--- a/S2ss_prop.thy Thu Sep 05 13:23:03 2013 +0800
+++ b/S2ss_prop.thy Thu Sep 12 13:50:22 2013 +0800
@@ -8,17 +8,29 @@
(* simpset for s2ss*)
+lemma co2sobj_some_case:
+ "\<lbrakk>co2sobj s obj = Some sobj; \<And> p. obj = O_proc p \<Longrightarrow> P (O_proc p);
+ \<And> f. obj = O_file f \<Longrightarrow> P (O_file f); \<And> h. obj = O_shm h \<Longrightarrow> P (O_shm h);
+ \<And> f. obj = O_dir f \<Longrightarrow> P (O_dir f); \<And> q. obj = O_msgq q \<Longrightarrow> P (O_msgq q)\<rbrakk>
+ \<Longrightarrow> P obj"
+by (case_tac obj, auto)
+
+lemma co2sobj_execve_alive:
+ "\<lbrakk>alive s obj; co2sobj s obj = Some x; valid (Execve p f fds # s)\<rbrakk>
+ \<Longrightarrow> alive (Execve p f fds # s) obj"
+apply (erule co2sobj_some_case)
+by (auto simp:alive_simps simp del:alive.simps)
lemma s2ss_execve:
"valid (Execve p f fds # s) \<Longrightarrow> s2ss (Execve p f fds # s) = (
if (\<exists> p'. p' \<noteq> p \<and> p' \<in> current_procs s \<and> co2sobj s (O_proc p') = co2sobj s (O_proc p))
then (case (cp2sproc (Execve p f fds # s) p) of
Some sp \<Rightarrow> s2ss s \<union> {S_proc sp (O_proc p \<in> Tainted s \<or> O_file f \<in> Tainted s)}
- | _ \<Rightarrow> s2ss s)
- else (case (cp2sproc (Execve p f fds # s) p) of
- Some sp \<Rightarrow> s2ss s - {S_proc sp (O_proc p \<in> Tainted s)}
- \<union> {S_proc sp (O_proc p \<in> Tainted s \<or> O_file f \<in> Tainted s)}
- | _ \<Rightarrow> s2ss s) )"
+ | _ \<Rightarrow> {} )
+ else (case (cp2sproc (Execve p f fds # s) p, cp2sproc s p) of
+ (Some sp, Some sp') \<Rightarrow> s2ss s - {S_proc sp' (O_proc p \<in> Tainted s)}
+ \<union> {S_proc sp (O_proc p \<in> Tainted s \<or> O_file f \<in> Tainted s)}
+ | _ \<Rightarrow> {} ) )"
apply (frule vd_cons, frule vt_grant_os, simp split:if_splits)
apply (rule conjI, rule impI, (erule exE|erule conjE)+)
@@ -39,41 +51,521 @@
apply (case_tac "x = S_proc sp (O_proc p \<in> tainted s)")
apply (rule_tac x = "O_proc p'" in exI)
apply (simp add:alive_execve co2sobj_execve tainted_eq_Tainted cp2sproc_execve)
+apply (case_tac "obj = O_proc p", simp)
+apply (frule co2sobj_execve_alive, simp, simp)
+apply (frule_tac obj = obj in co2sobj_execve, simp)
+apply (rule_tac x = obj in exI, simp, simp)
-apply (case_tac obj, simp_all add:co2sobj_execve alive_simps)
-thm alive_execve
-thm co2sobj_execve
+apply (erule conjE, frule current_proc_has_sp, simp, erule exE, rule impI, simp)
+apply (subgoal_tac "p \<in> current_procs (Execve p f fds # s)")
+apply (drule_tac p = p and s = "Execve p f fds # s" in current_proc_has_sp, simp)
+apply (erule exE, erule conjE, simp)
+apply (simp add:s2ss_def, rule set_eqI, rule iffI)
+apply (drule CollectD, (erule exE|erule conjE)+)
+apply (case_tac "obj = O_proc p", simp add:tainted_eq_Tainted)
+apply (rule disjI1, simp split:if_splits)
apply (simp add:co2sobj_execve, rule disjI2)
-apply (rule_tac x = obj in exI, simp split:t_object.splits)
-thm co2sobj_execve
-apply (case_tac obj, auto simp:co2sobj_execve alive_simps tainted_eq_Tainted split:option.splits dest!:current_proc_has_sp')[1]
-apply (simp add:co2sobj_execve)
-apply clarsimp
-thm current_proc_has_sp
+apply (rule conjI,rule_tac x = obj in exI, simp add:alive_simps split:t_object.splits)
+apply (rule notI, simp, case_tac obj)
+apply (erule_tac x = nat in allE, simp add:tainted_eq_Tainted, (simp split:option.splits)+)
+apply (erule disjE, simp)
+apply (rule_tac x = "O_proc p" in exI, simp add:tainted_eq_Tainted)
+apply (erule exE|erule conjE)+
+apply (rule_tac x = obj in exI)
+apply (drule_tac obj = obj in co2sobj_execve_alive, simp+)
+apply (frule_tac obj = obj in co2sobj_execve, simp, simp)
+apply (rule impI, simp add:tainted_eq_Tainted, simp)
+done
