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 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 ncf_notin_curf: "valid \<tau> \<Longrightarrow> new_childf f \<tau> \<notin> current_files \<tau>"

apply (drule finite_cf)

apply (simp add:new_childf_def next_fname_def all_fname_under_dir_def)

apply (rule notI)

apply (subgoal_tac "(CHR ''a'' # fname_all_a (Max (fname_length_set {fn. fn # f \<in> current_files \<tau>}))) \<in> {fn. fn # f \<in> current_files \<tau>}")

defer apply simp

apply (subgoal_tac "length (CHR ''a'' # fname_all_a (Max (fname_length_set {fn. fn # f \<in> current_files \<tau>}))) \<in> fname_length_set {fn. fn # f \<in> current_files \<tau>}")

defer apply (auto simp:fname_length_set_def image_def)[1]

apply (subgoal_tac "finite (fname_length_set {fn. fn # f \<in> current_files \<tau>})")  

defer

apply (simp add:fname_length_set_def) 

apply (rule 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 []) ` current_files \<tau>)" in finite_subset)

unfolding image_def

apply (clarsimp  split:if_splits)

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

apply (drule_tac s = "(fname_length_set {fn. fn # f \<in> current_files \<tau>})" in nn_notin_aux)

apply (erule_tac x = "length (CHR ''a'' # fname_all_a (Max (fname_length_set {fn. fn # f \<in> current_files \<tau>})))" in ballE)

apply (simp add:len_fname_all, simp)

done

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

by (simp add:new_childf_def)

end

end