6
+ − 1 theory MyhillNerode
23
+ − 2 imports "Main" "List_Prefix"
6
+ − 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
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+ − 10 "L1 ; L2 = {s1 @ s2 | s1 s2. s1 \<in> L1 \<and> s2 \<in> L2}"
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+ − 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)"
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+ − 36 unfolding lang_seq_def
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+ − 37 apply(auto)
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+ − 38 apply(metis)
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+ − 39 by (metis append_assoc)
6
+ − 40
+ − 41
+ − 42 section {* Kleene star for languages defined as least fixed point *}
+ − 43
+ − 44 inductive_set
+ − 45 Star :: "string set \<Rightarrow> string set" ("_\<star>" [101] 102)
+ − 46 for L :: "string set"
+ − 47 where
+ − 48 start[intro]: "[] \<in> L\<star>"
+ − 49 | step[intro]: "\<lbrakk>s1 \<in> L; s2 \<in> L\<star>\<rbrakk> \<Longrightarrow> s1@s2 \<in> L\<star>"
+ − 50
+ − 51 lemma lang_star_empty:
+ − 52 shows "{}\<star> = {[]}"
+ − 53 by (auto elim: Star.cases)
+ − 54
+ − 55 lemma lang_star_cases:
+ − 56 shows "L\<star> = {[]} \<union> L ; L\<star>"
+ − 57 proof
+ − 58 { fix x
+ − 59 have "x \<in> L\<star> \<Longrightarrow> x \<in> {[]} \<union> L ; L\<star>"
+ − 60 unfolding lang_seq_def
+ − 61 by (induct rule: Star.induct) (auto)
+ − 62 }
+ − 63 then show "L\<star> \<subseteq> {[]} \<union> L ; L\<star>" by auto
+ − 64 next
+ − 65 show "{[]} \<union> L ; L\<star> \<subseteq> L\<star>"
+ − 66 unfolding lang_seq_def by auto
+ − 67 qed
+ − 68
+ − 69 lemma lang_star_cases':
+ − 70 shows "L\<star> = {[]} \<union> L\<star> ; L"
+ − 71 proof
+ − 72 { fix x
+ − 73 have "x \<in> L\<star> \<Longrightarrow> x \<in> {[]} \<union> L\<star> ; L"
+ − 74 unfolding lang_seq_def
+ − 75 apply (induct rule: Star.induct)
+ − 76 apply simp
+ − 77 apply simp
+ − 78 apply (erule disjE)
+ − 79 apply (auto)[1]
+ − 80 apply (erule exE | erule conjE)+
+ − 81 apply (rule disjI2)
+ − 82 apply (rule_tac x = "s1 @ s1a" in exI)
+ − 83 by auto
+ − 84 }
+ − 85 then show "L\<star> \<subseteq> {[]} \<union> L\<star> ; L" by auto
+ − 86 next
+ − 87 show "{[]} \<union> L\<star> ; L \<subseteq> L\<star>"
+ − 88 unfolding lang_seq_def
+ − 89 apply auto
+ − 90 apply (erule Star.induct)
+ − 91 apply auto
+ − 92 apply (drule step[of _ _ "[]"])
+ − 93 by (auto intro:start)
+ − 94 qed
+ − 95
+ − 96 lemma lang_star_simple:
+ − 97 shows "L \<subseteq> L\<star>"
+ − 98 by (subst lang_star_cases)
+ − 99 (auto simp only: lang_seq_def)
+ − 100
+ − 101 lemma lang_star_prop0_aux:
+ − 102 "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)"
+ − 103 apply (erule Star.induct)
+ − 104 apply (clarify, rule_tac x = "[]" in exI, rule_tac x = s1 in exI, simp add:start)
+ − 105 apply (clarify, drule_tac x = s1 in spec)
+ − 106 apply (drule mp, simp, clarify)
+ − 107 apply (rule_tac x = "s1a @ s3" in exI, rule_tac x = s4 in exI)
+ − 108 by auto
+ − 109
+ − 110 lemma lang_star_prop0:
+ − 111 "\<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"
+ − 112 by (auto dest:lang_star_prop0_aux)
+ − 113
+ − 114 lemma lang_star_prop1:
+ − 115 assumes asm: "L1; L2 \<subseteq> L2"
+ − 116 shows "L1\<star>; L2 \<subseteq> L2"
+ − 117 proof -
+ − 118 { fix s1 s2
+ − 119 assume minor: "s2 \<in> L2"
+ − 120 assume major: "s1 \<in> L1\<star>"
+ − 121 then have "s1@s2 \<in> L2"
+ − 122 proof(induct rule: Star.induct)
+ − 123 case start
+ − 124 show "[]@s2 \<in> L2" using minor by simp
+ − 125 next
+ − 126 case (step s1 s1')
+ − 127 have "s1 \<in> L1" by fact
+ − 128 moreover
+ − 129 have "s1'@s2 \<in> L2" by fact
+ − 130 ultimately have "s1@(s1'@s2) \<in> L1; L2" by (auto simp add: lang_seq_def)
+ − 131 with asm have "s1@(s1'@s2) \<in> L2" by auto
+ − 132 then show "(s1@s1')@s2 \<in> L2" by simp
+ − 133 qed
+ − 134 }
+ − 135 then show "L1\<star>; L2 \<subseteq> L2" by (auto simp add: lang_seq_def)
+ − 136 qed
+ − 137
+ − 138 lemma lang_star_prop2_aux:
+ − 139 "s2 \<in> L2\<star> \<Longrightarrow> \<forall> s1. s1 \<in> L1 \<and> L1 ; L2 \<subseteq> L1 \<longrightarrow> s1 @ s2 \<in> L1"
+ − 140 apply (erule Star.induct, simp)
+ − 141 apply (clarify, drule_tac x = "s1a @ s1" in spec)
+ − 142 by (auto simp:lang_seq_def)
+ − 143
+ − 144 lemma lang_star_prop2:
+ − 145 "L1; L2 \<subseteq> L1 \<Longrightarrow> L1 ; L2\<star> \<subseteq> L1"
+ − 146 by (auto dest!:lang_star_prop2_aux simp:lang_seq_def)
+ − 147
+ − 148 lemma lang_star_seq_subseteq:
+ − 149 shows "L ; L\<star> \<subseteq> L\<star>"
+ − 150 using lang_star_cases by blast
+ − 151
+ − 152 lemma lang_star_double:
+ − 153 shows "L\<star>; L\<star> = L\<star>"
+ − 154 proof
+ − 155 show "L\<star> ; L\<star> \<subseteq> L\<star>"
+ − 156 using lang_star_prop1 lang_star_seq_subseteq by blast
+ − 157 next
+ − 158 have "L\<star> \<subseteq> L\<star> \<union> L\<star>; (L; L\<star>)" by auto
+ − 159 also have "\<dots> = L\<star>;{[]} \<union> L\<star>; (L; L\<star>)" by (simp add: lang_seq_empty)
+ − 160 also have "\<dots> = L\<star>; ({[]} \<union> L; L\<star>)" by (simp only: lang_seq_union)
+ − 161 also have "\<dots> = L\<star>; L\<star>" using lang_star_cases by simp
+ − 162 finally show "L\<star> \<subseteq> L\<star> ; L\<star>" by simp
+ − 163 qed
+ − 164
+ − 165 lemma lang_star_seq_subseteq':
+ − 166 shows "L\<star>; L \<subseteq> L\<star>"
+ − 167 proof -
+ − 168 have "L \<subseteq> L\<star>" by (rule lang_star_simple)
+ − 169 then have "L\<star>; L \<subseteq> L\<star>; L\<star>" by (auto simp add: lang_seq_def)
+ − 170 then show "L\<star>; L \<subseteq> L\<star>" using lang_star_double by blast
+ − 171 qed
+ − 172
+ − 173 lemma
+ − 174 shows "L\<star> \<subseteq> L\<star>\<star>"
+ − 175 by (rule lang_star_simple)
+ − 176
+ − 177
+ − 178 section {* regular expressions *}
+ − 179
+ − 180 datatype rexp =
+ − 181 NULL
+ − 182 | EMPTY
+ − 183 | CHAR char
+ − 184 | SEQ rexp rexp
+ − 185 | ALT rexp rexp
+ − 186 | STAR rexp
+ − 187
+ − 188
+ − 189 consts L:: "'a \<Rightarrow> string set"
+ − 190
+ − 191 overloading L_rexp \<equiv> "L:: rexp \<Rightarrow> string set"
+ − 192 begin
+ − 193 fun
+ − 194 L_rexp :: "rexp \<Rightarrow> string set"
+ − 195 where
+ − 196 "L_rexp (NULL) = {}"
+ − 197 | "L_rexp (EMPTY) = {[]}"
+ − 198 | "L_rexp (CHAR c) = {[c]}"
+ − 199 | "L_rexp (SEQ r1 r2) = (L_rexp r1) ; (L_rexp r2)"
+ − 200 | "L_rexp (ALT r1 r2) = (L_rexp r1) \<union> (L_rexp r2)"
+ − 201 | "L_rexp (STAR r) = (L_rexp r)\<star>"
+ − 202 end
+ − 203
+ − 204
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+ − 205 text{* ************ now is the codes writen by chunhan ************************************* *}
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+ − 206
+ − 207 section {* Arden's Lemma revised *}
+ − 208
+ − 209 lemma arden_aux1:
+ − 210 assumes a: "X \<subseteq> X ; A \<union> B"
+ − 211 and b: "[] \<notin> A"
+ − 212 shows "x \<in> X \<Longrightarrow> x \<in> B ; A\<star>"
+ − 213 apply (induct x taking:length rule:measure_induct)
+ − 214 apply (subgoal_tac "x \<in> X ; A \<union> B")
+ − 215 defer
+ − 216 using a
+ − 217 apply (auto)[1]
+ − 218 apply simp
+ − 219 apply (erule disjE)
+ − 220 defer
+ − 221 apply (auto simp add:lang_seq_def) [1]
+ − 222 apply (subgoal_tac "\<exists> x1 x2. x = x1 @ x2 \<and> x1 \<in> X \<and> x2 \<in> A")
+ − 223 defer
+ − 224 apply (auto simp add:lang_seq_def) [1]
+ − 225 apply (erule exE | erule conjE)+
+ − 226 apply simp
+ − 227 apply (drule_tac x = x1 in spec)
+ − 228 apply (simp)
+ − 229 using b
+ − 230 apply -
+ − 231 apply (auto)[1]
+ − 232 apply (subgoal_tac "x1 @ x2 \<in> (B ; A\<star>) ; A")
+ − 233 defer
+ − 234 apply (auto simp add:lang_seq_def)[1]
+ − 235 by (metis Un_absorb1 lang_seq_assoc lang_seq_union(2) lang_star_double lang_star_simple mem_def sup1CI)
+ − 236
+ − 237 theorem ardens_revised:
+ − 238 assumes nemp: "[] \<notin> A"
+ − 239 shows "(X = X ; A \<union> B) \<longleftrightarrow> (X = B ; A\<star>)"
+ − 240 apply(rule iffI)
+ − 241 defer
+ − 242 apply(simp)
+ − 243 apply(subst lang_star_cases')
+ − 244 apply(subst lang_seq_union)
+ − 245 apply(simp add: lang_seq_empty)
+ − 246 apply(simp add: lang_seq_assoc)
+ − 247 apply(auto)[1]
+ − 248 proof -
+ − 249 assume "X = X ; A \<union> B"
+ − 250 then have as1: "X ; A \<union> B \<subseteq> X" and as2: "X \<subseteq> X ; A \<union> B" by simp_all
+ − 251 from as1 have a: "X ; A \<subseteq> X" and b: "B \<subseteq> X" by simp_all
+ − 252 from b have "B; A\<star> \<subseteq> X ; A\<star>" by (auto simp add: lang_seq_def)
+ − 253 moreover
+ − 254 from a have "X ; A\<star> \<subseteq> X"
+ − 255
+ − 256 by (rule lang_star_prop2)
+ − 257 ultimately have f1: "B ; A\<star> \<subseteq> X" by simp
+ − 258 from as2 nemp
+ − 259 have f2: "X \<subseteq> B; A\<star>" using arden_aux1 by auto
+ − 260 from f1 f2 show "X = B; A\<star>" by auto
+ − 261 qed
+ − 262
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+ − 263
+ − 264
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+ − 265 section {* equiv class' definition *}
+ − 266
+ − 267 definition
+ − 268 equiv_str :: "string \<Rightarrow> string set \<Rightarrow> string \<Rightarrow> bool" ("_ \<equiv>_\<equiv> _" [100, 100, 100] 100)
+ − 269 where
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+ − 270 "x \<equiv>Lang\<equiv> y \<longleftrightarrow> (\<forall>z. x @ z \<in> Lang \<longleftrightarrow> y @ z \<in> Lang)"
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+ − 271
+ − 272 definition
+ − 273 equiv_class :: "string \<Rightarrow> (string set) \<Rightarrow> string set" ("\<lbrakk>_\<rbrakk>_" [100, 100] 100)
+ − 274 where
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+ − 275 "\<lbrakk>x\<rbrakk>Lang \<equiv> {y. x \<equiv>Lang\<equiv> y}"
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+ − 276
+ − 277 text {* Chunhan modifies Q to Quo *}
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+ − 278
6
+ − 279 definition
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+ − 280 quot :: "string set \<Rightarrow> string set \<Rightarrow> (string set) set" ("_ Quo _" [100, 100] 100)
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+ − 281 where
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+ − 282 "L1 Quo L2 \<equiv> { \<lbrakk>x\<rbrakk>L2 | x. x \<in> L1}"
+ − 283
23
+ − 284
6
+ − 285 lemma lang_eqs_union_of_eqcls:
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+ − 286 "Lang = \<Union> {X. X \<in> (UNIV Quo Lang) \<and> (\<forall> x \<in> X. x \<in> Lang)}"
6
+ − 287 proof
+ − 288 show "Lang \<subseteq> \<Union>{X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang}"
+ − 289 proof
+ − 290 fix x
+ − 291 assume "x \<in> Lang"
+ − 292 thus "x \<in> \<Union>{X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang}"
+ − 293 proof (simp add:quot_def)
+ − 294 assume "(1)": "x \<in> Lang"
+ − 295 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")
+ − 296 proof -
+ − 297 have "?P (\<lbrakk>x\<rbrakk>Lang)" using "(1)"
+ − 298 by (auto simp:equiv_class_def equiv_str_def dest: spec[where x = "[]"])
+ − 299 thus ?thesis by blast
+ − 300 qed
+ − 301 qed
+ − 302 qed
+ − 303 next
+ − 304 show "\<Union>{X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang} \<subseteq> Lang"
+ − 305 by auto
+ − 306 qed
+ − 307
+ − 308 lemma empty_notin_CS: "{} \<notin> UNIV Quo Lang"
