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1 |
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2 theory Re |
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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 lemma POSIX_SEQ: |
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131 assumes "POSIX (Seq v1 v2) (SEQ r1 r2)" "\<turnstile> v1 : r1" "\<turnstile> v2 : r2" |
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132 shows "POSIX v1 r1 \<and> POSIX v2 r2" |
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133 using assms |
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134 unfolding POSIX_def |
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135 apply(auto) |
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136 apply(drule_tac x="Seq v' v2" in spec) |
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137 apply(simp) |
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138 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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139 apply(drule_tac x="Seq v1 v'" in spec) |
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140 apply(simp) |
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141 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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142 *) |
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143 |
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144 (* |
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145 lemma POSIX_SEQ_I: |
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146 assumes "POSIX v1 r1" "POSIX v2 r2" |
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147 shows "POSIX (Seq v1 v2) (SEQ r1 r2)" |
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148 using assms |
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149 unfolding POSIX_def |
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150 apply(auto) |
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151 apply(rotate_tac 2) |
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152 apply(erule Prf.cases) |
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153 apply(simp_all)[5] |
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154 apply(auto)[1] |
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155 apply(rule ValOrd.intros) |
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156 |
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157 apply(auto) |
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158 done |
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159 *) |
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160 |
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161 lemma POSIX_ALT: |
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162 assumes "POSIX (Left v1) (ALT r1 r2)" |
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163 shows "POSIX v1 r1" |
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164 using assms |
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165 unfolding POSIX_def |
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166 apply(auto) |
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167 apply(drule_tac x="Left v'" in spec) |
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168 apply(simp) |
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169 apply(drule mp) |
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170 apply(rule Prf.intros) |
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171 apply(auto) |
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172 apply(erule ValOrd.cases) |
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173 apply(simp_all) |
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174 done |
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175 |
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176 lemma POSIX_ALT1a: |
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177 assumes "POSIX (Right v2) (ALT r1 r2)" |
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178 shows "POSIX v2 r2" |
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179 using assms |
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180 unfolding POSIX_def |
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181 apply(auto) |
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182 apply(drule_tac x="Right v'" in spec) |
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183 apply(simp) |
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184 apply(drule mp) |
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185 apply(rule Prf.intros) |
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186 apply(auto) |
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187 apply(erule ValOrd.cases) |
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188 apply(simp_all) |
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189 done |
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190 |
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191 |
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192 lemma POSIX_ALT1b: |
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193 assumes "POSIX (Right v2) (ALT r1 r2)" |
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194 shows "(\<forall>v'. (\<turnstile> v' : r2 \<and> flat v' = flat v2) \<longrightarrow> v2 \<succ>r2 v')" |
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195 using assms |
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196 apply(drule_tac POSIX_ALT1a) |
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197 unfolding POSIX_def |
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198 apply(auto) |
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199 done |
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200 |
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201 lemma POSIX_ALT_I1: |
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202 assumes "POSIX v1 r1" |
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203 shows "POSIX (Left v1) (ALT r1 r2)" |
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204 using assms |
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205 unfolding POSIX_def |
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206 apply(auto) |
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207 apply(rotate_tac 3) |
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208 apply(erule Prf.cases) |
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209 apply(simp_all)[5] |
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210 apply(auto) |
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211 apply(rule ValOrd.intros) |
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212 apply(auto) |
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213 apply(rule ValOrd.intros) |
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214 by simp |
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215 |
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216 lemma POSIX_ALT_I2: |
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217 assumes "POSIX v2 r2" "\<forall>v'. \<turnstile> v' : r1 \<longrightarrow> length (flat v2) > length (flat v')" |
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218 shows "POSIX (Right v2) (ALT r1 r2)" |
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219 using assms |
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220 unfolding POSIX_def |
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221 apply(auto) |
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222 apply(rotate_tac 3) |
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223 apply(erule Prf.cases) |
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224 apply(simp_all)[5] |
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225 apply(auto) |
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226 apply(rule ValOrd.intros) |
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227 apply metis |
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228 done |
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229 |
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230 |
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231 section {* The ordering is reflexive *} |
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232 |
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233 lemma ValOrd_refl: |
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234 assumes "\<turnstile> v : r" |
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235 shows "v \<succ>r v" |
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236 using assms |
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237 apply(induct) |
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238 apply(auto intro: ValOrd.intros) |
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239 done |
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240 |
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241 |
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242 section {* The Matcher *} |
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243 |
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244 fun |
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245 nullable :: "rexp \<Rightarrow> bool" |
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246 where |
