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1 theory Myhill_2 |
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2 imports Myhill_1 Prefix_subtract |
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3 "~~/src/HOL/Library/List_Prefix" |
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4 begin |
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5 |
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6 section {* Direction @{text "regular language \<Rightarrow>finite partition"} *} |
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7 |
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8 definition |
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9 str_eq :: "string \<Rightarrow> lang \<Rightarrow> string \<Rightarrow> bool" ("_ \<approx>_ _") |
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10 where |
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11 "x \<approx>A y \<equiv> (x, y) \<in> (\<approx>A)" |
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12 |
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13 definition |
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14 tag_eq_rel :: "(string \<Rightarrow> 'b) \<Rightarrow> (string \<times> string) set" ("=_=") |
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15 where |
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16 "=tag= \<equiv> {(x, y) | x y. tag x = tag y}" |
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17 |
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18 lemma finite_eq_tag_rel: |
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19 assumes rng_fnt: "finite (range tag)" |
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20 shows "finite (UNIV // =tag=)" |
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21 proof - |
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22 let "?f" = "\<lambda>X. tag ` X" and ?A = "(UNIV // =tag=)" |
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23 have "finite (?f ` ?A)" |
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24 proof - |
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25 have "range ?f \<subseteq> (Pow (range tag))" unfolding Pow_def by auto |
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26 moreover |
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27 have "finite (Pow (range tag))" using rng_fnt by simp |
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28 ultimately |
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29 have "finite (range ?f)" unfolding image_def by (blast intro: finite_subset) |
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30 moreover |
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31 have "?f ` ?A \<subseteq> range ?f" by auto |
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32 ultimately show "finite (?f ` ?A)" by (rule rev_finite_subset) |
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33 qed |
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34 moreover |
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35 have "inj_on ?f ?A" |
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36 proof - |
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37 { fix X Y |
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38 assume X_in: "X \<in> ?A" |
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39 and Y_in: "Y \<in> ?A" |
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40 and tag_eq: "?f X = ?f Y" |
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41 then obtain x y |
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42 where "x \<in> X" "y \<in> Y" "tag x = tag y" |
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43 unfolding quotient_def Image_def image_def tag_eq_rel_def |
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44 by (simp) (blast) |
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45 with X_in Y_in |
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46 have "X = Y" |
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47 unfolding quotient_def tag_eq_rel_def by auto |
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48 } |
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49 then show "inj_on ?f ?A" unfolding inj_on_def by auto |
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50 qed |
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51 ultimately show "finite (UNIV // =tag=)" by (rule finite_imageD) |
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52 qed |
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53 |
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54 lemma refined_partition_finite: |
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55 assumes fnt: "finite (UNIV // R1)" |
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56 and refined: "R1 \<subseteq> R2" |
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57 and eq1: "equiv UNIV R1" and eq2: "equiv UNIV R2" |
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58 shows "finite (UNIV // R2)" |
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59 proof - |
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60 let ?f = "\<lambda>X. {R1 `` {x} | x. x \<in> X}" |
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61 and ?A = "UNIV // R2" and ?B = "UNIV // R1" |
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62 have "?f ` ?A \<subseteq> Pow ?B" |
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63 unfolding image_def Pow_def quotient_def by auto |
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64 moreover |
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65 have "finite (Pow ?B)" using fnt by simp |
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66 ultimately |
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67 have "finite (?f ` ?A)" by (rule finite_subset) |
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68 moreover |
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69 have "inj_on ?f ?A" |
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70 proof - |
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71 { fix X Y |
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72 assume X_in: "X \<in> ?A" and Y_in: "Y \<in> ?A" and eq_f: "?f X = ?f Y" |
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73 from quotientE [OF X_in] |
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74 obtain x where "X = R2 `` {x}" by blast |
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75 with equiv_class_self[OF eq2] have x_in: "x \<in> X" by simp |
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76 then have "R1 ``{x} \<in> ?f X" by auto |
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77 with eq_f have "R1 `` {x} \<in> ?f Y" by simp |
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78 then obtain y |
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79 where y_in: "y \<in> Y" and eq_r1_xy: "R1 `` {x} = R1 `` {y}" by auto |
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80 with eq_equiv_class[OF _ eq1] |
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81 have "(x, y) \<in> R1" by blast |
