# HG changeset patch
# User Christian Urban <urbanc@in.tum.de>
# Date 1500472546 -3600
# Node ID fff2e1b40dfc11de30c8cc687b3286abf030c3d3
# Parent  d36be1e356c0ee01b9885e6fbf5f394af9c1e101
updated

diff -r d36be1e356c0 -r fff2e1b40dfc thys/Exercises.thy
--- a/thys/Exercises.thy	Tue Jul 18 18:39:20 2017 +0100
+++ b/thys/Exercises.thy	Wed Jul 19 14:55:46 2017 +0100
@@ -1,5 +1,5 @@
 theory Exercises
-  imports Lexer "~~/src/HOL/Library/Infinite_Set"
+  imports Spec "~~/src/HOL/Library/Infinite_Set"
 begin
 
 section {* Some Fun Facts *}
diff -r d36be1e356c0 -r fff2e1b40dfc thys/Journal/Paper.thy
--- a/thys/Journal/Paper.thy	Tue Jul 18 18:39:20 2017 +0100
+++ b/thys/Journal/Paper.thy	Wed Jul 19 14:55:46 2017 +0100
@@ -52,6 +52,7 @@
   injval ("inj _ _ _" [79,77,79] 76) and 
   mkeps ("mkeps _" [79] 76) and 
   length ("len _" [73] 73) and
+  intlen ("len _" [73] 73) and
  
   Prf ("_ : _" [75,75] 75) and  
   Posix ("'(_, _') \<rightarrow> _" [63,75,75] 75) and
@@ -71,7 +72,7 @@
   lex_list ("_ \<prec>\<^bsub>lex\<^esub> _") and
   PosOrd ("_ \<prec>\<^bsub>_\<^esub> _" [77,77,77] 77) and
   PosOrd_ex ("_ \<prec> _" [77,77] 77) and
-  PosOrd_ex1 ("_ \<^raw:\mbox{$\preccurlyeq$}> _" [77,77] 77) and
+  PosOrd_ex_eq ("_ \<^raw:\mbox{$\preccurlyeq$}> _" [77,77] 77) and
   pflat_len ("\<parallel>_\<parallel>\<^bsub>_\<^esub>") and
   nprec ("_ \<^raw:\mbox{$\not\prec$}> _" [77,77] 77) 
 
@@ -430,9 +431,8 @@
   @{thm[mode=Axiom] Prf.intros(5)[of "c"]}\\[4mm]
   @{thm[mode=Rule] Prf.intros(2)[of "v\<^sub>1" "r\<^sub>1" "r\<^sub>2"]} \qquad 
   @{thm[mode=Rule] Prf.intros(3)[of "v\<^sub>2" "r\<^sub>1" "r\<^sub>2"]}\\[4mm]
-  @{thm[mode=Rule] Prf.intros(1)[of "v\<^sub>1" "r\<^sub>1" "v\<^sub>2" "r\<^sub>2"]}\\[4mm]
-  @{thm[mode=Axiom] Prf.intros(6)[of "r"]} \qquad  
-  @{thm[mode=Rule] Prf.intros(7)[of "v" "r" "vs"]}
+  @{thm[mode=Rule] Prf.intros(1)[of "v\<^sub>1" "r\<^sub>1" "v\<^sub>2" "r\<^sub>2"]} \qquad
+  @{thm[mode=Rule] Prf.intros(6)[of "vs"]}
   \end{tabular}
   \end{center}
 
@@ -1098,7 +1098,7 @@
 
  @{thm PosOrd_ex_def[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2"]}
 
- @{thm PosOrd_ex1_def[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2"]}
+ @{thm PosOrd_ex_eq_def[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2"]}
 
  Lemma 1
 
diff -r d36be1e356c0 -r fff2e1b40dfc thys/Lexer.thy
--- a/thys/Lexer.thy	Tue Jul 18 18:39:20 2017 +0100
+++ b/thys/Lexer.thy	Wed Jul 19 14:55:46 2017 +0100
@@ -1,349 +1,10 @@
    
