theory Dynamic_static

imports Main Flask Static Init_prop Valid_prop Tainted_prop Delete_prop Co2sobj_prop S2ss_prop S2ss_prop2

begin

context tainting_s begin

definition init_ss_eq:: "t_static_state \<Rightarrow> t_static_state \<Rightarrow> bool" (infix "\<doteq>" 100)

where

  "ss \<doteq> ss' \<equiv> ss \<subseteq> ss' \<and> {sobj. is_init_sobj sobj \<and> sobj \<in> ss'} \<subseteq> ss"

lemma [simp]: "ss \<doteq> ss"

by (auto simp:init_ss_eq_def)

definition init_ss_in:: "t_static_state \<Rightarrow> t_static_state set \<Rightarrow> bool" (infix "\<propto>" 101)

where

  "ss \<propto> sss \<equiv> \<exists> ss' \<in> sss. ss \<doteq> ss'"

lemma s2ss_included_sobj:

  "\<lbrakk>alive s obj; co2sobj s obj= Some sobj\<rbrakk> \<Longrightarrow> sobj \<in> (s2ss s)"

by (simp add:s2ss_def, rule_tac x = obj in exI, simp)

lemma init_ss_in_prop:

  "\<lbrakk>s2ss s \<propto> static; co2sobj s obj = Some sobj; alive s obj; init_obj_related sobj obj\<rbrakk>

   \<Longrightarrow> \<exists> ss \<in> static. sobj \<in> ss"

apply (simp add:init_ss_in_def init_ss_eq_def)

apply (erule bexE, erule conjE)

apply (rule_tac x = ss' in bexI, auto dest!:s2ss_included_sobj)

done

definition enrich:: "t_state \<Rightarrow> t_object set \<Rightarrow> t_state \<Rightarrow> bool"

where

  "enrich s objs s' \<equiv> \<forall> obj \<in> objs. \<exists> obj'. obj' \<notin> objs \<and> alive s' obj \<and> co2sobj s' obj' = co2sobj s' obj"

definition reserve:: "t_state \<Rightarrow> t_object set \<Rightarrow> t_state \<Rightarrow> bool"

where

  "reserve s objs s' \<equiv> \<forall> obj. alive s obj \<longrightarrow> alive s' obj \<and> co2sobj s' obj = co2sobj s obj"

definition enrichable :: "t_state \<Rightarrow> t_object set \<Rightarrow> bool"

where

  "enrichable s objs \<equiv> \<exists> s'. valid s' \<and> s2ss s' = s2ss s \<and> enrich s objs s' \<and> reserve s objs s'"

definition is_created :: "t_state \<Rightarrow> t_object \<Rightarrow> bool"

where

  "is_created s obj \<equiv> init_alive obj \<longrightarrow> deleted obj s"

definition is_inited :: "t_state \<Rightarrow> t_object \<Rightarrow> bool"

where

  "is_inited s obj \<equiv> init_alive obj \<and> \<not> deleted obj s"

lemma is_inited_eq_not_created:

  "is_inited s obj = (\<not> is_created s obj)"

by (auto simp:is_created_def is_inited_def)

(* recorded in our static world *)

fun recorded :: "t_object \<Rightarrow> bool"

where

  "recorded (O_proc p)     = True"

| "recorded (O_file f)     = True"

| "recorded (O_dir  f)     = True"

| "recorded (O_node n)     = False" (* cause socket is temperary not considered *)

| "recorded (O_shm  h)     = True"

| "recorded (O_msgq q)     = True"

| "recorded _              = False"

lemma d2s_main_execve:

  "valid (Execve p f fds # s) \<Longrightarrow> s2ss (Execve p f fds # s) \<in> static"

apply (frule vd_cons, frule vt_grant_os, clarsimp simp:s2ss_execve)

sorry

lemma d2s_main:

  "valid s \<Longrightarrow> s2ss s \<propto> static"

apply (induct s, simp add:s2ss_nil_prop init_ss_in_def)

apply (rule_tac x = "init_static_state" in bexI, simp, simp add:s_init)

apply (frule vd_cons, frule vt_grant_os, simp)

apply (case_tac a) 

apply (clarsimp simp add:s2ss_execve)

apply (rule conjI, rule impI)

sorry

lemma many_sq_imp_sms:

  "\<lbrakk>S_msgq (Create, sec, sms) \<in> ss; ss \<in> static\<rbrakk> \<Longrightarrow> \<forall> sm \<in> (set sms). is_many_smsg sm"

sorry

lemma enrichability: 

  "\<lbrakk>valid s; \<forall> obj \<in> objs. alive s obj \<and> is_created s obj \<and> recorded obj\<rbrakk>

   \<Longrightarrow> enrichable s objs"

proof (induct s arbitrary:objs)

  case Nil

  hence "objs = {}" 

    apply (auto simp:is_created_def)

    apply (erule_tac x = x in ballE)

    apply (auto simp:init_alive_prop)

    done

  thus ?case using Nil unfolding enrichable_def enrich_def reserve_def

    by (rule_tac x = "[]" in exI, auto)

next

  case (Cons e s)

  hence p1: "\<And> objs. \<forall> obj \<in> objs. alive s obj \<and> is_created s obj \<and> recorded obj \<Longrightarrow> enrichable s objs"

    and p2: "valid (e # s)" and p3: "\<forall>obj\<in>objs. alive (e # s) obj \<and> is_created (e # s) obj \<and> recorded obj"

    and os: "os_grant s e" and se: "grant s e" and vd: "valid s"

    by (auto dest:vt_grant_os vd_cons vt_grant)

  show ?case sorry (*

  proof (cases e)

    case (Execve p f fds)

    hence p4: "e = Execve p f fds" by simp

    from p3 have p5: "is_inited s (O_proc p) \<Longrightarrow> (O_proc p) \<notin> objs"

      by (auto simp:is_created_def is_inited_def p4 elim!:ballE[where x = "O_proc p"])

    show "enrichable (e # s) objs"

    proof (case "is_inited s (O_proc p)")

      apply (simp add:enrichable_def p4)

    apply auto

    apply (auto simp:enrichable_def)

apply (induct s)

done

*)

qed

lemma s2d_main:

  "ss \<in> static \<Longrightarrow> \<exists> s. valid s \<and> s2ss s = ss"

apply (erule static.induct)

apply (rule_tac x = "[]" in exI, simp add:s2ss_nil_prop valid.intros)

apply (erule exE|erule conjE)+

apply (simp add:update_ss_def)

sorry

end

end