+
+lemma s2ss_clone_alive:
+ "\<lbrakk>co2sobj s obj = Some x; alive s obj; obj \<noteq> O_proc p'; valid (Clone p p' fds shms # s)\<rbrakk>
+ \<Longrightarrow> alive (Clone p p' fds shms # s) obj"
+by (erule co2sobj_some_case, auto simp:alive_clone)
+
+lemma s2ss_clone:
+ "valid (Clone p p' fds shms # s) \<Longrightarrow> s2ss (Clone p p' fds shms # s) = (
+ case (cp2sproc (Clone p p' fds shms # s) p') of
+ Some sp \<Rightarrow> s2ss s \<union> {S_proc sp (O_proc p \<in> Tainted s)}
+ | _ \<Rightarrow> {})"
+apply (frule vd_cons, frule vt_grant_os, split option.splits)
+apply (rule conjI, rule impI, drule current_proc_has_sp', simp, simp)
+apply (rule allI, rule impI, simp add:s2ss_def)
+apply (rule set_eqI, rule iffI, drule CollectD, (erule exE|erule conjE)+)
+apply (case_tac "obj = O_proc p'", simp add:tainted_eq_Tainted)
+apply (case_tac "O_proc p' \<in> Tainted s", drule Tainted_in_current, simp+)
+apply (rule disjI1, simp split:if_splits)
+apply (simp add:tainted_eq_Tainted, rule disjI2)
+apply (frule co2sobj_clone, simp)
+apply (rule_tac x = obj in exI, simp add:alive_simps split:t_object.splits)
+
+apply (simp, erule disjE, simp)
+apply (rule_tac x = "O_proc p'" in exI, simp add:tainted_eq_Tainted)
+apply (rule impI, rule notI, drule Tainted_in_current, simp+)
+apply (erule exE| erule conjE)+
+apply (case_tac "obj = O_proc p'", simp)
+apply (rule_tac x = obj in exI)
+apply (frule s2ss_clone_alive, simp+)
+apply (auto simp:co2sobj_clone alive_simps)
+done
+
+definition update_s2ss_shm:: "t_state \<Rightarrow> t_process \<Rightarrow> t_static_state"
+where
+ "update_s2ss_shm s pfrom \<equiv> s2ss s
+ \<union> {S_proc sp True| sp p. info_flow_shm s pfrom p \<and> cp2sproc s p = Some sp}
+ - {S_proc sp False | sp p. info_flow_shm s pfrom p \<and> cp2sproc s p = Some sp \<and>
+ (\<not> (\<exists> p'. \<not> info_flow_shm s pfrom p' \<and> p' \<in> current_procs s \<and>
+ cp2sproc s p' = Some sp \<and> O_proc p' \<notin> Tainted s))}"
+
+lemma Tainted_ptrace':
+ "valid s \<Longrightarrow> Tainted (Ptrace p p' # s) =
+ (if (O_proc p \<in> Tainted s \<and> O_proc p' \<notin> Tainted s)
+ then Tainted s \<union> {O_proc p'' | p''. info_flow_shm s p' p''}
+ else if (O_proc p' \<in> Tainted s \<and> O_proc p \<notin> Tainted s)
+ then Tainted s \<union> {O_proc p'' | p''. info_flow_shm s p p''}
+ else Tainted s)"
+using info_flow_shm_Tainted by auto
+
+lemma co2sobj_some_caseD:
+ "\<lbrakk>co2sobj s obj = Some sobj; \<And> p. \<lbrakk>co2sobj s obj = Some sobj; obj = O_proc p\<rbrakk> \<Longrightarrow> P (O_proc p);
+ \<And> f. \<lbrakk>co2sobj s obj = Some sobj; obj = O_file f\<rbrakk> \<Longrightarrow> P (O_file f);
+ \<And> h. \<lbrakk>co2sobj s obj = Some sobj; obj = O_shm h\<rbrakk> \<Longrightarrow> P (O_shm h);
+ \<And> f. \<lbrakk>co2sobj s obj = Some sobj; obj = O_dir f\<rbrakk> \<Longrightarrow> P (O_dir f);
+ \<And> q. \<lbrakk>co2sobj s obj = Some sobj; obj = O_msgq q\<rbrakk> \<Longrightarrow> P (O_msgq q)\<rbrakk>
+ \<Longrightarrow> P obj"
+by (case_tac obj, auto)
+
+lemma s2ss_ptrace1_aux: "x \<notin> {x. P x} \<Longrightarrow> \<not> P x" by simp
+
+lemma s2ss_ptrace1:
+ "\<lbrakk>valid (Ptrace p p' # s); O_proc p \<in> Tainted s; O_proc p' \<notin> Tainted s\<rbrakk>
+ \<Longrightarrow> s2ss (Ptrace p p' # s) = update_s2ss_shm s p'"
+unfolding update_s2ss_shm_def s2ss_def
+apply (frule vd_cons, rule set_eqI, rule iffI)
+apply (drule CollectD, (erule exE|erule conjE)+)
+apply (erule co2sobj_some_caseD)
+apply (rule DiffI)
+apply (case_tac "O_proc pa \<in> Tainted s")
+apply (rule UnI1, simp, rule_tac x = "O_proc pa" in exI)
+apply (clarsimp simp:Tainted_ptrace' cp2sproc_other tainted_eq_Tainted split:option.splits)