+ − 309 apply (clarsimp simp:quot_def equiv_class_def)
+ − 310 by (rule_tac x = x in exI, auto simp:equiv_str_def)
+ − 311
+ − 312 lemma no_two_cls_inters:
+ − 313 "\<lbrakk>X \<in> UNIV Quo Lang; Y \<in> UNIV Quo Lang; X \<noteq> Y\<rbrakk> \<Longrightarrow> X \<inter> Y = {}"
+ − 314 by (auto simp:quot_def equiv_class_def equiv_str_def)
+ − 315
+ − 316 text {* equiv_class transition *}
+ − 317 definition
+ − 318 CT :: "string set \<Rightarrow> char \<Rightarrow> string set \<Rightarrow> bool" ("_-_\<rightarrow>_" [99,99]99)
+ − 319 where
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+ − 320 "X-c\<rightarrow>Y \<equiv> ((X;{[c]}) \<subseteq> Y)"
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+ − 321
+ − 322 types t_equa_rhs = "(string set \<times> rexp) set"
+ − 323
+ − 324 types t_equa = "(string set \<times> t_equa_rhs)"
+ − 325
+ − 326 types t_equas = "t_equa set"
+ − 327
8
+ − 328 text {*
+ − 329 "empty_rhs" generates "\<lambda>" for init-state, just like "\<lambda>" for final states
+ − 330 in Brzozowski method. But if the init-state is "{[]}" ("\<lambda>" itself) then
+ − 331 empty set is returned, see definition of "equation_rhs"
+ − 332 *}
+ − 333
6
+ − 334 definition
+ − 335 empty_rhs :: "string set \<Rightarrow> t_equa_rhs"
+ − 336 where
+ − 337 "empty_rhs X \<equiv> if ([] \<in> X) then {({[]}, EMPTY)} else {}"
+ − 338
+ − 339 definition
+ − 340 folds :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b"
+ − 341 where
+ − 342 "folds f z S \<equiv> SOME x. fold_graph f z S x"
+ − 343
+ − 344 definition
+ − 345 equation_rhs :: "(string set) set \<Rightarrow> string set \<Rightarrow> t_equa_rhs"
+ − 346 where
+ − 347 "equation_rhs CS X \<equiv> if (X = {[]}) then {({[]}, EMPTY)}
23
+ − 348 else {(S, folds ALT NULL {CHAR c| c. S-c\<rightarrow>X} )| S. S \<in> CS} \<union> empty_rhs X"
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+ − 349
+ − 350 definition
+ − 351 equations :: "(string set) set \<Rightarrow> t_equas"
+ − 352 where
+ − 353 "equations CS \<equiv> {(X, equation_rhs CS X) | X. X \<in> CS}"
+ − 354
+ − 355 overloading L_rhs \<equiv> "L:: t_equa_rhs \<Rightarrow> string set"
+ − 356 begin
+ − 357 fun L_rhs:: "t_equa_rhs \<Rightarrow> string set"
+ − 358 where
+ − 359 "L_rhs rhs = {x. \<exists> X r. (X, r) \<in> rhs \<and> x \<in> X;(L r)}"
+ − 360 end
+ − 361
+ − 362 definition
+ − 363 distinct_rhs :: "t_equa_rhs \<Rightarrow> bool"
+ − 364 where
+ − 365 "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"
+ − 366
+ − 367 definition
+ − 368 distinct_equas_rhs :: "t_equas \<Rightarrow> bool"
+ − 369 where
+ − 370 "distinct_equas_rhs equas \<equiv> \<forall> X rhs. (X, rhs) \<in> equas \<longrightarrow> distinct_rhs rhs"
+ − 371
+ − 372 definition
+ − 373 distinct_equas :: "t_equas \<Rightarrow> bool"
+ − 374 where
+ − 375 "distinct_equas equas \<equiv> \<forall> X rhs rhs'. (X, rhs) \<in> equas \<and> (X, rhs') \<in> equas \<longrightarrow> rhs = rhs'"
+ − 376
+ − 377 definition
+ − 378 seq_rhs_r :: "t_equa_rhs \<Rightarrow> rexp \<Rightarrow> t_equa_rhs"
+ − 379 where
+ − 380 "seq_rhs_r rhs r \<equiv> (\<lambda>(X, reg). (X, SEQ reg r)) ` rhs"
+ − 381
+ − 382 definition
+ − 383 del_x_paired :: "('a \<times> 'b) set \<Rightarrow> 'a \<Rightarrow> ('a \<times> 'b) set"
+ − 384 where
+ − 385 "del_x_paired S x \<equiv> S - {X. X \<in> S \<and> fst X = x}"
+ − 386
+ − 387 definition
+ − 388 arden_variate :: "string set \<Rightarrow> rexp \<Rightarrow> t_equa_rhs \<Rightarrow> t_equa_rhs"
+ − 389 where
+ − 390 "arden_variate X r rhs \<equiv> seq_rhs_r (del_x_paired rhs X) (STAR r)"
+ − 391
+ − 392 definition
+ − 393 no_EMPTY_rhs :: "t_equa_rhs \<Rightarrow> bool"
+ − 394 where
+ − 395 "no_EMPTY_rhs rhs \<equiv> \<forall> X r. (X, r) \<in> rhs \<and> X \<noteq> {[]} \<longrightarrow> [] \<notin> L r"
+ − 396
+ − 397 definition
+ − 398 no_EMPTY_equas :: "t_equas \<Rightarrow> bool"
+ − 399 where
+ − 400 "no_EMPTY_equas equas \<equiv> \<forall> X rhs. (X, rhs) \<in> equas \<longrightarrow> no_EMPTY_rhs rhs"
+ − 401
+ − 402 lemma fold_alt_null_eqs:
+ − 403 "finite rS \<Longrightarrow> x \<in> L (folds ALT NULL rS) = (\<exists> r \<in> rS. x \<in> L r)"
+ − 404 apply (simp add:folds_def)
+ − 405 apply (rule someI2_ex)
+ − 406 apply (erule finite_imp_fold_graph)
+ − 407 apply (erule fold_graph.induct)
+ − 408 by auto (*??? how do this be in Isar ?? *)
+ − 409
+ − 410 lemma seq_rhs_r_prop1:
+ − 411 "L (seq_rhs_r rhs r) = (L rhs);(L r)"
+ − 412 apply (auto simp:seq_rhs_r_def image_def lang_seq_def)
+ − 413 apply (rule_tac x = "s1 @ s1a" in exI, rule_tac x = "s2a" in exI, simp)
+ − 414 apply (rule_tac x = a in exI, rule_tac x = b in exI, simp)
+ − 415 apply (rule_tac x = s1 in exI, rule_tac x = s1a in exI, simp)
+ − 416 apply (rule_tac x = X in exI, rule_tac x = "SEQ ra r" in exI, simp)
+ − 417 apply (rule conjI)
+ − 418 apply (rule_tac x = "(X, ra)" in bexI, simp+)
+ − 419 apply (rule_tac x = s1a in exI, rule_tac x = "s2a @ s2" in exI, simp)
+ − 420 apply (simp add:lang_seq_def)
+ − 421 by (rule_tac x = s2a in exI, rule_tac x = s2 in exI, simp)
+ − 422
+ − 423 lemma del_x_paired_prop1:
+ − 424 "\<lbrakk>distinct_rhs rhs; (X, r) \<in> rhs\<rbrakk> \<Longrightarrow> X ; L r \<union> L (del_x_paired rhs X) = L rhs"
7
86167563a1ed
deleted the matcher ate the beginning; made it to work with stable Isabelle and the development version
urbanc
diff
changeset
+ − 425 apply (simp add:del_x_paired_def)
86167563a1ed
deleted the matcher ate the beginning; made it to work with stable Isabelle and the development version
urbanc
diff
changeset
+ − 426 apply (simp add: distinct_rhs_def)
86167563a1ed
deleted the matcher ate the beginning; made it to work with stable Isabelle and the development version
urbanc
diff
changeset
+ − 427 apply(auto simp add: lang_seq_def)
86167563a1ed
deleted the matcher ate the beginning; made it to work with stable Isabelle and the development version
urbanc
diff
changeset
+ − 428 apply(metis)
86167563a1ed
deleted the matcher ate the beginning; made it to work with stable Isabelle and the development version
urbanc
diff
changeset
+ − 429 done
6
+ − 430
+ − 431 lemma arden_variate_prop:
+ − 432 assumes "(X, rx) \<in> rhs"
+ − 433 shows "(\<forall> Y. Y \<noteq> X \<longrightarrow> (\<exists> r. (Y, r) \<in> rhs) = (\<exists> r. (Y, r) \<in> (arden_variate X rx rhs)))"
+ − 434 proof (rule allI, rule impI)
+ − 435 fix Y
+ − 436 assume "(1)": "Y \<noteq> X"
+ − 437 show "(\<exists>r. (Y, r) \<in> rhs) = (\<exists>r. (Y, r) \<in> arden_variate X rx rhs)"
+ − 438 proof
+ − 439 assume "(1_1)": "\<exists>r. (Y, r) \<in> rhs"
+ − 440 show "\<exists>r. (Y, r) \<in> arden_variate X rx rhs" (is "\<exists>r. ?P r")
+ − 441 proof -
+ − 442 from "(1_1)" obtain r where "(Y, r) \<in> rhs" ..
+ − 443 hence "?P (SEQ r (STAR rx))"
+ − 444 proof (simp add:arden_variate_def image_def)
+ − 445 have "(Y, r) \<in> rhs \<Longrightarrow> (Y, r) \<in> del_x_paired rhs X"
+ − 446 by (auto simp:del_x_paired_def "(1)")
+ − 447 thus "(Y, r) \<in> rhs \<Longrightarrow> (Y, SEQ r (STAR rx)) \<in> seq_rhs_r (del_x_paired rhs X) (STAR rx)"
+ − 448 by (auto simp:seq_rhs_r_def)
+ − 449 qed
+ − 450 thus ?thesis by blast
+ − 451 qed
+ − 452 next
+ − 453 assume "(2_1)": "\<exists>r. (Y, r) \<in> arden_variate X rx rhs"
+ − 454 thus "\<exists>r. (Y, r) \<in> rhs" (is "\<exists> r. ?P r")
+ − 455 by (auto simp:arden_variate_def del_x_paired_def seq_rhs_r_def image_def)
+ − 456 qed
+ − 457 qed
+ − 458
+ − 459 text {*
11
+ − 460 arden_variate_valid: proves variation from
+ − 461
+ − 462 "X = X;r + Y;ry + \<dots>" to "X = Y;(SEQ ry (STAR r)) + \<dots>"
+ − 463
+ − 464 holds the law of "language of left equiv language of right"
6
+ − 465 *}
+ − 466 lemma arden_variate_valid:
+ − 467 assumes X_not_empty: "X \<noteq> {[]}"
+ − 468 and l_eq_r: "X = L rhs"
+ − 469 and dist: "distinct_rhs rhs"
+ − 470 and no_empty: "no_EMPTY_rhs rhs"
+ − 471 and self_contained: "(X, r) \<in> rhs"
+ − 472 shows "X = L (arden_variate X r rhs)"
+ − 473 proof -
+ − 474 have "[] \<notin> L r" using no_empty X_not_empty self_contained
+ − 475 by (auto simp:no_EMPTY_rhs_def)
+ − 476 hence ardens: "X = X;(L r) \<union> (L (del_x_paired rhs X)) \<longleftrightarrow> X = (L (del_x_paired rhs X)) ; (L r)\<star>"
+ − 477 by (rule ardens_revised)
+ − 478 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
+ − 479 by (auto dest!:del_x_paired_prop1)
+ − 480 show ?thesis
+ − 481 proof
+ − 482 show "X \<subseteq> L (arden_variate X r rhs)"
+ − 483 proof
+ − 484 fix x
+ − 485 assume "(1_1)": "x \<in> X" with l_eq_r ardens del_x
+ − 486 show "x \<in> L (arden_variate X r rhs)"
+ − 487 by (simp add:arden_variate_def seq_rhs_r_prop1 del:L_rhs.simps)
+ − 488 qed
+ − 489 next
+ − 490 show "L (arden_variate X r rhs) \<subseteq> X"
+ − 491 proof
+ − 492 fix x
+ − 493 assume "(2_1)": "x \<in> L (arden_variate X r rhs)" with ardens del_x l_eq_r
+ − 494 show "x \<in> X"
+ − 495 by (simp add:arden_variate_def seq_rhs_r_prop1 del:L_rhs.simps)
+ − 496 qed
+ − 497 qed
+ − 498 qed
+ − 499
11
+ − 500 text {*
23
+ − 501 merge_rhs {(x1, r1), (x2, r2}, (x4, r4), \<dots>} {(x1, r1'), (x3, r3'), \<dots>} =
11
+ − 502 {(x1, ALT r1 r1'}, (x2, r2), (x3, r3'), (x4, r4), \<dots>} *}
6
+ − 503 definition
+ − 504 merge_rhs :: "t_equa_rhs \<Rightarrow> t_equa_rhs \<Rightarrow> t_equa_rhs"
+ − 505 where
+ − 506 "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>
+ − 507 (\<exists> r1. (X, r1) \<in> rhs \<and> (\<not> (\<exists> r2. (X, r2) \<in> rhs')) \<and> r = r1) \<or>
23
+ − 508 (\<exists> r2. (X, r2) \<in> rhs' \<and> (\<not> (\<exists> r1. (X, r1) \<in> rhs)) \<and> r = r2) }"
6
+ − 509
+ − 510
+ − 511 text {* rhs_subst rhs X=xrhs r: substitude all occurence of X in rhs((X,r) \<in> rhs) with xrhs *}
+ − 512 definition
+ − 513 rhs_subst :: "t_equa_rhs \<Rightarrow> string set \<Rightarrow> t_equa_rhs \<Rightarrow> rexp \<Rightarrow> t_equa_rhs"
+ − 514 where
+ − 515 "rhs_subst rhs X xrhs r \<equiv> merge_rhs (del_x_paired rhs X) (seq_rhs_r xrhs r)"
+ − 516
+ − 517 definition
+ − 518 equas_subst_f :: "string set \<Rightarrow> t_equa_rhs \<Rightarrow> t_equa \<Rightarrow> t_equa"
+ − 519 where
+ − 520 "equas_subst_f X xrhs equa \<equiv> let (Y, rhs) = equa in
+ − 521 if (\<exists> r. (X, r) \<in> rhs)
+ − 522 then (Y, rhs_subst rhs X xrhs (SOME r. (X, r) \<in> rhs))
+ − 523 else equa"
+ − 524
+ − 525 definition
+ − 526 equas_subst :: "t_equas \<Rightarrow> string set \<Rightarrow> t_equa_rhs \<Rightarrow> t_equas"
+ − 527 where
+ − 528 "equas_subst ES X xrhs \<equiv> del_x_paired (equas_subst_f X xrhs ` ES) X"
+ − 529
+ − 530 lemma lang_seq_prop1:
+ − 531 "x \<in> X ; L r \<Longrightarrow> x \<in> X ; (L r \<union> L r')"
+ − 532 by (auto simp:lang_seq_def)
+ − 533
+ − 534 lemma lang_seq_prop1':
+ − 535 "x \<in> X; L r \<Longrightarrow> x \<in> X ; (L r' \<union> L r)"
+ − 536 by (auto simp:lang_seq_def)
+ − 537
+ − 538 lemma lang_seq_prop2:
+ − 539 "x \<in> X; (L r \<union> L r') \<Longrightarrow> x \<in> X;L r \<or> x \<in> X;L r'"
+ − 540 by (auto simp:lang_seq_def)
+ − 541
+ − 542 lemma merge_rhs_prop1:
+ − 543 shows "L (merge_rhs rhs rhs') = L rhs \<union> L rhs' "
+ − 544 apply (auto simp add:merge_rhs_def dest!:lang_seq_prop2 intro:lang_seq_prop1)
+ − 545 apply (rule_tac x = X in exI, rule_tac x = r1 in exI, simp)
+ − 546 apply (case_tac "\<exists> r2. (X, r2) \<in> rhs'")