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247 "nullable (NULL) = False" |
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248 | "nullable (EMPTY) = True" |
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249 | "nullable (CHAR c) = False" |
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250 | "nullable (ALT r1 r2) = (nullable r1 \<or> nullable r2)" |
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251 | "nullable (SEQ r1 r2) = (nullable r1 \<and> nullable r2)" |
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252 |
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253 lemma nullable_correctness: |
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254 shows "nullable r \<longleftrightarrow> [] \<in> (L r)" |
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255 apply (induct r) |
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256 apply(auto simp add: Sequ_def) |
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257 done |
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258 |
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259 fun mkeps :: "rexp \<Rightarrow> val" |
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260 where |
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261 "mkeps(EMPTY) = Void" |
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262 | "mkeps(SEQ r1 r2) = Seq (mkeps r1) (mkeps r2)" |
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263 | "mkeps(ALT r1 r2) = (if nullable(r1) then Left (mkeps r1) else Right (mkeps r2))" |
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264 |
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265 lemma mkeps_nullable: |
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266 assumes "nullable(r)" shows "\<turnstile> mkeps r : r" |
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267 using assms |
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268 apply(induct rule: nullable.induct) |
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269 apply(auto intro: Prf.intros) |
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270 done |
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271 |
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272 lemma mkeps_flat: |
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273 assumes "nullable(r)" shows "flat (mkeps r) = []" |
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274 using assms |
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275 apply(induct rule: nullable.induct) |
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276 apply(auto) |
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277 done |
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278 |
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279 text {* |
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280 The value mkeps returns is always the correct POSIX |
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281 value. |
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282 *} |
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283 |
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284 lemma mkeps_POSIX: |
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285 assumes "nullable r" |
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286 shows "POSIX (mkeps r) r" |
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287 using assms |
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288 apply(induct r) |
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289 apply(auto)[1] |
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290 apply(simp add: POSIX_def) |
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291 apply(auto)[1] |
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292 apply(erule Prf.cases) |
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293 apply(simp_all)[5] |
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294 apply (metis ValOrd.intros) |
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295 apply(simp add: POSIX_def) |
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296 apply(auto)[1] |
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297 apply(simp add: POSIX_def) |
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298 apply(auto)[1] |
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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 (simp add: ValOrd.intros(2) mkeps_flat) |
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303 apply(simp add: POSIX_def) |
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304 apply(auto)[1] |
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305 apply(erule Prf.cases) |
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306 apply(simp_all)[5] |
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307 apply(auto) |
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308 apply (simp add: ValOrd.intros(6)) |
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309 apply (simp add: ValOrd.intros(3)) |
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310 apply(simp add: POSIX_def) |
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311 apply(auto)[1] |
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312 apply(erule Prf.cases) |
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313 apply(simp_all)[5] |
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314 apply(auto) |
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315 apply (simp add: ValOrd.intros(6)) |
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316 apply (simp add: ValOrd.intros(3)) |
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317 apply(simp add: POSIX_def) |
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318 apply(auto)[1] |
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319 apply(erule Prf.cases) |
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320 apply(simp_all)[5] |
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321 apply(auto) |
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322 apply (metis Prf_flat_L mkeps_flat nullable_correctness) |
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323 by (simp add: ValOrd.intros(5)) |
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324 |
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325 |
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326 section {* Derivatives *} |
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327 |
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328 fun |
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329 der :: "char \<Rightarrow> rexp \<Rightarrow> rexp" |
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330 where |
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331 "der c (NULL) = NULL" |
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332 | "der c (EMPTY) = NULL" |
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333 | "der c (CHAR c') = (if c = c' then EMPTY else NULL)" |
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334 | "der c (ALT r1 r2) = ALT (der c r1) (der c r2)" |
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335 | "der c (SEQ r1 r2) = |
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336 (if nullable r1 |
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337 then ALT (SEQ (der c r1) r2) (der c r2) |
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338 else SEQ (der c r1) r2)" |
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339 |
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340 fun |
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341 ders :: "string \<Rightarrow> rexp \<Rightarrow> rexp" |
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342 where |
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343 "ders [] r = r" |
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344 | "ders (c # s) r = ders s (der c r)" |
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345 |
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346 section {* Injection function *} |
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347 |
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348 fun injval :: "rexp \<Rightarrow> char \<Rightarrow> val \<Rightarrow> val" |
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349 where |
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350 "injval (CHAR d) c Void = Char d" |
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351 | "injval (ALT r1 r2) c (Left v1) = Left(injval r1 c v1)" |
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352 | "injval (ALT r1 r2) c (Right v2) = Right(injval r2 c v2)" |
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353 | "injval (SEQ r1 r2) c (Seq v1 v2) = Seq (injval r1 c v1) v2" |
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354 | "injval (SEQ r1 r2) c (Left (Seq v1 v2)) = Seq (injval r1 c v1) v2" |
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355 | "injval (SEQ r1 r2) c (Right v2) = Seq (mkeps r1) (injval r2 c v2)" |
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356 |
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357 section {* Projection function *} |
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358 |
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359 fun projval :: "rexp \<Rightarrow> char \<Rightarrow> val \<Rightarrow> val" |
