theory ReStar

  imports "Main" 

begin

section {* Sequential Composition of Sets *}

definition

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

where 

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

fun spow where

  "spow s 0 = []"

| "spow s (Suc n) = s @ spow s n"

text {* Two Simple Properties about Sequential Composition *}

lemma seq_empty [simp]:

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

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

by (simp_all add: Sequ_def)

lemma seq_null [simp]:

  shows "A ;; {} = {}"

  and   "{} ;; A = {}"

by (simp_all add: Sequ_def)

definition

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

where

  "Der c A \<equiv> {s. [c] @ s \<in> A}"

lemma Der_null [simp]:

  shows "Der c {} = {}"

unfolding Der_def

by auto

lemma Der_empty [simp]:

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

unfolding Der_def

by auto

lemma Der_char [simp]:

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

unfolding Der_def

by auto

lemma Der_union [simp]:

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

unfolding Der_def

by auto

lemma Der_seq [simp]:

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

unfolding Der_def Sequ_def

apply (auto simp add: Cons_eq_append_conv)

done

lemma seq_image:

  assumes "\<forall>s1 s2. f (s1 @ s2) = (f s1) @ (f s2)"

  shows "f ` (A ;; B) = (f ` A) ;; (f ` B)"

apply(auto simp add: Sequ_def image_def)

apply(rule_tac x="f s1" in exI)

apply(rule_tac x="f s2" in exI)

using assms

apply(auto)

apply(rule_tac x="xa @ xb" in exI)

using assms

apply(auto)

done

section {* Kleene Star for Sets *}

inductive_set

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

  for A :: "string set"

where

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

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

lemma star_cases:

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

unfolding Sequ_def

by (auto) (metis Star.simps)

fun 

  pow :: "string set \<Rightarrow> nat \<Rightarrow> string set" ("_ \<up> _" [100,100] 100)

where

  "A \<up> 0 = {[]}"

| "A \<up> (Suc n) = A ;; (A \<up> n)"  

lemma star1: 

  shows "s \<in> A\<star> \<Longrightarrow> \<exists>n. s \<in> A \<up> n"

  apply(induct rule: Star.induct)

  apply (metis pow.simps(1) insertI1)

  apply(auto)

  apply(rule_tac x="Suc n" in exI)

  apply(auto simp add: Sequ_def)

  done

lemma star2:

  shows "s \<in> A \<up> n \<Longrightarrow> s \<in> A\<star>"

  apply(induct n arbitrary: s)

  apply (metis pow.simps(1) Star.simps empty_iff insertE)

  apply(auto simp add: Sequ_def)

  done

lemma star3:

  shows "A\<star> = (\<Union>i. A \<up> i)"

using star1 star2

apply(auto)

done

lemma star4:

  shows "s \<in> A \<up> n \<Longrightarrow> \<exists>ss. s = concat ss \<and> (\<forall>s' \<in> set ss. s' \<in> A)"

  apply(induct n arbitrary: s)

  apply(auto simp add: Sequ_def)

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

  apply(auto)

  apply(drule_tac x="s2" in meta_spec)

  apply(auto)

by (metis concat.simps(2) insertE set_simps(2))

lemma star5:

  assumes "f [] = []"

  assumes "\<forall>s1 s2. f (s1 @ s2) = (f s1) @ (f s2)"

  shows "(f ` A) \<up> n = f ` (A \<up> n)"

apply(induct n)

apply(simp add: assms)

apply(simp)

apply(subst seq_image[OF assms(2)])

apply(simp)

done

lemma star6:

  assumes "f [] = []"

  assumes "\<forall>s1 s2. f (s1 @ s2) = (f s1) @ (f s2)"

  shows "(f ` A)\<star> = f ` (A\<star>)"

apply(simp add: star3)

apply(simp add: image_UN)

apply(subst star5[OF assms])

apply(simp)

done

lemma star_decomp: 

  assumes a: "c # x \<in> A\<star>" 

  shows "\<exists>a b. x = a @ b \<and> c # a \<in> A \<and> b \<in> A\<star>"

using a

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

   (auto simp add: append_eq_Cons_conv)

lemma Der_star [simp]:

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

proof -    

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

    by (simp only: star_cases[symmetric])

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

    by (simp only: Der_union Der_empty) (simp)

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

    by simp

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

    unfolding Sequ_def Der_def

    by (auto dest: star_decomp)

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

qed

section {* Regular Expressions *}

datatype rexp =

  NULL

| EMPTY

| CHAR char

| SEQ rexp rexp

| ALT rexp rexp

| STAR rexp

section {* Semantics of Regular Expressions *}

fun

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

where

  "L (NULL) = {}"

| "L (EMPTY) = {[]}"

