1 theory ExtS |
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2 imports Prio |
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3 begin |
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4 |
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5 locale highest_set = |
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6 fixes s' th prio fixes s |
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7 defines s_def : "s \<equiv> (Set th prio#s')" |
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8 assumes vt_s: "vt step s" |
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9 and highest: "preced th s = Max ((cp s)`threads s)" |
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10 |
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11 context highest_set |
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12 begin |
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13 |
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14 |
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15 lemma vt_s': "vt step s'" |
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16 by (insert vt_s, unfold s_def, drule_tac step_back_vt, simp) |
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17 |
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18 lemma step_set: "step s' (Set th prio)" |
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19 by (insert vt_s, unfold s_def, drule_tac step_back_step, simp) |
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20 |
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21 lemma step_set_elim: |
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22 "\<lbrakk>\<lbrakk>th \<in> runing s'\<rbrakk> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q" |
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23 by (insert step_set, ind_cases "step s' (Set th prio)", auto) |
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24 |
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25 |
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26 lemma threads_s: "th \<in> threads s" |
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27 by (rule step_set_elim, unfold runing_def readys_def, auto simp:s_def) |
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28 |
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29 lemma same_depend: "depend s = depend s'" |
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30 by (insert depend_set_unchanged, unfold s_def, simp) |
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31 |
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32 lemma same_dependents: |
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33 "dependents (wq s) th = dependents (wq s') th" |
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34 apply (unfold cs_dependents_def) |
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35 by (unfold eq_depend same_depend, simp) |
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36 |
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37 lemma eq_cp_s_th: "cp s th = preced th s" |
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38 proof - |
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39 from highest and max_cp_eq[OF vt_s] |
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40 have is_max: "preced th s = Max ((\<lambda>th. preced th s) ` threads s)" by simp |
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41 have sbs: "({th} \<union> dependents (wq s) th) \<subseteq> threads s" |
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42 proof - |
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43 from threads_s and dependents_threads[OF vt_s, of th] |
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44 show ?thesis by auto |
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45 qed |
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46 show ?thesis |
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47 proof(unfold cp_eq_cpreced cpreced_def, rule Max_eqI) |
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48 show "preced th s \<in> (\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th)" by simp |
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49 next |
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50 fix y |
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51 assume "y \<in> (\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th)" |
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52 then obtain th1 where th1_in: "th1 \<in> ({th} \<union> dependents (wq s) th)" |
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53 and eq_y: "y = preced th1 s" by auto |
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54 show "y \<le> preced th s" |
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55 proof(unfold is_max, rule Max_ge) |
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56 from finite_threads[OF vt_s] |
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57 show "finite ((\<lambda>th. preced th s) ` threads s)" by simp |
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58 next |
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59 from sbs th1_in and eq_y |
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60 show "y \<in> (\<lambda>th. preced th s) ` threads s" by auto |
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61 qed |
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62 next |
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63 from sbs and finite_threads[OF vt_s] |
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64 show "finite ((\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th))" |
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65 by (auto intro:finite_subset) |
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66 qed |
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67 qed |
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68 |
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69 lemma highest_cp_preced: "cp s th = Max ((\<lambda> th'. preced th' s) ` threads s)" |
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70 by (fold max_cp_eq[OF vt_s], unfold eq_cp_s_th, insert highest, simp) |
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71 |
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72 lemma highest_preced_thread: "preced th s = Max ((\<lambda> th'. preced th' s) ` threads s)" |
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73 by (fold eq_cp_s_th, unfold highest_cp_preced, simp) |
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74 |
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75 lemma is_ready: "th \<in> readys s" |
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76 proof - |
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77 have "\<forall>cs. \<not> waiting s th cs" |
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78 apply (rule step_set_elim, unfold s_def, insert depend_set_unchanged[of th prio s']) |
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79 apply (unfold s_depend_def, unfold runing_def readys_def) |
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80 apply (auto, fold s_def) |
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81 apply (erule_tac x = cs in allE, auto simp:waiting_eq) |
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82 proof - |
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83 fix cs |
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84 assume h: |
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85 "{(Th t, Cs c) |t c. waiting (wq s) t c} \<union> {(Cs c, Th t) |c t. holding (wq s) t c} = |
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86 {(Th t, Cs c) |t c. waiting (wq s') t c} \<union> {(Cs c, Th t) |c t. holding (wq s') t c}" |
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87 (is "?L = ?R") |
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88 and wt: "waiting (wq s) th cs" and nwt: "\<not> waiting (wq s') th cs" |
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89 from wt have "(Th th, Cs cs) \<in> ?L" by auto |
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90 with h have "(Th th, Cs cs) \<in> ?R" by simp |
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91 hence "waiting (wq s') th cs" by auto with nwt |
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92 show False by auto |
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93 qed |
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94 with threads_s show ?thesis |
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95 by (unfold readys_def, auto) |
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96 qed |
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97 |
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98 lemma highest': "cp s th = Max (cp s ` threads s)" |
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99 proof - |
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100 from highest_cp_preced max_cp_eq[OF vt_s, symmetric] |
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101 show ?thesis by simp |
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102 qed |
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103 |