+ − 547 apply (clarify, rule_tac x = X in exI, rule_tac x = "ALT r r2" in exI, simp add:lang_seq_prop1)
+ − 548 apply (rule_tac x = X in exI, rule_tac x = r in exI, simp)
+ − 549 apply (case_tac "\<exists> r1. (X, r1) \<in> rhs")
+ − 550 apply (clarify, rule_tac x = X in exI, rule_tac x = "ALT r1 r" in exI, simp add:lang_seq_prop1')
+ − 551 apply (rule_tac x = X in exI, rule_tac x = r in exI, simp)
+ − 552 done
+ − 553
+ − 554 lemma no_EMPTY_rhss_imp_merge_no_EMPTY:
+ − 555 "\<lbrakk>no_EMPTY_rhs rhs; no_EMPTY_rhs rhs'\<rbrakk> \<Longrightarrow> no_EMPTY_rhs (merge_rhs rhs rhs')"
+ − 556 apply (simp add:no_EMPTY_rhs_def merge_rhs_def)
+ − 557 apply (clarify, (erule conjE | erule exE | erule disjE)+)
+ − 558 by auto
+ − 559
+ − 560 lemma distinct_rhs_prop:
+ − 561 "\<lbrakk>distinct_rhs rhs; (X, r1) \<in> rhs; (X, r2) \<in> rhs\<rbrakk> \<Longrightarrow> r1 = r2"
+ − 562 by (auto simp:distinct_rhs_def)
+ − 563
+ − 564 lemma merge_rhs_prop2:
+ − 565 assumes dist_rhs: "distinct_rhs rhs"
+ − 566 and dist_rhs':"distinct_rhs rhs'"
+ − 567 shows "distinct_rhs (merge_rhs rhs rhs')"
+ − 568 apply (auto simp:merge_rhs_def distinct_rhs_def)
+ − 569 using dist_rhs
+ − 570 apply (drule distinct_rhs_prop, simp+)
+ − 571 using dist_rhs'
+ − 572 apply (drule distinct_rhs_prop, simp+)
+ − 573 using dist_rhs
+ − 574 apply (drule distinct_rhs_prop, simp+)
+ − 575 using dist_rhs'
+ − 576 apply (drule distinct_rhs_prop, simp+)
+ − 577 done
+ − 578
+ − 579 lemma seq_rhs_r_holds_distinct:
+ − 580 "distinct_rhs rhs \<Longrightarrow> distinct_rhs (seq_rhs_r rhs r)"
+ − 581 by (auto simp:distinct_rhs_def seq_rhs_r_def)
+ − 582
+ − 583 lemma seq_rhs_r_prop0:
+ − 584 assumes l_eq_r: "X = L xrhs"
+ − 585 shows "L (seq_rhs_r xrhs r) = X ; L r "
+ − 586 using l_eq_r
+ − 587 by (auto simp only:seq_rhs_r_prop1)
+ − 588
+ − 589 lemma rhs_subst_prop1:
+ − 590 assumes l_eq_r: "X = L xrhs"
+ − 591 and dist: "distinct_rhs rhs"
+ − 592 shows "(X, r) \<in> rhs \<Longrightarrow> L rhs = L (rhs_subst rhs X xrhs r)"
+ − 593 apply (simp add:rhs_subst_def merge_rhs_prop1 del:L_rhs.simps)
+ − 594 using l_eq_r
+ − 595 apply (drule_tac r = r in seq_rhs_r_prop0, simp del:L_rhs.simps)
+ − 596 using dist
+ − 597 by (auto dest!:del_x_paired_prop1 simp del:L_rhs.simps)
+ − 598
+ − 599 lemma del_x_paired_holds_distinct_rhs:
+ − 600 "distinct_rhs rhs \<Longrightarrow> distinct_rhs (del_x_paired rhs x)"
+ − 601 by (auto simp:distinct_rhs_def del_x_paired_def)
+ − 602
+ − 603 lemma rhs_subst_holds_distinct_rhs:
+ − 604 "\<lbrakk>distinct_rhs rhs; distinct_rhs xrhs\<rbrakk> \<Longrightarrow> distinct_rhs (rhs_subst rhs X xrhs r)"
+ − 605 apply (drule_tac r = r and rhs = xrhs in seq_rhs_r_holds_distinct)
+ − 606 apply (drule_tac x = X in del_x_paired_holds_distinct_rhs)
+ − 607 by (auto dest:merge_rhs_prop2[where rhs = "del_x_paired rhs X"] simp:rhs_subst_def)
+ − 608
+ − 609 section {* myhill-nerode theorem *}
+ − 610
+ − 611 definition left_eq_cls :: "t_equas \<Rightarrow> (string set) set"
+ − 612 where
+ − 613 "left_eq_cls ES \<equiv> {X. \<exists> rhs. (X, rhs) \<in> ES} "
+ − 614
+ − 615 definition right_eq_cls :: "t_equas \<Rightarrow> (string set) set"
+ − 616 where
+ − 617 "right_eq_cls ES \<equiv> {Y. \<exists> X rhs r. (X, rhs) \<in> ES \<and> (Y, r) \<in> rhs }"
+ − 618
+ − 619 definition rhs_eq_cls :: "t_equa_rhs \<Rightarrow> (string set) set"
+ − 620 where
+ − 621 "rhs_eq_cls rhs \<equiv> {Y. \<exists> r. (Y, r) \<in> rhs}"
+ − 622
+ − 623 definition ardenable :: "t_equa \<Rightarrow> bool"
+ − 624 where
+ − 625 "ardenable equa \<equiv> let (X, rhs) = equa in
+ − 626 distinct_rhs rhs \<and> no_EMPTY_rhs rhs \<and> X = L rhs"
+ − 627
+ − 628 text {*
+ − 629 Inv: Invairance of the equation-system, during the decrease of the equation-system, Inv holds.
+ − 630 *}
+ − 631 definition Inv :: "string set \<Rightarrow> t_equas \<Rightarrow> bool"
+ − 632 where
+ − 633 "Inv X ES \<equiv> finite ES \<and> (\<exists> rhs. (X, rhs) \<in> ES) \<and> distinct_equas ES \<and>
23
+ − 634 (\<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))"
6
+ − 635
+ − 636 text {*
+ − 637 TCon: Termination Condition of the equation-system decreasion.
+ − 638 *}
+ − 639 definition TCon:: "'a set \<Rightarrow> bool"
+ − 640 where
+ − 641 "TCon ES \<equiv> card ES = 1"
+ − 642
+ − 643
+ − 644 text {*
+ − 645 The following is a iteration principle, and is the main framework for the proof:
23
+ − 646 1: We can form the initial equation-system using "equations" defined above, and prove it has invariance Inv by lemma "init_ES_satisfy_Inv";
+ − 647 2: We can decrease the number of the equation-system using ardens_lemma_revised and substitution ("equas_subst", defined above),
+ − 648 and prove it holds the property "step" in "wf_iter" by lemma "iteration_step"
+ − 649 and finally using property Inv and TCon to prove the myhill-nerode theorem
6
+ − 650
+ − 651 *}
+ − 652 lemma wf_iter [rule_format]:
+ − 653 fixes f
+ − 654 assumes step: "\<And> e. \<lbrakk>P e; \<not> Q e\<rbrakk> \<Longrightarrow> (\<exists> e'. P e' \<and> (f(e'), f(e)) \<in> less_than)"
+ − 655 shows pe: "P e \<longrightarrow> (\<exists> e'. P e' \<and> Q e')"
+ − 656 proof(induct e rule: wf_induct
+ − 657 [OF wf_inv_image[OF wf_less_than, where f = "f"]], clarify)
+ − 658 fix x
+ − 659 assume h [rule_format]:
+ − 660 "\<forall>y. (y, x) \<in> inv_image less_than f \<longrightarrow> P y \<longrightarrow> (\<exists>e'. P e' \<and> Q e')"
+ − 661 and px: "P x"
+ − 662 show "\<exists>e'. P e' \<and> Q e'"
+ − 663 proof(cases "Q x")
+ − 664 assume "Q x" with px show ?thesis by blast
+ − 665 next
+ − 666 assume nq: "\<not> Q x"
+ − 667 from step [OF px nq]
+ − 668 obtain e' where pe': "P e'" and ltf: "(f e', f x) \<in> less_than" by auto
+ − 669 show ?thesis
+ − 670 proof(rule h)
+ − 671 from ltf show "(e', x) \<in> inv_image less_than f"
+ − 672 by (simp add:inv_image_def)
+ − 673 next
+ − 674 from pe' show "P e'" .
+ − 675 qed
+ − 676 qed
+ − 677 qed
+ − 678
+ − 679
11
+ − 680 text {* ******BEGIN: proving the initial equation-system satisfies Inv ****** *}
6
+ − 681
+ − 682 lemma distinct_rhs_equations:
+ − 683 "(X, xrhs) \<in> equations (UNIV Quo Lang) \<Longrightarrow> distinct_rhs xrhs"
+ − 684 by (auto simp: equations_def equation_rhs_def distinct_rhs_def empty_rhs_def dest:no_two_cls_inters)
+ − 685
+ − 686 lemma every_nonempty_eqclass_has_strings:
+ − 687 "\<lbrakk>X \<in> (UNIV Quo Lang); X \<noteq> {[]}\<rbrakk> \<Longrightarrow> \<exists> clist. clist \<in> X \<and> clist \<noteq> []"
+ − 688 by (auto simp:quot_def equiv_class_def equiv_str_def)
+ − 689
+ − 690 lemma every_eqclass_is_derived_from_empty:
+ − 691 assumes not_empty: "X \<noteq> {[]}"
+ − 692 shows "X \<in> (UNIV Quo Lang) \<Longrightarrow> \<exists> clist. {[]};{clist} \<subseteq> X \<and> clist \<noteq> []"
+ − 693 using not_empty
+ − 694 apply (drule_tac every_nonempty_eqclass_has_strings, simp)
+ − 695 by (auto intro:exI[where x = clist] simp:lang_seq_def)
+ − 696
+ − 697 lemma equiv_str_in_CS:
+ − 698 "\<lbrakk>clist\<rbrakk>Lang \<in> (UNIV Quo Lang)"
+ − 699 by (auto simp:quot_def)
+ − 700
+ − 701 lemma has_str_imp_defined_by_str:
+ − 702 "\<lbrakk>str \<in> X; X \<in> UNIV Quo Lang\<rbrakk> \<Longrightarrow> X = \<lbrakk>str\<rbrakk>Lang"
+ − 703 by (auto simp:quot_def equiv_class_def equiv_str_def)
+ − 704
+ − 705 lemma every_eqclass_has_ascendent:
+ − 706 assumes has_str: "clist @ [c] \<in> X"
+ − 707 and in_CS: "X \<in> UNIV Quo Lang"
+ − 708 shows "\<exists> Y. Y \<in> UNIV Quo Lang \<and> Y-c\<rightarrow>X \<and> clist \<in> Y" (is "\<exists> Y. ?P Y")
+ − 709 proof -
+ − 710 have "?P (\<lbrakk>clist\<rbrakk>Lang)"
+ − 711 proof -
+ − 712 have "\<lbrakk>clist\<rbrakk>Lang \<in> UNIV Quo Lang"
+ − 713 by (simp add:quot_def, rule_tac x = clist in exI, simp)
+ − 714 moreover have "\<lbrakk>clist\<rbrakk>Lang-c\<rightarrow>X"
+ − 715 proof -
+ − 716 have "X = \<lbrakk>(clist @ [c])\<rbrakk>Lang" using has_str in_CS
+ − 717 by (auto intro!:has_str_imp_defined_by_str)
+ − 718 moreover have "\<forall> sl. sl \<in> \<lbrakk>clist\<rbrakk>Lang \<longrightarrow> sl @ [c] \<in> \<lbrakk>(clist @ [c])\<rbrakk>Lang"
+ − 719 by (auto simp:equiv_class_def equiv_str_def)
+ − 720 ultimately show ?thesis unfolding CT_def lang_seq_def
+ − 721 by auto
+ − 722 qed
+ − 723 moreover have "clist \<in> \<lbrakk>clist\<rbrakk>Lang"
+ − 724 by (auto simp:equiv_str_def equiv_class_def)
+ − 725 ultimately show "?P (\<lbrakk>clist\<rbrakk>Lang)" by simp
+ − 726 qed
+ − 727 thus ?thesis by blast
+ − 728 qed
+ − 729
+ − 730 lemma finite_charset_rS:
+ − 731 "finite {CHAR c |c. Y-c\<rightarrow>X}"
+ − 732 by (rule_tac A = UNIV and f = CHAR in finite_surj, auto)
+ − 733
+ − 734 lemma l_eq_r_in_equations:
+ − 735 assumes X_in_equas: "(X, xrhs) \<in> equations (UNIV Quo Lang)"
+ − 736 shows "X = L xrhs"
+ − 737 proof (cases "X = {[]}")
+ − 738 case True
18
+ − 739 thus ?thesis using X_in_equas
+ − 740 by (simp add:equations_def equation_rhs_def lang_seq_def)
6
+ − 741 next
+ − 742 case False
+ − 743 show ?thesis
+ − 744 proof
+ − 745 show "X \<subseteq> L xrhs"
+ − 746 proof
+ − 747 fix x
+ − 748 assume "(1)": "x \<in> X"
+ − 749 show "x \<in> L xrhs"
+ − 750 proof (cases "x = []")
+ − 751 assume empty: "x = []"
+ − 752 hence "x \<in> L (empty_rhs X)" using "(1)"
18
+ − 753 by (auto simp:empty_rhs_def lang_seq_def)
6
+ − 754 thus ?thesis using X_in_equas False empty "(1)"
+ − 755 unfolding equations_def equation_rhs_def by auto
+ − 756 next
+ − 757 assume not_empty: "x \<noteq> []"
18
+ − 758 hence "\<exists> clist c. x = clist @ [c]" by (case_tac x rule:rev_cases, auto)
+ − 759 then obtain clist c where decom: "x = clist @ [c]" by blast
+ − 760 moreover have "\<And> Y. \<lbrakk>Y \<in> UNIV Quo Lang; Y-c\<rightarrow>X; clist \<in> Y\<rbrakk>
+ − 761 \<Longrightarrow> [c] \<in> L (folds ALT NULL {CHAR c |c. Y-c\<rightarrow>X})"
6
+ − 762 proof -
+ − 763 fix Y
18
+ − 764 assume Y_is_eq_cl: "Y \<in> UNIV Quo Lang"
+ − 765 and Y_CT_X: "Y-c\<rightarrow>X"
+ − 766 and clist_in_Y: "clist \<in> Y"
6
+ − 767 with finite_charset_rS
+ − 768 show "[c] \<in> L (folds ALT NULL {CHAR c |c. Y-c\<rightarrow>X})"
18
+ − 769 by (auto simp :fold_alt_null_eqs)
6
+ − 770 qed
+ − 771 hence "\<exists>Xa. Xa \<in> UNIV Quo Lang \<and> clist @ [c] \<in> Xa ; L (folds ALT NULL {CHAR c |c. Xa-c\<rightarrow>X})"
+ − 772 using X_in_equas False not_empty "(1)" decom
18
+ − 773 by (auto dest!:every_eqclass_has_ascendent simp:equations_def equation_rhs_def lang_seq_def)
11
+ − 774 then obtain Xa where
18
+ − 775 "Xa \<in> UNIV Quo Lang \<and> clist @ [c] \<in> Xa ; L (folds ALT NULL {CHAR c |c. Xa-c\<rightarrow>X})" by blast
11
+ − 776 hence "x \<in> L {(S, folds ALT NULL {CHAR c |c. S-c\<rightarrow>X}) |S. S \<in> UNIV Quo Lang}"
+ − 777 using X_in_equas "(1)" decom
6
+ − 778 by (auto simp add:equations_def equation_rhs_def intro!:exI[where x = Xa])
18
+ − 779 thus "x \<in> L xrhs" using X_in_equas False not_empty unfolding equations_def equation_rhs_def
6
+ − 780 by auto
+ − 781 qed
+ − 782 qed
+ − 783 next
+ − 784 show "L xrhs \<subseteq> X"
+ − 785 proof
+ − 786 fix x
+ − 787 assume "(2)": "x \<in> L xrhs"
+ − 788 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"
+ − 789 using finite_charset_rS
18
+ − 790 by (auto simp:CT_def lang_seq_def fold_alt_null_eqs)
6
+ − 791 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"
