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 value "L(CHAR c)" |
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46 value "L(SEQ(CHAR c)(CHAR b))" |
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47 |
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48 |
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49 section {* Values *} |
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50 |
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51 datatype val = |
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52 Void |
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53 | Char char |
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54 | Seq val val |
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55 | Right val |
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56 | Left val |
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57 |
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58 section {* Relation between values and regular expressions *} |
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59 |
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60 inductive Prf :: "val \<Rightarrow> rexp \<Rightarrow> bool" ("\<turnstile> _ : _" [100, 100] 100) |
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61 where |
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62 "\<lbrakk>\<turnstile> v1 : r1; \<turnstile> v2 : r2\<rbrakk> \<Longrightarrow> \<turnstile> Seq v1 v2 : SEQ r1 r2" |
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63 | "\<turnstile> v1 : r1 \<Longrightarrow> \<turnstile> Left v1 : ALT r1 r2" |
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64 | "\<turnstile> v2 : r2 \<Longrightarrow> \<turnstile> Right v2 : ALT r1 r2" |
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65 | "\<turnstile> Void : EMPTY" |
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66 | "\<turnstile> Char c : CHAR c" |
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67 |
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68 section {* The string behind a value *} |
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69 |
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70 fun flat :: "val \<Rightarrow> string" |
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71 where |
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72 "flat(Void) = []" |
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73 | "flat(Char c) = [c]" |
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74 | "flat(Left v) = flat(v)" |
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75 | "flat(Right v) = flat(v)" |
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76 | "flat(Seq v1 v2) = flat(v1) @ flat(v2)" |
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77 |
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78 value "flat(Seq(Char c)(Char b))" |
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79 value "flat(Right(Void))" |
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80 |
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81 fun flats :: "val \<Rightarrow> string list" |
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82 where |
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83 "flats(Void) = [[]]" |
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84 | "flats(Char c) = [[c]]" |
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85 | "flats(Left v) = flats(v)" |
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86 | "flats(Right v) = flats(v)" |
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87 | "flats(Seq v1 v2) = (flats v1) @ (flats v2)" |
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88 |
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89 value "flats(Seq(Char c)(Char b))" |
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90 |
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91 lemma Prf_flat_L: |
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92 assumes "\<turnstile> v : r" shows "flat v \<in> L r" |
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93 using assms |
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94 apply(induct) |
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95 apply(auto simp add: Sequ_def) |
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96 done |
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97 |
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98 lemma L_flat_Prf: |
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99 "L(r) = {flat v | v. \<turnstile> v : r}" |
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100 apply(induct r) |
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101 apply(auto dest: Prf_flat_L simp add: Sequ_def) |
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102 apply (metis Prf.intros(4) flat.simps(1)) |
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103 apply (metis Prf.intros(5) flat.simps(2)) |
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104 apply (metis Prf.intros(1) flat.simps(5)) |
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105 apply (metis Prf.intros(2) flat.simps(3)) |
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106 apply (metis Prf.intros(3) flat.simps(4)) |
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107 apply(erule Prf.cases) |
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108 apply(auto) |
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109 done |
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110 |
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111 definition definition prefix :: :: "string \<Rightarrow> string \<Rightarrow> bool" ("_ \<sqsubset> _" [100, 100] 100) |
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112 where |
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113 "s1 \<sqsubset> s2 \<equiv> \<exists>s3. s1 @ s3 = s2" |
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114 |
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115 section {* Ordering of values *} |
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116 |
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117 inductive ValOrd :: "val \<Rightarrow> rexp \<Rightarrow> val \<Rightarrow> bool" ("_ \<succ>_ _" [100, 100, 100] 100) |
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118 where |
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119 "\<lbrakk>v1 = v1'; v2 \<succ>r2 v2'\<rbrakk> \<Longrightarrow> (Seq v1 v2) \<succ>(SEQ r1 r2) (Seq v1' v2')" |
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120 | "v1 \<succ>r1 v1' \<Longrightarrow> (Seq v1 v2) \<succ>(SEQ r1 r2) (Seq v1' v2')" |
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121 | "length (flat v1) \<ge> length (flat v2) \<Longrightarrow> (Left v1) \<succ>(ALT r1 r2) (Right v2)" |
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122 | "length (flat v2) > length (flat v1) \<Longrightarrow> (Right v2) \<succ>(ALT r1 r2) (Left v1)" |
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123 | "v2 \<succ>r2 v2' \<Longrightarrow> (Right v2) \<succ>(ALT r1 r2) (Right v2')" |
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124 | "v1 \<succ>r1 v1' \<Longrightarrow> (Left v1) \<succ>(ALT r1 r2) (Left v1')" |
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125 | "Void \<succ>EMPTY Void" |
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126 | "(Char c) \<succ>(CHAR c) (Char c)" |
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127 |
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128 section {* The ordering is reflexive *} |
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129 |
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130 lemma ValOrd_refl: |
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131 assumes "\<turnstile> v : r" |
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132 shows "v \<succ>r v" |
