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1 |
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2 theory Blexer |
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3 imports "Lexer" |
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4 begin |
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5 |
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6 section \<open>Bit-Encodings\<close> |
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7 |
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8 datatype bit = Z | S |
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9 |
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10 fun code :: "val \<Rightarrow> bit list" |
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11 where |
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12 "code Void = []" |
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13 | "code (Char c) = []" |
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14 | "code (Left v) = Z # (code v)" |
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15 | "code (Right v) = S # (code v)" |
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16 | "code (Seq v1 v2) = (code v1) @ (code v2)" |
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17 | "code (Stars []) = [S]" |
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18 | "code (Stars (v # vs)) = (Z # code v) @ code (Stars vs)" |
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19 |
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20 fun sz where |
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21 "sz ZERO = 0" |
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22 | "sz ONE = 0" |
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23 | "sz (CH _) = 0" |
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24 | "sz (SEQ r1 r2) = 1 + sz r1 + sz r2" |
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25 | "sz (ALT r1 r2) = 1 + sz r1 + sz r2" |
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26 | "sz (STAR r) = 1 + sz r" |
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27 | "sz (NTIMES r n) = 1 + n + sz r" |
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28 |
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29 |
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30 fun |
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31 Stars_add :: "val \<Rightarrow> val \<Rightarrow> val" |
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32 where |
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33 "Stars_add v (Stars vs) = Stars (v # vs)" |
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34 |
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35 function (sequential) |
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36 decode' :: "bit list \<Rightarrow> rexp \<Rightarrow> (val * bit list)" |
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37 where |
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38 "decode' bs ZERO = (undefined, bs)" |
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39 | "decode' bs ONE = (Void, bs)" |
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40 | "decode' bs (CH d) = (Char d, bs)" |
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41 | "decode' [] (ALT r1 r2) = (Void, [])" |
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42 | "decode' (Z # bs) (ALT r1 r2) = (let (v, bs') = decode' bs r1 in (Left v, bs'))" |
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43 | "decode' (S # bs) (ALT r1 r2) = (let (v, bs') = decode' bs r2 in (Right v, bs'))" |
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44 | "decode' bs (SEQ r1 r2) = (let (v1, bs') = decode' bs r1 in |
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45 let (v2, bs'') = decode' bs' r2 in (Seq v1 v2, bs''))" |
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46 | "decode' [] (STAR r) = (Void, [])" |
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47 | "decode' (S # bs) (STAR r) = (Stars [], bs)" |
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48 | "decode' (Z # bs) (STAR r) = (let (v, bs') = decode' bs r in |
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49 let (vs, bs'') = decode' bs' (STAR r) |
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50 in (Stars_add v vs, bs''))" |
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51 | "decode' [] (NTIMES r n) = (Void, [])" |
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52 | "decode' (S # bs) (NTIMES r n) = (Stars [], bs)" |
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53 (*| "decode' (Z # bs) (NTIMES r 0) = (undefined, bs)"*) |
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54 | "decode' (Z # bs) (NTIMES r n) = (let (v, bs') = decode' bs r in |
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55 let (vs, bs'') = decode' bs' (NTIMES r (n - 1)) |
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56 in (Stars_add v vs, bs''))" |
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57 by pat_completeness auto |
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58 |
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59 lemma decode'_smaller: |
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60 assumes "decode'_dom (bs, r)" |
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61 shows "length (snd (decode' bs r)) \<le> length bs" |
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62 using assms |
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63 apply(induct bs r) |
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64 apply(auto simp add: decode'.psimps split: prod.split) |
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65 using dual_order.trans apply blast |
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66 apply (meson dual_order.trans le_SucI) |
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67 apply (meson le_SucI le_trans) |
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68 done |
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69 |
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70 termination "decode'" |
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71 apply(relation "inv_image (measure(%cs. sz cs) <*lex*> measure(%s. size s)) (%(ds,r). (r,ds))") |
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72 apply(auto dest!: decode'_smaller) |
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73 apply (metis less_Suc_eq_le snd_conv) |
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74 by (metis less_Suc_eq_le snd_conv) |
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75 |
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76 definition |
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77 decode :: "bit list \<Rightarrow> rexp \<Rightarrow> val option" |
