# HG changeset patch
# User cu
# Date 1507410976 -3600
# Node ID a3134f7de0654036b1387c8021929a10f44bc1c9
# Parent  deea42c83c9e541ca8649ea0302299a1fa8976ea
updated

diff -r deea42c83c9e -r a3134f7de065 thys/LexerExt.thy
--- a/thys/LexerExt.thy	Thu Oct 05 12:45:13 2017 +0100
+++ b/thys/LexerExt.thy	Sat Oct 07 22:16:16 2017 +0100
@@ -1,613 +1,23 @@
    
 theory LexerExt
-  imports Main
+  imports SpecExt 
 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 [simp]:
-  shows "A ;; {[]} = A"
-  and   "{[]} ;; A = A"
-by (simp_all add: Sequ_def)
-
-lemma Sequ_null [simp]:
-  shows "A ;; {} = {}"
-  and   "{} ;; A = {}"
-by (simp_all add: Sequ_def)
-
-lemma Sequ_assoc: 
-  shows "A ;; (B ;; C) = (A ;; B) ;; C"
-apply(auto simp add: Sequ_def)
-apply(metis append_assoc)
-apply(metis)
-done
-
-lemma Sequ_UNION: 
-  shows "(\<Union>x\<in>A. B ;; C x) = B ;; (\<Union>x\<in>A. C x)"
-by (auto simp 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_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)
-
-lemma Der_union [simp]:
-  shows "Der c (A \<union> B) = Der c A \<union> Der c B"
-unfolding Der_def
-by auto
-
-lemma Der_UNION: 
-  shows "Der c (\<Union>x\<in>A. B x) = (\<Union>x\<in>A. Der c (B x))"
-by (auto simp add: Der_def)
-
-
-section {* Power operation for Sets *}
-
-fun 
-  Pow :: "string set \<Rightarrow> nat \<Rightarrow> string set" ("_ \<up> _" [101, 102] 101)
-where
-   "A \<up> 0 = {[]}"
-|  "A \<up> (Suc n) = A ;; (A \<up> n)"
-
-lemma Pow_empty [simp]:
-  shows "[] \<in> A \<up> n \<longleftrightarrow> (n = 0 \<or> [] \<in> A)"
-by(induct n) (auto simp add: Sequ_def)
-
-lemma Der_Pow [simp]:
-  shows "Der c (A \<up> (Suc n)) = 
-     (Der c A) ;; (A \<up> n) \<union> (if [] \<in> A then Der c (A \<up> n) else {})"
-unfolding Der_def Sequ_def 
-by(auto simp add: Cons_eq_append_conv Sequ_def)
-
-lemma Der_Pow_subseteq:
-  assumes "[] \<in> A"  
-  shows "Der c (A \<up> n) \<subseteq> (Der c A) ;; (A \<up> n)"
-using assms
-apply(induct n)
-apply(simp add: Sequ_def Der_def)
-apply(simp)
-apply(rule conjI)
-apply (smt Sequ_def append_Nil2 mem_Collect_eq Sequ_assoc subsetI)
-apply(subgoal_tac "((Der c A) ;; (A \<up> n)) \<subseteq> ((Der c A) ;; (A ;; (A \<up> n)))")
-apply(auto)[1]
-by (smt Sequ_def append_Nil2 mem_Collect_eq Sequ_assoc subsetI)
-
-lemma test:
-  shows "(\<Union>x\<le>Suc n. Der c (A \<up> x)) = (\<Union>x\<le>n. Der c A ;; A \<up> x)"
-apply(induct n)
-apply(simp)
-apply(auto)[1]
-apply(case_tac xa)
-apply(simp)
-apply(simp)
-apply(auto)[1]
-apply(case_tac "[] \<in> A")
-apply(simp)
-apply(simp)
-by (smt Der_Pow Der_Pow_subseteq UN_insert atMost_Suc sup.orderE sup_bot.right_neutral)
-
-lemma test2:
-  shows "(\<Union>x\<in>(Suc ` A). Der c (B \<up> x)) = (\<Union>x\<in>A. Der c B ;; B \<up> x)"
-apply(auto)[1]
-apply(case_tac "[] \<in> B")
-apply(simp)
-using Der_Pow_subseteq apply blast
-apply(simp)
-done
-
-
-section {* Kleene Star for Languages *}
-
-definition 
