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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 \<open>The Lexer Functions by Sulzmann and Lu (without simplification)\<close> |
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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 | "mkeps(NTIMES r n) = Stars (replicate n (mkeps r))" |
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16 |
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17 fun injval :: "rexp \<Rightarrow> char \<Rightarrow> val \<Rightarrow> val" |
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18 where |
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19 "injval (CH d) c Void = Char d" |
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20 | "injval (ALT r1 r2) c (Left v1) = Left(injval r1 c v1)" |
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21 | "injval (ALT r1 r2) c (Right v2) = Right(injval r2 c v2)" |
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22 | "injval (SEQ r1 r2) c (Seq v1 v2) = Seq (injval r1 c v1) v2" |
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23 | "injval (SEQ r1 r2) c (Left (Seq v1 v2)) = Seq (injval r1 c v1) v2" |
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24 | "injval (SEQ r1 r2) c (Right v2) = Seq (mkeps r1) (injval r2 c v2)" |
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25 | "injval (STAR r) c (Seq v (Stars vs)) = Stars ((injval r c v) # vs)" |
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26 | "injval (NTIMES r n) c (Seq v (Stars vs)) = Stars ((injval r c v) # vs)" |
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27 |
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28 fun |
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29 lexer :: "rexp \<Rightarrow> string \<Rightarrow> val option" |
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30 where |
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31 "lexer r [] = (if nullable r then Some(mkeps r) else None)" |
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32 | "lexer r (c#s) = (case (lexer (der c r) s) of |
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33 None \<Rightarrow> None |
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34 | Some(v) \<Rightarrow> Some(injval r c v))" |
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35 |
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36 |
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37 |
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38 section \<open>Mkeps, Injval Properties\<close> |
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39 |
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40 lemma mkeps_flat: |
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41 assumes "nullable(r)" |
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42 shows "flat (mkeps r) = []" |
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43 using assms |
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44 by (induct rule: mkeps.induct) (auto) |
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45 |
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46 lemma Prf_NTimes_empty: |
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47 assumes "\<forall>v \<in> set vs. \<Turnstile> v : r \<and> flat v = []" |
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48 and "length vs = n" |
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49 shows "\<Turnstile> Stars vs : NTIMES r n" |
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50 using assms |
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51 by (metis Prf.intros(7) empty_iff eq_Nil_appendI list.set(1)) |
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52 |
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53 |
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54 lemma mkeps_nullable: |
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55 assumes "nullable(r)" |
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56 shows "\<Turnstile> mkeps r : r" |
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57 using assms |
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58 apply (induct rule: mkeps.induct) |
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59 apply(auto intro: Prf.intros split: if_splits) |
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60 apply (metis Prf.intros(7) append_is_Nil_conv empty_iff list.set(1) list.size(3)) |
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61 apply(rule Prf_NTimes_empty) |
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62 apply(auto simp add: mkeps_flat) |
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63 done |
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64 |
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65 lemma Prf_injval_flat: |
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66 assumes "\<Turnstile> v : der c r" |
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67 shows "flat (injval r c v) = c # (flat v)" |
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68 using assms |
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69 apply(induct c r arbitrary: v rule: der.induct) |
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70 apply(auto elim!: Prf_elims intro: mkeps_flat split: if_splits) |
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71 done |
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72 |
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73 lemma Prf_injval: |
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74 assumes "\<Turnstile> v : der c r" |
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75 shows "\<Turnstile> (injval r c v) : r" |
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76 using assms |
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77 apply(induct r arbitrary: c v rule: rexp.induct) |
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78 apply(auto intro!: Prf.intros mkeps_nullable elim!: Prf_elims split: if_splits) |
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79 (* Star *) |
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80 apply(simp add: Prf_injval_flat) |
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81 (* NTimes *) |
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82 apply(case_tac x2) |
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83 apply(simp) |
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84 apply(simp) |
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85 apply(subst append.simps(2)[symmetric]) |
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86 apply(rule Prf.intros) |
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87 apply(auto simp add: Prf_injval_flat) |
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88 done |
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89 |
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90 |
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91 text \<open>Mkeps and injval produce, or preserve, Posix values.\<close> |
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92 lemma mkepsPosixSeq_pf2: |
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93 shows " \<And>x1 x2 s v. \<lbrakk>\<And>s v. s \<in> der c x1 \<rightarrow> v \<Longrightarrow> (c # s) \<in> x1 \<rightarrow> injval x1 c v; |
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94 \<And>s v. s \<in> der c x2 \<rightarrow> v \<Longrightarrow> (c # s) \<in> x2 \<rightarrow> injval x2 c v; |
