2
+ − 1 theory MyhilNerode
0
+ − 2 imports "Main"
+ − 3 begin
+ − 4
+ − 5 text {* sequential composition of languages *}
+ − 6
+ − 7 definition
+ − 8 lang_seq :: "string set \<Rightarrow> string set \<Rightarrow> string set" ("_ ; _" [100,100] 100)
+ − 9 where
+ − 10 "L1 ; L2 = {s1@s2 | s1 s2. s1 \<in> L1 \<and> s2 \<in> L2}"
+ − 11
+ − 12 lemma lang_seq_empty:
+ − 13 shows "{[]} ; L = L"
+ − 14 and "L ; {[]} = L"
+ − 15 unfolding lang_seq_def by auto
+ − 16
+ − 17 lemma lang_seq_null:
+ − 18 shows "{} ; L = {}"
+ − 19 and "L ; {} = {}"
+ − 20 unfolding lang_seq_def by auto
+ − 21
+ − 22 lemma lang_seq_append:
+ − 23 assumes a: "s1 \<in> L1"
+ − 24 and b: "s2 \<in> L2"
+ − 25 shows "s1@s2 \<in> L1 ; L2"
+ − 26 unfolding lang_seq_def
+ − 27 using a b by auto
+ − 28
+ − 29 lemma lang_seq_union:
+ − 30 shows "(L1 \<union> L2); L3 = (L1; L3) \<union> (L2; L3)"
+ − 31 and "L1; (L2 \<union> L3) = (L1; L2) \<union> (L1; L3)"
+ − 32 unfolding lang_seq_def by auto
+ − 33
+ − 34 lemma lang_seq_assoc:
+ − 35 shows "(L1 ; L2) ; L3 = L1 ; (L2 ; L3)"
+ − 36 by (simp add: lang_seq_def Collect_def mem_def expand_fun_eq)
+ − 37 (metis append_assoc)
+ − 38
+ − 39 lemma lang_seq_minus:
+ − 40 shows "(L1; L2) - {[]} =
+ − 41 (if [] \<in> L1 then L2 - {[]} else {}) \<union>
+ − 42 (if [] \<in> L2 then L1 - {[]} else {}) \<union> ((L1 - {[]}); (L2 - {[]}))"
+ − 43 apply(auto simp add: lang_seq_def)
+ − 44 apply(metis mem_def self_append_conv)
+ − 45 apply(metis mem_def self_append_conv2)
+ − 46 apply(metis mem_def self_append_conv2)
+ − 47 apply(metis mem_def self_append_conv)
+ − 48 done
+ − 49
+ − 50 section {* Kleene star for languages defined as least fixed point *}
+ − 51
+ − 52 inductive_set
+ − 53 Star :: "string set \<Rightarrow> string set" ("_\<star>" [101] 102)
+ − 54 for L :: "string set"
+ − 55 where
+ − 56 start[intro]: "[] \<in> L\<star>"
+ − 57 | step[intro]: "\<lbrakk>s1 \<in> L; s2 \<in> L\<star>\<rbrakk> \<Longrightarrow> s1@s2 \<in> L\<star>"
+ − 58
+ − 59 lemma lang_star_empty:
+ − 60 shows "{}\<star> = {[]}"
+ − 61 by (auto elim: Star.cases)
+ − 62
+ − 63 lemma lang_star_cases:
+ − 64 shows "L\<star> = {[]} \<union> L ; L\<star>"
+ − 65 proof
+ − 66 { fix x
+ − 67 have "x \<in> L\<star> \<Longrightarrow> x \<in> {[]} \<union> L ; L\<star>"
+ − 68 unfolding lang_seq_def
+ − 69 by (induct rule: Star.induct) (auto)
+ − 70 }
+ − 71 then show "L\<star> \<subseteq> {[]} \<union> L ; L\<star>" by auto
+ − 72 next
+ − 73 show "{[]} \<union> L ; L\<star> \<subseteq> L\<star>"
+ − 74 unfolding lang_seq_def by auto
+ − 75 qed
+ − 76
+ − 77 lemma lang_star_cases':
+ − 78 shows "L\<star> = {[]} \<union> L\<star> ; L"
+ − 79 proof
+ − 80 { fix x
+ − 81 have "x \<in> L\<star> \<Longrightarrow> x \<in> {[]} \<union> L\<star> ; L"
+ − 82 unfolding lang_seq_def
+ − 83 apply (induct rule: Star.induct)
+ − 84 apply simp
+ − 85 apply simp
+ − 86 apply (erule disjE)
+ − 87 apply (auto)[1]
+ − 88 apply (erule exE | erule conjE)+
+ − 89 apply (rule disjI2)
+ − 90 apply (rule_tac x = "s1 @ s1a" in exI)
+ − 91 by auto
+ − 92 }
+ − 93 then show "L\<star> \<subseteq> {[]} \<union> L\<star> ; L" by auto
+ − 94 next
+ − 95 show "{[]} \<union> L\<star> ; L \<subseteq> L\<star>"
+ − 96 unfolding lang_seq_def
+ − 97 apply auto
+ − 98 apply (erule Star.induct)
+ − 99 apply auto
+ − 100 apply (drule step[of _ _ "[]"])
+ − 101 by (auto intro:start)
+ − 102 qed
+ − 103
+ − 104 lemma lang_star_simple:
+ − 105 shows "L \<subseteq> L\<star>"
+ − 106 by (subst lang_star_cases)
+ − 107 (auto simp only: lang_seq_def)
+ − 108
+ − 109 lemma lang_star_prop0_aux:
+ − 110 "s2 \<in> L\<star> \<Longrightarrow> \<forall> s1. s1 \<in> L \<longrightarrow> (\<exists> s3 s4. s3 \<in> L\<star> \<and> s4 \<in> L \<and> s1 @ s2 = s3 @ s4)"
+ − 111 apply (erule Star.induct)
+ − 112 apply (clarify, rule_tac x = "[]" in exI, rule_tac x = s1 in exI, simp add:start)
+ − 113 apply (clarify, drule_tac x = s1 in spec)
+ − 114 apply (drule mp, simp, clarify)
+ − 115 apply (rule_tac x = "s1a @ s3" in exI, rule_tac x = s4 in exI)
+ − 116 by auto
+ − 117
+ − 118 lemma lang_star_prop0:
+ − 119 "\<lbrakk>s1 \<in> L; s2 \<in> L\<star>\<rbrakk> \<Longrightarrow> \<exists> s3 s4. s3 \<in> L\<star> \<and> s4 \<in> L \<and> s1 @ s2 = s3 @ s4"
+ − 120 by (auto dest:lang_star_prop0_aux)
+ − 121
+ − 122 lemma lang_star_prop1:
+ − 123 assumes asm: "L1; L2 \<subseteq> L2"
+ − 124 shows "L1\<star>; L2 \<subseteq> L2"
+ − 125 proof -
+ − 126 { fix s1 s2
+ − 127 assume minor: "s2 \<in> L2"
+ − 128 assume major: "s1 \<in> L1\<star>"
+ − 129 then have "s1@s2 \<in> L2"
+ − 130 proof(induct rule: Star.induct)
+ − 131 case start
+ − 132 show "[]@s2 \<in> L2" using minor by simp
+ − 133 next
+ − 134 case (step s1 s1')
+ − 135 have "s1 \<in> L1" by fact
+ − 136 moreover
+ − 137 have "s1'@s2 \<in> L2" by fact
+ − 138 ultimately have "s1@(s1'@s2) \<in> L1; L2" by (auto simp add: lang_seq_def)
+ − 139 with asm have "s1@(s1'@s2) \<in> L2" by auto
+ − 140 then show "(s1@s1')@s2 \<in> L2" by simp
+ − 141 qed
+ − 142 }
+ − 143 then show "L1\<star>; L2 \<subseteq> L2" by (auto simp add: lang_seq_def)
+ − 144 qed
+ − 145
+ − 146 lemma lang_star_prop2_aux:
+ − 147 "s2 \<in> L2\<star> \<Longrightarrow> \<forall> s1. s1 \<in> L1 \<and> L1 ; L2 \<subseteq> L1 \<longrightarrow> s1 @ s2 \<in> L1"
+ − 148 apply (erule Star.induct, simp)
+ − 149 apply (clarify, drule_tac x = "s1a @ s1" in spec)
+ − 150 by (auto simp:lang_seq_def)
+ − 151
+ − 152 lemma lang_star_prop2:
+ − 153 "L1; L2 \<subseteq> L1 \<Longrightarrow> L1 ; L2\<star> \<subseteq> L1"
+ − 154 by (auto dest!:lang_star_prop2_aux simp:lang_seq_def)
+ − 155
+ − 156 lemma lang_star_seq_subseteq:
+ − 157 shows "L ; L\<star> \<subseteq> L\<star>"
+ − 158 using lang_star_cases by blast
+ − 159
+ − 160 lemma lang_star_double:
+ − 161 shows "L\<star>; L\<star> = L\<star>"
+ − 162 proof
+ − 163 show "L\<star> ; L\<star> \<subseteq> L\<star>"
+ − 164 using lang_star_prop1 lang_star_seq_subseteq by blast
+ − 165 next
+ − 166 have "L\<star> \<subseteq> L\<star> \<union> L\<star>; (L; L\<star>)" by auto
+ − 167 also have "\<dots> = L\<star>;{[]} \<union> L\<star>; (L; L\<star>)" by (simp add: lang_seq_empty)
+ − 168 also have "\<dots> = L\<star>; ({[]} \<union> L; L\<star>)" by (simp only: lang_seq_union)
+ − 169 also have "\<dots> = L\<star>; L\<star>" using lang_star_cases by simp
+ − 170 finally show "L\<star> \<subseteq> L\<star> ; L\<star>" by simp
+ − 171 qed
+ − 172
+ − 173 lemma lang_star_seq_subseteq':
+ − 174 shows "L\<star>; L \<subseteq> L\<star>"
+ − 175 proof -
+ − 176 have "L \<subseteq> L\<star>" by (rule lang_star_simple)
+ − 177 then have "L\<star>; L \<subseteq> L\<star>; L\<star>" by (auto simp add: lang_seq_def)
+ − 178 then show "L\<star>; L \<subseteq> L\<star>" using lang_star_double by blast
+ − 179 qed
+ − 180
+ − 181 lemma
+ − 182 shows "L\<star> \<subseteq> L\<star>\<star>"
+ − 183 by (rule lang_star_simple)
+ − 184
+ − 185 section {* tricky section *}
+ − 186
+ − 187 lemma k1:
+ − 188 assumes b: "s \<in> L\<star>"
+ − 189 and a: "s \<noteq> []"
+ − 190 shows "s \<in> (L - {[]}); L\<star>"
+ − 191 using b a
+ − 192 apply(induct rule: Star.induct)
+ − 193 apply(simp)
+ − 194 apply(case_tac "s1=[]")
+ − 195 apply(simp)
+ − 196 apply(simp add: lang_seq_def)
+ − 197 apply(blast)
+ − 198 done
+ − 199
+ − 200 section {* (relies on lemma k1) *}
+ − 201
+ − 202 lemma zzz:
+ − 203 shows "{s. c#s \<in> L1\<star>} = {s. c#s \<in> L1} ; (L1\<star>)"
+ − 204 apply(auto simp add: lang_seq_def Cons_eq_append_conv)
+ − 205 apply(drule k1)
+ − 206 apply(auto)[1]
+ − 207 apply(auto simp add: lang_seq_def)[1]
+ − 208 apply(rule_tac x="tl s1" in exI)
+ − 209 apply(rule_tac x="s2" in exI)
+ − 210 apply(auto)[1]
+ − 211 apply(auto simp add: Cons_eq_append_conv)[2]
+ − 212 apply(drule lang_seq_append)
+ − 213 apply(assumption)
+ − 214 apply(rotate_tac 1)
+ − 215 apply(drule rev_subsetD)
+ − 216 apply(rule lang_star_seq_subseteq)
+ − 217 apply(simp)
+ − 218 done
+ − 219
+ − 220
+ − 221
+ − 222 section {* regular expressions *}
+ − 223
+ − 224 datatype rexp =
+ − 225 NULL
+ − 226 | EMPTY
+ − 227 | CHAR char
+ − 228 | SEQ rexp rexp
+ − 229 | ALT rexp rexp
+ − 230 | STAR rexp
+ − 231
+ − 232
+ − 233 consts L:: "'a \<Rightarrow> string set"
+ − 234
+ − 235 fun
+ − 236 L_rexp :: "rexp \<Rightarrow> string set"
+ − 237 where
+ − 238 "L_rexp (NULL) = {}"
+ − 239 | "L_rexp (EMPTY) = {[]}"
+ − 240 | "L_rexp (CHAR c) = {[c]}"
+ − 241 | "L_rexp (SEQ r1 r2) = (L_rexp r1) ; (L_rexp r2)"
+ − 242 | "L_rexp (ALT r1 r2) = (L_rexp r1) \<union> (L_rexp r2)"
+ − 243 | "L_rexp (STAR r) = (L_rexp r)\<star>"
+ − 244
+ − 245 defs (overloaded)
+ − 246 l_rexp_abs: "L rexp \<equiv> L_rexp rexp"
+ − 247
+ − 248 declare L_rexp.simps [simp del] L_rexp.simps [folded l_rexp_abs, simp add]
+ − 249
+ − 250 definition
+ − 251 Ls :: "rexp set \<Rightarrow> string set"
+ − 252 where
+ − 253 "Ls R = (\<Union>r\<in>R. (L r))"
+ − 254
+ − 255 lemma Ls_union:
+ − 256 "Ls (R1 \<union> R2) = (Ls R1) \<union> (Ls R2)"
+ − 257 unfolding Ls_def by auto
+ − 258
+ − 259 text {* helper function for termination proofs *}
+ − 260 fun
+ − 261 Left :: "rexp \<Rightarrow> rexp"
+ − 262 where
+ − 263 "Left (SEQ r1 r2) = r1"
+ − 264
+ − 265 text {* dagger function *}
+ − 266
+ − 267 function
+ − 268 dagger :: "rexp \<Rightarrow> char \<Rightarrow> rexp list" ("_ \<dagger> _")
+ − 269 where
+ − 270 c1: "(NULL \<dagger> c) = []"
+ − 271 | c2: "(EMPTY) \<dagger> c = []"
+ − 272 | c3: "(CHAR c') \<dagger> c = (if c = c' then [EMPTY] else [])"
+ − 273 | c4: "(ALT r1 r2) \<dagger> c = r1 \<dagger> c @ r2 \<dagger> c"
+ − 274 | c5: "(SEQ NULL r2) \<dagger> c = []"
+ − 275 | c6: "(SEQ EMPTY r2) \<dagger> c = r2 \<dagger> c"
+ − 276 | c7: "(SEQ (CHAR c') r2) \<dagger> c = (if c = c' then [r2] else [])"
+ − 277 | c8: "(SEQ (SEQ r11 r12) r2) \<dagger> c = (SEQ r11 (SEQ r12 r2)) \<dagger> c"
+ − 278 | c9: "(SEQ (ALT r11 r12) r2) \<dagger> c = (SEQ r11 r2) \<dagger> c @ (SEQ r12 r2) \<dagger> c"
+ − 279 | c10: "(SEQ (STAR r1) r2) \<dagger> c = r2 \<dagger> c @ [SEQ r' (SEQ (STAR r1) r2). r' \<leftarrow> r1 \<dagger> c]"
+ − 280 | c11: "(STAR r) \<dagger> c = [SEQ r' (STAR r) . r' \<leftarrow> r \<dagger> c]"
+ − 281 by (pat_completeness) (auto)
+ − 282
+ − 283 termination dagger
+ − 284 by (relation "measures [\<lambda>(r, c). size r, \<lambda>(r, c). size (Left r)]") (simp_all)
+ − 285
+ − 286 lemma dagger_correctness:
+ − 287 "Ls (set r \<dagger> c) = {s. c#s \<in> L r}"
+ − 288 proof (induct rule: dagger.induct)
+ − 289 case (1 c)
+ − 290 show "Ls (set NULL \<dagger> c) = {s. c#s \<in> L NULL}" by (simp add: Ls_def)
