1    

     2 theory SpecAlts

     3   imports Main "~~/src/HOL/Library/Sublist"

     4 begin

     5 

     6 section {* Sequential Composition of Languages *}

     7 

     8 definition

     9   Sequ :: "string set \<Rightarrow> string set \<Rightarrow> string set" ("_ ;; _" [100,100] 100)

    10 where 

    11   "A ;; B = {s1 @ s2 | s1 s2. s1 \<in> A \<and> s2 \<in> B}"

    12 

    13 text {* Two Simple Properties about Sequential Composition *}

    14 

    15 lemma Sequ_empty_string [simp]:

    16   shows "A ;; {[]} = A"

    17   and   "{[]} ;; A = A"

    18 by (simp_all add: Sequ_def)

    19 

    20 lemma Sequ_empty [simp]:

    21   shows "A ;; {} = {}"

    22   and   "{} ;; A = {}"

    23 by (simp_all add: Sequ_def)

    24 

    25 

    26 section {* Semantic Derivative (Left Quotient) of Languages *}

    27 

    28 definition

    29   Der :: "char \<Rightarrow> string set \<Rightarrow> string set"

    30 where

    31   "Der c A \<equiv> {s. c # s \<in> A}"

    32 

    33 definition

    34   Ders :: "string \<Rightarrow> string set \<Rightarrow> string set"

    35 where

    36   "Ders s A \<equiv> {s'. s @ s' \<in> A}"

    37 

    38 lemma Der_null [simp]:

    39   shows "Der c {} = {}"

    40 unfolding Der_def

    41 by auto

    42 

    43 lemma Der_empty [simp]:

    44   shows "Der c {[]} = {}"

    45 unfolding Der_def

    46 by auto

    47 

    48 lemma Der_char [simp]:

    49   shows "Der c {[d]} = (if c = d then {[]} else {})"

    50 unfolding Der_def

    51 by auto

    52 

    53 lemma Der_union [simp]:

    54   shows "Der c (A \<union> B) = Der c A \<union> Der c B"

    55 unfolding Der_def

    56   by auto

    57 

    58 lemma Der_Union [simp]:

    59   shows "Der c (\<Union>B. A) = (\<Union>B. Der c A)"

    60 unfolding Der_def

    61 by auto

    62 

    63 lemma Der_Sequ [simp]:

    64   shows "Der c (A ;; B) = (Der c A) ;; B \<union> (if [] \<in> A then Der c B else {})"

    65 unfolding Der_def Sequ_def

    66 by (auto simp add: Cons_eq_append_conv)

    67 

    68 

    69 section {* Kleene Star for Languages *}

    70 

    71 inductive_set

    72   Star :: "string set \<Rightarrow> string set" ("_\<star>" [101] 102)

    73   for A :: "string set"

    74 where

    75   start[intro]: "[] \<in> A\<star>"

    76 | step[intro]:  "\<lbrakk>s1 \<in> A; s2 \<in> A\<star>\<rbrakk> \<Longrightarrow> s1 @ s2 \<in> A\<star>"

    77 

    78 (* Arden's lemma *)

    79 

    80 lemma Star_cases:

    81   shows "A\<star> = {[]} \<union> A ;; A\<star>"

    82 unfolding Sequ_def

    83 by (auto) (metis Star.simps)

    84 

    85 lemma Star_decomp: 

    86   assumes "c # x \<in> A\<star>" 

    87   shows "\<exists>s1 s2. x = s1 @ s2 \<and> c # s1 \<in> A \<and> s2 \<in> A\<star>"

    88 using assms

    89 by (induct x\<equiv>"c # x" rule: Star.induct) 

    90    (auto simp add: append_eq_Cons_conv)

    91 

    92 lemma Star_Der_Sequ: 

    93   shows "Der c (A\<star>) \<subseteq> (Der c A) ;; A\<star>"

    94 unfolding Der_def Sequ_def

    95 by(auto simp add: Star_decomp)

    96 

    97 

    98 lemma Der_star [simp]:

    99   shows "Der c (A\<star>) = (Der c A) ;; A\<star>"

   100 proof -    

   101   have "Der c (A\<star>) = Der c ({[]} \<union> A ;; A\<star>)"  

   102     by (simp only: Star_cases[symmetric])

   103   also have "... = Der c (A ;; A\<star>)"

   104     by (simp only: Der_union Der_empty) (simp)

   105   also have "... = (Der c A) ;; A\<star> \<union> (if [] \<in> A then Der c (A\<star>) else {})"

   106     by simp

   107   also have "... =  (Der c A) ;; A\<star>"

   108     using Star_Der_Sequ by auto

   109   finally show "Der c (A\<star>) = (Der c A) ;; A\<star>" .

