theory Spec

  imports RegLangs

begin

section {* "Plain" Values *}

datatype val = 

  Void

| Char char

| Seq val val

| Right val

| Left val

| Stars "val list"

section {* The string behind a value *}

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 {* Lexical Values *}

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 : CHAR 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 : CHAR 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 {* Sets of Lexical Values *}

text {*

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

*}

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 (CHAR 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 (CHAR c s)

  show "finite (LV (CHAR 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 {* Our inductive POSIX Definition *}

inductive 

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

where

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

| Posix_CHAR: "[c] \<in> (CHAR 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> CHAR 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 {*

  For a give value and string, our Posix definition 

  determines a unique value.

*}

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_CHAR c v2)

  have "[c] \<in> CHAR 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 {*

  Our POSIX values are lexical values.

*}

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