+apply (case_tac "info_flow_shm s p' pa")
+apply (rule UnI2, rule CollectI, simp only:co2sobj.simps split:option.splits)
+apply (drule current_proc_has_sp', simp, simp)
+apply (rule_tac x = a in exI, rule_tac x = pa in exI)
+apply (clarsimp simp:Tainted_ptrace' cp2sproc_other tainted_eq_Tainted split:option.splits)
+apply (rule UnI1, simp)
+apply (rule_tac x = "O_proc pa" in exI)
+apply (clarsimp simp:Tainted_ptrace' cp2sproc_other tainted_eq_Tainted split:option.splits)
+apply (rule notI, clarsimp simp:cp2sproc_other tainted_eq_Tainted split:option.splits)
+apply (erule_tac x = pa in allE, simp)
+
+apply (rule DiffI, rule UnI1, rule CollectI, rule_tac x = obj in exI,
+ auto split:option.splits simp:co2sobj_ptrace alive_simps simp del:co2sobj.simps)[1]
+apply (simp)
+apply (rule DiffI, rule UnI1, rule CollectI, rule_tac x = obj in exI,
+ auto split:option.splits simp:co2sobj_ptrace alive_simps simp del:co2sobj.simps)[1]
+apply (simp split:option.splits)
+apply (rule DiffI, rule UnI1, rule CollectI, rule_tac x = obj in exI,
+ auto split:option.splits simp:co2sobj_ptrace alive_simps simp del:co2sobj.simps)[1]
+apply (simp split:option.splits)
+apply (rule DiffI, rule UnI1, rule CollectI, rule_tac x = obj in exI,
+ auto split:option.splits simp:co2sobj_ptrace alive_simps simp del:co2sobj.simps)[1]
+apply (simp split:option.splits)
+
+apply (erule DiffE, drule s2ss_ptrace1_aux, erule UnE)
+apply (erule CollectE, (erule exE|erule conjE)+, rule CollectI)
+apply (erule co2sobj_some_caseD)
+apply (case_tac "O_proc pa \<in> Tainted s")
+apply (rule_tac x = "O_proc pa" in exI)
+apply (clarsimp simp add:tainted_eq_Tainted cp2sproc_other split:option.splits)
+apply (case_tac "info_flow_shm s p' pa", simp only:co2sobj.simps split:option.splits)
+apply (drule current_proc_has_sp', simp, simp)
+apply (drule Meson.not_exD, erule_tac x = a in allE, drule Meson.not_exD, erule_tac x = pa in allE)
+apply (simp add:tainted_eq_Tainted, (erule exE|erule conjE)+)
+apply (rule_tac x = "O_proc p'a" in exI)
+apply (clarsimp simp:Tainted_ptrace' cp2sproc_other tainted_eq_Tainted split:option.splits)
+apply (rule_tac x = "O_proc pa" in exI)
+apply (clarsimp simp:Tainted_ptrace' cp2sproc_other tainted_eq_Tainted split:option.splits)
+
+apply (rule_tac x = obj in exI,
+ auto split:option.splits simp:co2sobj_ptrace alive_simps simp del:co2sobj.simps)[1]
+apply (rule_tac x = obj in exI,
+ auto split:option.splits simp:co2sobj_ptrace alive_simps simp del:co2sobj.simps)[1]
+apply (rule_tac x = obj in exI,
+ auto split:option.splits simp:co2sobj_ptrace alive_simps simp del:co2sobj.simps)[1]
+apply (rule_tac x = obj in exI,
+ auto split:option.splits simp:co2sobj_ptrace alive_simps simp del:co2sobj.simps)[1]
+
+apply (erule CollectE, (erule exE|erule conjE)+, rule CollectI)
+apply (rule_tac x = "O_proc pa" in exI)
+apply (clarsimp simp:Tainted_ptrace' cp2sproc_other tainted_eq_Tainted info_shm_flow_in_procs)
+done
+
+lemma s2ss_ptrace2:
+ "\<lbrakk>valid (Ptrace p p' # s); O_proc p' \<in> Tainted s; O_proc p \<notin> Tainted s\<rbrakk>
+ \<Longrightarrow> s2ss (Ptrace p p' # s) = update_s2ss_shm s p"
+unfolding update_s2ss_shm_def s2ss_def
+apply (frule vd_cons, rule set_eqI, rule iffI)
+apply (drule CollectD, (erule exE|erule conjE)+)
+apply (erule co2sobj_some_caseD)
+apply (rule DiffI)
+apply (case_tac "O_proc pa \<in> Tainted s")
+apply (rule UnI1, simp, rule_tac x = "O_proc pa" in exI)
+apply (clarsimp simp:Tainted_ptrace' cp2sproc_other tainted_eq_Tainted split:option.splits)
+apply (case_tac "info_flow_shm s p pa")
+apply (rule UnI2, rule CollectI, simp only:co2sobj.simps split:option.splits)
+apply (drule current_proc_has_sp', simp, simp)
