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1 theory Dynamic_static |
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2 imports Main Flask Static Init_prop Valid_prop Tainted_prop Delete_prop Co2sobj_prop S2ss_prop S2ss_prop2 |
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3 Temp Enrich |
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4 begin |
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5 |
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6 context tainting_s begin |
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7 |
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8 |
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9 |
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10 lemma "alive s obj \<Longrightarrow> alive (enrich_proc s p p') obj" |
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11 apply (induct s, simp) |
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12 apply (case_tac a, case_tac[!] obj) sorry (* |
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13 apply (auto simp:is_file_def is_dir_def split:option.splits t_inode_tag.splits) |
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14 thm is_file_other |
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15 *) |
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16 lemma enrich_proc_valid: |
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17 "\<lbrakk>p \<in> current_procs s; valid s; p \<in> init_procs \<longrightarrow> deleted (O_proc p) s; p' \<notin> current_procs s\<rbrakk> \<Longrightarrow> valid (enrich_proc s p p')" (* |
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18 apply (induct s, simp) |
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19 apply (frule vd_cons, frule vt_grant, frule vt_grant_os, case_tac a) |
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20 apply (auto intro!:valid.intros(2)) |
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21 prefer 28 |
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22 |
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23 |
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24 |
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25 end |
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26 *) |
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27 sorry |
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28 |
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29 |
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30 |
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31 |
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32 |
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33 (* for any created obj, we can enrich trace with events that create new objs with the same static-properties *) |
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34 definition enriched:: "t_state \<Rightarrow> t_object set \<Rightarrow> t_state \<Rightarrow> bool" |
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35 where |
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36 "enriched s objs s' \<equiv> \<forall> obj \<in> objs. \<exists> obj'. \<not> alive s obj' \<and> obj' \<notin> objs \<and> |
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37 alive s' obj' \<and> co2sobj s' obj' = co2sobj s' obj" |
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38 |
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39 definition reserved:: "t_state \<Rightarrow> t_object set \<Rightarrow> t_state \<Rightarrow> bool" |
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40 where |
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41 "reserved s objs s' \<equiv> \<forall> obj. alive s obj \<longrightarrow> alive s' obj \<and> co2sobj s' obj = co2sobj s obj" |
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42 |
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43 definition enrichable :: "t_state \<Rightarrow> t_object set \<Rightarrow> bool" |
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44 where |
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45 "enrichable s objs \<equiv> \<exists> s'. valid s' \<and> s2ss s' = s2ss s \<and> enriched s objs s' \<and> reserved s objs s'" |
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46 |
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47 fun is_created :: "t_state \<Rightarrow> t_object \<Rightarrow> bool" |
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48 where |
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49 "is_created s (O_file f) = (\<forall> f' \<in> same_inode_files s f. init_alive (O_file f') \<longrightarrow> deleted (O_file f') s)" |
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50 | "is_created s obj = (init_alive obj \<longrightarrow> deleted obj s)" |
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51 |
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52 definition is_inited :: "t_state \<Rightarrow> t_object \<Rightarrow> bool" |
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53 where |
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54 "is_inited s obj \<equiv> init_alive obj \<and> \<not> deleted obj s" |
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55 |
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56 (* |
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57 lemma is_inited_eq_not_created: |
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58 "is_inited s obj = (\<not> is_created s obj)" |
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59 by (auto simp:is_created_def is_inited_def) |
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60 *) |
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61 |
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62 lemma many_sq_imp_sms: |
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63 "\<lbrakk>S_msgq (Create, sec, sms) \<in> ss; ss \<in> static\<rbrakk> \<Longrightarrow> \<forall> sm \<in> (set sms). is_many_smsg sm" |
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64 sorry |
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65 |
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66 (* recorded in our static world *) |
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67 fun recorded :: "t_object \<Rightarrow> bool" |
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68 where |
