diff -r f493a20feeb3 -r 04b5e904a220 thys3/PosixSpec.thy
--- a/thys3/PosixSpec.thy	Sat Apr 30 00:50:08 2022 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,380 +0,0 @@
-   
-theory PosixSpec
-  imports RegLangs
-begin
-
-section \<open>"Plain" Values\<close>
-
-datatype val = 
-  Void
-| Char char
-| Seq val val
-| Right val
-| Left val
-| Stars "val list"
-
-
-section \<open>The string behind a value\<close>
-
-fun 
-  flat :: "val \<Rightarrow> string"
-where
-  "flat (Void) = []"
-| "flat (Char c) = [c]"
-| "flat (Left v) = flat v"
-| "flat (Right v) = flat v"
-| "flat (Seq v1 v2) = (flat v1) @ (flat v2)"
-| "flat (Stars []) = []"
-| "flat (Stars (v#vs)) = (flat v) @ (flat (Stars vs))" 
-
-abbreviation
-  "flats vs \<equiv> concat (map flat vs)"
-
-lemma flat_Stars [simp]:
- "flat (Stars vs) = flats vs"
-by (induct vs) (auto)
-
-
-section \<open>Lexical Values\<close>
-
-inductive 
-  Prf :: "val \<Rightarrow> rexp \<Rightarrow> bool" ("\<Turnstile> _ : _" [100, 100] 100)
-where
- "\<lbrakk>\<Turnstile> v1 : r1; \<Turnstile> v2 : r2\<rbrakk> \<Longrightarrow> \<Turnstile>  Seq v1 v2 : SEQ r1 r2"
-| "\<Turnstile> v1 : r1 \<Longrightarrow> \<Turnstile> Left v1 : ALT r1 r2"
-| "\<Turnstile> v2 : r2 \<Longrightarrow> \<Turnstile> Right v2 : ALT r1 r2"
-| "\<Turnstile> Void : ONE"
-| "\<Turnstile> Char c : CH c"
-| "\<forall>v \<in> set vs. \<Turnstile> v : r \<and> flat v \<noteq> [] \<Longrightarrow> \<Turnstile> Stars vs : STAR r"
-
-inductive_cases Prf_elims:
-  "\<Turnstile> v : ZERO"
-  "\<Turnstile> v : SEQ r1 r2"
-  "\<Turnstile> v : ALT r1 r2"
-  "\<Turnstile> v : ONE"
-  "\<Turnstile> v : CH c"
-  "\<Turnstile> vs : STAR r"
-
-lemma Prf_Stars_appendE:
-  assumes "\<Turnstile> Stars (vs1 @ vs2) : STAR r"
-  shows "\<Turnstile> Stars vs1 : STAR r \<and> \<Turnstile> Stars vs2 : STAR r" 
-using assms
-by (auto intro: Prf.intros elim!: Prf_elims)
-
-
-lemma flats_Prf_value:
-  assumes "\<forall>s\<in>set ss. \<exists>v. s = flat v \<and> \<Turnstile> v : r"
-  shows "\<exists>vs. flats vs = concat ss \<and> (\<forall>v\<in>set vs. \<Turnstile> v : r \<and> flat v \<noteq> [])"
-using assms
-apply(induct ss)
-apply(auto)
-apply(rule_tac x="[]" in exI)
-apply(simp)
-apply(case_tac "flat v = []")
-apply(rule_tac x="vs" in exI)
-apply(simp)
-apply(rule_tac x="v#vs" in exI)
-apply(simp)
-done
-
-
-lemma L_flat_Prf1:
-  assumes "\<Turnstile> v : r" 
-  shows "flat v \<in> L r"
-using assms
-by (induct) (auto simp add: Sequ_def Star_concat)
-
-lemma L_flat_Prf2:
-  assumes "s \<in> L r" 
-  shows "\<exists>v. \<Turnstile> v : r \<and> flat v = s"
-using assms
-proof(induct r arbitrary: s)
-  case (STAR r s)
-  have IH: "\<And>s. s \<in> L r \<Longrightarrow> \<exists>v. \<Turnstile> v : r \<and> flat v = s" by fact
-  have "s \<in> L (STAR r)" by fact
-  then obtain ss where "concat ss = s" "\<forall>s \<in> set ss. s \<in> L r \<and> s \<noteq> []"
-  using Star_split by auto  
-  then obtain vs where "flats vs = s" "\<forall>v\<in>set vs. \<Turnstile> v : r \<and> flat v \<noteq> []"
-  using IH flats_Prf_value by metis 
-  then show "\<exists>v. \<Turnstile> v : STAR r \<and> flat v = s"
-  using Prf.intros(6) flat_Stars by blast
-next 
-  case (SEQ r1 r2 s)
-  then show "\<exists>v. \<Turnstile> v : SEQ r1 r2 \<and> flat v = s"
-  unfolding Sequ_def L.simps by (fastforce intro: Prf.intros)
-next
-  case (ALT r1 r2 s)
-  then show "\<exists>v. \<Turnstile> v : ALT r1 r2 \<and> flat v = s"
-  unfolding L.simps by (fastforce intro: Prf.intros)
-qed (auto intro: Prf.intros)
-
-
-lemma L_flat_Prf:
-  shows "L(r) = {flat v | v. \<Turnstile> v : r}"
-using L_flat_Prf1 L_flat_Prf2 by blast
-
-
-
-section \<open>Sets of Lexical Values\<close>
-
-text \<open>
-  Shows that lexical values are finite for a given regex and string.
