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 *)+
−
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    +
−