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69 "recorded (O_proc p) = True" |
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70 | "recorded (O_file f) = True" |
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71 | "recorded (O_dir f) = True" |
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72 | "recorded (O_node n) = False" (* cause socket is temperary not considered *) |
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73 | "recorded (O_shm h) = True" |
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74 | "recorded (O_msgq q) = True" |
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75 | "recorded _ = False" |
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76 |
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77 lemma cf2sfile_fi_init_file: |
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78 "\<lbrakk>cf2sfile s f = Some (Init f', sec, psec, asecs); is_file s f; valid s\<rbrakk> |
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79 \<Longrightarrow> is_init_file f \<and> \<not> deleted (O_file f) s" |
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80 apply (simp add:cf2sfile_def sroot_def split:option.splits if_splits) |
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81 apply (case_tac f, simp, drule root_is_dir', simp+) |
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82 done |
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83 |
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84 lemma root_not_deleted: |
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85 "valid s \<Longrightarrow> \<not> deleted (O_dir []) s" |
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86 apply (induct s, simp) |
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87 apply (frule vd_cons, frule vt_grant_os, case_tac a) |
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88 by auto |
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89 |
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90 lemma cf2sfile_fi_init_dir: |
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91 "\<lbrakk>cf2sfile s f = Some (Init f', sec, psec, asecs); is_dir s f; valid s\<rbrakk> |
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92 \<Longrightarrow> is_init_dir f \<and> \<not> deleted (O_dir f) s" |
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93 apply (simp add:cf2sfile_def sroot_def split:option.splits if_splits) |
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94 apply (case_tac f, simp add:root_is_init_dir root_not_deleted, simp) |
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95 apply (drule file_dir_conflict, simp+) |
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96 done |
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97 |
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98 lemma is_created_imp_many: |
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99 "\<lbrakk>is_created s obj; co2sobj s obj = Some sobj; alive s obj; valid s\<rbrakk> \<Longrightarrow> is_many sobj" |
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100 apply (case_tac obj, auto simp:co2sobj.simps split:option.splits) |
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101 apply (case_tac [!] a) |
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102 apply (auto simp:cp2sproc_def ch2sshm_def cq2smsgq_def cf2sfiles_def same_inode_files_def |
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103 split:option.splits if_splits) |
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104 apply (frule cf2sfile_fi_init_file, simp add:is_file_def, simp) |
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105 apply (erule_tac x = f' in allE, simp) |
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106 apply (frule cf2sfile_fi_init_dir, simp+)+ |
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107 done |
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108 |
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109 lemma anotherp_imp_manysp: |
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110 "\<lbrakk>cp2sproc s p = Some sp; co2sobj s (O_proc p') = co2sobj s (O_proc p); p' \<noteq> p; |
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111 p' \<in> current_procs s; p \<in> current_procs s\<rbrakk> |
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112 \<Longrightarrow> is_many_sproc sp" |
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113 by (case_tac sp, auto simp:cp2sproc_def co2sobj.simps split:option.splits if_splits) |
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114 |
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115 lemma is_file_has_sfs: |
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116 "\<lbrakk>is_file s f; valid s; cf2sfile s f = Some sf\<rbrakk> |
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117 \<Longrightarrow> \<exists> sfs. co2sobj s (O_file f) = Some (S_file sfs (O_file f \<in> Tainted s)) \<and> sf \<in> sfs" |
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118 apply (rule_tac x = "{sf' | f' sf'. cf2sfile s f' = Some sf' \<and> f' \<in> same_inode_files s f}" in exI) |
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119 apply (auto simp:co2sobj.simps cf2sfiles_def tainted_eq_Tainted) |
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120 apply (rule_tac x = f in exI, simp add:same_inode_files_prop9) |
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121 done |
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122 |
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123 declare Product_Type.split_paired_Ex Product_Type.split_paired_All [simp del] |
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124 |
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125 lemma current_proc_in_s2ss: |
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126 "\<lbrakk>cp2sproc s p = Some sp; p \<in> current_procs s; valid s\<rbrakk> |
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127 \<Longrightarrow> S_proc sp (O_proc p \<in> Tainted s) \<in> s2ss s" |
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128 apply (simp add:s2ss_def, rule_tac x = "O_proc p" in exI) |
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129 apply (auto simp:co2sobj.simps tainted_eq_Tainted) |