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104 lemma is_runing: "th \<in> runing s" |
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105 proof - |
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106 have "Max (cp s ` threads s) = Max (cp s ` readys s)" |
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107 proof - |
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108 have " Max (cp s ` readys s) = cp s th" |
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109 proof(rule Max_eqI) |
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110 from finite_threads[OF vt_s] readys_threads finite_subset |
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111 have "finite (readys s)" by blast |
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112 thus "finite (cp s ` readys s)" by auto |
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113 next |
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114 from is_ready show "cp s th \<in> cp s ` readys s" by auto |
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115 next |
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116 fix y |
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117 assume "y \<in> cp s ` readys s" |
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118 then obtain th1 where |
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119 eq_y: "y = cp s th1" and th1_in: "th1 \<in> readys s" by auto |
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120 show "y \<le> cp s th" |
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121 proof - |
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122 have "y \<le> Max (cp s ` threads s)" |
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123 proof(rule Max_ge) |
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124 from eq_y and th1_in |
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125 show "y \<in> cp s ` threads s" |
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126 by (auto simp:readys_def) |
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127 next |
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128 from finite_threads[OF vt_s] |
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129 show "finite (cp s ` threads s)" by auto |
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130 qed |
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131 with highest' show ?thesis by auto |
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132 qed |
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133 qed |
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134 with highest' show ?thesis by auto |
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135 qed |
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136 thus ?thesis |
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137 by (unfold runing_def, insert highest' is_ready, auto) |
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138 qed |
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139 |
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140 end |
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141 |
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142 locale extend_highest_set = highest_set + |
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143 fixes t |
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144 assumes vt_t: "vt step (t@s)" |
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145 and create_low: "Create th' prio' \<in> set t \<Longrightarrow> prio' \<le> prio" |
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146 and set_diff_low: "Set th' prio' \<in> set t \<Longrightarrow> th' \<noteq> th \<and> prio' \<le> prio" |
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147 and exit_diff: "Exit th' \<in> set t \<Longrightarrow> th' \<noteq> th" |
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148 |
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149 lemma step_back_vt_app: |
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150 assumes vt_ts: "vt cs (t@s)" |
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151 shows "vt cs s" |
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152 proof - |
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153 from vt_ts show ?thesis |
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154 proof(induct t) |
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155 case Nil |
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156 from Nil show ?case by auto |
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157 next |
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158 case (Cons e t) |
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159 assume ih: " vt cs (t @ s) \<Longrightarrow> vt cs s" |
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160 and vt_et: "vt cs ((e # t) @ s)" |
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161 show ?case |
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162 proof(rule ih) |
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163 show "vt cs (t @ s)" |
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164 proof(rule step_back_vt) |
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165 from vt_et show "vt cs (e # t @ s)" by simp |
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166 qed |
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167 qed |
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168 qed |
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169 qed |
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170 |
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171 context extend_highest_set |
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172 begin |
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173 |
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174 lemma red_moment: |
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175 "extend_highest_set s' th prio (moment i t)" |
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176 apply (insert extend_highest_set_axioms, subst (asm) (1) moment_restm_s [of i t, symmetric]) |
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177 apply (unfold extend_highest_set_def extend_highest_set_axioms_def, clarsimp) |
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178 by (unfold highest_set_def, auto dest:step_back_vt_app) |
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179 |
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180 lemma ind [consumes 0, case_names Nil Cons, induct type]: |
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181 assumes |
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182 h0: "R []" |
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183 and h2: "\<And> e t. \<lbrakk>vt step (t@s); step (t@s) e; |
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184 extend_highest_set s' th prio t; |
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185 extend_highest_set s' th prio (e#t); R t\<rbrakk> \<Longrightarrow> R (e#t)" |
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186 shows "R t" |
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187 proof - |
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188 from vt_t extend_highest_set_axioms show ?thesis |
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189 proof(induct t) |
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190 from h0 show "R []" . |
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191 next |
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192 case (Cons e t') |
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193 assume ih: "\<lbrakk>vt step (t' @ s); extend_highest_set s' th prio t'\<rbrakk> \<Longrightarrow> R t'" |
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194 and vt_e: "vt step ((e # t') @ s)" |
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195 and et: "extend_highest_set s' th prio (e # t')" |
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196 from vt_e and step_back_step have stp: "step (t'@s) e" by auto |
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197 from vt_e and step_back_vt have vt_ts: "vt step (t'@s)" by auto |
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198 show ?case |
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199 proof(rule h2 [OF vt_ts stp _ _ _ ]) |
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200 show "R t'" |
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201 proof(rule ih) |
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202 from et show ext': "extend_highest_set s' th prio t'" |
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203 by (unfold extend_highest_set_def extend_highest_set_axioms_def, auto dest:step_back_vt) |
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204 next |
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205 from vt_ts show "vt step (t' @ s)" . |
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206 qed |
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207 next |
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208 from et show "extend_highest_set s' th prio (e # t')" . |
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209 next |