+apply (rule_tac x = a in exI, rule_tac x = pa in exI)
+apply (clarsimp simp:Tainted_ptrace' cp2sproc_other tainted_eq_Tainted split:option.splits)
+apply (rule UnI1, simp)
+apply (rule_tac x = "O_proc pa" in exI)
+apply (clarsimp simp:Tainted_ptrace' cp2sproc_other tainted_eq_Tainted split:option.splits)
+apply (rule notI, clarsimp simp:cp2sproc_other tainted_eq_Tainted split:option.splits)
+apply (erule_tac x = pa in allE, simp)
+
+apply (rule DiffI, rule UnI1, rule CollectI, rule_tac x = obj in exI,
+ auto split:option.splits simp:co2sobj_ptrace alive_simps simp del:co2sobj.simps)[1]
+apply (simp)
+apply (rule DiffI, rule UnI1, rule CollectI, rule_tac x = obj in exI,
+ auto split:option.splits simp:co2sobj_ptrace alive_simps simp del:co2sobj.simps)[1]
+apply (simp split:option.splits)
+apply (rule DiffI, rule UnI1, rule CollectI, rule_tac x = obj in exI,
+ auto split:option.splits simp:co2sobj_ptrace alive_simps simp del:co2sobj.simps)[1]
+apply (simp split:option.splits)
+apply (rule DiffI, rule UnI1, rule CollectI, rule_tac x = obj in exI,
+ auto split:option.splits simp:co2sobj_ptrace alive_simps simp del:co2sobj.simps)[1]
+apply (simp split:option.splits)
+
+apply (erule DiffE, drule s2ss_ptrace1_aux, erule UnE)
+apply (erule CollectE, (erule exE|erule conjE)+, rule CollectI)
+apply (erule co2sobj_some_caseD)
+apply (case_tac "O_proc pa \<in> Tainted s")
+apply (rule_tac x = "O_proc pa" in exI)
+apply (clarsimp simp add:tainted_eq_Tainted cp2sproc_other split:option.splits)
+apply (case_tac "info_flow_shm s p pa", simp only:co2sobj.simps split:option.splits)
+apply (drule current_proc_has_sp', simp, simp)
+apply (drule Meson.not_exD, erule_tac x = a in allE, drule Meson.not_exD, erule_tac x = pa in allE)
+apply (simp add:tainted_eq_Tainted, (erule exE|erule conjE)+)
+apply (rule_tac x = "O_proc p'a" in exI)
+apply (clarsimp simp:Tainted_ptrace' cp2sproc_other tainted_eq_Tainted split:option.splits)
+apply (rule_tac x = "O_proc pa" in exI)
+apply (clarsimp simp:Tainted_ptrace' cp2sproc_other tainted_eq_Tainted split:option.splits)
+
+apply (rule_tac x = obj in exI,
+ auto split:option.splits simp:co2sobj_ptrace alive_simps simp del:co2sobj.simps)[1]
+apply (rule_tac x = obj in exI,
+ auto split:option.splits simp:co2sobj_ptrace alive_simps simp del:co2sobj.simps)[1]
+apply (rule_tac x = obj in exI,
+ auto split:option.splits simp:co2sobj_ptrace alive_simps simp del:co2sobj.simps)[1]
+apply (rule_tac x = obj in exI,
+ auto split:option.splits simp:co2sobj_ptrace alive_simps simp del:co2sobj.simps)[1]
+
+apply (erule CollectE, (erule exE|erule conjE)+, rule CollectI)
+apply (rule_tac x = "O_proc pa" in exI)
+apply (clarsimp simp:Tainted_ptrace' cp2sproc_other tainted_eq_Tainted info_shm_flow_in_procs)
+done
+
+lemma s2ss_ptrace3:
+ "\<lbrakk>valid (Ptrace p p' # s); (O_proc p' \<in> Tainted s) = (O_proc p \<in> Tainted s)\<rbrakk>
+ \<Longrightarrow> s2ss (Ptrace p p' # s) = s2ss s"
+unfolding s2ss_def
+apply (frule vd_cons, rule set_eqI, rule iffI)
+apply (erule CollectE, (erule exE|erule conjE)+, rule CollectI)
+apply (rule_tac x = obj in exI)
+apply (frule alive_other, simp+)
+apply (frule_tac obj = obj in co2sobj_ptrace, simp)
+apply (auto simp add:tainted_eq_Tainted split:t_object.splits option.splits if_splits
+ intro:info_flow_shm_Tainted)[1]
+
+apply (erule CollectE, (erule exE|erule conjE)+, rule CollectI)
+apply (rule_tac x = obj in exI)
+apply (frule alive_other, simp+)
+apply (frule_tac obj = obj in co2sobj_ptrace, simp)
+apply (auto simp add:tainted_eq_Tainted split:t_object.splits option.splits if_splits
+ intro:info_flow_shm_Tainted)
+done
+
+lemma s2ss_ptrace:
+ "valid (Ptrace p p' # s) \<Longrightarrow> s2ss (Ptrace p p' # s) = (
