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1 |
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2 theory Lexer |
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3 imports PosixSpec |
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4 begin |
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5 |
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6 section {* The Lexer Functions by Sulzmann and Lu (without simplification) *} |
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7 |
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8 fun |
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9 mkeps :: "rexp \<Rightarrow> val" |
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10 where |
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11 "mkeps(ONE) = Void" |
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12 | "mkeps(SEQ r1 r2) = Seq (mkeps r1) (mkeps r2)" |
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13 | "mkeps(ALT r1 r2) = (if nullable(r1) then Left (mkeps r1) else Right (mkeps r2))" |
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14 | "mkeps(STAR r) = Stars []" |
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15 |
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16 fun injval :: "rexp \<Rightarrow> char \<Rightarrow> val \<Rightarrow> val" |
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17 where |
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18 "injval (CH d) c Void = Char d" |
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19 | "injval (ALT r1 r2) c (Left v1) = Left(injval r1 c v1)" |
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20 | "injval (ALT r1 r2) c (Right v2) = Right(injval r2 c v2)" |
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21 | "injval (SEQ r1 r2) c (Seq v1 v2) = Seq (injval r1 c v1) v2" |
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22 | "injval (SEQ r1 r2) c (Left (Seq v1 v2)) = Seq (injval r1 c v1) v2" |
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23 | "injval (SEQ r1 r2) c (Right v2) = Seq (mkeps r1) (injval r2 c v2)" |
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24 | "injval (STAR r) c (Seq v (Stars vs)) = Stars ((injval r c v) # vs)" |
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25 |
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26 fun |
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27 lexer :: "rexp \<Rightarrow> string \<Rightarrow> val option" |
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28 where |
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29 "lexer r [] = (if nullable r then Some(mkeps r) else None)" |
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30 | "lexer r (c#s) = (case (lexer (der c r) s) of |
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31 None \<Rightarrow> None |
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32 | Some(v) \<Rightarrow> Some(injval r c v))" |
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33 |
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34 |
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35 |
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36 section {* Mkeps, Injval Properties *} |
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37 |
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38 lemma mkeps_nullable: |
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39 assumes "nullable(r)" |
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40 shows "\<Turnstile> mkeps r : r" |
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41 using assms |
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42 by (induct rule: nullable.induct) |
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43 (auto intro: Prf.intros) |
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44 |
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45 lemma mkeps_flat: |
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46 assumes "nullable(r)" |
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47 shows "flat (mkeps r) = []" |
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48 using assms |
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49 by (induct rule: nullable.induct) (auto) |
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50 |
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51 lemma Prf_injval_flat: |
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52 assumes "\<Turnstile> v : der c r" |
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53 shows "flat (injval r c v) = c # (flat v)" |
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54 using assms |
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55 apply(induct c r arbitrary: v rule: der.induct) |
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56 apply(auto elim!: Prf_elims intro: mkeps_flat split: if_splits) |
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57 done |
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58 |
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59 lemma Prf_injval: |
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60 assumes "\<Turnstile> v : der c r" |
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61 shows "\<Turnstile> (injval r c v) : r" |
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62 using assms |
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63 apply(induct r arbitrary: c v rule: rexp.induct) |
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64 apply(auto intro!: Prf.intros mkeps_nullable elim!: Prf_elims split: if_splits) |
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65 apply(simp add: Prf_injval_flat) |
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66 done |
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67 |
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68 |
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69 |
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70 text {* |
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71 Mkeps and injval produce, or preserve, Posix values. |
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72 *} |
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73 |
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74 lemma Posix_mkeps: |
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75 assumes "nullable r" |
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76 shows "[] \<in> r \<rightarrow> mkeps r" |
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77 using assms |