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133 using assms |
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134 apply(induct) |
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135 apply(auto intro: ValOrd.intros) |
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136 done |
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137 |
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138 lemma ValOrd_flats: |
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139 assumes "v1 \<succ>r v2" |
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140 shows "hd (flats v2) = hd (flats v1)" |
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141 using assms |
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142 apply(induct) |
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143 apply(auto) |
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144 oops |
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145 |
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146 |
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147 section {* Posix definition *} |
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148 |
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149 definition POSIX :: "val \<Rightarrow> rexp \<Rightarrow> bool" |
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150 where |
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151 "POSIX v r \<equiv> (\<forall>v'. (\<turnstile> v' : r \<and> flat v = flat v') \<longrightarrow> v \<succ>r v')" |
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152 |
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153 (* |
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154 an alternative definition: might cause problems |
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155 with theorem mkeps_POSIX |
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156 *) |
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157 |
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158 definition POSIX2 :: "val \<Rightarrow> rexp \<Rightarrow> bool" |
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159 where |
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160 "POSIX2 v r \<equiv> \<turnstile> v : r \<and> (\<forall>v'. \<turnstile> v' : r \<longrightarrow> v \<succ>r v')" |
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161 |
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162 definition POSIX3 :: "val \<Rightarrow> rexp \<Rightarrow> bool" |
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163 where |
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164 "POSIX3 v r \<equiv> \<turnstile> v : r \<and> (\<forall>v'. (\<turnstile> v' : r \<and> length (flat v') \<le> length(flat v)) \<longrightarrow> v \<succ>r v')" |
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165 |
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166 |
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167 lemma POSIX_SEQ: |
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168 assumes "POSIX (Seq v1 v2) (SEQ r1 r2)" "\<turnstile> v1 : r1" "\<turnstile> v2 : r2" |
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169 shows "POSIX v1 r1 \<and> POSIX v2 r2" |
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170 using assms |
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171 unfolding POSIX_def |
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172 apply(auto) |
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173 apply(drule_tac x="Seq v' v2" in spec) |
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174 apply(simp) |
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175 apply(erule impE) |
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176 apply(rule Prf.intros) |
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177 apply(simp) |
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178 apply(simp) |
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179 apply(erule ValOrd.cases) |
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180 apply(simp_all) |
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181 apply(clarify) |
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182 defer |
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183 apply(drule_tac x="Seq v1 v'" in spec) |
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184 apply(simp) |
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185 apply(erule impE) |
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186 apply(rule Prf.intros) |
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187 apply(simp) |
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188 apply(simp) |
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189 apply(erule ValOrd.cases) |
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190 apply(simp_all) |
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191 apply(clarify) |
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192 oops (*not true*) |
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193 |
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194 lemma POSIX_SEQ_I: |
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195 assumes "POSIX v1 r1" "POSIX v2 r2" |
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196 shows "POSIX (Seq v1 v2) (SEQ r1 r2)" |
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197 using assms |
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198 unfolding POSIX_def |
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199 apply(auto) |
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200 apply(rotate_tac 2) |
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201 apply(erule Prf.cases) |
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202 apply(simp_all)[5] |
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203 apply(auto)[1] |
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204 apply(rule ValOrd.intros) |
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205 oops (* maybe also not true *) |
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206 |
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207 lemma POSIX3_SEQ_I: |
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208 assumes "POSIX3 v1 r1" "POSIX3 v2 r2" |
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209 shows "POSIX3 (Seq v1 v2) (SEQ r1 r2)" |
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210 using assms |
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211 unfolding POSIX3_def |
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212 apply(auto) |
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213 apply (metis Prf.intros(1)) |
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214 apply(rotate_tac 4) |
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215 apply(erule Prf.cases) |
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216 apply(simp_all)[5] |
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217 apply(auto)[1] |
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218 apply(case_tac "v1 = v1a") |
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219 apply(auto) |
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220 apply (metis ValOrd.intros(1)) |
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221 apply (rule ValOrd.intros(2)) |
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222 oops |
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223 |
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224 lemma POSIX_ALT2: |
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225 assumes "POSIX (Left v1) (ALT r1 r2)" |
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226 shows "POSIX v1 r1" |
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227 using assms |
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228 unfolding POSIX_def |
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229 apply(auto) |
