theory ReStar
imports "Main"
begin
section {* Sequential Composition of Languages *}
definition
Sequ :: "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: Sequ_def)
lemma seq_null [simp]:
shows "A ;; {} = {}"
and "{} ;; A = {}"
by (simp_all add: Sequ_def)
section {* Semantic Derivative of Languages *}
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_Sequ [simp]:
shows "Der c (A ;; B) = (Der c A) ;; B \<union> (if [] \<in> A then Der c B else {})"
unfolding Der_def Sequ_def
by (auto simp add: Cons_eq_append_conv)
section {* Kleene Star for Languages *}
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>"
lemma star_cases:
shows "A\<star> = {[]} \<union> A ;; A\<star>"
unfolding Sequ_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)
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 Sequ_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 =
ZERO
| ONE
| CHAR char
| SEQ rexp rexp
| ALT rexp rexp
| STAR rexp
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 {* Nullable, Derivatives *}
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"
fun
der :: "char \<Rightarrow> rexp \<Rightarrow> rexp"
where
"der c (ZERO) = ZERO"
| "der c (ONE) = ZERO"
| "der c (CHAR 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)"
lemma nullable_correctness:
shows "nullable r \<longleftrightarrow> [] \<in> (L r)"
by (induct r) (auto simp add: Sequ_def)
lemma der_correctness:
shows "L (der c r) = Der c (L r)"
apply(induct r)
apply(simp_all add: nullable_correctness)
done
section {* Values *}
datatype val =
Void
| Char char
| Seq val val
| Right val
| Left val
| Stars "val list"
datatype_compat val
section {* The string behind a value *}
fun
flat :: "val \<Rightarrow> string"
where
"flat (Void) = []"
| "flat (Char c) = [c]"
| "flat (Left v) = flat v"
| "flat (Right v) = flat v"
| "flat (Seq v1 v2) = (flat v1) @ (flat v2)"
| "flat (Stars []) = []"
| "flat (Stars (v#vs)) = (flat v) @ (flat (Stars vs))"
lemma flat_Stars [simp]:
"flat (Stars vs) = concat (map flat vs)"
by (induct vs) (auto)
section {* Relation between values and regular expressions *}
inductive
Prf :: "val \<Rightarrow> rexp \<Rightarrow> bool" ("\<turnstile> _ : _" [100, 100] 100)
where
"\<lbrakk>\<turnstile> v1 : r1; \<turnstile> v2 : r2\<rbrakk> \<Longrightarrow> \<turnstile> Seq v1 v2 : SEQ r1 r2"
| "\<turnstile> v1 : r1 \<Longrightarrow> \<turnstile> Left v1 : ALT r1 r2"
| "\<turnstile> v2 : r2 \<Longrightarrow> \<turnstile> Right v2 : ALT r1 r2"
| "\<turnstile> Void : ONE"
| "\<turnstile> Char c : CHAR c"
| "\<turnstile> Stars [] : STAR r"
| "\<lbrakk>\<turnstile> v : r; \<turnstile> Stars vs : STAR r\<rbrakk> \<Longrightarrow> \<turnstile> Stars (v # vs) : STAR r"
lemma not_nullable_flat:
assumes "\<turnstile> v : r" "\<not> nullable r"
shows "flat v \<noteq> []"
using assms
by (induct) (auto)
lemma Prf_flat_L:
assumes "\<turnstile> v : r" shows "flat v \<in> L r"
using assms
apply(induct v r rule: Prf.induct)
apply(auto simp add: Sequ_def)
done
lemma Prf_Stars:
assumes "\<forall>v \<in> set vs. \<turnstile> v : r"
shows "\<turnstile> Stars vs : STAR r"
using assms