+ − 291 next
+ − 292 case (2 c)
+ − 293 show "Ls (set EMPTY \<dagger> c) = {s. c#s \<in> L EMPTY}" by (simp add: Ls_def)
+ − 294 next
+ − 295 case (3 c' c)
+ − 296 show "Ls (set CHAR c' \<dagger> c) = {s. c#s \<in> L (CHAR c')}" by (simp add: Ls_def)
+ − 297 next
+ − 298 case (4 r1 r2 c)
+ − 299 have ih1: "Ls (set r1 \<dagger> c) = {s. c#s \<in> L r1}" by fact
+ − 300 have ih2: "Ls (set r2 \<dagger> c) = {s. c#s \<in> L r2}" by fact
+ − 301 show "Ls (set ALT r1 r2 \<dagger> c) = {s. c#s \<in> L (ALT r1 r2)}"
+ − 302 by (simp add: Ls_union ih1 ih2 Collect_disj_eq)
+ − 303 next
+ − 304 case (5 r2 c)
+ − 305 show "Ls (set SEQ NULL r2 \<dagger> c) = {s. c#s \<in> L (SEQ NULL r2)}" by (simp add: Ls_def lang_seq_null)
+ − 306 next
+ − 307 case (6 r2 c)
+ − 308 have ih: "Ls (set r2 \<dagger> c) = {s. c#s \<in> L r2}" by fact
+ − 309 show "Ls (set SEQ EMPTY r2 \<dagger> c) = {s. c#s \<in> L (SEQ EMPTY r2)}"
+ − 310 by (simp add: ih lang_seq_empty)
+ − 311 next
+ − 312 case (7 c' r2 c)
+ − 313 show "Ls (set SEQ (CHAR c') r2 \<dagger> c) = {s. c#s \<in> L (SEQ (CHAR c') r2)}"
+ − 314 by (simp add: Ls_def lang_seq_def)
+ − 315 next
+ − 316 case (8 r11 r12 r2 c)
+ − 317 have ih: "Ls (set SEQ r11 (SEQ r12 r2) \<dagger> c) = {s. c#s \<in> L (SEQ r11 (SEQ r12 r2))}" by fact
+ − 318 show "Ls (set SEQ (SEQ r11 r12) r2 \<dagger> c) = {s. c#s \<in> L (SEQ (SEQ r11 r12) r2)}"
+ − 319 by (simp add: ih lang_seq_assoc)
+ − 320 next
+ − 321 case (9 r11 r12 r2 c)
+ − 322 have ih1: "Ls (set SEQ r11 r2 \<dagger> c) = {s. c#s \<in> L (SEQ r11 r2)}" by fact
+ − 323 have ih2: "Ls (set SEQ r12 r2 \<dagger> c) = {s. c#s \<in> L (SEQ r12 r2)}" by fact
+ − 324 show "Ls (set SEQ (ALT r11 r12) r2 \<dagger> c) = {s. c#s \<in> L (SEQ (ALT r11 r12) r2)}"
+ − 325 by (simp add: Ls_union ih1 ih2 lang_seq_union Collect_disj_eq)
+ − 326 next
+ − 327 case (10 r1 r2 c)
+ − 328 have ih2: "Ls (set r2 \<dagger> c) = {s. c#s \<in> L r2}" by fact
+ − 329 have ih1: "Ls (set r1 \<dagger> c) = {s. c#s \<in> L r1}" by fact
+ − 330 have "Ls (set SEQ (STAR r1) r2 \<dagger> c) = Ls (set r2 \<dagger> c) \<union> (Ls (set r1 \<dagger> c); ((L r1)\<star> ; L r2))"
+ − 331 by (auto simp add: lang_seq_def Ls_def)
+ − 332 also have "\<dots> = {s. c#s \<in> L r2} \<union> ({s. c#s \<in> L r1} ; ((L r1)\<star> ; L r2))" using ih1 ih2 by simp
+ − 333 also have "\<dots> = {s. c#s \<in> L r2} \<union> ({s. c#s \<in> L r1} ; (L r1)\<star>) ; L r2" by (simp add: lang_seq_assoc)
+ − 334 also have "\<dots> = {s. c#s \<in> L r2} \<union> {s. c#s \<in> (L r1)\<star>} ; L r2" by (simp add: zzz)
+ − 335 also have "\<dots> = {s. c#s \<in> L r2} \<union> {s. c#s \<in> (L r1)\<star> ; L r2}"
+ − 336 by (auto simp add: lang_seq_def Cons_eq_append_conv)
+ − 337 also have "\<dots> = {s. c#s \<in> (L r1)\<star> ; L r2}"
+ − 338 by (force simp add: lang_seq_def)
+ − 339 finally show "Ls (set SEQ (STAR r1) r2 \<dagger> c) = {s. c#s \<in> L (SEQ (STAR r1) r2)}" by simp
+ − 340 next
+ − 341 case (11 r c)
+ − 342 have ih: "Ls (set r \<dagger> c) = {s. c#s \<in> L r}" by fact
+ − 343 have "Ls (set (STAR r) \<dagger> c) = Ls (set r \<dagger> c) ; (L r)\<star>"
+ − 344 by (auto simp add: lang_seq_def Ls_def)
+ − 345 also have "\<dots> = {s. c#s \<in> L r} ; (L r)\<star>" using ih by simp
+ − 346 also have "\<dots> = {s. c#s \<in> (L r)\<star>}" using zzz by simp
+ − 347 finally show "Ls (set (STAR r) \<dagger> c) = {s. c#s \<in> L (STAR r)}" by simp
+ − 348 qed
+ − 349
+ − 350
+ − 351 text {* matcher function (based on the "list"-dagger function) *}
+ − 352 fun
+ − 353 first_True :: "bool list \<Rightarrow> bool"
+ − 354 where
+ − 355 "first_True [] = False"
+ − 356 | "first_True (x#xs) = (if x then True else first_True xs)"
+ − 357
+ − 358 lemma not_first_True[simp]:
+ − 359 shows "(\<not>(first_True xs)) = (\<forall>x \<in> set xs. \<not>x)"
+ − 360 by (induct xs) (auto)
+ − 361
+ − 362 lemma first_True:
+ − 363 shows "(first_True xs) = (\<exists>x \<in> set xs. x)"
+ − 364 by (induct xs) (auto)
+ − 365
+ − 366 text {* matcher function *}
+ − 367
+ − 368 function
+ − 369 matcher :: "rexp \<Rightarrow> string \<Rightarrow> bool" ("_ ! _")
+ − 370 where
+ − 371 "NULL ! s = False"
+ − 372 | "EMPTY ! s = (s =[])"
+ − 373 | "CHAR c ! s = (s = [c])"
+ − 374 | "ALT r1 r2 ! s = (r1 ! s \<or> r2 ! s)"
+ − 375 | "STAR r ! [] = True"
+ − 376 | "STAR r ! c#s = first_True [SEQ (r') (STAR r) ! s. r' \<leftarrow> r \<dagger> c]"
+ − 377 | "SEQ r1 r2 ! [] = (r1 ! [] \<and> r2 ! [])"
+ − 378 | "SEQ NULL r2 ! (c#s) = False"
+ − 379 | "SEQ EMPTY r2 ! (c#s) = (r2 ! c#s)"
+ − 380 | "SEQ (CHAR c') r2 ! (c#s) = (if c'=c then r2 ! s else False)"
+ − 381 | "SEQ (SEQ r11 r12) r2 ! (c#s) = (SEQ r11 (SEQ r12 r2) ! c#s)"
+ − 382 | "SEQ (ALT r11 r12) r2 ! (c#s) = ((SEQ r11 r2) ! (c#s) \<or> (SEQ r12 r2) ! (c#s))"
+ − 383 | "SEQ (STAR r1) r2 ! (c#s) = (r2 ! (c#s) \<or> first_True [SEQ r' (SEQ (STAR r1) r2) ! s. r' \<leftarrow> r1 \<dagger> c])"
+ − 384 by (pat_completeness) (auto)
+ − 385
+ − 386 termination matcher
+ − 387 by(relation "measures [\<lambda>(r,s). length s, \<lambda>(r,s). size r, \<lambda>(r,s). size (Left r)]") (simp_all)
+ − 388
+ − 389 text {* positive correctness of the matcher *}
+ − 390 lemma matcher1:
+ − 391 shows "r ! s \<Longrightarrow> s \<in> L r"
+ − 392 proof (induct rule: matcher.induct)
+ − 393 case (1 s)
+ − 394 have "NULL ! s" by fact
+ − 395 then show "s \<in> L NULL" by simp
+ − 396 next
+ − 397 case (2 s)
+ − 398 have "EMPTY ! s" by fact
+ − 399 then show "s \<in> L EMPTY" by simp
+ − 400 next
+ − 401 case (3 c s)
+ − 402 have "CHAR c ! s" by fact
+ − 403 then show "s \<in> L (CHAR c)" by simp
+ − 404 next
+ − 405 case (4 r1 r2 s)
+ − 406 have ih1: "r1 ! s \<Longrightarrow> s \<in> L r1" by fact
+ − 407 have ih2: "r2 ! s \<Longrightarrow> s \<in> L r2" by fact
+ − 408 have "ALT r1 r2 ! s" by fact
+ − 409 with ih1 ih2 show "s \<in> L (ALT r1 r2)" by auto
+ − 410 next
+ − 411 case (5 r)
+ − 412 have "STAR r ! []" by fact
+ − 413 then show "[] \<in> L (STAR r)" by auto
+ − 414 next
+ − 415 case (6 r c s)
+ − 416 have ih1: "\<And>rx. \<lbrakk>rx \<in> set r \<dagger> c; SEQ rx (STAR r) ! s\<rbrakk> \<Longrightarrow> s \<in> L (SEQ rx (STAR r))" by fact
+ − 417 have as: "STAR r ! c#s" by fact
+ − 418 then obtain r' where imp1: "r' \<in> set r \<dagger> c" and imp2: "SEQ r' (STAR r) ! s"
+ − 419 by (auto simp add: first_True)
+ − 420 from imp2 imp1 have "s \<in> L (SEQ r' (STAR r))" using ih1 by simp
+ − 421 then have "s \<in> L r' ; (L r)\<star>" by simp
+ − 422 then have "s \<in> Ls (set r \<dagger> c) ; (L r)\<star>" using imp1 by (auto simp add: Ls_def lang_seq_def)
+ − 423 then have "s \<in> {s. c#s \<in> L r} ; (L r)\<star>" by (auto simp add: dagger_correctness)
+ − 424 then have "s \<in> {s. c#s \<in> (L r)\<star>}" by (simp add: zzz)
+ − 425 then have "c#s \<in> {[c]}; {s. c#s \<in> (L r)\<star>}" by (auto simp add: lang_seq_def)
+ − 426 then have "c#s \<in> (L r)\<star>" by (auto simp add: lang_seq_def)
+ − 427 then show "c#s \<in> L (STAR r)" by simp
+ − 428 next
+ − 429 case (7 r1 r2)
+ − 430 have ih1: "r1 ! [] \<Longrightarrow> [] \<in> L r1" by fact
+ − 431 have ih2: "r2 ! [] \<Longrightarrow> [] \<in> L r2" by fact
+ − 432 have as: "SEQ r1 r2 ! []" by fact
+ − 433 then have "r1 ! [] \<and> r2 ! []" by simp
+ − 434 then show "[] \<in> L (SEQ r1 r2)" using ih1 ih2 by (simp add: lang_seq_def)
+ − 435 next
+ − 436 case (8 r2 c s)
+ − 437 have "SEQ NULL r2 ! c#s" by fact
+ − 438 then show "c#s \<in> L (SEQ NULL r2)" by simp
+ − 439 next
+ − 440 case (9 r2 c s)
+ − 441 have ih1: "r2 ! c#s \<Longrightarrow> c#s \<in> L r2" by fact
+ − 442 have "SEQ EMPTY r2 ! c#s" by fact
+ − 443 then show "c#s \<in> L (SEQ EMPTY r2)" using ih1 by (simp add: lang_seq_def)
+ − 444 next
+ − 445 case (10 c' r2 c s)
+ − 446 have ih1: "\<lbrakk>c' = c; r2 ! s\<rbrakk> \<Longrightarrow> s \<in> L r2" by fact
+ − 447 have "SEQ (CHAR c') r2 ! c#s" by fact
+ − 448 then show "c#s \<in> L (SEQ (CHAR c') r2)"
+ − 449 using ih1 by (auto simp add: lang_seq_def split: if_splits)
+ − 450 next
+ − 451 case (11 r11 r12 r2 c s)
+ − 452 have ih1: "SEQ r11 (SEQ r12 r2) ! c#s \<Longrightarrow> c#s \<in> L (SEQ r11 (SEQ r12 r2))" by fact
+ − 453 have "SEQ (SEQ r11 r12) r2 ! c#s" by fact
+ − 454 then have "c#s \<in> L (SEQ r11 (SEQ r12 r2))" using ih1 by simp
+ − 455 then show "c#s \<in> L (SEQ (SEQ r11 r12) r2)" by (simp add: lang_seq_assoc)
+ − 456 next
+ − 457 case (12 r11 r12 r2 c s)
+ − 458 have ih1: "SEQ r11 r2 ! c#s \<Longrightarrow> c#s \<in> L (SEQ r11 r2)" by fact
+ − 459 have ih2: "SEQ r12 r2 ! c#s \<Longrightarrow> c#s \<in> L (SEQ r12 r2)" by fact
+ − 460 have "SEQ (ALT r11 r12) r2 ! c#s" by fact
+ − 461 then show "c#s \<in> L (SEQ (ALT r11 r12) r2)"
+ − 462 using ih1 ih2 by (auto simp add: lang_seq_union)
+ − 463 next
+ − 464 case (13 r1 r2 c s)
+ − 465 have ih1: "r2 ! c#s \<Longrightarrow> c#s \<in> L r2" by fact
+ − 466 have ih2: "\<And>r'. \<lbrakk>r' \<in> set r1 \<dagger> c; SEQ r' (SEQ (STAR r1) r2) ! s\<rbrakk> \<Longrightarrow>
+ − 467 s \<in> L (SEQ r' (SEQ (STAR r1) r2))" by fact
+ − 468 have "SEQ (STAR r1) r2 ! c#s" by fact
+ − 469 then have "(r2 ! c#s) \<or> (\<exists>r' \<in> set r1 \<dagger> c. SEQ r' (SEQ (STAR r1) r2) ! s)" by (auto simp add: first_True)
+ − 470 moreover
+ − 471 { assume "r2 ! c#s"
+ − 472 with ih1 have "c#s \<in> L r2" by simp
+ − 473 then have "c # s \<in> L r1\<star> ; L r2"
+ − 474 by (auto simp add: lang_seq_def)
+ − 475 then have "c#s \<in> L (SEQ (STAR r1) r2)" by simp
+ − 476 }
+ − 477 moreover
+ − 478 { assume "\<exists>r' \<in> set r1 \<dagger> c. SEQ r' (SEQ (STAR r1) r2) ! s"
+ − 479 then obtain r' where imp1: "r' \<in> set r1 \<dagger> c" and imp2: "SEQ r' (SEQ (STAR r1) r2) ! s" by blast
+ − 480 from imp2 imp1 have "s \<in> L (SEQ r' (SEQ (STAR r1) r2))" using ih2 by simp
+ − 481 then have "s \<in> L r' ; ((L r1)\<star> ; L r2)" by simp
+ − 482 then have "s \<in> Ls (set r1 \<dagger> c) ; ((L r1)\<star> ; L r2)" using imp1 by (auto simp add: Ls_def lang_seq_def)
+ − 483 then have "s \<in> {s. c#s \<in> L r1} ; ((L r1)\<star> ; L r2)" by (simp add: dagger_correctness)
+ − 484 then have "s \<in> ({s. c#s \<in> L r1} ; (L r1)\<star>) ; L r2" by (simp add: lang_seq_assoc)
+ − 485 then have "s \<in> {s. c#s \<in> (L r1)\<star>} ; L r2" by (simp add: zzz)
+ − 486 then have "c#s \<in> {[c]}; ({s. c#s \<in> (L r1)\<star>}; L r2)" by (auto simp add: lang_seq_def)
+ − 487 then have "c#s \<in> ({[c]}; {s. c#s \<in> (L r1)\<star>}) ; L r2" by (simp add: lang_seq_assoc)
+ − 488 then have "c#s \<in> (L r1)\<star>; L r2" by (auto simp add: lang_seq_def)