 theory Lexer
-  imports Main 
+  imports Spec 
 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 Sequ_empty_string [simp]:
-  shows "A ;; {[]} = A"
-  and   "{[]} ;; A = A"
-by (simp_all add: Sequ_def)
-
-lemma Sequ_empty [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}"
-
-definition
-  Ders :: "string \<Rightarrow> string set \<Rightarrow> string set"
-where
-  "Ders s A \<equiv> {s'. s @ 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>"
-
-(* Arden's lemma *)
-
-lemma Star_cases:
-  shows "A\<star> = {[]} \<union> A ;; A\<star>"
-unfolding Sequ_def
-by (auto) (metis Star.simps)
-
-lemma Star_decomp: 
-  assumes "c # x \<in> A\<star>" 
-  shows "\<exists>s1 s2. x = s1 @ s2 \<and> c # s1 \<in> A \<and> s2 \<in> A\<star>"
-using assms
-by (induct x\<equiv>"c # x" rule: Star.induct) 
-   (auto simp add: append_eq_Cons_conv)
-
-lemma Star_Der_Sequ: 
-  shows "Der c (A\<star>) \<subseteq> (Der c A) ;; A\<star>"
-unfolding Der_def
-apply(rule subsetI)
-apply(simp)
-unfolding Sequ_def
-apply(simp)
-by(auto simp add: Sequ_def Star_decomp)
-
-
-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>"
-    using Star_Der_Sequ by auto
-  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)
-
-lemma ders_correctness:
-  shows "L (ders s r) = Ders s (L r)"
-apply(induct s arbitrary: r)
-apply(simp_all add: Ders_def der_correctness Der_def)
-done
-
-
-section {* Lemmas about ders *}
-
-lemma ders_ZERO:
-  shows "ders s (ZERO) = ZERO"
-apply(induct s)
-apply(simp_all)
-done
-
-lemma ders_ONE:
-  shows "ders s (ONE) = (if s = [] then ONE else ZERO)"
-apply(induct s)
-apply(simp_all add: ders_ZERO)
-done
-
-lemma ders_CHAR:
-  shows "ders s (CHAR c) = 
-           (if s = [c] then ONE else 
-           (if s = [] then (CHAR c) else ZERO))"
-apply(induct s)
-apply(simp_all add: ders_ZERO ders_ONE)
-done
-
-lemma  ders_ALT:
-  shows "ders s (ALT r1 r2) = ALT (ders s r1) (ders s r2)"
-apply(induct s arbitrary: r1 r2)
-apply(simp_all)
-done
-
-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"
-| "\<forall>v \<in> set vs. \<turnstile> v : r \<Longrightarrow> \<turnstile> Stars 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 Star_concat:
-  assumes "\<forall>s \<in> set ss. s \<in> A"  
-  shows "concat ss \<in> A\<star>"
-using assms by (induct ss) (auto)
-
-
-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 Star_concat)
-
-lemma Prf_Stars_append:
-  assumes "\<turnstile> Stars vs1 : STAR r" "\<turnstile> Stars vs2 : STAR r"
-  shows "\<turnstile> Stars (vs1 @ vs2) : STAR r"
-using assms
-by (auto intro!: Prf.intros elim!: Prf_elims)
-
-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_Prf1:
-  assumes "\<turnstile> v : r" 
-  shows "flat v \<in> L r"
-using assms
-by (induct) (auto simp add: Sequ_def Star_concat)
-
-lemma L_flat_Prf2:
-  assumes "s \<in> L r" 
-  shows "\<exists>v. \<turnstile> v : r \<and> flat v = s"
-using assms
-proof(induct r arbitrary: s)
-  case (STAR r s)
-  have IH: "\<And>s. s \<in> L r \<Longrightarrow> \<exists>v. \<turnstile> v : r \<and> flat v = s" by fact
-  have "s \<in> L (STAR r)" by fact
-  then obtain ss where "concat ss = s" "\<forall>s \<in> set ss. s \<in> L r"
-  using Star_string by auto
-  then obtain vs where "concat (map flat vs) = s" "\<forall>v\<in>set vs. \<turnstile> v : r"
-  using IH Star_val by blast
-  then show "\<exists>v. \<turnstile> v : STAR r \<and> flat v = s"
-  using Prf.intros(6) flat_Stars by blast
-next 
-  case (SEQ r1 r2 s)
-  then show "\<exists>v. \<turnstile> v : SEQ r1 r2 \<and> flat v = s"
-  unfolding Sequ_def L.simps by (fastforce intro: Prf.intros)
-next
-  case (ALT r1 r2 s)
-  then show "\<exists>v. \<turnstile> v : ALT r1 r2 \<and> flat v = s"
-  unfolding L.simps by (fastforce intro: Prf.intros)
-qed (auto intro: Prf.intros)
-
-lemma L_flat_Prf:
-  "L(r) = {flat v | v. \<turnstile> v : r}"
-using L_flat_Prf1 L_flat_Prf2 by blast
-
-(*
-lemma Star_values_exists:
-  assumes "s \<in> (L r)\<star>"
-  shows "\<exists>vs. concat (map flat vs) = s \<and> \<turnstile> Stars vs : STAR r"
-using assms
-apply(drule_tac Star_string)
-apply(auto)
-by (metis L_flat_Prf2 Prf.intros(6) Star_val)
-*)
-
-
-section {* Sulzmann and Lu functions *}
+section {* The Lexer Functions by Sulzmann and Lu  *}
 
 fun 
   mkeps :: "rexp \<Rightarrow> val"
@@ -363,8 +24,17 @@
 | "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)" 
 
+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))"
 
-section {* Mkeps, injval *}
+
+
+section {* Mkeps, Injval Properties *}
 
 lemma mkeps_nullable:
   assumes "nullable(r)" 
@@ -379,7 +49,6 @@
 using assms
 by (induct rule: nullable.induct) (auto)
 
-
 lemma Prf_injval:
   assumes "\<turnstile> v : der c r" 
   shows "\<turnstile> (injval r c v) : r"
@@ -396,49 +65,9 @@
 apply(auto elim!: Prf_elims intro: mkeps_flat split: if_splits)
 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 elim!: Prf_elims)
-done
-
+text {*
+  Mkeps and injval produce, or preserve, Posix values.
+*}
 
 lemma Posix_mkeps:
   assumes "nullable r"
@@ -451,74 +80,6 @@
 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)"
@@ -669,15 +230,7 @@
 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))"
+section {* Lexer Correctness *}
 
 
 lemma lexer_correct_None:
diff -r d36be1e356c0 -r fff2e1b40dfc thys/Positions.thy
--- a/thys/Positions.thy	Tue Jul 18 18:39:20 2017 +0100
+++ b/thys/Positions.thy	Wed Jul 19 14:55:46 2017 +0100
@@ -1,6 +1,6 @@
    
 theory Positions
-  imports "Lexer" 
+  imports "Spec" 
 begin
 
 section {* Positions in Values *}
@@ -163,26 +163,6 @@
 
 
 
-definition pflat_len2 :: "val \<Rightarrow> nat list => (bool * nat)"
-where
-  "pflat_len2 v p \<equiv> (if p \<in> Pos v then (True, length (flat (at v p))) else (False, 0))"
-
-instance prod :: (ord, ord) ord
-  by (rule Orderings.class.Orderings.ord.of_class.intro)
-
-
-lemma "(0, 0) < (3::nat, 2::nat)"
-
-
-
-
-definition PosOrd2:: "val \<Rightarrow> nat list \<Rightarrow> val \<Rightarrow> bool" ("_ \<sqsubset>val2 _ _" [60, 60, 59] 60)
-where
-  "v1 \<sqsubset>val2 p v2 \<equiv> (fst (pflat_len2 v1 p) > fst (pflat_len2 v2 p) \<or>
-                     snd (pflat_len2 v1 p) > fst (pflat_len2 v2 p)) \<and>
-                    (\<forall>q \<in> Pos v1 \<union> Pos v2. q \<sqsubset>lex p \<longrightarrow> pflat_len v1 q = pflat_len v2 q)"
-
-
 definition PosOrd_ex:: "val \<Rightarrow> val \<Rightarrow> bool" ("_ :\<sqsubset>val _" [60, 59] 60)
 where
   "v1 :\<sqsubset>val v2 \<equiv> \<exists>p. v1 \<sqsubset>val p v2"
@@ -567,287 +547,6 @@
 by (metis PosOrd_SeqI1 PosOrd_shorterI WW1 antisym_conv3 append_eq_append_conv assms(2))
 