apply(induct vs)
apply (metis Prf.intros(6))
by (metis Prf.intros(7) insert_iff set_simps(2))
lemma Star_string:
assumes "s \<in> A\<star>"
shows "\<exists>ss. concat ss = s \<and> (\<forall>s \<in> set ss. s \<in> A)"
using assms
apply(induct rule: Star.induct)
apply(auto)
apply(rule_tac x="[]" in exI)
apply(simp)
apply(rule_tac x="s1#ss" in exI)
apply(simp)
done
lemma Star_val:
assumes "\<forall>s\<in>set ss. \<exists>v. s = flat v \<and> \<turnstile> v : r"
shows "\<exists>vs. concat (map flat vs) = concat ss \<and> (\<forall>v\<in>set vs. \<turnstile> v : r)"
using assms
apply(induct ss)
apply(auto)
apply (metis empty_iff list.set(1))
by (metis concat.simps(2) list.simps(9) set_ConsD)
lemma L_flat_Prf:
"L(r) = {flat v | v. \<turnstile> v : r}"
apply(induct r)
apply(auto dest: Prf_flat_L simp add: Sequ_def)
apply (metis Prf.intros(4) flat.simps(1))
apply (metis Prf.intros(5) flat.simps(2))
apply (metis Prf.intros(1) flat.simps(5))
apply (metis Prf.intros(2) flat.simps(3))
apply (metis Prf.intros(3) flat.simps(4))
apply(erule Prf.cases)
apply(auto)
apply(subgoal_tac "\<exists>vs::val list. concat (map flat vs) = x \<and> (\<forall>v \<in> set vs. \<turnstile> v : r)")
apply(auto)[1]
apply(rule_tac x="Stars vs" in exI)
apply(simp)
apply(rule Prf_Stars)
apply(simp)
apply(drule Star_string)
apply(auto)
apply(rule Star_val)
apply(simp)
done
section {* Sulzmann functions *}
fun
mkeps :: "rexp \<Rightarrow> val"
where
"mkeps(ONE) = Void"
| "mkeps(SEQ r1 r2) = Seq (mkeps r1) (mkeps r2)"
| "mkeps(ALT r1 r2) = (if nullable(r1) then Left (mkeps r1) else Right (mkeps r2))"
| "mkeps(STAR r) = Stars []"
fun injval :: "rexp \<Rightarrow> char \<Rightarrow> val \<Rightarrow> val"
where
"injval (CHAR d) c Void = Char d"
| "injval (ALT r1 r2) c (Left v1) = Left(injval r1 c v1)"
| "injval (ALT r1 r2) c (Right v2) = Right(injval r2 c v2)"
| "injval (SEQ r1 r2) c (Seq v1 v2) = Seq (injval r1 c v1) v2"
| "injval (SEQ r1 r2) c (Left (Seq v1 v2)) = Seq (injval r1 c v1) v2"
| "injval (SEQ r1 r2) c (Right v2) = Seq (mkeps r1) (injval r2 c v2)"
| "injval (STAR r) c (Seq v (Stars vs)) = Stars ((injval r c v) # vs)"
section {* Matcher *}
fun
matcher :: "rexp \<Rightarrow> string \<Rightarrow> val option"
where
"matcher r [] = (if nullable r then Some(mkeps r) else None)"
| "matcher r (c#s) = (case (matcher (der c r) s) of
None \<Rightarrow> None
| Some(v) \<Rightarrow> Some(injval r c v))"
fun
matcher2 :: "rexp \<Rightarrow> string \<Rightarrow> val"
where
"matcher2 r [] = mkeps r"
| "matcher2 r (c#s) = injval r c (matcher2 (der c r) s)"
section {* Mkeps, injval *}
lemma mkeps_nullable:
assumes "nullable(r)"
shows "\<turnstile> mkeps r : r"
using assms
apply(induct rule: nullable.induct)
apply(auto intro: Prf.intros)
done
lemma mkeps_flat:
assumes "nullable(r)"
shows "flat (mkeps r) = []"
using assms
apply(induct rule: nullable.induct)
apply(auto)
done
lemma Prf_injval:
assumes "\<turnstile> v : der c r"
shows "\<turnstile> (injval r c v) : r"
using assms
apply(induct r arbitrary: c v rule: rexp.induct)
apply(simp_all)
(* ZERO *)
apply(erule Prf.cases)
apply(simp_all)[7]