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82 with refined have "(x, y) \<in> R2" by auto |
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83 with quotient_eqI [OF eq2 X_in Y_in x_in y_in] |
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84 have "X = Y" . |
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85 } |
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86 then show "inj_on ?f ?A" unfolding inj_on_def by blast |
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87 qed |
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88 ultimately show "finite (UNIV // R2)" by (rule finite_imageD) |
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89 qed |
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90 |
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91 lemma tag_finite_imageD: |
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92 assumes rng_fnt: "finite (range tag)" |
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93 and same_tag_eqvt: "\<And>m n. tag m = tag n \<Longrightarrow> m \<approx>A n" |
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94 shows "finite (UNIV // \<approx>A)" |
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95 proof (rule_tac refined_partition_finite [of "=tag="]) |
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96 show "finite (UNIV // =tag=)" by (rule finite_eq_tag_rel[OF rng_fnt]) |
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97 next |
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98 from same_tag_eqvt |
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99 show "=tag= \<subseteq> \<approx>A" unfolding tag_eq_rel_def str_eq_def |
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100 by auto |
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101 next |
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102 show "equiv UNIV =tag=" |
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103 unfolding equiv_def tag_eq_rel_def refl_on_def sym_def trans_def |
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104 by auto |
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105 next |
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106 show "equiv UNIV (\<approx>A)" |
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107 unfolding equiv_def str_eq_rel_def sym_def refl_on_def trans_def |
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108 by blast |
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109 qed |
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110 |
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111 |
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112 subsection {* The proof *} |
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113 |
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114 subsubsection {* The base case for @{const "NULL"} *} |
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115 |
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116 lemma quot_null_eq: |
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117 shows "UNIV // \<approx>{} = {UNIV}" |
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118 unfolding quotient_def Image_def str_eq_rel_def by auto |
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119 |
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120 lemma quot_null_finiteI [intro]: |
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121 shows "finite (UNIV // \<approx>{})" |
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122 unfolding quot_null_eq by simp |
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123 |
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124 |
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125 subsubsection {* The base case for @{const "EMPTY"} *} |
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126 |
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127 lemma quot_empty_subset: |
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128 shows "UNIV // \<approx>{[]} \<subseteq> {{[]}, UNIV - {[]}}" |
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129 proof |
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130 fix x |
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131 assume "x \<in> UNIV // \<approx>{[]}" |
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132 then obtain y where h: "x = {z. (y, z) \<in> \<approx>{[]}}" |
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133 unfolding quotient_def Image_def by blast |
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134 show "x \<in> {{[]}, UNIV - {[]}}" |
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135 proof (cases "y = []") |
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136 case True with h |
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137 have "x = {[]}" by (auto simp: str_eq_rel_def) |
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138 thus ?thesis by simp |
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139 next |
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140 case False with h |
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141 have "x = UNIV - {[]}" by (auto simp: str_eq_rel_def) |
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142 thus ?thesis by simp |
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143 qed |
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144 qed |
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145 |
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146 lemma quot_empty_finiteI [intro]: |
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147 shows "finite (UNIV // \<approx>{[]})" |
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148 by (rule finite_subset[OF quot_empty_subset]) (simp) |
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149 |
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150 |
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151 subsubsection {* The base case for @{const "CHAR"} *} |
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152 |
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153 lemma quot_char_subset: |
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154 "UNIV // (\<approx>{[c]}) \<subseteq> {{[]},{[c]}, UNIV - {[], [c]}}" |
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155 proof |
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156 fix x |
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157 assume "x \<in> UNIV // \<approx>{[c]}" |
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158 then obtain y where h: "x = {z. (y, z) \<in> \<approx>{[c]}}" |
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159 unfolding quotient_def Image_def by blast |
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160 show "x \<in> {{[]},{[c]}, UNIV - {[], [c]}}" |
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161 proof - |
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162 { assume "y = []" hence "x = {[]}" using h |
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163 by (auto simp:str_eq_rel_def) } |
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164 moreover |