   110 qed

   111 

   112 

   113 section {* Regular Expressions *}

   114 

   115 datatype rexp =

   116   ZERO

   117 | ONE

   118 | CHAR char

   119 | SEQ rexp rexp

   120 | ALTS "rexp list"

   121 | STAR rexp

   122 

   123 section {* Semantics of Regular Expressions *}

   124  

   125 fun

   126   L :: "rexp \<Rightarrow> string set"

   127 where

   128   "L (ZERO) = {}"

   129 | "L (ONE) = {[]}"

   130 | "L (CHAR c) = {[c]}"

   131 | "L (SEQ r1 r2) = (L r1) ;; (L r2)"

   132 | "L (ALTS rs) = (\<Union>r \<in> set rs. L r)"

   133 | "L (STAR r) = (L r)\<star>"

   134 

   135 

   136 section {* Nullable, Derivatives *}

   137 

   138 fun

   139  nullable :: "rexp \<Rightarrow> bool"

   140 where

   141   "nullable (ZERO) = False"

   142 | "nullable (ONE) = True"

   143 | "nullable (CHAR c) = False"

   144 | "nullable (ALTS rs) = (\<exists>r \<in> set rs. nullable r)"

   145 | "nullable (SEQ r1 r2) = (nullable r1 \<and> nullable r2)"

   146 | "nullable (STAR r) = True"

   147 

   148 

   149 fun

   150  der :: "char \<Rightarrow> rexp \<Rightarrow> rexp"

   151 where

   152   "der c (ZERO) = ZERO"

   153 | "der c (ONE) = ZERO"

   154 | "der c (CHAR d) = (if c = d then ONE else ZERO)"

   155 | "der c (ALTS rs) = ALTS (map (der c) rs)"

   156 | "der c (SEQ r1 r2) = 

   157      (if nullable r1

   158       then ALTS [SEQ (der c r1) r2, der c r2]

   159       else SEQ (der c r1) r2)"

   160 | "der c (STAR r) = SEQ (der c r) (STAR r)"

   161 

   162 fun 

   163  ders :: "string \<Rightarrow> rexp \<Rightarrow> rexp"

   164 where

   165   "ders [] r = r"

   166 | "ders (c # s) r = ders s (der c r)"

   167 

   168 

   169 lemma nullable_correctness:

   170   shows "nullable r  \<longleftrightarrow> [] \<in> (L r)"

   171 by (induct r) (auto simp add: Sequ_def) 

   172 

   173 lemma der_correctness:

   174   shows "L (der c r) = Der c (L r)"

   175   apply(induct r) 

   176        apply(simp_all add: nullable_correctness)

   177   apply(auto simp add: Der_def)

   178   done

   179 

   180 lemma ders_correctness:

   181   shows "L (ders s r) = Ders s (L r)"

   182 by (induct s arbitrary: r)

   183    (simp_all add: Ders_def der_correctness Der_def)

   184 

   185 fun flats :: "rexp list \<Rightarrow> rexp list"

   186   where

   187   "flats [] = []"

   188 | "flats (ZERO # rs1) = flats(rs1)"

   189 | "flats ((ALTS rs1) #rs2) = rs1 @ (flats rs2)"

   190 | "flats (r1 # rs2) = r1 # flats rs2"

   191 

   192 fun simp_SEQ where

   193   "simp_SEQ ONE r\<^sub>2 = r\<^sub>2"

   194 | "simp_SEQ r\<^sub>1 ONE = r\<^sub>1"

   195 | "simp_SEQ r\<^sub>1 r\<^sub>2 = SEQ r\<^sub>1 r\<^sub>2"  

   196  

   197 fun 

   198   simp :: "rexp \<Rightarrow> rexp"

   199 where

   200   "simp (ALTS rs) = ALTS (remdups (flats (map simp rs)))" 

   201 | "simp (SEQ r1 r2) = simp_SEQ (simp r1) (simp r2)" 

   202 | "simp r = r"

   203 

   204 lemma simp_SEQ_correctness:

   205   shows "L (simp_SEQ r1 r2) = L (SEQ r1 r2)"

   206   apply(induct r1 r2 rule: simp_SEQ.induct)

   207   apply(simp_all)