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360 where |
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361 "projval (CHAR d) c _ = Void" |
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362 | "projval (ALT r1 r2) c (Left v1) = Left(projval r1 c v1)" |
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363 | "projval (ALT r1 r2) c (Right v2) = Right(projval r2 c v2)" |
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364 | "projval (SEQ r1 r2) c (Seq v1 v2) = |
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365 (if flat v1 = [] then Right(projval r2 c v2) |
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366 else if nullable r1 then Left (Seq (projval r1 c v1) v2) |
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367 else Seq (projval r1 c v1) v2)" |
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368 |
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369 text {* |
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370 Injection value is related to r |
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371 *} |
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372 |
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373 lemma v3: |
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374 assumes "\<turnstile> v : der c r" shows "\<turnstile> (injval r c v) : r" |
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375 using assms |
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376 apply(induct arbitrary: v rule: der.induct) |
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377 apply(simp) |
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378 apply(erule Prf.cases) |
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379 apply(simp_all)[5] |
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380 apply(erule Prf.cases) |
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381 apply(simp_all)[5] |
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382 apply(case_tac "c = c'") |
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383 apply(simp) |
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384 apply(erule Prf.cases) |
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385 apply(simp_all)[5] |
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386 apply (metis Prf.intros(5)) |
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387 apply(erule Prf.cases) |
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388 apply(simp_all)[5] |
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389 apply(erule Prf.cases) |
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390 apply(simp_all)[5] |
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391 apply (metis Prf.intros(2)) |
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392 apply (metis Prf.intros(3)) |
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393 apply(simp) |
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394 apply(case_tac "nullable r1") |
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395 apply(simp) |
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396 apply(erule Prf.cases) |
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397 apply(simp_all)[5] |
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398 apply(auto)[1] |
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399 apply(erule Prf.cases) |
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400 apply(simp_all)[5] |
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401 apply(auto)[1] |
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402 apply (metis Prf.intros(1)) |
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403 apply(auto)[1] |
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404 apply (metis Prf.intros(1) mkeps_nullable) |
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405 apply(simp) |
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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)[1] |
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409 apply(rule Prf.intros) |
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410 apply(auto)[2] |
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411 done |
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412 |
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413 text {* |
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414 The string behin the injection value is an added c |
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415 *} |
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416 |
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417 lemma v4: |
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418 assumes "\<turnstile> v : der c r" shows "flat (injval r c v) = c # (flat v)" |
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419 using assms |
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420 apply(induct arbitrary: v rule: der.induct) |
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421 apply(simp) |
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422 apply(erule Prf.cases) |
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423 apply(simp_all)[5] |
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424 apply(simp) |
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425 apply(erule Prf.cases) |
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426 apply(simp_all)[5] |
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427 apply(simp) |
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428 apply(case_tac "c = c'") |
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429 apply(simp) |
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430 apply(auto)[1] |
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431 apply(erule Prf.cases) |
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432 apply(simp_all)[5] |
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433 apply(simp) |
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434 apply(erule Prf.cases) |
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435 apply(simp_all)[5] |
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436 apply(simp) |
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437 apply(erule Prf.cases) |
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438 apply(simp_all)[5] |
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439 apply(simp) |
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440 apply(case_tac "nullable r1") |
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441 apply(simp) |
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442 apply(erule Prf.cases) |
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443 apply(simp_all)[5] |
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444 apply(auto)[1] |
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445 apply(erule Prf.cases) |
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446 apply(simp_all)[5] |
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447 apply(auto)[1] |
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448 apply (metis mkeps_flat) |
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449 apply(simp) |
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450 apply(erule Prf.cases) |
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451 apply(simp_all)[5] |
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452 done |
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453 |
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454 text {* |
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455 Injection followed by projection is the identity. |
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456 *} |
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457 |
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458 lemma proj_inj_id: |
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459 assumes "\<turnstile> v : der c r" |
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460 shows "projval r c (injval r c v) = v" |
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461 using assms |
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462 apply(induct r arbitrary: c v rule: rexp.induct) |
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463 apply(simp) |
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464 apply(erule Prf.cases) |
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465 apply(simp_all)[5] |
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466 apply(simp) |
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467 apply(erule Prf.cases) |
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468 apply(simp_all)[5] |
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469 apply(simp) |
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470 apply(case_tac "c = char") |
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471 apply(simp) |
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472 apply(erule Prf.cases) |
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473 apply(simp_all)[5] |
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474 apply(simp) |
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475 apply(erule Prf.cases) |
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476 apply(simp_all)[5] |
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477 defer |