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210 from et show ext': "extend_highest_set s' th prio t'" |
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211 by (unfold extend_highest_set_def extend_highest_set_axioms_def, auto dest:step_back_vt) |
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212 qed |
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213 qed |
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214 qed |
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215 |
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216 lemma th_kept: "th \<in> threads (t @ s) \<and> |
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217 preced th (t@s) = preced th s" (is "?Q t") |
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218 proof - |
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219 show ?thesis |
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220 proof(induct rule:ind) |
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221 case Nil |
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222 from threads_s |
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223 show "th \<in> threads ([] @ s) \<and> preced th ([] @ s) = preced th s" |
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224 by auto |
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225 next |
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226 case (Cons e t) |
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227 show ?case |
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228 proof(cases e) |
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229 case (Create thread prio) |
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230 assume eq_e: " e = Create thread prio" |
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231 show ?thesis |
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232 proof - |
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233 from Cons and eq_e have "step (t@s) (Create thread prio)" by auto |
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234 hence "th \<noteq> thread" |
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235 proof(cases) |
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236 assume "thread \<notin> threads (t @ s)" |
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237 with Cons show ?thesis by auto |
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238 qed |
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239 hence "preced th ((e # t) @ s) = preced th (t @ s)" |
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240 by (unfold eq_e, auto simp:preced_def) |
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241 moreover note Cons |
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242 ultimately show ?thesis |
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243 by (auto simp:eq_e) |
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244 qed |
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245 next |
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246 case (Exit thread) |
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247 assume eq_e: "e = Exit thread" |
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248 from Cons have "extend_highest_set s' th prio (e # t)" by auto |
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249 from extend_highest_set.exit_diff [OF this] and eq_e |
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250 have neq_th: "thread \<noteq> th" by auto |
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251 with Cons |
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252 show ?thesis |
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253 by (unfold eq_e, auto simp:preced_def) |
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254 next |
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255 case (P thread cs) |
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256 assume eq_e: "e = P thread cs" |
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257 with Cons |
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258 show ?thesis |
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259 by (auto simp:eq_e preced_def) |
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260 next |
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261 case (V thread cs) |
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262 assume eq_e: "e = V thread cs" |
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263 with Cons |
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264 show ?thesis |
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265 by (auto simp:eq_e preced_def) |
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266 next |
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267 case (Set thread prio') |
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268 assume eq_e: " e = Set thread prio'" |
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269 show ?thesis |
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270 proof - |
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271 from Cons have "extend_highest_set s' th prio (e # t)" by auto |
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272 from extend_highest_set.set_diff_low[OF this] and eq_e |
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273 have "th \<noteq> thread" by auto |
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274 hence "preced th ((e # t) @ s) = preced th (t @ s)" |
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275 by (unfold eq_e, auto simp:preced_def) |
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276 moreover note Cons |
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277 ultimately show ?thesis |
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278 by (auto simp:eq_e) |
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279 qed |
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280 qed |
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281 qed |
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282 qed |
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283 |
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284 lemma max_kept: "Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s))) = preced th s" |
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285 proof(induct rule:ind) |
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286 case Nil |
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287 from highest_preced_thread |
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288 show "Max ((\<lambda>th'. preced th' ([] @ s)) ` threads ([] @ s)) = preced th s" |
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289 by simp |
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290 next |
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291 case (Cons e t) |
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292 show ?case |
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293 proof(cases e) |
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294 case (Create thread prio') |
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295 assume eq_e: " e = Create thread prio'" |
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296 from Cons and eq_e have stp: "step (t@s) (Create thread prio')" by auto |
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297 hence neq_thread: "thread \<noteq> th" |
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298 proof(cases) |
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299 assume "thread \<notin> threads (t @ s)" |
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300 moreover have "th \<in> threads (t@s)" |
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301 proof - |
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302 from Cons have "extend_highest_set s' th prio t" by auto |
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303 from extend_highest_set.th_kept[OF this] show ?thesis by (simp add:s_def) |
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304 qed |
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305 ultimately show ?thesis by auto |
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306 qed |
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307 from Cons have "extend_highest_set s' th prio t" by auto |
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308 from extend_highest_set.th_kept[OF this] |
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309 have h': " th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" |
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310 by (auto simp:s_def) |
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311 from stp |
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312 have thread_ts: "thread \<notin> threads (t @ s)" |
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313 by (cases, auto) |
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314 show ?thesis (is "Max (?f ` ?A) = ?t") |
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315 proof - |
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316 have "Max (?f ` ?A) = Max(insert (?f thread) (?f ` (threads (t@s))))" |