+ if (O_proc p \<in> Tainted s \<and> O_proc p' \<notin> Tainted s)
+ then update_s2ss_shm s p'
+ else if (O_proc p' \<in> Tainted s \<and> O_proc p \<notin> Tainted s)
+ then update_s2ss_shm s p
+ else s2ss s )"
+apply (frule vt_grant_os, frule vd_cons)
+apply (simp add:s2ss_ptrace2 s2ss_ptrace1 split:if_splits)
+by (auto dest:s2ss_ptrace3)
-apply (auto simp:s2ss_def co2sobj_execve split:option.splits)
+lemma s2ss_kill:
+ "valid (Kill p p' # s) \<Longrightarrow> s2ss (Kill p p' # s) = (
+ if (\<exists> p''. p'' \<in> current_procs s \<and> p'' \<noteq> p' \<and> co2sobj s (O_proc p'') = co2sobj s (O_proc p'))
+ then s2ss s
+ else (case (co2sobj s (O_proc p')) of
+ Some sp \<Rightarrow> s2ss s - {sp}
+ | _ \<Rightarrow> {}))"
+apply (frule vt_grant_os, frule vd_cons)
+unfolding s2ss_def
+apply (simp split:if_splits, rule conjI)
+apply (rule impI, (erule exE|erule conjE)+)
+apply (split option.splits)
+apply (drule current_proc_has_sp', simp, simp)
+apply (simp split: option.splits, (erule conjE)+)
+apply (rule set_eqI, rule iffI, erule CollectE, (erule exE|erule conjE)+, rule CollectI)
+apply (rule_tac x = obj in exI)
+apply (simp add:co2sobj_kill tainted_eq_Tainted alive_kill split:t_object.splits if_splits)
+apply (erule CollectE, erule exE, erule conjE, rule CollectI)
+apply (erule co2sobj_some_caseD)
+apply (case_tac "pa = p'")
+apply (rule_tac x = "O_proc p''" in exI)
+apply (simp add:cp2sproc_kill tainted_eq_Tainted alive_kill
+ split:t_object.splits if_splits option.splits)
+apply (rule_tac x = "O_proc pa" in exI)
+apply (clarsimp simp add:cp2sproc_kill tainted_eq_Tainted alive_kill
+ split:t_object.splits if_splits option.splits)
+apply (rule_tac x = obj in exI, frule alive_kill, simp add:co2sobj_kill del:co2sobj.simps)+
+
+apply (rule impI, erule conjE, frule current_proc_has_sp, simp, erule exE, simp)
+apply (rule set_eqI, rule iffI)
+apply (erule CollectE, erule exE, erule conjE, rule DiffI)
+apply (rule CollectI, rule_tac x = obj in exI)
+apply (simp add:co2sobj_kill tainted_eq_Tainted alive_kill split:t_object.splits if_splits)
+apply (rule notI, simp, case_tac obj)
+apply (erule_tac x = nat in allE)
+apply (simp add:co2sobj_kill cp2sproc_kill tainted_eq_Tainted split:option.splits)
+apply (simp split:option.splits)+
+apply (erule co2sobj_some_caseD)
+apply (case_tac "pa = p'")
+apply (rule_tac x = "O_proc p''" in exI)
+apply (simp add:cp2sproc_kill tainted_eq_Tainted alive_kill
+ split:t_object.splits if_splits option.splits)
+apply (rule_tac x = "O_proc pa" in exI)
+apply (clarsimp simp add:cp2sproc_kill tainted_eq_Tainted alive_kill
+ split:t_object.splits if_splits option.splits)
+apply (rule_tac x = obj in exI, frule alive_kill, simp add:co2sobj_kill del:co2sobj.simps)+
+done
+
+lemma s2ss_exit:
+ "valid (Exit p # s) \<Longrightarrow> s2ss (Exit p # s) = (
+ if (\<exists> p'. p' \<in> current_procs s \<and> p' \<noteq> p \<and> co2sobj s (O_proc p') = co2sobj s (O_proc p))
+ then s2ss s
+ else (case (co2sobj s (O_proc p)) of
+ Some sp \<Rightarrow> s2ss s - {sp}
+ | _ \<Rightarrow> {}))"
+apply (frule vt_grant_os, frule vd_cons)
+unfolding s2ss_def
+apply (simp split:if_splits, rule conjI)
+apply (rule impI, (erule exE|erule conjE)+)
+apply (split option.splits)
+apply (drule current_proc_has_sp', simp, simp)
+apply (simp split: option.splits, (erule conjE)+)
+apply (rule set_eqI, rule iffI, erule CollectE, (erule exE|erule conjE)+, rule CollectI)
+apply (rule_tac x = obj in exI)
+apply (simp add:co2sobj_exit tainted_eq_Tainted alive_exit split:t_object.splits if_splits)
+apply (erule CollectE, erule exE, erule conjE, rule CollectI)
+apply (erule co2sobj_some_caseD)
