theory Matcher

  imports "Main" 

begin

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 {* Left-Quotient of a Set *}

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 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

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>"

section {* The Matcher *}

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"

fun

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

where

  "der c (NULL) = NULL"

| "der c (EMPTY) = NULL"

| "der c (CHAR c') = (if c = c' then EMPTY else NULL)"

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

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

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

fun 

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

where

  "derivative [] r = r"

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

fun

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

where

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

section {* Correctness Proof of the Matcher *}

lemma nullable_correctness:

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

by (induct r) (auto simp add: Seq_def) 

lemma der_correctness:

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

by (induct r) 

   (simp_all add: nullable_correctness)

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)

section {* Examples *}

definition 

  "CHRA \<equiv> CHAR (CHR ''a'')"

definition 

  "ALT1 \<equiv> ALT CHRA EMPTY"

definition 

  "SEQ3 \<equiv> SEQ (SEQ ALT1 ALT1) ALT1"

value "matcher SEQ3 ''aaa''"

value "matcher NULL []"

value "matcher (CHAR (CHR ''a'')) [CHR ''a'']"

value "matcher (CHAR a) [a,a]"

value "matcher (STAR (CHAR a)) []"

value "matcher (STAR (CHAR a))  [a,a]"

value "matcher (SEQ (CHAR (CHR ''a'')) (SEQ (STAR (CHAR (CHR ''b''))) (CHAR (CHR ''c'')))) ''abbbbc''"

value "matcher (SEQ (CHAR (CHR ''a'')) (SEQ (STAR (CHAR (CHR ''b''))) (CHAR (CHR ''c'')))) ''abbcbbc''"

section {* Incorrect Matcher - fun-definition rejected *}

fun

  match :: "rexp list \<Rightarrow> string \<Rightarrow> bool"

where

  "match [] [] = True"

| "match [] (c # s) = False"

| "match (NULL # rs) s = False"  

| "match (EMPTY # rs) s = match rs s"

| "match (CHAR c # rs) [] = False"

| "match (CHAR c # rs) (d # s) = (if c = d then match rs s else False)"         

| "match (ALT r1 r2 # rs) s = (match (r1 # rs) s \<or> match (r2 # rs) s)" 

| "match (SEQ r1 r2 # rs) s = match (r1 # r2 # rs) s"

| "match (STAR r # rs) s = (match rs s \<or> match (r # (STAR r) # rs) s)"

end