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317 by (unfold eq_e, simp) |
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318 moreover have "\<dots> = max (?f thread) (Max (?f ` (threads (t@s))))" |
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319 proof(rule Max_insert) |
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320 from Cons have "vt step (t @ s)" by auto |
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321 from finite_threads[OF this] |
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322 show "finite (?f ` (threads (t@s)))" by simp |
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323 next |
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324 from h' show "(?f ` (threads (t@s))) \<noteq> {}" by auto |
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325 qed |
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326 moreover have "(Max (?f ` (threads (t@s)))) = ?t" |
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327 proof - |
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328 have "(\<lambda>th'. preced th' ((e # t) @ s)) ` threads (t @ s) = |
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329 (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)" (is "?f1 ` ?B = ?f2 ` ?B") |
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330 proof - |
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331 { fix th' |
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332 assume "th' \<in> ?B" |
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333 with thread_ts eq_e |
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334 have "?f1 th' = ?f2 th'" by (auto simp:preced_def) |
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335 } thus ?thesis |
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336 apply (auto simp:Image_def) |
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337 proof - |
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338 fix th' |
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339 assume h: "\<And>th'. th' \<in> threads (t @ s) \<Longrightarrow> |
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340 preced th' (e # t @ s) = preced th' (t @ s)" |
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341 and h1: "th' \<in> threads (t @ s)" |
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342 show "preced th' (t @ s) \<in> (\<lambda>th'. preced th' (e # t @ s)) ` threads (t @ s)" |
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343 proof - |
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344 from h1 have "?f1 th' \<in> ?f1 ` ?B" by auto |
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345 moreover from h[OF h1] have "?f1 th' = ?f2 th'" by simp |
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346 ultimately show ?thesis by simp |
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347 qed |
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348 qed |
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349 qed |
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350 with Cons show ?thesis by auto |
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351 qed |
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352 moreover have "?f thread < ?t" |
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353 proof - |
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354 from Cons have " extend_highest_set s' th prio (e # t)" by auto |
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355 from extend_highest_set.create_low[OF this] and eq_e |
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356 have "prio' \<le> prio" by auto |
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357 thus ?thesis |
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358 by (unfold eq_e, auto simp:preced_def s_def precedence_less_def) |
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359 qed |
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360 ultimately show ?thesis by (auto simp:max_def) |
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361 qed |
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362 next |
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363 case (Exit thread) |
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364 assume eq_e: "e = Exit thread" |
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365 from Cons have vt_e: "vt step (e#(t @ s))" by auto |
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366 from Cons and eq_e have stp: "step (t@s) (Exit thread)" by auto |
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367 from stp have thread_ts: "thread \<in> threads (t @ s)" |
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368 by(cases, unfold runing_def readys_def, auto) |
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369 from Cons have "extend_highest_set s' th prio (e # t)" by auto |
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370 from extend_highest_set.exit_diff[OF this] and eq_e |
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371 have neq_thread: "thread \<noteq> th" by auto |
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372 from Cons have "extend_highest_set s' th prio t" by auto |
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373 from extend_highest_set.th_kept[OF this, folded s_def] |
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374 have h': "th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" . |
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375 show ?thesis (is "Max (?f ` ?A) = ?t") |
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376 proof - |
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377 have "threads (t@s) = insert thread ?A" |
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378 by (insert stp thread_ts, unfold eq_e, auto) |
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379 hence "Max (?f ` (threads (t@s))) = Max (?f ` \<dots>)" by simp |
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380 also from this have "\<dots> = Max (insert (?f thread) (?f ` ?A))" by simp |
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381 also have "\<dots> = max (?f thread) (Max (?f ` ?A))" |
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382 proof(rule Max_insert) |
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383 from finite_threads [OF vt_e] |
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384 show "finite (?f ` ?A)" by simp |
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385 next |
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386 from Cons have "extend_highest_set s' th prio (e # t)" by auto |
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387 from extend_highest_set.th_kept[OF this] |
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388 show "?f ` ?A \<noteq> {}" by (auto simp:s_def) |
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389 qed |
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390 finally have "Max (?f ` (threads (t@s))) = max (?f thread) (Max (?f ` ?A))" . |
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391 moreover have "Max (?f ` (threads (t@s))) = ?t" |
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392 proof - |
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393 from Cons show ?thesis |
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394 by (unfold eq_e, auto simp:preced_def) |
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395 qed |
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396 ultimately have "max (?f thread) (Max (?f ` ?A)) = ?t" by simp |
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397 moreover have "?f thread < ?t" |
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398 proof(unfold eq_e, simp add:preced_def, fold preced_def) |
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399 show "preced thread (t @ s) < ?t" |
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400 proof - |
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401 have "preced thread (t @ s) \<le> ?t" |
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402 proof - |
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403 from Cons |
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404 have "?t = Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" |
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405 (is "?t = Max (?g ` ?B)") by simp |
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406 moreover have "?g thread \<le> \<dots>" |
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407 proof(rule Max_ge) |
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408 have "vt step (t@s)" by fact |
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409 from finite_threads [OF this] |
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410 show "finite (?g ` ?B)" by simp |
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411 next |
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412 from thread_ts |
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413 show "?g thread \<in> (?g ` ?B)" by auto |
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414 qed |
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415 ultimately show ?thesis by auto |