+apply (case_tac "pa = p")
+apply (rule_tac x = "O_proc p'" in exI)
+apply (simp add:cp2sproc_exit tainted_eq_Tainted alive_exit
+ split:t_object.splits if_splits option.splits)
+apply (rule_tac x = "O_proc pa" in exI)
+apply (clarsimp simp add:cp2sproc_exit tainted_eq_Tainted alive_exit
+ split:t_object.splits if_splits option.splits)
+apply (rule_tac x = obj in exI, frule alive_exit, simp add:co2sobj_exit del:co2sobj.simps)+
+
+apply (rule impI, frule current_proc_has_sp, simp, erule exE, simp)
+apply (rule set_eqI, rule iffI)
+apply (erule CollectE, erule exE, erule conjE, rule DiffI)
+apply (rule CollectI, rule_tac x = obj in exI)
+apply (simp add:co2sobj_exit tainted_eq_Tainted alive_exit split:t_object.splits if_splits)
+apply (rule notI, simp, case_tac obj)
+apply (erule_tac x = nat in allE)
+apply (simp add:co2sobj_exit cp2sproc_exit tainted_eq_Tainted split:option.splits)
+apply (simp split:option.splits)+
+apply (erule co2sobj_some_caseD)
+apply (case_tac "pa = p")
+apply (rule_tac x = "O_proc p'" in exI)
+apply (simp add:cp2sproc_exit tainted_eq_Tainted alive_exit
+ split:t_object.splits if_splits option.splits)
+apply (rule_tac x = "O_proc pa" in exI)
+apply (clarsimp simp add:cp2sproc_exit tainted_eq_Tainted alive_exit
+ split:t_object.splits if_splits option.splits)
+apply (rule_tac x = obj in exI, frule alive_exit, simp add:co2sobj_exit del:co2sobj.simps)+
+done
+
+lemma alive_has_sobj':
+ "\<lbrakk>co2sobj s obj = None; valid s\<rbrakk> \<Longrightarrow> \<not> alive s obj"
+apply (case_tac obj)
+apply (auto split:option.splits)
+oops
+
+declare co2sobj.simps [simp del]
+
+lemma co2sobj_open_none:
+ "\<lbrakk>valid (Open p f flag fd None # s); alive s obj\<rbrakk> \<Longrightarrow> co2sobj (Open p f flag fd None # s) obj = (
+ if (obj = O_proc p)
+ then (case (cp2sproc (Open p f flag fd None # s) p) of
+ Some sp \<Rightarrow> Some (S_proc sp (O_proc p \<in> Tainted s))
+ | _ \<Rightarrow> None)
+ else co2sobj s obj)"
+apply (frule vt_grant_os, frule vd_cons)
+apply (frule_tac obj = obj in co2sobj_open, simp add:alive_open)
+apply (auto split:t_object.splits option.splits dest!:current_proc_has_sp')
+done
+
+lemma co2sobj_open_some:
+ "\<lbrakk>valid (Open p f flag fd (Some i) # s); alive s obj\<rbrakk> \<Longrightarrow> co2sobj (Open p f flag fd (Some i) # s) obj = (
+ if (obj = O_proc p)
+ then (case (cp2sproc (Open p f flag fd (Some i) # s) p) of
+ Some sp \<Rightarrow> Some (S_proc sp (O_proc p \<in> Tainted s))
+ | _ \<Rightarrow> None)
+ else if (obj = O_file f)
+ then (case (cf2sfile (Open p f flag fd (Some i) # s) f) of
+ Some sf \<Rightarrow> Some (S_file {sf} (O_proc p \<in> Tainted s))
+ | _ \<Rightarrow> None)
+ else co2sobj s obj)"
+apply (frule vt_grant_os, frule vd_cons)
+apply (frule_tac obj = obj in co2sobj_open, simp add:alive_open)
+apply (auto split:t_object.splits option.splits dest!:current_proc_has_sp')
+done
+lemma alive_co2sobj_some_open_none:
+ "\<lbrakk>co2sobj s obj = Some sobj; alive s obj; valid (Open p f flag fd None # s)\<rbrakk>
+ \<Longrightarrow> alive (Open p f flag fd None # s) obj"
+by (erule co2sobj_some_caseD, auto simp:alive_simps is_file_simps is_dir_simps)
+
+lemma alive_co2sobj_some_open_none':
+ "\<lbrakk>co2sobj (Open p f flag fd None # s) obj = Some sobj; alive (Open p f flag fd None # s) obj;
+ valid (Open p f flag fd None # s)\<rbrakk> \<Longrightarrow> alive s obj"
+by (erule co2sobj_some_caseD, auto simp:alive_simps is_file_simps is_dir_simps)
+
+lemma co2sobj_proc_obj:
+ "\<lbrakk>co2sobj s obj = Some x; co2sobj s (O_proc p) = Some x\<rbrakk>
+ \<Longrightarrow> \<exists> p'. obj = O_proc p'"
+by (case_tac obj, auto simp:co2sobj.simps split:option.splits)
+