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230 apply(drule_tac x="Left v'" in spec) |
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231 apply(simp) |
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232 apply(drule mp) |
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233 apply(rule Prf.intros) |
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234 apply(auto) |
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235 apply(erule ValOrd.cases) |
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236 apply(simp_all) |
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237 done |
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238 |
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239 lemma POSIX2_ALT: |
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240 assumes "POSIX2 (Left v1) (ALT r1 r2)" |
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241 shows "POSIX2 v1 r1" |
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242 using assms |
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243 unfolding POSIX2_def |
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244 apply(auto) |
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245 oops |
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246 |
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247 lemma POSIX_ALT: |
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248 assumes "POSIX (Left v1) (ALT r1 r2)" |
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249 shows "POSIX v1 r1" |
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250 using assms |
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251 unfolding POSIX_def |
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252 apply(auto) |
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253 apply(drule_tac x="Left v'" in spec) |
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254 apply(simp) |
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255 apply(drule mp) |
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256 apply(rule Prf.intros) |
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257 apply(auto) |
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258 apply(erule ValOrd.cases) |
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259 apply(simp_all) |
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260 done |
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261 |
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262 lemma POSIX2_ALT: |
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263 assumes "POSIX2 (Left v1) (ALT r1 r2)" |
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264 shows "POSIX2 v1 r1" |
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265 using assms |
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266 apply(simp add: POSIX2_def) |
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267 apply(auto) |
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268 apply(erule Prf.cases) |
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269 apply(simp_all)[5] |
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270 apply(drule_tac x="Left v'" in spec) |
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271 apply(drule mp) |
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272 apply(rule Prf.intros) |
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273 apply(auto) |
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274 apply(erule ValOrd.cases) |
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275 apply(simp_all) |
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276 done |
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277 |
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278 |
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279 lemma POSIX_ALT1a: |
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280 assumes "POSIX (Right v2) (ALT r1 r2)" |
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281 shows "POSIX v2 r2" |
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282 using assms |
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283 unfolding POSIX_def |
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284 apply(auto) |
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285 apply(drule_tac x="Right v'" in spec) |
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286 apply(simp) |
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287 apply(drule mp) |
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288 apply(rule Prf.intros) |
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289 apply(auto) |
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290 apply(erule ValOrd.cases) |
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291 apply(simp_all) |
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292 done |
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293 |
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294 lemma POSIX2_ALT1a: |
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295 assumes "POSIX2 (Right v2) (ALT r1 r2)" |
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296 shows "POSIX2 v2 r2" |
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297 using assms |
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298 unfolding POSIX2_def |
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299 apply(auto) |
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300 apply(erule Prf.cases) |
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301 apply(simp_all)[5] |
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302 apply(drule_tac x="Right v'" in spec) |
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303 apply(drule mp) |
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304 apply(rule Prf.intros) |
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305 apply(auto) |
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306 apply(erule ValOrd.cases) |
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307 apply(simp_all) |
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308 done |
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309 |
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310 |
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311 lemma POSIX_ALT1b: |
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312 assumes "POSIX (Right v2) (ALT r1 r2)" |
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313 shows "(\<forall>v'. (\<turnstile> v' : r2 \<and> flat v' = flat v2) \<longrightarrow> v2 \<succ>r2 v')" |
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314 using assms |
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315 apply(drule_tac POSIX_ALT1a) |
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316 unfolding POSIX_def |
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317 apply(auto) |
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318 done |
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319 |
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320 lemma POSIX_ALT_I1: |
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321 assumes "POSIX v1 r1" |
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322 shows "POSIX (Left v1) (ALT r1 r2)" |
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323 using assms |
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324 unfolding POSIX_def |
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325 apply(auto) |
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326 apply(rotate_tac 3) |
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327 apply(erule Prf.cases) |
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328 apply(simp_all)[5] |
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329 apply(auto) |
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330 apply(rule ValOrd.intros) |
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331 apply(auto) |