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416 qed |
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417 moreover have "preced thread (t @ s) \<noteq> ?t" |
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418 proof |
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419 assume "preced thread (t @ s) = preced th s" |
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420 with h' have "preced thread (t @ s) = preced th (t@s)" by simp |
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421 from preced_unique [OF this] have "thread = th" |
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422 proof |
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423 from h' show "th \<in> threads (t @ s)" by simp |
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424 next |
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425 from thread_ts show "thread \<in> threads (t @ s)" . |
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426 qed(simp) |
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427 with neq_thread show "False" by simp |
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428 qed |
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429 ultimately show ?thesis by auto |
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430 qed |
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431 qed |
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432 ultimately show ?thesis |
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433 by (auto simp:max_def split:if_splits) |
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434 qed |
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435 next |
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436 case (P thread cs) |
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437 with Cons |
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438 show ?thesis by (auto simp:preced_def) |
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439 next |
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440 case (V thread cs) |
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441 with Cons |
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442 show ?thesis by (auto simp:preced_def) |
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443 next |
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444 case (Set thread prio') |
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445 show ?thesis (is "Max (?f ` ?A) = ?t") |
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446 proof - |
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447 let ?B = "threads (t@s)" |
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448 from Cons have "extend_highest_set s' th prio (e # t)" by auto |
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449 from extend_highest_set.set_diff_low[OF this] and Set |
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450 have neq_thread: "thread \<noteq> th" and le_p: "prio' \<le> prio" by auto |
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451 from Set have "Max (?f ` ?A) = Max (?f ` ?B)" by simp |
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452 also have "\<dots> = ?t" |
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453 proof(rule Max_eqI) |
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454 fix y |
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455 assume y_in: "y \<in> ?f ` ?B" |
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456 then obtain th1 where |
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457 th1_in: "th1 \<in> ?B" and eq_y: "y = ?f th1" by auto |
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458 show "y \<le> ?t" |
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459 proof(cases "th1 = thread") |
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460 case True |
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461 with neq_thread le_p eq_y s_def Set |
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462 show ?thesis |
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463 by (auto simp:preced_def precedence_le_def) |
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464 next |
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465 case False |
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466 with Set eq_y |
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467 have "y = preced th1 (t@s)" |
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468 by (simp add:preced_def) |
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469 moreover have "\<dots> \<le> ?t" |
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470 proof - |
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471 from Cons |
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472 have "?t = Max ((\<lambda> th'. preced th' (t@s)) ` (threads (t@s)))" |
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473 by auto |
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474 moreover have "preced th1 (t@s) \<le> \<dots>" |
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475 proof(rule Max_ge) |
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476 from th1_in |
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477 show "preced th1 (t @ s) \<in> (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)" |
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478 by simp |
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479 next |
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480 show "finite ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" |
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481 proof - |
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482 from Cons have "vt step (t @ s)" by auto |
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483 from finite_threads[OF this] show ?thesis by auto |
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484 qed |
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485 qed |
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486 ultimately show ?thesis by auto |
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487 qed |
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488 ultimately show ?thesis by auto |
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489 qed |
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490 next |
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491 from Cons and finite_threads |
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492 show "finite (?f ` ?B)" by auto |
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493 next |
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494 from Cons have "extend_highest_set s' th prio t" by auto |
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495 from extend_highest_set.th_kept [OF this, folded s_def] |
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496 have h: "th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" . |
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497 show "?t \<in> (?f ` ?B)" |
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498 proof - |
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499 from neq_thread Set h |
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500 have "?t = ?f th" by (auto simp:preced_def) |
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501 with h show ?thesis by auto |
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502 qed |
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503 qed |
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504 finally show ?thesis . |
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505 qed |
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506 qed |
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507 qed |
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508 |
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509 lemma max_preced: "preced th (t@s) = Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s)))" |
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510 by (insert th_kept max_kept, auto) |
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511 |
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512 lemma th_cp_max_preced: "cp (t@s) th = Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s)))" |
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513 (is "?L = ?R") |
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514 proof - |
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515 have "?L = cpreced (t@s) (wq (t@s)) th" |
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516 by (unfold cp_eq_cpreced, simp) |
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517 also have "\<dots> = ?R" |
|
518 proof(unfold cpreced_def) |
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519 show "Max ((\<lambda>th. preced th (t @ s)) ` ({th} \<union> dependents (wq (t @ s)) th)) = |
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520 Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" |
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521 (is "Max (?f ` ({th} \<union> ?A)) = Max (?f ` ?B)") |
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522 proof(cases "?A = {}") |
|
523 case False |
