_dir_not_file)apply (drule current_file_has_sfile, simp)apply (auto simp:cf2sfile_def split:if_splits option.splits)donelemma is_dir_has_sdir': "\<lbrakk>is_dir s f; valid s\<rbrakk> \<Longrightarrow> \<exists> sf. cf2sfile s f = Some sf"apply (case_tac f)apply (rule_tac x = sroot in exI)apply (simp add:sroot_only)apply (drule is_dir_has_sfile, auto)donelemma sroot_set: "valid s \<Longrightarrow> \<exists> sec. sroot = (Init [], sec, None, {}) \<and> sectxt_of_obj s (O_dir []) = Some sec"apply (frule root_is_dir)apply (drule is_dir_has_sec, simp)apply (auto simp:sroot_def sec_of_root_def sectxt_of_obj_def type_of_obj.simps root_type_remains root_user_remains dest!:root_has_type' root_has_user' root_has_init_type' root_has_init_user' split:option.splits)donelemma cf2sfile_path_file: "\<lbrakk>is_file s (f # pf); valid s\<rbrakk> \<Longrightarrow> cf2sfile s (f # pf) = ( case (cf2sfile s pf) of Some (pfi, pfsec, psec, asecs) \<Rightarrow> (case (sectxt_of_obj s (O_file (f # pf))) of Some fsec \<Rightarrow> Some (if (\<not> deleted (O_file (f # pf)) s \<and> is_init_file (f # pf)) then Init (f # pf) else Created, fsec, Some pfsec, asecs \<union> {pfsec}) | None \<Rightarrow> None) | _ \<Rightarrow> None)"apply (frule is_file_in_current, drule parentf_is_dir'', simp)apply (frule is_dir_has_sfile, simp, frule is_file_has_sfile, simp)apply (frule sroot_set)apply (case_tac pf, (clarsimp simp:get_parentfs_ctxts'_def sectxt_of_pf_def)+) donelemma cf2sfile_path_dir: "\<lbrakk>is_dir s (f # pf); valid s\<rbrakk> \<Longrightarrow> cf2sfile s (f # pf) = ( case (cf2sfile s pf) of Some (pfi, pfsec, psec, asecs) \<Rightarrow> (case (sectxt_of_obj s (O_dir (f # pf))) of Some fsec \<Rightarrow> Some (if (\<not> deleted (O_dir (f # pf)) s \<and> is_init_dir (f # pf)) then Init (f # pf) else Created, fsec, Some pfsec, asecs \<union> {pfsec}) | None \<Rightarrow> None) | _ \<Rightarrow> None)"apply (frule is_dir_in_current, drule parentf_is_dir'', simp)apply (frule_tac f = "f # pf" in is_dir_has_sfile, simp)apply (frule_tac f = "pf" in is_dir_has_sfile, simp)apply (frule sroot_set)apply (case_tac pf, (clarsimp simp:get_parentfs_ctxts'_def sectxt_of_pf_def)+) done lemma cf2sfile_path: "\<lbrakk>f # pf \<in> current_files s; valid s\<rbrakk> \<Longrightarrow> cf2sfile s (f # pf) = ( case (cf2sfile s pf) of Some (pfi, pfsec, psec, asecs) \<Rightarrow> (if (is_file s (f # pf)) then (case (sectxt_of_obj s (O_file (f # pf))) of Some fsec \<Rightarrow> Some (if (\<not> deleted (O_file (f # pf)) s \<and> is_init_file (f # pf)) then Init (f # pf) else Created, fsec, Some pfsec, asecs \<union> {pfsec}) | None \<Rightarrow> None) else (case (sectxt_of_obj s (O_dir (f # pf))) of Some fsec \<Rightarrow> Some (if (\<not> deleted (O_dir (f # pf)) s \<and> is_init_dir (f # pf)) then Init (f # pf) else Created, fsec, Some pfsec, asecs \<union> {pfsec}) | None \<Rightarrow> None) ) | None \<Rightarrow> None)"apply (drule is_file_or_dir, simp)apply (erule disjE)apply (frule cf2sfile_path_file, simp) deferapply (frule cf2sfile_path_dir, simp, drule is_dir_not_file)apply (auto split:option.splits)donelemma cf2sfile_path_file_prop1: "\<lbrakk>is_file s (f # pf); cf2sfile s pf = Some (pfi, pfsec, psec, asecs); valid s\<rbrakk> \<Longrightarrow> \<exists> fsec. cf2sfile s (f # pf) = Some (if (\<not> deleted (O_file (f # pf)) s \<and> is_init_file (f # pf)) then Init (f # pf) else Created, fsec, Some pfsec, asecs \<union> {pfsec}) \<and> sectxt_of_obj s (O_file (f # pf)) = Some fsec"apply (frule is_file_has_sfile, simp)by (auto simp:cf2sfile_path_file)lemma cf2sfile_path_file_prop2: "\<lbrakk>is_file s (f # pf); cf2sfile s pf = Some (pfi, pfsec, psec, asecs); sectxt_of_obj s (O_file (f # pf)) = Some fsec; valid s\<rbrakk> \<Longrightarrow> cf2sfile s (f # pf) = Some (if (\<not> deleted (O_file (f # pf)) s \<and> is_init_file (f # pf)) then Init (f # pf) else Created, fsec, Some pfsec, asecs \<union> {pfsec})"by (drule cf2sfile_path_file_prop1, auto)lemma cf2sfile_path_dir_prop1: "\<lbrakk>is_dir s (f # pf); cf2sfile s pf = Some (pfi, pfsec, psec, asecs); valid s\<rbrakk> \<Longrightarrow> \<exists> fsec. cf2sfile s (f # pf) = Some (if (\<not> deleted (O_dir (f # pf)) s \<and> is_init_dir (f # pf)) then Init (f # pf) else Created, fsec, Some pfsec, asecs \<union> {pfsec}) \<and> sectxt_of_obj s (O_dir (f # pf)) = Some fsec"apply (frule is_dir_has_sfile, simp)by (auto simp:cf2sfile_path_dir)lemma cf2sfile_path_dir_prop2: "\<lbrakk>is_dir s (f # pf); cf2sfile s pf = Some (pfi, pfsec, psec, asecs); sectxt_of_obj s (O_dir (f # pf)) = Some fsec; valid s\<rbrakk> \<Longrightarrow> cf2sfile s (f # pf) = Some (if (\<not> deleted (O_dir (f # pf)) s \<and> is_init_dir (f # pf)) then Init (f # pf) else Created, fsec, Some pfsec, asecs \<union> {pfsec})"by (drule cf2sfile_path_dir_prop1, auto)(**************** cf2sfile event list simpset ****************)lemma cf2sfile_open_none': "valid (Open p f flag fd None # s) \<Longrightarrow> cf2sfile (Open p f flag fd None # s) f'= cf2sfile s f'"apply (frule vd_cons, frule vt_grant_os)apply (induct f', simp add:cf2sfile_def)apply (simp add:cf2sfile_def is_file_simps is_dir_simps current_files_simps sectxt_of_obj_simps get_parentfs_ctxts_simps)donelemma cf2sfile_open_none: "valid (Open p f flag fd None # s) \<Longrightarrow> cf2sfile (Open p f flag fd None # s) = cf2sfile s"apply (rule ext)by (simp add:cf2sfile_open_none')lemma cf2sfile_open_some1: "\<lbrakk>valid (Open p f flag fd (Some inum) # s); f' \<in> current_files s\<rbrakk> \<Longrightarrow> cf2sfile (Open p f flag fd (Some inum) # s) f' = cf2sfile s f'"apply (frule vd_cons, frule vt_grant_os, frule noroot_events)apply (case_tac "f = f'", simp)apply (induct f', simp add:sroot_only, simp)apply (frule parentf_in_current', simp+)apply (simp add:cf2sfile_def is_file_simps is_dir_simps current_files_simps sectxt_of_obj_simps get_parentfs_ctxts_simps)donelemma cf2sfile_open_some2: "\<lbrakk>valid (Open p f flag fd (Some inum) # s); is_file s f'\<rbrakk> \<Longrightarrow> cf2sfile (Open p f flag fd (Some inum) # s) f' = cf2sfile s f'"apply (frule vd_cons, drule is_file_in_current)by (simp add:cf2sfile_open_some1)lemma cf2sfile_open_some3: "\<lbrakk>valid (Open p f flag fd (Some inum) # s); is_dir s f'\<rbrakk> \<Longrightarrow> cf2sfile (Open p f flag fd (Some inum) # s) f' = cf2sfile s f'"apply (frule vd_cons, drule is_dir_in_current)by (simp add:cf2sfile_open_some1)lemma cf2sfile_open_some4: "valid (Open p f flag fd (Some inum) # s) \<Longrightarrow> cf2sfile (Open p f flag fd (Some inum) # s) f = ( case (parent f) of Some pf \<Rightarrow> (case (sectxt_of_obj (Open p f flag fd (Some inum) # s) (O_file f), sectxt_of_obj s (O_dir pf), get_parentfs_ctxts s pf) of (Some sec, Some psec, Some asecs) \<Rightarrow> Some (Created, sec, Some psec, set asecs) | _ \<Rightarrow> None) | None \<Rightarrow> None)"apply (frule vd_cons, frule vt_grant_os, frule noroot_events)apply (case_tac f, simp)apply (simp add:cf2sfile_def is_file_simps is_dir_simps current_files_simps sectxt_of_obj_simps get_parentfs_ctxts_simps)apply (rule impI, (erule conjE)+)apply (drule not_deleted_init_file, simp+)apply (simp add:is_file_in_current)donelemma cf2sfile_open: "\<lbrakk>valid (Open p f flag fd opt # s); f' \<in> current_files (Open p f flag fd opt # s)\<rbrakk> \<Longrightarrow> cf2sfile (Open p f flag fd opt # s) f' = ( if (opt = None) then cf2sfile s f' else if (f' = f) then (case (parent f) of Some pf \<Rightarrow> (case (sectxt_of_obj (Open p f flag fd opt # s) (O_file f), sectxt_of_obj s (O_dir pf), get_parentfs_ctxts s pf) of (Some sec, Some psec, Some asecs) \<Rightarrow> Some (Created, sec, Some psec, set asecs) | _ \<Rightarrow> None) | None \<Rightarrow> None) else cf2sfile s f')"apply (case_tac opt)apply (simp add:cf2sfile_open_none)apply (case_tac "f = f'")apply (simp add:cf2sfile_open_some4 split:option.splits)apply (simp add:cf2sfile_open_some1 current_files_simps)donelemma cf2sfile_mkdir1: "\<lbrakk>valid (Mkdir p f i # s); f' \<in> current_files s\<rbrakk> \<Longrightarrow> cf2sfile (Mkdir p f i # s) f' = cf2sfile s f'"apply (frule vd_cons, frule vt_grant_os, frule noroot_events)apply (case_tac "f = f'", simp)apply (induct f', simp add:sroot_only, simp)apply (frule parentf_in_current', simp+)apply (case_tac "f = f'", simp)apply (simp add:cf2sfile_path is_file_simps is_dir_simps current_files_simps sectxt_of_obj_simps get_parentfs_ctxts_simps split:if_splits option.splits)donelemma cf2sfile_mkdir2: "\<lbrakk>valid (Mkdir p f i # s); is_file s f'\<rbrakk> \<Longrightarrow> cf2sfile (Mkdir p f i # s) f' = cf2sfile s f'"apply (frule vd_cons, drule is_file_in_current)by (simp add:cf2sfile_mkdir1)lemma cf2sfile_mkdir3: "\<lbrakk>valid (Mkdir p f i # s); is_dir s f'\<rbrakk> \<Longrightarrow> cf2sfile (Mkdir p f i # s) f' = cf2sfile s f'"apply (frule vd_cons, drule is_dir_in_current)by (simp add:cf2sfile_mkdir1)lemma cf2sfile_mkdir4: "valid (Mkdir p f i # s) \<Longrightarrow> cf2sfile (Mkdir p f i # s) f = (case (parent f) of Some pf \<Rightarrow> (case (sectxt_of_obj (Mkdir p f i # s) (O_dir f), sectxt_of_obj s (O_dir pf), get_parentfs_ctxts s pf) of (Some sec, Some psec, Some asecs) \<Rightarrow> Some (Created, sec, Some psec, set asecs) | _ \<Rightarrow> None) | None \<Rightarrow> None)"apply (frule vd_cons, frule vt_grant_os, frule noroot_events)apply (case_tac f, simp)apply (clarsimp simp:os_grant.simps)apply (simp add:sectxt_of_obj_simps)apply (frule current_proc_has_sec, simp)apply (frule is_dir_has_sec, simp)apply (frule get_pfs_secs_prop, simp)apply (frule is_dir_not_file)apply (auto simp add:cf2sfile_def is_file_simps is_dir_simps current_files_simps sectxt_of_obj_simps get_parentfs_ctxts_simps split:option.splits if_splits dest:not_deleted_init_dir is_dir_in_current not_deleted_init_file is_file_in_current)donelemma cf2sfile_mkdir: "\<lbrakk>valid (Mkdir p f i # s); f' \<in> current_files (Mkdir p f i # s)\<rbrakk> \<Longrightarrow> cf2sfile (Mkdir p f i # s) f' = ( if (f' = f) then (case (parent f) of Some pf \<Rightarrow> (case (sectxt_of_obj (Mkdir p f i # s) (O_dir f), sectxt_of_obj s (O_dir pf), get_parentfs_ctxts s pf) of (Some sec, Some psec, Some asecs) \<Rightarrow> Some (Created, sec, Some psec, set asecs) | _ \<Rightarrow> None) | None \<Rightarrow> None) else cf2sfile s f')"apply (case_tac "f = f'")apply (simp add:cf2sfile_mkdir4 split:option.splits)apply (simp add:cf2sfile_mkdir1 current_files_simps)donelemma cf2sfile_linkhard1: "\<lbrakk>valid (LinkHard p oldf f # s); f' \<in> current_files s\<rbrakk> \<Longrightarrow> cf2sfile (LinkHard p oldf f# s) f' = cf2sfile s f'"apply (frule vd_cons, frule vt_grant_os, frule noroot_events)apply (case_tac "f = f'", simp)apply (induct f', simp add:sroot_only, simp)apply (frule parentf_in_current', simp+)apply (simp add:cf2sfile_def is_file_simps is_dir_simps current_files_simps sectxt_of_obj_simps get_parentfs_ctxts_simps split:if_splits option.splits)donelemma cf2sfile_linkhard2: "\<lbrakk>valid (LinkHard p oldf f # s); is_file s f'\<rbrakk> \<Longrightarrow> cf2sfile (LinkHard p oldf f # s) f' = cf2sfile s f'"apply (frule vd_cons, drule is_file_in_current)by (simp add:cf2sfile_linkhard1)lemma cf2sfile_linkhard3: "\<lbrakk>valid (LinkHard p oldf f # s); is_dir