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478 apply(simp) |
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479 apply(erule Prf.cases) |
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480 apply(simp_all)[5] |
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481 apply(simp) |
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482 apply(case_tac "nullable rexp1") |
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483 apply(simp) |
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484 apply(erule Prf.cases) |
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485 apply(simp_all)[5] |
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486 apply(auto)[1] |
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487 apply(erule Prf.cases) |
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488 apply(simp_all)[5] |
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489 apply(auto)[1] |
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490 apply (metis list.distinct(1) v4) |
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491 apply(auto)[1] |
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492 apply (metis mkeps_flat) |
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493 apply(auto) |
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494 apply(erule Prf.cases) |
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495 apply(simp_all)[5] |
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496 apply(auto)[1] |
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497 apply(simp add: v4) |
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498 done |
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499 |
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500 lemma "\<exists>v. POSIX v r" |
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501 apply(induct r) |
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502 apply(rule exI) |
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503 apply(simp add: POSIX_def) |
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504 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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505 apply(rule_tac x = "Void" in exI) |
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506 apply(simp add: POSIX_def) |
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507 apply (metis POSIX_def flat.simps(1) mkeps.simps(1) mkeps_POSIX nullable.simps(2)) |
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508 apply(rule_tac x = "Char char" in exI) |
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509 apply(simp add: POSIX_def) |
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510 apply(auto) [1] |
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511 apply(erule Prf.cases) |
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512 apply(simp_all) [5] |
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513 apply (metis ValOrd.intros(8)) |
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514 defer |
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515 apply(auto) |
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516 apply (metis POSIX_ALT_I1) |
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517 apply(rule_tac x = "Seq v va" in exI ) |
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518 apply(simp (no_asm) add: POSIX_def) |
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519 apply(auto) |
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520 apply(erule Prf.cases) |
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521 apply(simp_all) |
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522 apply(case_tac "v \<succ>r1a v1") |
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523 apply (metis ValOrd.intros(2)) |
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524 apply(simp add: POSIX_def) |
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525 apply(case_tac "flat v = flat v1") |
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526 apply(auto)[1] |
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527 apply(simp only: append_eq_append_conv2) |
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528 apply(auto) |
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529 thm append_eq_append_conv2 |
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530 |
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531 text {* |
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532 |
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533 HERE: Crucial lemma that does not go through in the sequence case. |
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534 |
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535 *} |
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536 lemma v5: |
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537 assumes "\<turnstile> v : der c r" "POSIX v (der c r)" |
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538 shows "POSIX (injval r c v) r" |
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539 using assms |
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540 apply(induct arbitrary: v rule: der.induct) |
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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(simp) |
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545 apply(erule Prf.cases) |
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546 apply(simp_all)[5] |
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547 apply(simp) |
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548 apply(case_tac "c = c'") |
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549 apply(auto simp add: POSIX_def)[1] |
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550 apply(erule Prf.cases) |
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551 apply(simp_all)[5] |
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552 apply(erule Prf.cases) |
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553 apply(simp_all)[5] |
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554 using ValOrd.simps apply blast |
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555 apply(auto) |
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556 apply(erule Prf.cases) |
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557 apply(simp_all)[5] |
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558 (* base cases done *) |
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559 (* ALT case *) |
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560 apply(erule Prf.cases) |
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561 apply(simp_all)[5] |
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562 using POSIX_ALT POSIX_ALT_I1 apply blast |
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563 apply(clarify) |
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564 apply(subgoal_tac "POSIX v2 (der c r2)") |
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565 prefer 2 |
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566 apply(auto simp add: POSIX_def)[1] |
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567 apply (metis POSIX_ALT1a POSIX_def flat.simps(4)) |
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568 apply(rotate_tac 1) |
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569 apply(drule_tac x="v2" in meta_spec) |
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570 apply(simp) |
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571 apply(subgoal_tac "\<turnstile> Right (injval r2 c v2) : (ALT r1 r2)") |
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572 prefer 2 |
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573 apply (metis Prf.intros(3) v3) |
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574 apply(rule ccontr) |
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575 apply(auto simp add: POSIX_def)[1] |
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576 |
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577 apply(rule allI) |
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578 apply(rule impI) |
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579 apply(erule conjE) |
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580 thm POSIX_ALT_I2 |
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581 apply(frule POSIX_ALT1a) |
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582 apply(drule POSIX_ALT1b) |
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583 apply(rule POSIX_ALT_I2) |
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584 apply auto[1] |
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585 apply(subst v4) |
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586 apply(auto)[2] |
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587 apply(rotate_tac 1) |
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588 apply(drule_tac x="v2" in meta_spec) |
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589 apply(simp) |
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590 apply(subst (asm) (4) POSIX_def) |
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591 apply(subst (asm) v4) |
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592 apply(auto)[2] |
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593 (* stuck in the ALT case *) |