theory Current_prop
imports Main Flask_type Flask My_list_prefix Init_prop Valid_prop Delete_prop
begin
(*<*)

context flask begin

lemma procs_of_shm_prop1: "\<lbrakk> p_flag \<in> procs_of_shm s h; valid s\<rbrakk> \<Longrightarrow> h \<in> current_shms s"
apply (induct s arbitrary:p_flag)
apply (case_tac p_flag, simp, drule init_procs_has_shm, simp)
apply (frule vd_cons, frule vt_grant_os)
apply (case_tac a, auto split:if_splits option.splits)
done

lemma procs_of_shm_prop2: "\<lbrakk>(p, flag) \<in> procs_of_shm s h; valid s\<rbrakk> \<Longrightarrow> p \<in> current_procs s"
apply (induct s arbitrary:p flag)
apply (simp, drule init_procs_has_shm, simp)
apply (frule vd_cons, frule vt_grant_os)
apply (case_tac a, auto split:if_splits option.splits)
done

lemma procs_of_shm_prop3: "\<lbrakk>(p, flag) \<in> procs_of_shm s h; (p, flag') \<in> procs_of_shm s h; valid s\<rbrakk>
  \<Longrightarrow> flag = flag'"
apply (induct s arbitrary:p flag flag')
apply (simp, drule_tac flag = flag in init_procs_has_shm, drule_tac flag = flag' in init_procs_has_shm, simp)
apply (frule vd_cons, frule vt_grant_os)
apply (case_tac a, auto split:if_splits option.splits dest:procs_of_shm_prop2)
done

lemma procs_of_shm_prop4: "\<lbrakk>(p, flag) \<in> procs_of_shm s h; valid s\<rbrakk> \<Longrightarrow> flag_of_proc_shm s p h = Some flag"
apply (induct s arbitrary:p flag)
apply (simp, drule init_procs_has_shm, simp)
apply (frule vd_cons, frule vt_grant_os)
apply (case_tac a, auto split:if_splits option.splits dest:procs_of_shm_prop2)
done

lemma procs_of_shm_prop4':
  "\<lbrakk>flag_of_proc_shm s p h = None; valid s\<rbrakk> \<Longrightarrow> \<forall> flag. (p, flag) \<notin> procs_of_shm s h"
by (auto dest:procs_of_shm_prop4)

lemma not_init_intro_proc:
  "\<lbrakk>p \<notin> current_procs s; valid s\<rbrakk> \<Longrightarrow> deleted (O_proc p) s \<or> p \<notin> init_procs"
using not_deleted_init_proc by auto

lemma not_init_intro_proc':
  "\<lbrakk>p \<notin> current_procs s; valid s\<rbrakk> \<Longrightarrow> \<not> (\<not> deleted (O_proc p) s \<and> p \<in> init_procs)"
using not_deleted_init_proc by auto

end

end