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165 { assume "y = [c]" hence "x = {[c]}" using h |
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166 by (auto dest!:spec[where x = "[]"] simp:str_eq_rel_def) } |
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167 moreover |
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168 { assume "y \<noteq> []" and "y \<noteq> [c]" |
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169 hence "\<forall> z. (y @ z) \<noteq> [c]" by (case_tac y, auto) |
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170 moreover have "\<And> p. (p \<noteq> [] \<and> p \<noteq> [c]) = (\<forall> q. p @ q \<noteq> [c])" |
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171 by (case_tac p, auto) |
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172 ultimately have "x = UNIV - {[],[c]}" using h |
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173 by (auto simp add:str_eq_rel_def) |
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174 } |
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175 ultimately show ?thesis by blast |
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176 qed |
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177 qed |
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178 |
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179 lemma quot_char_finiteI [intro]: |
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180 shows "finite (UNIV // \<approx>{[c]})" |
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181 by (rule finite_subset[OF quot_char_subset]) (simp) |
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182 |
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183 |
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184 subsubsection {* The inductive case for @{const ALT} *} |
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185 |
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186 definition |
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187 tag_str_ALT :: "lang \<Rightarrow> lang \<Rightarrow> string \<Rightarrow> (lang \<times> lang)" |
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188 where |
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189 "tag_str_ALT A B \<equiv> (\<lambda>x. (\<approx>A `` {x}, \<approx>B `` {x}))" |
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190 |
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191 lemma quot_union_finiteI [intro]: |
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192 fixes L1 L2::"lang" |
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193 assumes finite1: "finite (UNIV // \<approx>A)" |
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194 and finite2: "finite (UNIV // \<approx>B)" |
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195 shows "finite (UNIV // \<approx>(A \<union> B))" |
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196 proof (rule_tac tag = "tag_str_ALT A B" in tag_finite_imageD) |
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197 have "finite ((UNIV // \<approx>A) \<times> (UNIV // \<approx>B))" |
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198 using finite1 finite2 by auto |
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199 then show "finite (range (tag_str_ALT A B))" |
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200 unfolding tag_str_ALT_def quotient_def |
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201 by (rule rev_finite_subset) (auto) |
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202 next |
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203 show "\<And>x y. tag_str_ALT A B x = tag_str_ALT A B y \<Longrightarrow> x \<approx>(A \<union> B) y" |
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204 unfolding tag_str_ALT_def |
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205 unfolding str_eq_def |
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206 unfolding str_eq_rel_def |
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207 by auto |
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208 qed |
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209 |
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210 |
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211 subsubsection {* The inductive case for @{text "SEQ"}*} |
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212 |
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213 definition |
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214 tag_str_SEQ :: "lang \<Rightarrow> lang \<Rightarrow> string \<Rightarrow> (lang \<times> lang set)" |
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215 where |
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216 "tag_str_SEQ L1 L2 \<equiv> |
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217 (\<lambda>x. (\<approx>L1 `` {x}, {(\<approx>L2 `` {x - xa}) | xa. xa \<le> x \<and> xa \<in> L1}))" |
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218 |
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219 lemma Seq_in_cases: |
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220 assumes "x @ z \<in> A ;; B" |
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221 shows "(\<exists> x' \<le> x. x' \<in> A \<and> (x - x') @ z \<in> B) \<or> |
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222 (\<exists> z' \<le> z. (x @ z') \<in> A \<and> (z - z') \<in> B)" |
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223 using assms |
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224 unfolding Seq_def prefix_def |
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225 by (auto simp add: append_eq_append_conv2) |
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226 |
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227 lemma tag_str_SEQ_injI: |
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228 assumes eq_tag: "tag_str_SEQ A B x = tag_str_SEQ A B y" |
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229 shows "x \<approx>(A ;; B) y" |
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230 proof - |
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231 { fix x y z |
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232 assume xz_in_seq: "x @ z \<in> A ;; B" |
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233 and tag_xy: "tag_str_SEQ A B x = tag_str_SEQ A B y" |
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234 have"y @ z \<in> A ;; B" |
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235 proof - |
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236 { (* first case with x' in A and (x - x') @ z in B *) |
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237 fix x' |
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238 assume h1: "x' \<le> x" and h2: "x' \<in> A" and h3: "(x - x') @ z \<in> B" |
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239 obtain y' |
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240 where "y' \<le> y" |
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241 and "y' \<in> A" |
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242 and "(y - y') @ z \<in> B" |