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78 where |
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79 "decode ds r \<equiv> (let (v, ds') = decode' ds r |
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80 in (if ds' = [] then Some v else None))" |
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81 |
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82 lemma decode'_code_Stars: |
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83 assumes "\<forall>v\<in>set vs. \<Turnstile> v : r \<and> (\<forall>x. decode' (code v @ x) r = (v, x)) \<and> flat v \<noteq> []" |
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84 shows "decode' (code (Stars vs) @ ds) (STAR r) = (Stars vs, ds)" |
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85 using assms |
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86 apply(induct vs) |
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87 apply(auto) |
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88 done |
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89 |
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90 lemma decode'_code_NTIMES: |
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91 assumes "\<forall>v\<in>set vs. \<Turnstile> v : r \<and> (\<forall>x. decode' (code v @ x) r = (v, x))" |
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92 shows "decode' (code (Stars vs) @ ds) (NTIMES r n) = (Stars vs, ds)" |
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93 using assms |
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94 apply(induct vs arbitrary: n r ds) |
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95 apply(auto) |
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96 done |
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97 |
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98 |
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99 lemma decode'_code: |
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100 assumes "\<Turnstile> v : r" |
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101 shows "decode' ((code v) @ ds) r = (v, ds)" |
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102 using assms |
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103 apply(induct v r arbitrary: ds) |
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104 apply(auto) |
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105 using decode'_code_Stars apply blast |
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106 by (metis Un_iff decode'_code_NTIMES set_append) |
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107 |
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108 lemma decode_code: |
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109 assumes "\<Turnstile> v : r" |
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110 shows "decode (code v) r = Some v" |
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111 using assms unfolding decode_def |
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112 by (smt append_Nil2 decode'_code old.prod.case) |
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113 |
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114 |
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115 section {* Annotated Regular Expressions *} |
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116 |
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117 datatype arexp = |
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118 AZERO |
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119 | AONE "bit list" |
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120 | ACHAR "bit list" char |
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121 | ASEQ "bit list" arexp arexp |
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122 | AALTs "bit list" "arexp list" |
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123 | ASTAR "bit list" arexp |
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124 | ANTIMES "bit list" arexp nat |
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125 |
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126 abbreviation |
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127 "AALT bs r1 r2 \<equiv> AALTs bs [r1, r2]" |
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128 |
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129 fun asize :: "arexp \<Rightarrow> nat" where |
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130 "asize AZERO = 1" |
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131 | "asize (AONE cs) = 1" |
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132 | "asize (ACHAR cs c) = 1" |
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133 | "asize (AALTs cs rs) = Suc (sum_list (map asize rs))" |
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134 | "asize (ASEQ cs r1 r2) = Suc (asize r1 + asize r2)" |
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135 | "asize (ASTAR cs r) = Suc (asize r)" |
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136 | "asize (ANTIMES cs r n) = Suc (asize r) + n" |
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137 |
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138 fun |
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139 erase :: "arexp \<Rightarrow> rexp" |
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140 where |
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141 "erase AZERO = ZERO" |
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142 | "erase (AONE _) = ONE" |
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143 | "erase (ACHAR _ c) = CH c" |
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144 | "erase (AALTs _ []) = ZERO" |
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145 | "erase (AALTs _ [r]) = (erase r)" |
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146 | "erase (AALTs bs (r#rs)) = ALT (erase r) (erase (AALTs bs rs))" |
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147 | "erase (ASEQ _ r1 r2) = SEQ (erase r1) (erase r2)" |
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148 | "erase (ASTAR _ r) = STAR (erase r)" |
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149 | "erase (ANTIMES _ r n) = NTIMES (erase r) n" |
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150 |
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151 |
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152 fun fuse :: "bit list \<Rightarrow> arexp \<Rightarrow> arexp" where |
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153 "fuse bs AZERO = AZERO" |
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154 | "fuse bs (AONE cs) = AONE (bs @ cs)" |
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155 | "fuse bs (ACHAR cs c) = ACHAR (bs @ cs) c" |
