1 |
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2 theory Re1 |
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3 imports "Main" |
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4 begin |
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5 |
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6 section {* Sequential Composition of Sets *} |
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7 |
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8 definition |
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9 Sequ :: "string set \<Rightarrow> string set \<Rightarrow> string set" ("_ ;; _" [100,100] 100) |
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10 where |
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11 "A ;; B = {s1 @ s2 | s1 s2. s1 \<in> A \<and> s2 \<in> B}" |
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12 |
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13 text {* Two Simple Properties about Sequential Composition *} |
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14 |
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15 lemma seq_empty [simp]: |
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16 shows "A ;; {[]} = A" |
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17 and "{[]} ;; A = A" |
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18 by (simp_all add: Sequ_def) |
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19 |
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20 lemma seq_null [simp]: |
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21 shows "A ;; {} = {}" |
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22 and "{} ;; A = {}" |
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23 by (simp_all add: Sequ_def) |
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24 |
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25 section {* Regular Expressions *} |
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26 |
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27 datatype rexp = |
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28 NULL |
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29 | EMPTY |
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30 | CHAR char |
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31 | SEQ rexp rexp |
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32 | ALT rexp rexp |
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33 |
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34 section {* Semantics of Regular Expressions *} |
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35 |
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36 fun |
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37 L :: "rexp \<Rightarrow> string set" |
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38 where |
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39 "L (NULL) = {}" |
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40 | "L (EMPTY) = {[]}" |
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41 | "L (CHAR c) = {[c]}" |
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42 | "L (SEQ r1 r2) = (L r1) ;; (L r2)" |
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43 | "L (ALT r1 r2) = (L r1) \<union> (L r2)" |
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44 |
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45 |
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46 section {* Values *} |
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47 |
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48 datatype val = |
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49 Void |
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50 | Char char |
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51 | Seq val val |
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52 | Right val |
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53 | Left val |
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54 |
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55 section {* Relation between values and regular expressions *} |
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56 |
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57 inductive Prf :: "val \<Rightarrow> rexp \<Rightarrow> bool" ("\<turnstile> _ : _" [100, 100] 100) |
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58 where |
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59 "\<lbrakk>\<turnstile> v1 : r1; \<turnstile> v2 : r2\<rbrakk> \<Longrightarrow> \<turnstile> Seq v1 v2 : SEQ r1 r2" |
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60 | "\<turnstile> v1 : r1 \<Longrightarrow> \<turnstile> Left v1 : ALT r1 r2" |
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61 | "\<turnstile> v2 : r2 \<Longrightarrow> \<turnstile> Right v2 : ALT r1 r2" |
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62 | "\<turnstile> Void : EMPTY" |
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63 | "\<turnstile> Char c : CHAR c" |
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64 |
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65 section {* The string behind a value *} |
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66 |
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67 fun flat :: "val \<Rightarrow> string" |
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68 where |
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69 "flat(Void) = []" |
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70 | "flat(Char c) = [c]" |
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71 | "flat(Left v) = flat(v)" |
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72 | "flat(Right v) = flat(v)" |
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73 | "flat(Seq v1 v2) = flat(v1) @ flat(v2)" |
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74 |
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75 |
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76 lemma Prf_flat_L: |
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77 assumes "\<turnstile> v : r" shows "flat v \<in> L r" |
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78 using assms |
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79 apply(induct) |
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80 apply(auto simp add: Sequ_def) |
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81 done |
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82 |
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83 lemma L_flat_Prf: |
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84 "L(r) = {flat v | v. \<turnstile> v : r}" |
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85 apply(induct r) |
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86 apply(auto dest: Prf_flat_L simp add: Sequ_def) |
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87 apply (metis Prf.intros(4) flat.simps(1)) |
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88 apply (metis Prf.intros(5) flat.simps(2)) |
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89 apply (metis Prf.intros(1) flat.simps(5)) |
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90 apply (metis Prf.intros(2) flat.simps(3)) |
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91 apply (metis Prf.intros(3) flat.simps(4)) |
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92 apply(erule Prf.cases) |
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93 apply(auto) |
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94 done |