|
524 have "Max (?f ` ({th} \<union> ?A)) = Max (insert (?f th) (?f ` ?A))" by simp |
|
525 moreover have "\<dots> = max (?f th) (Max (?f ` ?A))" |
|
526 proof(rule Max_insert) |
|
527 show "finite (?f ` ?A)" |
|
528 proof - |
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529 from dependents_threads[OF vt_t] |
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530 have "?A \<subseteq> threads (t@s)" . |
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531 moreover from finite_threads[OF vt_t] have "finite \<dots>" . |
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532 ultimately show ?thesis |
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533 by (auto simp:finite_subset) |
|
534 qed |
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535 next |
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536 from False show "(?f ` ?A) \<noteq> {}" by simp |
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537 qed |
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538 moreover have "\<dots> = Max (?f ` ?B)" |
|
539 proof - |
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540 from max_preced have "?f th = Max (?f ` ?B)" . |
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541 moreover have "Max (?f ` ?A) \<le> \<dots>" |
|
542 proof(rule Max_mono) |
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543 from False show "(?f ` ?A) \<noteq> {}" by simp |
|
544 next |
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545 show "?f ` ?A \<subseteq> ?f ` ?B" |
|
546 proof - |
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547 have "?A \<subseteq> ?B" by (rule dependents_threads[OF vt_t]) |
|
548 thus ?thesis by auto |
|
549 qed |
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550 next |
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551 from finite_threads[OF vt_t] |
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552 show "finite (?f ` ?B)" by simp |
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553 qed |
|
554 ultimately show ?thesis |
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555 by (auto simp:max_def) |
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556 qed |
|
557 ultimately show ?thesis by auto |
|
558 next |
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559 case True |
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560 with max_preced show ?thesis by auto |
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561 qed |
|
562 qed |
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563 finally show ?thesis . |
|
564 qed |
|
565 |
|
566 lemma th_cp_max: "cp (t@s) th = Max (cp (t@s) ` threads (t@s))" |
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567 by (unfold max_cp_eq[OF vt_t] th_cp_max_preced, simp) |
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568 |
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569 lemma th_cp_preced: "cp (t@s) th = preced th s" |
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570 by (fold max_kept, unfold th_cp_max_preced, simp) |
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571 |
|
572 lemma preced_less': |
|
573 fixes th' |
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574 assumes th'_in: "th' \<in> threads s" |
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575 and neq_th': "th' \<noteq> th" |
|
576 shows "preced th' s < preced th s" |
|
577 proof - |
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578 have "preced th' s \<le> Max ((\<lambda>th'. preced th' s) ` threads s)" |
|
579 proof(rule Max_ge) |
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580 from finite_threads [OF vt_s] |
|
581 show "finite ((\<lambda>th'. preced th' s) ` threads s)" by simp |
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582 next |
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583 from th'_in show "preced th' s \<in> (\<lambda>th'. preced th' s) ` threads s" |
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584 by simp |
|
585 qed |
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586 moreover have "preced th' s \<noteq> preced th s" |
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587 proof |
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588 assume "preced th' s = preced th s" |
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589 from preced_unique[OF this th'_in] neq_th' is_ready |
|
590 show "False" by (auto simp:readys_def) |
|
591 qed |
|
592 ultimately show ?thesis using highest_preced_thread |
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593 by auto |
|
594 qed |
|
595 |
|
596 lemma pv_blocked: |
|
597 fixes th' |
|
598 assumes th'_in: "th' \<in> threads (t@s)" |
|
599 and neq_th': "th' \<noteq> th" |
|
600 and eq_pv: "cntP (t@s) th' = cntV (t@s) th'" |
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601 shows "th' \<notin> runing (t@s)" |
|
602 proof |
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603 assume "th' \<in> runing (t@s)" |
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604 hence "cp (t@s) th' = Max (cp (t@s) ` readys (t@s))" |
|
605 by (auto simp:runing_def) |
|
606 with max_cp_readys_threads [OF vt_t] |
|
607 have "cp (t @ s) th' = Max (cp (t@s) ` threads (t@s))" |
|
608 by auto |
|
609 moreover from th_cp_max have "cp (t @ s) th = \<dots>" by simp |
|
610 ultimately have "cp (t @ s) th' = cp (t @ s) th" by simp |
|
611 moreover from th_cp_preced and th_kept have "\<dots> = preced th (t @ s)" |
|
612 by simp |
|
613 finally have h: "cp (t @ s) th' = preced th (t @ s)" . |
|
614 show False |
|
615 proof - |
|
616 have "dependents (wq (t @ s)) th' = {}" |
|
617 by (rule count_eq_dependents [OF vt_t eq_pv]) |
|
618 moreover have "preced th' (t @ s) \<noteq> preced th (t @ s)" |
|
619 proof |
|
620 assume "preced th' (t @ s) = preced th (t @ s)" |
|
621 hence "th' = th" |
|
622 proof(rule preced_unique) |
|
623 from th_kept show "th \<in> threads (t @ s)" by simp |
|
624 next |
|
625 from th'_in show "th' \<in> threads (t @ s)" by simp |
|
626 qed |
|
627 with assms show False by simp |
|
628 qed |
|
629 ultimately show ?thesis |
|
630 by (insert h, unfold cp_eq_cpreced cpreced_def, simp) |
|
631 qed |
|
632 qed |
|
633 |
|
634 lemma runing_precond_pre: |
|
635 fixes th' |
|
636 assumes th'_in: "th' \<in> threads s" |
|
637 and eq_pv: "cntP s th' = cntV s th'" |
|
638 and neq_th': "th' \<noteq> th" |
|
639 shows "th' \<in> threads (t@s) \<and> |
|
640 cntP (t@s) th' = cntV (t@s) th'" |
|
641 proof - |
|
642 show ?thesis |
|
643 proof(induct rule:ind) |
|
644 case (Cons e t) |
|
645 from Cons |
|
646 have in_thread: "th' \<in> threads (t @ s)" |
|
647 and not_holding: "cntP (t @ s) th' = cntV (t @ s) th'" by auto |
|
648 have "extend_highest_set s' th prio t" by fact |
|
649 from extend_highest_set.pv_blocked |
|
650 [OF this, folded s_def, OF in_thread neq_th' not_holding] |
|
651 have not_runing: "th' \<notin> runing (t @ s)" . |
|
652 show ?case |
|
653 proof(cases e) |
|
654 case (V thread cs) |
|
655 from Cons and V have vt_v: "vt step (V thread cs#(t@s))" by auto |
|
656 |
|
657 show ?thesis |
|
658 proof - |
|
659 from Cons and V have "step (t@s) (V thread cs)" by auto |
|
660 hence neq_th': "thread \<noteq> th'" |
|
661 proof(cases) |
|
662 assume "thread \<in> runing (t@s)" |
|
663 moreover have "th' \<notin> runing (t@s)" by fact |
|
664 ultimately show ?thesis by auto |
|
665 qed |
|
666 with not_holding have cnt_eq: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'" |
|
667 by (unfold V, simp add:cntP_def cntV_def count_def) |
|
668 moreover from in_thread |
|
669 have in_thread': "th' \<in> threads ((e # t) @ s)" by (unfold V, simp) |
|
670 ultimately show ?thesis by auto |
|
671 qed |
|
672 next |
|
673 case (P thread cs) |
|
674 from Cons and P have "step (t@s) (P thread cs)" by auto |
|
675 hence neq_th': "thread \<noteq> th'" |
|
676 proof(cases) |
|
677 assume "thread \<in> runing (t@s)" |
|
678 moreover note not_runing |
|
679 ultimately show ?thesis by auto |
|
680 qed |
|
681 with Cons and P have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'" |
|
682 by (auto simp:cntP_def cntV_def count_def) |
|
683 moreover from Cons and P have in_thread': "th' \<in> threads ((e # t) @ s)" |
|
684 by auto |
|
685 ultimately show ?thesis by auto |
|
686 next |
|
687 case (Create thread prio') |
|
688 from Cons and Create have "step (t@s) (Create thread prio')" by auto |
|
689 hence neq_th': "thread \<noteq> th'" |
|
690 proof(cases) |
|
691 assume "thread \<notin> threads (t @ s)" |
|
692 moreover have "th' \<in> threads (t@s)" by fact |
|
693 ultimately show ?thesis by auto |
|
694 qed |
|
695 with Cons and Create |
|
696 have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'" |
|
697 by (auto simp:cntP_def cntV_def count_def) |
|
698 moreover from Cons and Create |
|
699 have in_thread': "th' \<in> threads ((e # t) @ s)" by auto |
|
700 ultimately show ?thesis by auto |
|
701 next |
|
702 case (Exit thread) |
|
703 from Cons and Exit have "step (t@s) (Exit thread)" by auto |
|
704 hence neq_th': "thread \<noteq> th'" |
|
705 proof(cases) |
|
706 assume "thread \<in> runing (t @ s)" |
|
707 moreover note not_runing |
|
708 ultimately show ?thesis by auto |
|
709 qed |
|