+ − 489 then have "c#s \<in> L (SEQ (STAR r1) r2)" by simp
+ − 490 }
+ − 491 ultimately show "c#s \<in> L (SEQ (STAR r1) r2)" by blast
+ − 492 qed
+ − 493
+ − 494 text {* negative correctness of the matcher *}
+ − 495 lemma matcher2:
+ − 496 shows "\<not> r ! s \<Longrightarrow> s \<notin> L r"
+ − 497 proof (induct rule: matcher.induct)
+ − 498 case (1 s)
+ − 499 have "\<not> NULL ! s" by fact
+ − 500 then show "s \<notin> L NULL" by simp
+ − 501 next
+ − 502 case (2 s)
+ − 503 have "\<not> EMPTY ! s" by fact
+ − 504 then show "s \<notin> L EMPTY" by simp
+ − 505 next
+ − 506 case (3 c s)
+ − 507 have "\<not> CHAR c ! s" by fact
+ − 508 then show "s \<notin> L (CHAR c)" by simp
+ − 509 next
+ − 510 case (4 r1 r2 s)
+ − 511 have ih2: "\<not> r1 ! s \<Longrightarrow> s \<notin> L r1" by fact
+ − 512 have ih4: "\<not> r2 ! s \<Longrightarrow> s \<notin> L r2" by fact
+ − 513 have "\<not> ALT r1 r2 ! s" by fact
+ − 514 then show "s \<notin> L (ALT r1 r2)" by (simp add: ih2 ih4)
+ − 515 next
+ − 516 case (5 r)
+ − 517 have "\<not> STAR r ! []" by fact
+ − 518 then show "[] \<notin> L (STAR r)" by simp
+ − 519 next
+ − 520 case (6 r c s)
+ − 521 have ih: "\<And>rx. \<lbrakk>rx \<in> set r \<dagger> c; \<not>SEQ rx (STAR r) ! s\<rbrakk> \<Longrightarrow> s \<notin> L (SEQ rx (STAR r))" by fact
+ − 522 have as: "\<not> STAR r ! c#s" by fact
+ − 523 then have "\<forall>r'\<in> set r \<dagger> c. \<not> (SEQ r' (STAR r) ! s)" by simp
+ − 524 then have "\<forall>r'\<in> set r \<dagger> c. s \<notin> L (SEQ r' (STAR r))" using ih by auto
+ − 525 then have "\<forall>r'\<in> set r \<dagger> c. s \<notin> L r' ; ((L r)\<star>)" by simp
+ − 526 then have "s \<notin> (Ls (set r \<dagger> c)) ; ((L r)\<star>)" by (auto simp add: Ls_def lang_seq_def)
+ − 527 then have "s \<notin> {s. c#s \<in> L r} ; ((L r)\<star>)" by (simp add: dagger_correctness)
+ − 528 then have "s \<notin> {s. c#s \<in> (L r)\<star>}" by (simp add: zzz)
+ − 529 then have "c#s \<notin> {[c]} ; {s. c#s \<in> (L r)\<star>}" by (auto simp add: lang_seq_assoc lang_seq_def)
+ − 530 then have "c#s \<notin> (L r)\<star>" by (simp add: lang_seq_def)
+ − 531 then show "c#s \<notin> L (STAR r)" by simp
+ − 532 next
+ − 533 case (7 r1 r2)
+ − 534 have ih2: "\<not> r1 ! [] \<Longrightarrow> [] \<notin> L r1" by fact
+ − 535 have ih4: "\<not> r2 ! [] \<Longrightarrow> [] \<notin> L r2" by fact
+ − 536 have "\<not> SEQ r1 r2 ! []" by fact
+ − 537 then have "\<not> r1 ! [] \<or> \<not> r2 ! []" by simp
+ − 538 then show "[] \<notin> L (SEQ r1 r2)" using ih2 ih4
+ − 539 by (auto simp add: lang_seq_def)
+ − 540 next
+ − 541 case (8 r2 c s)
+ − 542 have "\<not> SEQ NULL r2 ! c#s" by fact
+ − 543 then show "c#s \<notin> L (SEQ NULL r2)" by (simp add: lang_seq_null)
+ − 544 next
+ − 545 case (9 r2 c s)
+ − 546 have ih1: "\<not> r2 ! c#s \<Longrightarrow> c#s \<notin> L r2" by fact
+ − 547 have "\<not> SEQ EMPTY r2 ! c#s" by fact
+ − 548 then show "c#s \<notin> L (SEQ EMPTY r2)"
+ − 549 using ih1 by (simp add: lang_seq_def)
+ − 550 next
+ − 551 case (10 c' r2 c s)
+ − 552 have ih2: "\<lbrakk>c' = c; \<not>r2 ! s\<rbrakk> \<Longrightarrow> s \<notin> L r2" by fact
+ − 553 have "\<not> SEQ (CHAR c') r2 ! c#s" by fact
+ − 554 then show "c#s \<notin> L (SEQ (CHAR c') r2)"
+ − 555 using ih2 by (auto simp add: lang_seq_def)
+ − 556 next
+ − 557 case (11 r11 r12 r2 c s)
+ − 558 have ih2: "\<not> SEQ r11 (SEQ r12 r2) ! c#s \<Longrightarrow> c#s \<notin> L (SEQ r11 (SEQ r12 r2))" by fact
+ − 559 have "\<not> SEQ (SEQ r11 r12) r2 ! c#s" by fact
+ − 560 then show "c#s \<notin> L (SEQ (SEQ r11 r12) r2)"
+ − 561 using ih2 by (auto simp add: lang_seq_def)
+ − 562 next
+ − 563 case (12 r11 r12 r2 c s)
+ − 564 have ih2: "\<not> SEQ r11 r2 ! c#s \<Longrightarrow> c#s \<notin> L (SEQ r11 r2)" by fact
+ − 565 have ih4: "\<not> SEQ r12 r2 ! c#s \<Longrightarrow> c#s \<notin> L (SEQ r12 r2)" by fact
+ − 566 have "\<not> SEQ (ALT r11 r12) r2 ! c#s" by fact
+ − 567 then show " c#s \<notin> L (SEQ (ALT r11 r12) r2)"
+ − 568 using ih2 ih4 by (simp add: lang_seq_union)
+ − 569 next
+ − 570 case (13 r1 r2 c s)
+ − 571 have ih1: "\<not>r2 ! c#s \<Longrightarrow> c#s \<notin> L r2" by fact
+ − 572 have ih2: "\<And>rx. \<lbrakk>rx \<in> set r1 \<dagger> c; \<not> SEQ rx (SEQ (STAR r1) r2) ! s\<rbrakk>
+ − 573 \<Longrightarrow> s \<notin> L (SEQ rx (SEQ (STAR r1) r2))" by fact
+ − 574 have as: "\<not> SEQ (STAR r1) r2 ! c#s" by fact
+ − 575 then have as1: "\<not>r2 ! c#s" and as2: "\<forall>r1'\<in>set r1 \<dagger> c. \<not> SEQ r1' (SEQ (STAR r1) r2) ! s" by simp_all
+ − 576 from as1 have bs1: "c#s \<notin> L r2" using ih1 by simp
+ − 577 from as2 have "\<forall>r1'\<in>set r1 \<dagger> c. \<not> SEQ r1' (SEQ (STAR r1) r2) ! s" by simp
+ − 578 then have "\<forall>r1'\<in>set r1 \<dagger> c. s \<notin> L (SEQ r1' (SEQ (STAR r1) r2))" using ih2 by simp
+ − 579 then have "\<forall>r1'\<in>set r1 \<dagger> c. s \<notin> L r1'; ((L r1)\<star>; L r2)" by simp
+ − 580 then have "s \<notin> (Ls (set r1 \<dagger> c)) ; ((L r1)\<star>; L r2)" by (auto simp add: Ls_def lang_seq_def)
+ − 581 then have "s \<notin> {s. c#s \<in> L r1} ; ((L r1)\<star>; L r2)" by (simp add: dagger_correctness)
+ − 582 then have "s \<notin> ({s. c#s \<in> L r1} ; (L r1)\<star>); L r2" by (simp add: lang_seq_assoc)
+ − 583 then have "s \<notin> {s. c#s \<in> (L r1)\<star>}; L r2" by (simp add: zzz)
+ − 584 then have "c#s \<notin> {[c]}; ({s. c#s \<in> (L r1)\<star>}; L r2)" by (auto simp add: lang_seq_def)
+ − 585 then have "c#s \<notin> (L r1)\<star>; L r2" using bs1 by (auto simp add: lang_seq_def Cons_eq_append_conv)
+ − 586 then show "c#s \<notin> L (SEQ (STAR r1) r2)" by simp
+ − 587 qed
+ − 588
+ − 589 section {* Questions *}
+ − 590
+ − 591 text {*
+ − 592 - Why was the key lemma k1 omitted; is there an easy, non-induction
+ − 593 way for obtaining this property?
+ − 594 - Why was False included in the definition of the STAR-clause in
+ − 595 the matcher? Has this something to do with executing the code?
+ − 596
+ − 597 *}
+ − 598
+ − 599 section {* Code *}
+ − 600
+ − 601 export_code dagger in SML module_name Dagger file -
+ − 602 export_code matcher in SML module_name Dagger file -
+ − 603
+ − 604 section {* Examples *}
+ − 605
+ − 606 text {* since now everything is based on lists, the evaluation is quite fast *}
+ − 607
+ − 608 value "NULL ! []"
+ − 609 value "(CHAR (CHR ''a'')) ! [CHR ''a'']"
+ − 610 value "((CHAR a) ! [a,a])"
+ − 611 value "(STAR (CHAR a)) ! []"
+ − 612 value "(STAR (CHAR a)) ! [a,a]"
+ − 613 value "(SEQ (CHAR (CHR ''a'')) (SEQ (STAR (CHAR (CHR ''b''))) (CHAR (CHR ''c'')))) ! ''abbbbc''"
+ − 614 value "(SEQ (CHAR (CHR ''a'')) (SEQ (STAR (CHAR (CHR ''b''))) (CHAR (CHR ''c'')))) ! ''abbcbbc''"
+ − 615
+ − 616 section {* Slind et al's matcher based on derivatives *}
+ − 617
+ − 618 fun
+ − 619 nullable :: "rexp \<Rightarrow> bool"
+ − 620 where
+ − 621 "nullable (NULL) = False"
+ − 622 | "nullable (EMPTY) = True"
+ − 623 | "nullable (CHAR c) = False"
+ − 624 | "nullable (ALT r1 r2) = ((nullable r1) \<or> (nullable r2))"
+ − 625 | "nullable (SEQ r1 r2) = ((nullable r1) \<and> (nullable r2))"
+ − 626 | "nullable (STAR r) = True"
+ − 627
+ − 628 lemma nullable:
+ − 629 shows "([] \<in> L r) = nullable r"
+ − 630 by (induct r)
+ − 631 (auto simp add: lang_seq_def)
+ − 632
+ − 633 fun
+ − 634 der :: "char \<Rightarrow> rexp \<Rightarrow> rexp"
+ − 635 where
+ − 636 "der c (NULL) = NULL"
+ − 637 | "der c (EMPTY) = NULL"
+ − 638 | "der c (CHAR c') = (if c=c' then EMPTY else NULL)"
+ − 639 | "der c (ALT r1 r2) = ALT (der c r1) (der c r2)"
+ − 640 | "der c (SEQ r1 r2) = ALT (SEQ (der c r1) r2) (if nullable r1 then der c r2 else NULL)"
+ − 641 | "der c (STAR r) = SEQ (der c r) (STAR r)"
+ − 642
+ − 643 lemma k2:
+ − 644 assumes b: "s \<in> L1\<star>"
+ − 645 and a: "s \<noteq> []"
+ − 646 shows "s \<in> (L1; (L1\<star>))"
+ − 647 using b a
+ − 648 apply(induct rule: Star.induct)
+ − 649 apply(simp)
+ − 650 apply(case_tac "s1=[]")
+ − 651 apply(simp)
+ − 652 apply(simp add: lang_seq_def)
+ − 653 apply(blast)
+ − 654 done
+ − 655
+ − 656
+ − 657 lemma der_correctness:
+ − 658 shows "(s \<in> L (der c r)) = ((c#s) \<in> L r)"
+ − 659 proof (induct r arbitrary: s)
+ − 660 case (NULL s)
+ − 661 show "(s \<in> L (der c NULL)) = (c#s \<in> L NULL)" by simp
+ − 662 next
+ − 663 case (EMPTY s)
+ − 664 show "(s \<in> L (der c EMPTY)) = (c#s \<in> L EMPTY)" by simp
+ − 665 next
+ − 666 case (CHAR c' s)
+ − 667 show "(s \<in> L (der c (CHAR c'))) = (c#s \<in> L (CHAR c'))" by simp
+ − 668 next
+ − 669 case (SEQ r1 r2 s)
+ − 670 have ih1: "\<And>s. (s \<in> L (der c r1)) = (c#s \<in> L r1)" by fact
+ − 671 have ih2: "\<And>s. (s \<in> L (der c r2)) = (c#s \<in> L r2)" by fact
+ − 672 show "(s \<in> L (der c (SEQ r1 r2))) = (c#s \<in> L (SEQ r1 r2))"
+ − 673 using ih1 ih2
+ − 674 by (auto simp add: nullable[symmetric] lang_seq_def Cons_eq_append_conv)
+ − 675 next
+ − 676 case (ALT r1 r2 s)
+ − 677 have ih1: "\<And>s. (s \<in> L (der c r1)) = (c#s \<in> L r1)" by fact
+ − 678 have ih2: "\<And>s. (s \<in> L (der c r2)) = (c#s \<in> L r2)" by fact
+ − 679 show "(s \<in> L (der c (ALT r1 r2))) = (c#s \<in> L (ALT r1 r2))"
+ − 680 using ih1 ih2 by (auto simp add: lang_seq_def)
+ − 681 next
+ − 682 case (STAR r s)
+ − 683 have ih1: "\<And>s. (s \<in> L (der c r)) = (c#s \<in> L r)" by fact
+ − 684 show "(s \<in> L (der c (STAR r))) = (c#s \<in> L (STAR r))"
+ − 685 using ih1
+ − 686 apply(simp)
+ − 687 apply(auto simp add: lang_seq_def)
+ − 688 apply(drule lang_seq_append)
+ − 689 apply(assumption)
+ − 690 apply(simp)
+ − 691 apply(subst lang_star_cases)
+ − 692 apply(simp)
+ − 693 thm k1
+ − 694 apply(drule k2)
+ − 695 apply(simp)
+ − 696 apply(simp add: lang_seq_def)
+ − 697 apply(erule exE)+
+ − 698 apply(erule conjE)+
+ − 699 apply(auto simp add: lang_seq_def Cons_eq_append_conv)
+ − 700 apply(drule k1)
+ − 701 apply(simp)
+ − 702 apply(simp add: lang_seq_def)
+ − 703 apply(erule exE)+
+ − 704 apply(erule conjE)+
+ − 705 apply(auto simp add: lang_seq_def Cons_eq_append_conv)
+ − 706 done
+ − 707 qed
+ − 708
+ − 709 fun
+ − 710 derivative :: "string \<Rightarrow> rexp \<Rightarrow> rexp"
+ − 711 where
+ − 712 "derivative [] r = r"
+ − 713 | "derivative (c#s) r = derivative s (der c r)"
+ − 714
+ − 715 fun
+ − 716 slind_matcher :: "rexp \<Rightarrow> string \<Rightarrow> bool"
+ − 717 where
+ − 718 "slind_matcher r s = nullable (derivative s r)"
+ − 719
+ − 720 lemma slind_matcher:
+ − 721 shows "slind_matcher r s = (s \<in> L r)"
+ − 722 by (induct s arbitrary: r)
+ − 723 (auto simp add: nullable der_correctness)
+ − 724