s f'\<rbrakk> \<Longrightarrow> cf2sfile (LinkHard p oldf f # s) f' = cf2sfile s f'"apply (frule vd_cons, drule is_dir_in_current)by (simp add:cf2sfile_linkhard1)lemma cf2sfile_linkhard4: "valid (LinkHard p oldf f # s) \<Longrightarrow> cf2sfile (LinkHard p oldf f # s) f = (case (parent f) of Some pf \<Rightarrow> (case (sectxt_of_obj (LinkHard p oldf f # s) (O_file f), sectxt_of_obj s (O_dir pf), get_parentfs_ctxts s pf) of (Some sec, Some psec, Some asecs) \<Rightarrow> Some (Created, sec, Some psec, set asecs) | _ \<Rightarrow> None) | None \<Rightarrow> None)"apply (frule vd_cons, frule vt_grant_os, frule noroot_events)apply (case_tac f, simp)apply (simp add:cf2sfile_def is_file_simps is_dir_simps current_files_simps sectxt_of_obj_simps get_parentfs_ctxts_simps)apply (rule impI, (erule conjE)+)apply (drule not_deleted_init_file, simp+)apply (simp add:is_file_in_current)donelemma cf2sfile_linkhard: "\<lbrakk>valid (LinkHard p oldf f # s); f' \<in> current_files (LinkHard p oldf f # s)\<rbrakk> \<Longrightarrow> cf2sfile (LinkHard p oldf f # s) f' = ( if (f' = f) then (case (parent f) of Some pf \<Rightarrow> (case (sectxt_of_obj (LinkHard p oldf f # s) (O_file f), sectxt_of_obj s (O_dir pf), get_parentfs_ctxts s pf) of (Some sec, Some psec, Some asecs) \<Rightarrow> Some (Created, sec, Some psec, set asecs) | _ \<Rightarrow> None) | None \<Rightarrow> None) else cf2sfile s f')"apply (case_tac "f = f'")apply (simp add:cf2sfile_linkhard4 split:option.splits)apply (simp add:cf2sfile_linkhard1 current_files_simps)donelemma cf2sfile_other: "\<lbrakk>ff \<in> current_files s; \<forall> p f flag fd opt. e \<noteq> Open p f flag fd opt; \<forall> p fd. e \<noteq> CloseFd p fd; \<forall> p f. e \<noteq> UnLink p f; \<forall> p f. e \<noteq> Rmdir p f; \<forall> p f i. e \<noteq> Mkdir p f i; \<forall> p f f'. e \<noteq> LinkHard p f f'; valid (e # s)\<rbrakk> \<Longrightarrow> cf2sfile (e # s) ff = cf2sfile s ff"apply (frule vd_cons, frule vt_grant_os)apply (induct ff, simp add:sroot_only)apply (frule parentf_in_current', simp+, case_tac e)apply (auto simp:current_files_simps is_file_simps is_dir_simps sectxt_of_obj_simps cf2sfile_path split:if_splits option.splits) done lemma cf2sfile_other': "\<lbrakk>valid (e # s); \<forall> p f flag fd opt. e \<noteq> Open p f flag fd opt; \<forall> p fd. e \<noteq> CloseFd p fd; \<forall> p f. e \<noteq> UnLink p f; \<forall> p f. e \<noteq> Rmdir p f; \<forall> p f i. e \<noteq> Mkdir p f i; \<forall> p f f'. e \<noteq> LinkHard p f f'; ff \<in> current_files s\<rbrakk> \<Longrightarrow> cf2sfile (e # s) ff = cf2sfile s ff"by (auto intro!:cf2sfile_other)lemma cf2sfile_unlink: "\<lbrakk>valid (UnLink p f # s); f' \<in> current_files (UnLink p f # s)\<rbrakk> \<Longrightarrow> cf2sfile (UnLink p f # s) f' = cf2sfile s f'"apply (frule vd_cons, frule vt_grant_os)apply (simp add:current_files_simps split:if_splits)apply (auto simp:cf2sfile_def sectxt_of_obj_simps get_parentfs_ctxts_simps is_file_simps is_dir_simps split:if_splits option.splits)donelemma cf2sfile_rmdir: "\<lbrakk>valid (Rmdir p f # s); f' \<in> current_files (Rmdir p f # s)\<rbrakk> \<Longrightarrow> cf2sfile (Rmdir p f # s) f' = cf2sfile s f'"apply (frule vd_cons, frule vt_grant_os)apply (simp add:current_files_simps split:if_splits)apply (auto simp:cf2sfile_def sectxt_of_obj_simps get_parentfs_ctxts_simps is_file_simps is_dir_simps split:if_splits option.splits)donelemma pfdof_simp5: "\<lbrakk>proc_fd_of_file s f = {(p, fd)}; file_of_proc_fd s p fd = None\<rbrakk> \<Longrightarrow> False"apply (subgoal_tac "(p, fd) \<in> proc_fd_of_file s f")by (simp add:pfdof_simp2, simp)lemma pfdof_simp6: "proc_fd_of_file s f = {(p, fd)} \<Longrightarrow> file_of_proc_fd s p fd = Some f"apply (subgoal_tac "(p, fd) \<in> proc_fd_of_file s f")by (simp add:pfdof_simp2, simp)lemma cf2sfile_closefd: "\<lbrakk>valid (CloseFd p fd # s); f \<in> current_files (CloseFd p fd # s)\<rbrakk> \<Longrightarrow> cf2sfile (CloseFd p fd # s) f = cf2sfile s f"apply (frule vd_cons, frule vt_grant_os)apply (simp add:current_files_simps split:if_splits option.splits) (* costs too much time, but solved *)apply (auto simp:cf2sfile_def sectxt_of_obj_simps get_parentfs_ctxts_simps is_file_simps is_dir_simps split:if_splits option.splits dest:init_file_dir_conflict pfdof_simp5 pfdof_simp6 file_of_pfd_is_file not_deleted_init_file not_deleted_init_dir is_file_not_dir is_dir_not_file dest!:current_has_sec')donelemmas cf2sfile_simps = cf2sfile_open cf2sfile_mkdir cf2sfile_linkhard cf2sfile_other cf2sfile_unlink cf2sfile_rmdir cf2sfile_closefd(*********** cfd2sfd simpset *********)lemma cfd2sfd_open1: "valid (Open p f flags fd opt # s) \<Longrightarrow> cfd2sfd (Open p f flags fd opt # s) p fd = (case (sectxt_of_obj (Open p f flags fd opt # s) (O_fd p fd), cf2sfile (Open p f flags fd opt # s) f) of (Some sec, Some sf) \<Rightarrow> Some (sec, flags, sf) | _ \<Rightarrow> None)"by (simp add:cfd2sfd_def sectxt_of_obj_simps split:if_splits)lemma cfd2sfd_open_some2: "\<lbrakk>valid (Open p f flags fd (Some inum) # s); file_of_proc_fd s p' fd' = Some f'\<rbrakk> \<Longrightarrow> cfd2sfd (Open p f flags fd (Some inum) # s) p' fd' = cfd2sfd s p' fd'"apply (frule vd_cons, frule vt_grant_os)apply (frule proc_fd_in_fds, simp)apply (frule file_of_proc_fd_in_curf, simp)apply (case_tac "f = f'", simp)apply (simp add:cfd2sfd_def sectxt_of_obj_simps cf2sfile_open_some1)apply (case_tac "p = p'", simp)apply (rule conjI, rule impI, simp)apply (drule cf2sfile_open_some1, simp)apply (auto split:option.splits)[1]apply simpapply (drule cf2sfile_open_some1, simp)apply (auto split:option.splits)[1]donelemma cfd2sfd_open_none2: "\<lbrakk>valid (Open p f flags fd None # s); file_of_proc_fd s p' fd' = Some f'\<rbrakk> \<Longrightarrow> cfd2sfd (Open p f flags fd None # s) p' fd' = cfd2sfd s p' fd'"apply (frule vd_cons, frule vt_grant_os)apply (frule proc_fd_in_fds, simp)apply (frule file_of_proc_fd_in_curf, simp)apply (simp add:cfd2sfd_def sectxt_of_obj_simps cf2sfile_open_none)apply (case_tac "p = p'", simp)apply (rule conjI, rule impI, simp)apply (drule cf2sfile_open_none)apply (auto split:option.splits)[1]apply simpapply (drule cf2sfile_open_none)apply (auto split:option.splits)[1]donelemma cfd2sfd_open2: "\<lbrakk>valid (Open p f flags fd opt # s); file_of_proc_fd s p' fd' = Some f'\<rbrakk> \<Longrightarrow> cfd2sfd (Open p f flags fd opt # s) p' fd' = cfd2sfd s p' fd'"apply (case_tac opt)apply (simp add:cfd2sfd_open_none2)apply (simp add:cfd2sfd_open_some2)donelemma cfd2sfd_open: "\<lbrakk>valid (Open p f flags fd opt # s); file_of_proc_fd (Open p f flags fd opt # s) p' fd' = Some f'\<rbrakk> \<Longrightarrow> cfd2sfd (Open p f flags fd opt # s) p' fd' = (if (p' = p \<and> fd' = fd) then (case (sectxt_of_obj (Open p f flags fd opt # s) (O_fd p fd), cf2sfile (Open p f flags fd opt # s) f) of (Some sec, Some sf) \<Rightarrow> Some (sec, flags, sf) | _ \<Rightarrow> None) else cfd2sfd s p' fd')"apply (simp split:if_splits)apply (simp add:cfd2sfd_open1 split:option.splits)apply (simp add:cfd2sfd_open2)apply (rule impI, simp)donelemma cfd2sfd_closefd: "\<lbrakk>valid (CloseFd p fd # s); file_of_proc_fd (CloseFd p fd # s) p' fd' = Some f\<rbrakk> \<Longrightarrow> cfd2sfd (CloseFd p fd # s) p' fd' = cfd2sfd s p' fd'"apply (frule vd_cons, frule vt_grant_os)apply (frule proc_fd_in_fds, simp)apply (frule file_of_proc_fd_in_curf, simp)apply (frule cf2sfile_closefd, simp)apply (simp add:cfd2sfd_def sectxt_of_obj_simps)apply (auto split:option.splits if_splits)donelemma cfd2sfd_clone: "\<lbrakk>valid (Clone p p' fds shms # s); file_of_proc_fd (Clone p p' fds shms # s) p'' fd' = Some f\<rbrakk> \<Longrightarrow> cfd2sfd (Clone p p' fds shms # s) p'' fd' = ( if (p'' = p') then cfd2sfd s p fd' else cfd2sfd s p'' fd')"apply (frule vd_cons, frule vt_grant_os)apply (frule proc_fd_in_fds, simp)apply (frule file_of_proc_fd_in_curf, simp, simp add:current_files_simps)apply (frule_tac cf2sfile_other', simp+)apply (simp add:cfd2sfd_def sectxt_of_obj_simps)apply (case_tac "p'' = p'", simp)apply (auto split:option.splits if_splits)[1]apply (simp)apply (auto split:option.splits if_splits)[1]donelemma cfd2sfd_execve: "\<lbrakk>valid (Execve p f fds # s); file_of_proc_fd (Execve p f fds # s) p' fd' = Some f'\<rbrakk> \<Longrightarrow> cfd2sfd (Execve p f fds # s) p' fd' = cfd2sfd s p' fd'"apply (frule vd_cons, frule vt_grant_os)apply (frule proc_fd_in_fds, simp)apply (frule file_of_proc_fd_in_curf, simp, simp add:current_files_simps)apply (frule_tac cf2sfile_other', simp+)apply (simp add:cfd2sfd_def sectxt_of_obj_simps)apply (case_tac "p' = p", simp)apply (auto split:option.splits if_splits)[1]apply (simp)apply (auto split:option.splits if_splits)[1]donelemma cfd2sfd_kill: "\<lbrakk>valid (Kill p p'' # s); file_of_proc_fd (Kill p p'' # s) p' fd' = Some f'\<rbrakk> \<Longrightarrow> cfd2sfd (Kill p p'' # s) p' fd' = cfd2sfd s p' fd'"apply (frule vd_cons, frule vt_grant_os)apply (frule proc_fd_in_fds, simp)apply (frule proc_fd_in_procs, simp)apply (frule file_of_proc_fd_in_curf, simp, simp add:current_files_simps)apply (frule_tac cf2sfile_other', simp+)apply (simp add:cfd2sfd_def sectxt_of_obj_simps)apply (auto split:option.splits if_splits)donelemma cfd2sfd_exit: "\<lbrakk>valid (Exit p # s); file_of_proc_fd (Exit p # s) p' fd' = Some f'\<rbrakk> \<Longrightarrow> cfd2sfd (Exit p # s) p' fd' = cfd2sfd s p' fd'"apply (frule vd_cons, frule vt_grant_os)apply (frule proc_fd_in_fds, simp)apply (frule proc_fd_in_procs, simp)apply (frule file_of_proc_fd_in_curf, simp, simp add:current_files_simps)apply (frule_tac cf2sfile_other', simp+)apply (simp add:cfd2sfd_def sectxt_of_obj_simps)apply (auto split:option.splits if_splits)donelemma cfd2sfd_other: "\<lbrakk>valid (e # s); file_of_proc_fd (e # s) p' fd' = Some f'; \<forall> p f flags fd opt. e \<noteq> Open p f flags fd opt; \<forall> p p'' fds shms. e \<noteq> Clone p p'' fds shms\<rbrakk> \<Longrightarrow> cfd2sfd (e # s) p' fd' = cfd2sfd s p' fd'"apply (frule vd_cons, frule vt_grant_os)apply (frule proc_fd_in_fds, simp)apply (frule proc_fd_in_procs, simp)apply (frule file_of_proc_fd_in_curf, simp)apply (case_tac e)apply (auto intro!:cfd2sfd_execve cfd2sfd_closefd cfd2sfd_kill cfd2sfd_exit)apply (auto simp:cfd2sfd_def sectxt_of_obj_simps current_files_simps cf2sfile_simps split:option.splits)apply (auto dest!:current_has_sec' dest:file_of_proc_fd_in_curf proc_fd_in_fds)donelemmas cfd2sfd_simps = cfd2sfd_open cfd2sfd_clone cfd2sfd_other(********** cpfd2sfds simpset **********)lemma current_filefd_has_flags: "\<lbrakk>file_of_proc_fd s p fd = Some f; valid s\<rbrakk> \<Longrightarrow> \<exists> flags. flags_of_proc_fd s p fd = Some flags"apply (induct s arbitrary:p)apply (simp only:flags_of_proc_fd.simps file_of_proc_fd.simps init_filefd_prop4)apply (frule vd_cons, frule vt_grant_os, case_tac a)apply (auto split:if_splits option.splits dest:proc_fd_in_fds)donelemma current_filefd_has_flags': "\<lbrakk>flags_of_proc_fd s p fd = None; valid s\<rbrakk> \<Longrightarrow> file_of_proc_fd s p fd = None"apply (case_tac "file_of_proc_fd s p fd")apply (simp, drule current_filefd_has_flags, simp+)donelemma current_file_has_sfile': "\<lbrakk>cf2sfile s f = None; valid s\<rbrakk> \<Longrightarrow> f \<notin> current_files s"by (rule notI, drule current_file_has_sfile, simp+)lemma current_filefd_has_sfd: "\<lbrakk>file_of_proc_fd s p fd = Some f; valid s\<rbrakk> \<Longrightarrow> \<exists>sfd. cfd2sfd s p fd = Some sfd"by (auto simp:cfd2sfd_def split:option.splits dest!:current_has_sec' current_file_has_sfile' dest:file_of_proc_fd_in_curf proc_fd_in_fds current_filefd_has_flags)lemma current_filefd_has_sfd': "\<lbrakk>cfd2sfd s p fd = None; valid s\<rbrakk> \<Longrightarrow> file_of_proc_fd s p fd = None"by (case_tac "file_of_proc_fd s p fd", auto dest:current_filefd_has_sfd)lemma cpfd2sfds_open1: "valid (Open p f flags fd opt # s) \<Longrightarrow> cpfd2sfds (Open p f flags fd opt # s) p = ( case (cfd2sfd (Open p f flags fd opt # s) p fd) of Some sfd \<Rightarrow> (cpfd2sfds s p) \<union> {sfd} | _ \<Rightarrow> cpfd2sfds s p)"apply (frule vd_cons, frule vt_grant_os)apply (split option.splits)apply (rule conjI, rule impI, drule current_filefd_has_sfd', simp, simp)apply (rule allI, rule impI)apply (rule set_eqI, rule iffI)apply (case_tac "x = a", simp)unfolding cpfd2sfds_defapply (erule CollectE, (erule conjE|erule bexE)+)apply (simp add:proc_file_fds_def split:if_splits)apply (erule exE, rule_tac x = fda in exI)apply (simp add:cfd2sfd_open2)apply (case_tac "x = a", simp add:proc_file_fds_def)apply (rule_tac x = fd in exI, simp+)apply (erule conjE|erule bexE)+apply (rule_tac x = fda in bexI)apply (simp add:proc_file_fds_def, erule exE)apply (simp add:cfd2sfd_open2)apply (simp add:proc_file_fds_def)donelemma cpfd2sfds_open1': "valid (Open p f flags fd opt # s) \<Longrightarrow> cpfd2sfds (Open p f flags fd opt # s) p = ( case (sectxt_of_obj (Open p f flags fd opt # s) (O_fd p fd), cf2sfile (Open p f flags fd opt # s) f) of (Some sec, Some sf) \<Rightarrow> (cpfd2sfds s p) \<union> {(sec, flags, sf)} | _ \<Rightarrow> cpfd2sfds s p)"apply (frule cfd2sfd_open1)apply (auto dest:cpfd2sfds_open1 split:option.splits)donelemma cpfd2sfds_open2: "\<lbrakk>valid (Open p f flags fd opt # s); p' \<noteq> p\<rbrakk> \<Longrightarrow> cpfd2sfds (Open p f flags fd opt # s) p' = cpfd2sfds s p'"apply (frule vt_grant_os, frule vd_cons)unfolding cpfd2sfds_defapply (rule set_eqI, rule iffI)apply (simp add:proc_file_fds_def)apply (erule exE|erule conjE)+apply (simp only:file_of_proc_fd.simps cfd2sfd_open2 split:if_splits)apply (rule_tac x = fda in exI, simp)apply (simp add:proc_file_fds_def)apply (erule exE|erule conjE)+apply (rule_tac x = fda in exI, simp add:cfd2sfd_open2)donelemma cpfd2sfds_open: "valid (Open p f flags fd opt # s) \<Longrightarrow> cpfd2sfds (Open p f flags fd opt # s) = (cpfd2sfds s) (p := ( case (sectxt_of_obj (Open p f flags fd opt # s) (O_fd p fd), cf2sfile (Open p f flags fd opt # s) f) of (Some sec, Some sf) \<Rightarrow> (cpfd2sfds s p) \<union> {(sec, flags, sf)} | _ \<Rightarrow> cpfd2sfds s p))"apply (rule ext)apply (case_tac "x \<noteq> p")apply (simp add:cpfd2sfds_open2)apply (simp add:cpfd2sfds_open1')donelemma cpfd2sfds_execve: "valid (Execve p f fds # s) \<Longrightarrow> cpfd2sfds (Execve p f fds # s) = (cpfd2sfds s) (p := {sfd. \<exists> fd \<in> fds. \<exists> f. file_of_proc_fd s p fd = Some f \<and> cfd2sfd s p fd = Some sfd})"apply (frule vd_cons, frule vt_grant_os)apply (rule ext)apply (rule set_eqI, rule iffI)unfolding cpfd2sfds_def proc_file_fds_defapply (erule CollectE| erule bexE| erule conjE| erule exE| rule conjI)+apply (simp split:if_splits)apply (frule_tac p' = p and fd' = fd in cfd2sfd_other, simp+)apply (rule_tac x = fd in bexI, simp+)apply (simp add:cpfd2sfds_def proc_file_fds_def)apply (frule_tac p' = x and fd' = fd in cfd2sfd_other, simp+)apply (rule_tac x = fd in exI, simp)apply (simp split:if_splits)apply (erule CollectE| erule bexE| erule conjE| erule exE| rule conjI)+apply (rule_tac x = fd in exI, simp)apply (frule_tac p' = x and fd' = fd in cfd2sfd_other, simp+)apply (simp add:cpfd2sfds_def proc_file_fds_def)apply (erule CollectE| erule bexE| erule conjE| erule exE| rule conjI)+apply (rule_tac x = fd in exI, simp)apply (frule_tac p' = x and fd' = fd in cfd2sfd_other, simp+)donelemma cpfd2sfds_clone: "valid (Clone p p' fds shms # s) \<Longrightarrow> cpfd2sfds (Clone p p' fds shms # s) = (cpfd2sfds s) (p' := {sfd. \<exists> fd \<in> fds. \<exists> f. file_of_proc_fd s p fd = Some f \<and> cfd2sfd s p fd = Some sfd})"apply (frule vd_cons, frule vt_grant_os)apply (rule ext)apply (rule set_eqI, rule iffI)unfolding cpfd2sfds_def proc_file_fds_defapply (erule CollectE| erule bexE| erule conjE| erule exE| rule conjI)+apply (simp split:if_splits)apply (frule_tac p'' = p' and fd' = fd in cfd2sfd_clone, simp+)apply (rule_tac x = fd in bexI, simp+)apply (simp add:cpfd2sfds_def proc_file_fds_def)apply (frule_tac p'' = x and fd' = fd in cfd2sfd_clone, simp+)apply (rule_tac x = fd in exI, simp)apply (simp split:if_splits)apply (erule CollectE| erule bexE| erule conjE| erule exE| rule conjI)+apply (rule_tac x = fd in exI, simp)apply (frule_tac p'' = p' and fd' = fd in cfd2sfd_clone, simp+)apply (simp add:cpfd2sfds_def proc_file_fds_def)apply (erule CollectE| erule bexE| erule conjE| erule exE| rule conjI)+apply (rule_tac x = fd in exI, simp)apply (frule_tac p'' = x and fd' = fd in cfd2sfd_clone, simp+)donelemma cpfd2sfds_other: "\<lbrakk>valid (e # s); \<forall> p f flags fd opt. e \<noteq> Open p f flags fd opt; \<forall> p f fds. e \<noteq> Execve p f fds; \<forall> p p'. e \<noteq> Kill p p'; \<forall> p. e \<noteq> Exit p; \<forall> p fd. e \<noteq> CloseFd p fd; \<forall> p p' fds shms. e \<noteq> Clone p p' fds shms\<rbrakk> \<Longrightarrow> cpfd2sfds (e # s) = cpfd2sfds s"apply (frule vd_cons, frule vt_grant_os)apply (rule ext)unfolding cpfd2sfds_def proc_file_fds_defapply (case_tac e)using cfd2sfd_otherby autolemma cpfd2sfds_kill: "valid (Kill p p' # s) \<Longrightarrow> cpfd2sfds (Kill p p' # s) = (cpfd2sfds s) (p' := {})"apply (frule vd_cons, frule vt_grant_os)apply (rule ext, rule set_eqI)unfolding cpfd2sfds_def proc_file_fds_defapply (rule iffI)apply (erule CollectE| erule bexE| erule conjE| erule exE| rule conjI)+apply (simp split:if_splits add: cpfd2sfds_def proc_file_fds_def)apply (rule_tac x = fd in exI, simp)apply (drule_tac p' = x and fd' = fd in cfd2sfd_other, simp+)apply (simp split:if_splits add: cpfd2sfds_def proc_file_fds_def)apply (erule CollectE| erule bexE| erule conjE| erule exE| rule conjI)+apply (rule_tac x = fd in exI, simp)apply (drule_tac p' = x and fd' = fd in cfd2sfd_other, simp+)donelemma cpfd2sfds_exit: "valid (Exit p # s) \<Longrightarrow> cpfd2sfds (Exit p # s) = (cpfd2sfds s) (p := {})"apply (frule vd_cons, frule vt_grant_os)apply (rule ext, rule set_eqI)unfolding cpfd2sfds_def proc_file_fds_defapply (rule iffI)apply (erule CollectE| erule bexE| erule conjE| erule exE| rule conjI)+apply (simp split:if_splits add: cpfd2sfds_def proc_file_fds_def)apply (rule_tac x = fd in exI, simp)apply (drule_tac p' = x and fd' = fd in cfd2sfd_other, simp+)apply (simp split:if_splits add: cpfd2sfds_def proc_file_fds_def)apply (erule CollectE| erule bexE| erule conjE| erule exE| rule conjI)+apply (rule_tac x = fd in exI, simp)apply (drule_tac p' = x and fd' = fd in cfd2sfd_other, simp+)donelemma cpfd2sfds_closefd: "valid (CloseFd p fd # s) \<Longrightarrow> cpfd2sfds (CloseFd p fd # s) = (cpfd2sfds s) (p := if (fd \<in> proc_file_fds s p) then (case (cfd2sfd s p fd) of Some sfd \<Rightarrow> (if (\<exists> fd' f'. fd' \<noteq> fd \<and> file_of_proc_fd s p fd' = Some f' \<and> cfd2sfd s p fd' = Some sfd) then cpfd2sfds s p else cpfd2sfds s p - {sfd}) | _ \<Rightarrow> cpfd2sfds s p) else cpfd2sfds s p)"apply (frule vd_cons)apply (rule ext, rule set_eqI, rule iffI)unfolding cpfd2sfds_def proc_file_fds_defapply (erule CollectE| erule bexE| erule conjE| erule exE| rule conjI)+apply (simp split:if_splits)apply (rule conjI, rule impI, rule conjI, rule impI, erule exE)apply (frule_tac p = p and fd = fd in current_filefd_has_sfd, simp)apply (erule exE, simp)apply (rule conjI, rule impI, (erule exE|erule conjE)+)apply (rule_tac x = fda in exI, simp, simp add:cfd2sfd_closefd)apply (rule impI, rule conjI)apply (rule_tac x = fda in exI, simp, simp add:cfd2sfd_closefd)apply (rule notI, simp)apply (erule_tac x = fda in allE, simp add:cfd2sfd_closefd)apply (rule impI, simp add:cpfd2sfds_def proc_file_fds_def)apply (erule exE, rule_tac x = fda in exI, simp add:cfd2sfd_closefd)apply (rule impI| rule conjI)+apply (rule_tac x = fda in exI, simp add:cfd2sfd_closefd)apply (rule impI, simp add:cpfd2sfds_def proc_file_fds_def)apply (rule_tac x = fda in exI, simp add:cfd2sfd_closefd)apply (simp split:if_splits)apply (frule_tac p = p and fd = fd in current_filefd_has_sfd, simp)apply (erule exE, simp)apply (case_tac "\<exists>fd'. fd' \<noteq> fd \<and> (\<exists>f'. file_of_proc_fd s p fd' = Some f') \<and> cfd2sfd s p fd' = Some sfd")apply simpapply (case_tac "xa = sfd")apply (erule exE|erule conjE)+apply (rule_tac x = fd' in exI, simp add:cfd2sfd_closefd)apply (erule exE|erule conjE)+apply (rule_tac x = fda in exI, simp add:cfd2sfd_closefd)apply (rule notI, simp)apply (simp, (erule exE|erule conjE)+)apply (rule_tac x = fda in exI, simp add:cfd2sfd_closefd)apply (rule notI, simp)apply (erule exE|erule conjE)+apply (rule_tac x = fda in exI, simp add:cfd2sfd_closefd)apply (rule notI, simp)apply (simp add:cpfd2sfds_def proc_file_fds_def)apply (erule exE|erule conjE)+apply (rule_tac x = fda in exI, simp add:cfd2sfd_closefd)donelemmas cpfd2sfds_simps = cpfd2sfds_open cpfd2sfds_execve cpfd2sfds_clone cpfd2sfds_kill cpfd2sfds_exit cpfd2sfds_closefd cpfd2sfds_other(********* ch2sshm simpset ********)lemma ch2sshm_createshm: "valid (CreateShM p h # s) \<Longrightarrow> ch2sshm (CreateShM p h # s) = (ch2sshm s) (h := (case (sectxt_of_obj (CreateShM p h # s) (O_shm h)) of Some sec \<Rightarrow> Some (if (\<not> deleted (O_shm h) s \<and> h \<in> init_shms) then Init h else Created, sec) | _ \<Rightarrow> None))"apply (frule vd_cons, frule vt_grant_os)apply (auto simp:ch2sshm_def sectxt_of_obj_simps dest!:current_has_sec' split:option.splits if_splits)donelemma ch2sshm_other: "\<lbrakk>valid (e # s); \<forall> p h. e \<noteq> CreateShM p h; h' \<in> current_shms (e # s)\<rbrakk> \<Longrightarrow> ch2sshm (e # s) h' = ch2sshm s h'"apply (frule vd_cons, frule vt_grant_os)apply (case_tac e)apply (auto simp:ch2sshm_def sectxt_of_obj_simps dest!:current_has_sec' split:option.splits if_splits)donelemmas ch2sshm_simps = ch2sshm_createshm ch2sshm_otherlemma current_shm_has_sh: "\<lbrakk>h \<in> current_shms s; valid s\<rbrakk> \<Longrightarrow> \<exists> sh. ch2sshm s h = Some sh"by (auto simp:ch2sshm_def split:option.splits dest!:current_has_sec')lemma current_shm_has_sh': "\<lbrakk>ch2sshm s h = None; valid s\<rbrakk> \<Longrightarrow> h \<notin> current_shms s"by (auto dest:current_shm_has_sh)(********** cph2spshs simpset **********)lemma cph2spshs_attach: "valid (Attach p h flag # s) \<Longrightarrow> cph2spshs (Attach p h flag # s) = (cph2spshs s) (p := (case (ch2sshm s h) of Some sh \<Rightarrow> cph2spshs s p \<union> {(sh, flag)} | _ \<Rightarrow> cph2spshs s p) )"apply (frule vd_cons, frule vt_grant_os, rule ext)using ch2sshm_other[where e = "Attach p h flag" and s = s]apply (auto split del:t_open_must_flag.splits t_open_option_flag.splits