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95 s \<in> der c (SEQ x1 x2) \<rightarrow> v\<rbrakk> \<Longrightarrow> (c # s) \<in> (SEQ x1 x2) \<rightarrow> (injval (SEQ x1 x2) c v) " |
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96 apply(case_tac "v ") |
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97 |
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98 apply (metis Posix1a Posix_elims(4) Prf_elims(2) der.simps(5) val.distinct(3) val.distinct(5) val.distinct(7)) |
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99 |
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100 apply (metis Posix1a Posix_elims(4) Prf_elims(2) der.simps(5) val.distinct(11) val.distinct(13) val.distinct(15)) |
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101 sorry |
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102 |
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103 lemma Posix_mkeps: |
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104 assumes "nullable r" |
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105 shows "[] \<in> r \<rightarrow> mkeps r" |
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106 using assms |
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107 apply(induct r rule: nullable.induct) |
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108 apply(auto intro: Posix.intros simp add: nullable_correctness Sequ_def) |
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109 apply(subst append.simps(1)[symmetric]) |
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110 apply(rule Posix.intros) |
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111 apply(auto) |
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112 by (simp add: Posix_NTIMES2 pow_empty_iff) |
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113 |
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114 lemma Posix_injval: |
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115 assumes "s \<in> (der c r) \<rightarrow> v" |
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116 shows "(c # s) \<in> r \<rightarrow> (injval r c v)" |
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117 using assms |
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118 proof(induct r arbitrary: s v rule: rexp.induct) |
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119 case ZERO |
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120 have "s \<in> der c ZERO \<rightarrow> v" by fact |
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121 then have "s \<in> ZERO \<rightarrow> v" by simp |
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122 then have "False" by cases |
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123 then show "(c # s) \<in> ZERO \<rightarrow> (injval ZERO c v)" by simp |
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124 next |
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125 case ONE |
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126 have "s \<in> der c ONE \<rightarrow> v" by fact |
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127 then have "s \<in> ZERO \<rightarrow> v" by simp |
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128 then have "False" by cases |
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129 then show "(c # s) \<in> ONE \<rightarrow> (injval ONE c v)" by simp |
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130 next |
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131 case (CH d) |
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132 consider (eq) "c = d" | (ineq) "c \<noteq> d" by blast |
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133 then show "(c # s) \<in> (CH d) \<rightarrow> (injval (CH d) c v)" |
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134 proof (cases) |
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135 case eq |
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136 have "s \<in> der c (CH d) \<rightarrow> v" by fact |
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137 then have "s \<in> ONE \<rightarrow> v" using eq by simp |
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138 then have eqs: "s = [] \<and> v = Void" by cases simp |
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139 show "(c # s) \<in> CH d \<rightarrow> injval (CH d) c v" using eq eqs |
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140 by (auto intro: Posix.intros) |
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141 next |
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142 case ineq |
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143 have "s \<in> der c (CH d) \<rightarrow> v" by fact |
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144 then have "s \<in> ZERO \<rightarrow> v" using ineq by simp |
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145 then have "False" by cases |
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146 then show "(c # s) \<in> CH d \<rightarrow> injval (CH d) c v" by simp |
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147 qed |
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148 next |
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149 case (ALT r1 r2) |
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150 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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151 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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152 have "s \<in> der c (ALT r1 r2) \<rightarrow> v" by fact |
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153 then have "s \<in> ALT (der c r1) (der c r2) \<rightarrow> v" by simp |
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154 then consider (left) v' where "v = Left v'" "s \<in> der c r1 \<rightarrow> v'" |
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155 | (right) v' where "v = Right v'" "s \<notin> L (der c r1)" "s \<in> der c r2 \<rightarrow> v'" |
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156 by cases auto |
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157 then show "(c # s) \<in> ALT r1 r2 \<rightarrow> injval (ALT r1 r2) c v" |
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158 proof (cases) |
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159 case left |
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160 have "s \<in> der c r1 \<rightarrow> v'" by fact |
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161 then have "(c # s) \<in> r1 \<rightarrow> injval r1 c v'" using IH1 by simp |
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162 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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163 then show "(c # s) \<in> ALT r1 r2 \<rightarrow> injval (ALT r1 r2) c v" using left by simp |
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164 next |
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165 case right |
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166 have "s \<notin> L (der c r1)" by fact |
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167 then have "c # s \<notin> L r1" by (simp add: der_correctness Der_def) |
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168 moreover |
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169 have "s \<in> der c r2 \<rightarrow> v'" by fact |
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170 then have "(c # s) \<in> r2 \<rightarrow> injval r2 c v'" using IH2 by simp |
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171 ultimately have "(c # s) \<in> ALT r1 r2 \<rightarrow> injval (ALT r1 r2) c (Right v')" |