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130 done |
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131 |
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132 lemma current_file_in_s2ss: |
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133 "\<lbrakk>co2sobj s (O_file f) = Some (S_file sfs tagf); is_file s f; valid s\<rbrakk> |
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134 \<Longrightarrow> S_file sfs tagf \<in> s2ss s" |
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135 by (simp add:s2ss_def, rule_tac x = "O_file f" in exI, simp) |
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136 |
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137 declare npctxt_execve.simps grant_execve.simps search_check.simps [simp del] |
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138 |
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139 |
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140 lemma npctxt_execve_eq_sec: |
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141 "\<lbrakk>sectxt_of_obj (Execve p f fds # s) (O_proc p) = Some sec'; sectxt_of_obj s (O_proc p) = Some sec; |
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142 sectxt_of_obj s (O_file f) = Some fsec; valid (Execve p f fds # s)\<rbrakk> |
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143 \<Longrightarrow> npctxt_execve sec fsec = Some sec'" |
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144 by (case_tac sec, case_tac fsec, auto simp:npctxt_execve.simps sectxt_of_obj_simps split:option.splits) |
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145 |
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146 lemma npctxt_execve_eq_cp2sproc: |
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147 "\<lbrakk>cp2sproc (Execve p f fds # s) p = Some (pi', sec', sfds', shms'); valid (Execve p f fds # s); |
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148 cp2sproc s p = Some (pi, sec, sfds, shms); cf2sfile s f = Some (fi, fsec, psec, asecs)\<rbrakk> |
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149 \<Longrightarrow> npctxt_execve sec fsec = Some sec'" |
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150 apply (frule vt_grant_os, frule vd_cons) |
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151 apply (rule npctxt_execve_eq_sec, auto simp:cp2sproc_def cf2sfile_def split:option.splits) |
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152 apply (case_tac f, auto dest:root_is_dir') |
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153 done |
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154 |
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155 lemma seach_check_eq_static: |
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156 "\<lbrakk>cf2sfile s f = Some sf; valid s; is_dir s f \<or> is_file s f\<rbrakk> |
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157 \<Longrightarrow> search_check_s sec sf (is_file s f) = search_check s sec f" |
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158 apply (case_tac sf) |
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159 apply (induct f) |
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160 apply (auto simp:search_check_s_def search_check.simps cf2sfile_def sroot_def |
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161 root_sec_remains init_sectxt_prop sec_of_root_valid |
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162 dest!:root_is_dir' current_has_sec' split:option.splits) |
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163 done |
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164 |
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165 lemma grant_execve_intro_execve: |
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166 "\<lbrakk>cp2sproc (Execve p f fds # s) p = Some (pi', sec', sfds', shms'); valid (Execve p f fds # s); |
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167 cp2sproc s p = Some (pi, sec, sfds, shms); cf2sfile s f = Some (fi, fsec, psec, asecs)\<rbrakk> |
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168 \<Longrightarrow> grant_execve sec fsec sec'" |
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169 apply (frule vt_grant_os, frule vd_cons, frule vt_grant) |
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170 apply (auto split:option.splits dest!:current_has_sec' simp del:grant_execve.simps simp add:cp2sproc_execve) |
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171 apply (erule_tac x = aba in allE, erule_tac x = aca in allE, erule_tac x = bb in allE) |
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172 apply (auto simp del:grant_execve.simps simp add:cp2sproc_def cf2sfile_def split:option.splits) |
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173 apply (case_tac f, simp, drule root_is_dir', simp, simp, simp) |
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174 apply (simp add:sectxt_of_obj_simps) |
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175 done |
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176 |
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177 lemma search_check_intro_execve: |
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178 "\<lbrakk>cp2sproc s p = Some (pi, sec, sfds, shms); valid (Execve p f fds # s)\<rbrakk> |
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179 \<Longrightarrow> search_check s sec f" |
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180 apply (frule vt_grant_os, frule vd_cons, frule vt_grant) |
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181 apply (auto split:option.splits dest!:current_has_sec' simp del:grant_execve.simps simp add:cp2sproc_execve) |
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182 apply (erule_tac x = aaa in allE, erule_tac x = ab in allE, erule_tac x = ba in allE) |
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183 apply (auto simp add:cp2sproc_def cf2sfile_def split:option.splits) |
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184 done |
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185 |
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186 lemma inherit_fds_check_intro_execve: |
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187 "\<lbrakk>cp2sproc (Execve p f fds # s) p = Some (pi', sec', sfds', shms'); valid (Execve p f fds # s)\<rbrakk> |