+ − 725 export_code slind_matcher in SML module_name Slind file -
+ − 726
+ − 727
+ − 728 (* ******************************************** now is the codes writen by chunhan ************************************* *)
+ − 729
+ − 730 section {* Arden's Lemma revised *}
+ − 731
+ − 732 lemma arden_aux1:
+ − 733 assumes a: "X \<subseteq> X ; A \<union> B"
+ − 734 and b: "[] \<notin> A"
+ − 735 shows "x \<in> X \<Longrightarrow> x \<in> B ; A\<star>"
+ − 736 apply (induct x taking:length rule:measure_induct)
+ − 737 apply (subgoal_tac "x \<in> X ; A \<union> B")
+ − 738 defer
+ − 739 using a
+ − 740 apply (auto)[1]
+ − 741 apply simp
+ − 742 apply (erule disjE)
+ − 743 defer
+ − 744 apply (auto simp add:lang_seq_def) [1]
+ − 745 apply (subgoal_tac "\<exists> x1 x2. x = x1 @ x2 \<and> x1 \<in> X \<and> x2 \<in> A")
+ − 746 defer
+ − 747 apply (auto simp add:lang_seq_def) [1]
+ − 748 apply (erule exE | erule conjE)+
+ − 749 apply simp
+ − 750 apply (drule_tac x = x1 in spec)
+ − 751 apply (simp)
+ − 752 using b
+ − 753 apply -
+ − 754 apply (auto)[1]
+ − 755 apply (subgoal_tac "x1 @ x2 \<in> (B ; A\<star>) ; A")
+ − 756 defer
+ − 757 apply (auto simp add:lang_seq_def)[1]
+ − 758 by (metis Un_absorb1 lang_seq_assoc lang_seq_union(2) lang_star_double lang_star_simple mem_def sup1CI)
+ − 759
+ − 760 theorem ardens_revised:
+ − 761 assumes nemp: "[] \<notin> A"
+ − 762 shows "(X = X ; A \<union> B) \<longleftrightarrow> (X = B ; A\<star>)"
+ − 763 apply(rule iffI)
+ − 764 defer
+ − 765 apply(simp)
+ − 766 apply(subst lang_star_cases')
+ − 767 apply(subst lang_seq_union)
+ − 768 apply(simp add: lang_seq_empty)
+ − 769 apply(simp add: lang_seq_assoc)
+ − 770 apply(auto)[1]
+ − 771 proof -
+ − 772 assume "X = X ; A \<union> B"
+ − 773 then have as1: "X ; A \<union> B \<subseteq> X" and as2: "X \<subseteq> X ; A \<union> B" by simp_all
+ − 774 from as1 have a: "X ; A \<subseteq> X" and b: "B \<subseteq> X" by simp_all
+ − 775 from b have "B; A\<star> \<subseteq> X ; A\<star>" by (auto simp add: lang_seq_def)
+ − 776 moreover
+ − 777 from a have "X ; A\<star> \<subseteq> X"
+ − 778
+ − 779 by (rule lang_star_prop2)
+ − 780 ultimately have f1: "B ; A\<star> \<subseteq> X" by simp
+ − 781 from as2 nemp
+ − 782 have f2: "X \<subseteq> B; A\<star>" using arden_aux1 by auto
+ − 783 from f1 f2 show "X = B; A\<star>" by auto
+ − 784 qed
+ − 785
+ − 786 section {* equiv class' definition *}
+ − 787
+ − 788 definition
+ − 789 equiv_str :: "string \<Rightarrow> string set \<Rightarrow> string \<Rightarrow> bool" ("_ \<equiv>_\<equiv> _" [100, 100, 100] 100)
+ − 790 where
+ − 791 "x \<equiv>L'\<equiv> y \<longleftrightarrow> (\<forall>z. x@z \<in> L' \<longleftrightarrow> y@z \<in> L')"
+ − 792
+ − 793 definition
+ − 794 equiv_class :: "string \<Rightarrow> (string set) \<Rightarrow> string set" ("\<lbrakk>_\<rbrakk>_" [100, 100] 100)
+ − 795 where
+ − 796 "\<lbrakk>x\<rbrakk>L' \<equiv> {y. x \<equiv>L'\<equiv> y}"
+ − 797
+ − 798 text {* Chunhan modifies Q to Quo *}
+ − 799 definition
+ − 800 quot :: "string set \<Rightarrow> (string set) \<Rightarrow> (string set) set" ("_ Quo _" [100, 100] 100)
+ − 801 where
+ − 802 "L' Quo R \<equiv> { \<lbrakk>x\<rbrakk>R | x. x \<in> L'}"
+ − 803
+ − 804 lemma lang_eqs_union_of_eqcls:
+ − 805 "Lang = \<Union> {X. X \<in> (UNIV Quo Lang) \<and> (\<forall> x \<in> X. x \<in> Lang)}"
+ − 806 proof
+ − 807 show "Lang \<subseteq> \<Union>{X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang}"
+ − 808 proof
+ − 809 fix x
+ − 810 assume "x \<in> Lang"
+ − 811 thus "x \<in> \<Union>{X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang}"
+ − 812 proof (simp add:quot_def)
+ − 813 assume "(1)": "x \<in> Lang"
+ − 814 show "\<exists>xa. (\<exists>x. xa = \<lbrakk>x\<rbrakk>Lang) \<and> (\<forall>x\<in>xa. x \<in> Lang) \<and> x \<in> xa" (is "\<exists>xa.?P xa")
+ − 815 proof -
+ − 816 have "?P (\<lbrakk>x\<rbrakk>Lang)" using "(1)"
+ − 817 by (auto simp:equiv_class_def equiv_str_def dest: spec[where x = "[]"])
+ − 818 thus ?thesis by blast
+ − 819 qed
+ − 820 qed
+ − 821 qed
+ − 822 next
+ − 823 show "\<Union>{X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang} \<subseteq> Lang"
+ − 824 by auto
+ − 825 qed
+ − 826
+ − 827 lemma empty_notin_CS: "{} \<notin> UNIV Quo Lang"
+ − 828 apply (clarsimp simp:quot_def equiv_class_def)
+ − 829 by (rule_tac x = x in exI, auto simp:equiv_str_def)
+ − 830
+ − 831 lemma no_two_cls_inters:
+ − 832 "\<lbrakk>X \<in> UNIV Quo Lang; Y \<in> UNIV Quo Lang; X \<noteq> Y\<rbrakk> \<Longrightarrow> X \<inter> Y = {}"
+ − 833 by (auto simp:quot_def equiv_class_def equiv_str_def)
+ − 834
+ − 835 text {* equiv_class transition *}
+ − 836 definition
+ − 837 CT :: "string set \<Rightarrow> char \<Rightarrow> string set \<Rightarrow> bool" ("_-_\<rightarrow>_" [99,99]99)
+ − 838 where
+ − 839 "X-c\<rightarrow>Y \<equiv> ((X;{[c]}) \<subseteq> Y)"
+ − 840
+ − 841 types t_equa_rhs = "(string set \<times> rexp) set"
+ − 842
+ − 843 types t_equa = "(string set \<times> t_equa_rhs)"
+ − 844
+ − 845 types t_equas = "t_equa set"
+ − 846
+ − 847 text {* "empty_rhs" generates "\<lambda>" for init-state, just like "\<lambda>" for final states in Brzozowski method.
+ − 848 But if the init-state is "{[]}" ("\<lambda>" itself) then empty set is returned, see definition of "equation_rhs" *}
+ − 849 definition
+ − 850 empty_rhs :: "string set \<Rightarrow> t_equa_rhs"
+ − 851 where
+ − 852 "empty_rhs X \<equiv> if ([] \<in> X) then {({[]}, EMPTY)} else {}"
+ − 853
+ − 854 definition
+ − 855 folds :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b"
+ − 856 where
+ − 857 "folds f z S \<equiv> SOME x. fold_graph f z S x"
+ − 858
+ − 859 definition
+ − 860 equation_rhs :: "(string set) set \<Rightarrow> string set \<Rightarrow> t_equa_rhs"
+ − 861 where
+ − 862 "equation_rhs CS X \<equiv> if (X = {[]}) then {({[]}, EMPTY)}
+ − 863 else {(S, folds ALT NULL {CHAR c| c. S-c\<rightarrow>X} )| S. S \<in> CS} \<union> empty_rhs X"
+ − 864
+ − 865 definition
+ − 866 equations :: "(string set) set \<Rightarrow> t_equas"
+ − 867 where
+ − 868 "equations CS \<equiv> {(X, equation_rhs CS X) | X. X \<in> CS}"
+ − 869
+ − 870 definition
+ − 871 L_rhs :: "t_equa_rhs \<Rightarrow> string set"
+ − 872 where
+ − 873 "L_rhs rhs \<equiv> {x. \<exists> X r. (X, r) \<in> rhs \<and> x \<in> X;(L r)}"
+ − 874
+ − 875 defs (overloaded)
+ − 876 l_rhs_abs: "L rhs \<equiv> L_rhs rhs"
+ − 877
+ − 878 lemmas L_def = L_rhs_def [folded l_rhs_abs] L_rexp.simps (* ??? is this OK ?? *)
+ − 879
+ − 880 definition
+ − 881 distinct_rhs :: "t_equa_rhs \<Rightarrow> bool"
+ − 882 where
+ − 883 "distinct_rhs rhs \<equiv> \<forall> X reg\<^isub>1 reg\<^isub>2. (X, reg\<^isub>1) \<in> rhs \<and> (X, reg\<^isub>2) \<in> rhs \<longrightarrow> reg\<^isub>1 = reg\<^isub>2"
+ − 884
+ − 885 definition
+ − 886 distinct_equas_rhs :: "t_equas \<Rightarrow> bool"
+ − 887 where
+ − 888 "distinct_equas_rhs equas \<equiv> \<forall> X rhs. (X, rhs) \<in> equas \<longrightarrow> distinct_rhs rhs"
+ − 889
+ − 890 definition
+ − 891 distinct_equas :: "t_equas \<Rightarrow> bool"
+ − 892 where
+ − 893 "distinct_equas equas \<equiv> \<forall> X rhs rhs'. (X, rhs) \<in> equas \<and> (X, rhs') \<in> equas \<longrightarrow> rhs = rhs'"
+ − 894
+ − 895 definition
+ − 896 seq_rhs_r :: "t_equa_rhs \<Rightarrow> rexp \<Rightarrow> t_equa_rhs"
+ − 897 where
+ − 898 "seq_rhs_r rhs r \<equiv> (\<lambda>(X, reg). (X, SEQ reg r)) ` rhs"
+ − 899
+ − 900 definition
+ − 901 del_x_paired :: "('a \<times> 'b) set \<Rightarrow> 'a \<Rightarrow> ('a \<times> 'b) set"
+ − 902 where
+ − 903 "del_x_paired S x \<equiv> S - {X. X \<in> S \<and> fst X = x}"
+ − 904
+ − 905 definition
+ − 906 arden_variate :: "string set \<Rightarrow> rexp \<Rightarrow> t_equa_rhs \<Rightarrow> t_equa_rhs"
+ − 907 where
+ − 908 "arden_variate X r rhs \<equiv> seq_rhs_r (del_x_paired rhs X) (STAR r)"
+ − 909
+ − 910 definition
+ − 911 no_EMPTY_rhs :: "t_equa_rhs \<Rightarrow> bool"
+ − 912 where
+ − 913 "no_EMPTY_rhs rhs \<equiv> \<forall> X r. (X, r) \<in> rhs \<and> X \<noteq> {[]} \<longrightarrow> [] \<notin> L r"
+ − 914
+ − 915 definition
+ − 916 no_EMPTY_equas :: "t_equas \<Rightarrow> bool"
+ − 917 where
+ − 918 "no_EMPTY_equas equas \<equiv> \<forall> X rhs. (X, rhs) \<in> equas \<longrightarrow> no_EMPTY_rhs rhs"
+ − 919
+ − 920 lemma fold_alt_null_eqs:
+ − 921 "finite rS \<Longrightarrow> x \<in> L (folds ALT NULL rS) = (\<exists> r \<in> rS. x \<in> L r)"
+ − 922 apply (simp add:folds_def)
+ − 923 apply (rule someI2_ex)
+ − 924 apply (erule finite_imp_fold_graph)
+ − 925 apply (erule fold_graph.induct)
+ − 926 by auto (*??? how do this be in Isar ?? *)
+ − 927
+ − 928 lemma seq_rhs_r_prop1:
+ − 929 "L (seq_rhs_r rhs r) = (L rhs);(L r)"
+ − 930 apply (rule set_ext, rule iffI)
+ − 931 apply (auto simp:L_def seq_rhs_r_def image_def lang_seq_def)
+ − 932 apply (rule_tac x = "s1 @ s1a" in exI, rule_tac x = "s2a" in exI, simp)
+ − 933 apply (rule_tac x = a in exI, rule_tac x = b in exI, simp)
+ − 934 apply (rule_tac x = s1 in exI, rule_tac x = s1a in exI, simp)
+ − 935 apply (rule_tac x = X in exI, rule_tac x = "SEQ ra r" in exI, simp)
+ − 936 apply (rule conjI)
+ − 937 apply (rule_tac x = "(X, ra)" in bexI, simp+)
+ − 938 apply (rule_tac x = s1a in exI, rule_tac x = "s2a @ s2" in exI, simp)
+ − 939 apply (simp add:lang_seq_def)
+ − 940 by (rule_tac x = s2a in exI, rule_tac x = s2 in exI, simp)
+ − 941
+ − 942 lemma del_x_paired_prop1:
+ − 943 "\<lbrakk>distinct_rhs rhs; (X, r) \<in> rhs\<rbrakk> \<Longrightarrow> X ; L r \<union> L (del_x_paired rhs X) = L rhs"
+ − 944 apply (simp add:L_def del_x_paired_def)
+ − 945 apply (rule set_ext, rule iffI, simp)
+ − 946 apply (erule disjE, rule_tac x = X in exI, rule_tac x = r in exI, simp)
+ − 947 apply (clarify, rule_tac x = Xa in exI, rule_tac x = ra in exI, simp)
+ − 948 apply (clarsimp, drule_tac x = Xa in spec, drule_tac x = ra in spec)
+ − 949 apply (auto simp:distinct_rhs_def)
+ − 950 done
+ − 951
+ − 952 lemma arden_variate_prop:
+ − 953 assumes "(X, rx) \<in> rhs"
+ − 954 shows "(\<forall> Y. Y \<noteq> X \<longrightarrow> (\<exists> r. (Y, r) \<in> rhs) = (\<exists> r. (Y, r) \<in> (arden_variate X rx rhs)))"
+ − 955 proof (rule allI, rule impI)
+ − 956 fix Y
+ − 957 assume "(1)": "Y \<noteq> X"
+ − 958 show "(\<exists>r. (Y, r) \<in> rhs) = (\<exists>r. (Y, r) \<in> arden_variate X rx rhs)"
+ − 959 proof
+ − 960 assume "(1_1)": "\<exists>r. (Y, r) \<in> rhs"
+ − 961 show "\<exists>r. (Y, r) \<in> arden_variate X rx rhs" (is "\<exists>r. ?P r")
+ − 962 proof -
+ − 963 from "(1_1)" obtain r where "(Y, r) \<in> rhs" ..