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95 |
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96 section {* Ordering of values *} |
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97 |
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98 inductive ValOrd :: "val \<Rightarrow> rexp \<Rightarrow> val \<Rightarrow> bool" ("_ \<succ>_ _" [100, 100, 100] 100) |
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99 where |
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100 "\<lbrakk>v1 = v1'; v2 \<succ>r2 v2'\<rbrakk> \<Longrightarrow> (Seq v1 v2) \<succ>(SEQ r1 r2) (Seq v1' v2')" |
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101 | "v1 \<succ>r1 v1' \<Longrightarrow> (Seq v1 v2) \<succ>(SEQ r1 r2) (Seq v1' v2')" |
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102 | "length (flat v1) \<ge> length (flat v2) \<Longrightarrow> (Left v1) \<succ>(ALT r1 r2) (Right v2)" |
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103 | "length (flat v2) > length (flat v1) \<Longrightarrow> (Right v2) \<succ>(ALT r1 r2) (Left v1)" |
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104 | "v2 \<succ>r2 v2' \<Longrightarrow> (Right v2) \<succ>(ALT r1 r2) (Right v2')" |
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105 | "v1 \<succ>r1 v1' \<Longrightarrow> (Left v1) \<succ>(ALT r1 r2) (Left v1')" |
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106 | "Void \<succ>EMPTY Void" |
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107 | "(Char c) \<succ>(CHAR c) (Char c)" |
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108 |
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109 (* |
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110 lemma |
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111 assumes "r = SEQ (ALT EMPTY EMPTY) (ALT EMPTY (CHAR c))" |
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112 shows "(Seq (Left Void) (Right (Char c))) \<succ>r (Seq (Left Void) (Left Void))" |
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113 using assms |
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114 apply(simp) |
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115 apply(rule ValOrd.intros) |
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116 apply(rule ValOrd.intros) |
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117 apply(rule ValOrd.intros) |
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118 apply(rule ValOrd.intros) |
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119 apply(simp) |
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120 done |
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121 *) |
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122 |
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123 section {* Posix definition *} |
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124 |
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125 definition POSIX :: "val \<Rightarrow> rexp \<Rightarrow> bool" |
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126 where |
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127 "POSIX v r \<equiv> (\<forall>v'. (\<turnstile> v' : r \<and> flat v = flat v') \<longrightarrow> v \<succ>r v')" |
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128 |
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129 (* |
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130 an alternative definition: might cause problems |
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131 with theorem mkeps_POSIX |
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132 *) |
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133 |
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134 definition POSIX2 :: "val \<Rightarrow> rexp \<Rightarrow> bool" |
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135 where |
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136 "POSIX2 v r \<equiv> \<turnstile> v : r \<and> (\<forall>v'. \<turnstile> v' : r \<longrightarrow> v \<succ>r v')" |
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137 |
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138 definition POSIX3 :: "val \<Rightarrow> rexp \<Rightarrow> bool" |
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139 where |
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140 "POSIX3 v r \<equiv> \<turnstile> v : r \<and> (\<forall>v'. (\<turnstile> v' : r \<and> flat v \<ge> flat v') \<longrightarrow> v \<succ>r v')" |
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141 |
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142 |
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143 (* |
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144 lemma POSIX_SEQ: |
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145 assumes "POSIX (Seq v1 v2) (SEQ r1 r2)" "\<turnstile> v1 : r1" "\<turnstile> v2 : r2" |
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146 shows "POSIX v1 r1 \<and> POSIX v2 r2" |
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147 using assms |
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148 unfolding POSIX_def |
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149 apply(auto) |
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150 apply(drule_tac x="Seq v' v2" in spec) |
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151 apply(simp) |
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152 apply (smt Prf.intros(1) ValOrd.simps assms(3) rexp.inject(2) val.distinct(15) val.distinct(17) val.distinct(3) val.distinct(9) val.inject(2)) |
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153 apply(drule_tac x="Seq v1 v'" in spec) |
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154 apply(simp) |
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155 by (smt Prf.intros(1) ValOrd.simps rexp.inject(2) val.distinct(15) val.distinct(17) val.distinct(3) val.distinct(9) val.inject(2)) |
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156 *) |
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157 |
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158 (* |
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159 lemma POSIX_SEQ_I: |
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160 assumes "POSIX v1 r1" "POSIX v2 r2" |
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161 shows "POSIX (Seq v1 v2) (SEQ r1 r2)" |
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162 using assms |
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163 unfolding POSIX_def |
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164 apply(auto) |
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165 apply(rotate_tac 2) |
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166 apply(erule Prf.cases) |
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167 apply(simp_all)[5] |
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168 apply(auto)[1] |
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169 apply(rule ValOrd.intros) |
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170 |
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171 apply(auto) |
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172 done |
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173 *) |
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174 |
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175 |
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176 |
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177 |
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178 lemma POSIX_ALT2: |