   208   done

   209 

   210 lemma flats_correctness:

   211   shows "(\<Union>r \<in> set (flats rs). L r) = L (ALTS rs)"

   212   apply(induct rs rule: flats.induct)

   213   apply(simp_all)

   214   done

   215 

   216 

   217 lemma simp_correctness:

   218   shows "L (simp r) = L r"

   219   apply(induct r)

   220   apply(simp_all)

   221   apply(simp add: simp_SEQ_correctness)

   222   apply(simp add: flats_correctness)

   223   done

   224 

   225 fun 

   226  ders2 :: "string \<Rightarrow> rexp \<Rightarrow> rexp"

   227 where

   228   "ders2 [] r = r"

   229 | "ders2 (c # s) r = ders2 s (simp (der c r))"

   230 

   231 lemma ders2_ZERO:

   232   shows "ders2 s ZERO = ZERO"

   233   apply(induct s)

   234   apply(simp_all)

   235   done

   236 

   237 lemma ders2_ONE:

   238   shows "ders2 s ONE \<in> {ZERO, ONE}"

   239   apply(induct s)

   240   apply(simp_all)

   241   apply(auto)

   242   apply(case_tac s)

   243   apply(auto)

   244   apply(case_tac s)

   245   apply(auto)

   246   done

   247 

   248 lemma ders2_CHAR:

   249   shows "ders2 s (CHAR c) \<in> {ZERO, ONE, CHAR c}"

   250   apply(induct s)

   251   apply(simp_all)

   252   apply(auto simp add: ders2_ZERO)

   253   apply(case_tac s)

   254   apply(auto simp add: ders2_ZERO)

   255   using ders2_ONE

   256   apply(auto)[1]

   257   using ders2_ONE

   258   apply(auto)[1]

   259   done

   260 

   261 lemma remdup_size:

   262   shows "size_list f (remdups rs) \<le> size_list f rs"

   263   apply(induct rs)

   264    apply(simp_all)

   265   done

   266 

   267 lemma flats_append:

   268   shows "flats (rs1 @ rs2) = (flats rs1) @ (flats rs2)"

   269   apply(induct rs1 arbitrary: rs2)

   270    apply(auto)

   271   apply(case_tac a)

   272        apply(auto)

   273   done

   274 

   275 lemma flats_Cons:

   276   shows "flats (r # rs) = (flats [r]) @ (flats rs)"

   277   apply(subst flats_append[symmetric])

   278   apply(simp)

   279   done

   280 

   281 lemma flats_size:

   282   shows "size_list (\<lambda>x. size (ders2 s x)) (flats rs) \<le> size_list (\<lambda>x. size (ders2 s x))  rs"

   283   apply(induct rs arbitrary: s rule: flats.induct)

   284    apply(simp_all)

   285    apply(simp add: ders2_ZERO)

   286    apply (simp add: le_SucI)

   287   

   288    apply(subst flats_Cons)

   289   apply(simp)

   290   apply(case_tac a)

   291        apply(auto)

   292    apply(simp add: ders2_ZERO)

   293    apply (simp add: le_SucI)

   294   sorry

   295 

   296 lemma ders2_ALTS:

   297   shows "size (ders2 s (ALTS rs)) \<le> size (ALTS (map (ders2 s) rs))"

   298   apply(induct s arbitrary: rs)

   299    apply(simp_all)

   300   thm size_list_pointwise

   301   apply (simp add: size_list_pointwise)

   302   apply(drule_tac x="remdups (flats (map (simp \<circ> der a) rs))" in meta_spec)

   303   apply(rule le_trans)

   304    apply(assumption)

   305   apply(simp)

   306   apply(rule le_trans)

   307    apply(rule remdup_size)

   308   apply(simp add: comp_def)

   309   apply(rule le_trans)

   310   apply(rule flats_size)

   311   by (simp add: size_list_pointwise)

   312 

   313 definition

   314  "derss2 A r = {ders2 s r | s. s \<in> A}"

   315 

   316 lemma

   317   "\<forall>rd \<in> derss2 (UNIV) r. size rd \<le> Suc (size r)"

   318   apply(induct r)

   319   apply(auto simp add: derss2_def ders2_ZERO)[1]

   320       apply(auto simp add: derss2_def ders2_ZERO)[1]

   321   using ders2_ONE

   322       apply(auto)[1]

   323   apply (metis rexp.size(7) rexp.size(8) zero_le)

   324   using ders2_CHAR

   325      apply(auto)[1]