split add:if_splits option.splitsdest!:current_shm_has_sh' dest: procs_of_shm_prop1 simp:cph2spshs_def)apply (erule_tac x = ha in allE, frule procs_of_shm_prop1, simp, simp)apply (erule_tac x = ha in allE, frule procs_of_shm_prop1, simp, simp)apply (erule_tac x = ha in allE, frule procs_of_shm_prop1, simp, simp)apply (erule_tac x = ha in allE, frule procs_of_shm_prop1, simp, simp)apply (erule_tac x = ha in allE, frule procs_of_shm_prop1, simp, simp)apply (case_tac "ha = h", simp, frule procs_of_shm_prop1, simp)apply (rule_tac x = ha in exI, simp)apply (rule_tac x = ha in exI, simp, drule procs_of_shm_prop1, simp, simp)apply (rule_tac x = ha in exI, simp)apply (frule procs_of_shm_prop1, simp, simp)apply (rule impI, simp)donelemma cph2spshs_detach: "valid (Detach p h # s) \<Longrightarrow> cph2spshs (Detach p h # s) = (cph2spshs s) (p := (case (ch2sshm s h, flag_of_proc_shm s p h) of (Some sh, Some flag) \<Rightarrow> if (\<exists> h'. h' \<noteq> h \<and> (p,flag) \<in> procs_of_shm s h' \<and> ch2sshm s h' = Some sh) then cph2spshs s p else cph2spshs s p - {(sh, flag)} | _ \<Rightarrow> cph2spshs s p) )"apply (frule vd_cons, frule vt_grant_os, rule ext)apply (case_tac "x = p") deferusing ch2sshm_other[where e = "Detach p h" and s = s] apply (auto split del:t_open_must_flag.splits t_open_option_flag.splits split add:if_splits option.splitsdest!:current_shm_has_sh' procs_of_shm_prop4' dest:procs_of_shm_prop1 procs_of_shm_prop3 simp:cph2spshs_def) [1]apply (rule_tac x = ha in exI, frule_tac h = ha in procs_of_shm_prop1, simp, simp)apply (rule_tac x = ha in exI, frule_tac h = ha in procs_of_shm_prop1, simp, simp)apply (rule impI, simp)apply (clarsimp)apply (frule current_shm_has_sh, simp, erule exE)apply (frule procs_of_shm_prop4, simp, simp)apply (rule allI | rule conjI| erule conjE | erule exE | rule impI)+apply (simp only:cph2spshs_def, rule set_eqI, rule iffI)apply (erule CollectE | erule exE| erule conjE| rule CollectI)+apply (case_tac "ha = h", simp)apply (rule_tac x = sha in exI, rule_tac x = flaga in exI, rule_tac x = ha in exI, simp)using ch2sshm_other[where e = "Detach p h" and s = s] apply (simp add:procs_of_shm_prop1)apply (erule CollectE | erule exE| erule conjE| rule CollectI)+apply (case_tac "ha = h", simp)apply (rule_tac x = h' in exI, simp)apply (frule_tac flag = flag and flag' = flaga in procs_of_shm_prop3, simp+)apply (frule_tac flag = flaga in procs_of_shm_prop4, simp+)using ch2sshm_other[where e = "Detach p h" and s = s] apply (simp add:procs_of_shm_prop1)apply (frule_tac h = h' in procs_of_shm_prop1, simp, simp)apply (rule_tac x = ha in exI, simp, frule_tac h = ha in procs_of_shm_prop1, simp+)using ch2sshm_other[where e = "Detach p h" and s = s] apply (simp add:procs_of_shm_prop1)apply (rule allI | rule conjI| erule conjE | erule exE | rule impI)+apply (simp only:cph2spshs_def, rule set_eqI, rule iffI)apply (erule CollectE | erule exE| erule conjE| rule DiffI |rule CollectI)+apply (simp split:if_splits)apply (rule_tac x = ha in exI, simp, frule_tac h = ha in procs_of_shm_prop1, simp+)using ch2sshm_other[where e = "Detach p h" and s = s] apply (simp add:procs_of_shm_prop1)apply (rule notI, simp split:if_splits)apply (erule_tac x = ha in allE, simp, frule_tac h = ha in procs_of_shm_prop1, simp+)using ch2sshm_other[where e = "Detach p h" and s = s] apply (simp add:procs_of_shm_prop1)apply (erule CollectE | erule exE| erule conjE| erule DiffE |rule CollectI)+apply (simp split:if_splits)apply (erule_tac x = ha in allE, simp, rule_tac x = ha in exI, simp)apply (case_tac "ha = h", simp add:procs_of_shm_prop3, frule_tac h = ha in procs_of_shm_prop1, simp+)using ch2sshm_other[where e = "Detach p h" and s = s] apply (simp add:procs_of_shm_prop1)donelemma cph2spshs_deleteshm: "valid (DeleteShM p h # s) \<Longrightarrow> cph2spshs (DeleteShM p h # s) = (\<lambda> p'. (case (ch2sshm s h, flag_of_proc_shm s p' h) of (Some sh, Some flag) \<Rightarrow> if (\<exists> h'. h' \<noteq> h \<and> (p', flag) \<in> procs_of_shm s h' \<and> ch2sshm s h' = Some sh) then cph2spshs s p' else cph2spshs s p' - {(sh, flag)} | _ \<Rightarrow> cph2spshs s p') )"apply (frule vd_cons, frule vt_grant_os, rule ext)apply (clarsimp)apply (frule current_shm_has_sh, simp, erule exE, simp, simp split:option.splits)apply (rule conjI, rule impI)using ch2sshm_other[where e = "DeleteShM p h" and s = s] apply (auto split del:t_open_must_flag.splits t_open_option_flag.splits split add:if_splits option.splitsdest!:current_shm_has_sh' procs_of_shm_prop4' dest:procs_of_shm_prop1 procs_of_shm_prop3 simp:cph2spshs_def) [1]apply (rule_tac x = ha in exI, frule_tac h = ha in procs_of_shm_prop1, simp+)apply (rule_tac x = ha in exI, simp)apply (case_tac "ha = h", simp+, frule_tac h = ha in procs_of_shm_prop1, simp+)apply (rule allI | rule conjI| erule conjE | erule exE | rule impI)+apply (simp only:cph2spshs_def, rule set_eqI, rule iffI)apply (erule CollectE | erule exE| erule conjE| rule CollectI)+apply (case_tac "ha = h", simp)apply (rule_tac x = sha in exI, rule_tac x = flag in exI, rule_tac x = ha in exI, simp)using ch2sshm_other[where e = "DeleteShM p h" and s = s] apply (simp add:procs_of_shm_prop1)apply (erule CollectE | erule exE| erule conjE| rule CollectI)+apply (case_tac "ha = h", simp)apply (rule_tac x = h' in exI, simp)apply (frule_tac flag = flag in procs_of_shm_prop4, simp+)using ch2sshm_other[where e = "DeleteShM p h" and s = s] apply (simp add:procs_of_shm_prop1)apply (frule_tac h = h' in procs_of_shm_prop1, simp, simp)apply (rule_tac x = ha in exI, simp, frule_tac h = ha in procs_of_shm_prop1, simp+)using ch2sshm_other[where e = "DeleteShM p h" and s = s] apply (simp add:procs_of_shm_prop1)apply (rule allI | rule conjI| erule conjE | erule exE | rule impI)+apply (simp only:cph2spshs_def, rule set_eqI, rule iffI)apply (erule CollectE | erule exE| erule conjE| rule DiffI |rule CollectI)+apply (simp split:if_splits)apply (rule_tac x = ha in exI, simp, frule_tac h = ha in procs_of_shm_prop1, simp+)using ch2sshm_other[where e = "DeleteShM p h" and s = s] apply (simp add:procs_of_shm_prop1)apply (rule notI, simp split:if_splits)apply (erule_tac x = ha in allE, simp, frule_tac h = ha in procs_of_shm_prop1, simp+)using ch2sshm_other[where e = "DeleteShM p h" and s = s] apply (simp add:procs_of_shm_prop1)apply (erule CollectE | erule exE| erule conjE| erule DiffE |rule CollectI)+apply (simp split:if_splits)apply (erule_tac x = ha in allE, simp, rule_tac x = ha in exI, simp)apply (case_tac "ha = h", simp add:procs_of_shm_prop4)apply (frule_tac h = ha in procs_of_shm_prop1, simp+)using ch2sshm_other[where e = "DeleteShM p h" and s = s] apply (simp add:procs_of_shm_prop1)donelemma flag_of_proc_shm_prop1: "\<lbrakk>flag_of_proc_shm s p h = Some flag; valid s\<rbrakk> \<Longrightarrow> (p, flag) \<in> procs_of_shm s h"apply (induct s arbitrary:p)apply (simp add:init_shmflag_has_proc)apply (frule vt_grant_os, frule vd_cons, case_tac a, auto split:if_splits option.splits)donelemma cph2spshs_clone: "valid (Clone p p' fds shms # s) \<Longrightarrow> cph2spshs (Clone p p' fds shms # s) = (cph2spshs s) (p' := {(sh, flag) | h sh flag. h \<in> shms \<and> ch2sshm s h = Some sh \<and> flag_of_proc_shm s p h = Some flag} )"apply (frule vd_cons, frule vt_grant_os, rule ext)using ch2sshm_other[where e = "Clone p p' fds shms" and s = s]apply (auto split del:t_open_must_flag.splits t_open_option_flag.splits split add:if_splits option.splitsdest!:current_shm_has_sh' dest: procs_of_shm_prop1 procs_of_shm_prop2 procs_of_shm_prop3 simp:cph2spshs_def)apply (erule_tac x = h in allE, frule procs_of_shm_prop1, simp, simp add:procs_of_shm_prop4)apply (rule_tac x = h in exI, simp, frule flag_of_proc_shm_prop1, simp+, simp add:procs_of_shm_prop1)apply (rule_tac x = h in exI, simp, frule procs_of_shm_prop1, simp+)+donelemma cph2spshs_execve: "valid (Execve p f fds # s) \<Longrightarrow> cph2spshs (Execve p f fds # s) = (cph2spshs s) (p := {})"apply (frule vd_cons, frule vt_grant_os, rule ext)using ch2sshm_other[where e = "Execve p f fds" and s = s]apply (auto split del:t_open_must_flag.splits t_open_option_flag.splits split add:if_splits option.splitsdest!:current_shm_has_sh' dest: procs_of_shm_prop1 procs_of_shm_prop2 procs_of_shm_prop3 simp:cph2spshs_def)apply (rule_tac x = h in exI, simp add:procs_of_shm_prop1)+donelemma cph2spshs_kill: "valid (Kill p p' # s) \<Longrightarrow> cph2spshs (Kill p p' # s) = (cph2spshs s) (p' := {})"apply (frule vd_cons, frule vt_grant_os, rule ext)using ch2sshm_other[where e = "Kill p p'" and s = s]apply (auto split del:t_open_must_flag.splits t_open_option_flag.splits split add:if_splits option.splitsdest!:current_shm_has_sh' dest: procs_of_shm_prop1 procs_of_shm_prop2 procs_of_shm_prop3 simp:cph2spshs_def)apply (rule_tac x = h in exI, simp add:procs_of_shm_prop1)+donelemma cph2spshs_exit: "valid (Exit p # s) \<Longrightarrow> cph2spshs (Exit p # s) = (cph2spshs s) (p := {})"apply (frule vd_cons, frule vt_grant_os, rule ext)using ch2sshm_other[where e = "Exit p" and s = s]apply (auto split del:t_open_must_flag.splits t_open_option_flag.splits split add:if_splits option.splitsdest!:current_shm_has_sh' dest: procs_of_shm_prop1 procs_of_shm_prop2 procs_of_shm_prop3 simp:cph2spshs_def)apply (rule_tac x = h in exI, simp add:procs_of_shm_prop1)+donelemma cph2spshs_createshm: "valid (CreateShM p h # s) \<Longrightarrow> cph2spshs (CreateShM p h # s) = cph2spshs s"apply (frule vt_grant_os, frule vd_cons, rule ext)apply (auto simp:cph2spshs_def)using ch2sshm_createshmapply (auto split del:t_open_must_flag.splits t_open_option_flag.splits split add:if_splits option.splitsdest!:current_shm_has_sh' dest: procs_of_shm_prop1 procs_of_shm_prop2 procs_of_shm_prop3 simp:cph2spshs_def)apply (rule_tac x = ha in exI, auto simp:procs_of_shm_prop1)donelemma cph2spshs_other: "\<lbrakk>valid (e # s); \<forall> p h flag. e \<noteq> Attach p h flag; \<forall> p h. e \<noteq> Detach p h; \<forall> p h. e \<noteq> DeleteShM p h; \<forall> p p' fds shms. e \<noteq> Clone p p' fds shms; \<forall> p f fds. e \<noteq> Execve p f fds; \<forall> p p'. e \<noteq> Kill p p'; \<forall> p. e \<noteq> Exit p\<rbrakk> \<Longrightarrow> cph2spshs (e # s) = cph2spshs s"apply (frule vt_grant_os, frule vd_cons, rule ext)unfolding cph2spshs_def apply (rule set_eqI, rule iffI)apply (erule CollectE | erule conjE| erule exE| rule conjI| rule impI| rule CollectI)+apply (frule procs_of_shm_prop1, simp, rule_tac x= sh in exI, rule_tac x = flag in exI, rule_tac x = h in exI)apply (case_tac e)using ch2sshm_other[where e = e and s = s] apply (auto split del:t_open_must_flag.splits t_open_option_flag.splits split add:if_splits option.splitsdest!:current_shm_has_sh' dest: procs_of_shm_prop1 procs_of_shm_prop2 procs_of_shm_prop3 simp:ch2sshm_createshm)apply (rule_tac x = h in exI, case_tac e)using ch2sshm_other[where e = e and s = s] apply (auto split del:t_open_must_flag.splits t_open_option_flag.splits split add:if_splits option.splitsdest!:current_shm_has_sh' dest: procs_of_shm_prop1 procs_of_shm_prop2 procs_of_shm_prop3 simp:ch2sshm_createshm)donelemmas cph2spshs_simps = cph2spshs_attach cph2spshs_detach cph2spshs_deleteshm cph2spshs_clone cph2spshs_execve cph2spshs_exit cph2spshs_kill cph2spshs_other(******** cp2sproc