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78 apply(induct r rule: nullable.induct) |
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79 apply(auto intro: Posix.intros simp add: nullable_correctness Sequ_def) |
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80 apply(subst append.simps(1)[symmetric]) |
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81 apply(rule Posix.intros) |
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82 apply(auto) |
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83 done |
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84 |
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85 lemma Posix_injval: |
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86 assumes "s \<in> (der c r) \<rightarrow> v" |
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87 shows "(c # s) \<in> r \<rightarrow> (injval r c v)" |
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88 using assms |
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89 proof(induct r arbitrary: s v rule: rexp.induct) |
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90 case ZERO |
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91 have "s \<in> der c ZERO \<rightarrow> v" by fact |
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92 then have "s \<in> ZERO \<rightarrow> v" by simp |
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93 then have "False" by cases |
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94 then show "(c # s) \<in> ZERO \<rightarrow> (injval ZERO c v)" by simp |
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95 next |
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96 case ONE |
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97 have "s \<in> der c ONE \<rightarrow> v" by fact |
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98 then have "s \<in> ZERO \<rightarrow> v" by simp |
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99 then have "False" by cases |
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100 then show "(c # s) \<in> ONE \<rightarrow> (injval ONE c v)" by simp |
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101 next |
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102 case (CH d) |
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103 consider (eq) "c = d" | (ineq) "c \<noteq> d" by blast |
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104 then show "(c # s) \<in> (CH d) \<rightarrow> (injval (CH d) c v)" |
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105 proof (cases) |
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106 case eq |
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107 have "s \<in> der c (CH d) \<rightarrow> v" by fact |
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108 then have "s \<in> ONE \<rightarrow> v" using eq by simp |
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109 then have eqs: "s = [] \<and> v = Void" by cases simp |
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110 show "(c # s) \<in> CH d \<rightarrow> injval (CH d) c v" using eq eqs |
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111 by (auto intro: Posix.intros) |
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112 next |
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113 case ineq |
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114 have "s \<in> der c (CH d) \<rightarrow> v" by fact |
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115 then have "s \<in> ZERO \<rightarrow> v" using ineq by simp |
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116 then have "False" by cases |
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117 then show "(c # s) \<in> CH d \<rightarrow> injval (CH d) c v" by simp |
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118 qed |
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119 next |
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120 case (ALT r1 r2) |
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121 have IH1: "\<And>s v. s \<in> der c r1 \<rightarrow> v \<Longrightarrow> (c # s) \<in> r1 \<rightarrow> injval r1 c v" by fact |
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122 have IH2: "\<And>s v. s \<in> der c r2 \<rightarrow> v \<Longrightarrow> (c # s) \<in> r2 \<rightarrow> injval r2 c v" by fact |
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123 have "s \<in> der c (ALT r1 r2) \<rightarrow> v" by fact |
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124 then have "s \<in> ALT (der c r1) (der c r2) \<rightarrow> v" by simp |
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125 then consider (left) v' where "v = Left v'" "s \<in> der c r1 \<rightarrow> v'" |
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126 | (right) v' where "v = Right v'" "s \<notin> L (der c r1)" "s \<in> der c r2 \<rightarrow> v'" |
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127 by cases auto |
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128 then show "(c # s) \<in> ALT r1 r2 \<rightarrow> injval (ALT r1 r2) c v" |
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129 proof (cases) |
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130 case left |
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131 have "s \<in> der c r1 \<rightarrow> v'" by fact |
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132 then have "(c # s) \<in> r1 \<rightarrow> injval r1 c v'" using IH1 by simp |
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133 then have "(c # s) \<in> ALT r1 r2 \<rightarrow> injval (ALT r1 r2) c (Left v')" by (auto intro: Posix.intros) |
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134 then show "(c # s) \<in> ALT r1 r2 \<rightarrow> injval (ALT r1 r2) c v" using left by simp |
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135 next |
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136 case right |
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137 have "s \<notin> L (der c r1)" by fact |
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138 then have "c # s \<notin> L r1" by (simp add: der_correctness Der_def) |
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139 moreover |
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140 have "s \<in> der c r2 \<rightarrow> v'" by fact |
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141 then have "(c # s) \<in> r2 \<rightarrow> injval r2 c v'" using IH2 by simp |
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142 ultimately have "(c # s) \<in> ALT r1 r2 \<rightarrow> injval (ALT r1 r2) c (Right v')" |