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179 assumes "POSIX (Left v1) (ALT r1 r2)" |
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180 shows "POSIX v1 r1" |
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181 using assms |
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182 unfolding POSIX_def |
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183 apply(auto) |
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184 apply(drule_tac x="Left v'" in spec) |
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185 apply(simp) |
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186 apply(drule mp) |
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187 apply(rule Prf.intros) |
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188 apply(auto) |
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189 apply(erule ValOrd.cases) |
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190 apply(simp_all) |
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191 done |
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192 |
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193 lemma POSIX2_ALT: |
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194 assumes "POSIX2 (Left v1) (ALT r1 r2)" |
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195 shows "POSIX2 v1 r1" |
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196 using assms |
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197 unfolding POSIX2_def |
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198 apply(auto) |
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199 |
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200 done |
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201 |
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202 lemma POSIX_ALT: |
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203 assumes "POSIX (Left v1) (ALT r1 r2)" |
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204 shows "POSIX v1 r1" |
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205 using assms |
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206 unfolding POSIX_def |
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207 apply(auto) |
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208 apply(drule_tac x="Left v'" in spec) |
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209 apply(simp) |
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210 apply(drule mp) |
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211 apply(rule Prf.intros) |
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212 apply(auto) |
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213 apply(erule ValOrd.cases) |
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214 apply(simp_all) |
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215 done |
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216 |
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217 lemma POSIX2_ALT: |
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218 assumes "POSIX2 (Left v1) (ALT r1 r2)" |
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219 shows "POSIX2 v1 r1" |
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220 using assms |
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221 apply(simp add: POSIX2_def) |
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222 apply(auto) |
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223 apply(erule Prf.cases) |
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224 apply(simp_all)[5] |
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225 apply(drule_tac x="Left v'" in spec) |
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226 apply(drule mp) |
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227 apply(rule Prf.intros) |
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228 apply(auto) |
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229 apply(erule ValOrd.cases) |
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230 apply(simp_all) |
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231 done |
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232 |
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233 |
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234 lemma POSIX_ALT1a: |
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235 assumes "POSIX (Right v2) (ALT r1 r2)" |
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236 shows "POSIX v2 r2" |
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237 using assms |
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238 unfolding POSIX_def |
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239 apply(auto) |
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240 apply(drule_tac x="Right v'" in spec) |
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241 apply(simp) |
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242 apply(drule mp) |
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243 apply(rule Prf.intros) |
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244 apply(auto) |
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245 apply(erule ValOrd.cases) |
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246 apply(simp_all) |
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247 done |
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248 |
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249 lemma POSIX2_ALT1a: |
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250 assumes "POSIX2 (Right v2) (ALT r1 r2)" |
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251 shows "POSIX2 v2 r2" |
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252 using assms |
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253 unfolding POSIX2_def |
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254 apply(auto) |
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255 apply(erule Prf.cases) |
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256 apply(simp_all)[5] |
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257 apply(drule_tac x="Right v'" in spec) |
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258 apply(drule mp) |
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259 apply(rule Prf.intros) |
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260 apply(auto) |
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261 apply(erule ValOrd.cases) |
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262 apply(simp_all) |
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263 done |
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264 |
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265 |
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266 lemma POSIX_ALT1b: |
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267 assumes "POSIX (Right v2) (ALT r1 r2)" |
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268 shows "(\<forall>v'. (\<turnstile> v' : r2 \<and> flat v' = flat v2) \<longrightarrow> v2 \<succ>r2 v')" |
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269 using assms |
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270 apply(drule_tac POSIX_ALT1a) |
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271 unfolding POSIX_def |
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272 apply(auto) |
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273 done |
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274 |
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275 lemma POSIX_ALT_I1: |
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276 assumes "POSIX v1 r1" |