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332 apply(rule ValOrd.intros) |
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333 by simp |
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334 |
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335 lemma POSIX2_ALT_I1: |
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336 assumes "POSIX2 v1 r1" |
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337 shows "POSIX2 (Left v1) (ALT r1 r2)" |
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338 using assms |
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339 unfolding POSIX2_def |
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340 apply(auto) |
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341 apply(rule Prf.intros) |
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342 apply(simp) |
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343 apply(rotate_tac 2) |
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344 apply(erule Prf.cases) |
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345 apply(simp_all)[5] |
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346 apply(auto) |
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347 apply(rule ValOrd.intros) |
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348 apply(auto) |
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349 apply(rule ValOrd.intros) |
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350 oops |
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351 |
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352 lemma POSIX_ALT_I2: |
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353 assumes "POSIX v2 r2" "\<forall>v'. \<turnstile> v' : r1 \<longrightarrow> length (flat v2) > length (flat v')" |
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354 shows "POSIX (Right v2) (ALT r1 r2)" |
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355 using assms |
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356 unfolding POSIX_def |
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357 apply(auto) |
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358 apply(rotate_tac 3) |
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359 apply(erule Prf.cases) |
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360 apply(simp_all)[5] |
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361 apply(auto) |
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362 apply(rule ValOrd.intros) |
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363 apply metis |
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364 done |
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365 |
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366 |
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367 |
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368 |
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369 |
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370 section {* The Matcher *} |
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371 |
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372 fun |
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373 nullable :: "rexp \<Rightarrow> bool" |
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374 where |
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375 "nullable (NULL) = False" |
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376 | "nullable (EMPTY) = True" |
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377 | "nullable (CHAR c) = False" |
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378 | "nullable (ALT r1 r2) = (nullable r1 \<or> nullable r2)" |
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379 | "nullable (SEQ r1 r2) = (nullable r1 \<and> nullable r2)" |
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380 |
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381 lemma nullable_correctness: |
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382 shows "nullable r \<longleftrightarrow> [] \<in> (L r)" |
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383 apply (induct r) |
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384 apply(auto simp add: Sequ_def) |
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385 done |
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386 |
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387 fun mkeps :: "rexp \<Rightarrow> val" |
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388 where |
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389 "mkeps(EMPTY) = Void" |
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390 | "mkeps(SEQ r1 r2) = Seq (mkeps r1) (mkeps r2)" |
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391 | "mkeps(ALT r1 r2) = (if nullable(r1) then Left (mkeps r1) else Right (mkeps r2))" |
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392 |
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393 lemma mkeps_nullable: |
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394 assumes "nullable(r)" shows "\<turnstile> mkeps r : r" |
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395 using assms |
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396 apply(induct rule: nullable.induct) |
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397 apply(auto intro: Prf.intros) |
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398 done |
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399 |
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400 lemma mkeps_flat: |
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401 assumes "nullable(r)" shows "flat (mkeps r) = []" |
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402 using assms |
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403 apply(induct rule: nullable.induct) |
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404 apply(auto) |
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405 done |
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406 |
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407 text {* |
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408 The value mkeps returns is always the correct POSIX |
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409 value. |
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410 *} |
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411 |
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412 lemma mkeps_POSIX2: |
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413 assumes "nullable r" |
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414 shows "POSIX2 (mkeps r) r" |
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415 using assms |
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416 apply(induct r) |
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417 apply(auto)[1] |
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418 apply(simp add: POSIX2_def) |
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419 oops |
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420 |
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421 lemma mkeps_POSIX3: |
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422 assumes "nullable r" |
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423 shows "POSIX3 (mkeps r) r" |
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424 using assms |
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425 apply(induct r) |
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426 apply(auto)[1] |
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427 apply(simp add: POSIX3_def) |
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428 apply(auto)[1] |
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429 apply (metis Prf.intros(4)) |
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430 apply(erule Prf.cases) |