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188 \<Longrightarrow> inherit_fds_check s sec' p fds" |
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189 apply (frule vt_grant_os, frule vd_cons, frule vt_grant) |
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190 apply (auto split:option.splits dest!:current_has_sec' simp add:cp2sproc_execve) |
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191 apply (erule_tac x = aba in allE, erule_tac x = aca in allE, erule_tac x = bb in allE) |
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192 apply (auto simp add:cp2sproc_def cf2sfile_def split:option.splits) |
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193 apply (simp add:sectxt_of_obj_simps) |
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194 done |
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195 |
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196 lemma execve_sfds_subset: |
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197 "\<lbrakk>cp2sproc (Execve p f fds # s) p = Some (pi', sec', sfds', shms'); valid (Execve p f fds # s); |
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198 cp2sproc s p = Some (pi, sec, sfds, shms)\<rbrakk> |
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199 \<Longrightarrow> sfds' \<subseteq> sfds" |
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200 apply (frule vt_grant_os) |
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201 apply (auto simp:cp2sproc_def cpfd2sfds_execve split:option.splits dest!:current_has_sec') |
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202 apply (simp add:cpfd2sfds_def) |
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203 apply (rule_tac x = fd in bexI, auto simp:proc_file_fds_def) |
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204 done |
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205 |
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206 lemma inherit_fds_check_imp_static: |
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207 "\<lbrakk>cp2sproc (Execve p f fds # s) p = Some (pi', sec', sfds', shms'); |
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208 inherit_fds_check s sec' p fds; valid (Execve p f fds # s)\<rbrakk> |
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209 \<Longrightarrow> inherit_fds_check_s sec' sfds'" |
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210 apply (frule vt_grant_os, frule vd_cons, frule vt_grant) |
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211 apply (auto simp:cp2sproc_def cpfd2sfds_execve inherit_fds_check_def inherit_fds_check_s_def split:option.splits) |
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212 sorry (* |
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213 apply (erule_tac x = "(ad, ae, bc)" in ballE, auto simp:sectxts_of_sfds_def sectxts_of_fds_def) |
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214 apply (erule_tac x = fd in ballE, auto simp:cfd2sfd_def split:option.splits) |
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215 done *) |
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216 |
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217 lemma d2s_main_execve_grant_aux: |
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218 "\<lbrakk>cp2sproc (Execve p f fds # s) p = Some (pi', sec', sfds', shms'); valid (Execve p f fds # s); |
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219 cp2sproc s p = Some (pi, sec, sfds, shms); cf2sfile s f = Some (fi, fsec, psec, asecs)\<rbrakk> |
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220 \<Longrightarrow> (npctxt_execve sec fsec = Some sec') \<and> grant_execve sec fsec sec' \<and> |
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221 search_check_s sec (fi, fsec, psec, asecs) (is_file s f) \<and> |
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222 inherit_fds_check_s sec' sfds' \<and> sfds' \<subseteq> sfds" |
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223 apply (rule conjI, erule_tac pi = pi and sec = sec and sfds = sfds and |
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224 shms = shms and fi = fi and fsec = fsec and psec = psec and |
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225 asecs = asecs in npctxt_execve_eq_cp2sproc, simp, simp, simp) |
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226 apply (rule conjI, erule_tac pi = pi and sec = sec and sfds = sfds and |
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227 shms = shms and fi = fi and fsec = fsec and psec = psec and |
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228 asecs = asecs in grant_execve_intro_execve, simp, simp, simp) |
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229 apply (rule conjI, drule_tac sec = sec in search_check_intro_execve, simp) |
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230 apply (frule vd_cons, frule vt_grant_os) |
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231 apply (drule_tac sec = sec in seach_check_eq_static, simp, simp, simp) |
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232 apply (rule conjI, rule inherit_fds_check_imp_static, simp) |
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233 apply (erule inherit_fds_check_intro_execve, simp, simp) |
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234 apply (erule_tac pi = pi and sfds = sfds and shms = shms in execve_sfds_subset, simp+) |
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235 done |
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236 |
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237 lemma d2s_main_execve: |
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238 "\<lbrakk>valid (Execve p f fds # s); s2ss s \<propto> static\<rbrakk> \<Longrightarrow> s2ss (Execve p f fds # s) \<propto> static" |
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239 apply (frule vd_cons, frule vt_grant_os, clarsimp) |
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240 apply (frule is_file_has_sfile', simp, erule exE, frule is_file_has_sfs, simp+, erule exE, erule conjE) |
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241 apply (auto simp:s2ss_execve split:if_splits option.splits dest:current_proc_has_sp') |
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242 apply (clarsimp simp add:init_ss_in_def init_ss_eq_def) |