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277 shows "POSIX (Left v1) (ALT r1 r2)" |
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278 using assms |
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279 unfolding POSIX_def |
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280 apply(auto) |
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281 apply(rotate_tac 3) |
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282 apply(erule Prf.cases) |
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283 apply(simp_all)[5] |
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284 apply(auto) |
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285 apply(rule ValOrd.intros) |
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286 apply(auto) |
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287 apply(rule ValOrd.intros) |
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288 by simp |
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289 |
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290 lemma POSIX2_ALT_I1: |
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291 assumes "POSIX2 v1 r1" |
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292 shows "POSIX2 (Left v1) (ALT r1 r2)" |
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293 using assms |
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294 unfolding POSIX2_def |
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295 apply(auto) |
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296 apply(rule Prf.intros) |
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297 apply(simp) |
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298 apply(rotate_tac 2) |
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299 apply(erule Prf.cases) |
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300 apply(simp_all)[5] |
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301 apply(auto) |
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302 apply(rule ValOrd.intros) |
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303 apply(auto) |
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304 apply(rule ValOrd.intros) |
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305 |
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306 |
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307 by simp |
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308 |
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309 lemma POSIX_ALT_I2: |
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310 assumes "POSIX v2 r2" "\<forall>v'. \<turnstile> v' : r1 \<longrightarrow> length (flat v2) > length (flat v')" |
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311 shows "POSIX (Right v2) (ALT r1 r2)" |
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312 using assms |
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313 unfolding POSIX_def |
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314 apply(auto) |
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315 apply(rotate_tac 3) |
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316 apply(erule Prf.cases) |
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317 apply(simp_all)[5] |
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318 apply(auto) |
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319 apply(rule ValOrd.intros) |
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320 apply metis |
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321 done |
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322 |
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323 |
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324 section {* The ordering is reflexive *} |
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325 |
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326 lemma ValOrd_refl: |
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327 assumes "\<turnstile> v : r" |
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328 shows "v \<succ>r v" |
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329 using assms |
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330 apply(induct) |
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331 apply(auto intro: ValOrd.intros) |
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332 done |
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333 |
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334 |
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335 section {* The Matcher *} |
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336 |
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337 fun |
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338 nullable :: "rexp \<Rightarrow> bool" |
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339 where |
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340 "nullable (NULL) = False" |
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341 | "nullable (EMPTY) = True" |
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342 | "nullable (CHAR c) = False" |
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343 | "nullable (ALT r1 r2) = (nullable r1 \<or> nullable r2)" |
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344 | "nullable (SEQ r1 r2) = (nullable r1 \<and> nullable r2)" |
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345 |
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346 lemma nullable_correctness: |
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347 shows "nullable r \<longleftrightarrow> [] \<in> (L r)" |
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348 apply (induct r) |
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349 apply(auto simp add: Sequ_def) |
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350 done |
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351 |
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352 fun mkeps :: "rexp \<Rightarrow> val" |
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353 where |
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354 "mkeps(EMPTY) = Void" |
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355 | "mkeps(SEQ r1 r2) = Seq (mkeps r1) (mkeps r2)" |
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356 | "mkeps(ALT r1 r2) = (if nullable(r1) then Left (mkeps r1) else Right (mkeps r2))" |
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357 |
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358 lemma mkeps_nullable: |
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359 assumes "nullable(r)" shows "\<turnstile> mkeps r : r" |
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360 using assms |
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361 apply(induct rule: nullable.induct) |
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362 apply(auto intro: Prf.intros) |
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363 done |
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364 |
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365 lemma mkeps_flat: |
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366 assumes "nullable(r)" shows "flat (mkeps r) = []" |
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367 using assms |
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368 apply(induct rule: nullable.induct) |
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369 apply(auto) |
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370 done |
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371 |
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372 text {* |