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431 apply(simp_all)[5] |
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432 apply (metis ValOrd.intros) |
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433 apply(simp add: POSIX3_def) |
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434 apply(auto)[1] |
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435 apply(simp add: POSIX3_def) |
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436 apply(auto)[1] |
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437 apply (metis mkeps.simps(2) mkeps_nullable nullable.simps(5)) |
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438 apply(rotate_tac 6) |
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439 apply(erule Prf.cases) |
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440 apply(simp_all)[5] |
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441 apply (metis ValOrd.intros(2) add_leE gen_length_code(1) gen_length_def mkeps_flat) |
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442 apply(auto) |
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443 apply(simp add: POSIX3_def) |
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444 apply(auto) |
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445 apply (metis Prf.intros(2)) |
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446 apply(rotate_tac 4) |
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447 apply(erule Prf.cases) |
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448 apply(simp_all)[5] |
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449 apply (metis ValOrd.intros(6)) |
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450 apply(auto)[1] |
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451 apply (metis ValOrd.intros(3)) |
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452 apply(simp add: POSIX3_def) |
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453 apply(auto) |
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454 apply (metis Prf.intros(2)) |
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455 apply(rotate_tac 6) |
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456 apply(erule Prf.cases) |
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457 apply(simp_all)[5] |
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458 apply (metis ValOrd.intros(6)) |
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459 apply (metis ValOrd.intros(3)) |
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460 apply(simp add: POSIX3_def) |
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461 apply(auto) |
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462 apply (metis Prf.intros(3)) |
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463 apply(rotate_tac 5) |
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464 apply(erule Prf.cases) |
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465 apply(simp_all)[5] |
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466 apply (metis Prf_flat_L drop_0 drop_all list.size(3) mkeps_flat nullable_correctness) |
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467 by (metis ValOrd.intros(5)) |
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468 |
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469 |
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470 lemma mkeps_POSIX: |
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471 assumes "nullable r" |
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472 shows "POSIX (mkeps r) r" |
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473 using assms |
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474 apply(induct r) |
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475 apply(auto)[1] |
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476 apply(simp add: POSIX_def) |
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477 apply(auto)[1] |
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478 apply(erule Prf.cases) |
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479 apply(simp_all)[5] |
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480 apply (metis ValOrd.intros) |
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481 apply(simp add: POSIX_def) |
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482 apply(auto)[1] |
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483 apply(simp add: POSIX_def) |
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484 apply(auto)[1] |
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485 apply(erule Prf.cases) |
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486 apply(simp_all)[5] |
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487 apply(auto) |
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488 apply (simp add: ValOrd.intros(2) mkeps_flat) |
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489 apply(simp add: POSIX_def) |
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490 apply(auto)[1] |
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491 apply(erule Prf.cases) |
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492 apply(simp_all)[5] |
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493 apply(auto) |
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494 apply (simp add: ValOrd.intros(6)) |
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495 apply (simp add: ValOrd.intros(3)) |
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496 apply(simp add: POSIX_def) |
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497 apply(auto)[1] |
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498 apply(erule Prf.cases) |
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499 apply(simp_all)[5] |
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500 apply(auto) |
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501 apply (simp add: ValOrd.intros(6)) |
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502 apply (simp add: ValOrd.intros(3)) |
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503 apply(simp add: POSIX_def) |
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504 apply(auto)[1] |
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505 apply(erule Prf.cases) |
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506 apply(simp_all)[5] |
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507 apply(auto) |
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508 apply (metis Prf_flat_L mkeps_flat nullable_correctness) |
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509 by (simp add: ValOrd.intros(5)) |
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510 |
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511 |
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512 lemma mkeps_POSIX2: |
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513 assumes "nullable r" |
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514 shows "POSIX2 (mkeps r) r" |
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515 using assms |
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516 apply(induct r) |
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517 apply(simp) |
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518 apply(simp) |
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519 apply(simp add: POSIX2_def) |
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520 apply(rule conjI) |
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521 apply(rule Prf.intros) |
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522 apply(auto)[1] |
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523 apply(erule Prf.cases) |
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524 apply(simp_all)[5] |
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525 apply(rule ValOrd.intros) |
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526 apply(simp) |
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527 apply(simp) |