   326   apply (smt derss2_def le_SucI le_zero_eq mem_Collect_eq rexp.size(7) rexp.size(8) rexp.size(9))

   327     defer  

   328     apply(auto simp add: derss2_def)

   329     apply(rule le_trans)

   330      apply(rule ders2_ALTS)

   331     apply(simp)

   332     apply(simp add: comp_def)

   333   

   334     apply(simp add: size_list_pointwise)

   335     apply(case_tac s)

   336      apply(simp)

   337   apply(simp only:)

   338   apply(auto)[1]

   339    

   340   apply(case_tac s)

   341         apply(simp)

   342   apply(simp)

   343 

   344 section {* Values *}

   345 

   346 datatype val = 

   347   Void

   348 | Char char

   349 | Seq val val

   350 | Nth nat val

   351 | Stars "val list"

   352 

   353 

   354 section {* The string behind a value *}

   355 

   356 fun 

   357   flat :: "val \<Rightarrow> string"

   358 where

   359   "flat (Void) = []"

   360 | "flat (Char c) = [c]"

   361 | "flat (Nth n v) = flat v"

   362 | "flat (Seq v1 v2) = (flat v1) @ (flat v2)"

   363 | "flat (Stars []) = []"

   364 | "flat (Stars (v#vs)) = (flat v) @ (flat (Stars vs))" 

   365 

   366 abbreviation

   367   "flats vs \<equiv> concat (map flat vs)"

   368 

   369 lemma flat_Stars [simp]:

   370  "flat (Stars vs) = flats vs"

   371 by (induct vs) (auto)

   372 

   373 lemma Star_concat:

   374   assumes "\<forall>s \<in> set ss. s \<in> A"  

   375   shows "concat ss \<in> A\<star>"

   376 using assms by (induct ss) (auto)

   377 

   378 lemma Star_cstring:

   379   assumes "s \<in> A\<star>"

   380   shows "\<exists>ss. concat ss = s \<and> (\<forall>s \<in> set ss. s \<in> A \<and> s \<noteq> [])"

   381 using assms

   382 apply(induct rule: Star.induct)

   383 apply(auto)[1]

   384 apply(rule_tac x="[]" in exI)

   385 apply(simp)

   386 apply(erule exE)

   387 apply(clarify)

   388 apply(case_tac "s1 = []")

   389 apply(rule_tac x="ss" in exI)

   390 apply(simp)

   391 apply(rule_tac x="s1#ss" in exI)

   392 apply(simp)

   393 done

   394 

   395 

   396 section {* Lexical Values *}

   397 

   398 inductive 

   399   Prf :: "val \<Rightarrow> rexp \<Rightarrow> bool" ("\<Turnstile> _ : _" [100, 100] 100)

   400 where

   401   "\<lbrakk>\<Turnstile> v1 : r1; \<Turnstile> v2 : r2\<rbrakk> \<Longrightarrow> \<Turnstile>  Seq v1 v2 : SEQ r1 r2"

   402 | "\<lbrakk>\<Turnstile> v1 : (nth rs n); n < length rs\<rbrakk> \<Longrightarrow> \<Turnstile> (Nth n v1) : ALTS rs"

   403 | "\<Turnstile> Void : ONE"

   404 | "\<Turnstile> Char c : CHAR c"

   405 | "\<forall>v \<in> set vs. \<Turnstile> v : r \<and> flat v \<noteq> [] \<Longrightarrow> \<Turnstile> Stars vs : STAR r"

   406 

   407 inductive_cases Prf_elims:

   408   "\<Turnstile> v : ZERO"

   409   "\<Turnstile> v : SEQ r1 r2"

   410   "\<Turnstile> v : ALTS rs"

   411   "\<Turnstile> v : ONE"

   412   "\<Turnstile> v : CHAR c"

   413   "\<Turnstile> vs : STAR r"

   414 

   415 lemma Prf_Stars_appendE:

   416   assumes "\<Turnstile> Stars (vs1 @ vs2) : STAR r"

   417   shows "\<Turnstile> Stars vs1 : STAR r \<and> \<Turnstile> Stars vs2 : STAR r" 

   418 using assms

   419 by (auto intro: Prf.intros elim!: Prf_elims)

   420 

   421 

   422 lemma Star_cval:

   423   assumes "\<forall>s\<in>set ss. \<exists>v. s = flat v \<and> \<Turnstile> v : r"

