theory Matcher

  imports "Main" 

begin

section {* Regular Expressions *}

datatype rexp =

  ZERO

| ONE

| CHAR char

| SEQ rexp rexp

| ALT rexp rexp

| STAR rexp

section {* Sequential Composition of Sets *}

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 {* Two Simple Properties about Sequential Composition *}

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 {* 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>"

text {* A Standard Property of Star *}

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 {* Semantics of Regular Expressions *}

fun

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

where

  "L (ZERO) = {}"

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

| "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>"

section {* The Matcher *}

fun

 nullable :: "rexp \<Rightarrow> bool"

where

  "nullable (ZERO) = False"

| "nullable (ONE) = 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"

section {* Correctness Proof for Nullable *}

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 *)

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

end