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528 apply(simp add: POSIX2_def) |
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529 apply(rule conjI) |
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530 apply(rule Prf.intros) |
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531 apply(simp add: mkeps_nullable) |
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532 apply(simp add: mkeps_nullable) |
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533 apply(auto)[1] |
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534 apply(rotate_tac 6) |
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535 apply(erule Prf.cases) |
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536 apply(simp_all)[5] |
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537 apply(rule ValOrd.intros(2)) |
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538 apply(simp) |
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539 apply(simp only: nullable.simps) |
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540 apply(erule disjE) |
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541 apply(simp) |
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542 thm POSIX2_ALT1a |
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543 apply(rule POSIX2_ALT) |
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544 apply(simp add: POSIX2_def) |
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545 apply(rule conjI) |
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546 apply(rule Prf.intros) |
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547 apply(simp add: mkeps_nullable) |
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548 oops |
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549 |
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550 |
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551 section {* Derivatives *} |
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552 |
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553 fun |
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554 der :: "char \<Rightarrow> rexp \<Rightarrow> rexp" |
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555 where |
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556 "der c (NULL) = NULL" |
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557 | "der c (EMPTY) = NULL" |
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558 | "der c (CHAR c') = (if c = c' then EMPTY else NULL)" |
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559 | "der c (ALT r1 r2) = ALT (der c r1) (der c r2)" |
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560 | "der c (SEQ r1 r2) = |
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561 (if nullable r1 |
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562 then ALT (SEQ (der c r1) r2) (der c r2) |
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563 else SEQ (der c r1) r2)" |
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564 |
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565 fun |
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566 ders :: "string \<Rightarrow> rexp \<Rightarrow> rexp" |
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567 where |
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568 "ders [] r = r" |
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569 | "ders (c # s) r = ders s (der c r)" |
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570 |
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571 section {* Injection function *} |
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572 |
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573 fun injval :: "rexp \<Rightarrow> char \<Rightarrow> val \<Rightarrow> val" |
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574 where |
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575 "injval (CHAR d) c Void = Char d" |
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576 | "injval (ALT r1 r2) c (Left v1) = Left(injval r1 c v1)" |
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577 | "injval (ALT r1 r2) c (Right v2) = Right(injval r2 c v2)" |
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578 | "injval (SEQ r1 r2) c (Seq v1 v2) = Seq (injval r1 c v1) v2" |
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579 | "injval (SEQ r1 r2) c (Left (Seq v1 v2)) = Seq (injval r1 c v1) v2" |
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580 | "injval (SEQ r1 r2) c (Right v2) = Seq (mkeps r1) (injval r2 c v2)" |
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581 |
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582 section {* Projection function *} |
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583 |
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584 fun projval :: "rexp \<Rightarrow> char \<Rightarrow> val \<Rightarrow> val" |
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585 where |
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586 "projval (CHAR d) c _ = Void" |
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587 | "projval (ALT r1 r2) c (Left v1) = Left(projval r1 c v1)" |
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588 | "projval (ALT r1 r2) c (Right v2) = Right(projval r2 c v2)" |
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589 | "projval (SEQ r1 r2) c (Seq v1 v2) = |
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590 (if flat v1 = [] then Right(projval r2 c v2) |
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591 else if nullable r1 then Left (Seq (projval r1 c v1) v2) |
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592 else Seq (projval r1 c v1) v2)" |
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593 |
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594 text {* |
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595 Injection value is related to r |
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596 *} |
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597 |
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598 lemma v3: |
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599 assumes "\<turnstile> v : der c r" shows "\<turnstile> (injval r c v) : r" |
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600 using assms |
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601 apply(induct arbitrary: v rule: der.induct) |
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602 apply(simp) |
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603 apply(erule Prf.cases) |
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604 apply(simp_all)[5] |
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605 apply(simp) |
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606 apply(erule Prf.cases) |
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607 apply(simp_all)[5] |
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608 apply(case_tac "c = c'") |
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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 (metis Prf.intros(5)) |
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613 apply(simp) |
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614 apply(erule Prf.cases) |
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615 apply(simp_all)[5] |
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616 apply(simp) |
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617 apply(erule Prf.cases) |
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618 apply(simp_all)[5] |
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619 apply (metis Prf.intros(2)) |
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620 apply (metis Prf.intros(3)) |
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621 apply(simp) |
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622 apply(case_tac "nullable r1") |