(* ONE *)
apply(erule Prf.cases)
apply(simp_all)[7]
(* CHAR *)
apply(case_tac "c = x")
apply(simp)
apply(erule Prf.cases)
apply(simp_all)[7]
apply(rule Prf.intros(5))
apply(simp)
apply(erule Prf.cases)
apply(simp_all)[7]
(* SEQ *)
apply(case_tac "nullable x1")
apply(simp)
apply(erule Prf.cases)
apply(simp_all)[7]
apply(auto)[1]
apply(erule Prf.cases)
apply(simp_all)[7]
apply(auto)[1]
apply(rule Prf.intros)
apply(auto)[2]
apply (metis Prf.intros(1) mkeps_nullable)
apply(simp)
apply(erule Prf.cases)
apply(simp_all)[7]
apply(auto)[1]
apply(rule Prf.intros)
apply(auto)[2]
(* ALT *)
apply(erule Prf.cases)
apply(simp_all)[7]
apply(clarify)
apply (metis Prf.intros(2))
apply (metis Prf.intros(3))
(* STAR *)
apply(erule Prf.cases)
apply(simp_all)[7]
apply(clarify)
apply(rotate_tac 2)
apply(erule Prf.cases)
apply(simp_all)[7]
apply(auto)
apply (metis Prf.intros(6) Prf.intros(7))
by (metis Prf.intros(7))
lemma Prf_injval_flat:
assumes "\<turnstile> v : der c r"
shows "flat (injval r c v) = c # (flat v)"
using assms
apply(induct arbitrary: v rule: der.induct)
apply(simp)
apply(erule Prf.cases)
apply(simp_all)[7]
apply(simp)
apply(erule Prf.cases)
apply(simp_all)[7]
apply(simp)
apply(case_tac "c = d")
apply(simp)
apply(auto)[1]
apply(erule Prf.cases)
apply(simp_all)[7]
apply(simp)
apply(erule Prf.cases)
apply(simp_all)[7]
apply(simp)
apply(erule Prf.cases)
apply(simp_all)[7]
apply(simp)
apply(case_tac "nullable r1")
apply(simp)
apply(erule Prf.cases)
apply(simp_all (no_asm_use))[7]
apply(auto)[1]
apply(erule Prf.cases)
apply(simp_all)[7]
apply(clarify)
apply(simp only: injval.simps flat.simps)
apply(auto)[1]
apply (metis mkeps_flat)
apply(simp)
apply(erule Prf.cases)
apply(simp_all)[7]
apply(simp)
apply(erule Prf.cases)
apply(simp_all)[7]
apply(auto)
apply(rotate_tac 2)
apply(erule Prf.cases)
apply(simp_all)[7]
done
section {* Our Alternative Posix definition *}
inductive
PMatch :: "string \<Rightarrow> rexp \<Rightarrow> val \<Rightarrow> bool" ("_ \<in> _ \<rightarrow> _" [100, 100, 100] 100)
where
"[] \<in> ONE \<rightarrow> Void"
| "[c] \<in> (CHAR c) \<rightarrow> (Char c)"
| "s \<in> r1 \<rightarrow> v \<Longrightarrow> s \<in> (ALT r1 r2) \<rightarrow> (Left v)"
| "\<lbrakk>s \<in> r2 \<rightarrow> v; s \<notin> L(r1)\<rbrakk> \<Longrightarrow> s \<in> (ALT r1 r2) \<rightarrow> (Right v)"
| "\<lbrakk>s1 \<in> r1 \<rightarrow> v1; s2 \<in> r2 \<rightarrow> v2;
\<not>(\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (s1 @ s\<^sub>3) \<in> L r1 \<and> s\<^sub>4 \<in> L r2)\<rbrakk> \<Longrightarrow>
(s1 @ s2) \<in> (SEQ r1 r2) \<rightarrow> (Seq v1 v2)"
| "\<lbrakk>s1 \<in> r \<rightarrow> v; s2 \<in> STAR r \<rightarrow> Stars vs; flat v \<noteq> [];
\<not>(\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (s1 @ s\<^sub>3) \<in> L r \<and> s\<^sub>4 \<in> L (STAR r))\<rbrakk>
\<Longrightarrow> (s1 @ s2) \<in> STAR r \<rightarrow> Stars (v # vs)"
| "[] \<in> STAR r \<rightarrow> Stars []"
lemma PMatch1:
assumes "s \<in> r \<rightarrow> v"
shows "s \<in> L r" "flat v = s"
using assms
apply(induct s r v rule: PMatch.induct)