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143 by (auto intro: Posix.intros) |
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144 then show "(c # s) \<in> ALT r1 r2 \<rightarrow> injval (ALT r1 r2) c v" using right by simp |
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145 qed |
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146 next |
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147 case (SEQ r1 r2) |
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148 have IH1: "\<And>s v. s \<in> der c r1 \<rightarrow> v \<Longrightarrow> (c # s) \<in> r1 \<rightarrow> injval r1 c v" by fact |
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149 have IH2: "\<And>s v. s \<in> der c r2 \<rightarrow> v \<Longrightarrow> (c # s) \<in> r2 \<rightarrow> injval r2 c v" by fact |
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150 have "s \<in> der c (SEQ r1 r2) \<rightarrow> v" by fact |
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151 then consider |
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152 (left_nullable) v1 v2 s1 s2 where |
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153 "v = Left (Seq v1 v2)" "s = s1 @ s2" |
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154 "s1 \<in> der c r1 \<rightarrow> v1" "s2 \<in> r2 \<rightarrow> v2" "nullable r1" |
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155 "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r1) \<and> s\<^sub>4 \<in> L r2)" |
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156 | (right_nullable) v1 s1 s2 where |
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157 "v = Right v1" "s = s1 @ s2" |
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158 "s \<in> der c r2 \<rightarrow> v1" "nullable r1" "s1 @ s2 \<notin> L (SEQ (der c r1) r2)" |
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159 | (not_nullable) v1 v2 s1 s2 where |
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160 "v = Seq v1 v2" "s = s1 @ s2" |
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161 "s1 \<in> der c r1 \<rightarrow> v1" "s2 \<in> r2 \<rightarrow> v2" "\<not>nullable r1" |
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162 "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r1) \<and> s\<^sub>4 \<in> L r2)" |
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163 by (force split: if_splits elim!: Posix_elims simp add: Sequ_def der_correctness Der_def) |
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164 then show "(c # s) \<in> SEQ r1 r2 \<rightarrow> injval (SEQ r1 r2) c v" |
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165 proof (cases) |
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166 case left_nullable |
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167 have "s1 \<in> der c r1 \<rightarrow> v1" by fact |
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168 then have "(c # s1) \<in> r1 \<rightarrow> injval r1 c v1" using IH1 by simp |
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169 moreover |
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170 have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r1) \<and> s\<^sub>4 \<in> L r2)" by fact |
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171 then have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (c # s1) @ s\<^sub>3 \<in> L r1 \<and> s\<^sub>4 \<in> L r2)" by (simp add: der_correctness Der_def) |
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172 ultimately have "((c # s1) @ s2) \<in> SEQ r1 r2 \<rightarrow> Seq (injval r1 c v1) v2" using left_nullable by (rule_tac Posix.intros) |
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173 then show "(c # s) \<in> SEQ r1 r2 \<rightarrow> injval (SEQ r1 r2) c v" using left_nullable by simp |
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174 next |
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175 case right_nullable |
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176 have "nullable r1" by fact |
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177 then have "[] \<in> r1 \<rightarrow> (mkeps r1)" by (rule Posix_mkeps) |
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178 moreover |
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179 have "s \<in> der c r2 \<rightarrow> v1" by fact |
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180 then have "(c # s) \<in> r2 \<rightarrow> (injval r2 c v1)" using IH2 by simp |
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181 moreover |
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182 have "s1 @ s2 \<notin> L (SEQ (der c r1) r2)" by fact |
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183 then have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = c # s \<and> [] @ s\<^sub>3 \<in> L r1 \<and> s\<^sub>4 \<in> L r2)" using right_nullable |
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184 by(auto simp add: der_correctness Der_def append_eq_Cons_conv Sequ_def) |
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185 ultimately have "([] @ (c # s)) \<in> SEQ r1 r2 \<rightarrow> Seq (mkeps r1) (injval r2 c v1)" |
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186 by(rule Posix.intros) |
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187 then show "(c # s) \<in> SEQ r1 r2 \<rightarrow> injval (SEQ r1 r2) c v" using right_nullable by simp |
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188 next |
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189 case not_nullable |
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190 have "s1 \<in> der c r1 \<rightarrow> v1" by fact |
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191 then have "(c # s1) \<in> r1 \<rightarrow> injval r1 c v1" using IH1 by simp |
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192 moreover |
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193 have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r1) \<and> s\<^sub>4 \<in> L r2)" by fact |
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194 then have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (c # s1) @ s\<^sub>3 \<in> L r1 \<and> s\<^sub>4 \<in> L r2)" by (simp add: der_correctness Der_def) |
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195 ultimately have "((c # s1) @ s2) \<in> SEQ r1 r2 \<rightarrow> Seq (injval r1 c v1) v2" using not_nullable |
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196 by (rule_tac Posix.intros) (simp_all) |
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197 then show "(c # s) \<in> SEQ r1 r2 \<rightarrow> injval (SEQ r1 r2) c v" using not_nullable by simp |
