theory New_obj_prop

imports Main Finite_current Flask_type Flask Static

begin

context tainting_s begin

lemma nn_notin_aux: "finite s \<Longrightarrow> \<forall> a \<in> s. Max s \<ge> a "

apply (erule finite.induct, simp)

apply (rule ballI)

apply (case_tac "aa = a", simp+)

done

lemma nn_notin: "finite s \<Longrightarrow> next_nat s \<notin> s"

apply (drule nn_notin_aux)

apply (simp add:next_nat_def)

by (auto)

(* lemmas of new created obj *)

lemma np_notin_curp: "valid \<tau> \<Longrightarrow> new_proc \<tau> \<notin> current_procs \<tau>" using finite_cp

by (simp add:new_proc_def nn_notin)

lemma np_notin_curp': "\<lbrakk>new_proc \<tau> \<in> current_procs \<tau>; valid \<tau>\<rbrakk> \<Longrightarrow> False"

apply (drule np_notin_curp, simp)

done

lemma ni_notin_curi: "valid \<tau> \<Longrightarrow> new_inode_num \<tau> \<notin> current_inode_nums \<tau>"

apply (drule finite_ci)

by (simp add:new_inode_num_def nn_notin)

lemma ni_notin_curi': "\<lbrakk>new_inode_num \<tau> \<in> current_inode_nums \<tau>; valid \<tau>\<rbrakk> \<Longrightarrow> False"

by (drule ni_notin_curi, simp)

lemma nm_notin_curm: "valid \<tau> \<Longrightarrow> new_msgq \<tau> \<notin> current_msgqs \<tau>" using finite_cm

by (simp add:new_msgq_def nn_notin)

lemma nm_notin_curm': "\<lbrakk>new_msgq \<tau> \<in> current_msgqs \<tau>; valid \<tau>\<rbrakk> \<Longrightarrow> False"

by (drule nm_notin_curm, simp)

(*

lemma nh_notin_curh: "valid \<tau> \<Longrightarrow> new_shm \<tau> \<notin> current_shms \<tau>" using finite_ch

by (simp add:new_shm_def nn_notin)

lemma nh_notin_curh': "\<lbrakk>new_shm \<tau> \<in> current_shms \<tau>; valid \<tau>\<rbrakk> \<Longrightarrow> False"

by (drule nh_notin_curh, simp)

*)

lemma nfd_notin_curfd: "valid \<tau> \<Longrightarrow> new_proc_fd \<tau> p \<notin> current_proc_fds \<tau> p" 

using finite_cfd[where p = p]

apply (simp add:new_proc_fd_def nn_notin)

done

lemma nfd_notin_curfd': "\<lbrakk>new_proc_fd \<tau> p \<in> current_proc_fds \<tau> p; valid \<tau>\<rbrakk> \<Longrightarrow> False"

by (drule nfd_notin_curfd[where p = p], simp)

lemma nim_notin_curim: "valid \<tau> \<Longrightarrow> new_inode_num \<tau> \<notin> current_inode_nums \<tau>"

by (drule finite_ci, simp add:new_inode_num_def nn_notin)

lemma nim_notin_curim': "\<lbrakk>new_inode_num \<tau> \<in> current_inode_nums \<tau>; valid \<tau>\<rbrakk> \<Longrightarrow> False"

by (drule nim_notin_curim, simp)

lemma len_fname_all: "length (fname_all_a len) = len"

by (induct len, auto simp:fname_all_a.simps)

lemma finite_fs_fnames:

  "finite (fs::t_file set) \<Longrightarrow> finite {fn. fn # f \<in> fs}"

thm finite_imageI

apply (drule_tac h = "\<lambda> f'. case f' of [] \<Rightarrow> '''' | fn # pf' \<Rightarrow> if (pf' = f) then fn else ''''" in finite_imageI)

apply (rule_tac B = "(list_case [] (\<lambda>fn pf'. if pf' = f then fn else []) ` fs)" in finite_subset)

unfolding image_def

apply (clarsimp  split:if_splits)

apply (rule_tac x = "x # f" in bexI, simp+) 

done

lemma ncf_notin_curf_aux:

  "finite (fs::t_file set) \<Longrightarrow> Suc (Max (fname_length_set {fn. fn # f \<in> fs})) \<notin> fname_length_set {fn. fn # f \<in> fs}"

apply (drule_tac f = f in finite_fs_fnames)

apply (drule_tac h = length in finite_imageI)

apply (simp add:fname_length_set_def)

apply (drule nn_notin_aux)

apply auto

done

lemma ncf_notin_curf_general:

proof-

    by (erule_tac ncf_notin_curf_aux)

  have a2: "\<And> f pf fs. f # pf \<in> fs \<Longrightarrow> f \<in> {fn. fn # pf \<in> fs}" by auto

  have a3: "\<And> f pf fs. f \<in> {fn. fn # pf \<in> fs} \<Longrightarrow> length f \<in> fname_length_set {fn. fn # pf \<in> fs}"

    by (auto simp:fname_length_set_def)

  show ?thesis

    apply (simp only:new_childf_general_def next_fname_def all_fname_under_dir_def)

    apply (rule notI)

    apply (drule a2, drule a3)

    apply (simp only:len_fname_all)

    using a1 by auto

qed

lemma ncf_notin_curf:

 "valid \<tau> \<Longrightarrow> new_childf f \<tau> \<notin> (current_files \<tau>)"

apply (simp add:new_childf_def)

done

by (simp add:new_childf_general_def)

lemma ncf_parent: "valid \<tau> \<Longrightarrow> parent (new_childf f \<tau>) = Some f"

by (simp add:new_childf_def ncf_parent_general)

end

end