apply(auto simp add: Sequ_def)
done
lemma PMatch1a:
assumes "s \<in> r \<rightarrow> v"
shows "\<turnstile> v : r"
using assms
apply(induct s r v rule: PMatch.induct)
apply(auto intro: Prf.intros)
done
lemma PMatch_mkeps:
assumes "nullable r"
shows "[] \<in> r \<rightarrow> mkeps r"
using assms
apply(induct r)
apply(auto intro: PMatch.intros simp add: nullable_correctness Sequ_def)
apply(subst append.simps(1)[symmetric])
apply (rule PMatch.intros)
apply(auto)
done
lemma PMatch_determ:
assumes "s \<in> r \<rightarrow> v1" "s \<in> r \<rightarrow> v2"
shows "v1 = v2"
using assms
apply(induct s r v1 arbitrary: v2 rule: PMatch.induct)
apply(erule PMatch.cases)
apply(simp_all)[7]
apply(erule PMatch.cases)
apply(simp_all)[7]
apply(rotate_tac 2)
apply(erule PMatch.cases)
apply(simp_all (no_asm_use))[7]
apply(clarify)
apply(force)
apply(clarify)
apply(drule PMatch1(1))
apply(simp)
apply(rotate_tac 3)
apply(erule PMatch.cases)
apply(simp_all (no_asm_use))[7]
apply(drule PMatch1(1))+
apply(simp)
apply(simp)
apply(rotate_tac 5)
apply(erule PMatch.cases)
apply(simp_all (no_asm_use))[7]
apply(clarify)
apply(subgoal_tac "s1 = s1a")
apply(blast)
apply(simp add: append_eq_append_conv2)
apply(clarify)
apply (metis PMatch1(1) append_self_conv)
apply(rotate_tac 6)
apply(erule PMatch.cases)
apply(simp_all (no_asm_use))[7]
apply(clarify)
apply(subgoal_tac "s1 = s1a")
apply(simp)
apply(blast)
apply(simp (no_asm_use) add: append_eq_append_conv2)
apply(clarify)
apply (metis L.simps(6) PMatch1(1) append_self_conv)
apply(clarify)
apply(rotate_tac 2)
apply(erule PMatch.cases)
apply(simp_all (no_asm_use))[7]
using PMatch1(2) apply auto[1]
using PMatch1(2) apply blast
apply(erule PMatch.cases)
apply(simp_all (no_asm_use))[7]
apply(clarify)
apply (simp add: PMatch1(2))
apply(simp)
done
(* a proof that does not need proj *)
lemma PMatch2_roy_version:
assumes "s \<in> (der c r) \<rightarrow> v"
shows "(c#s) \<in> r \<rightarrow> (injval r c v)"
using assms
proof(induct r arbitrary: s v rule: rexp.induct)
case ZERO
have "s \<in> der c ZERO \<rightarrow> v" by fact
then have "s \<in> ZERO \<rightarrow> v" by simp
then have "False" by cases
then show "(c#s) \<in> ZERO \<rightarrow> (injval ZERO c v)" by simp
next
case ONE
have "s \<in> der c ONE \<rightarrow> v" by fact
then have "s \<in> ZERO \<rightarrow> v" by simp
then have "False" by cases
then show "(c#s) \<in> ONE \<rightarrow> (injval ONE c v)" by simp
next
case (CHAR d)
consider (eq) "c = d" | (ineq) "c \<noteq> d" by blast
then show "(c#s) \<in> (CHAR d) \<rightarrow> (injval (CHAR d) c v)"
proof (cases)
case eq
have "s \<in> der c (CHAR d) \<rightarrow> v" by fact
then have "s \<in> ONE \<rightarrow> v" using eq by simp
then have eqs: "s = [] \<and> v = Void" by cases simp
show "(c # s) \<in> CHAR d \<rightarrow> injval (CHAR d) c v" using eq eqs by (auto intro: PMatch.intros)
next
case ineq
have "s \<in> der c (CHAR d) \<rightarrow> v" by fact
then have "s \<in> ZERO \<rightarrow> v" using ineq by simp
then have "False" by cases
then show "(c # s) \<in> CHAR d \<rightarrow> injval (CHAR d) c v" by simp
qed
next
case (ALT r1 r2)