-  Star :: "string set \<Rightarrow> string set" ("_\<star>" [101] 102) where
-  "A\<star> = (\<Union>n. A \<up> n)"
-
-lemma Star_empty [intro]:
-  shows "[] \<in> A\<star>"
-unfolding Star_def
-by auto
-
-lemma Star_step [intro]:
-  assumes "s1 \<in> A" "s2 \<in> A\<star>"
-  shows "s1 @ s2 \<in> A\<star>"
-proof -
-  from assms obtain n where "s1 \<in>A" "s2 \<in> A \<up> n"
-  unfolding Star_def by auto
-  then have "s1 @ s2 \<in> A ;; (A \<up> n)"
-  by (auto simp add: Sequ_def)
-  then have "s1 @ s2 \<in> A \<up> (Suc n)"
-  by simp
-  then show "s1 @ s2 \<in> A\<star>" 
-  unfolding Star_def 
-  by (auto simp del: Pow.simps)
-qed
-
-lemma star_cases:
-  shows "A\<star> = {[]} \<union> A ;; A\<star>"
-unfolding Star_def
-apply(simp add: Sequ_def)
-apply(auto)
-apply(case_tac xa)
-apply(auto simp add: Sequ_def)
-apply(rule_tac x="Suc xa" in exI)
-apply(auto simp add: Sequ_def)
-done
-
-lemma Der_Star1:
-  shows "Der c (A ;; A\<star>) = (Der c A) ;; A\<star>"
-proof -
-  have "(Der c A) ;; A\<star> = (Der c A) ;; (\<Union>n. A \<up> n)"
-  unfolding Star_def by simp
-  also have "... = (\<Union>n. Der c A ;; A \<up> n)"
-  unfolding Sequ_UNION by simp
-  also have "... = (\<Union>x\<in>(Suc ` UNIV). Der c (A \<up> x))"
-  unfolding test2 by simp
-  also have "... = (\<Union>n. Der c (A \<up> (Suc n)))"
-  by (simp)
-  also have "... = Der c (\<Union>n. A \<up> (Suc n))"
-  unfolding Der_UNION by simp
-  also have "... = Der c (A ;; (\<Union>n. A \<up> n))"
-  by (simp only: Pow.simps Sequ_UNION)
-  finally show "Der c (A ;; A\<star>) = (Der c A) ;; A\<star>"
-  unfolding Star_def[symmetric] by blast 
-qed
-
-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>"
-    using Der_Star1 by simp
-  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
-| UPNTIMES rexp nat
-| NTIMES rexp nat
-| FROMNTIMES rexp nat
-| NMTIMES rexp nat nat
-| PLUS 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>"
-| "L (UPNTIMES r n) = (\<Union>i\<in> {..n} . (L r) \<up> i)"
-| "L (NTIMES r n) = (L r) \<up> n"
-| "L (FROMNTIMES r n) = (\<Union>i\<in> {n..} . (L r) \<up> i)"
-| "L (NMTIMES r n m) = (\<Union>i\<in>{n..m} . (L r) \<up> i)" 
-| "L (PLUS r) = (\<Union>i\<in> {1..} . (L r) \<up> i)"
-
-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"
-| "nullable (UPNTIMES r n) = True"
-| "nullable (NTIMES r n) = (if n = 0 then True else nullable r)"
-| "nullable (FROMNTIMES r n) = (if n = 0 then True else nullable r)"
-| "nullable (NMTIMES r n m) = (if m < n then False else (if n = 0 then True else nullable r))"
-| "nullable (PLUS r) = (nullable r)"
-
-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)"
-| "der c (UPNTIMES r 0) = ZERO"
-| "der c (UPNTIMES r (Suc n)) = SEQ (der c r) (UPNTIMES r n)"
-| "der c (NTIMES r 0) = ZERO"
-| "der c (NTIMES r (Suc n)) = SEQ (der c r) (NTIMES r n)"
-| "der c (FROMNTIMES r 0) = SEQ (der c r) (FROMNTIMES r 0)"
-| "der c (FROMNTIMES r (Suc n)) = SEQ (der c r) (FROMNTIMES r n)"
-| "der c (NMTIMES r n m) = 
-     (if m < n then ZERO 