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373 The value mkeps returns is always the correct POSIX |
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374 value. |
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375 *} |
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376 |
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377 lemma mkeps_POSIX2: |
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378 assumes "nullable r" |
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379 shows "POSIX2 (mkeps r) r" |
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380 using assms |
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381 apply(induct r) |
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382 apply(auto)[1] |
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383 apply(simp add: POSIX2_def) |
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384 |
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385 lemma mkeps_POSIX: |
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386 assumes "nullable r" |
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387 shows "POSIX (mkeps r) r" |
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388 using assms |
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389 apply(induct r) |
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390 apply(auto)[1] |
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391 apply(simp add: POSIX_def) |
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392 apply(auto)[1] |
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393 apply(erule Prf.cases) |
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394 apply(simp_all)[5] |
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395 apply (metis ValOrd.intros) |
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396 apply(simp add: POSIX_def) |
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397 apply(auto)[1] |
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398 apply(simp add: POSIX_def) |
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399 apply(auto)[1] |
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400 apply(erule Prf.cases) |
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401 apply(simp_all)[5] |
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402 apply(auto) |
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403 apply (simp add: ValOrd.intros(2) mkeps_flat) |
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404 apply(simp add: POSIX_def) |
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405 apply(auto)[1] |
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406 apply(erule Prf.cases) |
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407 apply(simp_all)[5] |
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408 apply(auto) |
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409 apply (simp add: ValOrd.intros(6)) |
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410 apply (simp add: ValOrd.intros(3)) |
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411 apply(simp add: POSIX_def) |
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412 apply(auto)[1] |
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413 apply(erule Prf.cases) |
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414 apply(simp_all)[5] |
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415 apply(auto) |
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416 apply (simp add: ValOrd.intros(6)) |
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417 apply (simp add: ValOrd.intros(3)) |
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418 apply(simp add: POSIX_def) |
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419 apply(auto)[1] |
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420 apply(erule Prf.cases) |
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421 apply(simp_all)[5] |
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422 apply(auto) |
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423 apply (metis Prf_flat_L mkeps_flat nullable_correctness) |
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424 by (simp add: ValOrd.intros(5)) |
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425 |
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426 |
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427 lemma mkeps_POSIX2: |
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428 assumes "nullable r" |
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429 shows "POSIX2 (mkeps r) r" |
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430 using assms |
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431 apply(induct r) |
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432 apply(simp) |
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433 apply(simp) |
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434 apply(simp add: POSIX2_def) |
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435 apply(rule conjI) |
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436 apply(rule Prf.intros) |
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437 apply(auto)[1] |
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438 apply(erule Prf.cases) |
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439 apply(simp_all)[5] |
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440 apply(rule ValOrd.intros) |
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441 apply(simp) |
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442 apply(simp) |
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443 apply(simp add: POSIX2_def) |
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444 apply(rule conjI) |
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445 apply(rule Prf.intros) |
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446 apply(simp add: mkeps_nullable) |
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447 apply(simp add: mkeps_nullable) |
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448 apply(auto)[1] |
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449 apply(rotate_tac 6) |
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450 apply(erule Prf.cases) |
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451 apply(simp_all)[5] |
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452 apply(rule ValOrd.intros(2)) |
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453 apply(simp) |
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454 apply(simp only: nullable.simps) |
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455 apply(erule disjE) |
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456 apply(simp) |
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457 thm POSIX2_ALT1a |
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458 apply(rule POSIX2_ALT) |
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459 apply(simp add: POSIX2_def) |
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460 apply(rule conjI) |
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461 apply(rule Prf.intros) |
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462 apply(simp add: mkeps_nullable) |
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463 apply(auto)[1] |
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464 apply(rotate_tac 4) |
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465 apply(erule Prf.cases) |
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466 apply(simp_all)[5] |
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467 apply(rule ValOrd.intros) |