have IH1: "\<And>s v. s \<in> der c r1 \<rightarrow> v \<Longrightarrow> (c # s) \<in> r1 \<rightarrow> injval r1 c v" by fact
have IH2: "\<And>s v. s \<in> der c r2 \<rightarrow> v \<Longrightarrow> (c # s) \<in> r2 \<rightarrow> injval r2 c v" by fact
have "s \<in> der c (ALT r1 r2) \<rightarrow> v" by fact
then have "s \<in> ALT (der c r1) (der c r2) \<rightarrow> v" by simp
then consider (left) v' where "v = Left v'" "s \<in> der c r1 \<rightarrow> v'"
| (right) v' where "v = Right v'" "s \<notin> L (der c r1)" "s \<in> der c r2 \<rightarrow> v'"
by cases auto
then show "(c # s) \<in> ALT r1 r2 \<rightarrow> injval (ALT r1 r2) c v"
proof (cases)
case left
have "s \<in> der c r1 \<rightarrow> v'" by fact
then have "(c # s) \<in> r1 \<rightarrow> injval r1 c v'" using IH1 by simp
then have "(c # s) \<in> ALT r1 r2 \<rightarrow> injval (ALT r1 r2) c (Left v')" by (auto intro: PMatch.intros)
then show "(c # s) \<in> ALT r1 r2 \<rightarrow> injval (ALT r1 r2) c v" using left by simp
next
case right
have "s \<notin> L (der c r1)" by fact
then have "c # s \<notin> L r1" by (simp add: der_correctness Der_def)
moreover
have "s \<in> der c r2 \<rightarrow> v'" by fact
then have "(c # s) \<in> r2 \<rightarrow> injval r2 c v'" using IH2 by simp
ultimately have "(c # s) \<in> ALT r1 r2 \<rightarrow> injval (ALT r1 r2) c (Right v')"
by (auto intro: PMatch.intros)
then show "(c # s) \<in> ALT r1 r2 \<rightarrow> injval (ALT r1 r2) c v" using right by simp
qed
next
case (SEQ r1 r2)
have IH1: "\<And>s v. s \<in> der c r1 \<rightarrow> v \<Longrightarrow> (c # s) \<in> r1 \<rightarrow> injval r1 c v" by fact
have IH2: "\<And>s v. s \<in> der c r2 \<rightarrow> v \<Longrightarrow> (c # s) \<in> r2 \<rightarrow> injval r2 c v" by fact
have "s \<in> der c (SEQ r1 r2) \<rightarrow> v" by fact
then consider
(left_nullable) v1 v2 s1 s2 where
"v = Left (Seq v1 v2)" "s = s1 @ s2"
"s1 \<in> der c r1 \<rightarrow> v1" "s2 \<in> r2 \<rightarrow> v2" "nullable r1"
"\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r1) \<and> s\<^sub>4 \<in> L r2)"
| (right_nullable) v1 s1 s2 where
"v = Right v1" "s = s1 @ s2"
"s \<in> der c r2 \<rightarrow> v1" "nullable r1" "s1 @ s2 \<notin> L (SEQ (der c r1) r2)"
| (not_nullable) v1 v2 s1 s2 where
"v = Seq v1 v2" "s = s1 @ s2"
"s1 \<in> der c r1 \<rightarrow> v1" "s2 \<in> r2 \<rightarrow> v2" "\<not>nullable r1"
"\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r1) \<and> s\<^sub>4 \<in> L r2)"
apply(auto split: if_splits simp add: Sequ_def) apply(erule PMatch.cases)
apply(auto elim: PMatch.cases simp add: Sequ_def der_correctness Der_def)
by fastforce
then show "(c # s) \<in> SEQ r1 r2 \<rightarrow> injval (SEQ r1 r2) c v"
proof (cases)
case left_nullable
have "s1 \<in> der c r1 \<rightarrow> v1" by fact
then have "(c # s1) \<in> r1 \<rightarrow> injval r1 c v1" using IH1 by simp
moreover
have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r1) \<and> s\<^sub>4 \<in> L r2)" by fact
then have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (c # s1) @ s\<^sub>3 \<in> L r1 \<and> s\<^sub>4 \<in> L r2)" by (simp add: der_correctness Der_def)