   424   shows "\<exists>vs. flats vs = concat ss \<and> (\<forall>v\<in>set vs. \<Turnstile> v : r \<and> flat v \<noteq> [])"

   425 using assms

   426 apply(induct ss)

   427 apply(auto)

   428 apply(rule_tac x="[]" in exI)

   429 apply(simp)

   430 apply(case_tac "flat v = []")

   431 apply(rule_tac x="vs" in exI)

   432 apply(simp)

   433 apply(rule_tac x="v#vs" in exI)

   434 apply(simp)

   435 done

   436 

   437 

   438 lemma L_flat_Prf1:

   439   assumes "\<Turnstile> v : r" 

   440   shows "flat v \<in> L r"

   441 using assms

   442   apply(induct) 

   443   apply(auto simp add: Sequ_def Star_concat)

   444   done  

   445 

   446 lemma L_flat_Prf2:

   447   assumes "s \<in> L r" 

   448   shows "\<exists>v. \<Turnstile> v : r \<and> flat v = s"

   449 using assms

   450 proof(induct r arbitrary: s)

   451   case (STAR r s)

   452   have IH: "\<And>s. s \<in> L r \<Longrightarrow> \<exists>v. \<Turnstile> v : r \<and> flat v = s" by fact

   453   have "s \<in> L (STAR r)" by fact

   454   then obtain ss where "concat ss = s" "\<forall>s \<in> set ss. s \<in> L r \<and> s \<noteq> []"

   455   using Star_cstring by auto  

   456   then obtain vs where "flats vs = s" "\<forall>v\<in>set vs. \<Turnstile> v : r \<and> flat v \<noteq> []"

   457   using IH Star_cval by metis 

   458   then show "\<exists>v. \<Turnstile> v : STAR r \<and> flat v = s"

   459   using Prf.intros(5) flat_Stars by blast

   460 next 

   461   case (SEQ r1 r2 s)

   462   then show "\<exists>v. \<Turnstile> v : SEQ r1 r2 \<and> flat v = s"

   463   unfolding Sequ_def L.simps by (fastforce intro: Prf.intros)

   464 next

   465   case (ALTS rs s)

   466   then show "\<exists>v. \<Turnstile> v : ALTS rs \<and> flat v = s"

   467     unfolding L.simps 

   468     apply(auto)

   469     apply(case_tac rs)

   470      apply(simp)

   471     apply(simp)

   472     apply(auto)

   473      apply(drule_tac x="a" in meta_spec)

   474      apply(simp)

   475      apply(drule_tac x="s" in meta_spec)

   476      apply(simp)

   477      apply(erule exE)

   478      apply(rule_tac x="Nth 0 v" in exI)

   479      apply(simp)

   480      apply(rule Prf.intros)

   481       apply(simp)

   482      apply(simp)

   483     apply(drule_tac x="x" in meta_spec)

   484     apply(simp)

   485     apply(drule_tac x="s" in meta_spec)

   486     apply(simp)

   487     apply(erule exE)

   488     apply(subgoal_tac "\<exists>n. nth list n = x \<and> n < length list")

   489     apply(erule exE)

   490      apply(rule_tac x="Nth (Suc n) v" in exI)

   491      apply(simp)

   492      apply(rule Prf.intros)

   493       apply(simp)

   494      apply(simp)

   495     by (meson in_set_conv_nth)

   496 qed (auto intro: Prf.intros)

   497 

   498 

   499 lemma L_flat_Prf:

   500   shows "L(r) = {flat v | v. \<Turnstile> v : r}"

   501 using L_flat_Prf1 L_flat_Prf2 by blast

   502 

   503 

   504 

   505 section {* Sets of Lexical Values *}

   506 

   507 text {*

   508   Shows that lexical values are finite for a given regex and string.

   509 *}

   510 

   511 definition

   512   LV :: "rexp \<Rightarrow> string \<Rightarrow> val set"

   513 where  "LV r s \<equiv> {v. \<Turnstile> v : r \<and> flat v = s}"

   514 

   515 lemma LV_simps:

   516   shows "LV ZERO s = {}"

   517   and   "LV ONE s = (if s = [] then {Void} else {})"

   518   and   "LV (CHAR c) s = (if s = [c] then {Char c} else {})"

   519 unfolding LV_def

   520 by (auto intro: Prf.intros elim: Prf.cases)

   521 

   522 

   523 abbreviation

   524   "Prefixes s \<equiv> {s'. prefix s' s}"

   525 

   526 abbreviation

   527   "Suffixes s \<equiv> {s'. suffix s' s}"