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198 qed |
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199 next |
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200 case (STAR r) |
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201 have IH: "\<And>s v. s \<in> der c r \<rightarrow> v \<Longrightarrow> (c # s) \<in> r \<rightarrow> injval r c v" by fact |
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202 have "s \<in> der c (STAR r) \<rightarrow> v" by fact |
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203 then consider |
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204 (cons) v1 vs s1 s2 where |
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205 "v = Seq v1 (Stars vs)" "s = s1 @ s2" |
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206 "s1 \<in> der c r \<rightarrow> v1" "s2 \<in> (STAR r) \<rightarrow> (Stars vs)" |
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207 "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r) \<and> s\<^sub>4 \<in> L (STAR r))" |
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208 apply(auto elim!: Posix_elims(1-5) simp add: der_correctness Der_def intro: Posix.intros) |
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209 apply(rotate_tac 3) |
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210 apply(erule_tac Posix_elims(6)) |
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211 apply (simp add: Posix.intros(6)) |
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212 using Posix.intros(7) by blast |
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213 then show "(c # s) \<in> STAR r \<rightarrow> injval (STAR r) c v" |
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214 proof (cases) |
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215 case cons |
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216 have "s1 \<in> der c r \<rightarrow> v1" by fact |
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217 then have "(c # s1) \<in> r \<rightarrow> injval r c v1" using IH by simp |
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218 moreover |
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219 have "s2 \<in> STAR r \<rightarrow> Stars vs" by fact |
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220 moreover |
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221 have "(c # s1) \<in> r \<rightarrow> injval r c v1" by fact |
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222 then have "flat (injval r c v1) = (c # s1)" by (rule Posix1) |
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223 then have "flat (injval r c v1) \<noteq> []" by simp |
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224 moreover |
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225 have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r) \<and> s\<^sub>4 \<in> L (STAR r))" by fact |
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226 then have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (c # s1) @ s\<^sub>3 \<in> L r \<and> s\<^sub>4 \<in> L (STAR r))" |
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227 by (simp add: der_correctness Der_def) |
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228 ultimately |
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229 have "((c # s1) @ s2) \<in> STAR r \<rightarrow> Stars (injval r c v1 # vs)" by (rule Posix.intros) |
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230 then show "(c # s) \<in> STAR r \<rightarrow> injval (STAR r) c v" using cons by(simp) |
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231 qed |
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232 qed |
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233 |
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234 |
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235 section {* Lexer Correctness *} |
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236 |
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237 |
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238 lemma lexer_correct_None: |
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239 shows "s \<notin> L r \<longleftrightarrow> lexer r s = None" |
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240 apply(induct s arbitrary: r) |
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241 apply(simp) |
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242 apply(simp add: nullable_correctness) |
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243 apply(simp) |
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244 apply(drule_tac x="der a r" in meta_spec) |
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245 apply(auto) |
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246 apply(auto simp add: der_correctness Der_def) |
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247 done |
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248 |
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249 lemma lexer_correct_Some: |
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250 shows "s \<in> L r \<longleftrightarrow> (\<exists>v. lexer r s = Some(v) \<and> s \<in> r \<rightarrow> v)" |
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251 apply(induct s arbitrary : r) |
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252 apply(simp only: lexer.simps) |
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253 apply(simp) |
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254 apply(simp add: nullable_correctness Posix_mkeps) |
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255 apply(drule_tac x="der a r" in meta_spec) |
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256 apply(simp (no_asm_use) add: der_correctness Der_def del: lexer.simps) |
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257 apply(simp del: lexer.simps) |
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258 apply(simp only: lexer.simps) |
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259 apply(case_tac "lexer (der a r) s = None") |
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260 apply(auto)[1] |
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261 apply(simp) |
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262 apply(erule exE) |
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263 apply(simp) |
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264 apply(rule iffI) |
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265 apply(simp add: Posix_injval) |