 
-section {* CPT and CPTpre *}
-
-
-inductive 
-  CPrf :: "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"
-| "\<forall>v \<in> set vs. \<Turnstile> v : r \<and> flat v \<noteq> [] \<Longrightarrow> \<Turnstile> Stars vs : STAR r"
-
-lemma Prf_CPrf:
-  assumes "\<Turnstile> v : r"
-  shows "\<turnstile> v : r"
-using assms
-by (induct)(auto intro: Prf.intros)
-
-lemma CPrf_stars:
-  assumes "\<Turnstile> Stars vs : STAR r"
-  shows "\<forall>v \<in> set vs. flat v \<noteq> [] \<and> \<Turnstile> v : r"
-using assms
-apply(erule_tac CPrf.cases)
-apply(simp_all)
-done
-
-lemma CPrf_Stars_appendE:
-  assumes "\<Turnstile> Stars (vs1 @ vs2) : STAR r"
-  shows "\<Turnstile> Stars vs1 : STAR r \<and> \<Turnstile> Stars vs2 : STAR r" 
-using assms
-apply(erule_tac CPrf.cases)
-apply(auto intro: CPrf.intros elim: Prf.cases)
-done
-
-definition PT :: "rexp \<Rightarrow> string \<Rightarrow> val set"
-where "PT r s \<equiv> {v. flat v = s \<and> \<turnstile> v : r}"
-
-definition
-  "CPT r s = {v. flat v = s \<and> \<Turnstile> v : r}"
-
-definition
-  "CPTpre r s = {v. \<exists>s'. flat v @ s' = s \<and> \<Turnstile> v : r}"
-
-lemma CPT_CPTpre_subset:
-  shows "CPT r s \<subseteq> CPTpre r s"
-by(auto simp add: CPT_def CPTpre_def)
-
-lemma CPT_simps:
-  shows "CPT ZERO s = {}"
-  and   "CPT ONE s = (if s = [] then {Void} else {})"
-  and   "CPT (CHAR c) s = (if s = [c] then {Char c} else {})"
-  and   "CPT (ALT r1 r2) s = Left ` CPT r1 s \<union> Right ` CPT r2 s"
-  and   "CPT (SEQ r1 r2) s = 
-          {Seq v1 v2 | v1 v2. flat v1 @ flat v2 = s \<and> v1 \<in> CPT r1 (flat v1) \<and> v2 \<in> CPT r2 (flat v2)}"
-  and   "CPT (STAR r) s = 
-          Stars ` {vs. concat (map flat vs) = s \<and> (\<forall>v \<in> set vs. v \<in> CPT r (flat v) \<and> flat v \<noteq> [])}"
-apply -
-apply(auto simp add: CPT_def intro: CPrf.intros)[1]
-apply(erule CPrf.cases)
-apply(simp_all)[6]
-apply(auto simp add: CPT_def intro: CPrf.intros)[1]
-apply(erule CPrf.cases)
-apply(simp_all)[6]
-apply(erule CPrf.cases)
-apply(simp_all)[6]
-apply(auto simp add: CPT_def intro: CPrf.intros)[1]
-apply(erule CPrf.cases)
-apply(simp_all)[6]
-apply(erule CPrf.cases)
-apply(simp_all)[6]
-apply(auto simp add: CPT_def intro: CPrf.intros)[1]
-apply(erule CPrf.cases)
-apply(simp_all)[6]
-apply(auto simp add: CPT_def intro: CPrf.intros)[1]
-apply(erule CPrf.cases)
-apply(simp_all)[6]
-(* STAR case *)
-apply(auto simp add: CPT_def intro: CPrf.intros)[1]
-apply(erule CPrf.cases)
-apply(simp_all)[6]
-done
-
-(*
-lemma CPTpre_STAR_finite:
-  assumes "\<And>s. finite (CPT r s)"
-  shows "finite (CPT (STAR r) s)"
-apply(induct s rule: length_induct)
-apply(case_tac xs)
-apply(simp)
-apply(simp add: CPT_simps)
-apply(auto)
-apply(rule finite_imageI)
-using assms
-thm finite_Un
-prefer 2
-apply(simp add: CPT_simps)
-apply(rule finite_imageI)
-apply(rule finite_subset)
-apply(rule CPTpre_subsets)
-apply(simp)
-apply(rule_tac B="(\<lambda>(v, vs). Stars (v#vs)) ` {(v, vs). v \<in> CPTpre r (a#list) \<and> flat v \<noteq> [] \<and> Stars vs \<in> CPTpre (STAR r) (drop (length (flat v)) (a#list))}" in finite_subset)
-apply(auto)[1]
-apply(rule finite_imageI)
-apply(simp add: Collect_case_prod_Sigma)
-apply(rule finite_SigmaI)
-apply(rule assms)
-apply(case_tac "flat v = []")
-apply(simp)
-apply(drule_tac x="drop (length (flat v)) (a # list)" in spec)
-apply(simp)
-apply(auto)[1]
-apply(rule test)
-apply(simp)
-done
-
-lemma CPTpre_subsets:
-  "CPTpre ZERO s = {}"
-  "CPTpre ONE s \<subseteq> {Void}"
-  "CPTpre (CHAR c) s \<subseteq> {Char c}"
-  "CPTpre (ALT r1 r2) s \<subseteq> Left ` CPTpre r1 s \<union> Right ` CPTpre r2 s"
-  "CPTpre (SEQ r1 r2) s \<subseteq> {Seq v1 v2 | v1 v2. v1 \<in> CPTpre r1 s \<and> v2 \<in> CPTpre r2 (drop (length (flat v1)) s)}"
-  "CPTpre (STAR r) s \<subseteq> {Stars []} \<union>
-    {Stars (v#vs) | v vs. v \<in> CPTpre r s \<and> flat v \<noteq> [] \<and> Stars vs \<in> CPTpre (STAR r) (drop (length (flat v)) s)}"
-  "CPTpre (STAR r) [] = {Stars []}"
-apply(auto simp add: CPTpre_def)
-apply(erule CPrf.cases)
-apply(simp_all)
-apply(erule CPrf.cases)
-apply(simp_all)
-apply(erule CPrf.cases)
-apply(simp_all)
-apply(erule CPrf.cases)
-apply(simp_all)