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468 apply(simp) |
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469 apply(rule ValOrd.intros) |
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470 |
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471 |
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472 section {* Derivatives *} |
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473 |
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474 fun |
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475 der :: "char \<Rightarrow> rexp \<Rightarrow> rexp" |
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476 where |
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477 "der c (NULL) = NULL" |
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478 | "der c (EMPTY) = NULL" |
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479 | "der c (CHAR c') = (if c = c' then EMPTY else NULL)" |
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480 | "der c (ALT r1 r2) = ALT (der c r1) (der c r2)" |
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481 | "der c (SEQ r1 r2) = |
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482 (if nullable r1 |
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483 then ALT (SEQ (der c r1) r2) (der c r2) |
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484 else SEQ (der c r1) r2)" |
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485 |
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486 fun |
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487 ders :: "string \<Rightarrow> rexp \<Rightarrow> rexp" |
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488 where |
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489 "ders [] r = r" |
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490 | "ders (c # s) r = ders s (der c r)" |
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491 |
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492 section {* Injection function *} |
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493 |
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494 fun injval :: "rexp \<Rightarrow> char \<Rightarrow> val \<Rightarrow> val" |
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495 where |
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496 "injval (CHAR d) c Void = Char d" |
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497 | "injval (ALT r1 r2) c (Left v1) = Left(injval r1 c v1)" |
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498 | "injval (ALT r1 r2) c (Right v2) = Right(injval r2 c v2)" |
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499 | "injval (SEQ r1 r2) c (Seq v1 v2) = Seq (injval r1 c v1) v2" |
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500 | "injval (SEQ r1 r2) c (Left (Seq v1 v2)) = Seq (injval r1 c v1) v2" |
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501 | "injval (SEQ r1 r2) c (Right v2) = Seq (mkeps r1) (injval r2 c v2)" |
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502 |
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503 section {* Projection function *} |
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504 |
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505 fun projval :: "rexp \<Rightarrow> char \<Rightarrow> val \<Rightarrow> val" |
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506 where |
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507 "projval (CHAR d) c _ = Void" |
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508 | "projval (ALT r1 r2) c (Left v1) = Left(projval r1 c v1)" |
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509 | "projval (ALT r1 r2) c (Right v2) = Right(projval r2 c v2)" |
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510 | "projval (SEQ r1 r2) c (Seq v1 v2) = |
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511 (if flat v1 = [] then Right(projval r2 c v2) |
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512 else if nullable r1 then Left (Seq (projval r1 c v1) v2) |
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513 else Seq (projval r1 c v1) v2)" |
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514 |
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515 text {* |
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516 Injection value is related to r |
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517 *} |
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518 |
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519 lemma v3: |
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520 assumes "\<turnstile> v : der c r" shows "\<turnstile> (injval r c v) : r" |
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521 using assms |
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522 apply(induct arbitrary: v rule: der.induct) |
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523 apply(simp) |
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524 apply(erule Prf.cases) |
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525 apply(simp_all)[5] |
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526 apply(erule Prf.cases) |
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527 apply(simp_all)[5] |
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528 apply(case_tac "c = c'") |
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529 apply(simp) |
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530 apply(erule Prf.cases) |
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531 apply(simp_all)[5] |
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532 apply (metis Prf.intros(5)) |
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533 apply(erule Prf.cases) |
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534 apply(simp_all)[5] |
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535 apply(erule Prf.cases) |
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536 apply(simp_all)[5] |
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537 apply (metis Prf.intros(2)) |
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538 apply (metis Prf.intros(3)) |
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539 apply(simp) |
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540 apply(case_tac "nullable r1") |
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541 apply(simp) |
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542 apply(erule Prf.cases) |
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543 apply(simp_all)[5] |
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544 apply(auto)[1] |
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545 apply(erule Prf.cases) |
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546 apply(simp_all)[5] |
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547 apply(auto)[1] |
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548 apply (metis Prf.intros(1)) |
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549 apply(auto)[1] |
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550 apply (metis Prf.intros(1) mkeps_nullable) |
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551 apply(simp) |
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552 apply(erule Prf.cases) |
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553 apply(simp_all)[5] |
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554 apply(auto)[1] |
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555 apply(rule Prf.intros) |
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556 apply(auto)[2] |