18
+ − 792 by (simp add:empty_rhs_def split:if_splits)
+ − 793 show "x \<in> X" using X_in_equas False "(2)"
+ − 794 by (auto intro:"(2_1)" "(2_2)" simp:equations_def equation_rhs_def lang_seq_def)
6
+ − 795 qed
+ − 796 qed
+ − 797 qed
+ − 798
11
+ − 799
6
+ − 800
+ − 801 lemma no_EMPTY_equations:
+ − 802 "(X, xrhs) \<in> equations CS \<Longrightarrow> no_EMPTY_rhs xrhs"
+ − 803 apply (clarsimp simp add:equations_def equation_rhs_def)
+ − 804 apply (simp add:no_EMPTY_rhs_def empty_rhs_def, auto)
23
+ − 805 apply (subgoal_tac "finite {CHAR c |c. Xa-c\<rightarrow>X}", drule_tac x = "[]" in fold_alt_null_eqs, clarsimp, rule finite_charset_rS)+
6
+ − 806 done
+ − 807
+ − 808 lemma init_ES_satisfy_ardenable:
+ − 809 "(X, xrhs) \<in> equations (UNIV Quo Lang) \<Longrightarrow> ardenable (X, xrhs)"
+ − 810 unfolding ardenable_def
+ − 811 by (auto intro:distinct_rhs_equations no_EMPTY_equations simp:l_eq_r_in_equations simp del:L_rhs.simps)
+ − 812
+ − 813 lemma init_ES_satisfy_Inv:
+ − 814 assumes finite_CS: "finite (UNIV Quo Lang)"
+ − 815 and X_in_eq_cls: "X \<in> UNIV Quo Lang"
+ − 816 shows "Inv X (equations (UNIV Quo Lang))"
+ − 817 proof -
+ − 818 have "finite (equations (UNIV Quo Lang))" using finite_CS
+ − 819 by (auto simp:equations_def)
+ − 820 moreover have "\<exists>rhs. (X, rhs) \<in> equations (UNIV Quo Lang)" using X_in_eq_cls
+ − 821 by (simp add:equations_def)
+ − 822 moreover have "distinct_equas (equations (UNIV Quo Lang))"
+ − 823 by (auto simp:distinct_equas_def equations_def)
+ − 824 moreover have "\<forall>X xrhs. (X, xrhs) \<in> equations (UNIV Quo Lang) \<longrightarrow>
+ − 825 rhs_eq_cls xrhs \<subseteq> insert {[]} (left_eq_cls (equations (UNIV Quo Lang)))"
+ − 826 apply (simp add:left_eq_cls_def equations_def rhs_eq_cls_def equation_rhs_def)
+ − 827 by (auto simp:empty_rhs_def split:if_splits)
+ − 828 moreover have "\<forall>X xrhs. (X, xrhs) \<in> equations (UNIV Quo Lang) \<longrightarrow> X \<noteq> {}"
+ − 829 by (clarsimp simp:equations_def empty_notin_CS intro:classical)
+ − 830 moreover have "\<forall>X xrhs. (X, xrhs) \<in> equations (UNIV Quo Lang) \<longrightarrow> ardenable (X, xrhs)"
+ − 831 by (auto intro!:init_ES_satisfy_ardenable)
+ − 832 ultimately show ?thesis by (simp add:Inv_def)
+ − 833 qed
+ − 834
+ − 835
11
+ − 836 text {* *********** END: proving the initial equation-system satisfies Inv ******* *}
6
+ − 837
+ − 838
11
+ − 839 text {* ****** BEGIN: proving every equation-system's iteration step satisfies Inv ***** *}
6
+ − 840
+ − 841 lemma not_T_aux: "\<lbrakk>\<not> TCon (insert a A); x = a\<rbrakk>
+ − 842 \<Longrightarrow> \<exists>y. x \<noteq> y \<and> y \<in> insert a A "
+ − 843 apply (case_tac "insert a A = {a}")
+ − 844 by (auto simp:TCon_def)
+ − 845
+ − 846 lemma not_T_atleast_2[rule_format]:
+ − 847 "finite S \<Longrightarrow> \<forall> x. x \<in> S \<and> (\<not> TCon S)\<longrightarrow> (\<exists> y. x \<noteq> y \<and> y \<in> S)"
+ − 848 apply (erule finite.induct, simp)
+ − 849 apply (clarify, case_tac "x = a")
+ − 850 by (erule not_T_aux, auto)
+ − 851
+ − 852 lemma exist_another_equa:
+ − 853 "\<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"
+ − 854 apply (drule not_T_atleast_2, simp)
+ − 855 apply (clarsimp simp:distinct_equas_def)
+ − 856 apply (drule_tac x= X in spec, drule_tac x = rhl in spec, drule_tac x = b in spec)
+ − 857 by auto
+ − 858
+ − 859 lemma Inv_mono_with_lambda:
+ − 860 assumes Inv_ES: "Inv X ES"
+ − 861 and X_noteq_Y: "X \<noteq> {[]}"
+ − 862 shows "Inv X (ES - {({[]}, yrhs)})"
+ − 863 proof -
+ − 864 have "finite (ES - {({[]}, yrhs)})" using Inv_ES
+ − 865 by (simp add:Inv_def)
+ − 866 moreover have "\<exists>rhs. (X, rhs) \<in> ES - {({[]}, yrhs)}" using Inv_ES X_noteq_Y
+ − 867 by (simp add:Inv_def)
+ − 868 moreover have "distinct_equas (ES - {({[]}, yrhs)})" using Inv_ES X_noteq_Y
+ − 869 apply (clarsimp simp:Inv_def distinct_equas_def)
+ − 870 by (drule_tac x = Xa in spec, simp)
+ − 871 moreover have "\<forall>X xrhs.(X, xrhs) \<in> ES - {({[]}, yrhs)} \<longrightarrow>
+ − 872 ardenable (X, xrhs) \<and> X \<noteq> {}" using Inv_ES
+ − 873 by (clarify, simp add:Inv_def)
+ − 874 moreover
+ − 875 have "insert {[]} (left_eq_cls (ES - {({[]}, yrhs)})) = insert {[]} (left_eq_cls ES)"
+ − 876 by (auto simp:left_eq_cls_def)
+ − 877 hence "\<forall>X xrhs.(X, xrhs) \<in> ES - {({[]}, yrhs)} \<longrightarrow>
+ − 878 rhs_eq_cls xrhs \<subseteq> insert {[]} (left_eq_cls (ES - {({[]}, yrhs)}))"
+ − 879 using Inv_ES by (auto simp:Inv_def)
+ − 880 ultimately show ?thesis by (simp add:Inv_def)
+ − 881 qed
+ − 882
+ − 883 lemma non_empty_card_prop:
+ − 884 "finite ES \<Longrightarrow> \<forall>e. e \<in> ES \<longrightarrow> card ES - Suc 0 < card ES"
+ − 885 apply (erule finite.induct, simp)
+ − 886 apply (case_tac[!] "a \<in> A")
+ − 887 by (auto simp:insert_absorb)
+ − 888
+ − 889 lemma ardenable_prop:
+ − 890 assumes not_lambda: "Y \<noteq> {[]}"
+ − 891 and ardable: "ardenable (Y, yrhs)"
23
+ − 892 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'")
6
+ − 893 proof (cases "(\<exists> reg. (Y, reg) \<in> yrhs)")
+ − 894 case True
+ − 895 thus ?thesis
+ − 896 proof
+ − 897 fix reg
+ − 898 assume self_contained: "(Y, reg) \<in> yrhs"
+ − 899 show ?thesis
+ − 900 proof -
+ − 901 have "?P (arden_variate Y reg yrhs)"
+ − 902 proof -
+ − 903 have "Y = L (arden_variate Y reg yrhs)"
+ − 904 using self_contained not_lambda ardable
+ − 905 by (rule_tac arden_variate_valid, simp_all add:ardenable_def)
+ − 906 moreover have "distinct_rhs (arden_variate Y reg yrhs)"
+ − 907 using ardable
+ − 908 by (auto simp:distinct_rhs_def arden_variate_def seq_rhs_r_def del_x_paired_def ardenable_def)
+ − 909 moreover have "rhs_eq_cls (arden_variate Y reg yrhs) = rhs_eq_cls yrhs - {Y}"
+ − 910 proof -
+ − 911 have "\<And> rhs r. rhs_eq_cls (seq_rhs_r rhs r) = rhs_eq_cls rhs"
+ − 912 apply (auto simp:rhs_eq_cls_def seq_rhs_r_def image_def)
+ − 913 by (rule_tac x = "SEQ ra r" in exI, rule_tac x = "(x, ra)" in bexI, simp+)
+ − 914 moreover have "\<And> rhs X. rhs_eq_cls (del_x_paired rhs X) = rhs_eq_cls rhs - {X}"
+ − 915 by (auto simp:rhs_eq_cls_def del_x_paired_def)
+ − 916 ultimately show ?thesis by (simp add:arden_variate_def)
+ − 917 qed
+ − 918 ultimately show ?thesis by simp
+ − 919 qed
+ − 920 thus ?thesis by (rule_tac x= "arden_variate Y reg yrhs" in exI, simp)
+ − 921 qed
+ − 922 qed
+ − 923 next
+ − 924 case False
+ − 925 hence "(2)": "rhs_eq_cls yrhs - {Y} = rhs_eq_cls yrhs"
+ − 926 by (auto simp:rhs_eq_cls_def)
+ − 927 show ?thesis
+ − 928 proof -
+ − 929 have "?P yrhs" using False ardable "(2)"
+ − 930 by (simp add:ardenable_def)
+ − 931 thus ?thesis by blast
+ − 932 qed
+ − 933 qed
+ − 934
+ − 935 lemma equas_subst_f_del_no_other:
+ − 936 assumes self_contained: "(Y, rhs) \<in> ES"
+ − 937 shows "\<exists> rhs'. (Y, rhs') \<in> (equas_subst_f X xrhs ` ES)" (is "\<exists> rhs'. ?P rhs'")
+ − 938 proof -
+ − 939 have "\<exists> rhs'. equas_subst_f X xrhs (Y, rhs) = (Y, rhs')"
+ − 940 by (auto simp:equas_subst_f_def)
+ − 941 then obtain rhs' where "equas_subst_f X xrhs (Y, rhs) = (Y, rhs')" by blast
+ − 942 hence "?P rhs'" unfolding image_def using self_contained
+ − 943 by (auto intro:bexI[where x = "(Y, rhs)"])
+ − 944 thus ?thesis by blast
+ − 945 qed
+ − 946
+ − 947 lemma del_x_paired_del_only_x:
+ − 948 "\<lbrakk>X \<noteq> Y; (X, rhs) \<in> ES\<rbrakk> \<Longrightarrow> (X, rhs) \<in> del_x_paired ES Y"
+ − 949 by (auto simp:del_x_paired_def)
+ − 950
+ − 951 lemma equas_subst_del_no_other:
+ − 952 "\<lbrakk>(X, rhs) \<in> ES; X \<noteq> Y\<rbrakk> \<Longrightarrow> (\<exists>rhs. (X, rhs) \<in> equas_subst ES Y rhs')"
+ − 953 unfolding equas_subst_def
+ − 954 apply (drule_tac X = Y and xrhs = rhs' in equas_subst_f_del_no_other)
+ − 955 by (erule exE, drule del_x_paired_del_only_x, auto)
+ − 956
+ − 957 lemma equas_subst_holds_distinct:
+ − 958 "distinct_equas ES \<Longrightarrow> distinct_equas (equas_subst ES Y rhs')"
+ − 959 apply (clarsimp simp add:equas_subst_def distinct_equas_def del_x_paired_def equas_subst_f_def)
+ − 960 by (auto split:if_splits)
+ − 961
+ − 962 lemma del_x_paired_dels:
+ − 963 "(X, rhs) \<in> ES \<Longrightarrow> {Y. Y \<in> ES \<and> fst Y = X} \<inter> ES \<noteq> {}"
+ − 964 by (auto)
+ − 965
+ − 966 lemma del_x_paired_subset:
+ − 967 "(X, rhs) \<in> ES \<Longrightarrow> ES - {Y. Y \<in> ES \<and> fst Y = X} \<subset> ES"
+ − 968 apply (drule del_x_paired_dels)
+ − 969 by auto
+ − 970
+ − 971 lemma del_x_paired_card_less:
+ − 972 "\<lbrakk>finite ES; (X, rhs) \<in> ES\<rbrakk> \<Longrightarrow> card (del_x_paired ES X) < card ES"
+ − 973 apply (simp add:del_x_paired_def)
+ − 974 apply (drule del_x_paired_subset)
+ − 975 by (auto intro:psubset_card_mono)
+ − 976
+ − 977 lemma equas_subst_card_less:
+ − 978 "\<lbrakk>finite ES; (Y, yrhs) \<in> ES\<rbrakk> \<Longrightarrow> card (equas_subst ES Y rhs') < card ES"
+ − 979 apply (simp add:equas_subst_def)
+ − 980 apply (frule_tac h = "equas_subst_f Y rhs'" in finite_imageI)
+ − 981 apply (drule_tac f = "equas_subst_f Y rhs'" in Finite_Set.card_image_le)
+ − 982 apply (drule_tac X = Y and xrhs = rhs' in equas_subst_f_del_no_other,erule exE)
+ − 983 by (drule del_x_paired_card_less, auto)
+ − 984
+ − 985 lemma equas_subst_holds_distinct_rhs:
+ − 986 assumes dist': "distinct_rhs yrhs'"
+ − 987 and history: "\<forall>X xrhs. (X, xrhs) \<in> ES \<longrightarrow> ardenable (X, xrhs)"
+ − 988 and X_in : "(X, xrhs) \<in> equas_subst ES Y yrhs'"
+ − 989 shows "distinct_rhs xrhs"
+ − 990 using X_in history
+ − 991 apply (clarsimp simp:equas_subst_def del_x_paired_def)
+ − 992 apply (drule_tac x = a in spec, drule_tac x = b in spec)
+ − 993 apply (simp add:ardenable_def equas_subst_f_def)
+ − 994 by (auto intro:rhs_subst_holds_distinct_rhs simp:dist' split:if_splits)
+ − 995
+ − 996 lemma r_no_EMPTY_imp_seq_rhs_r_no_EMPTY:
+ − 997 "[] \<notin> L r \<Longrightarrow> no_EMPTY_rhs (seq_rhs_r rhs r)"
+ − 998 by (auto simp:no_EMPTY_rhs_def seq_rhs_r_def lang_seq_def)
+ − 999
+ − 1000 lemma del_x_paired_holds_no_EMPTY:
+ − 1001 "no_EMPTY_rhs yrhs \<Longrightarrow> no_EMPTY_rhs (del_x_paired yrhs Y)"
+ − 1002 by (auto simp:no_EMPTY_rhs_def del_x_paired_def)
+ − 1003
+ − 1004 lemma rhs_subst_holds_no_EMPTY:
+ − 1005 "\<lbrakk>no_EMPTY_rhs yrhs; (Y, r) \<in> yrhs; Y \<noteq> {[]}\<rbrakk> \<Longrightarrow> no_EMPTY_rhs (rhs_subst yrhs Y rhs' r)"
+ − 1006 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)
+ − 1007 by (auto simp:no_EMPTY_rhs_def)
+ − 1008
+ − 1009 lemma equas_subst_holds_no_EMPTY:
+ − 1010 assumes substor: "Y \<noteq> {[]}"
+ − 1011 and history: "\<forall>X xrhs. (X, xrhs) \<in> ES \<longrightarrow> ardenable (X, xrhs)"
+ − 1012 and X_in:"(X, xrhs) \<in> equas_subst ES Y yrhs'"
+ − 1013 shows "no_EMPTY_rhs xrhs"
+ − 1014 proof-
+ − 1015 from X_in have "\<exists> (Z, zrhs) \<in> ES. (X, xrhs) = equas_subst_f Y yrhs' (Z, zrhs)"
+ − 1016 by (auto simp add:equas_subst_def del_x_paired_def)
+ − 1017 then obtain Z zrhs where Z_in: "(Z, zrhs) \<in> ES"
+ − 1018 and X_Z: "(X, xrhs) = equas_subst_f Y yrhs' (Z, zrhs)" by blast
+ − 1019 hence dist_zrhs: "distinct_rhs zrhs" using history
+ − 1020 by (auto simp:ardenable_def)
+ − 1021 show ?thesis
+ − 1022 proof (cases "\<exists> r. (Y, r) \<in> zrhs")
+ − 1023 case True
+ − 1024 then obtain r where Y_in_zrhs: "(Y, r) \<in> zrhs" ..
