theory Matcher

  imports "Main" 

begin

section \<open>Regular Expressions\<close>

datatype rexp =

  ZERO

| ONE

| CH char

| SEQ rexp rexp

| ALT rexp rexp

| STAR rexp

section \<open>Sequential Composition of Sets of Strings\<close>

definition

  Seq :: "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}"

text \<open>Two Simple Properties about Sequential Composition\<close>

lemma seq_empty [simp]:

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

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

by (simp_all add: Seq_def)

lemma seq_null [simp]:

  shows "A ;; {} = {}"

  and   "{} ;; A = {}"

by (simp_all add: Seq_def)

section \<open>Kleene Star for Sets\<close>

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>"

text \<open>A Standard Property of Star\<close>

lemma star_cases:

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

unfolding Seq_def

by (auto) (metis Star.simps)

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)

section \<open>Meaning of Regular Expressions\<close>

fun

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

where

  "L (ZERO) = {}"

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

| "L (CH 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>"

section \<open>The Matcher\<close>

fun

 nullable :: "rexp \<Rightarrow> bool"

where

  "nullable (ZERO) = False"

| "nullable (ONE) = True"

| "nullable (CH c) = False"

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

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

| "nullable (STAR r) = True"

section \<open>Correctness Proof for Nullable\<close>

lemma nullable_correctness:

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

apply(induct r)

(* ZERO case *)

apply(simp only: nullable.simps)

apply(simp only: L.simps)

apply(simp)

(* ONE case *)

apply(simp only: nullable.simps)

apply(simp only: L.simps)

apply(simp)

(* CHAR case *)

apply(simp only: nullable.simps)

apply(simp only: L.simps)

apply(simp)

prefer 2

(* ALT case *)

apply(simp (no_asm) only: nullable.simps)

apply(simp only:)

apply(simp only: L.simps)

apply(simp)

(* SEQ case *)

apply(simp only: L.simps)

apply(simp)

oops

lemma nullable_correctness:

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

apply(induct r)

apply(simp_all)

(* all easy subgoals are proved except the last 2 *)

(* where the definition of Seq needs to be unfolded. *)

oops

lemma nullable_correctness:

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

apply(induct r)

apply(simp_all add: Seq_def)

(* except the star case every thing is proved *)

(* we need to use the rule for Star.start *)

oops

lemma nullable_correctness:

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

apply(induct r)

apply(simp_all add: Seq_def Star.start)

done

section \<open>Derivative Operation\<close>

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

where

  "der c (ZERO) = ZERO"

| "der c (ONE) = ZERO"

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

| "der c (ALT r1 r2) = ALT (der c r1) (der c r2)"

| "der c (SEQ r1 r2) = (if nullable r1 then ALT (SEQ (der c r1) r2) (der c r2)

                                       else SEQ (der c r1) r2)"

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

fun 

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

where

  "ders [] r = r"

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

fun

  matcher :: "rexp \<Rightarrow> string \<Rightarrow> bool"

where

  "matcher r s = nullable (ders s r)"

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_insert_nil [simp]:

  shows "Der c (insert [] A) = Der c A"

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 Seq_def

by (auto simp add: Cons_eq_append_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 Seq_def Der_def

    by (auto dest: star_decomp)

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

qed

lemma der_correctness:

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

  apply(induct rule: der.induct) 

  apply(auto simp add: nullable_correctness)

  done

lemma matcher_correctness:

  shows "matcher r s \<longleftrightarrow> s \<in> L r"

by (induct s arbitrary: r)

   (simp_all add: nullable_correctness der_correctness Der_def)

end