-\<close>
-
-definition
-  LV :: "rexp \<Rightarrow> string \<Rightarrow> val set"
-where  "LV r s \<equiv> {v. \<Turnstile> v : r \<and> flat v = s}"
-
-lemma LV_simps:
-  shows "LV ZERO s = {}"
-  and   "LV ONE s = (if s = [] then {Void} else {})"
-  and   "LV (CH c) s = (if s = [c] then {Char c} else {})"
-  and   "LV (ALT r1 r2) s = Left ` LV r1 s \<union> Right ` LV r2 s"
-unfolding LV_def
-by (auto intro: Prf.intros elim: Prf.cases)
-
-
-abbreviation
-  "Prefixes s \<equiv> {s'. prefix s' s}"
-
-abbreviation
-  "Suffixes s \<equiv> {s'. suffix s' s}"
-
-abbreviation
-  "SSuffixes s \<equiv> {s'. strict_suffix s' s}"
-
-lemma Suffixes_cons [simp]:
-  shows "Suffixes (c # s) = Suffixes s \<union> {c # s}"
-by (auto simp add: suffix_def Cons_eq_append_conv)
-
-
-lemma finite_Suffixes: 
-  shows "finite (Suffixes s)"
-by (induct s) (simp_all)
-
-lemma finite_SSuffixes: 
-  shows "finite (SSuffixes s)"
-proof -
-  have "SSuffixes s \<subseteq> Suffixes s"
-   unfolding strict_suffix_def suffix_def by auto
-  then show "finite (SSuffixes s)"
-   using finite_Suffixes finite_subset by blast
-qed
-
-lemma finite_Prefixes: 
-  shows "finite (Prefixes s)"
-proof -
-  have "finite (Suffixes (rev s))" 
-    by (rule finite_Suffixes)
-  then have "finite (rev ` Suffixes (rev s))" by simp
-  moreover
-  have "rev ` (Suffixes (rev s)) = Prefixes s"
-  unfolding suffix_def prefix_def image_def
-   by (auto)(metis rev_append rev_rev_ident)+
-  ultimately show "finite (Prefixes s)" by simp
-qed
-
-lemma LV_STAR_finite:
-  assumes "\<forall>s. finite (LV r s)"
-  shows "finite (LV (STAR r) s)"
-proof(induct s rule: length_induct)
-  fix s::"char list"
-  assume "\<forall>s'. length s' < length s \<longrightarrow> finite (LV (STAR r) s')"
-  then have IH: "\<forall>s' \<in> SSuffixes s. finite (LV (STAR r) s')"
-    by (force simp add: strict_suffix_def suffix_def) 
-  define f where "f \<equiv> \<lambda>(v, vs). Stars (v # vs)"
-  define S1 where "S1 \<equiv> \<Union>s' \<in> Prefixes s. LV r s'"
-  define S2 where "S2 \<equiv> \<Union>s2 \<in> SSuffixes s. Stars -` (LV (STAR r) s2)"
-  have "finite S1" using assms
-    unfolding S1_def by (simp_all add: finite_Prefixes)
-  moreover 
-  with IH have "finite S2" unfolding S2_def
-    by (auto simp add: finite_SSuffixes inj_on_def finite_vimageI)
-  ultimately 
-  have "finite ({Stars []} \<union> f ` (S1 \<times> S2))" by simp
-  moreover 
-  have "LV (STAR r) s \<subseteq> {Stars []} \<union> f ` (S1 \<times> S2)" 
-  unfolding S1_def S2_def f_def
-  unfolding LV_def image_def prefix_def strict_suffix_def 
-  apply(auto)
-  apply(case_tac x)
-  apply(auto elim: Prf_elims)
-  apply(erule Prf_elims)
-  apply(auto)
-  apply(case_tac vs)
-  apply(auto intro: Prf.intros)  
-  apply(rule exI)
-  apply(rule conjI)
-  apply(rule_tac x="flat a" in exI)
-  apply(rule conjI)
-  apply(rule_tac x="flats list" in exI)
-  apply(simp)
-   apply(blast)
-  apply(simp add: suffix_def)