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156 | "fuse bs (AALTs cs rs) = AALTs (bs @ cs) rs" |
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157 | "fuse bs (ASEQ cs r1 r2) = ASEQ (bs @ cs) r1 r2" |
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158 | "fuse bs (ASTAR cs r) = ASTAR (bs @ cs) r" |
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159 | "fuse bs (ANTIMES cs r n) = ANTIMES (bs @ cs) r n" |
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160 |
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161 lemma fuse_append: |
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162 shows "fuse (bs1 @ bs2) r = fuse bs1 (fuse bs2 r)" |
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163 apply(induct r) |
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164 apply(auto) |
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165 done |
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166 |
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167 |
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168 fun intern :: "rexp \<Rightarrow> arexp" where |
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169 "intern ZERO = AZERO" |
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170 | "intern ONE = AONE []" |
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171 | "intern (CH c) = ACHAR [] c" |
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172 | "intern (ALT r1 r2) = AALT [] (fuse [Z] (intern r1)) |
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173 (fuse [S] (intern r2))" |
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174 | "intern (SEQ r1 r2) = ASEQ [] (intern r1) (intern r2)" |
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175 | "intern (STAR r) = ASTAR [] (intern r)" |
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176 | "intern (NTIMES r n) = ANTIMES [] (intern r) n" |
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177 |
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178 |
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179 fun retrieve :: "arexp \<Rightarrow> val \<Rightarrow> bit list" where |
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180 "retrieve (AONE bs) Void = bs" |
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181 | "retrieve (ACHAR bs c) (Char d) = bs" |
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182 | "retrieve (AALTs bs [r]) v = bs @ retrieve r v" |
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183 | "retrieve (AALTs bs (r#rs)) (Left v) = bs @ retrieve r v" |
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184 | "retrieve (AALTs bs (r#rs)) (Right v) = bs @ retrieve (AALTs [] rs) v" |
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185 | "retrieve (ASEQ bs r1 r2) (Seq v1 v2) = bs @ retrieve r1 v1 @ retrieve r2 v2" |
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186 | "retrieve (ASTAR bs r) (Stars []) = bs @ [S]" |
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187 | "retrieve (ASTAR bs r) (Stars (v#vs)) = |
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188 bs @ [Z] @ retrieve r v @ retrieve (ASTAR [] r) (Stars vs)" |
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189 | "retrieve (ANTIMES bs r 0) (Stars []) = bs @ [S]" |
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190 | "retrieve (ANTIMES bs r (Suc n)) (Stars (v#vs)) = |
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191 bs @ [Z] @ retrieve r v @ retrieve (ANTIMES [] r n) (Stars vs)" |
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192 |
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193 |
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194 fun |
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195 bnullable :: "arexp \<Rightarrow> bool" |
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196 where |
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197 "bnullable (AZERO) = False" |
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198 | "bnullable (AONE bs) = True" |
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199 | "bnullable (ACHAR bs c) = False" |
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200 | "bnullable (AALTs bs rs) = (\<exists>r \<in> set rs. bnullable r)" |
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201 | "bnullable (ASEQ bs r1 r2) = (bnullable r1 \<and> bnullable r2)" |
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202 | "bnullable (ASTAR bs r) = True" |
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203 | "bnullable (ANTIMES bs r n) = (if n = 0 then True else bnullable r)" |
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204 |
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205 abbreviation |
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206 bnullables :: "arexp list \<Rightarrow> bool" |
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207 where |
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208 "bnullables rs \<equiv> (\<exists>r \<in> set rs. bnullable r)" |
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209 |
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210 function (sequential) |
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211 bmkeps :: "arexp \<Rightarrow> bit list" |
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212 where |
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213 "bmkeps(AONE bs) = bs" |
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214 | "bmkeps(ASEQ bs r1 r2) = bs @ (bmkeps r1) @ (bmkeps r2)" |
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215 | "bmkeps(AALTs bs (r#rs)) = |
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216 (if bnullable(r) then (bs @ bmkeps r) else (bmkeps (AALTs bs rs)))" |
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217 | "bmkeps(ASTAR bs r) = bs @ [S]" |
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218 | "bmkeps(ANTIMES bs r 0) = bs @ [S]" |
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219 | "bmkeps(ANTIMES bs r (Suc n)) = bs @ [Z] @ (bmkeps r) @ bmkeps(ANTIMES [] r n)" |
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220 apply(pat_completeness) |
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221 apply(auto) |
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222 done |
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223 |
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224 termination "bmkeps" |
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225 apply(relation "measure asize") |
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226 apply(auto) |
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227 using asize.elims by force |
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228 |
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229 fun |
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230 bmkepss :: "arexp list \<Rightarrow> bit list" |