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172 by (auto intro: Posix.intros) |
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173 then show "(c # s) \<in> ALT r1 r2 \<rightarrow> injval (ALT r1 r2) c v" using right by simp |
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174 qed |
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175 next |
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176 case (SEQ r1 r2) |
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177 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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178 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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179 have "s \<in> der c (SEQ r1 r2) \<rightarrow> v" by fact |
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180 then consider |
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181 (left_nullable) v1 v2 s1 s2 where |
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182 "v = Left (Seq v1 v2)" "s = s1 @ s2" |
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183 "s1 \<in> der c r1 \<rightarrow> v1" "s2 \<in> r2 \<rightarrow> v2" "nullable r1" |
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184 "\<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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185 | (right_nullable) v1 s1 s2 where |
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186 "v = Right v1" "s = s1 @ s2" |
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187 "s \<in> der c r2 \<rightarrow> v1" "nullable r1" "s1 @ s2 \<notin> L (SEQ (der c r1) r2)" |
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188 | (not_nullable) v1 v2 s1 s2 where |
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189 "v = Seq v1 v2" "s = s1 @ s2" |
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190 "s1 \<in> der c r1 \<rightarrow> v1" "s2 \<in> r2 \<rightarrow> v2" "\<not>nullable r1" |
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191 "\<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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192 by (force split: if_splits elim!: Posix_elims simp add: Sequ_def der_correctness Der_def) |
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193 then show "(c # s) \<in> SEQ r1 r2 \<rightarrow> injval (SEQ r1 r2) c v" |
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194 proof (cases) |
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195 case left_nullable |
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196 have "s1 \<in> der c r1 \<rightarrow> v1" by fact |
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197 then have "(c # s1) \<in> r1 \<rightarrow> injval r1 c v1" using IH1 by simp |
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198 moreover |
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199 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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200 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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201 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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202 then show "(c # s) \<in> SEQ r1 r2 \<rightarrow> injval (SEQ r1 r2) c v" using left_nullable by simp |
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203 next |
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204 case right_nullable |
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205 have "nullable r1" by fact |
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206 then have "[] \<in> r1 \<rightarrow> (mkeps r1)" by (rule Posix_mkeps) |
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207 moreover |
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208 have "s \<in> der c r2 \<rightarrow> v1" by fact |
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209 then have "(c # s) \<in> r2 \<rightarrow> (injval r2 c v1)" using IH2 by simp |
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210 moreover |
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211 have "s1 @ s2 \<notin> L (SEQ (der c r1) r2)" by fact |
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212 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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213 by(auto simp add: der_correctness Der_def append_eq_Cons_conv Sequ_def) |
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214 ultimately have "([] @ (c # s)) \<in> SEQ r1 r2 \<rightarrow> Seq (mkeps r1) (injval r2 c v1)" |
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215 by(rule Posix.intros) |
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216 then show "(c # s) \<in> SEQ r1 r2 \<rightarrow> injval (SEQ r1 r2) c v" using right_nullable by simp |
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217 next |
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218 case not_nullable |
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219 have "s1 \<in> der c r1 \<rightarrow> v1" by fact |
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220 then have "(c # s1) \<in> r1 \<rightarrow> injval r1 c v1" using IH1 by simp |
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221 moreover |
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222 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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223 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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224 ultimately have "((c # s1) @ s2) \<in> SEQ r1 r2 \<rightarrow> Seq (injval r1 c v1) v2" using not_nullable |
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225 by (rule_tac Posix.intros) (simp_all) |
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226 then show "(c # s) \<in> SEQ r1 r2 \<rightarrow> injval (SEQ r1 r2) c v" using not_nullable by simp |
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227 qed |
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228 next |
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229 case (STAR r) |
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230 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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231 have "s \<in> der c (STAR r) \<rightarrow> v" by fact |
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232 then consider |
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233 (cons) v1 vs s1 s2 where |
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234 "v = Seq v1 (Stars vs)" "s = s1 @ s2" |
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235 "s1 \<in> der c r \<rightarrow> v1" "s2 \<in> (STAR r) \<rightarrow> (Stars vs)" |
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236 "\<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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237 apply(auto elim!: Posix_elims(1-5) simp add: der_correctness Der_def intro: Posix.intros) |
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238 apply(rotate_tac 3) |
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239 apply(erule_tac Posix_elims(6)) |
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240 apply (simp add: Posix.intros(6)) |
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241 using Posix.intros(7) by blast |
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242 then show "(c # s) \<in> STAR r \<rightarrow> injval (STAR r) c v" |
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243 proof (cases) |