710 with Cons and Exit |
|
711 have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'" |
|
712 by (auto simp:cntP_def cntV_def count_def) |
|
713 moreover from Cons and Exit and neq_th' |
|
714 have in_thread': "th' \<in> threads ((e # t) @ s)" |
|
715 by auto |
|
716 ultimately show ?thesis by auto |
|
717 next |
|
718 case (Set thread prio') |
|
719 with Cons |
|
720 show ?thesis |
|
721 by (auto simp:cntP_def cntV_def count_def) |
|
722 qed |
|
723 next |
|
724 case Nil |
|
725 with assms |
|
726 show ?case by auto |
|
727 qed |
|
728 qed |
|
729 |
|
730 (* |
|
731 lemma runing_precond: |
|
732 fixes th' |
|
733 assumes th'_in: "th' \<in> threads s" |
|
734 and eq_pv: "cntP s th' = cntV s th'" |
|
735 and neq_th': "th' \<noteq> th" |
|
736 shows "th' \<notin> runing (t@s)" |
|
737 proof - |
|
738 from runing_precond_pre[OF th'_in eq_pv neq_th'] |
|
739 have h1: "th' \<in> threads (t @ s)" and h2: "cntP (t @ s) th' = cntV (t @ s) th'" by auto |
|
740 from pv_blocked[OF h1 neq_th' h2] |
|
741 show ?thesis . |
|
742 qed |
|
743 *) |
|
744 |
|
745 lemma runing_precond: |
|
746 fixes th' |
|
747 assumes th'_in: "th' \<in> threads s" |
|
748 and neq_th': "th' \<noteq> th" |
|
749 and is_runing: "th' \<in> runing (t@s)" |
|
750 shows "cntP s th' > cntV s th'" |
|
751 proof - |
|
752 have "cntP s th' \<noteq> cntV s th'" |
|
753 proof |
|
754 assume eq_pv: "cntP s th' = cntV s th'" |
|
755 from runing_precond_pre[OF th'_in eq_pv neq_th'] |
|
756 have h1: "th' \<in> threads (t @ s)" |
|
757 and h2: "cntP (t @ s) th' = cntV (t @ s) th'" by auto |
|
758 from pv_blocked[OF h1 neq_th' h2] have " th' \<notin> runing (t @ s)" . |
|
759 with is_runing show "False" by simp |
|
760 qed |
|
761 moreover from cnp_cnv_cncs[OF vt_s, of th'] |
|
762 have "cntV s th' \<le> cntP s th'" by auto |
|
763 ultimately show ?thesis by auto |
|
764 qed |
|
765 |
|
766 lemma moment_blocked_pre: |
|
767 assumes neq_th': "th' \<noteq> th" |
|
768 and th'_in: "th' \<in> threads ((moment i t)@s)" |
|
769 and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'" |
|
770 shows "cntP ((moment (i+j) t)@s) th' = cntV ((moment (i+j) t)@s) th' \<and> |
|
771 th' \<in> threads ((moment (i+j) t)@s)" |
|
772 proof(induct j) |
|
773 case (Suc k) |
|
774 show ?case |
|
775 proof - |
|
776 { assume True: "Suc (i+k) \<le> length t" |
|
777 from moment_head [OF this] |
|
778 obtain e where |
|
779 eq_me: "moment (Suc(i+k)) t = e#(moment (i+k) t)" |
|
780 by blast |
|
781 from red_moment[of "Suc(i+k)"] |
|
782 and eq_me have "extend_highest_set s' th prio (e # moment (i + k) t)" by simp |
|
783 hence vt_e: "vt step (e#(moment (i + k) t)@s)" |
|
784 by (unfold extend_highest_set_def extend_highest_set_axioms_def |
|
785 highest_set_def s_def, auto) |
|
786 have not_runing': "th' \<notin> runing (moment (i + k) t @ s)" |
|
787 proof(unfold s_def) |
|
788 show "th' \<notin> runing (moment (i + k) t @ Set th prio # s')" |
|
789 proof(rule extend_highest_set.pv_blocked) |
|
790 from Suc show "th' \<in> threads (moment (i + k) t @ Set th prio # s')" |
|
791 by (simp add:s_def) |
|
792 next |
|
793 from neq_th' show "th' \<noteq> th" . |
|
794 next |
|
795 from red_moment show "extend_highest_set s' th prio (moment (i + k) t)" . |
|
796 next |
|
797 from Suc show "cntP (moment (i + k) t @ Set th prio # s') th' = |
|
798 cntV (moment (i + k) t @ Set th prio # s') th'" |
|
799 by (auto simp:s_def) |
|
800 qed |
|
801 qed |
|
802 from step_back_step[OF vt_e] |
|
803 have "step ((moment (i + k) t)@s) e" . |
|
804 hence "cntP (e#(moment (i + k) t)@s) th' = cntV (e#(moment (i + k) t)@s) th' \<and> |
|
805 th' \<in> threads (e#(moment (i + k) t)@s) |
|
806 " |
|
807 proof(cases) |
|
808 case (thread_create thread prio) |
|
809 with Suc show ?thesis by (auto simp:cntP_def cntV_def count_def) |
|
810 next |
|
811 case (thread_exit thread) |
|
812 moreover have "thread \<noteq> th'" |
|
813 proof - |
|
814 have "thread \<in> runing (moment (i + k) t @ s)" by fact |
|
815 moreover note not_runing' |
|
816 ultimately show ?thesis by auto |
|
817 qed |
|
818 moreover note Suc |
|
819 ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def) |
|
820 next |
|
821 case (thread_P thread cs) |
|
822 moreover have "thread \<noteq> th'" |
|
823 proof - |
|
824 have "thread \<in> runing (moment (i + k) t @ s)" by fact |
|
825 moreover note not_runing' |
|
826 ultimately show ?thesis by auto |
|
827 qed |
|
828 moreover note Suc |
|
829 ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def) |
|
830 next |
|
831 case (thread_V thread cs) |
|
832 moreover have "thread \<noteq> th'" |
|
833 proof - |
|
834 have "thread \<in> runing (moment (i + k) t @ s)" by fact |
|
835 moreover note not_runing' |
|
836 ultimately show ?thesis by auto |
|
837 qed |
|
838 moreover note Suc |
|
839 ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def) |
|
840 next |
|
841 case (thread_set thread prio') |
|
842 with Suc show ?thesis |
|
843 by (auto simp:cntP_def cntV_def count_def) |
|
844 qed |
|
845 with eq_me have ?thesis using eq_me by auto |
|
846 } note h = this |
|
847 show ?thesis |
|
848 proof(cases "Suc (i+k) \<le> length t") |
|
849 case True |
|
850 from h [OF this] show ?thesis . |
|
851 next |
|
852 case False |
|
853 with moment_ge |
|
854 have eq_m: "moment (i + Suc k) t = moment (i+k) t" by auto |
|
855 with Suc show ?thesis by auto |
|
856 qed |
|
857 qed |
|
858 next |
|
859 case 0 |
|
860 from assms show ?case by auto |
|
861 qed |
|
862 |
|
863 lemma moment_blocked: |
|
864 assumes neq_th': "th' \<noteq> th" |
|
865 and th'_in: "th' \<in> threads ((moment i t)@s)" |
|
866 and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'" |
|
867 and le_ij: "i \<le> j" |
|
868 shows "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th' \<and> |
|
869 th' \<in> threads ((moment j t)@s) \<and> |
|
870 th' \<notin> runing ((moment j t)@s)" |
|
871 proof - |
|
872 from moment_blocked_pre [OF neq_th' th'_in eq_pv, of "j-i"] and le_ij |
|
873 have h1: "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th'" |
|
874 and h2: "th' \<in> threads ((moment j t)@s)" by auto |
|
875 with extend_highest_set.pv_blocked [OF red_moment [of j], folded s_def, OF h2 neq_th' h1] |
|
876 show ?thesis by auto |
|
877 qed |
|
878 |
|
879 lemma runing_inversion_1: |
|
880 assumes neq_th': "th' \<noteq> th" |
|
881 and runing': "th' \<in> runing (t@s)" |
|
882 shows "th' \<in> threads s \<and> cntV s th' < cntP s th'" |
|
883 proof(cases "th' \<in> threads s") |
|
884 case True |
|
885 with runing_precond [OF this neq_th' runing'] show ?thesis by simp |
|
886 next |
|
887 case False |
|
888 let ?Q = "\<lambda> t. th' \<in> threads (t@s)" |
|
889 let ?q = "moment 0 t" |
|
890 from moment_eq and False have not_thread: "\<not> ?Q ?q" by simp |
|
891 from runing' have "th' \<in> threads (t@s)" by (simp add:runing_def readys_def) |
|
892 from p_split_gen [of ?Q, OF this not_thread] |
|
893 obtain i where lt_its: "i < length t" |
|
894 and le_i: "0 \<le> i" |
|
895 and pre: " th' \<notin> threads (moment i t @ s)" (is "th' \<notin> threads ?pre") |
|
896 and post: "(\<forall>i'>i. th' \<in> threads (moment i' t @ s))" by auto |
|
897 from lt_its have "Suc i \<le> length t" by auto |
|
898 from moment_head[OF this] obtain e where |
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899 eq_me: "moment (Suc i) t = e # moment i t" by blast |
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900 from red_moment[of "Suc i"] and eq_me |
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901 have "extend_highest_set s' th prio (e # moment i t)" by simp |
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902 hence vt_e: "vt step (e#(moment i t)@s)" |
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903 by (unfold extend_highest_set_def extend_highest_set_axioms_def |
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904 highest_set_def s_def, auto) |
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905 from step_back_step[OF this] have stp_i: "step (moment i t @ s) e" . |
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906 from post[rule_format, of "Suc i"] and eq_me |
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907 have not_in': "th' \<in> threads (e # moment i t@s)" by auto |
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908 from create_pre[OF stp_i pre this] |
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909 obtain prio where eq_e: "e = Create th' prio" . |
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910 have "cntP (moment i t@s) th' = cntV (moment i t@s) th'" |
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911 proof(rule cnp_cnv_eq) |
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912 from step_back_vt [OF vt_e] |
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913 show "vt step (moment i t @ s)" . |
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914 next |
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915 from eq_e and stp_i |
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916 have "step (moment i t @ s) (Create th' prio)" by simp |
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917 thus "th' \<notin> threads (moment i t @ s)" by (cases, simp) |