   528 

   529 abbreviation

   530   "SSuffixes s \<equiv> {s'. strict_suffix s' s}"

   531 

   532 lemma Suffixes_cons [simp]:

   533   shows "Suffixes (c # s) = Suffixes s \<union> {c # s}"

   534 by (auto simp add: suffix_def Cons_eq_append_conv)

   535 

   536 

   537 lemma finite_Suffixes: 

   538   shows "finite (Suffixes s)"

   539 by (induct s) (simp_all)

   540 

   541 lemma finite_SSuffixes: 

   542   shows "finite (SSuffixes s)"

   543 proof -

   544   have "SSuffixes s \<subseteq> Suffixes s"

   545    unfolding strict_suffix_def suffix_def by auto

   546   then show "finite (SSuffixes s)"

   547    using finite_Suffixes finite_subset by blast

   548 qed

   549 

   550 lemma finite_Prefixes: 

   551   shows "finite (Prefixes s)"

   552 proof -

   553   have "finite (Suffixes (rev s))" 

   554     by (rule finite_Suffixes)

   555   then have "finite (rev ` Suffixes (rev s))" by simp

   556   moreover

   557   have "rev ` (Suffixes (rev s)) = Prefixes s"

   558   unfolding suffix_def prefix_def image_def

   559    by (auto)(metis rev_append rev_rev_ident)+

   560   ultimately show "finite (Prefixes s)" by simp

   561 qed

   562 

   563 lemma LV_STAR_finite:

   564   assumes "\<forall>s. finite (LV r s)"

   565   shows "finite (LV (STAR r) s)"

   566 proof(induct s rule: length_induct)

   567   fix s::"char list"

   568   assume "\<forall>s'. length s' < length s \<longrightarrow> finite (LV (STAR r) s')"

   569   then have IH: "\<forall>s' \<in> SSuffixes s. finite (LV (STAR r) s')"

   570     by (force simp add: strict_suffix_def suffix_def) 

   571   define f where "f \<equiv> \<lambda>(v, vs). Stars (v # vs)"

   572   define S1 where "S1 \<equiv> \<Union>s' \<in> Prefixes s. LV r s'"

   573   define S2 where "S2 \<equiv> \<Union>s2 \<in> SSuffixes s. Stars -` (LV (STAR r) s2)"

   574   have "finite S1" using assms

   575     unfolding S1_def by (simp_all add: finite_Prefixes)

   576   moreover 

   577   with IH have "finite S2" unfolding S2_def

   578     by (auto simp add: finite_SSuffixes inj_on_def finite_vimageI)

   579   ultimately 

   580   have "finite ({Stars []} \<union> f ` (S1 \<times> S2))" by simp

   581   moreover 

   582   have "LV (STAR r) s \<subseteq> {Stars []} \<union> f ` (S1 \<times> S2)" 

   583   unfolding S1_def S2_def f_def

   584   unfolding LV_def image_def prefix_def strict_suffix_def

   585   apply(auto)

   586   apply(case_tac x)

   587   apply(auto elim: Prf_elims)

   588   apply(erule Prf_elims)

   589   apply(auto)

   590   apply(case_tac vs)

   591   apply(auto intro: Prf.intros)  

   592   apply(rule exI)

   593   apply(rule conjI)

   594   apply(rule_tac x="flat a" in exI)

   595   apply(rule conjI)

   596   apply(rule_tac x="flats list" in exI)

   597   apply(simp)

   598    apply(blast)

   599   apply(simp add: suffix_def)

   600   using Prf.intros(5) by blast  

   601   ultimately

   602   show "finite (LV (STAR r) s)" by (simp add: finite_subset)

   603 qed  

   604     

   605 

   606 lemma LV_finite:

   607   shows "finite (LV r s)"

   608 proof(induct r arbitrary: s)

   609   case (ZERO s) 

   610   show "finite (LV ZERO s)" by (simp add: LV_simps)

   611 next

   612   case (ONE s)

   613   show "finite (LV ONE s)" by (simp add: LV_simps)

   614 next

   615   case (CHAR c s)

   616   show "finite (LV (CHAR c) s)" by (simp add: LV_simps)

   617 next 

   618   case (ALTS rs s)

   619   then show "finite (LV (ALTS rs) s)" 

   620     sorry

   621 next 

   622   case (SEQ r1 r2 s)

   623   define f where "f \<equiv> \<lambda>(v1, v2). Seq v1 v2"