+ − 1025 hence some: "(SOME r. (Y, r) \<in> zrhs) = r" using Z_in dist_zrhs
+ − 1026 by (auto simp:distinct_rhs_def)
+ − 1027 hence "no_EMPTY_rhs (rhs_subst zrhs Y yrhs' r)"
+ − 1028 using substor Y_in_zrhs history Z_in
+ − 1029 by (rule_tac rhs_subst_holds_no_EMPTY, auto simp:ardenable_def)
+ − 1030 thus ?thesis using X_Z True some
+ − 1031 by (simp add:equas_subst_def equas_subst_f_def)
+ − 1032 next
+ − 1033 case False
+ − 1034 hence "(X, xrhs) = (Z, zrhs)" using Z_in X_Z
+ − 1035 by (simp add:equas_subst_f_def)
+ − 1036 thus ?thesis using history Z_in
+ − 1037 by (auto simp:ardenable_def)
+ − 1038 qed
+ − 1039 qed
+ − 1040
+ − 1041 lemma equas_subst_f_holds_left_eq_right:
+ − 1042 assumes substor: "Y = L rhs'"
+ − 1043 and history: "\<forall>X xrhs. (X, xrhs) \<in> ES \<longrightarrow> distinct_rhs xrhs \<and> X = L xrhs"
+ − 1044 and subst: "(X, xrhs) = equas_subst_f Y rhs' (Z, zrhs)"
+ − 1045 and self_contained: "(Z, zrhs) \<in> ES"
+ − 1046 shows "X = L xrhs"
+ − 1047 proof (cases "\<exists> r. (Y, r) \<in> zrhs")
+ − 1048 case True
+ − 1049 from True obtain r where "(1)":"(Y, r) \<in> zrhs" ..
+ − 1050 show ?thesis
+ − 1051 proof -
+ − 1052 from history self_contained
+ − 1053 have dist: "distinct_rhs zrhs" by auto
+ − 1054 hence "(SOME r. (Y, r) \<in> zrhs) = r" using self_contained "(1)"
+ − 1055 using distinct_rhs_def by (auto intro:some_equality)
+ − 1056 moreover have "L zrhs = L (rhs_subst zrhs Y rhs' r)" using substor dist "(1)" self_contained
+ − 1057 by (rule_tac rhs_subst_prop1, auto simp:distinct_equas_rhs_def)
+ − 1058 ultimately show ?thesis using subst history self_contained
+ − 1059 by (auto simp:equas_subst_f_def split:if_splits)
+ − 1060 qed
+ − 1061 next
+ − 1062 case False
+ − 1063 thus ?thesis using history subst self_contained
+ − 1064 by (auto simp:equas_subst_f_def)
+ − 1065 qed
+ − 1066
+ − 1067 lemma equas_subst_holds_left_eq_right:
+ − 1068 assumes history: "\<forall>X xrhs. (X, xrhs) \<in> ES \<longrightarrow> ardenable (X, xrhs)"
+ − 1069 and substor: "Y = L rhs'"
+ − 1070 and X_in : "(X, xrhs) \<in> equas_subst ES Y yrhs'"
+ − 1071 shows "\<forall>X xrhs. (X, xrhs) \<in> equas_subst ES Y rhs' \<longrightarrow> X = L xrhs"
+ − 1072 apply (clarsimp simp add:equas_subst_def del_x_paired_def)
+ − 1073 using substor
+ − 1074 apply (drule_tac equas_subst_f_holds_left_eq_right)
+ − 1075 using history
+ − 1076 by (auto simp:ardenable_def)
+ − 1077
+ − 1078 lemma equas_subst_holds_ardenable:
+ − 1079 assumes substor: "Y = L yrhs'"
+ − 1080 and history: "\<forall>X xrhs. (X, xrhs) \<in> ES \<longrightarrow> ardenable (X, xrhs)"
+ − 1081 and X_in:"(X, xrhs) \<in> equas_subst ES Y yrhs'"
+ − 1082 and dist': "distinct_rhs yrhs'"
+ − 1083 and not_lambda: "Y \<noteq> {[]}"
+ − 1084 shows "ardenable (X, xrhs)"
+ − 1085 proof -
+ − 1086 have "distinct_rhs xrhs" using history X_in dist'
+ − 1087 by (auto dest:equas_subst_holds_distinct_rhs)
+ − 1088 moreover have "no_EMPTY_rhs xrhs" using history X_in not_lambda
+ − 1089 by (auto intro:equas_subst_holds_no_EMPTY)
+ − 1090 moreover have "X = L xrhs" using history substor X_in
+ − 1091 by (auto dest: equas_subst_holds_left_eq_right simp del:L_rhs.simps)
+ − 1092 ultimately show ?thesis using ardenable_def by simp
+ − 1093 qed
+ − 1094
+ − 1095 lemma equas_subst_holds_cls_defined:
+ − 1096 assumes X_in: "(X, xrhs) \<in> equas_subst ES Y yrhs'"
+ − 1097 and Inv_ES: "Inv X' ES"
+ − 1098 and subst: "(Y, yrhs) \<in> ES"
+ − 1099 and cls_holds_but_Y: "rhs_eq_cls yrhs' = rhs_eq_cls yrhs - {Y}"
+ − 1100 shows "rhs_eq_cls xrhs \<subseteq> insert {[]} (left_eq_cls (equas_subst ES Y yrhs'))"
+ − 1101 proof-
+ − 1102 have tac: "\<lbrakk> A \<subseteq> B; C \<subseteq> D; E \<subseteq> A \<union> B\<rbrakk> \<Longrightarrow> E \<subseteq> B \<union> D" by auto
+ − 1103 from X_in have "\<exists> (Z, zrhs) \<in> ES. (X, xrhs) = equas_subst_f Y yrhs' (Z, zrhs)"
+ − 1104 by (auto simp add:equas_subst_def del_x_paired_def)
+ − 1105 then obtain Z zrhs where Z_in: "(Z, zrhs) \<in> ES"
+ − 1106 and X_Z: "(X, xrhs) = equas_subst_f Y yrhs' (Z, zrhs)" by blast
+ − 1107 hence "rhs_eq_cls zrhs \<subseteq> insert {[]} (left_eq_cls ES)" using Inv_ES
+ − 1108 by (auto simp:Inv_def)
+ − 1109 moreover have "rhs_eq_cls yrhs' \<subseteq> insert {[]} (left_eq_cls ES) - {Y}"
+ − 1110 using Inv_ES subst cls_holds_but_Y
+ − 1111 by (auto simp:Inv_def)
+ − 1112 moreover have "rhs_eq_cls xrhs \<subseteq> rhs_eq_cls zrhs \<union> rhs_eq_cls yrhs' - {Y}"
+ − 1113 using X_Z cls_holds_but_Y
+ − 1114 apply (clarsimp simp add:equas_subst_f_def rhs_subst_def split:if_splits)
+ − 1115 by (auto simp:rhs_eq_cls_def merge_rhs_def del_x_paired_def seq_rhs_r_def)
+ − 1116 moreover have "left_eq_cls (equas_subst ES Y yrhs') = left_eq_cls ES - {Y}" using subst
+ − 1117 by (auto simp: left_eq_cls_def equas_subst_def del_x_paired_def equas_subst_f_def
+ − 1118 dest: equas_subst_f_del_no_other
+ − 1119 split: if_splits)
+ − 1120 ultimately show ?thesis by blast
+ − 1121 qed
+ − 1122
+ − 1123 lemma iteration_step:
+ − 1124 assumes Inv_ES: "Inv X ES"
+ − 1125 and not_T: "\<not> TCon ES"
+ − 1126 shows "(\<exists> ES'. Inv X ES' \<and> (card ES', card ES) \<in> less_than)"
+ − 1127 proof -
+ − 1128 from Inv_ES not_T have another: "\<exists>Y yrhs. (Y, yrhs) \<in> ES \<and> X \<noteq> Y" unfolding Inv_def
+ − 1129 by (clarify, rule_tac exist_another_equa[where X = X], auto)
+ − 1130 then obtain Y yrhs where subst: "(Y, yrhs) \<in> ES" and not_X: " X \<noteq> Y" by blast
+ − 1131 show ?thesis (is "\<exists> ES'. ?P ES'")
+ − 1132 proof (cases "Y = {[]}")
+ − 1133 case True
+ − 1134 --"in this situation, we pick a \"\<lambda>\" equation, thus directly remove it from the equation-system"
+ − 1135 have "?P (ES - {(Y, yrhs)})"
+ − 1136 proof
+ − 1137 show "Inv X (ES - {(Y, yrhs)})" using True not_X
+ − 1138 by (simp add:Inv_ES Inv_mono_with_lambda)
+ − 1139 next
+ − 1140 show "(card (ES - {(Y, yrhs)}), card ES) \<in> less_than" using Inv_ES subst
+ − 1141 by (auto elim:non_empty_card_prop[rule_format] simp:Inv_def)
+ − 1142 qed
+ − 1143 thus ?thesis by blast
+ − 1144 next
+ − 1145 case False
11
+ − 1146 --"in this situation, we pick a equation and using ardenable to get a
+ − 1147 rhs without itself in it, then use equas_subst to form a new equation-system"
+ − 1148 hence "\<exists> yrhs'. Y = L yrhs' \<and> distinct_rhs yrhs' \<and> rhs_eq_cls yrhs' = rhs_eq_cls yrhs - {Y}"
+ − 1149 using subst Inv_ES
6
+ − 1150 by (auto intro:ardenable_prop simp add:Inv_def simp del:L_rhs.simps)
+ − 1151 then obtain yrhs' where Y'_l_eq_r: "Y = L yrhs'"
+ − 1152 and dist_Y': "distinct_rhs yrhs'"
+ − 1153 and cls_holds_but_Y: "rhs_eq_cls yrhs' = rhs_eq_cls yrhs - {Y}" by blast
+ − 1154 hence "?P (equas_subst ES Y yrhs')"
+ − 1155 proof -
+ − 1156 have finite_del: "\<And> S x. finite S \<Longrightarrow> finite (del_x_paired S x)"
+ − 1157 apply (rule_tac A = "del_x_paired S x" in finite_subset)
+ − 1158 by (auto simp:del_x_paired_def)
+ − 1159 have "finite (equas_subst ES Y yrhs')" using Inv_ES
+ − 1160 by (auto intro!:finite_del simp:equas_subst_def Inv_def)
+ − 1161 moreover have "\<exists>rhs. (X, rhs) \<in> equas_subst ES Y yrhs'" using Inv_ES not_X
+ − 1162 by (auto intro:equas_subst_del_no_other simp:Inv_def)
+ − 1163 moreover have "distinct_equas (equas_subst ES Y yrhs')" using Inv_ES
+ − 1164 by (auto intro:equas_subst_holds_distinct simp:Inv_def)
+ − 1165 moreover have "\<forall>X xrhs. (X, xrhs) \<in> equas_subst ES Y yrhs' \<longrightarrow> ardenable (X, xrhs)"
+ − 1166 using Inv_ES dist_Y' False Y'_l_eq_r
+ − 1167 apply (clarsimp simp:Inv_def)
+ − 1168 by (rule equas_subst_holds_ardenable, simp_all)
+ − 1169 moreover have "\<forall>X xrhs. (X, xrhs) \<in> equas_subst ES Y yrhs' \<longrightarrow> X \<noteq> {}" using Inv_ES
+ − 1170 by (clarsimp simp:equas_subst_def Inv_def del_x_paired_def equas_subst_f_def split:if_splits, auto)
+ − 1171 moreover have "\<forall>X xrhs. (X, xrhs) \<in> equas_subst ES Y yrhs' \<longrightarrow>
+ − 1172 rhs_eq_cls xrhs \<subseteq> insert {[]} (left_eq_cls (equas_subst ES Y yrhs'))"
+ − 1173 using Inv_ES subst cls_holds_but_Y
+ − 1174 apply (rule_tac impI | rule_tac allI)+
+ − 1175 by (erule equas_subst_holds_cls_defined, auto)
+ − 1176 moreover have "(card (equas_subst ES Y yrhs'), card ES) \<in> less_than"using Inv_ES subst
+ − 1177 by (simp add:equas_subst_card_less Inv_def)
+ − 1178 ultimately show "?P (equas_subst ES Y yrhs')" by (auto simp:Inv_def)
+ − 1179 qed
+ − 1180 thus ?thesis by blast
+ − 1181 qed
+ − 1182 qed
+ − 1183
11
+ − 1184 text {* ***** END: proving every equation-system's iteration step satisfies Inv ************** *}
6
+ − 1185
+ − 1186 lemma iteration_conc:
+ − 1187 assumes history: "Inv X ES"
+ − 1188 shows "\<exists> ES'. Inv X ES' \<and> TCon ES'" (is "\<exists> ES'. ?P ES'")
+ − 1189 proof (cases "TCon ES")
+ − 1190 case True
+ − 1191 hence "?P ES" using history by simp
+ − 1192 thus ?thesis by blast
+ − 1193 next
+ − 1194 case False
+ − 1195 thus ?thesis using history iteration_step
+ − 1196 by (rule_tac f = card in wf_iter, simp_all)
+ − 1197 qed
+ − 1198
+ − 1199 lemma eqset_imp_iff': "A = B \<Longrightarrow> \<forall> x. x \<in> A \<longleftrightarrow> x \<in> B"
+ − 1200 apply (auto simp:mem_def)
+ − 1201 done
+ − 1202
+ − 1203 lemma set_cases2:
+ − 1204 "\<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"
+ − 1205 apply (case_tac "A = {}", simp)
+ − 1206 by (case_tac "\<exists> x. A = {x}", clarsimp, blast)
+ − 1207
+ − 1208 lemma rhs_aux:"\<lbrakk>distinct_rhs rhs; {Y. \<exists>r. (Y, r) \<in> rhs} = {X}\<rbrakk> \<Longrightarrow> (\<exists>r. rhs = {(X, r)})"
+ − 1209 apply (rule_tac A = rhs in set_cases2, simp)
+ − 1210 apply (drule_tac x = X in eqset_imp_iff, clarsimp)
+ − 1211 apply (drule eqset_imp_iff',clarsimp)
+ − 1212 apply (frule_tac x = a in spec, drule_tac x = aa in spec)
+ − 1213 by (auto simp:distinct_rhs_def)
+ − 1214
+ − 1215 lemma every_eqcl_has_reg:
+ − 1216 assumes finite_CS: "finite (UNIV Quo Lang)"
+ − 1217 and X_in_CS: "X \<in> (UNIV Quo Lang)"
+ − 1218 shows "\<exists> (reg::rexp). L reg = X" (is "\<exists> r. ?E r")
+ − 1219 proof-
+ − 1220 have "\<exists>ES'. Inv X ES' \<and> TCon ES'" using finite_CS X_in_CS
+ − 1221 by (auto intro:init_ES_satisfy_Inv iteration_conc)
+ − 1222 then obtain ES' where Inv_ES': "Inv X ES'"
+ − 1223 and TCon_ES': "TCon ES'" by blast
+ − 1224 from Inv_ES' TCon_ES'
+ − 1225 have "\<exists> rhs. ES' = {(X, rhs)}"
+ − 1226 apply (clarsimp simp:Inv_def TCon_def)
+ − 1227 apply (rule_tac x = rhs in exI)
+ − 1228 by (auto dest!:card_Suc_Diff1 simp:card_eq_0_iff)
+ − 1229 then obtain rhs where ES'_single_equa: "ES' = {(X, rhs)}" ..