-      else (if n = 0 then (if m = 0 then ZERO else 
-                           SEQ (der c r) (UPNTIMES r (m - 1))) else 
-                           SEQ (der c r) (NMTIMES r (n - 1) (m - 1))))" 
-| "der c (PLUS 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)"
-apply(induct r) 
-apply(auto simp add: Sequ_def) 
-done
-
-lemma Der_Pow_notin:
-  assumes "[] \<notin> A"
-  shows "Der c (A \<up> (Suc n)) = (Der c A) ;; (A \<up> n)"
-using assms
-by(simp)
-
-lemma der_correctness:
-  shows "L (der c r) = Der c (L r)"
-apply(induct c r rule: der.induct) 
-apply(simp_all add: nullable_correctness)[7]
-apply(simp only: der.simps L.simps)
-apply(simp only: Der_UNION)
-apply(simp only: Sequ_UNION[symmetric])
-apply(simp add: test)
-apply(simp)
-(* NTIMES *)
-apply(simp only: L.simps der.simps)
-apply(simp)
-apply(rule impI)
-apply (simp add: Der_Pow_subseteq sup_absorb1)
-(* FROMNTIMES *)
-apply(simp only: der.simps)
-apply(simp only: L.simps)
-apply(simp)
-using Der_star Star_def apply auto[1]
-apply(simp only: der.simps)
-apply(simp only: L.simps)
-apply(simp add: Der_UNION)
-apply(subst Sequ_UNION[symmetric])
-apply(subst test2[symmetric])
-apply(subgoal_tac "(Suc ` {n..}) = {Suc n..}")
-apply simp
-apply(auto simp add: image_def)[1]
-using Suc_le_D apply blast
-(* NMTIMES *)
-apply(case_tac "n \<le> m")
-prefer 2
-apply(simp only: der.simps if_True)
-apply(simp only: L.simps)
-apply(simp)
-apply(auto)
-apply(subst (asm) Sequ_UNION[symmetric])
-apply(subst (asm) test[symmetric])
-apply(auto simp add: Der_UNION)[1]
-apply(auto simp add: Der_UNION)[1]
-apply(subst Sequ_UNION[symmetric])
-apply(subst test[symmetric])
-apply(auto)[1]
-apply(subst (asm) Sequ_UNION[symmetric])
-apply(subst (asm) test2[symmetric])
-apply(auto simp add: Der_UNION)[1]
-apply(subst Sequ_UNION[symmetric])
-apply(subst test2[symmetric])
-apply(auto simp add: Der_UNION)[1]
-(* PLUS *)
-apply(auto simp add: Sequ_def Star_def)[1]
-apply(simp add: Der_UNION)
-apply(rule_tac x="Suc xa" in bexI)
-apply(auto simp add: Sequ_def Der_def)[2]
-apply (metis append_Cons)
-apply(simp add: Der_UNION)
-apply(erule_tac bexE)
-apply(case_tac xa)
-apply(simp)
-apply(simp)
-apply(auto)
-apply(auto simp add: Sequ_def Der_def)[1]
-using Star_def apply auto[1]
-apply(case_tac "[] \<in> L r")
-apply(auto)
-using Der_UNION Der_star Star_def by fastforce
-
-
-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
-
-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"
-| "\<lbrakk>\<forall>v \<in> set vs. \<turnstile> v : r\<rbrakk> \<Longrightarrow> \<turnstile> Stars vs : STAR r"
-| "\<lbrakk>\<forall>v \<in> set vs. \<turnstile> v : r; length vs \<le> n\<rbrakk> \<Longrightarrow> \<turnstile> Stars vs : UPNTIMES r n"
-| "\<lbrakk>\<forall>v \<in> set vs. \<turnstile> v : r; length vs = n\<rbrakk> \<Longrightarrow> \<turnstile> Stars vs : NTIMES r n"
-| "\<lbrakk>\<forall>v \<in> set vs. \<turnstile> v : r; length vs \<ge> n\<rbrakk> \<Longrightarrow> \<turnstile> Stars vs : FROMNTIMES r n"