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243 proof - |
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244 have "{\<approx>B `` {x - x'} |x'. x' \<le> x \<and> x' \<in> A} = |
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245 {\<approx>B `` {y - y'} |y'. y' \<le> y \<and> y' \<in> A}" (is "?Left = ?Right") |
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246 using tag_xy unfolding tag_str_SEQ_def by simp |
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247 moreover |
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248 have "\<approx>B `` {x - x'} \<in> ?Left" using h1 h2 by auto |
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249 ultimately |
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250 have "\<approx>B `` {x - x'} \<in> ?Right" by simp |
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251 then obtain y' |
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252 where eq_xy': "\<approx>B `` {x - x'} = \<approx>B `` {y - y'}" |
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253 and pref_y': "y' \<le> y" and y'_in: "y' \<in> A" |
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254 by simp blast |
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255 |
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256 have "(x - x') \<approx>B (y - y')" using eq_xy' |
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257 unfolding Image_def str_eq_rel_def str_eq_def by auto |
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258 with h3 have "(y - y') @ z \<in> B" |
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259 unfolding str_eq_rel_def str_eq_def by simp |
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260 with pref_y' y'_in |
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261 show ?thesis using that by blast |
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262 qed |
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263 then have "y @ z \<in> A ;; B" by (erule_tac prefixE) (auto simp: Seq_def) |
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264 } |
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265 moreover |
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266 { (* second case with x @ z' in A and z - z' in B *) |
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267 fix z' |
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268 assume h1: "z' \<le> z" and h2: "(x @ z') \<in> A" and h3: "z - z' \<in> B" |
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269 have "\<approx>A `` {x} = \<approx>A `` {y}" |
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270 using tag_xy unfolding tag_str_SEQ_def by simp |
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271 with h2 have "y @ z' \<in> A" |
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272 unfolding Image_def str_eq_rel_def str_eq_def by auto |
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273 with h1 h3 have "y @ z \<in> A ;; B" |
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274 unfolding prefix_def Seq_def |
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275 by (auto) (metis append_assoc) |
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276 } |
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277 ultimately show "y @ z \<in> A ;; B" |
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278 using Seq_in_cases [OF xz_in_seq] by blast |
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279 qed |
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280 } |
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281 from this [OF _ eq_tag] and this [OF _ eq_tag [THEN sym]] |
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282 show "x \<approx>(A ;; B) y" unfolding str_eq_def str_eq_rel_def by blast |
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283 qed |
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284 |
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285 lemma quot_seq_finiteI [intro]: |
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286 fixes L1 L2::"lang" |
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287 assumes fin1: "finite (UNIV // \<approx>L1)" |
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288 and fin2: "finite (UNIV // \<approx>L2)" |
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289 shows "finite (UNIV // \<approx>(L1 ;; L2))" |
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290 proof (rule_tac tag = "tag_str_SEQ L1 L2" in tag_finite_imageD) |
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291 show "\<And>x y. tag_str_SEQ L1 L2 x = tag_str_SEQ L1 L2 y \<Longrightarrow> x \<approx>(L1 ;; L2) y" |
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292 by (rule tag_str_SEQ_injI) |
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293 next |
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294 have *: "finite ((UNIV // \<approx>L1) \<times> (Pow (UNIV // \<approx>L2)))" |
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295 using fin1 fin2 by auto |
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296 show "finite (range (tag_str_SEQ L1 L2))" |
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297 unfolding tag_str_SEQ_def |
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298 apply(rule finite_subset[OF _ *]) |
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299 unfolding quotient_def |
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300 by auto |
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301 qed |
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302 |
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303 |
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304 subsubsection {* The inductive case for @{const "STAR"} *} |
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305 |
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306 definition |
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307 tag_str_STAR :: "lang \<Rightarrow> string \<Rightarrow> lang set" |
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308 where |
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309 "tag_str_STAR L1 \<equiv> (\<lambda>x. {\<approx>L1 `` {x - xa} | xa. xa < x \<and> xa \<in> L1\<star>})" |
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310 |
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311 text {* A technical lemma. *} |
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312 lemma finite_set_has_max: "\<lbrakk>finite A; A \<noteq> {}\<rbrakk> \<Longrightarrow> |
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313 (\<exists> max \<in> A. \<forall> a \<in> A. f a <= (f max :: nat))" |
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314 proof (induct rule:finite.induct) |
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315 case emptyI thus ?case by simp |
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316 next |
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317 case (insertI A a) |
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318 show ?case |