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623 apply(simp) |
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624 apply(erule Prf.cases) |
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625 apply(simp_all)[5] |
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626 apply(auto)[1] |
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627 apply(erule Prf.cases) |
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628 apply(simp_all)[5] |
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629 apply(auto)[1] |
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630 apply (metis Prf.intros(1)) |
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631 apply(auto)[1] |
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632 apply (metis Prf.intros(1) mkeps_nullable) |
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633 apply(simp) |
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634 apply(erule Prf.cases) |
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635 apply(simp_all)[5] |
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636 apply(auto)[1] |
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637 apply(rule Prf.intros) |
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638 apply(auto)[2] |
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639 done |
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640 |
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641 text {* |
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642 The string behin the injection value is an added c |
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643 *} |
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644 |
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645 lemma v4: |
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646 assumes "\<turnstile> v : der c r" shows "flat (injval r c v) = c # (flat v)" |
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647 using assms |
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648 apply(induct arbitrary: v rule: der.induct) |
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649 apply(simp) |
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650 apply(erule Prf.cases) |
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651 apply(simp_all)[5] |
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652 apply(simp) |
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653 apply(erule Prf.cases) |
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654 apply(simp_all)[5] |
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655 apply(simp) |
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656 apply(case_tac "c = c'") |
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657 apply(simp) |
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658 apply(auto)[1] |
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659 apply(erule Prf.cases) |
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660 apply(simp_all)[5] |
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661 apply(simp) |
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662 apply(erule Prf.cases) |
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663 apply(simp_all)[5] |
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664 apply(simp) |
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665 apply(erule Prf.cases) |
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666 apply(simp_all)[5] |
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667 apply(simp) |
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668 apply(case_tac "nullable r1") |
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669 apply(simp) |
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670 apply(erule Prf.cases) |
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671 apply(simp_all)[5] |
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672 apply(auto)[1] |
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673 apply(erule Prf.cases) |
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674 apply(simp_all)[5] |
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675 apply(auto)[1] |
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676 apply (metis mkeps_flat) |
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677 apply(simp) |
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678 apply(erule Prf.cases) |
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679 apply(simp_all)[5] |
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680 done |
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681 |
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682 text {* |
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683 Injection followed by projection is the identity. |
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684 *} |
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685 |
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686 lemma proj_inj_id: |
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687 assumes "\<turnstile> v : der c r" |
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688 shows "projval r c (injval r c v) = v" |
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689 using assms |
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690 apply(induct r arbitrary: c v rule: rexp.induct) |
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691 apply(simp) |
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692 apply(erule Prf.cases) |
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693 apply(simp_all)[5] |
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694 apply(simp) |
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695 apply(erule Prf.cases) |
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696 apply(simp_all)[5] |
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697 apply(simp) |
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698 apply(case_tac "c = char") |
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699 apply(simp) |
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700 apply(erule Prf.cases) |
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701 apply(simp_all)[5] |
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702 apply(simp) |
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703 apply(erule Prf.cases) |
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704 apply(simp_all)[5] |
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705 defer |
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706 apply(simp) |
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707 apply(erule Prf.cases) |
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708 apply(simp_all)[5] |
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709 apply(simp) |
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710 apply(case_tac "nullable rexp1") |
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711 apply(simp) |
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712 apply(erule Prf.cases) |
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713 apply(simp_all)[5] |
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714 apply(auto)[1] |
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715 apply(erule Prf.cases) |
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716 apply(simp_all)[5] |
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717 apply(auto)[1] |
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718 apply (metis list.distinct(1) v4) |
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719 apply(auto)[1] |
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720 apply (metis mkeps_flat) |
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721 apply(auto) |
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722 apply(erule Prf.cases) |
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723 apply(simp_all)[5] |