-apply(erule CPrf.cases)
-apply(simp_all)
-
-apply(erule CPrf.cases)
-apply(simp_all)
-apply(erule CPrf.cases)
-apply(simp_all)
-apply(rule CPrf.intros)
-done
-
-
-lemma CPTpre_simps:
-  shows "CPTpre ONE s = {Void}"
-  and   "CPTpre (CHAR c) (d#s) = (if c = d then {Char c} else {})"
-  and   "CPTpre (ALT r1 r2) s = Left ` CPTpre r1 s \<union> Right ` CPTpre r2 s"
-  and   "CPTpre (SEQ r1 r2) s = 
-          {Seq v1 v2 | v1 v2. v1 \<in> CPTpre r1 s \<and> v2 \<in> CPTpre r2 (drop (length (flat v1)) s)}"
-apply -
-apply(rule subset_antisym)
-apply(rule CPTpre_subsets)
-apply(auto simp add: CPTpre_def intro: "CPrf.intros")[1]
-apply(case_tac "c = d")
-apply(simp)
-apply(rule subset_antisym)
-apply(rule CPTpre_subsets)
-apply(auto simp add: CPTpre_def intro: CPrf.intros)[1]
-apply(simp)
-apply(auto simp add: CPTpre_def intro: CPrf.intros)[1]
-apply(erule CPrf.cases)
-apply(simp_all)
-apply(rule subset_antisym)
-apply(rule CPTpre_subsets)
-apply(auto simp add: CPTpre_def intro: CPrf.intros)[1]
-apply(rule subset_antisym)
-apply(rule CPTpre_subsets)
-apply(auto simp add: CPTpre_def intro: CPrf.intros)[1]
-done
-
-lemma CPT_simps:
-  shows "CPT ONE s = (if s = [] then {Void} else {})"
-  and   "CPT (CHAR c) [d] = (if c = d then {Char c} else {})"
-  and   "CPT (ALT r1 r2) s = Left ` CPT r1 s \<union> Right ` CPT r2 s"
-  and   "CPT (SEQ r1 r2) s = 
-          {Seq v1 v2 | v1 v2 s1 s2. s1 @ s2 = s \<and> v1 \<in> CPT r1 s1 \<and> v2 \<in> CPT r2 s2}"
-apply -
-apply(rule subset_antisym)
-apply(auto simp add: CPT_def)[1]
-apply(erule CPrf.cases)
-apply(simp_all)[7]
-apply(erule CPrf.cases)
-apply(simp_all)[7]
-apply(auto simp add: CPT_def intro: CPrf.intros)[1]
-apply(auto simp add: CPT_def intro: CPrf.intros)[1]
-apply(erule CPrf.cases)
-apply(simp_all)[7]
-apply(erule CPrf.cases)
-apply(simp_all)[7]
-apply(auto simp add: CPT_def image_def intro: CPrf.intros)[1]
-apply(erule CPrf.cases)
-apply(simp_all)[7]
-apply(clarify)
-apply blast
-apply(auto simp add: CPT_def image_def intro: CPrf.intros)[1]
-apply(erule CPrf.cases)
-apply(simp_all)[7]
-done
-
-lemma test: 
-  assumes "finite A"
-  shows "finite {vs. Stars vs \<in> A}"
-using assms
-apply(induct A)
-apply(simp)
-apply(auto)
-apply(case_tac x)
-apply(simp_all)
-done
-
-lemma CPTpre_STAR_finite:
-  assumes "\<And>s. finite (CPTpre r s)"
-  shows "finite (CPTpre (STAR r) s)"
-apply(induct s rule: length_induct)
-apply(case_tac xs)
-apply(simp)
-apply(simp add: CPTpre_subsets)
-apply(rule finite_subset)
-apply(rule CPTpre_subsets)
-apply(simp)
-apply(rule_tac B="(\<lambda>(v, vs). Stars (v#vs)) ` {(v, vs). v \<in> CPTpre r (a#list) \<and> flat v \<noteq> [] \<and> Stars vs \<in> CPTpre (STAR r) (drop (length (flat v)) (a#list))}" in finite_subset)
-apply(auto)[1]
-apply(rule finite_imageI)
-apply(simp add: Collect_case_prod_Sigma)
-apply(rule finite_SigmaI)
-apply(rule assms)
-apply(case_tac "flat v = []")
-apply(simp)
-apply(drule_tac x="drop (length (flat v)) (a # list)" in spec)
-apply(simp)
-apply(auto)[1]
-apply(rule test)
-apply(simp)
-done
-
-lemma CPTpre_finite:
-  shows "finite (CPTpre r s)"
-apply(induct r arbitrary: s)
-apply(simp add: CPTpre_subsets)
-apply(rule finite_subset)
-apply(rule CPTpre_subsets)
-apply(simp)
-apply(rule finite_subset)
-apply(rule CPTpre_subsets)
-apply(simp)
-apply(rule finite_subset)
-apply(rule CPTpre_subsets)
-apply(rule_tac B="(\<lambda>(v1, v2). Seq v1 v2) ` {(v1, v2).  v1 \<in> CPTpre r1 s \<and> v2 \<in> CPTpre r2 (drop (length (flat v1)) s)}" in finite_subset)
-apply(auto)[1]
-apply(rule finite_imageI)
-apply(simp add: Collect_case_prod_Sigma)
-apply(rule finite_subset)
-apply(rule CPTpre_subsets)
-apply(simp)
-by (simp add: CPTpre_STAR_finite)
-
-
-lemma CPT_finite:
-  shows "finite (CPT r s)"
-apply(rule finite_subset)
-apply(rule CPT_CPTpre_subset)
-apply(rule CPTpre_finite)
-done
-*)
-
-lemma Posix_CPT:
-  assumes "s \<in> r \<rightarrow> v"
-  shows "v \<in> CPT r s"
-using assms
-apply(induct rule: Posix.induct)
-apply(auto simp add: CPT_def intro: CPrf.intros elim: CPrf.cases)
-apply(rotate_tac 5)
-apply(erule CPrf.cases)
-apply(simp_all)
-apply(rule CPrf.intros)
-apply(simp_all)
-done
-
-
 section {* The Posix Value is smaller than any other Value *}
 