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231 where |
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232 "bmkepss (r # rs) = (if bnullable(r) then (bmkeps r) else (bmkepss rs))" |
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233 |
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234 |
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235 lemma bmkepss1: |
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236 assumes "\<not> bnullables rs1" |
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237 shows "bmkepss (rs1 @ rs2) = bmkepss rs2" |
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238 using assms |
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239 by(induct rs1) (auto) |
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240 |
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241 |
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242 lemma bmkepss2: |
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243 assumes "bnullables rs1" |
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244 shows "bmkepss (rs1 @ rs2) = bmkepss rs1" |
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245 using assms |
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246 by (induct rs1) (auto) |
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247 |
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248 |
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249 fun |
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250 bder :: "char \<Rightarrow> arexp \<Rightarrow> arexp" |
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251 where |
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252 "bder c (AZERO) = AZERO" |
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253 | "bder c (AONE bs) = AZERO" |
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254 | "bder c (ACHAR bs d) = (if c = d then AONE bs else AZERO)" |
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255 | "bder c (AALTs bs rs) = AALTs bs (map (bder c) rs)" |
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256 | "bder c (ASEQ bs r1 r2) = |
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257 (if bnullable r1 |
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258 then AALT bs (ASEQ [] (bder c r1) r2) (fuse (bmkeps r1) (bder c r2)) |
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259 else ASEQ bs (bder c r1) r2)" |
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260 | "bder c (ASTAR bs r) = ASEQ (bs @ [Z]) (bder c r) (ASTAR [] r)" |
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261 | "bder c (ANTIMES bs r n) = (if n = 0 then AZERO else ASEQ (bs @ [Z]) (bder c r) (ANTIMES [] r (n - 1)))" |
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262 |
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263 fun |
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264 bders :: "arexp \<Rightarrow> string \<Rightarrow> arexp" |
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265 where |
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266 "bders r [] = r" |
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267 | "bders r (c#s) = bders (bder c r) s" |
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268 |
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269 lemma bders_append: |
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270 "bders c (s1 @ s2) = bders (bders c s1) s2" |
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271 apply(induct s1 arbitrary: c s2) |
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272 apply(simp_all) |
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273 done |
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274 |
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275 lemma bnullable_correctness: |
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276 shows "nullable (erase r) = bnullable r" |
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277 apply(induct r rule: erase.induct) |
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278 apply(simp_all) |
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279 done |
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280 |
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281 lemma erase_fuse: |
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282 shows "erase (fuse bs r) = erase r" |
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283 apply(induct r rule: erase.induct) |
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284 apply(simp_all) |
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285 done |
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286 |
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287 lemma erase_intern [simp]: |
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288 shows "erase (intern r) = r" |
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289 apply(induct r) |
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290 apply(simp_all add: erase_fuse) |
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291 done |
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292 |
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293 lemma erase_bder [simp]: |
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294 shows "erase (bder a r) = der a (erase r)" |
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295 apply(induct r rule: erase.induct) |
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296 apply(simp_all add: erase_fuse bnullable_correctness) |
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297 done |
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298 |
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299 lemma erase_bders [simp]: |
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300 shows "erase (bders r s) = ders s (erase r)" |
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301 apply(induct s arbitrary: r ) |
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302 apply(simp_all) |
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303 done |
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304 |
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305 lemma bnullable_fuse: |
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306 shows "bnullable (fuse bs r) = bnullable r" |
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307 apply(induct r arbitrary: bs) |
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308 apply(auto) |
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309 done |
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310 |
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311 lemma retrieve_encode_STARS: |
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312 assumes "\<forall>v\<in>set vs. \<Turnstile> v : r \<and> code v = retrieve (intern r) v" |
