diff -r a42c773ec8ab -r 841f7b9c0a6a thys/ReStar.thy
--- a/thys/ReStar.thy	Tue May 17 14:28:22 2016 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,639 +0,0 @@
-   
-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 (Left Quotient) 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)"
-by (induct r) (simp_all add: nullable_correctness)
-
-
-section {* Values *}
-
-datatype val = 
-  Void
-| Char char
-| Seq val val
-| Right val
-| Left val
-| Stars "val list"
-
-
-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"
-
-inductive_cases Prf_elims:
-  "\<turnstile> v : ZERO"
-  "\<turnstile> v : SEQ r1 r2"
-  "\<turnstile> v : ALT r1 r2"
-  "\<turnstile> v : ONE"
-  "\<turnstile> v : CHAR c"
-(*  "\<turnstile> vs : STAR r"*)
-
-lemma Prf_flat_L:
-  assumes "\<turnstile> v : r" shows "flat v \<in> L r"
-using assms
-by(induct v r rule: Prf.induct)
-  (auto simp add: Sequ_def)
-
-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(auto elim!: Prf_elims)
-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 and Lu 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 {* Mkeps, injval *}
-
-lemma mkeps_nullable:
-  assumes "nullable(r)" 
-  shows "\<turnstile> mkeps r : r"
-using assms
-by (induct rule: nullable.induct) 
-   (auto intro: Prf.intros)
-
-lemma mkeps_flat:
-  assumes "nullable(r)" 
-  shows "flat (mkeps r) = []"
-using assms
-by (induct rule: nullable.induct) (auto)
-
-
-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(auto intro!: Prf.intros mkeps_nullable elim!: Prf_elims split: if_splits)
-(* STAR *)
-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(auto elim!: Prf_elims split: if_splits)
-apply(metis mkeps_flat)
-apply(rotate_tac 2)
-apply(erule Prf.cases)
-apply(simp_all)[7]
-done
-
-
-
-section {* Our Alternative Posix definition *}
-
-inductive 
-  Posix :: "string \<Rightarrow> rexp \<Rightarrow> val \<Rightarrow> bool" ("_ \<in> _ \<rightarrow> _" [100, 100, 100] 100)
-where
-  Posix_ONE: "[] \<in> ONE \<rightarrow> Void"
-| Posix_CHAR: "[c] \<in> (CHAR c) \<rightarrow> (Char c)"
-| Posix_ALT1: "s \<in> r1 \<rightarrow> v \<Longrightarrow> s \<in> (ALT r1 r2) \<rightarrow> (Left v)"
-| Posix_ALT2: "\<lbrakk>s \<in> r2 \<rightarrow> v; s \<notin> L(r1)\<rbrakk> \<Longrightarrow> s \<in> (ALT r1 r2) \<rightarrow> (Right v)"
-| Posix_SEQ: "\<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)"
-| Posix_STAR1: "\<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)"
-| Posix_STAR2: "[] \<in> STAR r \<rightarrow> Stars []"
-
-inductive_cases Posix_elims:
-  "s \<in> ZERO \<rightarrow> v"
-  "s \<in> ONE \<rightarrow> v"
-  "s \<in> CHAR c \<rightarrow> v"
-  "s \<in> ALT r1 r2 \<rightarrow> v"
-  "s \<in> SEQ r1 r2 \<rightarrow> v"
-  "s \<in> STAR r \<rightarrow> v"
-
-lemma Posix1:
-  assumes "s \<in> r \<rightarrow> v"
-  shows "s \<in> L r" "flat v = s"
-using assms
-by (induct s r v rule: Posix.induct)
-   (auto simp add: Sequ_def)
-
-
-lemma Posix1a:
-  assumes "s \<in> r \<rightarrow> v"
-  shows "\<turnstile> v : r"
-using assms
-apply(induct s r v rule: Posix.induct)
-apply(auto intro: Prf.intros)
-done
-
-lemma Posix_mkeps:
-  assumes "nullable r"
-  shows "[] \<in> r \<rightarrow> mkeps r"
-using assms
-apply(induct r)
-apply(auto intro: Posix.intros simp add: nullable_correctness Sequ_def)
-apply(subst append.simps(1)[symmetric])
-apply(rule Posix.intros)
-apply(auto)
-done
-
-
-lemma Posix_determ:
-  assumes "s \<in> r \<rightarrow> v1" "s \<in> r \<rightarrow> v2"
-  shows "v1 = v2"
-using assms
-proof (induct s r v1 arbitrary: v2 rule: Posix.induct)
-  case (Posix_ONE v2)
-  have "[] \<in> ONE \<rightarrow> v2" by fact
-  then show "Void = v2" by cases auto
-next 
-  case (Posix_CHAR c v2)
-  have "[c] \<in> CHAR c \<rightarrow> v2" by fact
-  then show "Char c = v2" by cases auto
-next 
-  case (Posix_ALT1 s r1 v r2 v2)
-  have "s \<in> ALT r1 r2 \<rightarrow> v2" by fact
-  moreover
-  have "s \<in> r1 \<rightarrow> v" by fact
-  then have "s \<in> L r1" by (simp add: Posix1)
-  ultimately obtain v' where eq: "v2 = Left v'" "s \<in> r1 \<rightarrow> v'" by cases auto 
-  moreover
-  have IH: "\<And>v2. s \<in> r1 \<rightarrow> v2 \<Longrightarrow> v = v2" by fact