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266 apply(simp add: Posix1(1)) |
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267 done |
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268 |
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269 lemma lexer_correctness: |
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270 shows "(lexer r s = Some v) \<longleftrightarrow> s \<in> r \<rightarrow> v" |
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271 and "(lexer r s = None) \<longleftrightarrow> \<not>(\<exists>v. s \<in> r \<rightarrow> v)" |
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272 using Posix1(1) Posix_determ lexer_correct_None lexer_correct_Some apply fastforce |
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273 using Posix1(1) lexer_correct_None lexer_correct_Some by blast |
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274 |
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275 |
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276 subsection {* A slight reformulation of the lexer algorithm using stacked functions*} |
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277 |
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278 fun flex :: "rexp \<Rightarrow> (val \<Rightarrow> val) => string \<Rightarrow> (val \<Rightarrow> val)" |
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279 where |
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280 "flex r f [] = f" |
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281 | "flex r f (c#s) = flex (der c r) (\<lambda>v. f (injval r c v)) s" |
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282 |
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283 lemma flex_fun_apply: |
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284 shows "g (flex r f s v) = flex r (g o f) s v" |
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285 apply(induct s arbitrary: g f r v) |
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286 apply(simp_all add: comp_def) |
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287 by meson |
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288 |
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289 lemma flex_fun_apply2: |
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290 shows "g (flex r id s v) = flex r g s v" |
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291 by (simp add: flex_fun_apply) |
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292 |
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293 |
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294 lemma flex_append: |
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295 shows "flex r f (s1 @ s2) = flex (ders s1 r) (flex r f s1) s2" |
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296 apply(induct s1 arbitrary: s2 r f) |
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297 apply(simp_all) |
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298 done |
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299 |
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300 lemma lexer_flex: |
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301 shows "lexer r s = (if nullable (ders s r) |
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302 then Some(flex r id s (mkeps (ders s r))) else None)" |
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303 apply(induct s arbitrary: r) |
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304 apply(simp_all add: flex_fun_apply) |
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305 done |
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306 |
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307 lemma Posix_flex: |
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308 assumes "s2 \<in> (ders s1 r) \<rightarrow> v" |
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309 shows "(s1 @ s2) \<in> r \<rightarrow> flex r id s1 v" |
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310 using assms |
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311 apply(induct s1 arbitrary: r v s2) |
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312 apply(simp) |
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313 apply(simp) |
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314 apply(drule_tac x="der a r" in meta_spec) |
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315 apply(drule_tac x="v" in meta_spec) |
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316 apply(drule_tac x="s2" in meta_spec) |
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317 apply(simp) |
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318 using Posix_injval |
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319 apply(drule_tac Posix_injval) |
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320 apply(subst (asm) (5) flex_fun_apply) |
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321 apply(simp) |
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322 done |
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323 |
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324 lemma injval_inj: |
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325 assumes "\<Turnstile> a : (der c r)" "\<Turnstile> v : (der c r)" "injval r c a = injval r c v" |
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326 shows "a = v" |
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327 using assms |
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328 apply(induct r arbitrary: a c v) |
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329 apply(auto) |
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330 using Prf_elims(1) apply blast |
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331 using Prf_elims(1) apply blast |
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332 apply(case_tac "c = x") |
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333 apply(auto) |
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334 using Prf_elims(4) apply auto[1] |
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335 using Prf_elims(1) apply blast |
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336 prefer 2 |
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337 apply (smt Prf_elims(3) injval.simps(2) injval.simps(3) val.distinct(25) val.inject(3) val.inject(4)) |
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338 apply(case_tac "nullable r1") |
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339 apply(auto) |
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340 apply(erule Prf_elims) |