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724 apply(auto)[1] |
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725 apply(simp add: v4) |
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726 done |
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727 |
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728 lemma "L r \<noteq> {} \<Longrightarrow> \<exists>v. POSIX3 v r" |
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729 apply(induct r) |
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730 apply(simp) |
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731 apply(simp add: POSIX3_def) |
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732 apply(rule_tac x="Void" in exI) |
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733 apply(auto)[1] |
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734 apply (metis Prf.intros(4)) |
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735 apply (metis POSIX3_def flat.simps(1) mkeps.simps(1) mkeps_POSIX3 nullable.simps(2) order_refl) |
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736 apply(simp add: POSIX3_def) |
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737 apply(rule_tac x="Char char" in exI) |
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738 apply(auto)[1] |
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739 apply (metis Prf.intros(5)) |
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740 apply(erule Prf.cases) |
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741 apply(simp_all)[5] |
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742 apply (metis ValOrd.intros(8)) |
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743 apply(simp add: Sequ_def) |
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744 apply(auto)[1] |
|
745 apply(drule meta_mp) |
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746 apply(auto)[2] |
|
747 apply(drule meta_mp) |
|
748 apply(auto)[2] |
|
749 apply(rule_tac x="Seq v va" in exI) |
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750 apply(simp (no_asm) add: POSIX3_def) |
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751 apply(auto)[1] |
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752 apply (metis POSIX3_def Prf.intros(1)) |
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753 apply(erule Prf.cases) |
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754 apply(simp_all)[5] |
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755 apply(clarify) |
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756 apply(case_tac "v \<succ>r1a v1") |
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757 apply(rule ValOrd.intros(2)) |
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758 apply(simp) |
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759 apply(case_tac "v = v1") |
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760 apply(rule ValOrd.intros(1)) |
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761 apply(simp) |
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762 apply(simp) |
|
763 apply (metis ValOrd_refl) |
|
764 apply(simp add: POSIX3_def) |
|
765 |
|
766 |
|
767 lemma "\<exists>v. POSIX v r" |
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768 apply(induct r) |
|
769 apply(rule exI) |
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770 apply(simp add: POSIX_def) |
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771 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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772 apply(rule_tac x = "Void" in exI) |
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773 apply(simp add: POSIX_def) |
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774 apply (metis POSIX_def flat.simps(1) mkeps.simps(1) mkeps_POSIX nullable.simps(2)) |
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775 apply(rule_tac x = "Char char" in exI) |
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776 apply(simp add: POSIX_def) |
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777 apply(auto) [1] |
|
778 apply(erule Prf.cases) |
|
779 apply(simp_all) [5] |
|
780 apply (metis ValOrd.intros(8)) |
|
781 defer |
|
782 apply(auto) |
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783 apply (metis POSIX_ALT_I1) |
|
784 (* maybe it is too early to instantiate this existential quantifier *) |
|
785 (* potentially this is the wrong POSIX value *) |
|
786 apply(case_tac "r1 = NULL") |
|
787 apply(simp add: POSIX_def) |
|
788 apply(auto)[1] |
|
789 apply (metis L.simps(1) L.simps(4) Prf_flat_L mkeps_flat nullable.simps(1) nullable.simps(2) nullable_correctness seq_null(2)) |
|
790 apply(case_tac "r1 = EMPTY") |
|
791 apply(rule_tac x = "Seq Void va" in exI ) |
|
792 apply(simp (no_asm) add: POSIX_def) |
|
793 apply(auto) |
|
794 apply(erule Prf.cases) |
|
795 apply(simp_all) |
|
796 apply(auto)[1] |
|
797 apply(erule Prf.cases) |
|
798 apply(simp_all) |
|
799 apply(rule ValOrd.intros(2)) |
|
800 apply(rule ValOrd.intros) |
|
801 apply(case_tac "\<exists>c. r1 = CHAR c") |
|
802 apply(auto) |
|
803 apply(rule_tac x = "Seq (Char c) va" in exI ) |
|
804 apply(simp (no_asm) add: POSIX_def) |
|
805 apply(auto) |
|
806 apply(erule Prf.cases) |
|
807 apply(simp_all) |
|
808 apply(auto)[1] |
|
809 apply(erule Prf.cases) |
|
810 apply(simp_all) |
|
811 apply(auto)[1] |
|
812 apply(rule ValOrd.intros(2)) |
|
813 apply(rule ValOrd.intros) |
|
814 apply(case_tac "\<exists>r1a r1b. r1 = ALT r1a r1b") |
|
815 apply(auto) |
|
816 oops (* not sure if this can be proved by induction *) |
|
817 |
|
818 text {* |
|
819 |
|
820 HERE: Crucial lemma that does not go through in the sequence case. |
|
821 |
|
822 *} |
|
823 lemma v5: |
|
824 assumes "\<turnstile> v : der c r" "POSIX v (der c r)" |
|
825 shows "POSIX (injval r c v) r" |
|
826 using assms |
|
827 apply(induct arbitrary: v rule: der.induct) |
|
828 apply(simp) |
|
829 apply(erule Prf.cases) |
|
830 apply(simp_all)[5] |
|
831 apply(simp) |
|
832 apply(erule Prf.cases) |
|
833 apply(simp_all)[5] |
|
834 apply(simp) |
|
835 apply(case_tac "c = c'") |
|
836 apply(auto simp add: POSIX_def)[1] |
|
837 apply(erule Prf.cases) |
|
838 apply(simp_all)[5] |
|
839 apply(erule Prf.cases) |
|
840 apply(simp_all)[5] |
|
841 using ValOrd.simps apply blast |
|
842 apply(auto) |
|
843 apply(erule Prf.cases) |
|
844 apply(simp_all)[5] |
|
845 (* base cases done *) |
|
846 (* ALT case *) |
|
847 apply(erule Prf.cases) |
|
848 apply(simp_all)[5] |
|
849 using POSIX_ALT POSIX_ALT_I1 apply blast |
|
850 apply(clarify) |
|
851 apply(subgoal_tac "POSIX v2 (der c r2)") |
|
852 prefer 2 |
|
853 apply(auto simp add: POSIX_def)[1] |
|
854 apply (metis POSIX_ALT1a POSIX_def flat.simps(4)) |
|
855 apply(frule POSIX_ALT1a) |
|
856 apply(drule POSIX_ALT1b) |
|
857 apply(rule POSIX_ALT_I2) |
|
858 apply(rotate_tac 1) |
|
859 apply(drule_tac x="v2" in meta_spec) |
|
860 apply(simp) |
|
861 apply(subgoal_tac "\<turnstile> Right (injval r2 c v2) : (ALT r1 r2)") |
|
862 prefer 2 |
|
863 apply (metis Prf.intros(3) v3) |
|
864 |
|
865 apply auto[1] |
|
866 apply(subst v4) |
|
867 apply(auto)[2] |
|
868 apply(subst (asm) (4) POSIX_def) |
|
869 apply(subst (asm) v4) |
|
870 apply(drule_tac x="v2" in meta_spec) |
|
871 apply(simp) |
|
872 |
|
873 apply(auto)[2] |
|
874 |
|
875 thm POSIX_ALT_I2 |
|
876 apply(rule POSIX_ALT_I2) |
|
877 |
|
878 apply(rule ccontr) |
|
879 apply(auto simp add: POSIX_def)[1] |
|
880 |
|
881 apply(rule allI) |
|
882 apply(rule impI) |
|
883 apply(erule conjE) |
|
884 thm POSIX_ALT_I2 |
|
885 apply(frule POSIX_ALT1a) |
|
886 apply(drule POSIX_ALT1b) |
|
887 apply(rule POSIX_ALT_I2) |
|
888 apply auto[1] |
|
889 apply(subst v4) |
|
890 apply(auto)[2] |
|
891 apply(rotate_tac 1) |
|
892 apply(drule_tac x="v2" in meta_spec) |
|
893 apply(simp) |
|
894 apply(subst (asm) (4) POSIX_def) |
|
895 apply(subst (asm) v4) |
|
896 apply(auto)[2] |
|
897 (* stuck in the ALT case *) |
|