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"

| "\<lbrakk>\<forall>v \<in> set vs1. \<Turnstile> v : r \<and> flat v \<noteq> []; 

    \<forall>v \<in> set vs2. \<Turnstile> v : r \<and> flat v = []; 

    length (vs1 @ vs2) = n\<rbrakk> \<Longrightarrow> \<Turnstile> Stars (vs1 @ vs2) : NTIMES r n"   

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"

  "\<Turnstile> vs : NTIMES r n"

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 Pow_cstring:

  fixes A::"string set"

  assumes "s \<in> A ^^ n"

  shows "\<exists>ss1 ss2. concat (ss1 @ ss2) = s \<and> length (ss1 @ ss2) = n \<and> 

         (\<forall>s \<in> set ss1. s \<in> A \<and> s \<noteq> []) \<and> (\<forall>s \<in> set ss2. s \<in> A \<and> s = [])"

using assms

apply(induct n arbitrary: s)

  apply(auto)[1]

  apply(auto simp add: Sequ_def)

  apply(drule_tac x="s2" in meta_spec)

  apply(simp)

apply(erule exE)+

  apply(clarify)

apply(case_tac "s1 = []")

apply(simp)

apply(rule_tac x="ss1" in exI)

apply(rule_tac x="s1 # ss2" in exI)

apply(simp)

apply(rule_tac x="s1 # ss1" in exI)

apply(rule_tac x="ss2" in exI)

  apply(simp)

  done

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 Aux:

  assumes "\<forall>s\<in>set ss. s = []"

  shows "concat ss = []"

using assms

by (induct ss) (auto)

lemma flats_cval:

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

  shows "\<exists>vs1 vs2. flats (vs1 @ vs2) = concat ss \<and> length (vs1 @ vs2) = length ss \<and> 

          (\<forall>v\<in>set vs1. \<Turnstile> v : r \<and> flat v \<noteq> []) \<and>

          (\<forall>v\<in>set vs2. \<Turnstile> v : r \<and> flat v = [])"

using assms

apply(induct ss rule: rev_induct)

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

apply(simp)

apply(simp)

apply(clarify)

apply(case_tac "flat v = []")

apply(rule_tac x="vs1" in exI)

apply(rule_tac x="v#vs2" in exI)

apply(simp)

apply(rule_tac x="vs1 @ [v]" in exI)

apply(rule_tac x="vs2" in exI)

apply(simp)

by (simp add: Aux)

lemma pow_Prf:

  assumes "\<forall>v\<in>set vs. \<Turnstile> v : r \<and> flat v \<in> A"

  shows "flats vs \<in> A ^^ (length vs)"

  using assms

  by (induct vs) (auto)

lemma L_flat_Prf1:

  assumes "\<Turnstile> v : r" 

  shows "flat v \<in> L r"

  using assms

  apply (induct v r rule: Prf.induct) 

  apply(auto simp add: Sequ_def Star_concat lang_pow_add)

  by (metis pow_Prf)

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)

next

  case (NTIMES r n)

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

  have "s \<in> L (NTIMES r n)" by fact

  then obtain ss1 ss2 where "concat (ss1 @ ss2) = s" "length (ss1 @ ss2) = n" 

    "\<forall>s \<in> set ss1. s \<in> L r \<and> s \<noteq> []" "\<forall>s \<in> set ss2. s \<in> L r \<and> s = []"

  using Pow_cstring by force

  then obtain vs1 vs2 where "flats (vs1 @ vs2) = s" "length (vs1 @ vs2) = n" 

      "\<forall>v\<in>set vs1. \<Turnstile> v : r \<and> flat v \<noteq> []" "\<forall>v\<in>set vs2. \<Turnstile> v : r \<and> flat v = []"

    using IH flats_cval 

  apply -

  apply(drule_tac x="ss1 @ ss2" in meta_spec)

  apply(drule_tac x="r" in meta_spec)

  apply(drule meta_mp)

  apply(simp)

  apply (metis Un_iff)

  apply(clarify)

  apply(drule_tac x="vs1" in meta_spec)

  apply(drule_tac x="vs2" in meta_spec)

  apply(simp)

  done

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

  using Prf.intros(7) flat_Stars by blast

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"

  and   "LV (NTIMES r 0) s = (if s = [] then {Stars []} else {})"

unfolding LV_def

  apply (auto intro: Prf.intros elim: Prf.cases)

  by (metis Prf.intros(7) append.right_neutral empty_iff list.set(1) list.size(3))