   624   define S1 where "S1 \<equiv> \<Union>s' \<in> Prefixes s. LV r1 s'"

   625   define S2 where "S2 \<equiv> \<Union>s' \<in> Suffixes s. LV r2 s'"

   626   have IHs: "\<And>s. finite (LV r1 s)" "\<And>s. finite (LV r2 s)" by fact+

   627   then have "finite S1" "finite S2" unfolding S1_def S2_def

   628     by (simp_all add: finite_Prefixes finite_Suffixes)

   629   moreover

   630   have "LV (SEQ r1 r2) s \<subseteq> f ` (S1 \<times> S2)"

   631     unfolding f_def S1_def S2_def 

   632     unfolding LV_def image_def prefix_def suffix_def

   633     apply (auto elim!: Prf_elims)

   634     by (metis (mono_tags, lifting) mem_Collect_eq)  

   635   ultimately 

   636   show "finite (LV (SEQ r1 r2) s)"

   637     by (simp add: finite_subset)

   638 next

   639   case (STAR r s)

   640   then show "finite (LV (STAR r) s)" by (simp add: LV_STAR_finite)

   641 qed

   642 

   643 

   644 (*

   645 section {* Our POSIX Definition *}

   646 

   647 inductive 

   648   Posix :: "string \<Rightarrow> rexp \<Rightarrow> val \<Rightarrow> bool" ("_ \<in> _ \<rightarrow> _" [100, 100, 100] 100)

   649 where

   650   Posix_ONE: "[] \<in> ONE \<rightarrow> Void"

   651 | Posix_CHAR: "[c] \<in> (CHAR c) \<rightarrow> (Char c)"

   652 | Posix_ALT1: "s \<in> r1 \<rightarrow> v \<Longrightarrow> s \<in> (ALT r1 r2) \<rightarrow> (Left v)"

   653 | Posix_ALT2: "\<lbrakk>s \<in> r2 \<rightarrow> v; s \<notin> L(r1)\<rbrakk> \<Longrightarrow> s \<in> (ALT r1 r2) \<rightarrow> (Right v)"

   654 | Posix_SEQ: "\<lbrakk>s1 \<in> r1 \<rightarrow> v1; s2 \<in> r2 \<rightarrow> v2;

   655     \<not>(\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (s1 @ s\<^sub>3) \<in> L r1 \<and> s\<^sub>4 \<in> L r2)\<rbrakk> \<Longrightarrow> 

   656     (s1 @ s2) \<in> (SEQ r1 r2) \<rightarrow> (Seq v1 v2)"

   657 | Posix_STAR1: "\<lbrakk>s1 \<in> r \<rightarrow> v; s2 \<in> STAR r \<rightarrow> Stars vs; flat v \<noteq> [];

   658     \<not>(\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (s1 @ s\<^sub>3) \<in> L r \<and> s\<^sub>4 \<in> L (STAR r))\<rbrakk>

   659     \<Longrightarrow> (s1 @ s2) \<in> STAR r \<rightarrow> Stars (v # vs)"

   660 | Posix_STAR2: "[] \<in> STAR r \<rightarrow> Stars []"

   661 

   662 inductive_cases Posix_elims:

   663   "s \<in> ZERO \<rightarrow> v"

   664   "s \<in> ONE \<rightarrow> v"

   665   "s \<in> CHAR c \<rightarrow> v"

   666   "s \<in> ALT r1 r2 \<rightarrow> v"

   667   "s \<in> SEQ r1 r2 \<rightarrow> v"

   668   "s \<in> STAR r \<rightarrow> v"

   669 

   670 lemma Posix1:

   671   assumes "s \<in> r \<rightarrow> v"

   672   shows "s \<in> L r" "flat v = s"

   673 using assms

   674 by (induct s r v rule: Posix.induct)

   675    (auto simp add: Sequ_def)

   676 

   677 text {*

   678   Our Posix definition determines a unique value.

   679 *}

   680 

   681 lemma Posix_determ:

   682   assumes "s \<in> r \<rightarrow> v1" "s \<in> r \<rightarrow> v2"

   683   shows "v1 = v2"

   684 using assms

   685 proof (induct s r v1 arbitrary: v2 rule: Posix.induct)

   686   case (Posix_ONE v2)

   687   have "[] \<in> ONE \<rightarrow> v2" by fact

   688   then show "Void = v2" by cases auto

   689 next 

   690   case (Posix_CHAR c v2)