 
@@ -1044,81 +743,6 @@
 by (metis Posix_PosOrd_stronger less_irrefl PosOrd_def 
     PosOrd_ex_eq_def PosOrd_ex_def PosOrd_trans)
 
-(*
-lemma PosOrd_Posix_Stars:
-  assumes "(Stars vs) \<in> CPT (STAR r) (flat (Stars vs))" "\<forall>v \<in> set vs. flat v \<in> r \<rightarrow> v"
-  and "\<not>(\<exists>vs2 \<in> PT (STAR r) (flat (Stars vs)). vs2 :\<sqsubset>val (Stars vs))"
-  shows "(flat (Stars vs)) \<in> (STAR r) \<rightarrow> Stars vs" 
-using assms
-apply(induct vs)
-apply(simp)
-apply(rule Posix.intros)
-apply(simp (no_asm))
-apply(rule Posix.intros)
-apply(auto)[1]
-apply(auto simp add: CPT_def PT_def)[1]
-defer
-apply(simp)
-apply(drule meta_mp)
-apply(auto simp add: CPT_def PT_def)[1]
-apply(erule CPrf.cases)
-apply(simp_all)
-apply(drule meta_mp)
-apply(auto simp add: CPT_def PT_def)[1]
-apply(erule Prf.cases)
-apply(simp_all)
-using CPrf_stars PosOrd_irrefl apply fastforce
-apply(clarify)
-apply(drule_tac x="Stars (a#v#vsa)" in spec)
-apply(simp)
-apply(drule mp)
-apply (meson CPrf_stars Prf.intros(7) Prf_CPrf list.set_intros(1))
-apply(subst (asm) (2) PosOrd_ex_def)
-apply(simp)
-apply (metis flat.simps(7) flat_Stars PosOrd_StarsI2 PosOrd_ex_def)
-apply(auto simp add: CPT_def PT_def)[1]
-using CPrf_stars apply auto[1]
-apply(auto)[1]
-apply(auto simp add: CPT_def PT_def)[1]
-apply(subgoal_tac "\<exists>vA. flat vA = flat a @ s\<^sub>3 \<and> \<turnstile> vA : r")
-prefer 2
-apply (meson L_flat_Prf2)
-apply(subgoal_tac "\<exists>vB. flat (Stars vB) = s\<^sub>4 \<and> \<turnstile> (Stars vB) : (STAR r)")
-apply(clarify)
-apply(drule_tac x="Stars (vA # vB)" in spec)
-apply(simp)
-apply(drule mp)
-using Prf.intros(7) apply blast
-apply(subst (asm) (2) PosOrd_ex_def)
-apply(simp)
-prefer 2
-apply(simp)
-using Star_values_exists apply blast
-prefer 2
-apply(drule meta_mp)
-apply(erule CPrf.cases)
-apply(simp_all)
-apply(drule meta_mp)
-apply(auto)[1]
-prefer 2
-apply(simp)
-apply(erule CPrf.cases)
-apply(simp_all)
-apply(clarify)
-apply(rotate_tac 3)
-apply(erule Prf.cases)
-apply(simp_all)
-apply (metis CPrf_stars intlen.cases less_irrefl list.set_intros(1) PosOrd_def PosOrd_ex_def)
-apply(drule_tac x="Stars (v#va#vsb)" in spec)
-apply(drule mp)
-apply (simp add: Posix1a Prf.intros(7))
-apply(simp)
-apply(subst (asm) (2) PosOrd_ex_def)
-apply(simp)
-apply (metis flat.simps(7) flat_Stars PosOrd_StarsI2 PosOrd_ex_def)
-by (metis PosOrd_StarsI PosOrd_ex_def PosOrd_spreI append_assoc append_self_conv flat.simps(7) flat_Stars prefix_list_def sprefix_list_def)
-*)
-
 