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243 apply (rule_tac x = "update_ss ss' (S_proc (ah, (ai, aj, bd), ak, be) (O_proc p \<in> Tainted s)) |
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244 (S_proc (ad, (ae, af, bb), ag, bc) (O_proc p \<in> Tainted s \<or> O_file f \<in> Tainted s))" in bexI) |
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245 apply (auto simp:update_ss_def elim:Set.subset_insertI2 simp:anotherp_imp_manysp)[1] |
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246 apply (case_tac "ah = ad", case_tac "bc = {}", simp) |
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247 apply (erule_tac sfs = sfs and fi = a and fsec = "(aa, ab,b)" and pfsec = ac and asecs = ba in s_execve, |
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248 auto intro:current_proc_in_s2ss current_file_in_s2ss split:option.splits dest:d2s_main_execve_grant_aux)[1] |
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249 apply (simp add:cp2sproc_execve split:option.splits) |
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250 apply (simp add:cp2sproc_def split:option.splits if_splits) |
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251 |
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252 apply (clarsimp simp add:init_ss_in_def init_ss_eq_def) |
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253 apply (rule_tac x = "update_ss ss' (S_proc (ah, (ai, aj, bd), ak, be) (O_proc p \<in> Tainted s)) |
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254 (S_proc (ad, (ae, af, bb), ag, bc) (O_proc p \<in> Tainted s \<or> O_file f \<in> Tainted s))" in bexI) |
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255 apply (rule conjI, simp add:update_ss_def) |
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256 apply (rule conjI, simp add:update_ss_def) |
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257 apply (auto)[1] |
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258 apply (simp add:update_ss_def) |
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259 apply (rule conjI, rule impI) |
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260 apply (rule subsetI, clarsimp) |
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261 apply (erule impE) |
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262 apply (erule set_mp, simp) |
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263 apply (case_tac ah, simp+) |
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264 apply (rule impI, rule subsetI, clarsimp) |
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265 apply (erule set_mp, simp) |
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266 |
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267 apply (case_tac "ah = ad", case_tac "bc = {}", simp) |
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268 apply (erule_tac sfs = sfs and fi = a and fsec = "(aa, ab,b)" and pfsec = ac and asecs = ba in s_execve, |
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269 auto intro:current_proc_in_s2ss current_file_in_s2ss split:option.splits dest:d2s_main_execve_grant_aux)[1] |
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270 apply (simp add:cp2sproc_execve split:option.splits) |
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271 apply (simp add:cp2sproc_def split:option.splits if_splits) |
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272 done |
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273 |
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274 lemma co2sobj_eq_alive_proc_imp: |
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275 "\<lbrakk>co2sobj s obj = co2sobj s (O_proc p); alive s (O_proc p); valid s\<rbrakk> |
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276 \<Longrightarrow> \<exists> p'. obj = O_proc p'" |
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277 by (auto simp add:co2sobj.simps split:option.splits dest:current_proc_has_sp' intro:co2sobj_sproc_imp) |
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278 |
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279 |
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280 lemma enrichable_execve: |
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281 assumes p1: "\<And> objs. \<forall> obj \<in> objs. alive s obj \<and> is_created s obj \<and> recorded obj |
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282 \<Longrightarrow> \<exists> s'. valid s' \<and> s2ss s' = s2ss s \<and> enriched s objs s' \<and> reserved s objs s'" |
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283 and p2: "valid (e # s)" and p3: "\<forall>obj\<in>objs. alive (e # s) obj \<and> is_created (e # s) obj \<and> recorded obj" |
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284 and p4: "e = Execve p f fds" |
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285 shows "enrichable (e # s) objs" |
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286 proof- |
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287 from p2 have os: "os_grant s e" and se: "grant s e" and vd: "valid s" |
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288 by (auto dest:vt_grant_os vd_cons vt_grant) |
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289 from p3 have recorded: "\<forall> obj \<in> objs. recorded obj" by auto |
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290 from p3 p4 p2 have p1': "\<forall> obj \<in> objs. alive s obj \<and> is_created s obj" |
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291 apply clarify |
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292 apply (erule_tac x = obj in ballE, simp add:alive_simps) |
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293 apply (case_tac obj, auto simp:same_inode_files_simps) |
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294 done |
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295 then obtain s' where a1: "valid s'" and a2: "s2ss s' = s2ss s" and a3: "enriched s objs s'" |
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296 and a4: "reserved s objs s'" |
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297 using p1 recorded by metis |
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298 show ?thesis |
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299 proof (cases "O_proc p \<in> objs") |