+ − 964 hence "?P (SEQ r (STAR rx))"
+ − 965 proof (simp add:arden_variate_def image_def)
+ − 966 have "(Y, r) \<in> rhs \<Longrightarrow> (Y, r) \<in> del_x_paired rhs X"
+ − 967 by (auto simp:del_x_paired_def "(1)")
+ − 968 thus "(Y, r) \<in> rhs \<Longrightarrow> (Y, SEQ r (STAR rx)) \<in> seq_rhs_r (del_x_paired rhs X) (STAR rx)"
+ − 969 by (auto simp:seq_rhs_r_def)
+ − 970 qed
+ − 971 thus ?thesis by blast
+ − 972 qed
+ − 973 next
+ − 974 assume "(2_1)": "\<exists>r. (Y, r) \<in> arden_variate X rx rhs"
+ − 975 thus "\<exists>r. (Y, r) \<in> rhs" (is "\<exists> r. ?P r")
+ − 976 by (auto simp:arden_variate_def del_x_paired_def seq_rhs_r_def image_def)
+ − 977 qed
+ − 978 qed
+ − 979
+ − 980 text {*
+ − 981 arden_variate_valid: proves variation from "X = X;r + Y;ry + \<dots>" to "X = Y;(SEQ ry (STAR r)) + \<dots>" holds the law of "language of left equiv language of right"
+ − 982 *}
+ − 983 lemma arden_variate_valid:
+ − 984 assumes X_not_empty: "X \<noteq> {[]}"
+ − 985 and l_eq_r: "X = L rhs"
+ − 986 and dist: "distinct_rhs rhs"
+ − 987 and no_empty: "no_EMPTY_rhs rhs"
+ − 988 and self_contained: "(X, r) \<in> rhs"
+ − 989 shows "X = L (arden_variate X r rhs)"
+ − 990 proof -
+ − 991 have "[] \<notin> L r" using no_empty X_not_empty self_contained
+ − 992 by (auto simp:no_EMPTY_rhs_def)
+ − 993 hence ardens: "X = X;(L r) \<union> (L (del_x_paired rhs X)) \<longleftrightarrow> X = (L (del_x_paired rhs X)) ; (L r)\<star>"
+ − 994 by (rule ardens_revised)
+ − 995 have del_x: "X = X ; L r \<union> L (del_x_paired rhs X) \<longleftrightarrow> X = L rhs" using dist l_eq_r self_contained
+ − 996 by (auto dest:del_x_paired_prop1)
+ − 997 show ?thesis
+ − 998 proof
+ − 999 show "X \<subseteq> L (arden_variate X r rhs)"
+ − 1000 proof
+ − 1001 fix x
+ − 1002 assume "(1_1)": "x \<in> X" with l_eq_r ardens del_x
+ − 1003 show "x \<in> L (arden_variate X r rhs)"
+ − 1004 by (simp add:arden_variate_def seq_rhs_r_prop1)
+ − 1005 qed
+ − 1006 next
+ − 1007 show "L (arden_variate X r rhs) \<subseteq> X"
+ − 1008 proof
+ − 1009 fix x
+ − 1010 assume "(2_1)": "x \<in> L (arden_variate X r rhs)" with ardens del_x l_eq_r
+ − 1011 show "x \<in> X"
+ − 1012 by (simp add:arden_variate_def seq_rhs_r_prop1)
+ − 1013 qed
+ − 1014 qed
+ − 1015 qed
+ − 1016
+ − 1017 text {* merge_rhs {(X1, r1), (x2, r2}, (x4, r4), \<dots>} {(x1, r1'), (x3, r3'), \<dots>} = {(x1, ALT r1 r1'}, (x2, r2), (x3, r3'), (x4, r4), \<dots>} *}
+ − 1018 definition
+ − 1019 merge_rhs :: "t_equa_rhs \<Rightarrow> t_equa_rhs \<Rightarrow> t_equa_rhs"
+ − 1020 where
+ − 1021 "merge_rhs rhs rhs' \<equiv> {(X, r). (\<exists> r1 r2. ((X,r1) \<in> rhs \<and> (X, r2) \<in> rhs') \<and> r = ALT r1 r2) \<or>
+ − 1022 (\<exists> r1. (X, r1) \<in> rhs \<and> (\<not> (\<exists> r2. (X, r2) \<in> rhs')) \<and> r = r1) \<or>
+ − 1023 (\<exists> r2. (X, r2) \<in> rhs' \<and> (\<not> (\<exists> r1. (X, r1) \<in> rhs)) \<and> r = r2) }"
+ − 1024
+ − 1025
+ − 1026 text {* rhs_subst rhs X=xrhs r: substitude all occurence of X in rhs((X,r) \<in> rhs) with xrhs *}
+ − 1027 definition
+ − 1028 rhs_subst :: "t_equa_rhs \<Rightarrow> string set \<Rightarrow> t_equa_rhs \<Rightarrow> rexp \<Rightarrow> t_equa_rhs"
+ − 1029 where
+ − 1030 "rhs_subst rhs X xrhs r \<equiv> merge_rhs (del_x_paired rhs X) (seq_rhs_r xrhs r)"
+ − 1031
+ − 1032 definition
+ − 1033 equas_subst_f :: "string set \<Rightarrow> t_equa_rhs \<Rightarrow> t_equa \<Rightarrow> t_equa"
+ − 1034 where
+ − 1035 "equas_subst_f X xrhs equa \<equiv> let (Y, rhs) = equa in
+ − 1036 if (\<exists> r. (X, r) \<in> rhs)
+ − 1037 then (Y, rhs_subst rhs X xrhs (SOME r. (X, r) \<in> rhs))
+ − 1038 else equa"
+ − 1039
+ − 1040 definition
+ − 1041 equas_subst :: "t_equas \<Rightarrow> string set \<Rightarrow> t_equa_rhs \<Rightarrow> t_equas"
+ − 1042 where
+ − 1043 "equas_subst ES X xrhs \<equiv> del_x_paired (equas_subst_f X xrhs ` ES) X"
+ − 1044
+ − 1045 lemma lang_seq_prop1:
+ − 1046 "x \<in> X ; L r \<Longrightarrow> x \<in> X ; (L r \<union> L r')"
+ − 1047 by (auto simp:lang_seq_def)
+ − 1048
+ − 1049 lemma lang_seq_prop1':
+ − 1050 "x \<in> X; L r \<Longrightarrow> x \<in> X ; (L r' \<union> L r)"
+ − 1051 by (auto simp:lang_seq_def)
+ − 1052
+ − 1053 lemma lang_seq_prop2:
+ − 1054 "x \<in> X; (L r \<union> L r') \<Longrightarrow> x \<in> X;L r \<or> x \<in> X;L r'"
+ − 1055 by (auto simp:lang_seq_def)
+ − 1056
+ − 1057 lemma merge_rhs_prop1:
+ − 1058 shows "L (merge_rhs rhs rhs') = L rhs \<union> L rhs' "
+ − 1059 apply (auto simp add:merge_rhs_def L_def dest!:lang_seq_prop2 intro:lang_seq_prop1)
+ − 1060 apply (rule_tac x = X in exI, rule_tac x = r1 in exI, simp)
+ − 1061 apply (case_tac "\<exists> r2. (X, r2) \<in> rhs'")
+ − 1062 apply (clarify, rule_tac x = X in exI, rule_tac x = "ALT r r2" in exI, simp add:lang_seq_prop1)
+ − 1063 apply (rule_tac x = X in exI, rule_tac x = r in exI, simp)
+ − 1064 apply (case_tac "\<exists> r1. (X, r1) \<in> rhs")
+ − 1065 apply (clarify, rule_tac x = X in exI, rule_tac x = "ALT r1 r" in exI, simp add:lang_seq_prop1')
+ − 1066 apply (rule_tac x = X in exI, rule_tac x = r in exI, simp)
+ − 1067 done
+ − 1068
+ − 1069 lemma no_EMPTY_rhss_imp_merge_no_EMPTY:
+ − 1070 "\<lbrakk>no_EMPTY_rhs rhs; no_EMPTY_rhs rhs'\<rbrakk> \<Longrightarrow> no_EMPTY_rhs (merge_rhs rhs rhs')"
+ − 1071 apply (simp add:no_EMPTY_rhs_def merge_rhs_def)
+ − 1072 apply (clarify, (erule conjE | erule exE | erule disjE)+)
+ − 1073 by auto
+ − 1074
+ − 1075 lemma distinct_rhs_prop:
+ − 1076 "\<lbrakk>distinct_rhs rhs; (X, r1) \<in> rhs; (X, r2) \<in> rhs\<rbrakk> \<Longrightarrow> r1 = r2"
+ − 1077 by (auto simp:distinct_rhs_def)
+ − 1078
+ − 1079 lemma merge_rhs_prop2:
+ − 1080 assumes dist_rhs: "distinct_rhs rhs"
+ − 1081 and dist_rhs':"distinct_rhs rhs'"
+ − 1082 shows "distinct_rhs (merge_rhs rhs rhs')"
+ − 1083 apply (auto simp:merge_rhs_def distinct_rhs_def)
+ − 1084 using dist_rhs
+ − 1085 apply (drule distinct_rhs_prop, simp+)
+ − 1086 using dist_rhs'
+ − 1087 apply (drule distinct_rhs_prop, simp+)
+ − 1088 using dist_rhs
+ − 1089 apply (drule distinct_rhs_prop, simp+)
+ − 1090 using dist_rhs'
+ − 1091 apply (drule distinct_rhs_prop, simp+)
+ − 1092 done
+ − 1093
+ − 1094 lemma seq_rhs_r_holds_distinct:
+ − 1095 "distinct_rhs rhs \<Longrightarrow> distinct_rhs (seq_rhs_r rhs r)"
+ − 1096 by (auto simp:distinct_rhs_def seq_rhs_r_def)
+ − 1097
+ − 1098 lemma seq_rhs_r_prop0:
+ − 1099 assumes l_eq_r: "X = L xrhs"
+ − 1100 shows "L (seq_rhs_r xrhs r) = X ; L r "
+ − 1101 using l_eq_r
+ − 1102 by (auto simp:seq_rhs_r_prop1)
+ − 1103
+ − 1104 lemma rhs_subst_prop1:
+ − 1105 assumes l_eq_r: "X = L xrhs"
+ − 1106 and dist: "distinct_rhs rhs"
+ − 1107 shows "(X, r) \<in> rhs \<Longrightarrow> L rhs = L (rhs_subst rhs X xrhs r)"
+ − 1108 apply (simp add:rhs_subst_def merge_rhs_prop1)
+ − 1109 using l_eq_r
+ − 1110 apply (drule_tac r = r in seq_rhs_r_prop0, simp)
+ − 1111 using dist
+ − 1112 apply (auto dest:del_x_paired_prop1)
+ − 1113 done
+ − 1114
+ − 1115 lemma del_x_paired_holds_distinct_rhs:
+ − 1116 "distinct_rhs rhs \<Longrightarrow> distinct_rhs (del_x_paired rhs x)"
+ − 1117 by (auto simp:distinct_rhs_def del_x_paired_def)
+ − 1118
+ − 1119 lemma rhs_subst_holds_distinct_rhs:
+ − 1120 "\<lbrakk>distinct_rhs rhs; distinct_rhs xrhs\<rbrakk> \<Longrightarrow> distinct_rhs (rhs_subst rhs X xrhs r)"
+ − 1121 apply (drule_tac r = r and rhs = xrhs in seq_rhs_r_holds_distinct)
+ − 1122 apply (drule_tac x = X in del_x_paired_holds_distinct_rhs)
+ − 1123 by (auto dest:merge_rhs_prop2[where rhs = "del_x_paired rhs X"] simp:rhs_subst_def)
+ − 1124
+ − 1125 section {* myhill-nerode theorem *}
+ − 1126
+ − 1127 definition left_eq_cls :: "t_equas \<Rightarrow> (string set) set"
+ − 1128 where
+ − 1129 "left_eq_cls ES \<equiv> {X. \<exists> rhs. (X, rhs) \<in> ES} "
+ − 1130
+ − 1131 definition right_eq_cls :: "t_equas \<Rightarrow> (string set) set"
+ − 1132 where
+ − 1133 "right_eq_cls ES \<equiv> {Y. \<exists> X rhs r. (X, rhs) \<in> ES \<and> (Y, r) \<in> rhs }"
+ − 1134
+ − 1135 definition rhs_eq_cls :: "t_equa_rhs \<Rightarrow> (string set) set"
+ − 1136 where
+ − 1137 "rhs_eq_cls rhs \<equiv> {Y. \<exists> r. (Y, r) \<in> rhs}"
+ − 1138
+ − 1139 definition ardenable :: "t_equa \<Rightarrow> bool"
+ − 1140 where
+ − 1141 "ardenable equa \<equiv> let (X, rhs) = equa in
+ − 1142 distinct_rhs rhs \<and> no_EMPTY_rhs rhs \<and> X = L rhs"
+ − 1143
+ − 1144 text {*
+ − 1145 Inv: Invairance of the equation-system, during the decrease of the equation-system, Inv holds.
+ − 1146 *}
+ − 1147 definition Inv :: "string set \<Rightarrow> t_equas \<Rightarrow> bool"
+ − 1148 where
+ − 1149 "Inv X ES \<equiv> finite ES \<and> (\<exists> rhs. (X, rhs) \<in> ES) \<and> distinct_equas ES \<and>
+ − 1150 (\<forall> X xrhs. (X, xrhs) \<in> ES \<longrightarrow> ardenable (X, xrhs) \<and> X \<noteq> {} \<and> rhs_eq_cls xrhs \<subseteq> insert {[]} (left_eq_cls ES))"
+ − 1151
+ − 1152 text {*
+ − 1153 TCon: Termination Condition of the equation-system decreasion.
+ − 1154 *}
+ − 1155 definition TCon:: "'a set \<Rightarrow> bool"
+ − 1156 where
+ − 1157 "TCon ES \<equiv> card ES = 1"
+ − 1158
+ − 1159
+ − 1160 text {*
+ − 1161 The following is a iteration principle, and is the main framework for the proof:
+ − 1162 1: We can form the initial equation-system using "equations" defined above, and prove it has invariance Inv by lemma "init_ES_satisfy_Inv";
+ − 1163 2: We can decrease the number of the equation-system using ardens_lemma_revised and substitution ("equas_subst", defined above),
+ − 1164 and prove it holds the property "step" in "wf_iter" by lemma "iteration_step"
+ − 1165 and finally using property P and Q to prove the myhill-nerode theorem
+ − 1166
+ − 1167 *}
+ − 1168 lemma wf_iter [rule_format]:
+ − 1169 fixes f
+ − 1170 assumes step: "\<And> e. \<lbrakk>P e; \<not> Q e\<rbrakk> \<Longrightarrow> (\<exists> e'. P e' \<and> (f(e'), f(e)) \<in> less_than)"
+ − 1171 shows pe: "P e \<longrightarrow> (\<exists> e'. P e' \<and> Q e')"
+ − 1172 proof(induct e rule: wf_induct
+ − 1173 [OF wf_inv_image[OF wf_less_than, where f = "f"]], clarify)
+ − 1174 fix x
+ − 1175 assume h [rule_format]:
+ − 1176 "\<forall>y. (y, x) \<in> inv_image less_than f \<longrightarrow> P y \<longrightarrow> (\<exists>e'. P e' \<and> Q e')"
+ − 1177 and px: "P x"
+ − 1178 show "\<exists>e'. P e' \<and> Q e'"
+ − 1179 proof(cases "Q x")
+ − 1180 assume "Q x" with px show ?thesis by blast
+ − 1181 next
+ − 1182 assume nq: "\<not> Q x"
+ − 1183 from step [OF px nq]
+ − 1184 obtain e' where pe': "P e'" and ltf: "(f e', f x) \<in> less_than" by auto
+ − 1185 show ?thesis
+ − 1186 proof(rule h)
+ − 1187 from ltf show "(e', x) \<in> inv_image less_than f"
+ − 1188 by (simp add:inv_image_def)
+ − 1189 next
+ − 1190 from pe' show "P e'" .