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557 done |
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558 |
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559 text {* |
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560 The string behin the injection value is an added c |
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561 *} |
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562 |
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563 lemma v4: |
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564 assumes "\<turnstile> v : der c r" shows "flat (injval r c v) = c # (flat v)" |
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565 using assms |
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566 apply(induct arbitrary: v rule: der.induct) |
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567 apply(simp) |
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568 apply(erule Prf.cases) |
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569 apply(simp_all)[5] |
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570 apply(simp) |
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571 apply(erule Prf.cases) |
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572 apply(simp_all)[5] |
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573 apply(simp) |
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574 apply(case_tac "c = c'") |
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575 apply(simp) |
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576 apply(auto)[1] |
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577 apply(erule Prf.cases) |
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578 apply(simp_all)[5] |
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579 apply(simp) |
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580 apply(erule Prf.cases) |
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581 apply(simp_all)[5] |
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582 apply(simp) |
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583 apply(erule Prf.cases) |
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584 apply(simp_all)[5] |
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585 apply(simp) |
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586 apply(case_tac "nullable r1") |
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587 apply(simp) |
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588 apply(erule Prf.cases) |
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589 apply(simp_all)[5] |
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590 apply(auto)[1] |
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591 apply(erule Prf.cases) |
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592 apply(simp_all)[5] |
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593 apply(auto)[1] |
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594 apply (metis mkeps_flat) |
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595 apply(simp) |
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596 apply(erule Prf.cases) |
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597 apply(simp_all)[5] |
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598 done |
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599 |
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600 text {* |
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601 Injection followed by projection is the identity. |
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602 *} |
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603 |
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604 lemma proj_inj_id: |
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605 assumes "\<turnstile> v : der c r" |
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606 shows "projval r c (injval r c v) = v" |
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607 using assms |
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608 apply(induct r arbitrary: c v rule: rexp.induct) |
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609 apply(simp) |
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610 apply(erule Prf.cases) |
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611 apply(simp_all)[5] |
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612 apply(simp) |
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613 apply(erule Prf.cases) |
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614 apply(simp_all)[5] |
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615 apply(simp) |
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616 apply(case_tac "c = char") |
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617 apply(simp) |
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618 apply(erule Prf.cases) |
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619 apply(simp_all)[5] |
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620 apply(simp) |
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621 apply(erule Prf.cases) |
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622 apply(simp_all)[5] |
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623 defer |
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624 apply(simp) |
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625 apply(erule Prf.cases) |
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626 apply(simp_all)[5] |
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627 apply(simp) |
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628 apply(case_tac "nullable rexp1") |
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629 apply(simp) |
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630 apply(erule Prf.cases) |
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631 apply(simp_all)[5] |
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632 apply(auto)[1] |
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633 apply(erule Prf.cases) |
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634 apply(simp_all)[5] |
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635 apply(auto)[1] |
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636 apply (metis list.distinct(1) v4) |
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637 apply(auto)[1] |
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638 apply (metis mkeps_flat) |
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639 apply(auto) |
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640 apply(erule Prf.cases) |
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641 apply(simp_all)[5] |
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642 apply(auto)[1] |
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643 apply(simp add: v4) |
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644 done |
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645 |
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646 lemma "\<exists>v. POSIX v r" |
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647 apply(induct r) |
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648 apply(rule exI) |
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649 apply(simp add: POSIX_def) |
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650 apply (metis (full_types) Prf_flat_L der.simps(1) der.simps(2) der.simps(3) flat.simps(1) nullable.simps(1) nullable_correctness proj_inj_id projval.simps(1) v3 v4) |
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651 apply(rule_tac x = "Void" in exI) |