+ − 1230 hence X_ardenable: "ardenable (X, rhs)" using Inv_ES'
+ − 1231 by (simp add:Inv_def)
+ − 1232
+ − 1233 from X_ardenable have X_l_eq_r: "X = L rhs"
+ − 1234 by (simp add:ardenable_def)
+ − 1235 hence rhs_not_empty: "rhs \<noteq> {}" using Inv_ES' ES'_single_equa
+ − 1236 by (auto simp:Inv_def ardenable_def)
+ − 1237 have rhs_eq_cls: "rhs_eq_cls rhs \<subseteq> {X, {[]}}"
+ − 1238 using Inv_ES' ES'_single_equa
+ − 1239 by (auto simp:Inv_def ardenable_def left_eq_cls_def)
+ − 1240 have X_not_empty: "X \<noteq> {}" using Inv_ES' ES'_single_equa
+ − 1241 by (auto simp:Inv_def)
+ − 1242 show ?thesis
+ − 1243 proof (cases "X = {[]}")
+ − 1244 case True
+ − 1245 hence "?E EMPTY" by (simp)
+ − 1246 thus ?thesis by blast
+ − 1247 next
+ − 1248 case False with X_ardenable
+ − 1249 have "\<exists> rhs'. X = L rhs' \<and> rhs_eq_cls rhs' = rhs_eq_cls rhs - {X} \<and> distinct_rhs rhs'"
+ − 1250 by (drule_tac ardenable_prop, auto)
+ − 1251 then obtain rhs' where X_eq_rhs': "X = L rhs'"
+ − 1252 and rhs'_eq_cls: "rhs_eq_cls rhs' = rhs_eq_cls rhs - {X}"
+ − 1253 and rhs'_dist : "distinct_rhs rhs'" by blast
+ − 1254 have "rhs_eq_cls rhs' \<subseteq> {{[]}}" using rhs_eq_cls False rhs'_eq_cls rhs_not_empty
+ − 1255 by blast
+ − 1256 hence "rhs_eq_cls rhs' = {{[]}}" using X_not_empty X_eq_rhs'
+ − 1257 by (auto simp:rhs_eq_cls_def)
+ − 1258 hence "\<exists> r. rhs' = {({[]}, r)}" using rhs'_dist
+ − 1259 by (auto intro:rhs_aux simp:rhs_eq_cls_def)
+ − 1260 then obtain r where "rhs' = {({[]}, r)}" ..
+ − 1261 hence "?E r" using X_eq_rhs' by (auto simp add:lang_seq_def)
+ − 1262 thus ?thesis by blast
+ − 1263 qed
+ − 1264 qed
+ − 1265
11
+ − 1266 text {* definition of a regular language *}
+ − 1267
+ − 1268 abbreviation
+ − 1269 reg :: "string set => bool"
+ − 1270 where
+ − 1271 "reg L1 \<equiv> (\<exists>r::rexp. L r = L1)"
+ − 1272
6
+ − 1273 theorem myhill_nerode:
+ − 1274 assumes finite_CS: "finite (UNIV Quo Lang)"
+ − 1275 shows "\<exists> (reg::rexp). Lang = L reg" (is "\<exists> r. ?P r")
+ − 1276 proof -
+ − 1277 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
+ − 1278 by (auto dest:every_eqcl_has_reg)
+ − 1279 have "\<exists> (rS::rexp set). finite rS \<and>
+ − 1280 (\<forall> C \<in> {X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang}. \<exists> r \<in> rS. C = L r) \<and>
+ − 1281 (\<forall> r \<in> rS. \<exists> C \<in> {X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang}. C = L r)"
+ − 1282 (is "\<exists> rS. ?Q rS")
+ − 1283 proof-
+ − 1284 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)"
+ − 1285 using has_r_each
+ − 1286 apply (erule_tac x = C in ballE, erule_tac exE)
+ − 1287 by (rule_tac a = r in someI2, simp+)
+ − 1288 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
+ − 1289 using finite_CS by auto
+ − 1290 thus ?thesis by blast
+ − 1291 qed
+ − 1292 then obtain rS where finite_rS : "finite rS"
+ − 1293 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"
+ − 1294 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
+ − 1295 have "?P (folds ALT NULL rS)"
+ − 1296 proof
+ − 1297 show "Lang \<subseteq> L (folds ALT NULL rS)" using finite_rS lang_eqs_union_of_eqcls[of Lang] has_r_each'
+ − 1298 apply (clarsimp simp:fold_alt_null_eqs) by blast
+ − 1299 next
+ − 1300 show "L (folds ALT NULL rS) \<subseteq> Lang" using finite_rS lang_eqs_union_of_eqcls[of Lang] has_cl_each
+ − 1301 by (clarsimp simp:fold_alt_null_eqs)
+ − 1302 qed
+ − 1303 thus ?thesis by blast
+ − 1304 qed
8
+ − 1305
+ − 1306
+ − 1307 text {* tests by Christian *}
+ − 1308
23
+ − 1309 (* Alternative definition for Quo *)
11
+ − 1310 definition
+ − 1311 QUOT :: "string set \<Rightarrow> (string set) set"
8
+ − 1312 where
11
+ − 1313 "QUOT Lang \<equiv> (\<Union>x. {\<lbrakk>x\<rbrakk>Lang})"
+ − 1314
23
+ − 1315 lemma test:
+ − 1316 "UNIV Quo Lang = QUOT Lang"
+ − 1317 by (auto simp add: quot_def QUOT_def)
13
+ − 1318
23
+ − 1319 lemma test1:
+ − 1320 assumes finite_CS: "finite (QUOT Lang)"
+ − 1321 shows "reg Lang"
+ − 1322 using finite_CS
+ − 1323 unfolding test[symmetric]
+ − 1324 by (auto dest: myhill_nerode)
18
+ − 1325
23
+ − 1326 lemma cons_one: "x @ y \<in> {z} \<Longrightarrow> x @ y = z"
+ − 1327 by simp
+ − 1328
12
+ − 1329 lemma quot_lambda: "QUOT {[]} = {{[]}, UNIV - {[]}}"
+ − 1330 proof
+ − 1331 show "QUOT {[]} \<subseteq> {{[]}, UNIV - {[]}}"
+ − 1332 proof
+ − 1333 fix x
+ − 1334 assume in_quot: "x \<in> QUOT {[]}"
+ − 1335 show "x \<in> {{[]}, UNIV - {[]}}"
+ − 1336 proof -
+ − 1337 from in_quot
+ − 1338 have "x = {[]} \<or> x = UNIV - {[]}"
+ − 1339 unfolding QUOT_def equiv_class_def
+ − 1340 proof
+ − 1341 fix xa
+ − 1342 assume in_univ: "xa \<in> UNIV"
+ − 1343 and in_eqiv: "x \<in> {{y. xa \<equiv>{[]}\<equiv> y}}"
+ − 1344 show "x = {[]} \<or> x = UNIV - {[]}"
+ − 1345 proof(cases "xa = []")
+ − 1346 case True
+ − 1347 hence "{y. xa \<equiv>{[]}\<equiv> y} = {[]}" using in_eqiv
+ − 1348 by (auto simp add:equiv_str_def)
+ − 1349 thus ?thesis using in_eqiv by (rule_tac disjI1, simp)
+ − 1350 next
+ − 1351 case False
+ − 1352 hence "{y. xa \<equiv>{[]}\<equiv> y} = UNIV - {[]}" using in_eqiv
+ − 1353 by (auto simp:equiv_str_def)
+ − 1354 thus ?thesis using in_eqiv by (rule_tac disjI2, simp)
+ − 1355 qed
+ − 1356 qed
+ − 1357 thus ?thesis by simp
+ − 1358 qed
+ − 1359 qed
+ − 1360 next
+ − 1361 show "{{[]}, UNIV - {[]}} \<subseteq> QUOT {[]}"
+ − 1362 proof
+ − 1363 fix x
+ − 1364 assume in_res: "x \<in> {{[]}, (UNIV::string set) - {[]}}"
+ − 1365 show "x \<in> (QUOT {[]})"
+ − 1366 proof -
+ − 1367 have "x = {[]} \<Longrightarrow> x \<in> QUOT {[]}"
+ − 1368 apply (simp add:QUOT_def equiv_class_def equiv_str_def)
+ − 1369 by (rule_tac x = "[]" in exI, auto)
+ − 1370 moreover have "x = UNIV - {[]} \<Longrightarrow> x \<in> QUOT {[]}"
+ − 1371 apply (simp add:QUOT_def equiv_class_def equiv_str_def)
+ − 1372 by (rule_tac x = "''a''" in exI, auto)
+ − 1373 ultimately show ?thesis using in_res by blast
+ − 1374 qed
+ − 1375 qed
+ − 1376 qed
+ − 1377
+ − 1378 lemma quot_single_aux: "\<lbrakk>x \<noteq> []; x \<noteq> [c]\<rbrakk> \<Longrightarrow> x @ z \<noteq> [c]"
+ − 1379 by (induct x, auto)
+ − 1380
+ − 1381 lemma quot_single: "\<And> (c::char). QUOT {[c]} = {{[]}, {[c]}, UNIV - {[], [c]}}"
+ − 1382 proof -
+ − 1383 fix c::"char"
+ − 1384 have exist_another: "\<exists> a. a \<noteq> c"
+ − 1385 apply (case_tac "c = CHR ''a''")
+ − 1386 apply (rule_tac x = "CHR ''b''" in exI, simp)
+ − 1387 by (rule_tac x = "CHR ''a''" in exI, simp)
+ − 1388 show "QUOT {[c]} = {{[]}, {[c]}, UNIV - {[], [c]}}"
+ − 1389 proof
+ − 1390 show "QUOT {[c]} \<subseteq> {{[]},{[c]}, UNIV - {[], [c]}}"
+ − 1391 proof
+ − 1392 fix x
+ − 1393 assume in_quot: "x \<in> QUOT {[c]}"
+ − 1394 show "x \<in> {{[]}, {[c]}, UNIV - {[],[c]}}"
+ − 1395 proof -
+ − 1396 from in_quot
+ − 1397 have "x = {[]} \<or> x = {[c]} \<or> x = UNIV - {[],[c]}"
+ − 1398 unfolding QUOT_def equiv_class_def
+ − 1399 proof
+ − 1400 fix xa
+ − 1401 assume in_eqiv: "x \<in> {{y. xa \<equiv>{[c]}\<equiv> y}}"
+ − 1402 show "x = {[]} \<or> x = {[c]} \<or> x = UNIV - {[], [c]}"
+ − 1403 proof-
+ − 1404 have "xa = [] \<Longrightarrow> x = {[]}" using in_eqiv
+ − 1405 by (auto simp add:equiv_str_def)
+ − 1406 moreover have "xa = [c] \<Longrightarrow> x = {[c]}"
+ − 1407 proof -
+ − 1408 have "xa = [c] \<Longrightarrow> {y. xa \<equiv>{[c]}\<equiv> y} = {[c]}" using in_eqiv
+ − 1409 apply (simp add:equiv_str_def)
23
+ − 1410 apply (rule set_ext, rule iffI, simp)
12
+ − 1411 apply (drule_tac x = "[]" in spec, auto)
+ − 1412 done
+ − 1413 thus "xa = [c] \<Longrightarrow> x = {[c]}" using in_eqiv by simp
+ − 1414 qed
+ − 1415 moreover have "\<lbrakk>xa \<noteq> []; xa \<noteq> [c]\<rbrakk> \<Longrightarrow> x = UNIV - {[],[c]}"
+ − 1416 proof -
+ − 1417 have "\<lbrakk>xa \<noteq> []; xa \<noteq> [c]\<rbrakk> \<Longrightarrow> {y. xa \<equiv>{[c]}\<equiv> y} = UNIV - {[],[c]}"
+ − 1418 apply (clarsimp simp add:equiv_str_def)
23
+ − 1419 apply (rule set_ext, rule iffI, simp)
12
+ − 1420 apply (rule conjI)
+ − 1421 apply (drule_tac x = "[c]" in spec, simp)
+ − 1422 apply (drule_tac x = "[]" in spec, simp)
+ − 1423 by (auto dest:quot_single_aux)
+ − 1424 thus "\<lbrakk>xa \<noteq> []; xa \<noteq> [c]\<rbrakk> \<Longrightarrow> x = UNIV - {[],[c]}" using in_eqiv by simp
+ − 1425 qed
+ − 1426 ultimately show ?thesis by blast
+ − 1427 qed
+ − 1428 qed
+ − 1429 thus ?thesis by simp
+ − 1430 qed
+ − 1431 qed
+ − 1432 next
+ − 1433 show "{{[]}, {[c]}, UNIV - {[],[c]}} \<subseteq> QUOT {[c]}"
+ − 1434 proof
+ − 1435 fix x
+ − 1436 assume in_res: "x \<in> {{[]},{[c]}, (UNIV::string set) - {[],[c]}}"
+ − 1437 show "x \<in> (QUOT {[c]})"
+ − 1438 proof -
+ − 1439 have "x = {[]} \<Longrightarrow> x \<in> QUOT {[c]}"
+ − 1440 apply (simp add:QUOT_def equiv_class_def equiv_str_def)
+ − 1441 by (rule_tac x = "[]" in exI, auto)
+ − 1442 moreover have "x = {[c]} \<Longrightarrow> x \<in> QUOT {[c]}"
+ − 1443 apply (simp add:QUOT_def equiv_class_def equiv_str_def)
+ − 1444 apply (rule_tac x = "[c]" in exI, simp)
23
+ − 1445 apply (rule set_ext, rule iffI, simp+)
12
+ − 1446 by (drule_tac x = "[]" in spec, simp)
+ − 1447 moreover have "x = UNIV - {[],[c]} \<Longrightarrow> x \<in> QUOT {[c]}"
+ − 1448 using exist_another
+ − 1449 apply (clarsimp simp add:QUOT_def equiv_class_def equiv_str_def)
+ − 1450 apply (rule_tac x = "[a]" in exI, simp)
23
+ − 1451 apply (rule set_ext, rule iffI, simp)
12
+ − 1452 apply (clarsimp simp:quot_single_aux, simp)
+ − 1453 apply (rule conjI)
+ − 1454 apply (drule_tac x = "[c]" in spec, simp)
+ − 1455 by (drule_tac x = "[]" in spec, simp)
+ − 1456 ultimately show ?thesis using in_res by blast
+ − 1457 qed
+ − 1458 qed
+ − 1459 qed
+ − 1460 qed
+ − 1461
23
+ − 1462 lemma eq_class_imp_eq_str:
+ − 1463 "\<lbrakk>x\<rbrakk>lang = \<lbrakk>y\<rbrakk>lang \<Longrightarrow> x \<equiv>lang\<equiv> y"
+ − 1464 by (auto simp:equiv_class_def equiv_str_def)
18
+ − 1465
+ − 1466 lemma finite_tag_image:
+ − 1467 "finite (range tag) \<Longrightarrow> finite (((op `) tag) ` S)"