-| "\<lbrakk>\<forall>v \<in> set vs. \<turnstile> v : r; length vs \<ge> n; length vs \<le> m\<rbrakk> \<Longrightarrow> \<turnstile> Stars vs : NMTIMES r n m"
-| "\<lbrakk>\<forall>v \<in> set vs. \<turnstile> v : r; length vs \<ge> 1\<rbrakk> \<Longrightarrow> \<turnstile> Stars vs : PLUS 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_STAR:
-   assumes "\<forall>v\<in>set vs. \<turnstile> v : r \<and> flat v \<in> L r"  
-   shows "concat (map flat vs) \<in> L r\<star>"
-using assms 
-apply(induct vs)
-apply(auto)
-done
-
-lemma Prf_Pow:
-  assumes "\<forall>v\<in>set vs. \<turnstile> v : r \<and> flat v \<in> L r" 
-  shows "concat (map flat vs) \<in> L r \<up> length vs"
-using assms
-apply(induct vs)
-apply(auto simp add: Sequ_def)
-done
-
-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)
-apply(rule Prf_STAR)
-apply(simp)
-apply(rule_tac x="length vs" in bexI)
-apply(rule Prf_Pow)
-apply(simp)
-apply(simp)
-apply(rule Prf_Pow)
-apply(simp)
-apply(rule_tac x="length vs" in bexI)
-apply(rule Prf_Pow)
-apply(simp)
-apply(simp)
-apply(rule_tac x="length vs" in bexI)
-apply(rule Prf_Pow)
-apply(simp)
-apply(simp)
-using Prf_Pow by blast
-
-lemma Star_Pow:
-  assumes "s \<in> A \<up> n"
-  shows "\<exists>ss. concat ss = s \<and> (\<forall>s \<in> set ss. s \<in> A) \<and> (length ss = n)"
-using assms
-apply(induct n arbitrary: s)
-apply(auto simp add: Sequ_def)
-apply(drule_tac x="s2" in meta_spec)
-apply(auto)
-apply(rule_tac x="s1#ss" in exI)
-apply(simp)
-done
-
-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(auto simp add: Star_def)
-using Star_Pow by blast
-
-
-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 Star_val_length:
-  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) \<and> (length vs) = (length ss)"
-using assms
-apply(induct ss)
-apply(auto)
-by (metis List.bind_def bind_simps(2) length_Suc_conv set_ConsD)
-
-
-
-
-
-lemma L_flat_Prf2:
-  assumes "s \<in> L r" shows "\<exists>v. \<turnstile> v : r \<and> flat v = s"
-using assms
-apply(induct r arbitrary: s)
-apply(auto simp add: Sequ_def intro: Prf.intros)
-using Prf.intros(1) flat.simps(5) apply blast
-using Prf.intros(2) flat.simps(3) apply blast
-using Prf.intros(3) flat.simps(4) apply blast
-apply(subgoal_tac "\<exists>vs::val list. concat (map flat vs) = s \<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.intros)
-apply(simp)
-using Star_string Star_val apply force
-apply(subgoal_tac "\<exists>vs::val list. concat (map flat vs) = s \<and> (\<forall>v \<in> set vs. \<turnstile> v : r) \<and> (length vs = x)")
-apply(auto)[1]
-apply(rule_tac x="Stars vs" in exI)
-apply(simp)
-apply(rule Prf.intros)
-apply(simp)
-apply(simp)
-using Star_Pow Star_val_length apply blast
-apply(subgoal_tac "\<exists>vs::val list. concat (map flat vs) = s \<and> (\<forall>v \<in> set vs. \<turnstile> v : r) \<and> (length vs = x2)")
-apply(auto)[1]
-apply(rule_tac x="Stars vs" in exI)
-apply(simp)
-apply(rule Prf.intros)
-apply(simp)
