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1 (* Title: HOL/UNITY/Mutex.thy |
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2 ID: $Id$ |
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3 Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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4 Copyright 1998 University of Cambridge |
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5 |
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6 Based on "A Family of 2-Process Mutual Exclusion Algorithms" by J Misra |
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7 *) |
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8 |
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9 open Mutex; |
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10 |
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11 val cmd_defs = [mutex_def, cmd0_def, cmd1u_def, cmd1v_def, |
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12 cmd2_def, cmd3_def, cmd4_def]; |
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13 |
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14 goalw thy [mutex_def] "id : mutex"; |
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15 by (Simp_tac 1); |
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16 qed "id_in_mutex"; |
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17 AddIffs [id_in_mutex]; |
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18 |
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19 |
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20 (** Constrains/Ensures tactics: NEED TO BE GENERALIZED OVER ALL PROGRAMS **) |
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21 |
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22 (*proves "constrains" properties when the program is specified*) |
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23 val constrains_tac = |
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24 SELECT_GOAL |
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25 (EVERY [rtac constrainsI 1, |
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26 rewtac mutex_def, |
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27 REPEAT_FIRST (eresolve_tac [insertE, emptyE]), |
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28 rewrite_goals_tac cmd_defs, |
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29 ALLGOALS (SELECT_GOAL Auto_tac)]); |
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30 |
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31 |
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32 (*proves "ensures" properties when the program is specified*) |
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33 fun ensures_tac sact = |
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34 SELECT_GOAL |
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35 (EVERY [REPEAT (resolve_tac [LeadsTo_Basis, leadsTo_Basis, ensuresI] 1), |
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36 res_inst_tac [("act", sact)] transient_mem 2, |
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37 Simp_tac 2, |
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38 constrains_tac 1, |
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39 rewrite_goals_tac cmd_defs, |
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40 Auto_tac]); |
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41 |
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42 |
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43 (*The booleans p, u, v are always either 0 or 1*) |
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44 goalw thy [stable_def, boolVars_def] |
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45 "stable mutex boolVars"; |
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46 by (constrains_tac 1); |
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47 by (auto_tac (claset() addSEs [less_SucE], simpset())); |
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48 qed "stable_boolVars"; |
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49 |
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50 goal thy "reachable MInit mutex <= boolVars"; |
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51 by (rtac strongest_invariant 1); |
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52 by (rtac stable_boolVars 2); |
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53 by (rewrite_goals_tac [MInit_def, boolVars_def]); |
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54 by Auto_tac; |
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55 qed "reachable_subset_boolVars"; |
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56 |
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57 val reachable_subset_boolVars' = |
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58 rewrite_rule [boolVars_def] reachable_subset_boolVars; |
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59 |
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60 goalw thy [stable_def, invariant_def] |
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61 "stable mutex (invariant 0 UU MM)"; |
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62 by (constrains_tac 1); |
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63 qed "stable_invar_0um"; |
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64 |
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65 goalw thy [stable_def, invariant_def] |
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66 "stable mutex (invariant 1 VV NN)"; |
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67 by (constrains_tac 1); |
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68 qed "stable_invar_1vn"; |
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69 |
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70 goalw thy [MInit_def, invariant_def] "MInit <= invariant 0 UU MM"; |
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71 by Auto_tac; |
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72 qed "MInit_invar_0um"; |
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73 |
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74 goalw thy [MInit_def, invariant_def] "MInit <= invariant 1 VV NN"; |
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75 by Auto_tac; |
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76 qed "MInit_invar_1vn"; |
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77 |
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78 (*The intersection is an invariant of the system*) |
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79 goal thy "reachable MInit mutex <= invariant 0 UU MM Int invariant 1 VV NN"; |
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80 by (simp_tac (simpset() addsimps |
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81 [strongest_invariant, Int_greatest, stable_Int, |
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82 stable_invar_0um, stable_invar_1vn, |
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83 MInit_invar_0um,MInit_invar_1vn]) 1); |
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84 qed "reachable_subset_invariant"; |
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85 |
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86 val reachable_subset_invariant' = |
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87 rewrite_rule [invariant_def] reachable_subset_invariant; |
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88 |
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89 |