+lemma s2ss_open_none:
+ "valid (Open p f flag fd None # s) \<Longrightarrow> s2ss (Open p f flag fd None # s) = (
+ case (co2sobj s (O_proc p), co2sobj (Open p f flag fd None # s) (O_proc p)) of
+ (Some sp, Some sp') \<Rightarrow>
+ if (\<exists> p'. p' \<in> current_procs s \<and> p' \<noteq> p \<and> co2sobj s (O_proc p') = Some sp)
+ then s2ss s \<union> {sp'}
+ else s2ss s - {sp} \<union> {sp'}
+ | _ \<Rightarrow> {} )"
+unfolding s2ss_def
+apply (frule vt_grant_os, frule vd_cons)
+apply (case_tac "co2sobj s (O_proc p)", simp add:co2sobj.simps split:option.splits)
+apply (drule current_proc_has_sp', simp, simp)
+apply (case_tac "co2sobj (Open p f flag fd None # s) (O_proc p)")
+apply (simp add:co2sobj.simps split:option.splits)
+apply (drule current_proc_has_sp', simp, simp)
+apply (rule set_eqI, rule iffI, erule CollectE, erule exE, erule conjE, simp)
+apply (frule_tac obj = obj in alive_co2sobj_some_open_none', simp, simp)
+apply (rule conjI, rule impI, erule exE, (erule conjE)+)
+apply (rule Meson.disj_comm, rule disjCI, case_tac "obj = O_proc p", simp)
+apply (rule_tac x = obj in exI, simp add:co2sobj_open_none)
+apply (rule impI)
+apply (case_tac "obj = O_proc p", simp)
+apply (rule Meson.disj_comm, rule disjCI, rule conjI)
+apply (rule_tac x = obj in exI, simp add:co2sobj_open_none)
+apply (simp add:co2sobj_open_none split:option.splits)
+apply (rule notI)
+apply (frule co2sobj_proc_obj, simp, erule exE)
+apply (erule_tac x = p' in allE, simp)
+
+apply (simp split:if_splits)
+apply (erule disjE, rule_tac x = "O_proc p" in exI, simp)
+apply (erule exE, erule conjE, case_tac "obj = O_proc p")
+apply (rule_tac x = "O_proc p'" in exI, simp add:co2sobj_open_none)
+apply (rule_tac x = obj in exI, simp add:co2sobj_open_none alive_co2sobj_some_open_none)
+apply (erule disjE, rule_tac x = "O_proc p" in exI, simp)
+apply (erule conjE, erule exE, erule conjE, case_tac "obj = O_proc p")
+apply (rule_tac x = "O_proc p'" in exI, simp add:co2sobj_open_none)
+apply (rule_tac x = obj in exI, simp add:co2sobj_open_none alive_co2sobj_some_open_none)
+done
+
+lemma alive_co2sobj_some_open_some:
+ "\<lbrakk>alive s obj; valid (Open p f flag fd (Some i) # s)\<rbrakk>
+ \<Longrightarrow> alive (Open p f flag fd (Some i) # s) obj"
+by (case_tac obj, auto simp:alive_simps is_file_simps is_dir_simps)
+
+lemma alive_co2sobj_some_open_some':
+ "\<lbrakk>co2sobj (Open p f flag fd (Some i) # s) obj = Some sobj; alive (Open p f flag fd (Some i) # s) obj;
+ valid (Open p f flag fd (Some i) # s); obj \<noteq> O_file f\<rbrakk> \<Longrightarrow> alive s obj"
+by (erule co2sobj_some_caseD, auto simp:alive_simps is_file_simps is_dir_simps)
+
+lemma s2ss_open_some:
+ "valid (Open p f flag fd (Some i) # s) \<Longrightarrow> s2ss (Open p f flag fd (Some i) # s) = (
+ case (co2sobj s (O_proc p), co2sobj (Open p f flag fd (Some i) # s) (O_proc p),
+ co2sobj (Open p f flag fd (Some i) # s) (O_file f)) of
+ (Some sp, Some sp', Some sf) \<Rightarrow>
+ if (\<exists> p'. p' \<in> current_procs s \<and> p' \<noteq> p \<and> co2sobj s (O_proc p') = Some sp)
+ then s2ss s \<union> {sp', sf}
+ else s2ss s - {sp} \<union> {sp', sf}
+ | _ \<Rightarrow> {} )"
+unfolding s2ss_def
+apply (frule vt_grant_os, frule vd_cons)
+apply (case_tac "co2sobj s (O_proc p)", simp add:co2sobj.simps split:option.splits)
+apply (drule current_proc_has_sp', simp, simp)
+apply (case_tac "co2sobj (Open p f flag fd (Some i) # s) (O_proc p)")
+apply (simp add:co2sobj.simps split:option.splits)
+apply (drule current_proc_has_sp', simp, simp)
+apply (case_tac "co2sobj (Open p f flag fd (Some i) # s) (O_file f)")
+apply (simp add:co2sobj.simps split:option.splits)
+apply (clarsimp split del:if_splits)