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918 qed |
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919 with eq_e |
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920 have "cntP ((e#moment i t)@s) th' = cntV ((e#moment i t)@s) th'" |
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921 by (simp add:cntP_def cntV_def count_def) |
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922 with eq_me[symmetric] |
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923 have h1: "cntP (moment (Suc i) t @ s) th' = cntV (moment (Suc i) t@ s) th'" |
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924 by simp |
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925 from eq_e have "th' \<in> threads ((e#moment i t)@s)" by simp |
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926 with eq_me [symmetric] |
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927 have h2: "th' \<in> threads (moment (Suc i) t @ s)" by simp |
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928 from moment_blocked [OF neq_th' h2 h1, of "length t"] and lt_its |
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929 and moment_ge |
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930 have "th' \<notin> runing (t @ s)" by auto |
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931 with runing' |
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932 show ?thesis by auto |
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933 qed |
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934 |
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935 lemma runing_inversion_2: |
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936 assumes runing': "th' \<in> runing (t@s)" |
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937 shows "th' = th \<or> (th' \<noteq> th \<and> th' \<in> threads s \<and> cntV s th' < cntP s th')" |
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938 proof - |
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939 from runing_inversion_1[OF _ runing'] |
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940 show ?thesis by auto |
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941 qed |
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942 |
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943 lemma live: "runing (t@s) \<noteq> {}" |
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944 proof(cases "th \<in> runing (t@s)") |
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945 case True thus ?thesis by auto |
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946 next |
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947 case False |
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948 then have not_ready: "th \<notin> readys (t@s)" |
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949 apply (unfold runing_def, |
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950 insert th_cp_max max_cp_readys_threads[OF vt_t, symmetric]) |
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951 by auto |
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952 from th_kept have "th \<in> threads (t@s)" by auto |
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953 from th_chain_to_ready[OF vt_t this] and not_ready |
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954 obtain th' where th'_in: "th' \<in> readys (t@s)" |
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955 and dp: "(Th th, Th th') \<in> (depend (t @ s))\<^sup>+" by auto |
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956 have "th' \<in> runing (t@s)" |
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957 proof - |
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958 have "cp (t @ s) th' = Max (cp (t @ s) ` readys (t @ s))" |
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959 proof - |
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960 have " Max ((\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')) = |
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961 preced th (t@s)" |
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962 proof(rule Max_eqI) |
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963 fix y |
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964 assume "y \<in> (\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')" |
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965 then obtain th1 where |
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966 h1: "th1 = th' \<or> th1 \<in> dependents (wq (t @ s)) th'" |
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967 and eq_y: "y = preced th1 (t@s)" by auto |
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968 show "y \<le> preced th (t @ s)" |
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969 proof - |
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970 from max_preced |
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971 have "preced th (t @ s) = Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" . |
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972 moreover have "y \<le> \<dots>" |
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973 proof(rule Max_ge) |
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974 from h1 |
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975 have "th1 \<in> threads (t@s)" |
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976 proof |
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977 assume "th1 = th'" |
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978 with th'_in show ?thesis by (simp add:readys_def) |
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979 next |
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980 assume "th1 \<in> dependents (wq (t @ s)) th'" |
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981 with dependents_threads [OF vt_t] |
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982 show "th1 \<in> threads (t @ s)" by auto |
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983 qed |
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984 with eq_y show " y \<in> (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)" by simp |
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985 next |
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986 from finite_threads[OF vt_t] |
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987 show "finite ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" by simp |
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988 qed |
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989 ultimately show ?thesis by auto |
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990 qed |
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991 next |
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992 from finite_threads[OF vt_t] dependents_threads [OF vt_t, of th'] |
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993 show "finite ((\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th'))" |
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994 by (auto intro:finite_subset) |
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995 next |
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996 from dp |
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997 have "th \<in> dependents (wq (t @ s)) th'" |
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998 by (unfold cs_dependents_def, auto simp:eq_depend) |
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999 thus "preced th (t @ s) \<in> |
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1000 (\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')" |
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1001 by auto |
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1002 qed |
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1003 moreover have "\<dots> = Max (cp (t @ s) ` readys (t @ s))" |
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1004 proof - |
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1005 from max_preced and max_cp_eq[OF vt_t, symmetric] |
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1006 have "preced th (t @ s) = Max (cp (t @ s) ` threads (t @ s))" by simp |
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1007 with max_cp_readys_threads[OF vt_t] show ?thesis by simp |
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1008 qed |
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1009 ultimately show ?thesis by (unfold cp_eq_cpreced cpreced_def, simp) |
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1010 qed |
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1011 with th'_in show ?thesis by (auto simp:runing_def) |
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1012 qed |
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1013 thus ?thesis by auto |
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1014 qed |
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1015 |
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1016 end |
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1017 |
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1018 end |
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1019 |
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