-  ultimately have "v = v'" by simp
-  then show "Left v = v2" using eq by simp
-next 
-  case (Posix_ALT2 s r2 v r1 v2)
-  have "s \<in> ALT r1 r2 \<rightarrow> v2" by fact
-  moreover
-  have "s \<notin> L r1" by fact
-  ultimately obtain v' where eq: "v2 = Right v'" "s \<in> r2 \<rightarrow> v'" 
-    by cases (auto simp add: Posix1) 
-  moreover
-  have IH: "\<And>v2. s \<in> r2 \<rightarrow> v2 \<Longrightarrow> v = v2" by fact
-  ultimately have "v = v'" by simp
-  then show "Right v = v2" using eq by simp
-next
-  case (Posix_SEQ s1 r1 v1 s2 r2 v2 v')
-  have "(s1 @ s2) \<in> SEQ r1 r2 \<rightarrow> v'" 
-       "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)" by fact+
-  then obtain v1' v2' where "v' = Seq v1' v2'" "s1 \<in> r1 \<rightarrow> v1'" "s2 \<in> r2 \<rightarrow> v2'"
-  apply(cases) apply (auto simp add: append_eq_append_conv2)
-  using Posix1(1) by fastforce+
-  moreover
-  have IHs: "\<And>v1'. s1 \<in> r1 \<rightarrow> v1' \<Longrightarrow> v1 = v1'"
-            "\<And>v2'. s2 \<in> r2 \<rightarrow> v2' \<Longrightarrow> v2 = v2'" by fact+
-  ultimately show "Seq v1 v2 = v'" by simp
-next
-  case (Posix_STAR1 s1 r v s2 vs v2)
-  have "(s1 @ s2) \<in> STAR r \<rightarrow> v2" 
-       "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))" by fact+
-  then obtain v' vs' where "v2 = Stars (v' # vs')" "s1 \<in> r \<rightarrow> v'" "s2 \<in> (STAR r) \<rightarrow> (Stars vs')"
-  apply(cases) apply (auto simp add: append_eq_append_conv2)
-  using Posix1(1) apply fastforce
-  apply (metis Posix1(1) Posix_STAR1.hyps(6) append_Nil append_Nil2)
-  using Posix1(2) by blast
-  moreover
-  have IHs: "\<And>v2. s1 \<in> r \<rightarrow> v2 \<Longrightarrow> v = v2"
-            "\<And>v2. s2 \<in> STAR r \<rightarrow> v2 \<Longrightarrow> Stars vs = v2" by fact+
-  ultimately show "Stars (v # vs) = v2" by auto
-next
-  case (Posix_STAR2 r v2)
-  have "[] \<in> STAR r \<rightarrow> v2" by fact
-  then show "Stars [] = v2" by cases (auto simp add: Posix1)
-qed
-
-
-lemma Posix_injval:
-  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: Posix.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: Posix.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: Posix.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)"
-        by (force split: if_splits elim!: Posix_elims simp add: Sequ_def der_correctness Der_def)   
-  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 Posix.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 Posix_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 Posix.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 Posix.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(auto elim!: Posix_elims(1-5) simp add: der_correctness Der_def intro: Posix.intros)
-        apply(rotate_tac 3)
-        apply(erule_tac Posix_elims(6))
-        apply (simp add: Posix.intros(6))
-        using Posix.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 Posix1)
-          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 Posix.intros)
-        then show "(c # s) \<in> STAR r \<rightarrow> injval (STAR r) c v" using cons by(simp)
-    qed
-qed
-
-
-section {* The Lexer by Sulzmann and Lu  *}
-
-fun 
-  lexer :: "rexp \<Rightarrow> string \<Rightarrow> val option"
-where
-  "lexer r [] = (if nullable r then Some(mkeps r) else None)"
-| "lexer r (c#s) = (case (lexer (der c r) s) of  
-                    None \<Rightarrow> None
-                  | Some(v) \<Rightarrow> Some(injval r c v))"
-
-
-lemma lexer_correct_None:
-  shows "s \<notin> L r \<longleftrightarrow> lexer r s = None"
-using assms
-apply(induct s arbitrary: r)
-apply(simp add: nullable_correctness)
-apply(drule_tac x="der a r" in meta_spec)
-apply(auto simp add: der_correctness Der_def)
-done
-
-lemma lexer_correct_Some:
-  shows "s \<in> L r \<longleftrightarrow> (\<exists>!v. lexer r s = Some(v) \<and> s \<in> r \<rightarrow> v)"
-using assms
-apply(induct s arbitrary: r)
-apply(auto simp add: Posix_mkeps nullable_correctness)[1]
-apply(drule_tac x="der a r" in meta_spec)
-apply(simp add: der_correctness Der_def)
-apply(rule iffI)
-apply(auto intro: Posix_injval simp add: Posix1(1))
-done 
-
-
-end
\ No newline at end of file