-  using Prf.intros(6) by blast  
-  ultimately
-  show "finite (LV (STAR r) s)" by (simp add: finite_subset)
-qed  
-    
-
-lemma LV_finite:
-  shows "finite (LV r s)"
-proof(induct r arbitrary: s)
-  case (ZERO s) 
-  show "finite (LV ZERO s)" by (simp add: LV_simps)
-next
-  case (ONE s)
-  show "finite (LV ONE s)" by (simp add: LV_simps)
-next
-  case (CH c s)
-  show "finite (LV (CH c) s)" by (simp add: LV_simps)
-next 
-  case (ALT r1 r2 s)
-  then show "finite (LV (ALT r1 r2) s)" by (simp add: LV_simps)
-next 
-  case (SEQ r1 r2 s)
-  define f where "f \<equiv> \<lambda>(v1, v2). Seq v1 v2"
-  define S1 where "S1 \<equiv> \<Union>s' \<in> Prefixes s. LV r1 s'"
-  define S2 where "S2 \<equiv> \<Union>s' \<in> Suffixes s. LV r2 s'"
-  have IHs: "\<And>s. finite (LV r1 s)" "\<And>s. finite (LV r2 s)" by fact+
-  then have "finite S1" "finite S2" unfolding S1_def S2_def
-    by (simp_all add: finite_Prefixes finite_Suffixes)
-  moreover
-  have "LV (SEQ r1 r2) s \<subseteq> f ` (S1 \<times> S2)"
-    unfolding f_def S1_def S2_def 
-    unfolding LV_def image_def prefix_def suffix_def
-    apply (auto elim!: Prf_elims)
-    by (metis (mono_tags, lifting) mem_Collect_eq)  
-  ultimately 
-  show "finite (LV (SEQ r1 r2) s)"
-    by (simp add: finite_subset)
-next
-  case (STAR r s)
-  then show "finite (LV (STAR r) s)" by (simp add: LV_STAR_finite)
-qed
-
-
-
-section \<open>Our inductive POSIX Definition\<close>
-
-inductive 
-  Posix :: "string \<Rightarrow> rexp \<Rightarrow> val \<Rightarrow> bool" ("_ \<in> _ \<rightarrow> _" [100, 100, 100] 100)
-where
-  Posix_ONE: "[] \<in> ONE \<rightarrow> Void"
-| Posix_CH: "[c] \<in> (CH c) \<rightarrow> (Char c)"
-| Posix_ALT1: "s \<in> r1 \<rightarrow> v \<Longrightarrow> s \<in> (ALT r1 r2) \<rightarrow> (Left v)"
-| Posix_ALT2: "\<lbrakk>s \<in> r2 \<rightarrow> v; s \<notin> L(r1)\<rbrakk> \<Longrightarrow> s \<in> (ALT r1 r2) \<rightarrow> (Right v)"
-| Posix_SEQ: "\<lbrakk>s1 \<in> r1 \<rightarrow> v1; s2 \<in> r2 \<rightarrow> v2;
-    \<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> 
-    (s1 @ s2) \<in> (SEQ r1 r2) \<rightarrow> (Seq v1 v2)"
-| Posix_STAR1: "\<lbrakk>s1 \<in> r \<rightarrow> v; s2 \<in> STAR r \<rightarrow> Stars vs; flat v \<noteq> [];
-    \<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>
-    \<Longrightarrow> (s1 @ s2) \<in> STAR r \<rightarrow> Stars (v # vs)"
-| Posix_STAR2: "[] \<in> STAR r \<rightarrow> Stars []"
-
-inductive_cases Posix_elims:
-  "s \<in> ZERO \<rightarrow> v"
-  "s \<in> ONE \<rightarrow> v"
-  "s \<in> CH c \<rightarrow> v"
-  "s \<in> ALT r1 r2 \<rightarrow> v"
-  "s \<in> SEQ r1 r2 \<rightarrow> v"
-  "s \<in> STAR r \<rightarrow> v"
-
-lemma Posix1:
-  assumes "s \<in> r \<rightarrow> v"
-  shows "s \<in> L r" "flat v = s"
-using assms
-  by(induct s r v rule: Posix.induct)
-    (auto simp add: Sequ_def)
-
-text \<open>
-  For a give value and string, our Posix definition 
-  determines a unique value.