+ − 1468 apply (rule_tac B = "Pow (tag ` UNIV)" in finite_subset)
+ − 1469 by (auto simp add:image_def Pow_def)
+ − 1470
+ − 1471 lemma str_inj_imps:
+ − 1472 assumes str_inj: "\<And> m n. tag m = tag (n::string) \<Longrightarrow> m \<equiv>lang\<equiv> n"
23
+ − 1473 shows "inj_on ((op `) tag) (QUOT lang)"
18
+ − 1474 proof (clarsimp simp add:inj_on_def QUOT_def)
+ − 1475 fix x y
+ − 1476 assume eq_tag: "tag ` \<lbrakk>x\<rbrakk>lang = tag ` \<lbrakk>y\<rbrakk>lang"
+ − 1477 show "\<lbrakk>x\<rbrakk>lang = \<lbrakk>y\<rbrakk>lang"
+ − 1478 proof -
+ − 1479 have aux1:"\<And>a b. a \<in> (\<lbrakk>b\<rbrakk>lang) \<Longrightarrow> (a \<equiv>lang\<equiv> b)"
+ − 1480 by (simp add:equiv_class_def equiv_str_def)
+ − 1481 have aux2: "\<And> A B f. \<lbrakk>f ` A = f ` B; A \<noteq> {}\<rbrakk> \<Longrightarrow> \<exists> a b. a \<in> A \<and> b \<in> B \<and> f a = f b"
+ − 1482 by auto
+ − 1483 have aux3: "\<And> a l. \<lbrakk>a\<rbrakk>l \<noteq> {}"
+ − 1484 by (auto simp:equiv_class_def equiv_str_def)
+ − 1485 show ?thesis using eq_tag
+ − 1486 apply (drule_tac aux2, simp add:aux3, clarsimp)
+ − 1487 apply (drule_tac str_inj, (drule_tac aux1)+)
+ − 1488 by (simp add:equiv_str_def equiv_class_def)
+ − 1489 qed
+ − 1490 qed
+ − 1491
23
+ − 1492 definition tag_str_ALT :: "string set \<Rightarrow> string set \<Rightarrow> string \<Rightarrow> (string set \<times> string set)"
18
+ − 1493 where
+ − 1494 "tag_str_ALT L\<^isub>1 L\<^isub>2 x \<equiv> (\<lbrakk>x\<rbrakk>L\<^isub>1, \<lbrakk>x\<rbrakk>L\<^isub>2)"
+ − 1495
+ − 1496 lemma tag_str_alt_range_finite:
+ − 1497 assumes finite1: "finite (QUOT L\<^isub>1)"
+ − 1498 and finite2: "finite (QUOT L\<^isub>2)"
+ − 1499 shows "finite (range (tag_str_ALT L\<^isub>1 L\<^isub>2))"
+ − 1500 proof -
+ − 1501 have "{y. \<exists>x. y = (\<lbrakk>x\<rbrakk>L\<^isub>1, \<lbrakk>x\<rbrakk>L\<^isub>2)} \<subseteq> (QUOT L\<^isub>1) \<times> (QUOT L\<^isub>2)"
23
+ − 1502 by (auto simp:QUOT_def)
18
+ − 1503 thus ?thesis using finite1 finite2
+ − 1504 by (auto simp: image_def tag_str_ALT_def UNION_def
+ − 1505 intro: finite_subset[where B = "(QUOT L\<^isub>1) \<times> (QUOT L\<^isub>2)"])
+ − 1506 qed
+ − 1507
+ − 1508 lemma tag_str_alt_inj:
+ − 1509 "tag_str_ALT L\<^isub>1 L\<^isub>2 x = tag_str_ALT L\<^isub>1 L\<^isub>2 y \<Longrightarrow> x \<equiv>(L\<^isub>1 \<union> L\<^isub>2)\<equiv> y"
+ − 1510 apply (simp add:tag_str_ALT_def equiv_class_def equiv_str_def)
+ − 1511 by blast
+ − 1512
+ − 1513 lemma quot_alt:
+ − 1514 assumes finite1: "finite (QUOT L\<^isub>1)"
+ − 1515 and finite2: "finite (QUOT L\<^isub>2)"
+ − 1516 shows "finite (QUOT (L\<^isub>1 \<union> L\<^isub>2))"
23
+ − 1517 proof (rule_tac f = "(op `) (tag_str_ALT L\<^isub>1 L\<^isub>2)" in finite_imageD)
+ − 1518 show "finite (op ` (tag_str_ALT L\<^isub>1 L\<^isub>2) ` QUOT (L\<^isub>1 \<union> L\<^isub>2))"
18
+ − 1519 using finite_tag_image tag_str_alt_range_finite finite1 finite2
+ − 1520 by auto
23
+ − 1521 next
+ − 1522 show "inj_on (op ` (tag_str_ALT L\<^isub>1 L\<^isub>2)) (QUOT (L\<^isub>1 \<union> L\<^isub>2))"
18
+ − 1523 apply (rule_tac str_inj_imps)
+ − 1524 by (erule_tac tag_str_alt_inj)
+ − 1525 qed
+ − 1526
+ − 1527 (* list_diff:: list substract, once different return tailer *)
+ − 1528 fun list_diff :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" (infix "-" 51)
+ − 1529 where
+ − 1530 "list_diff [] xs = []" |
+ − 1531 "list_diff (x#xs) [] = x#xs" |
+ − 1532 "list_diff (x#xs) (y#ys) = (if x = y then list_diff xs ys else (x#xs))"
+ − 1533
23
+ − 1534 lemma [simp]: "(x @ y) - x = y"
+ − 1535 apply (induct x)
+ − 1536 by (case_tac y, simp+)
+ − 1537
+ − 1538 lemma [simp]: "x - x = []"
+ − 1539 by (induct x, auto)
+ − 1540
+ − 1541 lemma [simp]: "x = xa @ y \<Longrightarrow> x - xa = y "
+ − 1542 by (induct x, auto)
+ − 1543
+ − 1544 lemma [simp]: "x - [] = x"
+ − 1545 by (induct x, auto)
+ − 1546
+ − 1547 lemma [simp]: "xa \<le> x \<Longrightarrow> (x @ y) - xa = (x - xa) @ y"
+ − 1548 by (auto elim:prefixE)
+ − 1549
+ − 1550 definition tag_str_SEQ:: "string set \<Rightarrow> string set \<Rightarrow> string \<Rightarrow> (string set \<times> string set set)"
18
+ − 1551 where
23
+ − 1552 "tag_str_SEQ L\<^isub>1 L\<^isub>2 x \<equiv> if (\<exists> xa \<le> x. xa \<in> L\<^isub>1)
+ − 1553 then (\<lbrakk>x\<rbrakk>L\<^isub>1, {\<lbrakk>(x - xa)\<rbrakk>L\<^isub>2 | xa. xa \<le> x \<and> xa \<in> L\<^isub>1})
+ − 1554 else (\<lbrakk>x\<rbrakk>L\<^isub>1, {})"
+ − 1555
+ − 1556 lemma tag_seq_eq_E:
+ − 1557 "tag_str_SEQ L\<^isub>1 L\<^isub>2 x = tag_str_SEQ L\<^isub>1 L\<^isub>2 y \<Longrightarrow>
+ − 1558 ((\<exists> xa \<le> x. xa \<in> L\<^isub>1) \<and> \<lbrakk>x\<rbrakk>L\<^isub>1 = \<lbrakk>y\<rbrakk>L\<^isub>1 \<and>
+ − 1559 {\<lbrakk>(x - xa)\<rbrakk>L\<^isub>2 | xa. xa \<le> x \<and> xa \<in> L\<^isub>1} = {\<lbrakk>(y - ya)\<rbrakk>L\<^isub>2 | ya. ya \<le> y \<and> ya \<in> L\<^isub>1} ) \<or>
+ − 1560 ((\<forall> xa \<le> x. xa \<notin> L\<^isub>1) \<and> \<lbrakk>x\<rbrakk>L\<^isub>1 = \<lbrakk>y\<rbrakk>L\<^isub>1)"
+ − 1561 by (simp add:tag_str_SEQ_def split:if_splits, blast)
18
+ − 1562
+ − 1563 lemma tag_str_seq_range_finite:
+ − 1564 assumes finite1: "finite (QUOT L\<^isub>1)"
+ − 1565 and finite2: "finite (QUOT L\<^isub>2)"
+ − 1566 shows "finite (range (tag_str_SEQ L\<^isub>1 L\<^isub>2))"
+ − 1567 proof -
23
+ − 1568 have "(range (tag_str_SEQ L\<^isub>1 L\<^isub>2)) \<subseteq> (QUOT L\<^isub>1) \<times> (Pow (QUOT L\<^isub>2))"
18
+ − 1569 by (auto simp:image_def tag_str_SEQ_def QUOT_def)
+ − 1570 thus ?thesis using finite1 finite2
23
+ − 1571 by (rule_tac B = "(QUOT L\<^isub>1) \<times> (Pow (QUOT L\<^isub>2))" in finite_subset, auto)
18
+ − 1572 qed
+ − 1573
23
+ − 1574 lemma app_in_seq_decom[rule_format]:
+ − 1575 "\<forall> x. x @ z \<in> L\<^isub>1 ; L\<^isub>2 \<longrightarrow> (\<exists> xa \<le> x. xa \<in> L\<^isub>1 \<and> (x - xa) @ z \<in> L\<^isub>2) \<or>
+ − 1576 (\<exists> za \<le> z. (x @ za) \<in> L\<^isub>1 \<and> (z - za) \<in> L\<^isub>2)"
+ − 1577 apply (induct z)
+ − 1578 apply (simp, rule allI, rule impI, rule disjI1)
+ − 1579 apply (clarsimp simp add:lang_seq_def)
+ − 1580 apply (rule_tac x = s1 in exI, simp)
+ − 1581 apply (rule allI | rule impI)+
+ − 1582 apply (drule_tac x = "x @ [a]" in spec, simp)
+ − 1583 apply (erule exE | erule conjE | erule disjE)+
+ − 1584 apply (rule disjI2, rule_tac x = "[a]" in exI, simp)
+ − 1585 apply (rule disjI1, rule_tac x = xa in exI, simp)
+ − 1586 apply (erule exE | erule conjE)+
+ − 1587 apply (rule disjI2, rule_tac x = "a # za" in exI, simp)
+ − 1588 done
+ − 1589
18
+ − 1590 lemma tag_str_seq_inj:
+ − 1591 assumes tag_eq: "tag_str_SEQ L\<^isub>1 L\<^isub>2 x = tag_str_SEQ L\<^isub>1 L\<^isub>2 y"
+ − 1592 shows "(x::string) \<equiv>(L\<^isub>1 ; L\<^isub>2)\<equiv> y"
23
+ − 1593 proof -
+ − 1594 have aux: "\<And> x y z. \<lbrakk>tag_str_SEQ L\<^isub>1 L\<^isub>2 x = tag_str_SEQ L\<^isub>1 L\<^isub>2 y; x @ z \<in> L\<^isub>1 ; L\<^isub>2\<rbrakk>
+ − 1595 \<Longrightarrow> y @ z \<in> L\<^isub>1 ; L\<^isub>2"
+ − 1596 proof (drule app_in_seq_decom, erule disjE)
+ − 1597 fix x y z
+ − 1598 assume tag_eq: "tag_str_SEQ L\<^isub>1 L\<^isub>2 x = tag_str_SEQ L\<^isub>1 L\<^isub>2 y"
+ − 1599 and x_gets_l2: "\<exists>xa\<le>x. xa \<in> L\<^isub>1 \<and> (x - xa) @ z \<in> L\<^isub>2"
+ − 1600 from x_gets_l2 have "\<exists> xa \<le> x. xa \<in> L\<^isub>1" by blast
+ − 1601 hence xy_l2:"{\<lbrakk>(x - xa)\<rbrakk>L\<^isub>2 | xa. xa \<le> x \<and> xa \<in> L\<^isub>1} = {\<lbrakk>(y - ya)\<rbrakk>L\<^isub>2 | ya. ya \<le> y \<and> ya \<in> L\<^isub>1}"
+ − 1602 using tag_eq tag_seq_eq_E by blast
+ − 1603 from x_gets_l2 obtain xa where xa_le_x: "xa \<le> x"
+ − 1604 and xa_in_l1: "xa \<in> L\<^isub>1"
+ − 1605 and rest_in_l2: "(x - xa) @ z \<in> L\<^isub>2" by blast
+ − 1606 hence "\<exists> ya. \<lbrakk>(x - xa)\<rbrakk>L\<^isub>2 = \<lbrakk>(y - ya)\<rbrakk>L\<^isub>2 \<and> ya \<le> y \<and> ya \<in> L\<^isub>1" using xy_l2 by auto
+ − 1607 then obtain ya where ya_le_x: "ya \<le> y"
+ − 1608 and ya_in_l1: "ya \<in> L\<^isub>1"
+ − 1609 and rest_eq: "\<lbrakk>(x - xa)\<rbrakk>L\<^isub>2 = \<lbrakk>(y - ya)\<rbrakk>L\<^isub>2" by blast
+ − 1610 from rest_eq rest_in_l2 have "(y - ya) @ z \<in> L\<^isub>2"
+ − 1611 by (auto simp:equiv_class_def equiv_str_def)
+ − 1612 hence "ya @ ((y - ya) @ z) \<in> L\<^isub>1 ; L\<^isub>2" using ya_in_l1
+ − 1613 by (auto simp:lang_seq_def)
+ − 1614 thus "y @ z \<in> L\<^isub>1 ; L\<^isub>2" using ya_le_x
+ − 1615 by (erule_tac prefixE, simp)
18
+ − 1616 next
23
+ − 1617 fix x y z
+ − 1618 assume tag_eq: "tag_str_SEQ L\<^isub>1 L\<^isub>2 x = tag_str_SEQ L\<^isub>1 L\<^isub>2 y"
+ − 1619 and x_gets_l1: "\<exists>za\<le>z. x @ za \<in> L\<^isub>1 \<and> z - za \<in> L\<^isub>2"
+ − 1620 from tag_eq tag_seq_eq_E have x_y_eq: "\<lbrakk>x\<rbrakk>L\<^isub>1 = \<lbrakk>y\<rbrakk>L\<^isub>1" by blast
+ − 1621 from x_gets_l1 obtain za where za_le_z: "za \<le> z"
+ − 1622 and x_za_in_l1: "(x @ za) \<in> L\<^isub>1"
+ − 1623 and rest_in_l2: "z - za \<in> L\<^isub>2" by blast
+ − 1624 from x_y_eq x_za_in_l1 have y_za_in_l1: "y @ za \<in> L\<^isub>1"
+ − 1625 by (auto simp:equiv_class_def equiv_str_def)
+ − 1626 hence "(y @ za) @ (z - za) \<in> L\<^isub>1 ; L\<^isub>2" using rest_in_l2
+ − 1627 apply (simp add:lang_seq_def)
+ − 1628 by (rule_tac x = "y @ za" in exI, rule_tac x = "z - za" in exI, simp)
+ − 1629 thus "y @ z \<in> L\<^isub>1 ; L\<^isub>2" using za_le_z
+ − 1630 by (erule_tac prefixE, simp)
18
+ − 1631 qed
23
+ − 1632 show ?thesis using tag_eq
+ − 1633 apply (simp add:equiv_str_def)
+ − 1634 by (auto intro:aux)
18
+ − 1635 qed
+ − 1636
+ − 1637 lemma quot_seq:
+ − 1638 assumes finite1: "finite (QUOT L\<^isub>1)"
+ − 1639 and finite2: "finite (QUOT L\<^isub>2)"
+ − 1640 shows "finite (QUOT (L\<^isub>1;L\<^isub>2))"
+ − 1641 proof (rule_tac f = "(op `) (tag_str_SEQ L\<^isub>1 L\<^isub>2)" in finite_imageD)
+ − 1642 show "finite (op ` (tag_str_SEQ L\<^isub>1 L\<^isub>2) ` QUOT (L\<^isub>1 ; L\<^isub>2))"
+ − 1643 using finite_tag_image tag_str_seq_range_finite finite1 finite2