simpset *********)lemma cp2sproc_clone: "valid (Clone p p' fds shms # s) \<Longrightarrow> cp2sproc (Clone p p' fds shms # s) = (cp2sproc s) (p' := case (sectxt_of_obj s (O_proc p)) of Some sec \<Rightarrow> Some (Created, sec, {sfd. \<exists> fd \<in> fds. \<exists> f. file_of_proc_fd s p fd = Some f \<and> cfd2sfd s p fd = Some sfd}, {(sh, flag) | h sh flag. h \<in> shms \<and> ch2sshm s h = Some sh \<and> flag_of_proc_shm s p h = Some flag}) | _ \<Rightarrow> None)"apply (frule cph2spshs_clone, frule cpfd2sfds_clone)apply (frule vd_cons, frule vt_grant_os, simp only:os_grant.simps)apply ((erule exE| erule conjE)+, frule not_init_intro_proc, simp, rule ext, case_tac "x = p'", simp)apply (auto simp:cp2sproc_def sectxt_of_obj_simps dest!:current_has_sec' dest:current_proc_has_sec split:option.splits if_splits)donelemma cp2sproc_other: "\<lbrakk>valid (e # s); \<forall> p f flags fd opt. e \<noteq> Open p f flags fd opt; \<forall> p fd. e \<noteq> CloseFd p fd; \<forall> p h flag. e \<noteq> Attach p h flag; \<forall> p h. e \<noteq> Detach p h; \<forall> p h. e \<noteq> DeleteShM p h; \<forall> p p' fds shms. e \<noteq> Clone p p' fds shms; \<forall> p f fds. e \<noteq> Execve p f fds; \<forall> p p'. e \<noteq> Kill p p'; \<forall> p. e \<noteq> Exit p\<rbrakk> \<Longrightarrow> cp2sproc (e # s) = cp2sproc s"apply (frule cph2spshs_other, simp+, frule cpfd2sfds_other, simp+)apply (frule vt_grant_os, frule vd_cons, rule ext, case_tac e, simp_all) apply (auto simp:cp2sproc_def sectxt_of_obj_simps dest!:current_has_sec' dest:current_proc_has_sec not_deleted_init_proc split:option.splits if_splits)donelemma cp2sproc_open: "valid (Open p f flags fd opt # s) \<Longrightarrow> cp2sproc (Open p f flags fd opt # s) = (cp2sproc s) (p := case (sectxt_of_obj s (O_proc p), sectxt_of_obj (Open p f flags fd opt # s) (O_fd p fd), cf2sfile (Open p f flags fd opt # s) f) of (Some sec, Some fdsec, Some sf) \<Rightarrow> Some (if (\<not> deleted (O_proc p) s \<and> p \<in> init_procs) then Init p else Created, sec, (cpfd2sfds s p) \<union> {(fdsec, flags, sf)}, cph2spshs s p) | _ \<Rightarrow> None)"apply (frule cph2spshs_other, simp, simp, simp, simp, simp, simp, simp)apply (frule cpfd2sfds_open, frule vt_grant_os, frule vd_cons, rule ext)apply (case_tac "x = p") apply (auto simp:cp2sproc_def sectxt_of_obj_simps current_files_simps dest!:current_has_sec' dest!:current_file_has_sfile' dest:is_file_in_current is_dir_in_current dest:current_proc_has_sec not_deleted_init_proc split:option.splits if_splits)donelemma cp2sproc_closefd: "valid (CloseFd p fd # s) \<Longrightarrow> cp2sproc (CloseFd p fd # s) = (cp2sproc s) (p := if (fd \<in> proc_file_fds s p) then (case (cfd2sfd s p fd) of Some sfd \<Rightarrow> (if (\<exists> fd' f'. fd' \<noteq> fd \<and> file_of_proc_fd s p fd' = Some f' \<and> cfd2sfd s p fd' = Some sfd) then cp2sproc s p else (case (sectxt_of_obj s (O_proc p)) of Some sec \<Rightarrow> Some (if (\<not> deleted (O_proc p) s \<and> p \<in> init_procs) then Init p else Created, sec, cpfd2sfds s p - {sfd}, cph2spshs s p) | _ \<Rightarrow> None)) | _ \<Rightarrow> cp2sproc s p) else cp2sproc s p)"apply (frule cpfd2sfds_closefd, frule cph2spshs_other, simp, simp, simp, simp, simp, simp, simp)apply (frule vt_grant_os, frule vd_cons, rule ext)apply (case_tac "x = p") apply (auto simp:cp2sproc_def sectxt_of_obj_simps current_files_simps dest!:current_has_sec' dest!:current_file_has_sfile' dest:is_file_in_current is_dir_in_current dest:current_proc_has_sec not_deleted_init_proc split:option.splits if_splits)donelemma cp2sproc_attach: "valid (Attach p h flag # s) \<Longrightarrow> cp2sproc (Attach p h flag # s) = (cp2sproc s) (p := (case (ch2sshm s h) of Some sh \<Rightarrow> (case (sectxt_of_obj s (O_proc p)) of Some sec \<Rightarrow> Some (if (\<not> deleted (O_proc p) s \<and> p \<in> init_procs) then Init p else Created, sec, cpfd2sfds s p, cph2spshs s p \<union> {(sh, flag)}) | _ \<Rightarrow> None) | _ \<Rightarrow> cp2sproc s p) )"apply (frule cph2spshs_attach, frule cpfd2sfds_other, simp, simp, simp, simp, simp, simp)apply (frule vt_grant_os, frule vd_cons, rule ext)apply (case_tac "x = p") apply (auto simp:cp2sproc_def sectxt_of_obj_simps current_files_simps dest!:current_has_sec' dest!:current_file_has_sfile' dest:is_file_in_current is_dir_in_current dest:current_proc_has_sec not_deleted_init_proc split:option.splits if_splits)donelemma cp2sproc_detach: "valid (Detach p h # s) \<Longrightarrow> cp2sproc (Detach p h # s) = (cp2sproc s) (p := (case (ch2sshm s h, flag_of_proc_shm s p h) of (Some sh, Some flag) \<Rightarrow> if (\<exists> h'. h' \<noteq> h \<and> (p,flag) \<in> procs_of_shm s h' \<and> ch2sshm s h' = Some sh) then cp2sproc s p else (case (sectxt_of_obj s (O_proc p)) of Some sec \<Rightarrow> Some (if (\<not> deleted (O_proc p) s \<and> p \<in> init_procs) then Init p else Created, sec, cpfd2sfds s p, cph2spshs s p - {(sh, flag)}) | None \<Rightarrow> None) | _ \<Rightarrow> cp2sproc s p) )"apply (frule cph2spshs_detach, frule cpfd2sfds_other, simp, simp, simp, simp, simp, simp)apply (frule vt_grant_os, frule vd_cons, rule ext)apply (case_tac "x = p") apply (auto simp:cp2sproc_def sectxt_of_obj_simps current_files_simps dest!:current_has_sec' dest!:current_file_has_sfile' dest:is_file_in_current is_dir_in_current dest:current_proc_has_sec not_deleted_init_proc split:option.splits if_splits)donelemma cp2sproc_deleteshm: "valid (DeleteShM p h # s) \<Longrightarrow> cp2sproc (DeleteShM p h # s) = (\<lambda> p'. (case (ch2sshm s h, flag_of_proc_shm s p' h) of (Some sh, Some flag) \<Rightarrow> if (\<exists> h'. h' \<noteq> h \<and> (p', flag) \<in> procs_of_shm s h' \<and> ch2sshm s h' = Some sh) then cp2sproc s p' else (case (sectxt_of_obj s (O_proc p')) of Some sec \<Rightarrow> Some (if (\<not> deleted (O_proc p') s \<and> p' \<in> init_procs) then Init p' else Created, sec, cpfd2sfds s p', cph2spshs s p' - {(sh, flag)}) | None \<Rightarrow> None) | _ \<Rightarrow> cp2sproc s p') )"apply (frule cph2spshs_deleteshm, frule cpfd2sfds_other, simp, simp, simp, simp, simp, simp)apply (frule vt_grant_os, frule vd_cons, rule ext)apply (auto simp:cp2sproc_def sectxt_of_obj_simps current_files_simps dest!:current_has_sec' dest!:current_file_has_sfile' dest:is_file_in_current is_dir_in_current dest:current_proc_has_sec not_deleted_init_proc split:option.splits if_splits)donelemma cp2sproc_execve: "valid (Execve p f fds # s) \<Longrightarrow> cp2sproc (Execve p f fds # s) = (cp2sproc s) (p := (case (sectxt_of_obj (Execve p f fds # s) (O_proc p)) of Some sec \<Rightarrow> Some (if (\<not> deleted (O_proc p) s \<and> p \<in> init_procs) then Init p else Created, sec, {sfd. \<exists> fd \<in> fds. \<exists> f. file_of_proc_fd s p fd = Some f \<and> cfd2sfd s p fd = Some sfd}, {}) | _ \<Rightarrow> None) )"apply (frule cph2spshs_execve, frule cpfd2sfds_execve)apply (frule vt_grant_os, frule vd_cons, rule ext)apply (auto simp:cp2sproc_def sectxt_of_obj_simps current_files_simps dest!:current_has_sec' dest!:current_file_has_sfile' dest:is_file_in_current is_dir_in_current dest:current_proc_has_sec not_deleted_init_proc split:option.splits if_splits)donelemma cp2sproc_kill: "\<lbrakk>valid (Kill p p' # s); p'' \<noteq> p'\<rbrakk> \<Longrightarrow> cp2sproc (Kill p p' # s) p'' = (cp2sproc s) p''"apply (frule cph2spshs_kill, frule cpfd2sfds_kill)apply (frule vt_grant_os, frule vd_cons)apply (auto simp:cp2sproc_def sectxt_of_obj_simps current_files_simps dest!:current_has_sec' dest!:current_file_has_sfile' dest:is_file_in_current is_dir_in_current dest:current_proc_has_sec not_deleted_init_proc split:option.splits if_splits)donelemma cp2sproc_kill': "\<lbrakk>valid (Kill p p' # s); p'' \<in> current_procs (Kill p p' # s)\<rbrakk> \<Longrightarrow> cp2sproc (Kill p p' # s) p'' = (cp2sproc s) p''"by (drule_tac p'' = p'' in cp2sproc_kill, simp+)lemma cp2sproc_exit: "\<lbrakk>valid (Exit p # s); p' \<noteq> p\<rbrakk> \<Longrightarrow> cp2sproc (Exit p # s) p' = (cp2sproc s) p'"apply (frule cph2spshs_exit, frule cpfd2sfds_exit)apply (frule vt_grant_os, frule vd_cons)apply (auto simp:cp2sproc_def sectxt_of_obj_simps current_files_simps dest!:current_has_sec' dest!:current_file_has_sfile' dest:is_file_in_current is_dir_in_current dest:current_proc_has_sec not_deleted_init_proc split:option.splits if_splits)donelemma cp2sproc_exit': "\<lbrakk>valid (Exit p # s); p' \<in> current_procs (Exit p # s)\<rbrakk> \<Longrightarrow> cp2sproc (Exit p # s) p' = (cp2sproc s) p'"by (drule_tac p'= p' in cp2sproc_exit, simp+)lemmas cp2sproc_simps = cp2sproc_open cp2sproc_closefd cp2sproc_attach cp2sproc_detach cp2sproc_deleteshm cp2sproc_clone cp2sproc_execve cp2sproc_kill cp2sproc_exit cp2sproc_other lemma current_proc_has_sp: "\<lbrakk>p \<in> current_procs s; valid s\<rbrakk> \<Longrightarrow> \<exists> sp. cp2sproc s p = Some sp"by (auto simp:cp2sproc_def split:option.splits dest!:current_has_sec')lemma current_proc_has_sp': "\<lbrakk>cp2sproc s p = None; valid s\<rbrakk> \<Longrightarrow> p \<notin> current_procs s"by (auto dest:current_proc_has_sp)(* simpset for cf2sfiles *)lemma cf2sfiles_open: "\<lbrakk>valid (Open p f flag fd opt # s); f' \<in> current_files (Open p f flag fd opt # s)\<rbrakk> \<Longrightarrow> cf2sfiles (Open p f flag fd opt # s) f' = ( if (f' = f \<and> opt \<noteq> None) then (case cf2sfile (Open p f flag fd opt # s) f of Some sf \<Rightarrow> {sf} | _ \<Rightarrow> {} ) else cf2sfiles s f')"apply (frule vt_grant_os, frule vd_cons)apply (auto simp:cf2sfiles_def cf2sfile_open_none cf2sfile_simps same_inode_files_open split:if_splits option.splits dest!:current_file_has_sfile' dest:cf2sfile_open)apply (rule_tac x = "f'a" in bexI, drule same_inode_files_prop1, simp add:cf2sfile_open_some1, simp)+donelemma cf2sfiles_other: "\<lbrakk>valid (e # s); \<forall> p f flag fd opt. e \<noteq> Open p f flag fd opt; \<forall> p fd. e \<noteq> CloseFd p fd; \<forall> p f. e \<noteq> UnLink p f; \<forall> p f f'. e \<noteq> LinkHard p f f'\<rbrakk> \<Longrightarrow> cf2sfiles (e # s) = cf2sfiles s"apply (frule vt_grant_os, frule vd_cons, rule ext)apply (simp add:cf2sfiles_def, rule set_eqI, rule iffI)apply (drule Set.CollectD, erule bexE, rule CollectI)apply (rule_tac x = f' in bexI, case_tac e)apply (auto simp:cf2sfiles_def cf2sfile_simps same_inode_files_simps current_files_simps split:if_splits option.splits dest!:current_file_has_sfile' dest:same_inode_files_prop1 cf2sfile_other')apply (drule_tac f' = f' in cf2sfile_rmdir)apply (simp add:current_files_simps same_inode_files_prop1 same_inode_files_prop3 dir_is_empty_def)+apply (rule_tac x = f' in bexI, case_tac e)apply (auto simp:cf2sfiles_def cf2sfile_simps same_inode_files_simps current_files_simps split:if_splits option.splits dest!:current_file_has_sfile' dest:same_inode_files_prop1 cf2sfile_other')apply (drule_tac f' = f' in cf2sfile_rmdir)apply (simp add:current_files_simps same_inode_files_prop1 same_inode_files_prop3 dir_is_empty_def)+donelemma cf2sfile_linkhard1': "\<lbrakk>valid (LinkHard p oldf f # s); f' \<in> same_inode_files s f''\<rbrakk> \<Longrightarrow> cf2sfile (LinkHard p oldf f# s) f' = cf2sfile s f'"apply (drule same_inode_files_prop1)by (simp add:cf2sfile_linkhard1)lemma cf2sfiles_linkhard: "valid (LinkHard p oldf f # s) \<Longrightarrow> cf2sfiles (LinkHard p oldf f # s) = (\<lambda> f'. if (f' = f \<or> f' \<in> same_inode_files s oldf) then (case (cf2sfile (LinkHard p oldf f # s) f) of Some sf \<Rightarrow> cf2sfiles s oldf \<union> {sf} | _ \<Rightarrow> {}) else cf2sfiles s f')"apply (frule vt_grant_os, frule vd_cons, rule ext)apply (auto simp:cf2sfiles_def cf2sfile_linkhard1' same_inode_files_linkhard current_files_linkhard split:if_splits option.splits dest!:current_file_has_sfile' current_has_sec' dest:same_inode_files_prop1)donelemma cf2sfile_unlink': "\<lbrakk>valid (UnLink p f # s); f' \<in> same_inode_files (UnLink p f # s) f''\<rbrakk> \<Longrightarrow> cf2sfile (UnLink p f # s) f' = cf2sfile s f'"apply (drule same_inode_files_prop1)by (simp add:cf2sfile_unlink)lemma cf2sfiles_unlink: "\<lbrakk>valid (UnLink p f # s); f' \<in> current_files (UnLink p f # s)\<rbrakk> \<Longrightarrow> cf2sfiles (UnLink p f # s) f' = ( if (f' \<in> same_inode_files s f \<and> proc_fd_of_file s f = {} \<and> (\<forall> f'' \<in> same_inode_files s f. f'' \<noteq> f \<longrightarrow> cf2sfile s f'' \<noteq> cf2sfile s f)) then (case (cf2sfile s f) of Some sf \<Rightarrow> cf2sfiles s f' - {sf} | _ \<Rightarrow> {}) else cf2sfiles s f')"apply (frule vt_grant_os, frule vd_cons, simp add:current_files_simps split:if_splits)apply (rule conjI, clarify, frule is_file_has_sfile', simp, erule exE, simp)apply (simp add:cf2sfiles_def, rule set_eqI, rule iffI, drule CollectD)apply (erule bexE, frule_tac f' = f' in same_inode_files_unlink)apply (simp add:current_files_unlink, simp, erule conjE)apply (erule_tac x = f'a in ballE, frule_tac f' = "f'a" in cf2sfile_unlink)apply (simp add:current_files_unlink same_inode_files_prop1, simp)apply (rule_tac x = f'a in bexI, simp, simp)apply (drule_tac f = f and f' = f' and f'' = f'a in same_inode_files_prop4, simp+)apply (erule conjE|erule exE|erule bexE)+apply (case_tac "f'a = f", simp)apply (frule_tac f' = f' in same_inode_files_unlink, simp add:current_files_unlink)apply (frule_tac f' = f'a in cf2sfile_unlink, simp add:current_files_unlink same_inode_files_prop1)apply (rule_tac x = f'a in bexI, simp, simp)apply (rule impI)+apply (simp add:cf2sfiles_def, rule set_eqI, rule iffI, drule CollectD)apply (erule bexE, frule_tac f' = f' in same_inode_files_unlink)apply (simp add:current_files_unlink, simp, (erule conjE)+)apply (rule_tac x = f'a in bexI, frule_tac f' = "f'a" in cf2sfile_unlink)apply (simp add:current_files_unlink same_inode_files_prop1, simp, simp)apply (drule CollectD, erule bexE, frule_tac f' = f' in same_inode_files_unlink)apply (simp add:current_files_unlink, simp)apply (case_tac "f'a = f", simp)apply (frule_tac f = f' and f' = f in same_inode_files_prop5, simp)apply (erule bexE, erule conjE)apply (rule_tac x = f'' in bexI)apply (drule_tac f' = f'' in cf2sfile_unlink, simp add:current_files_unlink same_inode_files_prop1)apply (simp, simp, erule same_inode_files_prop4, simp)apply (rule_tac x = f'a in bexI)apply (drule_tac f' = f'a in cf2sfile_unlink, simp add:current_files_unlink same_inode_files_prop1)apply (simp, simp)apply (simp add:cf2sfiles_def, rule set_eqI, rule iffI, drule CollectD)apply (erule bexE, frule_tac f' = f' in same_inode_files_unlink)apply (simp add:current_files_unlink, simp)apply (rule_tac x = f'a in bexI)apply (frule_tac f' = f'a in cf2sfile_unlink)apply (simp add:same_inode_files_prop1 current_files_unlink, simp, simp)apply (drule CollectD, erule bexE, frule_tac f' = f' in same_inode_files_unlink)apply (simp add:current_files_unlink, simp)apply (rule_tac x = f'a in bexI)apply (frule_tac f' = f'a in cf2sfile_unlink)apply (simp add:same_inode_files_prop1 current_files_unlink, simp, simp)donelemma cf2sfiles_closefd: "\<lbrakk>valid (CloseFd p fd # s); f' \<in> current_files (CloseFd p fd # s)\<rbrakk> \<Longrightarrow> cf2sfiles (CloseFd p fd # s) f' = ( case (file_of_proc_fd s p fd) of Some f \<Rightarrow> if (f' \<in> same_inode_files s f \<and> proc_fd_of_file s f = {(p, fd)} \<and> f \<in> files_hung_by_del s \<and> (\<forall> f'' \<in> same_inode_files s f. f'' \<noteq> f \<longrightarrow> cf2sfile s f'' \<noteq> cf2sfile s f)) then (case (cf2sfile s f) of Some sf \<Rightarrow> cf2sfiles s f' - {sf} | _ \<Rightarrow> {}) else cf2sfiles s f' | _ \<Rightarrow> cf2sfiles s f')"apply (frule vt_grant_os, frule vd_cons, case_tac "file_of_proc_fd s p fd")apply (simp_all add:current_files_simps split:if_splits)apply (simp add:cf2sfiles_def, rule set_eqI, rule iffI, drule CollectD)apply (erule bexE, frule_tac f' = f' in same_inode_files_closefd)apply (simp add:current_files_closefd, simp)apply (rule_tac x = f'a in bexI)apply (frule_tac f = f'a in cf2sfile_closefd)apply (simp add:same_inode_files_prop1 current_files_closefd, simp, simp)apply (drule CollectD, erule bexE, frule_tac f' = f' in same_inode_files_closefd)apply (simp add:current_files_closefd, simp)apply (rule_tac x = f'a in bexI)apply (frule_tac f = f'a in cf2sfile_closefd)apply (simp add:same_inode_files_prop1 current_files_closefd, simp, simp)apply (rule conjI, clarify, frule file_of_pfd_is_file, simp)apply (frule is_file_has_sfile', simp, erule exE, simp)apply (simp add:cf2sfiles_def, rule set_eqI, rule iffI, drule CollectD)apply (erule bexE, frule_tac f' = f' in same_inode_files_closefd)apply (simp add:current_files_closefd, simp, erule conjE)apply (erule_tac x = f'a in ballE, frule_tac f = "f'a" in cf2sfile_closefd)apply (simp add:current_files_closefd same_inode_files_prop1, simp)apply (rule_tac x = f'a in bexI, simp, simp)apply (drule_tac f = a and f' = f' and f'' = f'a in same_inode_files_prop4, simp+)apply (erule conjE|erule exE|erule bexE)+apply (case_tac "f'a = a", simp)apply (frule_tac f' = f' in same_inode_files_closefd, simp add:current_files_closefd, simp)apply (frule_tac f = f'a in cf2sfile_closefd, simp add:current_files_closefd same_inode_files_prop1)apply (rule_tac x = f'a in bexI, simp, simp)apply (rule impI)+apply (simp add:cf2sfiles_def, rule set_eqI, rule iffI, drule CollectD)apply (erule bexE, frule_tac f' = f' in same_inode_files_closefd)apply (simp add:current_files_closefd, simp, (erule conjE)+)apply (rule_tac x = f'a in bexI, frule_tac f = f'a in cf2sfile_closefd)apply (simp add:current_files_closefd same_inode_files_prop1, simp, simp)apply (drule CollectD, erule bexE, frule_tac f' = f' in same_inode_files_closefd)apply (simp add:current_files_closefd, simp)apply (case_tac "f'a = a", simp)apply (frule_tac f = f' and f' = a in same_inode_files_prop5, simp)apply (erule bexE, erule conjE)apply (rule_tac x = f'' in bexI)apply (drule_tac f = f'' in cf2sfile_closefd, simp add:current_files_closefd same_inode_files_prop1)apply (simp, simp, erule same_inode_files_prop4, simp)apply (rule_tac x = f'a in bexI)apply (drule_tac f = f'a in cf2sfile_closefd, simp add:current_files_closefd same_inode_files_prop1)apply (simp, simp)apply (rule conjI, clarify)apply (rule impI)apply (case_tac "a \<in> files_hung_by_del s", simp_all)apply (simp add:cf2sfiles_def, rule set_eqI, rule iffI, drule CollectD)apply (erule bexE, frule_tac f' = f' in same_inode_files_closefd)apply (simp add:current_files_closefd, simp)apply (rule_tac x = f'a in bexI)apply (frule_tac f = f'a in cf2sfile_closefd)apply (simp add:same_inode_files_prop1 current_files_closefd, simp, simp)apply (drule CollectD, erule bexE, frule_tac f' = f' in same_inode_files_closefd)apply (simp add:current_files_closefd, simp)apply (rule_tac x = f'a in bexI)apply (frule_tac f = f'a in cf2sfile_closefd)apply (simp add:same_inode_files_prop1 current_files_closefd, simp, simp)apply (simp add:cf2sfiles_def, rule set_eqI, rule iffI, drule CollectD)apply (erule bexE, frule_tac f' = f' in same_inode_files_closefd)apply (simp add:current_files_closefd, simp)apply (rule_tac x = f'a in bexI)apply (frule_tac f = f'a in cf2sfile_closefd)apply (simp add:same_inode_files_prop1 current_files_closefd, simp, simp)apply (drule CollectD, erule bexE, frule_tac f' = f' in same_inode_files_closefd)apply (simp add:current_files_closefd, simp)apply (rule_tac x = f'a in bexI)apply (frule_tac f = f'a in cf2sfile_closefd)apply (simp add:same_inode_files_prop1 current_files_closefd, simp, simp)donelemmas cf2sfiles_simps = cf2sfiles_open cf2sfiles_linkhard cf2sfiles_other cf2sfiles_unlink cf2sfiles_closefd(* simpset for co2sobj *)lemma co2sobj_execve: "\<lbrakk>valid (Execve p f fds # s); alive (Execve p f fds # s) obj\<rbrakk> \<Longrightarrow> co2sobj (Execve p f fds # s) obj = ( if (obj = O_proc p) then (case (cp2sproc (Execve p f fds # s) p) of Some sp \<Rightarrow> Some (S_proc sp (O_proc p \<in> Tainted s \<or> O_file f \<in> Tainted s)) | _ \<Rightarrow> None) else co2sobj s obj )"apply (frule vt_grant_os, frule vd_cons, case_tac obj)apply (simp_all add:current_files_simps cf2sfiles_other ch2sshm_other cq2smsgq_other tainted_eq_Tainted)apply (case_tac "cp2sproc (Execve p f fds # s) p")apply (drule current_proc_has_sp', simp, simp)apply (simp (no_asm_simp) add:cp2sproc_execve tainted_eq_Tainted split:option.splits)apply (simp add:is_dir_simps, frule_tac s = s in is_dir_has_sdir', simp, erule exE, simp)apply (frule_tac ff = list in cf2sfile_other', simp_all)apply (simp add:is_dir_in_current)donelemma co2sobj_execve': "\<lbrakk>valid (Execve p f fds # s); alive s obj\<rbrakk> \<Longrightarrow> co2sobj (Execve p f fds # s) obj = ( if (obj = O_proc p) then (case (cp2sproc (Execve p f fds # s) p) of Some sp \<Rightarrow> Some (S_proc sp (O_proc p \<in> Tainted s \<or> O_file f \<in> Tainted s)) | _ \<Rightarrow> None) else co2sobj s obj )"apply (frule vt_grant_os, frule vd_cons, case_tac obj)apply (simp_all add:current_files_simps cf2sfiles_other ch2sshm_other cq2smsgq_other tainted_eq_Tainted)apply (case_tac "cp2sproc (Execve p f fds # s) p")apply (drule current_proc_has_sp', simp, simp)apply (simp (no_asm_simp) add:cp2sproc_execve tainted_eq_Tainted split:option.splits)apply (frule_tac s = s in is_dir_has_sdir', simp, erule exE, simp)apply (frule_tac ff = list in cf2sfile_other', simp_all)apply (simp add:is_dir_in_current)donelemma co2sobj_clone': "\<lbrakk>valid (Clone p p' fds shms # s); alive s obj\<rbrakk> \<Longrightarrow> co2sobj (Clone p p' fds shms # s) obj = ( if (obj = O_proc p') then (case (cp2sproc (Clone p p' fds shms # 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, case_tac obj)apply (simp_all add:current_files_simps cf2sfiles_other ch2sshm_other cq2smsgq_other tainted_eq_Tainted)apply (case_tac "cp2sproc (Clone p p' fds shms # s) p'")apply (drule current_proc_has_sp', simp, simp)apply ((erule conjE)+, frule_tac p = p in current_proc_has_sec, simp, erule exE, simp)apply (rule conjI, rule impI, simp)apply (simp (no_asm_simp) add:cp2sproc_clone tainted_eq_Tainted split:option.splits)apply (frule_tac s = s in is_dir_has_sdir', simp, erule exE, simp)apply (frule_tac ff = list in cf2sfile_other', simp_all)apply (simp add:is_dir_in_current)donelemma co2sobj_clone: "\<lbrakk>valid (Clone p p' fds shms # s); alive (Clone p p' fds shms # s) obj\<rbrakk> \<Longrightarrow> co2sobj (Clone p p' fds shms # s) obj = ( if (obj = O_proc p') then (case (cp2sproc (Clone p p' fds shms # 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, case_tac obj)apply (simp_all add:current_files_simps