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

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

| "L (ALT r1 r2) = (L r1) \<union> (L r2)"

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

fun

 nullable :: "rexp \<Rightarrow> bool"

where

  "nullable (NULL) = False"

| "nullable (EMPTY) = True"

| "nullable (CHAR c) = False"

| "nullable (ALT r1 r2) = (nullable r1 \<or> nullable r2)"

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

| "nullable (STAR r) = True"

lemma nullable_correctness:

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

apply (induct r) 

apply(auto simp add: Sequ_def) 

done

section {* 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))" 

lemma [simp]:

 "flat (Stars vs) = concat (map flat vs)"

apply(induct vs)

apply(auto)

done

section {* Relation between values and regular expressions *}

inductive 

  NPrf :: "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 : EMPTY"

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

| "\<Turnstile> Stars [] : STAR r"

| "\<lbrakk>\<Turnstile> v : r; \<Turnstile> Stars vs : STAR r; flat v \<noteq> []\<rbrakk> \<Longrightarrow> \<Turnstile> Stars (v # vs) : STAR r"

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 : EMPTY"

| "\<turnstile> Char c : CHAR c"

| "\<turnstile> Stars [] : STAR r"

| "\<lbrakk>\<turnstile> v : r; \<turnstile> Stars vs : STAR r\<rbrakk> \<Longrightarrow> \<turnstile> Stars (v # vs) : STAR r"

lemma NPrf_imp_Prf:

  assumes "\<Turnstile> v : r" 

  shows "\<turnstile> v : r"

using assms

apply(induct)

apply(auto intro: Prf.intros)

done

lemma NPrf_Prf_val:

  shows "\<turnstile> v : r \<Longrightarrow> \<exists>v'. flat v' = flat v \<and> \<Turnstile> v' : r"

  and   "\<turnstile> Stars vs : r \<Longrightarrow> \<exists>vs'. flat (Stars vs') = flat (Stars vs) \<and> \<Turnstile> Stars vs' : r"

using assms

apply(induct v and vs arbitrary: r and r rule: val.inducts)

apply(auto)[1]

apply(erule Prf.cases)

apply(simp_all)[7]

apply(rule_tac x="Void" in exI)

apply(simp)

apply(rule NPrf.intros)

apply(erule Prf.cases)

apply(simp_all)[7]

apply(rule_tac x="Char c" in exI)

apply(simp)

apply(rule NPrf.intros)

apply(erule Prf.cases)

apply(simp_all)[7]

apply(auto)[1]

apply(drule_tac x="r1" in meta_spec)

apply(drule_tac x="r2" in meta_spec)

apply(simp)

apply(auto)[1]

apply(rule_tac x="Seq v' v'a" in exI)

apply(simp)

apply (metis NPrf.intros(1))

apply(erule Prf.cases)

apply(simp_all)[7]

apply(clarify)

apply(drule_tac x="r2" in meta_spec)

apply(simp)

apply(auto)[1]

apply(rule_tac x="Right v'" in exI)

apply(simp)

apply (metis NPrf.intros)

apply(erule Prf.cases)

apply(simp_all)[7]

apply(clarify)

apply(drule_tac x="r1" in meta_spec)

apply(simp)

apply(auto)[1]

apply(rule_tac x="Left v'" in exI)

apply(simp)

apply (metis NPrf.intros)

apply(drule_tac x="r" in meta_spec)

apply(simp)

apply(auto)[1]

apply(rule_tac x="Stars vs'" in exI)

apply(simp)

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

apply(simp)

apply(erule Prf.cases)

apply(simp_all)[7]

apply (metis NPrf.intros(6))

apply(erule Prf.cases)

apply(simp_all)[7]

apply(auto)[1]

apply(drule_tac x="ra" in meta_spec)

apply(simp)

apply(drule_tac x="STAR ra" in meta_spec)

apply(simp)

apply(auto)

apply(case_tac "flat v = []")

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

apply(simp)

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

apply(simp)

apply(rule NPrf.intros)

apply(auto)

done

lemma NPrf_Prf:

  shows "{flat v | v. \<turnstile> v : r} = {flat v | v. \<Turnstile> v : r}"

apply(auto)

apply (metis NPrf_Prf_val(1))

by (metis NPrf_imp_Prf)

lemma not_nullable_flat:

  assumes "\<turnstile> v : r" "\<not>nullable r"

  shows "flat v \<noteq> []"

using assms

apply(induct)

apply(auto)

done

lemma Prf_flat_L:

  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)

done

lemma NPrf_flat_L:

  assumes "\<Turnstile> v : r" shows "flat v \<in> L r"

using assms

by (metis NPrf_imp_Prf Prf_flat_L)

lemma Prf_Stars:

  assumes "\<forall>v \<in> set vs. \<turnstile> v : r"

  shows "\<turnstile> Stars vs : STAR r"

using assms

apply(induct vs)

apply (metis Prf.intros(6))

by (metis Prf.intros(7) insert_iff set_simps(2))

lemma Star_string:

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

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

using assms

apply(induct rule: Star.induct)

apply(auto)

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

apply(simp)

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

apply(simp)

done

lemma Star_val:

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

  shows "\<exists>vs. concat (map flat vs) = concat ss \<and> (\<forall>v\<in>set vs. \<turnstile> v : r)"

using assms

apply(induct ss)

apply(auto)

apply (metis empty_iff list.set(1))

by (metis concat.simps(2) list.simps(9) set_ConsD)

lemma Star_valN:

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

  shows "\<exists>vs. concat (map flat vs) = concat ss \<and> (\<forall>v\<in>set vs. \<Turnstile> v : r)"

using assms

apply(induct ss)

apply(auto)

apply (metis empty_iff list.set(1))

by (metis concat.simps(2) list.simps(9) set_ConsD)

lemma L_flat_Prf:

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

apply(induct r)

apply(auto dest: Prf_flat_L simp add: Sequ_def)

apply (metis Prf.intros(4) flat.simps(1))

apply (metis Prf.intros(5) flat.simps(2))

apply (metis Prf.intros(1) flat.simps(5))

apply (metis Prf.intros(2) flat.simps(3))

apply (metis Prf.intros(3) flat.simps(4))

apply(erule Prf.cases)

apply(auto)

apply(subgoal_tac "\<exists>vs::val list. concat (map flat vs) = x \<and> (\<forall>v \<in> set vs. \<turnstile> v : r)")

apply(auto)[1]

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

apply(simp)

apply(rule Prf_Stars)

apply(simp)

apply(drule Star_string)

apply(auto)

apply(rule Star_val)

apply(simp)

done

lemma L_flat_NPrf:

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

by (metis L_flat_Prf NPrf_Prf)

text {* nicer proofs by Fahad *}

lemma Prf_Star_flat_L:

  assumes "\<turnstile> v : STAR r" shows "flat v \<in> (L r)\<star>"

using assms

apply(induct v r\<equiv>"STAR r" arbitrary: r rule: Prf.induct)

apply(auto)

apply(simp add: star3)

apply(auto)

apply(rule_tac x="Suc x" in exI)

apply(auto simp add: Sequ_def)

apply(rule_tac x="flat v" in exI)

apply(rule_tac x="flat (Stars vs)" in exI)

apply(auto)

by (metis Prf_flat_L)

lemma L_flat_Prf2:

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

apply(induct r)

apply(auto)

using L.simps(1) Prf_flat_L 

apply(blast)

using Prf.intros(4) 

apply(force)

using L.simps(2) Prf_flat_L 

apply(blast)

using Prf.intros(5) apply force

using L.simps(3) Prf_flat_L apply blast

using L_flat_Prf apply auto[1]

apply (smt L.simps(4) Sequ_def mem_Collect_eq)

using Prf_flat_L 

apply(fastforce)

apply(metis Prf.intros(2) flat.simps(3))

apply(metis Prf.intros(3) flat.simps(4))

apply(erule Prf.cases)

apply(simp)

apply(simp)

apply(auto)

using L_flat_Prf apply auto[1]

apply (smt Collect_cong L.simps(6) mem_Collect_eq)

using Prf_Star_flat_L 

apply(fastforce)

done

section {* Values Sets *}

definition prefix :: "string \<Rightarrow> string \<Rightarrow> bool" ("_ \<sqsubseteq> _" [100, 100] 100)

where

  "s1 \<sqsubseteq> s2 \<equiv> \<exists>s3. s1 @ s3 = s2"

definition sprefix :: "string \<Rightarrow> string \<Rightarrow> bool" ("_ \<sqsubset> _" [100, 100] 100)

where

  "s1 \<sqsubset> s2 \<equiv> (s1 \<sqsubseteq> s2 \<and> s1 \<noteq> s2)"

lemma length_sprefix:

  "s1 \<sqsubset> s2 \<Longrightarrow> length s1 < length s2"

unfolding sprefix_def prefix_def

by (auto)

definition Prefixes :: "string \<Rightarrow> string set" where

  "Prefixes s \<equiv> {sp. sp \<sqsubseteq> s}"

definition Suffixes :: "string \<Rightarrow> string set" where

  "Suffixes s \<equiv> rev ` (Prefixes (rev s))"