   691   have "[c] \<in> CHAR c \<rightarrow> v2" by fact

   692   then show "Char c = v2" by cases auto

   693 next 

   694   case (Posix_ALT1 s r1 v r2 v2)

   695   have "s \<in> ALT r1 r2 \<rightarrow> v2" by fact

   696   moreover

   697   have "s \<in> r1 \<rightarrow> v" by fact

   698   then have "s \<in> L r1" by (simp add: Posix1)

   699   ultimately obtain v' where eq: "v2 = Left v'" "s \<in> r1 \<rightarrow> v'" by cases auto 

   700   moreover

   701   have IH: "\<And>v2. s \<in> r1 \<rightarrow> v2 \<Longrightarrow> v = v2" by fact

   702   ultimately have "v = v'" by simp

   703   then show "Left v = v2" using eq by simp

   704 next 

   705   case (Posix_ALT2 s r2 v r1 v2)

   706   have "s \<in> ALT r1 r2 \<rightarrow> v2" by fact

   707   moreover

   708   have "s \<notin> L r1" by fact

   709   ultimately obtain v' where eq: "v2 = Right v'" "s \<in> r2 \<rightarrow> v'" 

   710     by cases (auto simp add: Posix1) 

   711   moreover

   712   have IH: "\<And>v2. s \<in> r2 \<rightarrow> v2 \<Longrightarrow> v = v2" by fact

   713   ultimately have "v = v'" by simp

   714   then show "Right v = v2" using eq by simp

   715 next

   716   case (Posix_SEQ s1 r1 v1 s2 r2 v2 v')

   717   have "(s1 @ s2) \<in> SEQ r1 r2 \<rightarrow> v'" 

   718        "s1 \<in> r1 \<rightarrow> v1" "s2 \<in> r2 \<rightarrow> v2"

   719        "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L r1 \<and> s\<^sub>4 \<in> L r2)" by fact+

   720   then obtain v1' v2' where "v' = Seq v1' v2'" "s1 \<in> r1 \<rightarrow> v1'" "s2 \<in> r2 \<rightarrow> v2'"

   721   apply(cases) apply (auto simp add: append_eq_append_conv2)

   722   using Posix1(1) by fastforce+

   723   moreover

   724   have IHs: "\<And>v1'. s1 \<in> r1 \<rightarrow> v1' \<Longrightarrow> v1 = v1'"

   725             "\<And>v2'. s2 \<in> r2 \<rightarrow> v2' \<Longrightarrow> v2 = v2'" by fact+

   726   ultimately show "Seq v1 v2 = v'" by simp

   727 next

   728   case (Posix_STAR1 s1 r v s2 vs v2)

   729   have "(s1 @ s2) \<in> STAR r \<rightarrow> v2" 

   730        "s1 \<in> r \<rightarrow> v" "s2 \<in> STAR r \<rightarrow> Stars vs" "flat v \<noteq> []"

   731        "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L r \<and> s\<^sub>4 \<in> L (STAR r))" by fact+

   732   then obtain v' vs' where "v2 = Stars (v' # vs')" "s1 \<in> r \<rightarrow> v'" "s2 \<in> (STAR r) \<rightarrow> (Stars vs')"

   733   apply(cases) apply (auto simp add: append_eq_append_conv2)

   734   using Posix1(1) apply fastforce

   735   apply (metis Posix1(1) Posix_STAR1.hyps(6) append_Nil append_Nil2)

   736   using Posix1(2) by blast

   737   moreover

   738   have IHs: "\<And>v2. s1 \<in> r \<rightarrow> v2 \<Longrightarrow> v = v2"

   739             "\<And>v2. s2 \<in> STAR r \<rightarrow> v2 \<Longrightarrow> Stars vs = v2" by fact+

   740   ultimately show "Stars (v # vs) = v2" by auto

   741 next

   742   case (Posix_STAR2 r v2)

   743   have "[] \<in> STAR r \<rightarrow> v2" by fact

   744   then show "Stars [] = v2" by cases (auto simp add: Posix1)

   745 qed

   746 

   747 

   748 text {*

   749   Our POSIX value is a lexical value.

   750 *}

   751 

   752 lemma Posix_LV:

   753   assumes "s \<in> r \<rightarrow> v"

   754   shows "v \<in> LV r s"

   755 using assms unfolding LV_def

   756 apply(induct rule: Posix.induct)

   757 apply(auto simp add: intro!: Prf.intros elim!: Prf_elims)

   758 done

   759 *)

   760 

   761 

   762 end