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341 apply(erule Prf_elims) |
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342 apply(erule Prf_elims) |
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343 apply(erule Prf_elims) |
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344 apply(auto) |
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345 apply (metis Prf_injval_flat list.distinct(1) mkeps_flat) |
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346 apply(erule Prf_elims) |
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347 apply(erule Prf_elims) |
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348 apply(auto) |
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349 using Prf_injval_flat mkeps_flat apply fastforce |
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350 apply(erule Prf_elims) |
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351 apply(erule Prf_elims) |
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352 apply(auto) |
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353 apply(erule Prf_elims) |
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354 apply(erule Prf_elims) |
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355 apply(auto) |
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356 apply (smt Prf_elims(6) injval.simps(7) list.inject val.inject(5)) |
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357 by (smt Prf_elims(6) injval.simps(7) list.inject val.inject(5)) |
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358 |
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359 |
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360 |
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361 lemma uu: |
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362 assumes "(c # s) \<in> r \<rightarrow> injval r c v" "\<Turnstile> v : (der c r)" |
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363 shows "s \<in> der c r \<rightarrow> v" |
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364 using assms |
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365 apply - |
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366 apply(subgoal_tac "lexer r (c # s) = Some (injval r c v)") |
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367 prefer 2 |
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368 using lexer_correctness(1) apply blast |
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369 apply(simp add: ) |
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370 apply(case_tac "lexer (der c r) s") |
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371 apply(simp) |
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372 apply(simp) |
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373 apply(case_tac "s \<in> der c r \<rightarrow> a") |
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374 prefer 2 |
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375 apply (simp add: lexer_correctness(1)) |
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376 apply(subgoal_tac "\<Turnstile> a : (der c r)") |
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377 prefer 2 |
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378 using Posix_Prf apply blast |
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379 using injval_inj by blast |
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380 |
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381 |
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382 lemma Posix_flex2: |
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383 assumes "(s1 @ s2) \<in> r \<rightarrow> flex r id s1 v" "\<Turnstile> v : ders s1 r" |
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384 shows "s2 \<in> (ders s1 r) \<rightarrow> v" |
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385 using assms |
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386 apply(induct s1 arbitrary: r v s2 rule: rev_induct) |
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387 apply(simp) |
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388 apply(simp) |
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389 apply(drule_tac x="r" in meta_spec) |
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390 apply(drule_tac x="injval (ders xs r) x v" in meta_spec) |
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391 apply(drule_tac x="x#s2" in meta_spec) |
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392 apply(simp add: flex_append ders_append) |
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393 using Prf_injval uu by blast |
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394 |
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395 lemma Posix_flex3: |
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396 assumes "s1 \<in> r \<rightarrow> flex r id s1 v" "\<Turnstile> v : ders s1 r" |
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397 shows "[] \<in> (ders s1 r) \<rightarrow> v" |
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398 using assms |
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399 by (simp add: Posix_flex2) |
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400 |
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401 lemma flex_injval: |
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402 shows "flex (der a r) (injval r a) s v = injval r a (flex (der a r) id s v)" |
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403 by (simp add: flex_fun_apply) |
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404 |
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405 lemma Prf_flex: |
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406 assumes "\<Turnstile> v : ders s r" |
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407 shows "\<Turnstile> flex r id s v : r" |
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408 using assms |
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409 apply(induct s arbitrary: v r) |
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410 apply(simp) |
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411 apply(simp) |
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412 by (simp add: Prf_injval flex_injval) |
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413 |
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414 |
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415 unused_thms |
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416 |
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417 end |