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90 (*The safety property (mutual exclusion) follows from the claimed invar_s*) |
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91 goalw thy [invariant_def] |
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92 "{s. s MM = 3 & s NN = 3} <= \ |
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93 \ Compl (invariant 0 UU MM Int invariant 1 VV NN)"; |
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94 by Auto_tac; |
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95 val lemma = result(); |
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96 |
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97 goal thy "{s. s MM = 3 & s NN = 3} <= Compl (reachable MInit mutex)"; |
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98 by (rtac ([lemma, reachable_subset_invariant RS Compl_anti_mono] MRS subset_trans) 1); |
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99 qed "mutual_exclusion"; |
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100 |
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101 |
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102 (*The bad invariant FAILS in cmd1v*) |
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103 goalw thy [stable_def, bad_invariant_def] |
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104 "stable mutex (bad_invariant 0 UU MM)"; |
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105 by (constrains_tac 1); |
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106 by (trans_tac 1); |
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107 by (safe_tac (claset() addSEs [le_SucE])); |
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108 by (Asm_full_simp_tac 1); |
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109 (*Resulting state: n=1, p=false, m=4, u=false. |
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110 Execution of cmd1v (the command of process v guarded by n=1) sets p:=true, |
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111 violating the invariant!*) |
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112 (*Check that subgoals remain: proof failed.*) |
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113 getgoal 1; |
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114 |
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115 |
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116 (*** Progress for U ***) |
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117 |
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118 goalw thy [unless_def] "unless mutex {s. s MM=2} {s. s MM=3}"; |
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119 by (constrains_tac 1); |
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120 qed "U_F0"; |
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121 |
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122 goal thy "LeadsTo MInit mutex {s. s MM=1} {s. s PP = s VV & s MM = 2}"; |
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123 by (rtac leadsTo_imp_LeadsTo 1); (*makes the proof much faster*) |
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124 by (ensures_tac "cmd1u" 1); |
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125 qed "U_F1"; |
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126 |
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127 goal thy "LeadsTo MInit mutex {s. s PP = 0 & s MM = 2} {s. s MM = 3}"; |
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128 by (cut_facts_tac [reachable_subset_invariant'] 1); |
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129 by (ensures_tac "cmd2 0 MM" 1); |
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130 qed "U_F2"; |
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131 |
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132 goalw thy [mutex_def] "LeadsTo MInit mutex {s. s MM = 3} {s. s PP = 1}"; |
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133 by (rtac leadsTo_imp_LeadsTo 1); |
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134 by (res_inst_tac [("B", "{s. s MM = 4}")] leadsTo_Trans 1); |
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135 by (ensures_tac "cmd4 1 MM" 2); |
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136 by (ensures_tac "cmd3 UU MM" 1); |
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137 qed "U_F3"; |
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138 |
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139 goal thy "LeadsTo MInit mutex {s. s MM = 2} {s. s PP = 1}"; |
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140 by (rtac ([LeadsTo_weaken_L, subset_refl RS subset_imp_LeadsTo] |
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141 MRS LeadsTo_Diff) 1); |
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142 by (rtac ([U_F2, U_F3] MRS LeadsTo_Trans) 1); |
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143 by (cut_facts_tac [reachable_subset_boolVars'] 1); |
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144 by (auto_tac (claset() addSEs [less_SucE], simpset())); |
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145 val lemma2 = result(); |
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146 |
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147 goal thy "LeadsTo MInit mutex {s. s MM = 1} {s. s PP = 1}"; |
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148 by (rtac ([U_F1 RS LeadsTo_weaken_R, lemma2] MRS LeadsTo_Trans) 1); |
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149 by (Blast_tac 1); |
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150 val lemma1 = result(); |
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151 |
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152 |
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153 goal thy "LeadsTo MInit mutex {s. 1 <= s MM & s MM <= 3} {s. s PP = 1}"; |
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154 by (simp_tac (simpset() addsimps [le_Suc_eq, conj_disj_distribL] |
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155 addcongs [rev_conj_cong]) 1); |
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156 by (simp_tac (simpset() addsimps [Collect_disj_eq, LeadsTo_Un_distrib, |
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157 lemma1, lemma2, U_F3] ) 1); |
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158 val lemma123 = result(); |
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159 |
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160 |
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161 (*Misra's F4*) |
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162 goal thy "LeadsTo MInit mutex {s. s UU = 1} {s. s PP = 1}"; |
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163 by (rtac LeadsTo_weaken_L 1); |
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164 by (rtac lemma123 1); |
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165 by (cut_facts_tac [reachable_subset_invariant'] 1); |
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166 by Auto_tac; |
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167 qed "u_leadsto_p"; |
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168 |
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169 |
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170 (*** Progress for V ***) |
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171 |
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172 |
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173 goalw thy [unless_def] "unless mutex {s. s NN=2} {s. s NN=3}"; |