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319 proof (cases "A = {}") |
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320 case True thus ?thesis by (rule_tac x = a in bexI, auto) |
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321 next |
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322 case False |
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323 with insertI.hyps and False |
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324 obtain max |
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325 where h1: "max \<in> A" |
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326 and h2: "\<forall>a\<in>A. f a \<le> f max" by blast |
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327 show ?thesis |
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328 proof (cases "f a \<le> f max") |
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329 assume "f a \<le> f max" |
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330 with h1 h2 show ?thesis by (rule_tac x = max in bexI, auto) |
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331 next |
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332 assume "\<not> (f a \<le> f max)" |
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333 thus ?thesis using h2 by (rule_tac x = a in bexI, auto) |
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334 qed |
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335 qed |
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336 qed |
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337 |
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338 |
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339 text {* The following is a technical lemma, which helps to show the range finiteness of tag function. *} |
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340 |
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341 lemma finite_strict_prefix_set: "finite {xa. xa < (x::string)}" |
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342 apply (induct x rule:rev_induct, simp) |
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343 apply (subgoal_tac "{xa. xa < xs @ [x]} = {xa. xa < xs} \<union> {xs}") |
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344 by (auto simp:strict_prefix_def) |
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345 |
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346 |
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347 lemma tag_str_STAR_injI: |
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348 assumes eq_tag: "tag_str_STAR L\<^isub>1 v = tag_str_STAR L\<^isub>1 w" |
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349 shows "v \<approx>(L\<^isub>1\<star>) w" |
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350 proof- |
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351 { fix x y z |
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352 assume xz_in_star: "x @ z \<in> L\<^isub>1\<star>" |
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353 and tag_xy: "tag_str_STAR L\<^isub>1 x = tag_str_STAR L\<^isub>1 y" |
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354 have "y @ z \<in> L\<^isub>1\<star>" |
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355 proof(cases "x = []") |
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356 case True |
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357 with tag_xy have "y = []" |
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358 by (auto simp add: tag_str_STAR_def strict_prefix_def) |
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359 thus ?thesis using xz_in_star True by simp |
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360 next |
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361 case False |
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362 let ?S = "{xa. xa < x \<and> xa \<in> L\<^isub>1\<star> \<and> (x - xa) @ z \<in> L\<^isub>1\<star>}" |
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363 have "finite ?S" |
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364 by (rule_tac B = "{xa. xa < x}" in finite_subset, |
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365 auto simp:finite_strict_prefix_set) |
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366 moreover have "?S \<noteq> {}" using False xz_in_star |
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367 by (simp, rule_tac x = "[]" in exI, auto simp:strict_prefix_def) |
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368 ultimately have "\<exists> xa_max \<in> ?S. \<forall> xa \<in> ?S. length xa \<le> length xa_max" |
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369 using finite_set_has_max by blast |
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370 then obtain xa_max |
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371 where h1: "xa_max < x" |
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372 and h2: "xa_max \<in> L\<^isub>1\<star>" |
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373 and h3: "(x - xa_max) @ z \<in> L\<^isub>1\<star>" |
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374 and h4:"\<forall> xa < x. xa \<in> L\<^isub>1\<star> \<and> (x - xa) @ z \<in> L\<^isub>1\<star> |
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375 \<longrightarrow> length xa \<le> length xa_max" |
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376 by blast |
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377 obtain ya |
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378 where h5: "ya < y" and h6: "ya \<in> L\<^isub>1\<star>" |
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379 and eq_xya: "(x - xa_max) \<approx>L\<^isub>1 (y - ya)" |
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380 proof- |
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381 from tag_xy have "{\<approx>L\<^isub>1 `` {x - xa} |xa. xa < x \<and> xa \<in> L\<^isub>1\<star>} = |
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382 {\<approx>L\<^isub>1 `` {y - xa} |xa. xa < y \<and> xa \<in> L\<^isub>1\<star>}" (is "?left = ?right") |
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383 by (auto simp:tag_str_STAR_def) |
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384 moreover have "\<approx>L\<^isub>1 `` {x - xa_max} \<in> ?left" using h1 h2 by auto |
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385 ultimately have "\<approx>L\<^isub>1 `` {x - xa_max} \<in> ?right" by simp |
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386 thus ?thesis using that |
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387 apply (simp add:Image_def str_eq_rel_def str_eq_def) by blast |
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388 qed |
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389 have "(y - ya) @ z \<in> L\<^isub>1\<star>" |
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390 proof- |
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391 obtain za zb where eq_zab: "z = za @ zb" |
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392 and l_za: "(y - ya)@za \<in> L\<^isub>1" and ls_zb: "zb \<in> L\<^isub>1\<star>" |