+ − 1644 by auto
+ − 1645 next
+ − 1646 show "inj_on (op ` (tag_str_SEQ L\<^isub>1 L\<^isub>2)) (QUOT (L\<^isub>1 ; L\<^isub>2))"
+ − 1647 apply (rule_tac str_inj_imps)
+ − 1648 by (erule_tac tag_str_seq_inj)
+ − 1649 qed
+ − 1650
23
+ − 1651 (****************** the STAR case ************************)
+ − 1652
+ − 1653 lemma app_eq_elim[rule_format]:
+ − 1654 "\<And> a. \<forall> b x y. a @ b = x @ y \<longrightarrow> (\<exists> aa ab. a = aa @ ab \<and> x = aa \<and> y = ab @ b) \<or>
+ − 1655 (\<exists> ba bb. b = ba @ bb \<and> x = a @ ba \<and> y = bb \<and> ba \<noteq> [])"
+ − 1656 apply (induct_tac a rule:List.induct, simp)
+ − 1657 apply (rule allI | rule impI)+
+ − 1658 by (case_tac x, auto)
+ − 1659
+ − 1660 definition tag_str_STAR:: "string set \<Rightarrow> string \<Rightarrow> string set set"
18
+ − 1661 where
23
+ − 1662 "tag_str_STAR L\<^isub>1 x \<equiv> if (x = [])
+ − 1663 then {}
+ − 1664 else {\<lbrakk>x\<^isub>2\<rbrakk>L\<^isub>1 | x\<^isub>1 x\<^isub>2. x = x\<^isub>1 @ x\<^isub>2 \<and> x\<^isub>1 \<in> L\<^isub>1\<star> \<and> x\<^isub>2 \<noteq> []}"
18
+ − 1665
+ − 1666 lemma tag_str_star_range_finite:
+ − 1667 assumes finite1: "finite (QUOT L\<^isub>1)"
+ − 1668 shows "finite (range (tag_str_STAR L\<^isub>1))"
+ − 1669 proof -
+ − 1670 have "range (tag_str_STAR L\<^isub>1) \<subseteq> Pow (QUOT L\<^isub>1)"
+ − 1671 by (auto simp:image_def tag_str_STAR_def QUOT_def)
+ − 1672 thus ?thesis using finite1
+ − 1673 by (rule_tac B = "Pow (QUOT L\<^isub>1)" in finite_subset, auto)
+ − 1674 qed
+ − 1675
23
+ − 1676 lemma star_prop[rule_format]: "x \<in> lang\<star> \<Longrightarrow> \<forall> y. y \<in> lang\<star> \<longrightarrow> x @ y \<in> lang\<star>"
+ − 1677 by (erule Star.induct, auto)
+ − 1678
+ − 1679 lemma star_prop2: "y \<in> lang \<Longrightarrow> y \<in> lang\<star>"
+ − 1680 by (drule step[of y lang "[]"], auto simp:start)
+ − 1681
+ − 1682 lemma star_prop3[rule_format]: "x \<in> lang\<star> \<Longrightarrow> \<forall>y . y \<in> lang \<longrightarrow> x @ y \<in> lang\<star>"
+ − 1683 by (erule Star.induct, auto intro:star_prop2)
+ − 1684
+ − 1685 lemma postfix_prop: "y >>= (x @ y) \<Longrightarrow> x = []"
+ − 1686 by (erule postfixE, induct x arbitrary:y, auto)
+ − 1687
+ − 1688 lemma inj_aux:
+ − 1689 "\<lbrakk>(m @ z) \<in> L\<^isub>1\<star>; m \<equiv>L\<^isub>1\<equiv> yb; xa @ m = x; xa \<in> L\<^isub>1\<star>; m \<noteq> [];
+ − 1690 \<forall> xa xb. x = xa @ xb \<and> xa \<in> L\<^isub>1\<star> \<and> xb \<noteq> [] \<and> xb @ z \<in> L\<^isub>1\<star> \<longrightarrow> xb >>= m\<rbrakk>
+ − 1691 \<Longrightarrow> (yb @ z) \<in> L\<^isub>1\<star>"
+ − 1692 proof-
+ − 1693 have "\<And>s. s \<in> L\<^isub>1\<star> \<Longrightarrow> \<forall> m z yb. (s = m @ z \<and> m \<equiv>L\<^isub>1\<equiv> yb \<and> x = xa @ m \<and> xa \<in> L\<^isub>1\<star> \<and> m \<noteq> [] \<and>
+ − 1694 (\<forall> xa xb. x = xa @ xb \<and> xa \<in> L\<^isub>1\<star> \<and> xb \<noteq> [] \<and> xb @ z \<in> L\<^isub>1\<star> \<longrightarrow> xb >>= m) \<longrightarrow> (yb @ z) \<in> L\<^isub>1\<star>)"
+ − 1695 apply (erule Star.induct, simp)
+ − 1696 apply (rule allI | rule impI | erule conjE)+
+ − 1697 apply (drule app_eq_elim)
+ − 1698 apply (erule disjE | erule exE | erule conjE)+
+ − 1699 apply simp
+ − 1700 apply (simp (no_asm) only:append_assoc[THEN sym])
+ − 1701 apply (rule step)
+ − 1702 apply (simp add:equiv_str_def)
+ − 1703 apply simp
+ − 1704
+ − 1705 apply (erule exE | erule conjE)+
+ − 1706 apply (rotate_tac 3)
+ − 1707 apply (frule_tac x = "xa @ s1" in spec)
+ − 1708 apply (rotate_tac 12)
+ − 1709 apply (drule_tac x = ba in spec)
+ − 1710 apply (erule impE)
+ − 1711 apply ( simp add:star_prop3)
+ − 1712 apply (simp)
+ − 1713 apply (drule postfix_prop)
+ − 1714 apply simp
+ − 1715 done
+ − 1716 thus "\<lbrakk>(m @ z) \<in> L\<^isub>1\<star>; m \<equiv>L\<^isub>1\<equiv> yb; xa @ m = x; xa \<in> L\<^isub>1\<star>; m \<noteq> [];
+ − 1717 \<forall> xa xb. x = xa @ xb \<and> xa \<in> L\<^isub>1\<star> \<and> xb \<noteq> [] \<and> xb @ z \<in> L\<^isub>1\<star> \<longrightarrow> xb >>= m\<rbrakk>
+ − 1718 \<Longrightarrow> (yb @ z) \<in> L\<^isub>1\<star>" by blast
+ − 1719 qed
+ − 1720
+ − 1721
+ − 1722 lemma min_postfix_exists[rule_format]:
+ − 1723 "finite A \<Longrightarrow> A \<noteq> {} \<and> (\<forall> a \<in> A. \<forall> b \<in> A. ((b >>= a) \<or> (a >>= b)))
+ − 1724 \<longrightarrow> (\<exists> min. (min \<in> A \<and> (\<forall> a \<in> A. a >>= min)))"
+ − 1725 apply (erule finite.induct)
+ − 1726 apply simp
+ − 1727 apply simp
+ − 1728 apply (case_tac "A = {}")
+ − 1729 apply simp
+ − 1730 apply clarsimp
+ − 1731 apply (case_tac "a >>= min")
+ − 1732 apply (rule_tac x = min in exI, simp)
+ − 1733 apply (rule_tac x = a in exI, simp)
+ − 1734 apply clarify
+ − 1735 apply (rotate_tac 5)
+ − 1736 apply (erule_tac x = aa in ballE) defer apply simp
+ − 1737 apply (erule_tac ys = min in postfix_trans)
+ − 1738 apply (erule_tac x = min in ballE)
+ − 1739 by simp+
+ − 1740
18
+ − 1741 lemma tag_str_star_inj:
+ − 1742 "tag_str_STAR L\<^isub>1 x = tag_str_STAR L\<^isub>1 (y::string) \<Longrightarrow> x \<equiv>L\<^isub>1\<star>\<equiv> y"
+ − 1743 proof -
23
+ − 1744 have aux: "\<And> x y z. \<lbrakk>tag_str_STAR L\<^isub>1 x = tag_str_STAR L\<^isub>1 y; x @ z \<in> L\<^isub>1\<star>\<rbrakk> \<Longrightarrow> y @ z \<in> L\<^isub>1\<star>"
+ − 1745 proof-
+ − 1746 fix x y z
+ − 1747 assume tag_eq: "tag_str_STAR L\<^isub>1 x = tag_str_STAR L\<^isub>1 y"
+ − 1748 and x_z: "x @ z \<in> L\<^isub>1\<star>"
+ − 1749 show "y @ z \<in> L\<^isub>1\<star>"
+ − 1750 proof (cases "x = []")
+ − 1751 case True
+ − 1752 with tag_eq have "y = []" by (simp add:tag_str_STAR_def split:if_splits, blast)
+ − 1753 thus ?thesis using x_z True by simp
+ − 1754 next
+ − 1755 case False
+ − 1756 hence not_empty: "{xb. \<exists> xa. x = xa @ xb \<and> xa \<in> L\<^isub>1\<star> \<and> xb \<noteq> [] \<and> xb @ z \<in> L\<^isub>1\<star>} \<noteq> {}" using x_z
+ − 1757 by (simp, rule_tac x = x in exI, rule_tac x = "[]" in exI, simp add:start)
+ − 1758 have finite_set: "finite {xb. \<exists> xa. x = xa @ xb \<and> xa \<in> L\<^isub>1\<star> \<and> xb \<noteq> [] \<and> xb @ z \<in> L\<^isub>1\<star>}"
+ − 1759 apply (rule_tac B = "{xb. \<exists> xa. x = xa @ xb}" in finite_subset)
+ − 1760 apply auto
+ − 1761 apply (induct x, simp)
+ − 1762 apply (subgoal_tac "{xb. \<exists>xa. a # x = xa @ xb} = insert (a # x) {xb. \<exists>xa. x = xa @ xb}")
+ − 1763 apply auto
+ − 1764 by (case_tac xaa, simp+)
+ − 1765 have comparable: "\<forall> a \<in> {xb. \<exists> xa. x = xa @ xb \<and> xa \<in> L\<^isub>1\<star> \<and> xb \<noteq> [] \<and> xb @ z \<in> L\<^isub>1\<star>}.
+ − 1766 \<forall> b \<in> {xb. \<exists> xa. x = xa @ xb \<and> xa \<in> L\<^isub>1\<star> \<and> xb \<noteq> [] \<and> xb @ z \<in> L\<^isub>1\<star>}.
+ − 1767 ((b >>= a) \<or> (a >>= b))"
+ − 1768 by (auto simp:postfix_def, drule app_eq_elim, blast)
+ − 1769 hence "\<exists> min. min \<in> {xb. \<exists> xa. x = xa @ xb \<and> xa \<in> L\<^isub>1\<star> \<and> xb \<noteq> [] \<and> xb @ z \<in> L\<^isub>1\<star>}
+ − 1770 \<and> (\<forall> xa xb. x = xa @ xb \<and> xa \<in> L\<^isub>1\<star> \<and> xb \<noteq> [] \<and> xb @ z \<in> L\<^isub>1\<star> \<longrightarrow> xb >>= min)"
+ − 1771 using finite_set not_empty comparable
+ − 1772 apply (drule_tac min_postfix_exists, simp)
+ − 1773 by (erule exE, rule_tac x = min in exI, auto)
+ − 1774 then obtain min xa where x_decom: "x = xa @ min \<and> xa \<in> L\<^isub>1\<star>"
+ − 1775 and min_not_empty: "min \<noteq> []"
+ − 1776 and min_z_in_star: "min @ z \<in> L\<^isub>1\<star>"
+ − 1777 and is_min: "\<forall> xa xb. x = xa @ xb \<and> xa \<in> L\<^isub>1\<star> \<and> xb \<noteq> [] \<and> xb @ z \<in> L\<^isub>1\<star> \<longrightarrow> xb >>= min" by blast
+ − 1778 from x_decom min_not_empty have "\<lbrakk>min\<rbrakk>L\<^isub>1 \<in> tag_str_STAR L\<^isub>1 x" by (auto simp:tag_str_STAR_def)
+ − 1779 hence "\<exists> yb. \<lbrakk>yb\<rbrakk>L\<^isub>1 \<in> tag_str_STAR L\<^isub>1 y \<and> \<lbrakk>min\<rbrakk>L\<^isub>1 = \<lbrakk>yb\<rbrakk>L\<^isub>1" using tag_eq by auto
+ − 1780 hence "\<exists> ya yb. y = ya @ yb \<and> ya \<in> L\<^isub>1\<star> \<and> min \<equiv>L\<^isub>1\<equiv> yb \<and> yb \<noteq> [] "
+ − 1781 by (simp add:tag_str_STAR_def equiv_class_def equiv_str_def split:if_splits, blast)
+ − 1782 then obtain ya yb where y_decom: "y = ya @ yb"
+ − 1783 and ya_in_star: "ya \<in> L\<^isub>1\<star>"
+ − 1784 and yb_not_empty: "yb \<noteq> []"
+ − 1785 and min_yb_eq: "min \<equiv>L\<^isub>1\<equiv> yb" by blast
+ − 1786 from min_z_in_star min_yb_eq min_not_empty is_min x_decom
+ − 1787 have "yb @ z \<in> L\<^isub>1\<star>"
+ − 1788 by (rule_tac x = x and xa = xa in inj_aux, simp+)
+ − 1789 thus ?thesis using ya_in_star y_decom
+ − 1790 by (auto dest:star_prop)
+ − 1791 qed
18
+ − 1792 qed
23
+ − 1793 show "tag_str_STAR L\<^isub>1 x = tag_str_STAR L\<^isub>1 (y::string) \<Longrightarrow> x \<equiv>L\<^isub>1\<star>\<equiv> y"
+ − 1794 by (auto intro:aux simp:equiv_str_def)
+ − 1795 qed
18
+ − 1796
+ − 1797 lemma quot_star:
+ − 1798 assumes finite1: "finite (QUOT L\<^isub>1)"
+ − 1799 shows "finite (QUOT (L\<^isub>1\<star>))"
+ − 1800 proof (rule_tac f = "(op `) (tag_str_STAR L\<^isub>1)" in finite_imageD)
+ − 1801 show "finite (op ` (tag_str_STAR L\<^isub>1) ` QUOT (L\<^isub>1\<star>))"
+ − 1802 using finite_tag_image tag_str_star_range_finite finite1
+ − 1803 by auto
+ − 1804 next
+ − 1805 show "inj_on (op ` (tag_str_STAR L\<^isub>1)) (QUOT (L\<^isub>1\<star>))"
+ − 1806 apply (rule_tac str_inj_imps)
+ − 1807 by (erule_tac tag_str_star_inj)
+ − 1808 qed
+ − 1809
+ − 1810 lemma other_direction:
+ − 1811 "Lang = L (r::rexp) \<Longrightarrow> finite (QUOT Lang)"
+ − 1812 apply (induct arbitrary:Lang rule:rexp.induct)
+ − 1813 apply (simp add:QUOT_def equiv_class_def equiv_str_def)
+ − 1814 by (simp_all add:quot_lambda quot_single quot_seq quot_alt quot_star)
+ − 1815
23
+ − 1816 end