+ − 1191 qed
+ − 1192 qed
+ − 1193 qed
+ − 1194
+ − 1195
+ − 1196 (* ********************************* BEGIN: proving the initial equation-system satisfies Inv **************************************** *)
+ − 1197
+ − 1198 lemma distinct_rhs_equations:
+ − 1199 "(X, xrhs) \<in> equations (UNIV Quo Lang) \<Longrightarrow> distinct_rhs xrhs"
+ − 1200 by (auto simp: equations_def equation_rhs_def distinct_rhs_def empty_rhs_def dest:no_two_cls_inters)
+ − 1201
+ − 1202 lemma every_nonempty_eqclass_has_strings:
+ − 1203 "\<lbrakk>X \<in> (UNIV Quo Lang); X \<noteq> {[]}\<rbrakk> \<Longrightarrow> \<exists> clist. clist \<in> X \<and> clist \<noteq> []"
+ − 1204 by (auto simp:quot_def equiv_class_def equiv_str_def)
+ − 1205
+ − 1206 lemma every_eqclass_is_derived_from_empty:
+ − 1207 assumes not_empty: "X \<noteq> {[]}"
+ − 1208 shows "X \<in> (UNIV Quo Lang) \<Longrightarrow> \<exists> clist. {[]};{clist} \<subseteq> X \<and> clist \<noteq> []"
+ − 1209 using not_empty
+ − 1210 apply (drule_tac every_nonempty_eqclass_has_strings, simp)
+ − 1211 by (auto intro:exI[where x = clist] simp:lang_seq_def)
+ − 1212
+ − 1213 lemma equiv_str_in_CS:
+ − 1214 "\<lbrakk>clist\<rbrakk>Lang \<in> (UNIV Quo Lang)"
+ − 1215 by (auto simp:quot_def)
+ − 1216
+ − 1217 lemma has_str_imp_defined_by_str:
+ − 1218 "\<lbrakk>str \<in> X; X \<in> UNIV Quo Lang\<rbrakk> \<Longrightarrow> X = \<lbrakk>str\<rbrakk>Lang"
+ − 1219 by (auto simp:quot_def equiv_class_def equiv_str_def)
+ − 1220
+ − 1221 lemma every_eqclass_has_ascendent:
+ − 1222 assumes has_str: "clist @ [c] \<in> X"
+ − 1223 and in_CS: "X \<in> UNIV Quo Lang"
+ − 1224 shows "\<exists> Y. Y \<in> UNIV Quo Lang \<and> Y-c\<rightarrow>X \<and> clist \<in> Y" (is "\<exists> Y. ?P Y")
+ − 1225 proof -
+ − 1226 have "?P (\<lbrakk>clist\<rbrakk>Lang)"
+ − 1227 proof -
+ − 1228 have "\<lbrakk>clist\<rbrakk>Lang \<in> UNIV Quo Lang"
+ − 1229 by (simp add:quot_def, rule_tac x = clist in exI, simp)
+ − 1230 moreover have "\<lbrakk>clist\<rbrakk>Lang-c\<rightarrow>X"
+ − 1231 proof -
+ − 1232 have "X = \<lbrakk>(clist @ [c])\<rbrakk>Lang" using has_str in_CS
+ − 1233 by (auto intro!:has_str_imp_defined_by_str)
+ − 1234 moreover have "\<forall> sl. sl \<in> \<lbrakk>clist\<rbrakk>Lang \<longrightarrow> sl @ [c] \<in> \<lbrakk>(clist @ [c])\<rbrakk>Lang"
+ − 1235 by (auto simp:equiv_class_def equiv_str_def)
+ − 1236 ultimately show ?thesis unfolding CT_def lang_seq_def
+ − 1237 by auto
+ − 1238 qed
+ − 1239 moreover have "clist \<in> \<lbrakk>clist\<rbrakk>Lang"
+ − 1240 by (auto simp:equiv_str_def equiv_class_def)
+ − 1241 ultimately show "?P (\<lbrakk>clist\<rbrakk>Lang)" by simp
+ − 1242 qed
+ − 1243 thus ?thesis by blast
+ − 1244 qed
+ − 1245
+ − 1246 lemma finite_charset_rS:
+ − 1247 "finite {CHAR c |c. Y-c\<rightarrow>X}"
+ − 1248 by (rule_tac A = UNIV and f = CHAR in finite_surj, auto)
+ − 1249
+ − 1250 lemma l_eq_r_in_equations:
+ − 1251 assumes X_in_equas: "(X, xrhs) \<in> equations (UNIV Quo Lang)"
+ − 1252 shows "X = L xrhs"
+ − 1253 proof (cases "X = {[]}")
+ − 1254 case True
+ − 1255 thus ?thesis using X_in_equas
+ − 1256 by (simp add:equations_def equation_rhs_def L_def lang_seq_def)
+ − 1257 next
+ − 1258 case False
+ − 1259 show ?thesis
+ − 1260 proof
+ − 1261 show "X \<subseteq> L xrhs"
+ − 1262 proof
+ − 1263 fix x
+ − 1264 assume "(1)": "x \<in> X"
+ − 1265 show "x \<in> L xrhs"
+ − 1266 proof (cases "x = []")
+ − 1267 assume empty: "x = []"
+ − 1268 hence "x \<in> L (empty_rhs X)" using "(1)"
+ − 1269 by (auto simp:empty_rhs_def L_def lang_seq_def)
+ − 1270 thus ?thesis using X_in_equas False empty "(1)"
+ − 1271 unfolding equations_def equation_rhs_def by (auto simp:L_def)
+ − 1272 next
+ − 1273 assume not_empty: "x \<noteq> []"
+ − 1274 hence "\<exists> clist c. x = clist @ [c]" by (case_tac x rule:rev_cases, auto)
+ − 1275 then obtain clist c where decom: "x = clist @ [c]" by blast
+ − 1276 moreover have "\<And> Y. \<lbrakk>Y \<in> UNIV Quo Lang; Y-c\<rightarrow>X; clist \<in> Y\<rbrakk>\<Longrightarrow> [c] \<in> L (folds ALT NULL {CHAR c |c. Y-c\<rightarrow>X})"
+ − 1277 proof -
+ − 1278 fix Y
+ − 1279 assume Y_is_eq_cl: "Y \<in> UNIV Quo Lang"
+ − 1280 and Y_CT_X: "Y-c\<rightarrow>X"
+ − 1281 and clist_in_Y: "clist \<in> Y"
+ − 1282 with finite_charset_rS
+ − 1283 show "[c] \<in> L (folds ALT NULL {CHAR c |c. Y-c\<rightarrow>X})"
+ − 1284 by (auto simp :fold_alt_null_eqs)
+ − 1285 qed
+ − 1286 hence "\<exists>Xa. Xa \<in> UNIV Quo Lang \<and> clist @ [c] \<in> Xa ; L (folds ALT NULL {CHAR c |c. Xa-c\<rightarrow>X})"
+ − 1287 using X_in_equas False not_empty "(1)" decom
+ − 1288 by (auto dest!:every_eqclass_has_ascendent simp:equations_def equation_rhs_def L_def lang_seq_def)
+ − 1289 then obtain Xa where "Xa \<in> UNIV Quo Lang \<and> clist @ [c] \<in> Xa ; L (folds ALT NULL {CHAR c |c. Xa-c\<rightarrow>X})" by blast
+ − 1290 hence "x \<in> L {(S, folds ALT NULL {CHAR c |c. S-c\<rightarrow>X}) |S. S \<in> UNIV Quo Lang}" using X_in_equas "(1)" decom
+ − 1291 by (auto simp add:L_def equations_def equation_rhs_def intro!:exI[where x = Xa])
+ − 1292 thus "x \<in> L xrhs" using X_in_equas False not_empty unfolding equations_def equation_rhs_def
+ − 1293 by (auto simp:L_def)
+ − 1294 qed
+ − 1295 qed
+ − 1296 next
+ − 1297 show "L xrhs \<subseteq> X"
+ − 1298 proof
+ − 1299 fix x
+ − 1300 assume "(2)": "x \<in> L xrhs"
+ − 1301 have "(2_1)": "\<And> s1 s2 r Xa. \<lbrakk>s1 \<in> Xa; s2 \<in> L (folds ALT NULL {CHAR c |c. Xa-c\<rightarrow>X})\<rbrakk> \<Longrightarrow> s1 @ s2 \<in> X"
+ − 1302 using finite_charset_rS
+ − 1303 by (auto simp:CT_def lang_seq_def fold_alt_null_eqs)
+ − 1304 have "(2_2)": "\<And> s1 s2 Xa r.\<lbrakk>s1 \<in> Xa; s2 \<in> L r; (Xa, r) \<in> empty_rhs X\<rbrakk> \<Longrightarrow> s1 @ s2 \<in> X"
+ − 1305 by (simp add:empty_rhs_def split:if_splits)
+ − 1306 show "x \<in> X" using X_in_equas False "(2)"
+ − 1307 by (auto intro:"(2_1)" "(2_2)" simp:equations_def equation_rhs_def L_def lang_seq_def)
+ − 1308 qed
+ − 1309 qed
+ − 1310 qed
+ − 1311
+ − 1312 lemma finite_CT_chars:
+ − 1313 "finite {CHAR c |c. Xa-c\<rightarrow>X}"
+ − 1314 by (auto)
+ − 1315
+ − 1316 lemma no_EMPTY_equations:
+ − 1317 "(X, xrhs) \<in> equations CS \<Longrightarrow> no_EMPTY_rhs xrhs"
+ − 1318 apply (clarsimp simp add:equations_def equation_rhs_def)
+ − 1319 apply (simp add:no_EMPTY_rhs_def empty_rhs_def, auto)
+ − 1320 apply (subgoal_tac "finite {CHAR c |c. Xa-c\<rightarrow>X}", drule_tac x = "[]" in fold_alt_null_eqs, clarsimp, rule finite_CT_chars)+
+ − 1321 done
+ − 1322
+ − 1323 lemma init_ES_satisfy_ardenable:
+ − 1324 "(X, xrhs) \<in> equations (UNIV Quo Lang) \<Longrightarrow> ardenable (X, xrhs)"
+ − 1325 unfolding ardenable_def
+ − 1326 by (auto intro:distinct_rhs_equations no_EMPTY_equations simp:l_eq_r_in_equations)
+ − 1327
+ − 1328 lemma init_ES_satisfy_Inv:
+ − 1329 assumes finite_CS: "finite (UNIV Quo Lang)"
+ − 1330 and X_in_eq_cls: "X \<in> UNIV Quo Lang"
+ − 1331 shows "Inv X (equations (UNIV Quo Lang))"
+ − 1332 proof -
+ − 1333 have "finite (equations (UNIV Quo Lang))" using finite_CS
+ − 1334 by (auto simp:equations_def)
+ − 1335 moreover have "\<exists>rhs. (X, rhs) \<in> equations (UNIV Quo Lang)" using X_in_eq_cls
+ − 1336 by (simp add:equations_def)
+ − 1337 moreover have "distinct_equas (equations (UNIV Quo Lang))"
+ − 1338 by (auto simp:distinct_equas_def equations_def)
+ − 1339 moreover have "\<forall>X xrhs. (X, xrhs) \<in> equations (UNIV Quo Lang) \<longrightarrow>
+ − 1340 rhs_eq_cls xrhs \<subseteq> insert {[]} (left_eq_cls (equations (UNIV Quo Lang)))"
+ − 1341 apply (simp add:left_eq_cls_def equations_def rhs_eq_cls_def equation_rhs_def)
+ − 1342 by (auto simp:empty_rhs_def split:if_splits)
+ − 1343 moreover have "\<forall>X xrhs. (X, xrhs) \<in> equations (UNIV Quo Lang) \<longrightarrow> X \<noteq> {}"
+ − 1344 by (clarsimp simp:equations_def empty_notin_CS intro:classical)
+ − 1345 moreover have "\<forall>X xrhs. (X, xrhs) \<in> equations (UNIV Quo Lang) \<longrightarrow> ardenable (X, xrhs)"
+ − 1346 by (auto intro!:init_ES_satisfy_ardenable)
+ − 1347 ultimately show ?thesis by (simp add:Inv_def)
+ − 1348 qed
+ − 1349
+ − 1350
+ − 1351 (* ********************************* END: proving the initial equation-system satisfies Inv **************************************** *)
+ − 1352
+ − 1353
+ − 1354 (* ***************************** BEGIN: proving every equation-system's iteration step satisfies Inv ******************************* *)
+ − 1355
+ − 1356 lemma not_T_aux: "\<lbrakk>\<not> TCon (insert a A); x = a\<rbrakk>
+ − 1357 \<Longrightarrow> \<exists>y. x \<noteq> y \<and> y \<in> insert a A "
+ − 1358 apply (case_tac "insert a A = {a}")
+ − 1359 by (auto simp:TCon_def)
+ − 1360
+ − 1361 lemma not_T_atleast_2[rule_format]:
+ − 1362 "finite S \<Longrightarrow> \<forall> x. x \<in> S \<and> (\<not> TCon S)\<longrightarrow> (\<exists> y. x \<noteq> y \<and> y \<in> S)"
+ − 1363 apply (erule finite.induct, simp)
+ − 1364 apply (clarify, case_tac "x = a")
+ − 1365 by (erule not_T_aux, auto)
+ − 1366
+ − 1367 lemma exist_another_equa:
+ − 1368 "\<lbrakk>\<not> TCon ES; finite ES; distinct_equas ES; (X, rhl) \<in> ES\<rbrakk> \<Longrightarrow> \<exists> Y yrhl. (Y, yrhl) \<in> ES \<and> X \<noteq> Y"
+ − 1369 apply (drule not_T_atleast_2, simp)
+ − 1370 apply (clarsimp simp:distinct_equas_def)
+ − 1371 apply (drule_tac x= X in spec, drule_tac x = rhl in spec, drule_tac x = b in spec)
+ − 1372 by auto
+ − 1373
+ − 1374 lemma Inv_mono_with_lambda:
+ − 1375 assumes Inv_ES: "Inv X ES"