ultimately have "((c # s1) @ s2) \<in> SEQ r1 r2 \<rightarrow> Seq (injval r1 c v1) v2" using left_nullable by (rule_tac PMatch.intros)
then show "(c # s) \<in> SEQ r1 r2 \<rightarrow> injval (SEQ r1 r2) c v" using left_nullable by simp
next
case right_nullable
have "nullable r1" by fact
then have "[] \<in> r1 \<rightarrow> (mkeps r1)" by (rule PMatch_mkeps)
moreover
have "s \<in> der c r2 \<rightarrow> v1" by fact
then have "(c # s) \<in> r2 \<rightarrow> (injval r2 c v1)" using IH2 by simp
moreover
have "s1 @ s2 \<notin> L (SEQ (der c r1) r2)" by fact
then have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = c # s \<and> [] @ s\<^sub>3 \<in> L r1 \<and> s\<^sub>4 \<in> L r2)" using right_nullable
by(auto simp add: der_correctness Der_def append_eq_Cons_conv Sequ_def)
ultimately have "([] @ (c # s)) \<in> SEQ r1 r2 \<rightarrow> Seq (mkeps r1) (injval r2 c v1)"
by(rule PMatch.intros)
then show "(c # s) \<in> SEQ r1 r2 \<rightarrow> injval (SEQ r1 r2) c v" using right_nullable by simp
next
case not_nullable
have "s1 \<in> der c r1 \<rightarrow> v1" by fact
then have "(c # s1) \<in> r1 \<rightarrow> injval r1 c v1" using IH1 by simp
moreover
have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r1) \<and> s\<^sub>4 \<in> L r2)" by fact
then have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (c # s1) @ s\<^sub>3 \<in> L r1 \<and> s\<^sub>4 \<in> L r2)" by (simp add: der_correctness Der_def)
ultimately have "((c # s1) @ s2) \<in> SEQ r1 r2 \<rightarrow> Seq (injval r1 c v1) v2" using not_nullable
by (rule_tac PMatch.intros) (simp_all)
then show "(c # s) \<in> SEQ r1 r2 \<rightarrow> injval (SEQ r1 r2) c v" using not_nullable by simp
qed
next
case (STAR r)
have IH: "\<And>s v. s \<in> der c r \<rightarrow> v \<Longrightarrow> (c # s) \<in> r \<rightarrow> injval r c v" by fact
have "s \<in> der c (STAR r) \<rightarrow> v" by fact
then consider
(cons) v1 vs s1 s2 where
"v = Seq v1 (Stars vs)" "s = s1 @ s2"
"s1 \<in> der c r \<rightarrow> v1" "s2 \<in> (STAR r) \<rightarrow> (Stars vs)"
"\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r) \<and> s\<^sub>4 \<in> L (STAR r))"
apply(erule_tac PMatch.cases)
apply(auto)
apply(rotate_tac 4)
apply(erule_tac PMatch.cases)
apply(auto)
apply (simp add: PMatch.intros(6))
using PMatch.intros(7) by blast
then show "(c # s) \<in> STAR r \<rightarrow> injval (STAR r) c v"
proof (cases)
case cons
have "s1 \<in> der c r \<rightarrow> v1" by fact
then have "(c # s1) \<in> r \<rightarrow> injval r c v1" using IH by simp
moreover
have "s2 \<in> STAR r \<rightarrow> Stars vs" by fact
moreover
have "(c # s1) \<in> r \<rightarrow> injval r c v1" by fact
then have "flat (injval r c v1) = (c # s1)" by (rule PMatch1)
then have "flat (injval r c v1) \<noteq> []" by simp
moreover
have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r) \<and> s\<^sub>4 \<in> L (STAR r))" by fact
then have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (c # s1) @ s\<^sub>3 \<in> L r \<and> s\<^sub>4 \<in> L (STAR r))"
by (simp add: der_correctness Der_def)
ultimately