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244 case cons |
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245 have "s1 \<in> der c r \<rightarrow> v1" by fact |
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246 then have "(c # s1) \<in> r \<rightarrow> injval r c v1" using IH by simp |
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247 moreover |
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248 have "s2 \<in> STAR r \<rightarrow> Stars vs" by fact |
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249 moreover |
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250 have "(c # s1) \<in> r \<rightarrow> injval r c v1" by fact |
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251 then have "flat (injval r c v1) = (c # s1)" by (rule Posix1) |
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252 then have "flat (injval r c v1) \<noteq> []" by simp |
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253 moreover |
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254 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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255 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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256 by (simp add: der_correctness Der_def) |
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257 ultimately |
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258 have "((c # s1) @ s2) \<in> STAR r \<rightarrow> Stars (injval r c v1 # vs)" by (rule Posix.intros) |
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259 then show "(c # s) \<in> STAR r \<rightarrow> injval (STAR r) c v" using cons by(simp) |
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260 qed |
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261 next |
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262 case (NTIMES r n) |
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263 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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264 have "s \<in> der c (NTIMES r n) \<rightarrow> v" by fact |
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265 then consider |
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266 (cons) v1 vs s1 s2 where |
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267 "v = Seq v1 (Stars vs)" "s = s1 @ s2" |
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268 "s1 \<in> der c r \<rightarrow> v1" "s2 \<in> (NTIMES r (n - 1)) \<rightarrow> (Stars vs)" "0 < n" |
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269 "\<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 (NTIMES r (n - 1)))" |
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270 |
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271 apply(auto elim: Posix_elims simp add: der_correctness Der_def intro: Posix.intros split: if_splits) |
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272 apply(erule Posix_elims) |
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273 apply(simp) |
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274 apply(subgoal_tac "\<exists>vss. v2 = Stars vss") |
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275 apply(clarify) |
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276 apply(drule_tac x="vss" in meta_spec) |
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277 apply(drule_tac x="s1" in meta_spec) |
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278 apply(drule_tac x="s2" in meta_spec) |
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279 apply(simp add: der_correctness Der_def) |
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280 apply(erule Posix_elims) |
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281 apply(auto) |
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282 done |
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283 then show "(c # s) \<in> (NTIMES r n) \<rightarrow> injval (NTIMES r n) c v" |
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284 proof (cases) |
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285 case cons |
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286 have "s1 \<in> der c r \<rightarrow> v1" by fact |
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287 then have "(c # s1) \<in> r \<rightarrow> injval r c v1" using IH by simp |
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288 moreover |
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289 have "s2 \<in> (NTIMES r (n - 1)) \<rightarrow> Stars vs" by fact |
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290 moreover |
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291 have "(c # s1) \<in> r \<rightarrow> injval r c v1" by fact |
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292 then have "flat (injval r c v1) = (c # s1)" by (rule Posix1) |
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293 then have "flat (injval r c v1) \<noteq> []" by simp |
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294 moreover |
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295 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 (NTIMES r (n - 1)))" by fact |
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296 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 (NTIMES r (n - 1)))" |
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297 by (simp add: der_correctness Der_def) |
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298 ultimately |
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299 have "((c # s1) @ s2) \<in> NTIMES r n \<rightarrow> Stars (injval r c v1 # vs)" |
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300 apply (rule_tac Posix.intros) |
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301 apply(simp_all) |
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302 apply(case_tac n) |
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303 apply(simp) |
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304 using Posix_elims(1) NTIMES.prems apply auto[1] |
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305 apply(simp) |
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306 done |
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307 then show "(c # s) \<in> NTIMES r n \<rightarrow> injval (NTIMES r n) c v" using cons by(simp) |
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308 qed |
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309 |
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310 qed |
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311 |
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312 |
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313 section \<open>Lexer Correctness\<close> |
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314 |
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315 |
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316 lemma lexer_correct_None: |
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317 shows "s \<notin> L r \<longleftrightarrow> lexer r s = None" |
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318 apply(induct s arbitrary: r) |
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319 apply(simp) |
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320 apply(simp add: nullable_correctness) |
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321 apply(simp) |
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322 apply(drule_tac x="der a r" in meta_spec) |
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323 apply(auto) |