 lemma test2: 
   assumes "\<forall>v \<in> set vs. flat v \<in> r \<rightarrow> v \<and> flat v \<noteq> []"
@@ -1163,7 +787,7 @@
         apply(simp_all)
         apply(drule_tac x="Stars (v # vs)" in bspec)
         apply(simp add: PT_def CPT_def)
-        using Posix1a Prf.intros(6) calculation
+        using Posix_Prf Prf.intros(6) calculation
         apply(rule_tac Prf.intros)
         apply(simp add:)
         apply (simp add: PosOrd_StarsI2)
diff -r d36be1e356c0 -r fff2e1b40dfc thys/ROOT
--- a/thys/ROOT	Tue Jul 18 18:39:20 2017 +0100
+++ b/thys/ROOT	Wed Jul 19 14:55:46 2017 +0100
@@ -1,7 +1,8 @@
 session "Lex" = HOL +
   theories [document = false]
-	"Lexer"
-        "LexerExt" 
+        "Spec"
+        "Lexer"
+        "LexerExt"
         "Simplifying"
         (*"Sulzmann"*) 
         "Positions"
diff -r d36be1e356c0 -r fff2e1b40dfc thys/Spec.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/thys/Spec.thy	Wed Jul 19 14:55:46 2017 +0100
@@ -0,0 +1,746 @@
+   
+theory Spec
+  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 Sequ_empty_string [simp]:
+  shows "A ;; {[]} = A"
+  and   "{[]} ;; A = A"
+by (simp_all add: Sequ_def)
+
+lemma Sequ_empty [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}"
+
+definition
+  Ders :: "string \<Rightarrow> string set \<Rightarrow> string set"
+where
+  "Ders s A \<equiv> {s'. s @ 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>"
+
+(* Arden's lemma *)
+
+lemma Star_cases:
+  shows "A\<star> = {[]} \<union> A ;; A\<star>"
+unfolding Sequ_def
+by (auto) (metis Star.simps)
+
+lemma Star_decomp: 
+  assumes "c # x \<in> A\<star>" 
+  shows "\<exists>s1 s2. x = s1 @ s2 \<and> c # s1 \<in> A \<and> s2 \<in> A\<star>"
+using assms
+by (induct x\<equiv>"c # x" rule: Star.induct) 
+   (auto simp add: append_eq_Cons_conv)
+
+lemma Star_Der_Sequ: 
+  shows "Der c (A\<star>) \<subseteq> (Der c A) ;; A\<star>"
+unfolding Der_def Sequ_def
+by(auto simp add: Star_decomp)
+
+
+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>"
+    using Star_Der_Sequ by auto
+  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)
+
+lemma ders_correctness:
+  shows "L (ders s r) = Ders s (L r)"
+apply(induct s arbitrary: r)
+apply(simp_all add: Ders_def der_correctness Der_def)
+done
+
+
+section {* Lemmas about ders *}
+
+lemma ders_ZERO:
+  shows "ders s (ZERO) = ZERO"
+apply(induct s)
+apply(simp_all)
+done
+
+lemma ders_ONE:
+  shows "ders s (ONE) = (if s = [] then ONE else ZERO)"
+apply(induct s)
+apply(simp_all add: ders_ZERO)
+done
+
+lemma ders_CHAR:
+  shows "ders s (CHAR c) = 
+           (if s = [c] then ONE else 
+           (if s = [] then (CHAR c) else ZERO))"
+apply(induct s)
+apply(simp_all add: ders_ZERO ders_ONE)
+done
+
+lemma  ders_ALT:
+  shows "ders s (ALT r1 r2) = ALT (ders s r1) (ders s r2)"
+apply(induct s arbitrary: r1 r2)
+apply(simp_all)
+done
+
+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"
+| "\<forall>v \<in> set vs. \<turnstile> v : r \<Longrightarrow> \<turnstile> Stars 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 Star_concat:
+  assumes "\<forall>s \<in> set ss. s \<in> A"  
+  shows "concat ss \<in> A\<star>"
+using assms by (induct ss) (auto)
+
+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(rule_tac x="[]" in exI)
+apply(simp)
+apply(rule_tac x="v#vs" in exI)
+apply(simp)
+done
+
+lemma Prf_Stars_append:
+  assumes "\<turnstile> Stars vs1 : STAR r" "\<turnstile> Stars vs2 : STAR r"
+  shows "\<turnstile> Stars (vs1 @ vs2) : STAR r"
+using assms
+by (auto intro!: Prf.intros elim!: Prf_elims)
+
+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 Star_concat)
+
+
+lemma L_flat_Prf1:
+  assumes "\<turnstile> v : r" 
+  shows "flat v \<in> L r"
+using assms
+by (induct) (auto simp add: Sequ_def Star_concat)
+
+lemma L_flat_Prf2:
+  assumes "s \<in> L r" 
+  shows "\<exists>v. \<turnstile> v : r \<and> flat v = s"
+using assms
+proof(induct r arbitrary: s)
+  case (STAR r s)
+  have IH: "\<And>s. s \<in> L r \<Longrightarrow> \<exists>v. \<turnstile> v : r \<and> flat v = s" by fact
+  have "s \<in> L (STAR r)" by fact
+  then obtain ss where "concat ss = s" "\<forall>s \<in> set ss. s \<in> L r"
+  using Star_string by auto
+  then obtain vs where "concat (map flat vs) = s" "\<forall>v\<in>set vs. \<turnstile> v : r"
+  using IH Star_val by blast
+  then show "\<exists>v. \<turnstile> v : STAR r \<and> flat v = s"
+  using Prf.intros(6) flat_Stars by blast
+next 
+  case (SEQ r1 r2 s)
+  then show "\<exists>v. \<turnstile> v : SEQ r1 r2 \<and> flat v = s"
+  unfolding Sequ_def L.simps by (fastforce intro: Prf.intros)
+next
+  case (ALT r1 r2 s)
+  then show "\<exists>v. \<turnstile> v : ALT r1 r2 \<and> flat v = s"