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300 case True |
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301 hence p_in: "p \<in> current_procs s'" using a4 os p4 |
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302 by (auto simp:reserved_def elim:allE[where x = "O_proc p"]) |
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303 with a1 a3 True obtain p' where b1: "\<not> alive s (O_proc p')" and b2: "O_proc p' \<notin> objs" |
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304 and b3: "alive s' (O_proc p')" and b4: "co2sobj s' (O_proc p') = co2sobj s' (O_proc p)" |
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305 apply (simp only:enriched_def) |
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306 apply (erule_tac x = "O_proc p" in ballE) |
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307 apply (erule exE|erule conjE)+ |
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308 apply (frule co2sobj_eq_alive_proc_imp, auto) |
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309 done |
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310 have "valid (Execve p' f fds # e # s')" |
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311 sorry |
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312 moreover have "s2ss (Execve p' f fds # e # s') = s2ss (e # s)" |
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313 sorry |
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314 moreover have "enriched (e # s) objs (Execve p' f fds # e # s')" |
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315 sorry |
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316 moreover have "reserved (e # s) objs (Execve p' f fds # e # s')" |
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317 sorry |
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318 ultimately show ?thesis |
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319 apply (simp add:enrichable_def) |
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320 apply (rule_tac x = "Execve p' f fds # e # s'" in exI) |
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321 by auto |
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322 next |
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323 case False |
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324 from a4 os p4 have "p \<in> current_procs s'" |
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325 apply (simp add:reserved_def) |
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326 by (erule_tac x = "O_proc p" in allE, auto) |
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327 moreover from a4 os p4 have "is_file s' f" |
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328 apply (simp add:reserved_def) |
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329 by (erule_tac x = "O_file f" in allE, auto) |
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330 moreover from a4 os p4 vd have "fds \<subseteq> proc_file_fds s' p" |
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331 apply (rule_tac subsetI, clarsimp simp:reserved_def current_proc_fds.simps) |
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332 apply (erule_tac x = "O_fd p x" in allE, erule impE) |
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333 sorry |
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334 ultimately have "os_grant s' e" |
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335 by (simp add:p4) |
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336 moreover have "grant s' e" |
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337 sorry |
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338 ultimately have "valid (e # s')" |
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339 using a1 by (erule_tac valid.intros(2), simp+) |
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340 thus ?thesis |
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341 apply (simp add:enrichable_def) |
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342 apply (rule_tac x = "e # s'" in exI) |
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343 apply (simp) |
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344 sorry |
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345 qed |
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346 qed |
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347 |
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348 lemma s2d_main_execve: |
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349 "\<lbrakk>grant_execve pctxt fsec pctxt'; ss \<in> static; S_proc (pi, pctxt, fds, shms) tagp \<in> ss; S_file sfs tagf \<in> ss; |
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350 (fi, fsec, pfsec, asecs) \<in> sfs; npctxt_execve pctxt fsec = Some pctxt'; |
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351 search_check_s pctxt (fi, fsec, pfsec, asecs) True; inherit_fds_check_s pctxt' fds'; fds' \<subseteq> fds; valid s; |
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352 s2ss s = ss\<rbrakk> \<Longrightarrow> \<exists>s. valid s \<and> |
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353 s2ss s = update_ss ss (S_proc (pi, pctxt, fds, shms) tagp) (S_proc (pi, pctxt', fds', {}) (tagp \<or> tagf))" |
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354 apply (simp add:update_ss_def) |
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355 thm update_ss_def |
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356 sorry |
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357 |
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358 (* |
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359 lemma s2d_main_execve: |
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360 "ss \<in> static \<Longrightarrow> \<exists> s. valid s \<and> s2ss s = ss" |
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361 apply (erule static.induct) |
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362 apply (rule_tac x = "[]" in exI, simp add:s2ss_nil_prop valid.intros) |
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363 apply (erule exE|erule conjE)+ |
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364 apply (rule s2d_main_execve, simp+) |
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365 |
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366 apply (erule exE|erule conjE)+ |