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313 shows "code (Stars vs) = retrieve (ASTAR [] (intern r)) (Stars vs)" |
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314 using assms |
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315 apply(induct vs) |
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316 apply(simp_all) |
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317 done |
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318 |
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319 lemma retrieve_encode_NTIMES: |
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320 assumes "\<forall>v\<in>set vs. \<Turnstile> v : r \<and> code v = retrieve (intern r) v" "length vs = n" |
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321 shows "code (Stars vs) = retrieve (ANTIMES [] (intern r) n) (Stars vs)" |
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322 using assms |
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323 apply(induct vs arbitrary: n) |
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324 apply(simp_all) |
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325 by force |
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326 |
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327 |
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328 lemma retrieve_fuse2: |
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329 assumes "\<Turnstile> v : (erase r)" |
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330 shows "retrieve (fuse bs r) v = bs @ retrieve r v" |
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331 using assms |
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332 apply(induct r arbitrary: v bs) |
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333 apply(auto elim: Prf_elims)[4] |
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334 apply(case_tac x2a) |
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335 apply(simp) |
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336 using Prf_elims(1) apply blast |
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337 apply(case_tac x2a) |
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338 apply(simp) |
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339 apply(simp) |
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340 apply(case_tac list) |
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341 apply(simp) |
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342 apply(simp) |
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343 apply (smt (verit, best) Prf_elims(3) append_assoc retrieve.simps(4) retrieve.simps(5)) |
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344 apply(simp) |
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345 using retrieve_encode_STARS |
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346 apply(auto elim!: Prf_elims)[1] |
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347 apply(case_tac vs) |
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348 apply(simp) |
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349 apply(simp) |
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350 (* NTIMES *) |
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351 apply(auto elim!: Prf_elims)[1] |
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352 apply(case_tac vs1) |
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353 apply(simp_all) |
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354 apply(case_tac vs2) |
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355 apply(simp_all) |
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356 done |
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357 |
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358 lemma retrieve_fuse: |
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359 assumes "\<Turnstile> v : r" |
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360 shows "retrieve (fuse bs (intern r)) v = bs @ retrieve (intern r) v" |
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361 using assms |
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362 by (simp_all add: retrieve_fuse2) |
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363 |
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364 |
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365 lemma retrieve_code: |
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366 assumes "\<Turnstile> v : r" |
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367 shows "code v = retrieve (intern r) v" |
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368 using assms |
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369 apply(induct v r ) |
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370 apply(simp_all add: retrieve_fuse retrieve_encode_STARS) |
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371 apply(subst retrieve_encode_NTIMES) |
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372 apply(auto) |
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373 done |
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374 |
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375 |
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376 |
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377 lemma retrieve_AALTs_bnullable1: |
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378 assumes "bnullable r" |
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379 shows "retrieve (AALTs bs (r # rs)) (mkeps (erase (AALTs bs (r # rs)))) |
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380 = bs @ retrieve r (mkeps (erase r))" |
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381 using assms |
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382 apply(case_tac rs) |
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383 apply(auto simp add: bnullable_correctness) |
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384 done |
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385 |
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386 lemma retrieve_AALTs_bnullable2: |
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387 assumes "\<not>bnullable r" "bnullables rs" |
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388 shows "retrieve (AALTs bs (r # rs)) (mkeps (erase (AALTs bs (r # rs)))) |
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389 = retrieve (AALTs bs rs) (mkeps (erase (AALTs bs rs)))" |
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390 using assms |
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391 apply(induct rs arbitrary: r bs) |
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392 apply(auto) |
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393 using bnullable_correctness apply blast |
