theory Matcher2+ −
imports "Main" + −
begin+ −
+ −
section {* Regular Expressions *}+ −
+ −
datatype rexp =+ −
NULL+ −
| EMPTY+ −
| CHAR char+ −
| SEQ rexp rexp+ −
| ALT rexp rexp+ −
| STAR rexp+ −
| NOT rexp+ −
| PLUS rexp+ −
| OPT rexp+ −
| NTIMES rexp nat+ −
| NMTIMES rexp nat nat+ −
+ −
fun M :: "rexp \<Rightarrow> nat"+ −
where+ −
"M (NULL) = 0"+ −
| "M (EMPTY) = 0"+ −
| "M (CHAR char) = 0"+ −
| "M (SEQ r1 r2) = Suc ((M r1) + (M r2))"+ −
| "M (ALT r1 r2) = Suc ((M r1) + (M r2))"+ −
| "M (STAR r) = Suc (M r)"+ −
| "M (NOT r) = Suc (M r)"+ −
| "M (PLUS r) = Suc (M r)"+ −
| "M (OPT r) = Suc (M r)"+ −
| "M (NTIMES r n) = Suc (M r) * 2 * (Suc n)"+ −
| "M (NMTIMES r n m) = Suc (M r) * 2 * (Suc n + Suc m)"+ −
+ −
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)+ −
+ −
lemma seq_union:+ −
shows "A ;; (B \<union> C) = A ;; B \<union> A ;; C"+ −
and "(B \<union> C) ;; A = B ;; A \<union> C ;; A"+ −
by (auto simp add: Seq_def)+ −
+ −
lemma seq_Union:+ −
shows "A ;; (\<Union>x\<in>B. C x) = (\<Union>x\<in>B. A ;; C x)"+ −
by (auto simp add: Seq_def)+ −
+ −
lemma seq_empty_in [simp]:+ −
"[] \<in> A ;; B \<longleftrightarrow> ([] \<in> A \<and> [] \<in> B)"+ −
by (simp add: Seq_def)+ −
+ −
lemma seq_assoc: + −
shows "A ;; (B ;; C) = (A ;; B) ;; C" + −
apply(auto simp add: Seq_def)+ −
apply(metis append_assoc)+ −
apply(metis)+ −
done+ −
+ −
+ −
section {* Power for Sets *}+ −
+ −
fun + −
pow :: "string set \<Rightarrow> nat \<Rightarrow> string set" ("_ \<up> _" [101, 102] 101)+ −
where+ −
"A \<up> 0 = {[]}"+ −
| "A \<up> (Suc n) = A ;; (A \<up> n)"+ −
+ −
lemma pow_empty [simp]:+ −
shows "[] \<in> A \<up> n \<longleftrightarrow> (n = 0 \<or> [] \<in> A)"+ −
by (induct n) (auto)+ −
+ −
lemma pow_plus:+ −
"A \<up> (n + m) = A \<up> n ;; A \<up> m"+ −
by (induct n) (simp_all add: seq_assoc)+ −
+ −
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_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 + −
using a+ −
by (induct x\<equiv>"c # x" rule: Star.induct) + −
(auto simp add: append_eq_Cons_conv)+ −
+ −
lemma star_cases:+ −
shows "A\<star> = {[]} \<union> A ;; A\<star>"+ −
unfolding Seq_def + −
by (auto) (metis Star.simps)+ −
+ −
lemma Star_in_Pow:+ −
assumes a: "s \<in> A\<star>"+ −
shows "\<exists>n. s \<in> A \<up> n"+ −
using a+ −
apply(induct)+ −
apply(auto)+ −
apply(rule_tac x="Suc n" in exI)+ −
apply(auto simp add: Seq_def)+ −
done+ −
+ −
lemma Pow_in_Star:+ −
assumes a: "s \<in> A \<up> n"+ −
shows "s \<in> A\<star>"+ −
using a+ −
by (induct n arbitrary: s) (auto simp add: Seq_def)+ −
+ −
+ −
lemma Star_def2: + −
shows "A\<star> = (\<Union>n. A \<up> n)"+ −
using Star_in_Pow Pow_in_Star+ −
by (auto)+ −
+ −
+ −
+ −
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>"+ −