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393 proof - |
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394 from h1 have "(x - xa_max) @ z \<noteq> []" |
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395 by (auto simp:strict_prefix_def elim:prefixE) |
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396 from star_decom [OF h3 this] |
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397 obtain a b where a_in: "a \<in> L\<^isub>1" |
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398 and a_neq: "a \<noteq> []" and b_in: "b \<in> L\<^isub>1\<star>" |
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399 and ab_max: "(x - xa_max) @ z = a @ b" by blast |
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400 let ?za = "a - (x - xa_max)" and ?zb = "b" |
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401 have pfx: "(x - xa_max) \<le> a" (is "?P1") |
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402 and eq_z: "z = ?za @ ?zb" (is "?P2") |
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403 proof - |
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404 have "((x - xa_max) \<le> a \<and> (a - (x - xa_max)) @ b = z) \<or> |
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405 (a < (x - xa_max) \<and> ((x - xa_max) - a) @ z = b)" |
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406 using append_eq_dest[OF ab_max] by (auto simp:strict_prefix_def) |
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407 moreover { |
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408 assume np: "a < (x - xa_max)" |
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409 and b_eqs: "((x - xa_max) - a) @ z = b" |
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410 have "False" |
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411 proof - |
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412 let ?xa_max' = "xa_max @ a" |
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413 have "?xa_max' < x" |
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414 using np h1 by (clarsimp simp:strict_prefix_def diff_prefix) |
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415 moreover have "?xa_max' \<in> L\<^isub>1\<star>" |
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416 using a_in h2 by (simp add:star_intro3) |
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417 moreover have "(x - ?xa_max') @ z \<in> L\<^isub>1\<star>" |
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418 using b_eqs b_in np h1 by (simp add:diff_diff_append) |
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419 moreover have "\<not> (length ?xa_max' \<le> length xa_max)" |
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420 using a_neq by simp |
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421 ultimately show ?thesis using h4 by blast |
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422 qed } |
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423 ultimately show ?P1 and ?P2 by auto |
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424 qed |
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425 hence "(x - xa_max)@?za \<in> L\<^isub>1" using a_in by (auto elim:prefixE) |
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426 with eq_xya have "(y - ya) @ ?za \<in> L\<^isub>1" |
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427 by (auto simp:str_eq_def str_eq_rel_def) |
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428 with eq_z and b_in |
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429 show ?thesis using that by blast |
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430 qed |
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431 have "((y - ya) @ za) @ zb \<in> L\<^isub>1\<star>" using l_za ls_zb by blast |
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432 with eq_zab show ?thesis by simp |
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433 qed |
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434 with h5 h6 show ?thesis |
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435 by (drule_tac star_intro1, auto simp:strict_prefix_def elim:prefixE) |
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436 qed |
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437 } |
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438 from this [OF _ eq_tag] and this [OF _ eq_tag [THEN sym]] |
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439 show ?thesis unfolding str_eq_def str_eq_rel_def by blast |
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440 qed |
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441 |
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442 lemma quot_star_finiteI [intro]: |
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443 assumes finite1: "finite (UNIV // \<approx>L1)" |
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444 shows "finite (UNIV // \<approx>(L1\<star>))" |
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445 proof (rule_tac tag = "tag_str_STAR L1" in tag_finite_imageD) |
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446 show "\<And>x y. tag_str_STAR L1 x = tag_str_STAR L1 y \<Longrightarrow> x \<approx>(L1\<star>) y" |
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447 by (rule tag_str_STAR_injI) |
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448 next |
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449 have *: "finite (Pow (UNIV // \<approx>L1))" |
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450 using finite1 by auto |
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451 show "finite (range (tag_str_STAR L1))" |
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452 unfolding tag_str_STAR_def |
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453 apply(rule finite_subset[OF _ *]) |
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454 unfolding quotient_def |
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455 by auto |
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456 qed |
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457 |
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458 subsubsection{* The conclusion *} |
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459 |
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460 lemma Myhill_Nerode2: |
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461 fixes r::"rexp" |
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462 shows "finite (UNIV // \<approx>(L r))" |
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463 by (induct r) (auto) |
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464 |
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465 |
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466 theorem Myhill_Nerode: |
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467 shows "(\<exists>r::rexp. A = L r) \<longleftrightarrow> finite (UNIV // \<approx>A)" |
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468 using Myhill_Nerode1 Myhill_Nerode2 by auto |
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469 |
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470 end |