cf2sfiles_other ch2sshm_other cq2smsgq_other tainted_eq_Tainted)apply (rule conjI, rule impI, simp)apply (case_tac "cp2sproc (Clone p p' fds shms # s) p'")apply (drule current_proc_has_sp', simp, simp)apply ((erule conjE)+, frule_tac p = p in current_proc_has_sec, simp, erule exE, simp)apply (simp add:tainted_eq_Tainted, rule impI, rule notI)apply (drule Tainted_in_current, simp+)apply (rule impI, simp)apply (drule current_proc_has_sp, simp, (erule exE|erule conjE)+)apply (simp (no_asm_simp) add:cp2sproc_clone tainted_eq_Tainted Tainted_in_current split:option.splits)apply (simp add:is_dir_simps, frule_tac s = s in is_dir_has_sdir', simp, erule exE, simp)apply (frule_tac ff = list in cf2sfile_other', simp_all)apply (simp add:is_dir_in_current)donelemma co2sobj_ptrace: "\<lbrakk>valid (Ptrace p p' # s); alive s obj\<rbrakk>\<Longrightarrow> co2sobj (Ptrace p p' # s) obj = ( case obj of O_proc p'' \<Rightarrow> if (info_flow_shm s p' p'') then (case (cp2sproc s p'') of Some sp \<Rightarrow> Some (S_proc sp (O_proc p'' \<in> Tainted s \<or> O_proc p \<in> Tainted s)) | _ \<Rightarrow> None) else if (info_flow_shm s p p'') then (case (cp2sproc s p'') of Some sp \<Rightarrow> Some (S_proc sp (O_proc p'' \<in> Tainted s \<or> O_proc p' \<in> Tainted s)) | _ \<Rightarrow> None) else co2sobj s (O_proc p'') | _ \<Rightarrow> co2sobj s obj)"apply (frule vt_grant_os, frule vd_cons, case_tac obj)apply (simp_all add:current_files_simps ch2sshm_other cq2smsgq_other cf2sfiles_other tainted_eq_Tainted)apply (auto simp:cp2sproc_other tainted_eq_Tainted split:option.splits dest!:current_proc_has_sec' current_proc_has_sp' intro:info_flow_shm_Tainted)[1]apply (frule_tac s = s in is_dir_has_sdir', simp, erule exE, simp)apply (frule_tac ff = list in cf2sfile_other', simp_all)apply (simp add:is_dir_in_current)donelemma co2sobj_open: "\<lbrakk>valid (Open p f flag fd opt # s); alive (Open p f flag fd opt # s) obj\<rbrakk> \<Longrightarrow> co2sobj (Open p f flag fd opt # s) obj = (case obj of O_file f' \<Rightarrow> if (f' = f \<and> opt \<noteq> None) then (case (cf2sfile (Open p f flag fd opt # s) f) of Some sf \<Rightarrow> Some (S_file {sf} (O_proc p \<in> Tainted s)) | _ \<Rightarrow> None) else co2sobj s (O_file f') | O_proc p' \<Rightarrow> if (p' = p) then (case (cp2sproc (Open p f flag fd opt # s) p) of Some sp \<Rightarrow> Some (S_proc sp (O_proc p \<in> Tainted s)) | _ \<Rightarrow> None) else co2sobj s (O_proc p') | _ \<Rightarrow> co2sobj s obj )"apply (frule vt_grant_os, frule vd_cons, case_tac obj)apply (auto simp:cp2sproc_simps tainted_eq_Tainted split:option.splits dest!:current_proc_has_sp' intro:info_flow_shm_Tainted)[1]apply (simp split:if_splits t_object.splits)apply (rule conjI, rule impI, erule conjE, erule exE, simp, (erule exE|erule conjE)+)apply (case_tac "cf2sfile (Open p f flag fd (Some y) # s) f")apply (drule current_file_has_sfile', simp, simp add:current_files_simps, simp)apply (frule_tac f' = f in cf2sfiles_open, simp add:current_files_simps)apply (simp add:tainted_eq_Tainted)apply (rule impI, rule notI, drule Tainted_in_current, simp, simp add:is_file_in_current)apply (rule impI, simp add:tainted_eq_Tainted cf2sfiles_open is_file_in_current split:option.splits)apply (simp_all add:current_files_simps is_dir_simps ch2sshm_other cq2smsgq_other tainted_eq_Tainted)apply (frule is_dir_in_current)apply (frule_tac f' = list in cf2sfile_open)apply (simp add:current_files_simps split:option.splits)apply (simp split:if_splits option.splits)donelemma co2sobj_readfile: "\<lbrakk>valid (ReadFile p fd # s); alive s obj\<rbrakk> \<Longrightarrow> co2sobj (ReadFile p fd # s) obj = ( case obj of O_proc p' \<Rightarrow> (case (file_of_proc_fd s p fd) of Some f \<Rightarrow> (if (info_flow_shm s p p' \<and> O_file f \<in> Tainted s) then (case (cp2sproc s p') of Some sp \<Rightarrow> Some (S_proc sp True) | _ \<Rightarrow> None) else co2sobj s obj) | _ \<Rightarrow> None) | _ \<Rightarrow> co2sobj s obj)"apply (frule vt_grant_os, frule vd_cons, case_tac obj)apply (auto simp:cp2sproc_simps tainted_eq_Tainted split:option.splits dest!:current_proc_has_sp' intro:info_flow_shm_Tainted)[1]apply (simp_all add:current_files_simps is_dir_simps ch2sshm_other cq2smsgq_other tainted_eq_Tainted)apply (auto split:if_splits option.splits dest!:current_file_has_sfile' simp:current_files_simps cf2sfiles_simps cf2sfile_simps dest:is_file_in_current is_dir_in_current)donelemma co2sobj_writefile: "\<lbrakk>valid (WriteFile p fd # s); alive s obj\<rbrakk> \<Longrightarrow> co2sobj (WriteFile p fd # s) obj = ( case obj of O_file f' \<Rightarrow> (case (file_of_proc_fd s p fd) of Some f \<Rightarrow> (if (f \<in> same_inode_files s f') then Some (S_file (cf2sfiles s f') (O_file f' \<in> Tainted s \<or> O_proc p \<in> Tainted s)) else co2sobj s obj) | _ \<Rightarrow> None) | _ \<Rightarrow> co2sobj s obj)"apply (frule vt_grant_os, frule vd_cons, case_tac obj)apply (simp_all add:current_files_simps is_dir_simps ch2sshm_other cq2smsgq_other tainted_eq_Tainted)apply (auto split:if_splits option.splits dest!:current_file_has_sfile' current_proc_has_sp' simp:current_files_simps cf2sfiles_simps cf2sfile_simps cp2sproc_simps tainted_eq_Tainted same_inode_files_prop6 dest:is_file_in_current is_dir_in_current)(* should delete has_same_inode !?!?*)by (auto simp:same_inode_files_def is_file_def has_same_inode_def split:if_splits option.splits)lemma co2sobj_closefd: "\<lbrakk>valid (CloseFd p fd # s); alive (CloseFd p fd # s) obj\<rbrakk> \<Longrightarrow> co2sobj (CloseFd p fd # s) obj = ( case obj of O_file f' \<Rightarrow> (case (file_of_proc_fd s p fd) of Some f \<Rightarrow> (if (f' \<in> same_inode_files s f \<and> proc_fd_of_file s f = {(p, fd)} \<and> f \<in> files_hung_by_del s \<and> (\<forall> f'' \<in> same_inode_files s f. f'' \<noteq> f \<longrightarrow> cf2sfile s f'' \<noteq> cf2sfile s f)) then (case (cf2sfile s f, co2sobj s (O_file f')) of (Some sf, Some (S_file sfs b)) \<Rightarrow> Some (S_file (sfs - {sf}) b) | _ \<Rightarrow> None) else co2sobj s obj) | _ \<Rightarrow> co2sobj s obj) | O_proc p' \<Rightarrow> (if (p = p') then (case (cp2sproc (CloseFd p fd # s) p) of Some sp \<Rightarrow> Some (S_proc sp (O_proc p \<in> Tainted s)) | _ \<Rightarrow> None) else co2sobj s obj) | _ \<Rightarrow> co2sobj s obj) "apply (frule vt_grant_os, frule vd_cons, case_tac obj)apply (simp_all add:current_files_simps ch2sshm_other cq2smsgq_other tainted_eq_Tainted)apply (auto simp:cp2sproc_simps tainted_eq_Tainted split:option.splits if_splits dest!:current_proc_has_sp' intro:info_flow_shm_Tainted)[1]apply (frule is_file_in_current)apply (case_tac "file_of_proc_fd s p fd")apply (simp add:tainted_eq_Tainted)apply (drule_tac f' = list in cf2sfiles_closefd, simp add:current_files_closefd, simp)apply (frule_tac f' = list in cf2sfiles_closefd, simp)apply (simp add:is_file_simps current_files_simps)apply (auto simp add:tainted_eq_Tainted cf2sfile_closefd split:if_splits option.splits dest!:current_file_has_sfile' dest:hung_file_in_current)[1]apply (simp add:is_dir_simps, frule is_dir_in_current)apply (frule_tac f = list in cf2sfile_closefd)apply (clarsimp simp:current_files_closefd split:option.splits)apply (drule file_of_pfd_is_file', simp+)donelemma co2sobj_unlink: "\<lbrakk>valid (UnLink p f # s); alive (UnLink p f # s) obj\<rbrakk> \<Longrightarrow> co2sobj (UnLink p f # s) obj = ( case obj of O_file f' \<Rightarrow> if (f' \<in> same_inode_files s f \<and> proc_fd_of_file s f = {} \<and> (\<forall> f'' \<in> same_inode_files s f. f'' \<noteq> f \<longrightarrow> cf2sfile s f'' \<noteq> cf2sfile s f)) then (case (cf2sfile s f, co2sobj s (O_file f')) of (Some sf, Some (S_file sfs b)) \<Rightarrow> Some (S_file (sfs - {sf}) b) | _ \<Rightarrow> None) else co2sobj s obj | _ \<Rightarrow> co2sobj s obj)"apply (frule vt_grant_os, frule vd_cons, case_tac obj)apply (simp_all add:current_files_simps ch2sshm_other cq2smsgq_other tainted_eq_Tainted)apply (auto simp:cp2sproc_simps tainted_eq_Tainted split:option.splits if_splits dest!:current_proc_has_sp' intro:info_flow_shm_Tainted)[1]apply (frule is_file_in_current)apply (frule_tac f' = list in cf2sfile_unlink, simp)apply (frule_tac f' = list in cf2sfiles_unlink, simp)apply (simp add:is_file_simps current_files_simps)apply (auto simp add:tainted_eq_Tainted is_file_in_current split:if_splits option.splits dest!:current_file_has_sfile')[1]apply (simp add:is_dir_simps, frule is_dir_in_current)apply (frule_tac f' = list in cf2sfile_unlink)apply (clarsimp simp:current_files_unlink split:option.splits)apply (drule file_dir_conflict, simp+)donelemma co2sobj_rmdir: "\<lbrakk>valid (Rmdir p f # s); alive (Rmdir p f # s) obj\<rbrakk> \<Longrightarrow> co2sobj (Rmdir p f # s) obj = co2sobj s obj"apply (frule vt_grant_os, frule vd_cons, case_tac obj)apply (simp_all add:current_files_simps ch2sshm_other cq2smsgq_other tainted_eq_Tainted)apply (auto simp:cp2sproc_simps tainted_eq_Tainted split:option.splits if_splits dest!:current_proc_has_sp' intro:info_flow_shm_Tainted)[1]apply (simp add:is_file_simps dir_is_empty_def)apply (case_tac "f = list", drule file_dir_conflict, simp, simp)apply (simp add:cf2sfiles_other)apply (auto simp:cf2sfile_simps dest:is_dir_in_current)donelemma co2sobj_mkdir: "\<lbrakk>valid (Mkdir p f i # s); alive (Mkdir p f i # s) obj\<rbrakk> \<Longrightarrow> co2sobj (Mkdir p f i # s) obj = ( if (obj = O_dir f) then (case (cf2sfile (Mkdir p f i # s) f) of Some sf \<Rightarrow> Some (S_dir sf) | _ \<Rightarrow> None) else co2sobj s obj)"apply (frule vt_grant_os, frule vd_cons, case_tac obj)apply (simp_all add:current_files_simps ch2sshm_other cq2smsgq_other tainted_eq_Tainted)apply (auto simp:cp2sproc_simps tainted_eq_Tainted split:option.splits if_splits dest!:current_proc_has_sp' intro:info_flow_shm_Tainted)[1]apply (frule_tac cf2sfiles_other, simp+)apply (frule is_dir_in_current, rule impI, simp add:current_files_mkdir)apply (frule current_file_has_sfile, simp, erule exE, simp)apply (drule cf2sfile_mkdir1, simp+)donelemma co2sobj_linkhard: "\<lbrakk>valid (LinkHard p oldf f # s); alive (LinkHard p oldf f # s) obj\<rbrakk> \<Longrightarrow> co2sobj (LinkHard p oldf f # s) obj = ( case obj of O_file f' \<Rightarrow> if (f' = f \<or> f' \<in> same_inode_files s oldf) then (case (cf2sfile (LinkHard p oldf f # s) f) of Some sf \<Rightarrow> Some (S_file (cf2sfiles s oldf \<union> {sf}) (O_file oldf \<in> Tainted s)) | _ \<Rightarrow> None) else co2sobj s obj | _ \<Rightarrow> co2sobj s obj)"apply (frule vt_grant_os, frule vd_cons, case_tac obj)apply (simp_all add:current_files_simps ch2sshm_other cq2smsgq_other tainted_eq_Tainted)apply (auto simp:cp2sproc_simps tainted_eq_Tainted split:option.splits if_splits dest!:current_proc_has_sp' intro:info_flow_shm_Tainted)[1]apply (frule_tac cf2sfiles_linkhard, simp+)apply (frule_tac f' = f in cf2sfile_linkhard, simp add:current_files_linkhard)apply (auto simp:is_file_simps sectxt_of_obj_simps current_files_simps is_file_in_current same_inodes_Tainted split:option.splits if_splits dest:Tainted_in_current dest!:current_has_sec' current_file_has_sfile')[1]apply (frule is_dir_in_current, simp add:current_files_linkhard is_dir_simps is_dir_in_current)apply (frule is_dir_in_current)apply (frule