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394 apply(case_tac rs) |
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395 apply(auto) |
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396 using bnullable_correctness apply blast |
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397 apply(case_tac rs) |
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398 apply(auto) |
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399 done |
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400 |
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401 lemma bmkeps_retrieve_AALTs: |
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402 assumes "\<forall>r \<in> set rs. bnullable r \<longrightarrow> bmkeps r = retrieve r (mkeps (erase r))" |
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403 "bnullables rs" |
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404 shows "bs @ bmkepss rs = retrieve (AALTs bs rs) (mkeps (erase (AALTs bs rs)))" |
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405 using assms |
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406 apply(induct rs arbitrary: bs) |
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407 apply(auto) |
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408 using retrieve_AALTs_bnullable1 apply presburger |
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409 apply (metis retrieve_AALTs_bnullable2) |
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410 apply (simp add: retrieve_AALTs_bnullable1) |
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411 by (metis retrieve_AALTs_bnullable2) |
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412 |
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413 lemma bmkeps_retrieve_ANTIMES: |
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414 assumes "if n = 0 then True else bmkeps r = retrieve r (mkeps (erase r))" |
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415 and "bnullable (ANTIMES bs r n)" |
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416 shows "bmkeps (ANTIMES bs r n) = retrieve (ANTIMES bs r n) (Stars (replicate n (mkeps (erase r))))" |
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417 using assms |
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418 apply(induct n arbitrary: r bs) |
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419 apply(auto)[1] |
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420 apply(simp) |
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421 done |
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422 |
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423 lemma bmkeps_retrieve: |
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424 assumes "bnullable r" |
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425 shows "bmkeps r = retrieve r (mkeps (erase r))" |
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426 using assms |
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427 apply(induct r rule: bmkeps.induct) |
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428 apply(auto) |
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429 apply (simp add: retrieve_AALTs_bnullable1) |
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430 using retrieve_AALTs_bnullable1 apply force |
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431 by (metis retrieve_AALTs_bnullable2) |
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432 |
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433 |
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434 lemma bder_retrieve: |
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435 assumes "\<Turnstile> v : der c (erase r)" |
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436 shows "retrieve (bder c r) v = retrieve r (injval (erase r) c v)" |
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437 using assms |
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438 apply(induct r arbitrary: v rule: erase.induct) |
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439 using Prf_elims(1) apply auto[1] |
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440 using Prf_elims(1) apply auto[1] |
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441 apply(auto)[1] |
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442 apply (metis Prf_elims(4) injval.simps(1) retrieve.simps(1) retrieve.simps(2)) |
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443 using Prf_elims(1) apply blast |
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444 (* AALTs case *) |
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445 apply(simp) |
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446 apply(erule Prf_elims) |
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447 apply(simp) |
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448 apply(simp) |
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449 apply(rename_tac "r\<^sub>1" "r\<^sub>2" rs v) |
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450 apply(erule Prf_elims) |
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451 apply(simp) |
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452 apply(simp) |
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453 apply(case_tac rs) |
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454 apply(simp) |
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455 apply(simp) |
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456 using Prf_elims(3) apply fastforce |
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457 (* ASEQ case *) |
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458 apply(simp) |
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459 apply(case_tac "nullable (erase r1)") |
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460 apply(simp) |
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461 apply(erule Prf_elims) |
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462 using Prf_elims(2) bnullable_correctness apply force |
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463 apply (simp add: bmkeps_retrieve bnullable_correctness retrieve_fuse2) |
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464 apply (simp add: bmkeps_retrieve bnullable_correctness retrieve_fuse2) |
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465 using Prf_elims(2) apply force |
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466 (* ASTAR case *) |
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467 apply(rename_tac bs r v) |
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468 apply(simp) |
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469 apply(erule Prf_elims) |
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470 apply(clarify) |