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174 by (constrains_tac 1); |
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175 qed "V_F0"; |
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176 |
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177 goal thy "LeadsTo MInit mutex {s. s NN=1} {s. s PP = 1 - s UU & s NN = 2}"; |
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178 by (rtac leadsTo_imp_LeadsTo 1); (*makes the proof much faster*) |
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179 by (ensures_tac "cmd1v" 1); |
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180 qed "V_F1"; |
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181 |
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182 goal thy "LeadsTo MInit mutex {s. s PP = 1 & s NN = 2} {s. s NN = 3}"; |
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183 by (cut_facts_tac [reachable_subset_invariant'] 1); |
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184 by (ensures_tac "cmd2 1 NN" 1); |
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185 qed "V_F2"; |
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186 |
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187 goalw thy [mutex_def] "LeadsTo MInit mutex {s. s NN = 3} {s. s PP = 0}"; |
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188 by (rtac leadsTo_imp_LeadsTo 1); |
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189 by (res_inst_tac [("B", "{s. s NN = 4}")] leadsTo_Trans 1); |
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190 by (ensures_tac "cmd4 0 NN" 2); |
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191 by (ensures_tac "cmd3 VV NN" 1); |
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192 qed "V_F3"; |
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193 |
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194 goal thy "LeadsTo MInit mutex {s. s NN = 2} {s. s PP = 0}"; |
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195 by (rtac ([LeadsTo_weaken_L, subset_refl RS subset_imp_LeadsTo] |
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196 MRS LeadsTo_Diff) 1); |
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197 by (rtac ([V_F2, V_F3] MRS LeadsTo_Trans) 1); |
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198 by (cut_facts_tac [reachable_subset_boolVars'] 1); |
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199 by (auto_tac (claset() addSEs [less_SucE], simpset())); |
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200 val lemma2 = result(); |
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201 |
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202 goal thy "LeadsTo MInit mutex {s. s NN = 1} {s. s PP = 0}"; |
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203 by (rtac ([V_F1 RS LeadsTo_weaken_R, lemma2] MRS LeadsTo_Trans) 1); |
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204 by (Blast_tac 1); |
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205 val lemma1 = result(); |
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206 |
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207 |
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208 goal thy "LeadsTo MInit mutex {s. 1 <= s NN & s NN <= 3} {s. s PP = 0}"; |
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209 by (simp_tac (simpset() addsimps [le_Suc_eq, conj_disj_distribL] |
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210 addcongs [rev_conj_cong]) 1); |
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211 by (simp_tac (simpset() addsimps [Collect_disj_eq, LeadsTo_Un_distrib, |
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212 lemma1, lemma2, V_F3] ) 1); |
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213 val lemma123 = result(); |
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214 |
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215 |
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216 (*Misra's F4*) |
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217 goal thy "LeadsTo MInit mutex {s. s VV = 1} {s. s PP = 0}"; |
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218 by (rtac LeadsTo_weaken_L 1); |
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219 by (rtac lemma123 1); |
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220 by (cut_facts_tac [reachable_subset_invariant'] 1); |
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221 by Auto_tac; |
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222 qed "v_leadsto_not_p"; |
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223 |
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224 |
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225 (** Absence of starvation **) |
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226 |
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227 (*Misra's F6*) |
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228 goal thy "LeadsTo MInit mutex {s. s MM = 1} {s. s MM = 3}"; |
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229 by (rtac LeadsTo_Un_duplicate 1); |
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230 by (rtac LeadsTo_cancel2 1); |
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231 by (rtac U_F2 2); |
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232 by (simp_tac (simpset() addsimps [Collect_conj_eq] ) 1); |
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233 by (stac Un_commute 1); |
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234 by (rtac LeadsTo_Un_duplicate 1); |
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235 by (rtac ([v_leadsto_not_p, U_F0] MRS R_PSP_unless RSN(2, LeadsTo_cancel2)) 1); |
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236 by (rtac (U_F1 RS LeadsTo_weaken_R) 1); |
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237 by (cut_facts_tac [reachable_subset_boolVars'] 1); |
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238 by (auto_tac (claset() addSEs [less_SucE], simpset())); |
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239 qed "m1_leadsto_3"; |
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240 |
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241 |
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242 (*The same for V*) |
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243 goal thy "LeadsTo MInit mutex {s. s NN = 1} {s. s NN = 3}"; |
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244 by (rtac LeadsTo_Un_duplicate 1); |
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245 by (rtac LeadsTo_cancel2 1); |
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246 by (rtac V_F2 2); |
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247 by (simp_tac (simpset() addsimps [Collect_conj_eq] ) 1); |
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248 by (stac Un_commute 1); |
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249 by (rtac LeadsTo_Un_duplicate 1); |
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250 by (rtac ([u_leadsto_p, V_F0] MRS R_PSP_unless RSN(2, LeadsTo_cancel2)) 1); |
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251 by (rtac (V_F1 RS LeadsTo_weaken_R) 1); |
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252 by (cut_facts_tac [reachable_subset_boolVars'] 1); |
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253 by (auto_tac (claset() addSEs [less_SucE], simpset())); |
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254 qed "n1_leadsto_3"; |