+  unfolding L.simps by (fastforce intro: Prf.intros)
+qed (auto intro: Prf.intros)
+
+lemma L_flat_Prf:
+  shows "L(r) = {flat v | v. \<turnstile> v : r}"
+using L_flat_Prf1 L_flat_Prf2 by blast
+
+section {* CPT and CPTpre *}
+
+
+inductive 
+  CPrf :: "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"
+| "\<forall>v \<in> set vs. \<Turnstile> v : r \<and> flat v \<noteq> [] \<Longrightarrow> \<Turnstile> Stars vs : STAR r"
+
+lemma Prf_CPrf:
+  assumes "\<Turnstile> v : r"
+  shows "\<turnstile> v : r"
+using assms
+by (induct)(auto intro: Prf.intros)
+
+lemma CPrf_stars:
+  assumes "\<Turnstile> Stars vs : STAR r"
+  shows "\<forall>v \<in> set vs. flat v \<noteq> [] \<and> \<Turnstile> v : r"
+using assms
+apply(erule_tac CPrf.cases)
+apply(simp_all)
+done
+
+lemma CPrf_Stars_appendE:
+  assumes "\<Turnstile> Stars (vs1 @ vs2) : STAR r"
+  shows "\<Turnstile> Stars vs1 : STAR r \<and> \<Turnstile> Stars vs2 : STAR r" 
+using assms
+apply(erule_tac CPrf.cases)
+apply(auto intro: CPrf.intros elim: Prf.cases)
+done
+
+definition PT :: "rexp \<Rightarrow> string \<Rightarrow> val set"
+where "PT r s \<equiv> {v. flat v = s \<and> \<turnstile> v : r}"
+
+definition
+  "CPT r s = {v. flat v = s \<and> \<Turnstile> v : r}"
+
+definition
+  "CPTpre r s = {v. \<exists>s'. flat v @ s' = s \<and> \<Turnstile> v : r}"
+
+lemma CPT_CPTpre_subset:
+  shows "CPT r s \<subseteq> CPTpre r s"
+by(auto simp add: CPT_def CPTpre_def)
+
+lemma CPT_simps:
+  shows "CPT ZERO s = {}"
+  and   "CPT ONE s = (if s = [] then {Void} else {})"
+  and   "CPT (CHAR c) s = (if s = [c] then {Char c} else {})"
+  and   "CPT (ALT r1 r2) s = Left ` CPT r1 s \<union> Right ` CPT r2 s"
+  and   "CPT (SEQ r1 r2) s = 
+          {Seq v1 v2 | v1 v2. flat v1 @ flat v2 = s \<and> v1 \<in> CPT r1 (flat v1) \<and> v2 \<in> CPT r2 (flat v2)}"
+  and   "CPT (STAR r) s = 
+          Stars ` {vs. concat (map flat vs) = s \<and> (\<forall>v \<in> set vs. v \<in> CPT r (flat v) \<and> flat v \<noteq> [])}"
+apply -
+apply(auto simp add: CPT_def intro: CPrf.intros)[1]
+apply(erule CPrf.cases)
+apply(simp_all)[6]
+apply(auto simp add: CPT_def intro: CPrf.intros)[1]
+apply(erule CPrf.cases)
+apply(simp_all)[6]
+apply(erule CPrf.cases)
+apply(simp_all)[6]
+apply(auto simp add: CPT_def intro: CPrf.intros)[1]
+apply(erule CPrf.cases)
+apply(simp_all)[6]
+apply(erule CPrf.cases)
+apply(simp_all)[6]
+apply(auto simp add: CPT_def intro: CPrf.intros)[1]
+apply(erule CPrf.cases)
+apply(simp_all)[6]
+apply(auto simp add: CPT_def intro: CPrf.intros)[1]
+apply(erule CPrf.cases)
+apply(simp_all)[6]
+(* STAR case *)
+apply(auto simp add: CPT_def intro: CPrf.intros)[1]
+apply(erule CPrf.cases)
+apply(simp_all)[6]
+done
+
+
+
+section {* Our 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 Posix_Prf:
+  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 elim!: Prf_elims)
+done
+
+text {*
+  Our Posix definition determines a unique value.
+*}
+
+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
+
+
+text {*
+  Our POSIX value is a canonical value.
+*}
+
+lemma Posix_CPT:
+  assumes "s \<in> r \<rightarrow> v"
+  shows "v \<in> CPT r s"
+using assms
+apply(induct rule: Posix.induct)
+apply(auto simp add: CPT_def intro: CPrf.intros elim: CPrf.cases)
+apply(rotate_tac 5)
+apply(erule CPrf.cases)
+apply(simp_all)
+apply(rule CPrf.intros)
+apply(simp_all)
+done
+
+
+
+(*
+lemma CPTpre_STAR_finite:
+  assumes "\<And>s. finite (CPT r s)"
+  shows "finite (CPT (STAR r) s)"
+apply(induct s rule: length_induct)
+apply(case_tac xs)
+apply(simp)
+apply(simp add: CPT_simps)
+apply(auto)
+apply(rule finite_imageI)
+using assms
+thm finite_Un
+prefer 2
+apply(simp add: CPT_simps)
+apply(rule finite_imageI)
+apply(rule finite_subset)
+apply(rule CPTpre_subsets)
+apply(simp)
+apply(rule_tac B="(\<lambda>(v, vs). Stars (v#vs)) ` {(v, vs). v \<in> CPTpre r (a#list) \<and> flat v \<noteq> [] \<and> Stars vs \<in> CPTpre (STAR r) (drop (length (flat v)) (a#list))}" in finite_subset)
+apply(auto)[1]
+apply(rule finite_imageI)
+apply(simp add: Collect_case_prod_Sigma)
+apply(rule finite_SigmaI)
+apply(rule assms)
+apply(case_tac "flat v = []")
+apply(simp)
+apply(drule_tac x="drop (length (flat v)) (a # list)" in spec)
+apply(simp)
+apply(auto)[1]
+apply(rule test)
+apply(simp)
+done
+
+lemma CPTpre_subsets:
+  "CPTpre ZERO s = {}"
+  "CPTpre ONE s \<subseteq> {Void}"
+  "CPTpre (CHAR c) s \<subseteq> {Char c}"
+  "CPTpre (ALT r1 r2) s \<subseteq> Left ` CPTpre r1 s \<union> Right ` CPTpre r2 s"
+  "CPTpre (SEQ r1 r2) s \<subseteq> {Seq v1 v2 | v1 v2. v1 \<in> CPTpre r1 s \<and> v2 \<in> CPTpre r2 (drop (length (flat v1)) s)}"