have "((c # s1) @ s2) \<in> STAR r \<rightarrow> Stars (injval r c v1 # vs)" by (rule PMatch.intros)
then show "(c # s) \<in> STAR r \<rightarrow> injval (STAR r) c v" using cons by(simp)
qed
qed
lemma lex_correct1:
assumes "s \<notin> L r"
shows "matcher r s = None"
using assms
apply(induct s arbitrary: r)
apply(simp)
apply (metis nullable_correctness)
apply(auto)
apply(drule_tac x="der a r" in meta_spec)
apply(drule meta_mp)
apply(auto)
apply(simp add: der_correctness Der_def)
done
lemma lex_correct1a:
shows "s \<notin> L r \<longleftrightarrow> matcher r s = None"
using assms
apply(induct s arbitrary: r)
apply(simp)
apply (metis nullable_correctness)
apply(auto)
apply(drule_tac x="der a r" in meta_spec)
apply(auto)
apply(simp add: der_correctness Der_def)
apply(drule_tac x="der a r" in meta_spec)
apply(auto)
apply(simp add: der_correctness Der_def)
done
lemma lex_correct2:
assumes "s \<in> L r"
shows "\<exists>v. matcher r s = Some(v) \<and> \<turnstile> v : r \<and> flat v = s"
using assms
apply(induct s arbitrary: r)
apply(simp)
apply (metis mkeps_flat mkeps_nullable nullable_correctness)
apply(drule_tac x="der a r" in meta_spec)
apply(drule meta_mp)
apply(simp add: der_correctness Der_def)
apply(auto)
apply (metis Prf_injval)
apply(rule Prf_injval_flat)
by simp
lemma lex_correct3:
assumes "s \<in> L r"
shows "\<exists>v. matcher r s = Some(v) \<and> s \<in> r \<rightarrow> v"
using assms
apply(induct s arbitrary: r)
apply(simp)
apply (metis PMatch_mkeps nullable_correctness)
apply(drule_tac x="der a r" in meta_spec)
apply(drule meta_mp)
apply(simp add: der_correctness Der_def)
apply(auto)
by (metis PMatch2_roy_version)
lemma lex_correct3a:
shows "s \<in> L r \<longleftrightarrow> (\<exists>v. matcher r s = Some(v) \<and> s \<in> r \<rightarrow> v)"
using assms
apply(induct s arbitrary: r)
apply(simp)
apply (metis PMatch_mkeps nullable_correctness)
apply(drule_tac x="der a r" in meta_spec)
apply(auto)
apply(metis PMatch2_roy_version)
apply(simp add: der_correctness Der_def)
using lex_correct1a
apply fastforce
apply(simp add: der_correctness Der_def)
by (simp add: lex_correct1a)
lemma lex_correct3b:
shows "s \<in> L r \<longleftrightarrow> (\<exists>!v. matcher r s = Some(v) \<and> s \<in> r \<rightarrow> v)"
using assms
apply(induct s arbitrary: r)
apply(simp)
apply (metis PMatch_mkeps nullable_correctness)
apply(drule_tac x="der a r" in meta_spec)
apply(simp add: der_correctness Der_def)
apply(case_tac "matcher (der a r) s = None")
apply(simp)
apply(simp)
apply(clarify)
apply(rule iffI)
apply(auto)
apply(rule PMatch2_roy_version)
apply(simp)
using PMatch1(1) by auto
lemma lex_correct5:
assumes "s \<in> L r"
shows "s \<in> r \<rightarrow> (matcher2 r s)"
using assms
apply(induct s arbitrary: r)
apply(simp)
apply (metis PMatch_mkeps nullable_correctness)
apply(simp)
apply(rule PMatch2_roy_version)
apply(drule_tac x="der a r" in meta_spec)
apply(drule meta_mp)
apply(simp add: der_correctness Der_def)
apply(auto)
done
lemma
"matcher2 (ALT (CHAR a) (ALT (CHAR b) (SEQ (CHAR a) (CHAR b)))) [a,b] = Right (Right (Seq (Char a) (Char b)))"
apply(simp)
done
end