+
+apply (rule set_eqI, rule iffI, erule CollectE, erule exE, erule conjE)
+apply (split if_splits, rule conjI, rule impI, erule exE, erule conjE, erule conjE)
+apply (case_tac "obj = O_proc p", simp, case_tac "obj = O_file f", simp)
+apply (rule UnI1, rule CollectI, rule_tac x = obj in exI)
+apply (frule_tac obj = obj in alive_co2sobj_some_open_some', simp+)
+apply (simp add:co2sobj_open split:t_object.splits)
+apply (rule impI, case_tac "obj = O_proc p", simp, case_tac "obj = O_file f", simp)
+apply (rule UnI1, rule DiffI, rule CollectI, rule_tac x = obj in exI)
+apply (frule_tac obj = obj in alive_co2sobj_some_open_some', simp+)
+apply (simp add:co2sobj_open split:t_object.splits)
+apply (frule_tac obj = obj in co2sobj_open_some, simp+)
+apply (simp add:alive_co2sobj_some_open_some', simp)
+apply (rule notI)
+apply (frule_tac obj = obj and p = p in co2sobj_proc_obj, simp+, erule exE)
+apply (erule_tac x = p' in allE, simp)
+
+apply (simp split:if_splits, erule disjE)
+apply (rule_tac x = "O_proc p" in exI, simp)
+apply (erule disjE, rule_tac x = "O_file f" in exI, simp add:is_file_simps)
+apply (erule exE, erule conjE)
+apply (case_tac "obj = O_proc p", simp)
+apply (rule_tac x = "O_proc p'" in exI, simp add:co2sobj_open_some)
+apply (case_tac "obj = O_file f", simp add:is_file_in_current)
+apply (rule_tac x = obj in exI, simp add:co2sobj_open_some alive_co2sobj_some_open_some)
+apply (erule disjE, rule_tac x = "O_proc p" in exI, simp)
+apply (erule disjE, rule_tac x = "O_file f" in exI, simp add:is_file_simps)
+apply (erule conjE, erule exE, erule conjE)
+apply (case_tac "obj = O_proc p", simp)
+apply (case_tac "obj = O_file f", simp add:is_file_in_current)
+apply (rule_tac x = obj in exI, simp add:co2sobj_open_some alive_co2sobj_some_open_some)
+done
+
+lemma s2ss_open:
+ "valid (Open p f flag fd opt # s) \<Longrightarrow> s2ss (Open p f flag fd opt # s) = (
+ if opt = None
+ then (case (co2sobj s (O_proc p), co2sobj (Open p f flag fd opt # s) (O_proc p)) of
+ (Some sp, Some sp') \<Rightarrow>
+ if (\<exists> p'. p' \<in> current_procs s \<and> p' \<noteq> p \<and> co2sobj s (O_proc p') = Some sp)
+ then s2ss s \<union> {sp'}
+ else s2ss s - {sp} \<union> {sp'}
+ | _ \<Rightarrow> {} )
+ else (case (co2sobj s (O_proc p), co2sobj (Open p f flag fd opt # s) (O_proc p),
+ co2sobj (Open p f flag fd opt # s) (O_file f)) of
+ (Some sp, Some sp', Some sf) \<Rightarrow>
+ if (\<exists> p'. p' \<in> current_procs s \<and> p' \<noteq> p \<and> co2sobj s (O_proc p') = Some sp)
+ then s2ss s \<union> {sp', sf}
+ else s2ss s - {sp} \<union> {sp', sf}
+ | _ \<Rightarrow> {} ) )"
+apply (case_tac opt)
+apply (simp add:s2ss_open_some s2ss_open_none)+
+done
+
+lemma
-apply (rule conjI, clarify)
-apply (frule_tac p = p in current_proc_has_sp, simp, erule exE)
-apply (frule_tac p = p' in current_proc_has_sp, simp, erule exE)
-apply (simp, (erule conjE)+)
-apply (split option.splits, rule conjI, rule impI, drule current_proc_has_sp', simp, simp)
-apply (rule allI, rule impI)
-apply (rule set_eqI, rule iffI)
+lemmas s2ss_simps = s2ss_execve s2ss_clone s2ss_ptrace s2ss_kill s2ss_exit s2ss_open
-apply (simp split:option.splits)
-apply (frule_tac p = p and s = "Execve p f fds # s" in current_proc_has_sp)
-thm current_proc_has_sp
-
-
-apply (simp split:option.splits)
-apply (drule current_proc_has_sp', simp, simp)
-apply (rule conjI, rule impI, drule current_proc_has_sp', simp, simp)
-apply (simp add:s2ss_def)
-apply (rule allI|rule impI)+
-apply (rule set_eqI, rule iffI)
-apply (auto simp:alive_simps)
-apply (case_tac obj, auto split:option.splits simp:cp2sproc_execve)
-apply (auto split:if_splits)
\ No newline at end of file