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324 apply(auto simp add: der_correctness Der_def) |
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325 done |
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326 |
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327 lemma lexer_correct_Some: |
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328 shows "s \<in> L r \<longleftrightarrow> (\<exists>v. lexer r s = Some(v) \<and> s \<in> r \<rightarrow> v)" |
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329 apply(induct s arbitrary : r) |
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330 apply(simp only: lexer.simps) |
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331 apply(simp) |
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332 apply(simp add: nullable_correctness Posix_mkeps) |
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333 apply(drule_tac x="der a r" in meta_spec) |
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334 apply(simp (no_asm_use) add: der_correctness Der_def del: lexer.simps) |
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335 apply(simp del: lexer.simps) |
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336 apply(simp only: lexer.simps) |
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337 apply(case_tac "lexer (der a r) s = None") |
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338 apply(auto)[1] |
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339 apply(simp) |
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340 apply(erule exE) |
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341 apply(simp) |
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342 apply(rule iffI) |
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343 apply(simp add: Posix_injval) |
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344 apply(simp add: Posix1(1)) |
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345 done |
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346 |
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347 lemma lexer_correctness: |
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348 shows "(lexer r s = Some v) \<longleftrightarrow> s \<in> r \<rightarrow> v" |
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349 and "(lexer r s = None) \<longleftrightarrow> \<not>(\<exists>v. s \<in> r \<rightarrow> v)" |
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350 using Posix1(1) Posix_determ lexer_correct_None lexer_correct_Some apply fastforce |
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351 using Posix1(1) lexer_correct_None lexer_correct_Some by blast |
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352 |
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353 |
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354 subsection {* A slight reformulation of the lexer algorithm using stacked functions*} |
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355 |
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356 fun flex :: "rexp \<Rightarrow> (val \<Rightarrow> val) => string \<Rightarrow> (val \<Rightarrow> val)" |
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357 where |
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358 "flex r f [] = f" |
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359 | "flex r f (c#s) = flex (der c r) (\<lambda>v. f (injval r c v)) s" |
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360 |
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361 lemma flex_fun_apply: |
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362 shows "g (flex r f s v) = flex r (g o f) s v" |
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363 apply(induct s arbitrary: g f r v) |
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364 apply(simp_all add: comp_def) |
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365 by meson |
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366 |
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367 lemma flex_fun_apply2: |
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368 shows "g (flex r id s v) = flex r g s v" |
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369 by (simp add: flex_fun_apply) |
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370 |
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371 |
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372 lemma flex_append: |
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373 shows "flex r f (s1 @ s2) = flex (ders s1 r) (flex r f s1) s2" |
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374 apply(induct s1 arbitrary: s2 r f) |
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375 apply(simp_all) |
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376 done |
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377 |
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378 lemma lexer_flex: |
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379 shows "lexer r s = (if nullable (ders s r) |
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380 then Some(flex r id s (mkeps (ders s r))) else None)" |
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381 apply(induct s arbitrary: r) |
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382 apply(simp_all add: flex_fun_apply) |
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383 done |
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384 |
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385 lemma Posix_flex: |
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386 assumes "s2 \<in> (ders s1 r) \<rightarrow> v" |
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387 shows "(s1 @ s2) \<in> r \<rightarrow> flex r id s1 v" |
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388 using assms |
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389 apply(induct s1 arbitrary: r v s2) |
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390 apply(simp) |
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391 apply(simp) |
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392 apply(drule_tac x="der a r" in meta_spec) |
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393 apply(drule_tac x="v" in meta_spec) |
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394 apply(drule_tac x="s2" in meta_spec) |
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395 apply(simp) |
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396 using Posix_injval |
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397 apply(drule_tac Posix_injval) |
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398 apply(subst (asm) (5) flex_fun_apply) |
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399 apply(simp) |
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400 done |
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401 |
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402 lemma injval_inj: |
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403 assumes "\<Turnstile> a : (der c r)" "\<Turnstile> v : (der c r)" "injval r c a = injval r c v" |
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404 shows "a = v" |
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405 using assms |
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406 apply(induct r arbitrary: a c v) |
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407 apply(auto) |
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408 using Prf_elims(1) apply blast |
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409 using Prf_elims(1) apply blast |