current_file_has_sfile, simp)apply (drule cf2sfile_linkhard1, simp+)donelemma co2sobj_truncate: "\<lbrakk>valid (Truncate p f len # s); alive s obj\<rbrakk> \<Longrightarrow> co2sobj (Truncate p f len # s) obj = ( case obj of O_file f' \<Rightarrow> if (f' \<in> same_inode_files s f \<and> len > 0) then Some (S_file (cf2sfiles s f') (O_file f' \<in> Tainted s \<or> O_proc p \<in> Tainted s)) else co2sobj s obj | _ \<Rightarrow> co2sobj s obj)"apply (frule vt_grant_os, frule vd_cons, case_tac obj)apply (simp_all add:current_files_simps is_dir_simps ch2sshm_other cq2smsgq_other tainted_eq_Tainted)apply (auto split:if_splits option.splits dest!:current_file_has_sfile' current_proc_has_sp' simp:current_files_simps cf2sfiles_simps cf2sfile_simps cp2sproc_simps tainted_eq_Tainted same_inode_files_prop6 dest:is_file_in_current is_dir_in_current)donelemma co2sobj_kill: "\<lbrakk>valid (Kill p p' # s); alive (Kill p p' # s) obj\<rbrakk> \<Longrightarrow> co2sobj (Kill p p' # s) obj = co2sobj s obj"apply (frule vt_grant_os, frule vd_cons, case_tac obj)apply (simp_all add:current_files_simps is_dir_simps ch2sshm_other cq2smsgq_other tainted_eq_Tainted)apply (auto split:if_splits option.splits dest!:current_file_has_sfile' current_proc_has_sp' simp:current_files_simps cf2sfiles_simps cf2sfile_simps cp2sproc_simps tainted_eq_Tainted same_inode_files_prop6 dest:is_file_in_current is_dir_in_current)donelemma co2sobj_exit: "\<lbrakk>valid (Exit p # s); alive (Exit p # s) obj\<rbrakk> \<Longrightarrow> co2sobj (Exit p # s) obj = co2sobj s obj"apply (frule vt_grant_os, frule vd_cons, case_tac obj)apply (simp_all add:current_files_simps is_dir_simps ch2sshm_other cq2smsgq_other tainted_eq_Tainted)apply (auto split:if_splits option.splits dest!:current_file_has_sfile' current_proc_has_sp' simp:current_files_simps cf2sfiles_simps cf2sfile_simps cp2sproc_simps tainted_eq_Tainted same_inode_files_prop6 dest:is_file_in_current is_dir_in_current)donelemma co2sobj_createmsgq: "\<lbrakk>valid (CreateMsgq p q # s); alive (CreateMsgq p q # s) obj\<rbrakk> \<Longrightarrow> co2sobj (CreateMsgq p q # s) obj = ( case obj of O_msgq q' \<Rightarrow> if (q' = q) then (case (cq2smsgq (CreateMsgq p q # s) q) of Some sq \<Rightarrow> Some (S_msgq sq) | _ \<Rightarrow> None) else co2sobj s obj | _ \<Rightarrow> co2sobj s obj)"apply (frule vt_grant_os, frule vd_cons, case_tac obj)apply (simp_all add:current_files_simps is_dir_simps ch2sshm_other cq2smsgq_other tainted_eq_Tainted)apply (auto split:if_splits option.splits dest!:current_file_has_sfile' current_proc_has_sp' simp:current_files_simps cf2sfiles_simps cf2sfile_simps cp2sproc_simps tainted_eq_Tainted same_inode_files_prop6 dest!:current_has_smsgq' dest:is_file_in_current is_dir_in_current cq2smsg_createmsgq)donelemma co2sobj_sendmsg: "\<lbrakk>valid (SendMsg p q m # s); alive (SendMsg p q m # s) obj\<rbrakk> \<Longrightarrow> co2sobj (SendMsg p q m # s) obj = ( case obj of O_msgq q' \<Rightarrow> if (q' = q) then (case (cq2smsgq (SendMsg p q m # s) q) of Some sq \<Rightarrow> Some (S_msgq sq) | _ \<Rightarrow> None) else co2sobj s obj | _ \<Rightarrow> co2sobj s obj)"apply (frule vt_grant_os, frule vd_cons, case_tac obj)apply (simp_all add:current_files_simps is_dir_simps ch2sshm_other cq2smsgq_other tainted_eq_Tainted)apply (auto split:if_splits option.splits dest!:current_file_has_sfile' current_proc_has_sp' simp:current_files_simps cf2sfiles_simps cf2sfile_simps cp2sproc_simps tainted_eq_Tainted same_inode_files_prop6 dest!:current_has_smsgq' dest:is_file_in_current is_dir_in_current cq2smsg_sendmsg)donelemma co2sobj_recvmsg: "\<lbrakk>valid (RecvMsg p q m # s); alive (RecvMsg p q m # s) obj\<rbrakk> \<Longrightarrow> co2sobj (RecvMsg p q m # s) obj = ( case obj of O_msgq q' \<Rightarrow> if (q' = q) then (case (cq2smsgq (RecvMsg p q m # s) q) of Some sq \<Rightarrow> Some (S_msgq sq) | _ \<Rightarrow> None) else co2sobj s obj | O_proc p' \<Rightarrow> if (info_flow_shm s p p' \<and> O_msg q m \<in> Tainted s) then (case (cp2sproc s p') of Some sp \<Rightarrow> Some (S_proc sp True) | _ \<Rightarrow> None) else co2sobj s obj | _ \<Rightarrow> co2sobj s obj)"apply (frule vt_grant_os, frule vd_cons, case_tac obj)apply (simp_all add:current_files_simps is_dir_simps ch2sshm_other cq2smsgq_other tainted_eq_Tainted)apply (auto split:if_splits option.splits dest!:current_file_has_sfile' current_proc_has_sp' simp:current_files_simps cf2sfiles_simps cf2sfile_simps cp2sproc_simps tainted_eq_Tainted same_inode_files_prop6 dest!:current_has_smsgq' dest:is_file_in_current is_dir_in_current cq2smsg_recvmsg)donelemma co2sobj_removemsgq: "\<lbrakk>valid (RemoveMsgq p q # s); alive (RemoveMsgq p q # s) obj\<rbrakk> \<Longrightarrow> co2sobj (RemoveMsgq p q # s) obj = co2sobj s obj"apply (frule vt_grant_os, frule vd_cons, case_tac obj)apply (simp_all add:current_files_simps is_dir_simps ch2sshm_other cq2smsgq_other tainted_eq_Tainted)apply (auto split:if_splits option.splits dest!:current_file_has_sfile' current_proc_has_sp' simp:current_files_simps cf2sfiles_simps cf2sfile_simps cp2sproc_simps tainted_eq_Tainted same_inode_files_prop6 dest!:current_has_smsgq' dest:is_file_in_current is_dir_in_current cq2smsg_removemsgq)donelemma co2sobj_createshm: "\<lbrakk>valid (CreateShM p h # s); alive (CreateShM p h # s) obj\<rbrakk> \<Longrightarrow> co2sobj (CreateShM p h # s) obj = ( case obj of O_shm h' \<Rightarrow> if (h' = h) then (case (ch2sshm (CreateShM p h # s) h) of Some sh \<Rightarrow> Some (S_shm sh) | _ \<Rightarrow> None) else co2sobj s obj | _ \<Rightarrow> co2sobj s obj)"apply (frule vt_grant_os, frule vd_cons, case_tac obj)apply (simp_all add:current_files_simps is_dir_simps ch2sshm_other cq2smsgq_other tainted_eq_Tainted)apply (auto split:if_splits option.splits dest!:current_file_has_sfile' current_proc_has_sp' simp:current_files_simps cf2sfiles_simps cf2sfile_simps cp2sproc_simps tainted_eq_Tainted same_inode_files_prop6 ch2sshm_simps dest!:current_shm_has_sh' dest:is_file_in_current is_dir_in_current)donelemma co2sobj_detach: "\<lbrakk>valid (Detach p h # s); alive s obj\<rbrakk> \<Longrightarrow> co2sobj (Detach p h # s) obj = ( case obj of O_proc p' \<Rightarrow> if (p' = p) then (case (cp2sproc (Detach p h # s) p) of Some sp \<Rightarrow> Some (S_proc sp (O_proc p \<in> Tainted s)) | _ \<Rightarrow> None) else co2sobj s obj | _ \<Rightarrow> co2sobj s obj)"apply (frule vt_grant_os, frule vd_cons, case_tac obj)apply (simp_all add:current_files_simps is_dir_simps ch2sshm_other cq2smsgq_other tainted_eq_Tainted)apply (auto split:if_splits option.splits dest!:current_file_has_sfile' current_proc_has_sp' simp:current_files_simps cf2sfiles_simps cf2sfile_simps cp2sproc_simps tainted_eq_Tainted same_inode_files_prop6 ch2sshm_simps dest!:current_shm_has_sh' dest:is_file_in_current is_dir_in_current)donelemma co2sobj_deleteshm: "\<lbrakk>valid (DeleteShM p h # s); alive (DeleteShM p h # s) obj\<rbrakk> \<Longrightarrow> co2sobj (DeleteShM p h # s) obj = ( case obj of O_proc p' \<Rightarrow> (case (cp2sproc (DeleteShM p h # s) p') of Some sp \<Rightarrow> Some (S_proc sp (O_proc p' \<in> Tainted s)) | _ \<Rightarrow> None) | _ \<Rightarrow> co2sobj s obj)"apply (frule vt_grant_os, frule vd_cons, case_tac obj)apply (simp_all add:current_files_simps is_dir_simps ch2sshm_other cq2smsgq_other tainted_eq_Tainted)apply (auto split:if_splits option.splits dest!:current_file_has_sfile' current_proc_has_sp' simp:current_files_simps cf2sfiles_simps cf2sfile_simps tainted_eq_Tainted same_inode_files_prop6 ch2sshm_simps dest!:current_shm_has_sh' dest:is_file_in_current is_dir_in_current)donedeclare Product_Type.split_paired_Ex Product_Type.split_paired_All [simp del]lemma info_flow_shm_prop1: "p \<in> current_procs s \<Longrightarrow> info_flow_shm s p p"by (simp add:info_flow_shm_def)lemma co2sobj_attach: "\<lbrakk>valid (Attach p h flag # s); alive s obj\<rbrakk> \<Longrightarrow> co2sobj (Attach p h flag # s) obj = ( case obj of O_proc p' \<Rightarrow> if (info_flow_shm s p p') then (case (cp2sproc (Attach p h flag # s) p') of Some sp \<Rightarrow> Some (S_proc sp (O_proc p' \<in> Tainted s \<or> (\<exists> p''. O_proc p'' \<in> Tainted s \<and> (p'', SHM_RDWR) \<in> procs_of_shm s h))) | _ \<Rightarrow> None) else if (\<exists> p'' flag'. (p'', flag') \<in> procs_of_shm s h \<and> flag = SHM_RDWR \<and> O_proc p \<in> Tainted s \<and> info_flow_shm s p'' p') then (case (cp2sproc s p') of Some sp \<Rightarrow> Some (S_proc sp True) | _ \<Rightarrow> None) else co2sobj s obj | _ \<Rightarrow> co2sobj s obj)"apply (frule vt_grant_os, frule vd_cons, case_tac obj)apply (simp_all add:current_files_simps is_dir_simps ch2sshm_other cq2smsgq_other tainted_eq_Tainted)apply (rule conjI|rule impI|erule exE)+apply (simp split:option.splits del:split_paired_Ex)apply (rule impI, frule current_proc_has_sp, simp)apply ((erule exE)+, auto simp:tainted_eq_Tainted intro:info_flow_shm_Tainted)[1]apply (rule impI, simp add:tainted_eq_Tainted split:option.splits del:split_paired_Ex)apply (auto simp:info_flow_shm_prop1 cp2sproc_attach dest!:current_proc_has_sp')[1]apply (case_tac "cp2sproc (Attach p h flag # s) nat")apply (drule current_proc_has_sp', simp+)apply (rule conjI|erule exE|erule conjE|rule impI)+apply (simp add:tainted_eq_Tainted)apply (auto simp:info_flow_shm_prop1 cp2sproc_attach intro:info_flow_shm_Tainted dest!:current_proc_has_sp')[1]apply (auto simp:info_flow_shm_prop1 cp2sproc_attach tainted_eq_Tainted intro:info_flow_shm_Tainted dest!:current_proc_has_sp'split:option.splits if_splits)[1]apply (auto split:if_splits option.splits dest!:current_file_has_sfile' simp:current_files_simps cf2sfiles_simps cf2sfile_simps tainted_eq_Tainted same_inode_files_prop6 dest:is_file_in_current is_dir_in_current)donelemma co2sobj_other: "\<lbrakk>valid (e # s); alive (e # s) obj; \<forall> p f fds. e \<noteq> Execve p f fds; \<forall> p p' fds shms. e \<noteq> Clone p p' fds shms; \<forall> p p'. e \<noteq> Ptrace p p'; \<forall> p f flags fd opt. e \<noteq> Open p f flags fd opt; \<forall> p fd. e \<noteq> ReadFile p fd; \<forall> p fd. e \<noteq> WriteFile p fd; \<forall> p fd. e \<noteq> CloseFd p fd; \<forall> p f. e \<noteq> UnLink p f; \<forall> p f. e \<noteq> Rmdir p f; \<forall> p f i. e \<noteq> Mkdir p f i; \<forall> p f f'. e \<noteq> LinkHard p f f'; \<forall> p f len. e \<noteq> Truncate p f len; \<forall> p q. e \<noteq> CreateMsgq p q; \<forall> p q m. e \<noteq> SendMsg p q m; \<forall> p q m. e \<noteq> RecvMsg p q m; \<forall> p q. e \<noteq> RemoveMsgq p q; \<forall> p h. e \<noteq> CreateShM p h; \<forall> p h flag. e \<noteq> Attach p h flag; \<forall> p h. e \<noteq> Detach p h; \<forall> p h. e \<noteq> DeleteShM p h \<rbrakk> \<Longrightarrow> co2sobj (e # s) obj = co2sobj s obj"apply (frule vt_grant, case_tac e)apply (auto intro:co2sobj_kill co2sobj_exit)donelemmas co2sobj_simps = co2sobj_execve co2sobj_clone co2sobj_ptrace co2sobj_open co2sobj_readfile co2sobj_writefile co2sobj_closefd co2sobj_unlink co2sobj_rmdir co2sobj_mkdir co2sobj_linkhard co2sobj_truncate co2sobj_kill co2sobj_exit co2sobj_createmsgq co2sobj_sendmsg co2sobj_recvmsg co2sobj_removemsgq co2sobj_attach co2sobj_detach co2sobj_createshm co2sobj_deleteshmend(*<*)end(*>*)