-\<close>
-
-lemma Posix_determ:
-  assumes "s \<in> r \<rightarrow> v1" "s \<in> r \<rightarrow> v2"
-  shows "v1 = v2"
-using assms
-proof (induct s r v1 arbitrary: v2 rule: Posix.induct)
-  case (Posix_ONE v2)
-  have "[] \<in> ONE \<rightarrow> v2" by fact
-  then show "Void = v2" by cases auto
-next 
-  case (Posix_CH c v2)
-  have "[c] \<in> CH c \<rightarrow> v2" by fact
-  then show "Char c = v2" by cases auto
-next 
-  case (Posix_ALT1 s r1 v r2 v2)
-  have "s \<in> ALT r1 r2 \<rightarrow> v2" by fact
-  moreover
-  have "s \<in> r1 \<rightarrow> v" by fact
-  then have "s \<in> L r1" by (simp add: Posix1)
-  ultimately obtain v' where eq: "v2 = Left v'" "s \<in> r1 \<rightarrow> v'" by cases auto 
-  moreover
-  have IH: "\<And>v2. s \<in> r1 \<rightarrow> v2 \<Longrightarrow> v = v2" by fact
-  ultimately have "v = v'" by simp
-  then show "Left v = v2" using eq by simp
-next 
-  case (Posix_ALT2 s r2 v r1 v2)
-  have "s \<in> ALT r1 r2 \<rightarrow> v2" by fact
-  moreover
-  have "s \<notin> L r1" by fact
-  ultimately obtain v' where eq: "v2 = Right v'" "s \<in> r2 \<rightarrow> v'" 
-    by cases (auto simp add: Posix1) 
-  moreover
-  have IH: "\<And>v2. s \<in> r2 \<rightarrow> v2 \<Longrightarrow> v = v2" by fact
-  ultimately have "v = v'" by simp
-  then show "Right v = v2" using eq by simp
-next
-  case (Posix_SEQ s1 r1 v1 s2 r2 v2 v')
-  have "(s1 @ s2) \<in> SEQ r1 r2 \<rightarrow> v'" 
-       "s1 \<in> r1 \<rightarrow> v1" "s2 \<in> r2 \<rightarrow> v2"
-       "\<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+
-  then obtain v1' v2' where "v' = Seq v1' v2'" "s1 \<in> r1 \<rightarrow> v1'" "s2 \<in> r2 \<rightarrow> v2'"
-  apply(cases) apply (auto simp add: append_eq_append_conv2)
-  using Posix1(1) by fastforce+
-  moreover
-  have IHs: "\<And>v1'. s1 \<in> r1 \<rightarrow> v1' \<Longrightarrow> v1 = v1'"
-            "\<And>v2'. s2 \<in> r2 \<rightarrow> v2' \<Longrightarrow> v2 = v2'" by fact+
-  ultimately show "Seq v1 v2 = v'" by simp
-next
-  case (Posix_STAR1 s1 r v s2 vs v2)
-  have "(s1 @ s2) \<in> STAR r \<rightarrow> v2" 
-       "s1 \<in> r \<rightarrow> v" "s2 \<in> STAR r \<rightarrow> Stars vs" "flat v \<noteq> []"
-       "\<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+
-  then obtain v' vs' where "v2 = Stars (v' # vs')" "s1 \<in> r \<rightarrow> v'" "s2 \<in> (STAR r) \<rightarrow> (Stars vs')"
-  apply(cases) apply (auto simp add: append_eq_append_conv2)
-  using Posix1(1) apply fastforce
-  apply (metis Posix1(1) Posix_STAR1.hyps(6) append_Nil append_Nil2)
-  using Posix1(2) by blast
-  moreover
-  have IHs: "\<And>v2. s1 \<in> r \<rightarrow> v2 \<Longrightarrow> v = v2"
-            "\<And>v2. s2 \<in> STAR r \<rightarrow> v2 \<Longrightarrow> Stars vs = v2" by fact+
-  ultimately show "Stars (v # vs) = v2" by auto
-next
-  case (Posix_STAR2 r v2)
-  have "[] \<in> STAR r \<rightarrow> v2" by fact
-  then show "Stars [] = v2" by cases (auto simp add: Posix1)
-qed
-
-
-text \<open>
-  Our POSIX values are lexical values.
-\<close>
-
-lemma Posix_LV:
-  assumes "s \<in> r \<rightarrow> v"
-  shows "v \<in> LV r s"
-  using assms unfolding LV_def
-  apply(induct rule: Posix.induct)
-  apply(auto simp add: intro!: Prf.intros elim!: Prf_elims)
-  done
-
-lemma Posix_Prf:
-  assumes "s \<in> r \<rightarrow> v"
-  shows "\<Turnstile> v : r"
-  using assms Posix_LV LV_def
-  by simp
-
-end