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410 apply(case_tac "c = x") |
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411 apply(auto) |
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412 using Prf_elims(4) apply auto[1] |
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413 using Prf_elims(1) apply blast |
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414 prefer 2 |
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415 apply (smt Prf_elims(3) injval.simps(2) injval.simps(3) val.distinct(25) val.inject(3) val.inject(4)) |
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416 apply(case_tac "nullable r1") |
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417 apply(auto) |
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418 apply(erule Prf_elims) |
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419 apply(erule Prf_elims) |
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420 apply(erule Prf_elims) |
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421 apply(erule Prf_elims) |
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422 apply(auto) |
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423 apply (metis Prf_injval_flat list.distinct(1) mkeps_flat) |
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424 apply(erule Prf_elims) |
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425 apply(erule Prf_elims) |
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426 apply(auto) |
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427 using Prf_injval_flat mkeps_flat apply fastforce |
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428 apply(erule Prf_elims) |
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429 apply(erule Prf_elims) |
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430 apply(auto) |
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431 apply(erule Prf_elims) |
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432 apply(erule Prf_elims) |
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433 apply(auto) |
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434 apply (smt Prf_elims(6) injval.simps(7) list.inject val.inject(5)) |
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435 apply (smt Prf_elims(6) injval.simps(7) list.inject val.inject(5)) |
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436 by (smt (verit, best) Prf_elims(1) Prf_elims(2) Prf_elims(7) injval.simps(8) list.inject val.simps(5)) |
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437 |
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438 |
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439 |
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440 lemma uu: |
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441 assumes "(c # s) \<in> r \<rightarrow> injval r c v" "\<Turnstile> v : (der c r)" |
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442 shows "s \<in> der c r \<rightarrow> v" |
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443 using assms |
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444 apply - |
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445 apply(subgoal_tac "lexer r (c # s) = Some (injval r c v)") |
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446 prefer 2 |
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447 using lexer_correctness(1) apply blast |
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448 apply(simp add: ) |
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449 apply(case_tac "lexer (der c r) s") |
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450 apply(simp) |
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451 apply(simp) |
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452 apply(case_tac "s \<in> der c r \<rightarrow> a") |
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453 prefer 2 |
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454 apply (simp add: lexer_correctness(1)) |
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455 apply(subgoal_tac "\<Turnstile> a : (der c r)") |
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456 prefer 2 |
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457 using Posix1a apply blast |
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458 using injval_inj by blast |
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459 |
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460 |
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461 lemma Posix_flex2: |
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462 assumes "(s1 @ s2) \<in> r \<rightarrow> flex r id s1 v" "\<Turnstile> v : ders s1 r" |
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463 shows "s2 \<in> (ders s1 r) \<rightarrow> v" |
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464 using assms |
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465 apply(induct s1 arbitrary: r v s2 rule: rev_induct) |
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466 apply(simp) |
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467 apply(simp) |
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468 apply(drule_tac x="r" in meta_spec) |
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469 apply(drule_tac x="injval (ders xs r) x v" in meta_spec) |
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470 apply(drule_tac x="x#s2" in meta_spec) |
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471 apply(simp add: flex_append ders_append) |
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472 using Prf_injval uu by blast |
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473 |
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474 lemma Posix_flex3: |
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475 assumes "s1 \<in> r \<rightarrow> flex r id s1 v" "\<Turnstile> v : ders s1 r" |
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476 shows "[] \<in> (ders s1 r) \<rightarrow> v" |
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477 using assms |
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478 by (simp add: Posix_flex2) |
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479 |
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480 lemma flex_injval: |
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481 shows "flex (der a r) (injval r a) s v = injval r a (flex (der a r) id s v)" |
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482 by (simp add: flex_fun_apply) |
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483 |
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484 lemma Prf_flex: |
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485 assumes "\<Turnstile> v : ders s r" |
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486 shows "\<Turnstile> flex r id s v : r" |
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487 using assms |
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488 apply(induct s arbitrary: v r) |
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489 apply(simp) |
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490 apply(simp) |
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491 by (simp add: Prf_injval flex_injval) |
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492 |
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493 |
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494 unused_thms |
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495 |
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496 end |