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367 |
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368 |
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369 sorry |
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370 *) |
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371 |
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372 |
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373 (*********************** uppest-level 3 theorems ***********************) |
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374 |
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375 lemma enrichability: |
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376 "\<lbrakk>valid s; \<forall> obj \<in> objs. alive s obj \<and> is_created s obj \<and> recorded obj\<rbrakk> |
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377 \<Longrightarrow> enrichable s objs" sorry (* |
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378 proof (induct s arbitrary:objs) |
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379 case Nil |
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380 hence "objs = {}" |
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381 apply (auto) |
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382 apply (erule_tac x = x in ballE) |
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383 apply (case_tac x) |
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384 apply (auto simp:init_alive_prop) |
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385 sorry (* done *) |
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386 thus ?case using Nil unfolding enrichable_def enriched_def reserved_def |
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387 by (rule_tac x = "[]" in exI, auto) |
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388 next |
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389 case (Cons e s) |
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390 hence p1: "\<And> objs. \<forall> obj \<in> objs. alive s obj \<and> is_created s obj \<and> recorded obj |
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391 \<Longrightarrow> \<exists> s'. valid s' \<and> s2ss s' = s2ss s \<and> enriched s objs s' \<and> reserved s objs s'" |
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392 and p2: "valid (e # s)" and p3: "\<forall>obj\<in>objs. alive (e # s) obj \<and> is_created (e # s) obj \<and> recorded obj" |
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393 and os: "os_grant s e" and se: "grant s e" and vd: "valid s" |
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394 by (auto dest:vt_grant_os vd_cons vt_grant simp:enrichable_def) |
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395 show ?case |
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396 proof (cases e) |
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397 case (Execve p f fds) |
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398 hence p4: "e = Execve p f fds" by simp |
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399 from p3 have p5: "is_inited s (O_proc p) \<Longrightarrow> (O_proc p) \<notin> objs" |
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400 by (auto simp:is_created_def is_inited_def p4 elim!:ballE[where x = "O_proc p"]) |
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401 show "enrichable (e # s) objs" |
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402 proof (cases "is_inited s (O_proc p)") |
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403 case True |
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404 with p5 have a1: "(O_proc p) \<notin> objs" by simp |
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405 with p3 p4 p2 have a2: "\<forall> obj \<in> objs. alive s obj \<and> is_created s obj" and a2': "\<forall> obj \<in> objs. recorded obj" |
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406 apply (auto simp:is_created_def alive_simps is_inited_def) |
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407 apply (erule_tac x = obj in ballE, auto simp:alive_simps split:t_object.splits) |
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408 done |
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409 then obtain s' where a3: "valid s'" and a4: "s2ss s' = s2ss s" |
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410 and a5: "enriched s objs s'" and a6: "reserved s objs s'" |
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411 using p1 apply (simp add:enrichable_def) sorry |
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412 from a5 p4 p2 a2' have a7: "enriched s objs (e # s')" |
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413 apply (clarsimp simp add:enriched_def co2sobj_execve) |
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414 apply (erule_tac x = obj in ballE, clarsimp) |
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415 apply (rule_tac x = obj' in exI, auto simp:co2sobj_execve alive_simps) |
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416 thm enriched_def |
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417 |
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418 |
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419 |
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420 obtain s' where p6:"enriched s objs s'" |
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421 apply (simp add: alive_simps enrichable_def) |
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422 apply auto apply (rule ballI, rule_tac x = obj in exI) |
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423 |
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424 have p6:"enriched (e # s) objs (e # s)" |
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425 apply (simp add:enriched_def alive_simps) |
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426 apply auto apply (rule ballI, rule_tac x = obj in exI) |
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427 |
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428 have "enrich (e # s) objs (e # s)" |
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429 apply (simp add:enrich_def p4) |
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430 sorry |
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431 moreover have "reserve (e # s) objs (e # s)" |