+ − 1376 and X_noteq_Y: "X \<noteq> {[]}"
+ − 1377 shows "Inv X (ES - {({[]}, yrhs)})"
+ − 1378 proof -
+ − 1379 have "finite (ES - {({[]}, yrhs)})" using Inv_ES
+ − 1380 by (simp add:Inv_def)
+ − 1381 moreover have "\<exists>rhs. (X, rhs) \<in> ES - {({[]}, yrhs)}" using Inv_ES X_noteq_Y
+ − 1382 by (simp add:Inv_def)
+ − 1383 moreover have "distinct_equas (ES - {({[]}, yrhs)})" using Inv_ES X_noteq_Y
+ − 1384 apply (clarsimp simp:Inv_def distinct_equas_def)
+ − 1385 by (drule_tac x = Xa in spec, simp)
+ − 1386 moreover have "\<forall>X xrhs.(X, xrhs) \<in> ES - {({[]}, yrhs)} \<longrightarrow>
+ − 1387 ardenable (X, xrhs) \<and> X \<noteq> {}" using Inv_ES
+ − 1388 by (clarify, simp add:Inv_def)
+ − 1389 moreover
+ − 1390 have "insert {[]} (left_eq_cls (ES - {({[]}, yrhs)})) = insert {[]} (left_eq_cls ES)"
+ − 1391 by (auto simp:left_eq_cls_def)
+ − 1392 hence "\<forall>X xrhs.(X, xrhs) \<in> ES - {({[]}, yrhs)} \<longrightarrow>
+ − 1393 rhs_eq_cls xrhs \<subseteq> insert {[]} (left_eq_cls (ES - {({[]}, yrhs)}))"
+ − 1394 using Inv_ES by (auto simp:Inv_def)
+ − 1395 ultimately show ?thesis by (simp add:Inv_def)
+ − 1396 qed
+ − 1397
+ − 1398 lemma non_empty_card_prop:
+ − 1399 "finite ES \<Longrightarrow> \<forall>e. e \<in> ES \<longrightarrow> card ES - Suc 0 < card ES"
+ − 1400 apply (erule finite.induct, simp)
+ − 1401 apply (case_tac[!] "a \<in> A")
+ − 1402 by (auto simp:insert_absorb)
+ − 1403
+ − 1404 lemma ardenable_prop:
+ − 1405 assumes not_lambda: "Y \<noteq> {[]}"
+ − 1406 and ardable: "ardenable (Y, yrhs)"
+ − 1407 shows "\<exists> yrhs'. Y = L yrhs' \<and> distinct_rhs yrhs' \<and> rhs_eq_cls yrhs' = rhs_eq_cls yrhs - {Y}" (is "\<exists> yrhs'. ?P yrhs'")
+ − 1408 proof (cases "(\<exists> reg. (Y, reg) \<in> yrhs)")
+ − 1409 case True
+ − 1410 thus ?thesis
+ − 1411 proof
+ − 1412 fix reg
+ − 1413 assume self_contained: "(Y, reg) \<in> yrhs"
+ − 1414 show ?thesis
+ − 1415 proof -
+ − 1416 have "?P (arden_variate Y reg yrhs)"
+ − 1417 proof -
+ − 1418 have "Y = L (arden_variate Y reg yrhs)"
+ − 1419 using self_contained not_lambda ardable
+ − 1420 by (rule_tac arden_variate_valid, simp_all add:ardenable_def)
+ − 1421 moreover have "distinct_rhs (arden_variate Y reg yrhs)"
+ − 1422 using ardable
+ − 1423 by (auto simp:distinct_rhs_def arden_variate_def seq_rhs_r_def del_x_paired_def ardenable_def)
+ − 1424 moreover have "rhs_eq_cls (arden_variate Y reg yrhs) = rhs_eq_cls yrhs - {Y}"
+ − 1425 proof -
+ − 1426 have "\<And> rhs r. rhs_eq_cls (seq_rhs_r rhs r) = rhs_eq_cls rhs"
+ − 1427 apply (auto simp:rhs_eq_cls_def seq_rhs_r_def image_def)
+ − 1428 by (rule_tac x = "SEQ ra r" in exI, rule_tac x = "(x, ra)" in bexI, simp+)
+ − 1429 moreover have "\<And> rhs X. rhs_eq_cls (del_x_paired rhs X) = rhs_eq_cls rhs - {X}"
+ − 1430 by (auto simp:rhs_eq_cls_def del_x_paired_def)
+ − 1431 ultimately show ?thesis by (simp add:arden_variate_def)
+ − 1432 qed
+ − 1433 ultimately show ?thesis by simp
+ − 1434 qed
+ − 1435 thus ?thesis by (rule_tac x= "arden_variate Y reg yrhs" in exI, simp)
+ − 1436 qed
+ − 1437 qed
+ − 1438 next
+ − 1439 case False
+ − 1440 hence "(2)": "rhs_eq_cls yrhs - {Y} = rhs_eq_cls yrhs"
+ − 1441 by (auto simp:rhs_eq_cls_def)
+ − 1442 show ?thesis
+ − 1443 proof -
+ − 1444 have "?P yrhs" using False ardable "(2)"
+ − 1445 by (simp add:ardenable_def)
+ − 1446 thus ?thesis by blast
+ − 1447 qed
+ − 1448 qed
+ − 1449
+ − 1450 lemma equas_subst_f_del_no_other:
+ − 1451 assumes self_contained: "(Y, rhs) \<in> ES"
+ − 1452 shows "\<exists> rhs'. (Y, rhs') \<in> (equas_subst_f X xrhs ` ES)" (is "\<exists> rhs'. ?P rhs'")
+ − 1453 proof -
+ − 1454 have "\<exists> rhs'. equas_subst_f X xrhs (Y, rhs) = (Y, rhs')"
+ − 1455 by (auto simp:equas_subst_f_def)
+ − 1456 then obtain rhs' where "equas_subst_f X xrhs (Y, rhs) = (Y, rhs')" by blast
+ − 1457 hence "?P rhs'" unfolding image_def using self_contained
+ − 1458 by (auto intro:bexI[where x = "(Y, rhs)"])
+ − 1459 thus ?thesis by blast
+ − 1460 qed
+ − 1461
+ − 1462 lemma del_x_paired_del_only_x:
+ − 1463 "\<lbrakk>X \<noteq> Y; (X, rhs) \<in> ES\<rbrakk> \<Longrightarrow> (X, rhs) \<in> del_x_paired ES Y"
+ − 1464 by (auto simp:del_x_paired_def)
+ − 1465
+ − 1466 lemma del_x_paired_del_only_x':
+ − 1467 "(X, rhs) \<in> del_x_paired ES Y \<Longrightarrow> X \<noteq> Y \<and> (X, rhs) \<in> ES"
+ − 1468 by (auto simp:del_x_paired_def)
+ − 1469
+ − 1470 lemma equas_subst_del_no_other:
+ − 1471 "\<lbrakk>(X, rhs) \<in> ES; X \<noteq> Y\<rbrakk> \<Longrightarrow> (\<exists>rhs. (X, rhs) \<in> equas_subst ES Y rhs')"
+ − 1472 unfolding equas_subst_def
+ − 1473 apply (drule_tac X = Y and xrhs = rhs' in equas_subst_f_del_no_other)
+ − 1474 by (erule exE, drule del_x_paired_del_only_x, auto)
+ − 1475
+ − 1476 lemma equas_subst_holds_distinct:
+ − 1477 "distinct_equas ES \<Longrightarrow> distinct_equas (equas_subst ES Y rhs')"
+ − 1478 apply (clarsimp simp add:equas_subst_def distinct_equas_def del_x_paired_def equas_subst_f_def)
+ − 1479 by (auto split:if_splits)
+ − 1480
+ − 1481 lemma del_x_paired_dels:
+ − 1482 "(X, rhs) \<in> ES \<Longrightarrow> {Y. Y \<in> ES \<and> fst Y = X} \<inter> ES \<noteq> {}"
+ − 1483 by (auto)
+ − 1484
+ − 1485 lemma del_x_paired_subset:
+ − 1486 "(X, rhs) \<in> ES \<Longrightarrow> ES - {Y. Y \<in> ES \<and> fst Y = X} \<subset> ES"
+ − 1487 apply (drule del_x_paired_dels)
+ − 1488 by auto
+ − 1489
+ − 1490 lemma del_x_paired_card_less:
+ − 1491 "\<lbrakk>finite ES; (X, rhs) \<in> ES\<rbrakk> \<Longrightarrow> card (del_x_paired ES X) < card ES"
+ − 1492 apply (simp add:del_x_paired_def)
+ − 1493 apply (drule del_x_paired_subset)
+ − 1494 by (auto intro:psubset_card_mono)
+ − 1495
+ − 1496 lemma equas_subst_card_less:
+ − 1497 "\<lbrakk>finite ES; (Y, yrhs) \<in> ES\<rbrakk> \<Longrightarrow> card (equas_subst ES Y rhs') < card ES"
+ − 1498 apply (simp add:equas_subst_def)
+ − 1499 apply (frule_tac h = "equas_subst_f Y rhs'" in finite_imageI)
+ − 1500 apply (drule_tac f = "equas_subst_f Y rhs'" in Finite_Set.card_image_le)
+ − 1501 apply (drule_tac X = Y and xrhs = rhs' in equas_subst_f_del_no_other,erule exE)
+ − 1502 by (drule del_x_paired_card_less, auto)
+ − 1503
+ − 1504 lemma equas_subst_holds_distinct_rhs:
+ − 1505 assumes dist': "distinct_rhs yrhs'"
+ − 1506 and history: "\<forall>X xrhs. (X, xrhs) \<in> ES \<longrightarrow> ardenable (X, xrhs)"
+ − 1507 and X_in : "(X, xrhs) \<in> equas_subst ES Y yrhs'"
+ − 1508 shows "distinct_rhs xrhs"
+ − 1509 using X_in history
+ − 1510 apply (clarsimp simp:equas_subst_def del_x_paired_def)
+ − 1511 apply (drule_tac x = a in spec, drule_tac x = b in spec)
+ − 1512 apply (simp add:ardenable_def equas_subst_f_def)
+ − 1513 by (auto intro:rhs_subst_holds_distinct_rhs simp:dist' split:if_splits)
+ − 1514
+ − 1515 lemma r_no_EMPTY_imp_seq_rhs_r_no_EMPTY:
+ − 1516 "[] \<notin> L r \<Longrightarrow> no_EMPTY_rhs (seq_rhs_r rhs r)"
+ − 1517 by (auto simp:no_EMPTY_rhs_def seq_rhs_r_def lang_seq_def)
+ − 1518
+ − 1519 lemma del_x_paired_holds_no_EMPTY:
+ − 1520 "no_EMPTY_rhs yrhs \<Longrightarrow> no_EMPTY_rhs (del_x_paired yrhs Y)"
+ − 1521 by (auto simp:no_EMPTY_rhs_def del_x_paired_def)
+ − 1522
+ − 1523 lemma rhs_subst_holds_no_EMPTY:
+ − 1524 "\<lbrakk>no_EMPTY_rhs yrhs; (Y, r) \<in> yrhs; Y \<noteq> {[]}\<rbrakk> \<Longrightarrow> no_EMPTY_rhs (rhs_subst yrhs Y rhs' r)"
+ − 1525 apply (auto simp:rhs_subst_def intro!:no_EMPTY_rhss_imp_merge_no_EMPTY r_no_EMPTY_imp_seq_rhs_r_no_EMPTY del_x_paired_holds_no_EMPTY)
+ − 1526 by (auto simp:no_EMPTY_rhs_def)
+ − 1527
+ − 1528 lemma equas_subst_holds_no_EMPTY:
+ − 1529 assumes substor: "Y \<noteq> {[]}"
+ − 1530 and history: "\<forall>X xrhs. (X, xrhs) \<in> ES \<longrightarrow> ardenable (X, xrhs)"
+ − 1531 and X_in:"(X, xrhs) \<in> equas_subst ES Y yrhs'"
+ − 1532 shows "no_EMPTY_rhs xrhs"
+ − 1533 proof-
+ − 1534 from X_in have "\<exists> (Z, zrhs) \<in> ES. (X, xrhs) = equas_subst_f Y yrhs' (Z, zrhs)"
+ − 1535 by (auto simp add:equas_subst_def del_x_paired_def)
+ − 1536 then obtain Z zrhs where Z_in: "(Z, zrhs) \<in> ES"
+ − 1537 and X_Z: "(X, xrhs) = equas_subst_f Y yrhs' (Z, zrhs)" by blast
+ − 1538 hence dist_zrhs: "distinct_rhs zrhs" using history
+ − 1539 by (auto simp:ardenable_def)
+ − 1540 show ?thesis
+ − 1541 proof (cases "\<exists> r. (Y, r) \<in> zrhs")
+ − 1542 case True
+ − 1543 then obtain r where Y_in_zrhs: "(Y, r) \<in> zrhs" ..
+ − 1544 hence some: "(SOME r. (Y, r) \<in> zrhs) = r" using Z_in dist_zrhs
+ − 1545 by (auto simp:distinct_rhs_def)
+ − 1546 hence "no_EMPTY_rhs (rhs_subst zrhs Y yrhs' r)"
+ − 1547 using substor Y_in_zrhs history Z_in
+ − 1548 by (rule_tac rhs_subst_holds_no_EMPTY, auto simp:ardenable_def)
+ − 1549 thus ?thesis using X_Z True some
+ − 1550 by (simp add:equas_subst_def equas_subst_f_def)
+ − 1551 next
+ − 1552 case False
+ − 1553 hence "(X, xrhs) = (Z, zrhs)" using Z_in X_Z
+ − 1554 by (simp add:equas_subst_f_def)
+ − 1555 thus ?thesis using history Z_in
+ − 1556 by (auto simp:ardenable_def)
+ − 1557 qed
+ − 1558 qed
+ − 1559
+ − 1560 lemma equas_subst_f_holds_left_eq_right:
+ − 1561 assumes substor: "Y = L rhs'"
+ − 1562 and history: "\<forall>X xrhs. (X, xrhs) \<in> ES \<longrightarrow> distinct_rhs xrhs \<and> X = L xrhs"
+ − 1563 and subst: "(X, xrhs) = equas_subst_f Y rhs' (Z, zrhs)"
+ − 1564 and self_contained: "(Z, zrhs) \<in> ES"
+ − 1565 shows "X = L xrhs"
+ − 1566 proof (cases "\<exists> r. (Y, r) \<in> zrhs")
+ − 1567 case True
+ − 1568 from True obtain r where "(1)":"(Y, r) \<in> zrhs" ..