-apply(simp)
-using Star_Pow Star_val_length apply blast
-apply(subgoal_tac "\<exists>vs::val list. concat (map flat vs) = s \<and> (\<forall>v \<in> set vs. \<turnstile> v : r) \<and> (length vs = x)")
-apply(auto)[1]
-apply(rule_tac x="Stars vs" in exI)
-apply(simp)
-apply(rule Prf.intros)
-apply(simp)
-apply(simp)
-using Star_Pow Star_val_length apply blast
-apply(subgoal_tac "\<exists>vs::val list. concat (map flat vs) = s \<and> (\<forall>v \<in> set vs. \<turnstile> v : r) \<and> (length vs = x)")
-apply(auto)[1]
-apply(rule_tac x="Stars vs" in exI)
-apply(simp)
-apply(rule Prf.intros)
-apply(simp)
-apply(simp)
-apply(simp)
-using Star_Pow Star_val_length apply blast
-apply(subgoal_tac "\<exists>vs::val list. concat (map flat vs) = s \<and> (\<forall>v \<in> set vs. \<turnstile> v : r) \<and> (length vs = x)")
-apply(auto)[1]
-apply(rule_tac x="Stars vs" in exI)
-apply(simp)
-apply(rule Prf.intros)
-apply(simp)
-apply(simp)
-using Star_Pow Star_val_length apply blast
-done
-
-lemma L_flat_Prf:
-  "L(r) = {flat v | v. \<turnstile> v : r}"
-using Prf_flat_L L_flat_Prf2 by blast
-
-
-section {* Sulzmann and Lu functions *}
+section {* The Lexer Functions by Sulzmann and Lu  *}
 
 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(ALT r1 r2) = (if nullable(r1) then Left (mkeps r1) else Right (mkeps r2))"
 | "mkeps(STAR r) = Stars []"
 | "mkeps(UPNTIMES r n) = Stars []"
 | "mkeps(NTIMES r n) = Stars (replicate n (mkeps r))"
 | "mkeps(FROMNTIMES r n) = Stars (replicate n (mkeps r))"
 | "mkeps(NMTIMES r n m) = Stars (replicate n (mkeps r))"
-| "mkeps(PLUS r) = Stars ([mkeps r])"
-
-
+  
 fun injval :: "rexp \<Rightarrow> char \<Rightarrow> val \<Rightarrow> val"
 where
   "injval (CHAR d) c Void = Char d"
@@ -617,364 +27,157 @@
 | "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)" 
-| "injval (UPNTIMES r n) c (Seq v (Stars vs)) = Stars ((injval r c v) # vs)"
-| "injval (NTIMES r n) c (Seq v (Stars vs)) = Stars ((injval r c v) # vs)"
-| "injval (FROMNTIMES r n) c (Seq v (Stars vs)) = Stars ((injval r c v) # vs)"
-| "injval (NMTIMES r n m) c (Seq v (Stars vs)) = Stars ((injval r c v) # vs)"
-| "injval (PLUS r) c (Seq v (Stars vs)) = Stars ((injval r c v) # vs)"
-
-section {* Mkeps, injval *}
+| "injval (NTIMES r n) c (Seq v (Stars vs)) = Stars ((injval r c v) # vs)" 
+| "injval (FROMNTIMES r n) c (Seq v (Stars vs)) = Stars ((injval r c v) # vs)" 
+| "injval (UPNTIMES r n) c (Seq v (Stars vs)) = Stars ((injval r c v) # vs)" 
+| "injval (NMTIMES r n m) 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))"
 
-lemma mkeps_nullable:
-  assumes "nullable(r)" 
-  shows "\<turnstile> mkeps r : r"
-using assms
-apply(induct r rule: mkeps.induct) 
-apply(auto intro: Prf.intros)
-by (metis Prf.intros(10) in_set_replicate length_replicate not_le order_refl)
+
+
+section {* Mkeps, Injval Properties *}
 
 lemma mkeps_flat:
   assumes "nullable(r)" 
   shows "flat (mkeps r) = []"
 using assms
-apply (induct rule: nullable.induct) 