+  "CPTpre (STAR r) s \<subseteq> {Stars []} \<union>
+    {Stars (v#vs) | v vs. v \<in> CPTpre r s \<and> flat v \<noteq> [] \<and> Stars vs \<in> CPTpre (STAR r) (drop (length (flat v)) s)}"
+  "CPTpre (STAR r) [] = {Stars []}"
+apply(auto simp add: CPTpre_def)
+apply(erule CPrf.cases)
+apply(simp_all)
+apply(erule CPrf.cases)
+apply(simp_all)
+apply(erule CPrf.cases)
+apply(simp_all)
+apply(erule CPrf.cases)
+apply(simp_all)
+apply(erule CPrf.cases)
+apply(simp_all)
+
+apply(erule CPrf.cases)
+apply(simp_all)
+apply(erule CPrf.cases)
+apply(simp_all)
+apply(rule CPrf.intros)
+done
+
+
+lemma CPTpre_simps:
+  shows "CPTpre ONE s = {Void}"
+  and   "CPTpre (CHAR c) (d#s) = (if c = d then {Char c} else {})"
+  and   "CPTpre (ALT r1 r2) s = Left ` CPTpre r1 s \<union> Right ` CPTpre r2 s"
+  and   "CPTpre (SEQ r1 r2) s = 
+          {Seq v1 v2 | v1 v2. v1 \<in> CPTpre r1 s \<and> v2 \<in> CPTpre r2 (drop (length (flat v1)) s)}"
+apply -
+apply(rule subset_antisym)
+apply(rule CPTpre_subsets)
+apply(auto simp add: CPTpre_def intro: "CPrf.intros")[1]
+apply(case_tac "c = d")
+apply(simp)
+apply(rule subset_antisym)
+apply(rule CPTpre_subsets)
+apply(auto simp add: CPTpre_def intro: CPrf.intros)[1]
+apply(simp)
+apply(auto simp add: CPTpre_def intro: CPrf.intros)[1]
+apply(erule CPrf.cases)
+apply(simp_all)
+apply(rule subset_antisym)
+apply(rule CPTpre_subsets)
+apply(auto simp add: CPTpre_def intro: CPrf.intros)[1]
+apply(rule subset_antisym)
+apply(rule CPTpre_subsets)
+apply(auto simp add: CPTpre_def intro: CPrf.intros)[1]
+done
+
+lemma CPT_simps:
+  shows "CPT ONE s = (if s = [] then {Void} else {})"
+  and   "CPT (CHAR c) [d] = (if c = d then {Char c} else {})"
+  and   "CPT (ALT r1 r2) s = Left ` CPT r1 s \<union> Right ` CPT r2 s"
+  and   "CPT (SEQ r1 r2) s = 
+          {Seq v1 v2 | v1 v2 s1 s2. s1 @ s2 = s \<and> v1 \<in> CPT r1 s1 \<and> v2 \<in> CPT r2 s2}"
+apply -
+apply(rule subset_antisym)
+apply(auto simp add: CPT_def)[1]
+apply(erule CPrf.cases)
+apply(simp_all)[7]
+apply(erule CPrf.cases)
+apply(simp_all)[7]
+apply(auto simp add: CPT_def intro: CPrf.intros)[1]
+apply(auto simp add: CPT_def intro: CPrf.intros)[1]
+apply(erule CPrf.cases)
+apply(simp_all)[7]
+apply(erule CPrf.cases)
+apply(simp_all)[7]
+apply(auto simp add: CPT_def image_def intro: CPrf.intros)[1]
+apply(erule CPrf.cases)
+apply(simp_all)[7]
+apply(clarify)
+apply blast
+apply(auto simp add: CPT_def image_def intro: CPrf.intros)[1]
+apply(erule CPrf.cases)
+apply(simp_all)[7]
+done
+
+lemma test: 
+  assumes "finite A"
+  shows "finite {vs. Stars vs \<in> A}"
+using assms
+apply(induct A)
+apply(simp)
+apply(auto)
+apply(case_tac x)
+apply(simp_all)
+done
+
+lemma CPTpre_STAR_finite:
+  assumes "\<And>s. finite (CPTpre r s)"
+  shows "finite (CPTpre (STAR r) s)"
+apply(induct s rule: length_induct)
+apply(case_tac xs)
+apply(simp)
+apply(simp add: CPTpre_subsets)
+apply(rule finite_subset)
+apply(rule CPTpre_subsets)
+apply(simp)
+apply(rule_tac B="(\<lambda>(v, vs). Stars (v#vs)) ` {(v, vs). v \<in> CPTpre r (a#list) \<and> flat v \<noteq> [] \<and> Stars vs \<in> CPTpre (STAR r) (drop (length (flat v)) (a#list))}" in finite_subset)
+apply(auto)[1]
+apply(rule finite_imageI)
+apply(simp add: Collect_case_prod_Sigma)
+apply(rule finite_SigmaI)
+apply(rule assms)
+apply(case_tac "flat v = []")
+apply(simp)
+apply(drule_tac x="drop (length (flat v)) (a # list)" in spec)
+apply(simp)
+apply(auto)[1]
+apply(rule test)
+apply(simp)
+done
+
+lemma CPTpre_finite:
+  shows "finite (CPTpre r s)"
+apply(induct r arbitrary: s)
+apply(simp add: CPTpre_subsets)
+apply(rule finite_subset)
+apply(rule CPTpre_subsets)
+apply(simp)
+apply(rule finite_subset)
+apply(rule CPTpre_subsets)
+apply(simp)
+apply(rule finite_subset)
+apply(rule CPTpre_subsets)
+apply(rule_tac B="(\<lambda>(v1, v2). Seq v1 v2) ` {(v1, v2).  v1 \<in> CPTpre r1 s \<and> v2 \<in> CPTpre r2 (drop (length (flat v1)) s)}" in finite_subset)
+apply(auto)[1]
+apply(rule finite_imageI)
+apply(simp add: Collect_case_prod_Sigma)
+apply(rule finite_subset)
+apply(rule CPTpre_subsets)
+apply(simp)
+by (simp add: CPTpre_STAR_finite)
+
+
+lemma CPT_finite:
+  shows "finite (CPT r s)"
+apply(rule finite_subset)
+apply(rule CPT_CPTpre_subset)
+apply(rule CPTpre_finite)
+done
+*)
+
+lemma test2: 
+  assumes "\<forall>v \<in> set vs. flat v \<in> r \<rightarrow> v \<and> flat v \<noteq> []"
+  shows "(Stars vs) \<in> CPT (STAR r) (flat (Stars vs))" 
+using assms
+apply(induct vs)
+apply(auto simp add: CPT_def)
+apply(rule CPrf.intros)
+apply(simp)
+apply(rule CPrf.intros)
+apply(simp_all)
+by (metis (no_types, lifting) CPT_def Posix_CPT mem_Collect_eq)
+
+
+end
\ No newline at end of file