+ − 1569 show ?thesis
+ − 1570 proof -
+ − 1571 from history self_contained
+ − 1572 have dist: "distinct_rhs zrhs" by auto
+ − 1573 hence "(SOME r. (Y, r) \<in> zrhs) = r" using self_contained "(1)"
+ − 1574 using distinct_rhs_def by (auto intro:some_equality)
+ − 1575 moreover have "L zrhs = L (rhs_subst zrhs Y rhs' r)" using substor dist "(1)" self_contained
+ − 1576 by (rule_tac rhs_subst_prop1, auto simp:distinct_equas_rhs_def)
+ − 1577 ultimately show ?thesis using subst history self_contained
+ − 1578 by (auto simp:equas_subst_f_def split:if_splits)
+ − 1579 qed
+ − 1580 next
+ − 1581 case False
+ − 1582 thus ?thesis using history subst self_contained
+ − 1583 by (auto simp:equas_subst_f_def)
+ − 1584 qed
+ − 1585
+ − 1586 lemma equas_subst_holds_left_eq_right:
+ − 1587 assumes history: "\<forall>X xrhs. (X, xrhs) \<in> ES \<longrightarrow> ardenable (X, xrhs)"
+ − 1588 and substor: "Y = L rhs'"
+ − 1589 and X_in : "(X, xrhs) \<in> equas_subst ES Y yrhs'"
+ − 1590 shows "\<forall>X xrhs. (X, xrhs) \<in> equas_subst ES Y rhs' \<longrightarrow> X = L xrhs"
+ − 1591 apply (clarsimp simp add:equas_subst_def del_x_paired_def)
+ − 1592 using substor
+ − 1593 apply (drule_tac equas_subst_f_holds_left_eq_right)
+ − 1594 using history
+ − 1595 by (auto simp:ardenable_def)
+ − 1596
+ − 1597 lemma equas_subst_holds_ardenable:
+ − 1598 assumes substor: "Y = L yrhs'"
+ − 1599 and history: "\<forall>X xrhs. (X, xrhs) \<in> ES \<longrightarrow> ardenable (X, xrhs)"
+ − 1600 and X_in:"(X, xrhs) \<in> equas_subst ES Y yrhs'"
+ − 1601 and dist': "distinct_rhs yrhs'"
+ − 1602 and not_lambda: "Y \<noteq> {[]}"
+ − 1603 shows "ardenable (X, xrhs)"
+ − 1604 proof -
+ − 1605 have "distinct_rhs xrhs" using history X_in dist'
+ − 1606 by (auto dest:equas_subst_holds_distinct_rhs)
+ − 1607 moreover have "no_EMPTY_rhs xrhs" using history X_in not_lambda
+ − 1608 by (auto intro:equas_subst_holds_no_EMPTY)
+ − 1609 moreover have "X = L xrhs" using history substor X_in
+ − 1610 by (auto dest: equas_subst_holds_left_eq_right)
+ − 1611 ultimately show ?thesis using ardenable_def by simp
+ − 1612 qed
+ − 1613
+ − 1614 lemma equas_subst_holds_cls_defined:
+ − 1615 assumes X_in: "(X, xrhs) \<in> equas_subst ES Y yrhs'"
+ − 1616 and Inv_ES: "Inv X' ES"
+ − 1617 and subst: "(Y, yrhs) \<in> ES"
+ − 1618 and cls_holds_but_Y: "rhs_eq_cls yrhs' = rhs_eq_cls yrhs - {Y}"
+ − 1619 shows "rhs_eq_cls xrhs \<subseteq> insert {[]} (left_eq_cls (equas_subst ES Y yrhs'))"
+ − 1620 proof-
+ − 1621 have tac: "\<lbrakk> A \<subseteq> B; C \<subseteq> D; E \<subseteq> A \<union> B\<rbrakk> \<Longrightarrow> E \<subseteq> B \<union> D" by auto
+ − 1622 from X_in have "\<exists> (Z, zrhs) \<in> ES. (X, xrhs) = equas_subst_f Y yrhs' (Z, zrhs)"
+ − 1623 by (auto simp add:equas_subst_def del_x_paired_def)
+ − 1624 then obtain Z zrhs where Z_in: "(Z, zrhs) \<in> ES"
+ − 1625 and X_Z: "(X, xrhs) = equas_subst_f Y yrhs' (Z, zrhs)" by blast
+ − 1626 hence "rhs_eq_cls zrhs \<subseteq> insert {[]} (left_eq_cls ES)" using Inv_ES
+ − 1627 by (auto simp:Inv_def)
+ − 1628 moreover have "rhs_eq_cls yrhs' \<subseteq> insert {[]} (left_eq_cls ES) - {Y}"
+ − 1629 using Inv_ES subst cls_holds_but_Y
+ − 1630 by (auto simp:Inv_def)
+ − 1631 moreover have "rhs_eq_cls xrhs \<subseteq> rhs_eq_cls zrhs \<union> rhs_eq_cls yrhs' - {Y}"
+ − 1632 using X_Z cls_holds_but_Y
+ − 1633 apply (clarsimp simp add:equas_subst_f_def rhs_subst_def split:if_splits)
+ − 1634 by (auto simp:rhs_eq_cls_def merge_rhs_def del_x_paired_def seq_rhs_r_def)
+ − 1635 moreover have "left_eq_cls (equas_subst ES Y yrhs') = left_eq_cls ES - {Y}" using subst
+ − 1636 by (auto simp: left_eq_cls_def equas_subst_def del_x_paired_def equas_subst_f_def
+ − 1637 dest: equas_subst_f_del_no_other
+ − 1638 split: if_splits)
+ − 1639 ultimately show ?thesis by blast
+ − 1640 qed
+ − 1641
+ − 1642 lemma iteration_step:
+ − 1643 assumes Inv_ES: "Inv X ES"
+ − 1644 and not_T: "\<not> TCon ES"
+ − 1645 shows "(\<exists> ES'. Inv X ES' \<and> (card ES', card ES) \<in> less_than)"
+ − 1646 proof -
+ − 1647 from Inv_ES not_T have another: "\<exists>Y yrhs. (Y, yrhs) \<in> ES \<and> X \<noteq> Y" unfolding Inv_def
+ − 1648 by (clarify, rule_tac exist_another_equa[where X = X], auto)
+ − 1649 then obtain Y yrhs where subst: "(Y, yrhs) \<in> ES" and not_X: " X \<noteq> Y" by blast
+ − 1650 show ?thesis (is "\<exists> ES'. ?P ES'")
+ − 1651 proof (cases "Y = {[]}")
+ − 1652 case True
+ − 1653 --"in this situation, we pick a \"\<lambda>\" equation, thus directly remove it from the equation-system"
+ − 1654 have "?P (ES - {(Y, yrhs)})"
+ − 1655 proof
+ − 1656 show "Inv X (ES - {(Y, yrhs)})" using True not_X
+ − 1657 by (simp add:Inv_ES Inv_mono_with_lambda)
+ − 1658 next
+ − 1659 show "(card (ES - {(Y, yrhs)}), card ES) \<in> less_than" using Inv_ES subst
+ − 1660 by (auto elim:non_empty_card_prop[rule_format] simp:Inv_def)
+ − 1661 qed
+ − 1662 thus ?thesis by blast
+ − 1663 next
+ − 1664 case False
+ − 1665 --"in this situation, we pick a equation and using ardenable to get a rhs without itself in it, then use equas_subst to form a new equation-system"
+ − 1666 hence "\<exists> yrhs'. Y = L yrhs' \<and> distinct_rhs yrhs' \<and> rhs_eq_cls yrhs' = rhs_eq_cls yrhs - {Y}" using subst Inv_ES
+ − 1667 by (auto intro:ardenable_prop simp:Inv_def)
+ − 1668 then obtain yrhs' where Y'_l_eq_r: "Y = L yrhs'"
+ − 1669 and dist_Y': "distinct_rhs yrhs'"
+ − 1670 and cls_holds_but_Y: "rhs_eq_cls yrhs' = rhs_eq_cls yrhs - {Y}" by blast
+ − 1671 hence "?P (equas_subst ES Y yrhs')"
+ − 1672 proof -
+ − 1673 have finite_del: "\<And> S x. finite S \<Longrightarrow> finite (del_x_paired S x)"
+ − 1674 apply (rule_tac A = "del_x_paired S x" in finite_subset)
+ − 1675 by (auto simp:del_x_paired_def)
+ − 1676 have "finite (equas_subst ES Y yrhs')" using Inv_ES
+ − 1677 by (auto intro!:finite_del simp:equas_subst_def Inv_def)
+ − 1678 moreover have "\<exists>rhs. (X, rhs) \<in> equas_subst ES Y yrhs'" using Inv_ES not_X
+ − 1679 by (auto intro:equas_subst_del_no_other simp:Inv_def)
+ − 1680 moreover have "distinct_equas (equas_subst ES Y yrhs')" using Inv_ES
+ − 1681 by (auto intro:equas_subst_holds_distinct simp:Inv_def)
+ − 1682 moreover have "\<forall>X xrhs. (X, xrhs) \<in> equas_subst ES Y yrhs' \<longrightarrow> ardenable (X, xrhs)"
+ − 1683 using Inv_ES dist_Y' False Y'_l_eq_r
+ − 1684 apply (clarsimp simp:Inv_def)
+ − 1685 by (rule equas_subst_holds_ardenable, simp_all)
+ − 1686 moreover have "\<forall>X xrhs. (X, xrhs) \<in> equas_subst ES Y yrhs' \<longrightarrow> X \<noteq> {}" using Inv_ES
+ − 1687 by (clarsimp simp:equas_subst_def Inv_def del_x_paired_def equas_subst_f_def split:if_splits, auto)
+ − 1688 moreover have "\<forall>X xrhs. (X, xrhs) \<in> equas_subst ES Y yrhs' \<longrightarrow>
+ − 1689 rhs_eq_cls xrhs \<subseteq> insert {[]} (left_eq_cls (equas_subst ES Y yrhs'))"
+ − 1690 using Inv_ES subst cls_holds_but_Y
+ − 1691 apply (rule_tac impI | rule_tac allI)+
+ − 1692 by (erule equas_subst_holds_cls_defined, auto)
+ − 1693 moreover have "(card (equas_subst ES Y yrhs'), card ES) \<in> less_than"using Inv_ES subst
+ − 1694 by (simp add:equas_subst_card_less Inv_def)
+ − 1695 ultimately show "?P (equas_subst ES Y yrhs')" by (auto simp:Inv_def)
+ − 1696 qed
+ − 1697 thus ?thesis by blast
+ − 1698 qed
+ − 1699 qed
+ − 1700
+ − 1701 (* ****************************** END: proving every equation-system's iteration step satisfies Inv ******************************* *)
+ − 1702
+ − 1703 lemma iteration_conc:
+ − 1704 assumes history: "Inv X ES"
+ − 1705 shows "\<exists> ES'. Inv X ES' \<and> TCon ES'" (is "\<exists> ES'. ?P ES'")
+ − 1706 proof (cases "TCon ES")
+ − 1707 case True
+ − 1708 hence "?P ES" using history by simp
+ − 1709 thus ?thesis by blast
+ − 1710 next
+ − 1711 case False
+ − 1712 thus ?thesis using history iteration_step
+ − 1713 by (rule_tac f = card in wf_iter, simp_all)
+ − 1714 qed
+ − 1715
+ − 1716 lemma eqset_imp_iff': "A = B \<Longrightarrow> \<forall> x. x \<in> A \<longleftrightarrow> x \<in> B"
+ − 1717 apply (auto simp:mem_def)
+ − 1718 done
+ − 1719
+ − 1720 lemma set_cases2:
+ − 1721 "\<lbrakk>(A = {} \<Longrightarrow> R A); \<And> x. (A = {x}) \<Longrightarrow> R A; \<And> x y. \<lbrakk>x \<noteq> y; x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> R A\<rbrakk> \<Longrightarrow> R A"
+ − 1722 apply (case_tac "A = {}", simp)
+ − 1723 by (case_tac "\<exists> x. A = {x}", clarsimp, blast)
+ − 1724
+ − 1725 lemma rhs_aux:"\<lbrakk>distinct_rhs rhs; {Y. \<exists>r. (Y, r) \<in> rhs} = {X}\<rbrakk> \<Longrightarrow> (\<exists>r. rhs = {(X, r)})"
+ − 1726 apply (rule_tac A = rhs in set_cases2, simp)
+ − 1727 apply (drule_tac x = X in eqset_imp_iff, clarsimp)
+ − 1728 apply (drule eqset_imp_iff',clarsimp)
+ − 1729 apply (frule_tac x = a in spec, drule_tac x = aa in spec)
+ − 1730 by (auto simp:distinct_rhs_def)
+ − 1731
+ − 1732 lemma every_eqcl_has_reg:
+ − 1733 assumes finite_CS: "finite (UNIV Quo Lang)"
+ − 1734 and X_in_CS: "X \<in> (UNIV Quo Lang)"
+ − 1735 shows "\<exists> (reg::rexp). L reg = X" (is "\<exists> r. ?E r")
+ − 1736 proof-
+ − 1737 have "\<exists>ES'. Inv X ES' \<and> TCon ES'" using finite_CS X_in_CS
+ − 1738 by (auto intro:init_ES_satisfy_Inv iteration_conc)
+ − 1739 then obtain ES' where Inv_ES': "Inv X ES'"
+ − 1740 and TCon_ES': "TCon ES'" by blast
+ − 1741 from Inv_ES' TCon_ES'
+ − 1742 have "\<exists> rhs. ES' = {(X, rhs)}"
+ − 1743 apply (clarsimp simp:Inv_def TCon_def)
+ − 1744 apply (rule_tac x = rhs in exI)
+ − 1745 by (auto dest!:card_Suc_Diff1 simp:card_eq_0_iff)
+ − 1746 then obtain rhs where ES'_single_equa: "ES' = {(X, rhs)}" ..
+ − 1747 hence X_ardenable: "ardenable (X, rhs)" using Inv_ES'
+ − 1748 by (simp add:Inv_def)
+ − 1749
+ − 1750 from X_ardenable have X_l_eq_r: "X = L rhs"
+ − 1751 by (simp add:ardenable_def)
+ − 1752 hence rhs_not_empty: "rhs \<noteq> {}" using Inv_ES' ES'_single_equa
+ − 1753 by (auto simp:Inv_def ardenable_def L_def)
+ − 1754 have rhs_eq_cls: "rhs_eq_cls rhs \<subseteq> {X, {[]}}"
+ − 1755 using Inv_ES' ES'_single_equa
+ − 1756 by (auto simp:Inv_def ardenable_def left_eq_cls_def)
+ − 1757 have X_not_empty: "X \<noteq> {}" using Inv_ES' ES'_single_equa
+ − 1758 by (auto simp:Inv_def)
+ − 1759 show ?thesis
+ − 1760 proof (cases "X = {[]}")
+ − 1761 case True
+ − 1762 hence "?E EMPTY" by (simp add:L_def)
+ − 1763 thus ?thesis by blast
+ − 1764 next
+ − 1765 case False with X_ardenable
+ − 1766 have "\<exists> rhs'. X = L rhs' \<and> rhs_eq_cls rhs' = rhs_eq_cls rhs - {X} \<and> distinct_rhs rhs'"
+ − 1767 by (drule_tac ardenable_prop, auto)
+ − 1768 then obtain rhs' where X_eq_rhs': "X = L rhs'"
+ − 1769 and rhs'_eq_cls: "rhs_eq_cls rhs' = rhs_eq_cls rhs - {X}"
+ − 1770 and rhs'_dist : "distinct_rhs rhs'" by blast
+ − 1771 have "rhs_eq_cls rhs' \<subseteq> {{[]}}" using rhs_eq_cls False rhs'_eq_cls rhs_not_empty
+ − 1772 by blast
+ − 1773 hence "rhs_eq_cls rhs' = {{[]}}" using X_not_empty X_eq_rhs'
+ − 1774 by (auto simp:L_def rhs_eq_cls_def)
+ − 1775 hence "\<exists> r. rhs' = {({[]}, r)}" using rhs'_dist
+ − 1776 by (auto intro:rhs_aux simp:rhs_eq_cls_def)
+ − 1777 then obtain r where "rhs' = {({[]}, r)}" ..
+ − 1778 hence "?E r" using X_eq_rhs' by (auto simp add:L_def lang_seq_def)
+ − 1779 thus ?thesis by blast
+ − 1780 qed
+ − 1781 qed
+ − 1782
+ − 1783 theorem myhill_nerode:
+ − 1784 assumes finite_CS: "finite (UNIV Quo Lang)"
+ − 1785 shows "\<exists> (reg::rexp). Lang = L reg" (is "\<exists> r. ?P r")
+ − 1786 proof -
+ − 1787 have has_r_each: "\<forall>C\<in>{X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang}. \<exists>(r::rexp). C = L r" using finite_CS
+ − 1788 by (auto dest:every_eqcl_has_reg)
+ − 1789 have "\<exists> (rS::rexp set). finite rS \<and>
+ − 1790 (\<forall> C \<in> {X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang}. \<exists> r \<in> rS. C = L r) \<and>
+ − 1791 (\<forall> r \<in> rS. \<exists> C \<in> {X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang}. C = L r)"
+ − 1792 (is "\<exists> rS. ?Q rS")
+ − 1793 proof-
+ − 1794 have "\<And> C. C \<in> {X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang} \<Longrightarrow> C = L (SOME (ra::rexp). C = L ra)"
+ − 1795 using has_r_each
+ − 1796 apply (erule_tac x = C in ballE, erule_tac exE)
+ − 1797 by (rule_tac a = r in someI2, simp+)
+ − 1798 hence "?Q ((\<lambda> C. SOME r. C = L r) ` {X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang})" using has_r_each
+ − 1799 using finite_CS by auto
+ − 1800 thus ?thesis by blast
+ − 1801 qed
+ − 1802 then obtain rS where finite_rS : "finite rS"
+ − 1803 and has_r_each': "\<forall> C \<in> {X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang}. \<exists> r \<in> (rS::rexp set). C = L r"
+ − 1804 and has_cl_each: "\<forall> r \<in> (rS::rexp set). \<exists> C \<in> {X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang}. C = L r" by blast
+ − 1805 have "?P (folds ALT NULL rS)"
+ − 1806 proof
+ − 1807 show "Lang \<subseteq> L (folds ALT NULL rS)" using finite_rS lang_eqs_union_of_eqcls[of Lang] has_r_each'
+ − 1808 apply (clarsimp simp:fold_alt_null_eqs) by blast
+ − 1809 next
+ − 1810 show "L (folds ALT NULL rS) \<subseteq> Lang" using finite_rS lang_eqs_union_of_eqcls[of Lang] has_cl_each
+ − 1811 by (clarsimp simp:fold_alt_null_eqs)
+ − 1812 qed
+ − 1813 thus ?thesis by blast
+ − 1814 qed
+ − 1815
+ − 1816 end
+ − 1817
+ − 1818
+ − 1819
+ − 1820
+ − 1821
+ − 1822
+ − 1823
+ − 1824
+ − 1825
+ − 1826