theory Finite_current

imports Main Valid_prop Flask Flask_type Proc_fd_of_file_prop

begin

context flask begin

lemma finite_cf: "valid \<tau> \<Longrightarrow> finite (current_files \<tau>)"

apply (induct \<tau>)

apply (simp add:current_files_def inum_of_file.simps)

apply (rule_tac B = "init_files" in finite_subset)

apply (clarsimp dest!:inof_has_file_tag, simp add:init_finite_sets)

apply (frule vt_grant_os, frule vd_cons, simp, case_tac a)

apply (auto simp:current_files_def os_grant.simps inum_of_file.simps split:if_splits option.splits)

apply (rule_tac B = "insert list {f. \<exists>i. inum_of_file \<tau> f = Some i}" in finite_subset, clarsimp, simp)

apply (rule_tac B = "{f. \<exists>i. inum_of_file \<tau> f = Some i}" in finite_subset, clarsimp, simp)

apply (rule_tac B = "{f. \<exists>i. inum_of_file \<tau> f = Some i}" in finite_subset, clarsimp, simp)

apply (rule_tac B = "{f. \<exists>i. inum_of_file \<tau> f = Some i}" in finite_subset, clarsimp, simp)

apply (rule_tac B = "insert list {f. \<exists>i. inum_of_file \<tau> f = Some i}" in finite_subset, clarsimp, simp)

apply (rule_tac B = "insert list2 {f. \<exists>i. inum_of_file \<tau> f = Some i}" in finite_subset, clarsimp, simp)

done

lemma finite_cp: "finite (current_procs \<tau>)"

apply (induct \<tau>)

apply (simp add:current_procs.simps init_finite_sets)

apply (case_tac a, auto simp:current_procs.simps)

done

lemma finite_cfd: "valid \<tau> \<Longrightarrow> finite (current_proc_fds \<tau> p)"

apply (induct \<tau> arbitrary:p)

apply (simp add:current_proc_fds.simps init_finite_sets)

apply (frule vd_cons, frule vt_grant_os, case_tac a, auto simp:current_proc_fds.simps)

apply (erule finite_subset)

apply (frule_tac s = \<tau> and p = nat in file_fds_subset_pfds)

apply (erule finite_subset, simp)

apply (erule finite_subset)

apply (frule_tac s = \<tau> and p = nat1 in file_fds_subset_pfds)

apply (erule finite_subset, simp)

done

lemma finite_pair: "\<lbrakk>finite A; finite B\<rbrakk> \<Longrightarrow> finite {(x, y). x \<in> A \<and> y \<in> B}"

by auto

lemma finite_UN_I': "\<lbrakk>finite X; \<forall> x. x \<in> X \<longrightarrow> finite (f x)\<rbrakk> \<Longrightarrow> finite {(x, y). x \<in> X \<and> y \<in> f x}"

apply (frule_tac B = f in finite_UN_I, simp)

apply (drule_tac finite_pair, simp)

apply (rule_tac B = "{(x, y). x \<in> X \<and> y \<in> (\<Union>a\<in>X. f a)}" in finite_subset, auto)

done

lemma finite_init_netobjs: "finite init_sockets"

apply (subgoal_tac "finite {(p, fd). p \<in> init_procs \<and> fd \<in> init_fds_of_proc p}")

apply (rule_tac B = "{(p, fd). p \<in> init_procs \<and> fd \<in> init_fds_of_proc p}" in finite_subset)

apply (clarsimp dest!:init_socket_has_inode, simp)

using init_finite_sets finite_UN_I'

by (metis Collect_mem_eq SetCompr_Sigma_eq internal_split_def) 

lemma finite_cn_aux: "valid \<tau> \<Longrightarrow> finite {s. \<exists>i. inum_of_socket \<tau> s = Some i}"

apply (induct \<tau>)

apply (rule_tac B = "init_sockets" in finite_subset)

apply (clarsimp simp:inum_of_socket.simps dest!:inos_has_sock_tag, simp add:finite_init_netobjs)

apply (frule vd_cons, frule vt_grant_os, simp, case_tac a)

apply (auto split:option.splits if_splits) 

apply (rule_tac B = "{s. \<exists>i. inum_of_socket \<tau> s = Some i}" in finite_subset, clarsimp split:if_splits, simp)

apply (rule_tac B = "{s. \<exists>i. inum_of_socket \<tau> s = Some i} \<union> {(p, fd). \<exists> i. inum_of_socket \<tau> (nat1, fd) = Some i \<and> p = nat2 \<and> fd \<in> set1}" in finite_subset, clarsimp split:if_splits)

apply (simp only:finite_Un, rule conjI, simp)

apply (rule_tac B = "{(p, fd). \<exists> i. inum_of_socket \<tau> (nat1, fd) = Some i \<and> p = nat2}" in finite_subset, clarsimp)

apply (drule_tac h = "\<lambda> (p, fd). if (p = nat1) then (nat2, fd) else (p, fd)" in finite_imageI)

apply (rule_tac B = "((\<lambda>(p, fd). if p = nat1 then (nat2, fd) else (p, fd)) ` {a. \<exists>i. inum_of_socket \<tau> a = Some i})" in finite_subset) 

apply (rule subsetI,erule CollectE, case_tac x, simp, (erule exE|erule conjE)+)

unfolding image_def

apply (rule CollectI, rule_tac x = "(nat1, b)" in bexI, simp+)

apply (rule_tac B = "{s. \<exists>i. inum_of_socket \<tau> s = Some i}" in finite_subset, clarsimp split:if_splits, simp)+

apply (rule_tac B = "insert (nat1, nat2) {s. \<exists>i. inum_of_socket \<tau> s = Some i}" in finite_subset, clarsimp, simp)

apply (rule_tac B = "insert (nat1, nat4) {s. \<exists>i. inum_of_socket \<tau> s = Some i}" in finite_subset, clarsimp, simp)

done

lemma finite_cn: "valid \<tau> \<Longrightarrow> finite (current_sockets \<tau>)"

apply (simp add:current_sockets_def inum_of_socket.simps)

using finite_cn_aux[where \<tau> = \<tau>] by auto

lemma finite_ch: "finite (current_shms \<tau>)"

apply (induct \<tau>) defer

apply (case_tac a, auto simp:current_shms.simps init_finite_sets)

done

lemma finite_cm: "finite (current_msgqs \<tau>)"

apply (induct \<tau>) defer

apply (case_tac a, auto simp: init_finite_sets)

done

lemma finite_option: "finite {x. \<exists> y. f x = Some y} \<Longrightarrow> finite {y. \<exists> x. f x = Some y}"

apply (drule_tac h = f in finite_imageI)

apply (clarsimp simp only:image_def)

apply (rule_tac f = Some in finite_imageD)

apply (rule_tac B = "{y. \<exists>x. (\<exists>y. f x = Some y) \<and> y = f x}" in finite_subset)

unfolding image_def

apply auto

done

lemma finite_ci: "valid \<tau> \<Longrightarrow> finite (current_inode_nums \<tau>)"

apply (simp add:current_inode_nums_def current_file_inums_def current_sock_inums_def)

apply (rule conjI, drule finite_cf, simp add:current_files_def, erule finite_option) 

using finite_cn[where \<tau> = \<tau>] 

apply (simp add:current_sockets_def, drule_tac finite_option, simp)

done

end

end