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432 sorry |
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433 ultimately show ?thesis using p2 |
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434 apply (simp add:enrichable_def) |
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435 by (rule_tac x = "e # s" in exI, simp) |
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436 next |
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437 |
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438 |
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439 thm enrichable_def |
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440 apply (simp add:enrichable_def p4) |
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441 |
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442 |
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443 |
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444 apply auto |
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445 apply (auto simp:enrichable_def) |
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446 apply (induct s) |
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447 |
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448 |
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449 |
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450 done |
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451 |
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452 qed |
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453 *) |
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454 |
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455 lemma d2s_main: |
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456 "valid s \<Longrightarrow> s2ss s \<propto> static" |
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457 apply (induct s, simp add:s2ss_nil_prop init_ss_in_def) |
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458 apply (rule_tac x = "init_static_state" in bexI, simp, simp add:s_init) |
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459 apply (frule vd_cons, frule vt_grant_os, simp) |
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460 apply (case_tac a) |
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461 apply (clarsimp simp add:s2ss_execve) |
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462 apply (rule conjI, rule impI) |
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463 |
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464 |
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465 |
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466 sorry |
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467 |
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468 lemma s2d_main: |
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469 "ss \<in> static \<Longrightarrow> \<exists> s. valid s \<and> s2ss s = ss" |
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470 apply (erule static.induct) |
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471 apply (rule_tac x = "[]" in exI, simp add:s2ss_nil_prop valid.intros) |
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472 |
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473 apply (erule exE|erule conjE)+ |
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474 |
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475 apply (simp add:update_ss_def) |
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476 |
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477 sorry |
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478 |
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479 lemma deleted_imp_deleted_s: |
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480 "\<lbrakk>deleted obj s; valid s\<rbrakk> \<Longrightarrow> \<exists> ss \<in> static. deleted_s ss obj " |
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481 sorry |
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482 |
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483 lemma deleted_s_imp_deleted: |
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484 "\<lbrakk>deleted_s ss obj; ss \<in> static\<rbrakk> \<Longrightarrow> \<exists> s. valid s \<and> deleted obj s" |
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485 sorry |
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486 |
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487 lemma updeletable_s_sound_comp: |
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488 "undeletable_s obj = undeletable obj" |
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489 apply (simp add:undeletable_def undeletable_s_def) |
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490 apply (auto dest:deleted_imp_deleted_s deleted_s_imp_deleted) |
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491 done |
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492 |
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493 |
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494 |
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495 end |
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496 |
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497 end |
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498 |
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499 (* |
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500 lemma s2d_main': |
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501 "ss \<in> static \<Longrightarrow> \<exists> s. valid s \<and> s2ss s \<doteq> ss" |
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502 apply (erule static.induct) |
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503 apply (rule_tac x = "[]" in exI, simp add:s2ss_nil_prop valid.intros) |
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504 |
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505 apply (erule exE|erule conjE)+ |
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506 |
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507 apply (simp add:update_ss_def) |
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508 |
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509 sorry |
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510 |
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511 *) |