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471 apply(erule Prf_elims) |
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472 apply(clarify) |
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473 apply (simp add: retrieve_fuse2) |
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474 (* ANTIMES case *) |
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475 apply(auto) |
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476 apply(erule Prf_elims) |
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477 apply(erule Prf_elims) |
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478 apply(clarify) |
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479 apply(erule Prf_elims) |
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480 apply(clarify) |
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481 by (metis (full_types) Suc_pred append_assoc injval.simps(8) retrieve.simps(10) retrieve.simps(6)) |
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482 |
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483 |
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484 lemma MAIN_decode: |
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485 assumes "\<Turnstile> v : ders s r" |
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486 shows "Some (flex r id s v) = decode (retrieve (bders (intern r) s) v) r" |
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487 using assms |
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488 proof (induct s arbitrary: v rule: rev_induct) |
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489 case Nil |
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490 have "\<Turnstile> v : ders [] r" by fact |
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491 then have "\<Turnstile> v : r" by simp |
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492 then have "Some v = decode (retrieve (intern r) v) r" |
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493 using decode_code retrieve_code by auto |
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494 then show "Some (flex r id [] v) = decode (retrieve (bders (intern r) []) v) r" |
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495 by simp |
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496 next |
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497 case (snoc c s v) |
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498 have IH: "\<And>v. \<Turnstile> v : ders s r \<Longrightarrow> |
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499 Some (flex r id s v) = decode (retrieve (bders (intern r) s) v) r" by fact |
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500 have asm: "\<Turnstile> v : ders (s @ [c]) r" by fact |
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501 then have asm2: "\<Turnstile> injval (ders s r) c v : ders s r" |
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502 by (simp add: Prf_injval ders_append) |
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503 have "Some (flex r id (s @ [c]) v) = Some (flex r id s (injval (ders s r) c v))" |
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504 by (simp add: flex_append) |
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505 also have "... = decode (retrieve (bders (intern r) s) (injval (ders s r) c v)) r" |
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506 using asm2 IH by simp |
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507 also have "... = decode (retrieve (bder c (bders (intern r) s)) v) r" |
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508 using asm by (simp_all add: bder_retrieve ders_append) |
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509 finally show "Some (flex r id (s @ [c]) v) = |
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510 decode (retrieve (bders (intern r) (s @ [c])) v) r" by (simp add: bders_append) |
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511 qed |
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512 |
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513 definition blexer where |
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514 "blexer r s \<equiv> if bnullable (bders (intern r) s) then |
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515 decode (bmkeps (bders (intern r) s)) r else None" |
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516 |
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517 lemma blexer_correctness: |
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518 shows "blexer r s = lexer r s" |
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519 proof - |
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520 { define bds where "bds \<equiv> bders (intern r) s" |
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521 define ds where "ds \<equiv> ders s r" |
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522 assume asm: "nullable ds" |
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523 have era: "erase bds = ds" |
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524 unfolding ds_def bds_def by simp |
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525 have mke: "\<Turnstile> mkeps ds : ds" |
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526 using asm by (simp add: mkeps_nullable) |
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527 have "decode (bmkeps bds) r = decode (retrieve bds (mkeps ds)) r" |
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528 using bmkeps_retrieve |
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529 using asm era |
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530 using bnullable_correctness by force |
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531 also have "... = Some (flex r id s (mkeps ds))" |
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532 using mke by (simp_all add: MAIN_decode ds_def bds_def) |
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533 finally have "decode (bmkeps bds) r = Some (flex r id s (mkeps ds))" |
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534 unfolding bds_def ds_def . |
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535 } |
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536 then show "blexer r s = lexer r s" |
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537 unfolding blexer_def lexer_flex |
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538 by (auto simp add: bnullable_correctness[symmetric]) |
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539 qed |
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540 |
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541 |
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542 unused_thms |
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543 |
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544 end |