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

definition

  "Stars_Append Vs1 Vs2 \<equiv> {Stars (vs1 @ vs2) | vs1 vs2. Stars vs1 \<in> Vs1 \<and> Stars vs2 \<in> Vs2}"

lemma finite_Stars_Append:

  assumes "finite Vs1" "finite Vs2"

  shows "finite (Stars_Append Vs1 Vs2)"

  using assms  

proof -

  define UVs1 where "UVs1 \<equiv> Stars -` Vs1"

  define UVs2 where "UVs2 \<equiv> Stars -` Vs2"  

  from assms have "finite UVs1" "finite UVs2"

    unfolding UVs1_def UVs2_def

    by(simp_all add: finite_vimageI inj_on_def) 

  then have "finite ((\<lambda>(vs1, vs2). Stars (vs1 @ vs2)) ` (UVs1 \<times> UVs2))"

    by simp

  moreover 

    have "Stars_Append Vs1 Vs2 = (\<lambda>(vs1, vs2). Stars (vs1 @ vs2)) ` (UVs1 \<times> UVs2)"

    unfolding Stars_Append_def UVs1_def UVs2_def by auto    

  ultimately show "finite (Stars_Append Vs1 Vs2)"   

    by simp

qed 

lemma LV_NTIMES_subset:

  "LV (NTIMES r n) s \<subseteq> Stars_Append (LV (STAR r) s) (\<Union>i\<le>n. LV (NTIMES r i) [])"

apply(auto simp add: LV_def)

apply(auto elim!: Prf_elims)

  apply(auto simp add: Stars_Append_def)

  apply(rule_tac x="vs1" in exI)

  apply(rule_tac x="vs2" in exI)  

  apply(auto)

    using Prf.intros(6) apply(auto)

      apply(rule_tac x="length vs2" in bexI)

    thm Prf.intros

      apply(subst append.simps(1)[symmetric])

    apply(rule Prf.intros)

      apply(auto)[1]

      apply(auto)[1]

     apply(simp)

    apply(simp)

    done

lemma LV_NTIMES_Suc_empty:

  shows "LV (NTIMES r (Suc n)) [] = 

     (\<lambda>(v, vs). Stars (v#vs)) ` (LV r [] \<times> (Stars -` (LV (NTIMES r n) [])))"

unfolding LV_def

apply(auto elim!: Prf_elims simp add: image_def)

apply(case_tac vs1)

apply(auto)

apply(case_tac vs2)

apply(auto)

apply(subst append.simps(1)[symmetric])

apply(rule Prf.intros)

apply(auto)

apply(subst append.simps(1)[symmetric])

apply(rule Prf.intros)

apply(auto)

  done 

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 finite_NTimes_empty:

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

  shows "finite (LV (NTIMES r n) [])"

  using assms

  apply(induct n)

   apply(auto simp add: LV_simps)

  apply(subst LV_NTIMES_Suc_empty)

  apply(rule finite_imageI)

  apply(rule finite_cartesian_product)

  using assms apply simp 

  apply(rule finite_vimageI)

  apply(simp)

  apply(simp add: inj_on_def)

  done

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)

next

  case (NTIMES r n s)

  have "\<And>s. finite (LV r s)" by fact

  then have "finite (Stars_Append (LV (STAR r) s) (\<Union>i\<le>n. LV (NTIMES r i) []))" 

    apply(rule_tac finite_Stars_Append)

     apply (simp add: LV_STAR_finite)

    using finite_NTimes_empty by blast

  then show "finite (LV (NTIMES r n) s)"

    by (metis LV_NTIMES_subset finite_subset)

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 []"

| Posix_NTIMES1: "\<lbrakk>s1 \<in> r \<rightarrow> v; s2 \<in> NTIMES r (n - 1) \<rightarrow> Stars vs; flat v \<noteq> []; 0 < n;

    \<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 (NTIMES r (n - 1)))\<rbrakk>

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

| Posix_NTIMES2: "\<lbrakk>\<forall>v \<in> set vs. [] \<in> r \<rightarrow> v; length vs = n\<rbrakk>

    \<Longrightarrow> [] \<in> NTIMES r n \<rightarrow> Stars vs"  

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"

  "s \<in> NTIMES r n \<rightarrow> v"

lemma Posix1:

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

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

using assms

  apply(induct s r v rule: Posix.induct)

  apply(auto simp add: pow_empty_iff)

  apply (metis Suc_pred concI lang_pow.simps(2))

  by (meson ex_in_conv set_empty)

lemma Posix1a:

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

  shows "\<Turnstile> v : r"

using assms

  apply(induct s r v rule: Posix.induct)

  apply(auto intro: Prf.intros)

  apply (metis Prf.intros(6) Prf_elims(6) set_ConsD val.inject(5))

  prefer 2

  apply (metis Posix1(2) Prf.intros(7) append_Nil empty_iff list.set(1))

  apply(erule Prf_elims)

  apply(auto)

  apply(subst append.simps(2)[symmetric])

  apply(rule Prf.intros)

  apply(auto)

  done

text \<open>

  For a give value and string, our Posix definition 

  determines a unique value.

\<close>

lemma List_eq_zipI:

  assumes "list_all2 (\<lambda>v1 v2. v1 = v2) vs1 vs2" 

  and "length vs1 = length vs2"

  shows "vs1 = vs2"  

 using assms

  apply(induct vs1 vs2 rule: list_all2_induct)

  apply(auto)

  done 

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