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652 apply(simp add: POSIX_def) |
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653 apply (metis POSIX_def flat.simps(1) mkeps.simps(1) mkeps_POSIX nullable.simps(2)) |
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654 apply(rule_tac x = "Char char" in exI) |
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655 apply(simp add: POSIX_def) |
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656 apply(auto) [1] |
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657 apply(erule Prf.cases) |
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658 apply(simp_all) [5] |
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659 apply (metis ValOrd.intros(8)) |
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660 defer |
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661 apply(auto) |
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662 apply (metis POSIX_ALT_I1) |
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663 (* maybe it is too early to instantiate this existential quantifier *) |
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664 (* potentially this is the wrong POSIX value *) |
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665 apply(rule_tac x = "Seq v va" in exI ) |
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666 apply(simp (no_asm) add: POSIX_def) |
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667 apply(auto) |
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668 apply(erule Prf.cases) |
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669 apply(simp_all) |
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670 apply(case_tac "v \<succ>r1a v1") |
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671 apply (metis ValOrd.intros(2)) |
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672 apply(simp add: POSIX_def) |
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673 apply(case_tac "flat v = flat v1") |
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674 apply(auto)[1] |
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675 apply(simp only: append_eq_append_conv2) |
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676 apply(auto) |
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677 thm append_eq_append_conv2 |
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678 |
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679 text {* |
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680 |
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681 HERE: Crucial lemma that does not go through in the sequence case. |
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682 |
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683 *} |
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684 lemma v5: |
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685 assumes "\<turnstile> v : der c r" "POSIX v (der c r)" |
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686 shows "POSIX (injval r c v) r" |
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687 using assms |
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688 apply(induct arbitrary: v rule: der.induct) |
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689 apply(simp) |
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690 apply(erule Prf.cases) |
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691 apply(simp_all)[5] |
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692 apply(simp) |
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693 apply(erule Prf.cases) |
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694 apply(simp_all)[5] |
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695 apply(simp) |
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696 apply(case_tac "c = c'") |
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697 apply(auto simp add: POSIX_def)[1] |
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698 apply(erule Prf.cases) |
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699 apply(simp_all)[5] |
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700 apply(erule Prf.cases) |
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701 apply(simp_all)[5] |
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702 using ValOrd.simps apply blast |
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703 apply(auto) |
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704 apply(erule Prf.cases) |
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705 apply(simp_all)[5] |
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706 (* base cases done *) |
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707 (* ALT case *) |
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708 apply(erule Prf.cases) |
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709 apply(simp_all)[5] |
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710 using POSIX_ALT POSIX_ALT_I1 apply blast |
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711 apply(clarify) |
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712 apply(subgoal_tac "POSIX v2 (der c r2)") |
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713 prefer 2 |
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714 apply(auto simp add: POSIX_def)[1] |
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715 apply (metis POSIX_ALT1a POSIX_def flat.simps(4)) |
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716 apply(rotate_tac 1) |
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717 apply(drule_tac x="v2" in meta_spec) |
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718 apply(simp) |
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719 apply(subgoal_tac "\<turnstile> Right (injval r2 c v2) : (ALT r1 r2)") |
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720 prefer 2 |
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721 apply (metis Prf.intros(3) v3) |
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722 apply(rule ccontr) |
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723 apply(auto simp add: POSIX_def)[1] |
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724 |
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725 apply(rule allI) |
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726 apply(rule impI) |
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727 apply(erule conjE) |
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728 thm POSIX_ALT_I2 |
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729 apply(frule POSIX_ALT1a) |
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730 apply(drule POSIX_ALT1b) |
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731 apply(rule POSIX_ALT_I2) |
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732 apply auto[1] |
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733 apply(subst v4) |
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734 apply(auto)[2] |
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735 apply(rotate_tac 1) |
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736 apply(drule_tac x="v2" in meta_spec) |
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737 apply(simp) |
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738 apply(subst (asm) (4) POSIX_def) |
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739 apply(subst (asm) v4) |
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740 apply(auto)[2] |
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741 (* stuck in the ALT case *) |
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