| "L (NOT r) = UNIV - (L r)"+ −
| "L (PLUS r) = (L r) ;; ((L r)\<star>)"+ −
| "L (OPT r) = (L r) \<union> {[]}"+ −
| "L (NTIMES r n) = (L r) \<up> n"+ −
| "L (NMTIMES r n m) = (\<Union>i\<in> {n..n+m} . ((L r) \<up> i))" + −
+ −
+ −
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"+ −
| "nullable (NOT r) = (\<not>(nullable r))"+ −
| "nullable (PLUS r) = (nullable r)"+ −
| "nullable (OPT r) = True"+ −
| "nullable (NTIMES r n) = (if n = 0 then True else nullable r)"+ −
| "nullable (NMTIMES r n m) = (if n = 0 then True else nullable r)"+ −
+ −
function+ −
der :: "char \<Rightarrow> rexp \<Rightarrow> rexp"+ −
where+ −
"der c (NULL) = NULL"+ −
| "der c (EMPTY) = NULL"+ −
| "der c (CHAR d) = (if c = d 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)"+ −
| "der c (NOT r) = NOT(der c r)"+ −
| "der c (PLUS r) = SEQ (der c r) (STAR r)"+ −
| "der c (OPT r) = der c r"+ −
| "der c (NTIMES r 0) = NULL"+ −
| "der c (NTIMES r (Suc n)) = der c (SEQ r (NTIMES r n))"+ −
| "der c (NMTIMES r 0 0) = NULL"+ −
| "der c (NMTIMES r 0 (Suc m)) = ALT (der c (NTIMES r (Suc m))) (der c (NMTIMES r 0 m))"+ −
| "der c (NMTIMES r (Suc n) m) = der c (SEQ r (NMTIMES r n m))"+ −
by pat_completeness auto+ −
+ −
termination der + −
by (relation "measure (\<lambda>(c, r). M r)") (simp_all)+ −
+ −
+ −
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)"+ −
+ −
+ −
section {* Correctness Proof of the Matcher *}+ −
+ −
lemma nullable_correctness:+ −
shows "nullable r \<longleftrightarrow> [] \<in> (L r)"+ −
by(induct r) (auto simp add: Seq_def) + −
+ −
+ −
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_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_UNIV [simp]:+ −
"Der c (UNIV - A) = UNIV - Der c A"+ −
unfolding Der_def+ −
by (auto)+ −
+ −
lemma Der_pow [simp]:+ −
shows "Der c (A \<up> (Suc n)) = (Der c A) ;; (A \<up> n) \<union> (if [] \<in> A then Der c (A \<up> n) else {})"+ −
unfolding Der_def + −
by(auto simp add: Cons_eq_append_conv Seq_def)+ −
+ −
lemma Der_UNION [simp]: + −
shows "Der c (\<Union>x\<in>A. B x) = (\<Union>x\<in>A. Der c (B x))"+ −
by (auto simp add: Der_def)+ −
+ −
lemma Suc_Union:+ −
"(\<Union> x\<le>Suc m. B x) = (B (Suc m) \<union> (\<Union> x\<le>m. B x))"+ −
by (metis UN_insert atMost_Suc)+ −
+ −
lemma Suc_reduce_Union:+ −
"(\<Union>x\<in>{Suc n..Suc m}. B x) = (\<Union>x\<in>{n..m}. B (Suc x))"+ −
by (metis UN_extend_simps(10) image_Suc_atLeastAtMost)+ −
+ −
+ −
lemma der_correctness:+ −
shows "L (der c r) = Der c (L r)"+ −
by (induct rule: der.induct) + −
(simp_all add: nullable_correctness + −
Suc_Union Suc_reduce_Union seq_Union atLeast0AtMost)+ −
+ −
+ −
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 + −