-apply(auto)
-by meson
-
-
-lemma Prf_injval:
-  assumes "\<turnstile> v : der c r" 
-  shows "\<turnstile> (injval r c v) : r"
+  apply(induct rule: nullable.induct) 
+         apply(auto)
+  by presburger  
+  
+  
+lemma mkeps_nullable:
+  assumes "nullable(r)" 
+  shows "\<Turnstile> mkeps r : 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)
-apply(auto)
-using Prf.intros(6) apply auto[1]
-(* UPNTIMES *)
-apply(case_tac x2)
-apply(simp)
-using Prf_elims(1) apply auto[1]
-apply(simp)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(clarify)
-apply(rotate_tac 2)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(clarify)
-using Prf.intros(7) apply auto[1]
-(* NTIMES *)
-apply(case_tac x2)
-apply(simp)
-using Prf_elims(1) apply auto[1]
-apply(simp)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(clarify)
-apply(rotate_tac 2)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(clarify)
-using Prf.intros(8) apply auto[1]
-(* FROMNTIMES *)
-apply(case_tac x2)
-apply(simp)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(clarify)
-apply(rotate_tac 2)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(clarify)
-using Prf.intros(9) apply auto[1]
-apply(rotate_tac 1)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(clarify)
-apply(rotate_tac 2)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(clarify)
-using Prf.intros(9) apply auto
-(* NMTIMES *)
-apply(rotate_tac 3)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(clarify)
-apply(rule Prf.intros)
-apply(auto)[2]
-apply simp
-apply(rotate_tac 4)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(clarify)
-apply(rule Prf.intros)
-apply(auto)[2]
-apply simp
-(* PLUS *)
-apply(rotate_tac 2)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(auto)
-using Prf.intros apply auto[1]
+apply(induct rule: nullable.induct) 
+         apply(auto intro: Prf.intros split: if_splits)
+  using Prf.intros(8) apply force
+     apply(subst append.simps(1)[symmetric])
+     apply(rule Prf.intros)
+       apply(simp)
+      apply(simp)
+       apply (simp add: mkeps_flat)
+      apply(simp)
+  using Prf.intros(9) apply force
+    apply(subst append.simps(1)[symmetric])
+     apply(rule Prf.intros)
+       apply(simp)
+      apply(simp)
+       apply (simp add: mkeps_flat)
+    apply(simp)
+  using Prf.intros(11) apply force
+    apply(subst append.simps(1)[symmetric])
+     apply(rule Prf.intros)
+       apply(simp)
+      apply(simp)
+    apply (simp add: mkeps_flat)
+   apply(simp)
+  apply(simp)
 done
-
+    
 
 lemma Prf_injval_flat:
-  assumes "\<turnstile> v : der c r" 
+  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)
-apply(rotate_tac 2)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(rotate_tac 2)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(rotate_tac 2)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(rotate_tac 2)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(rotate_tac 3)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(rotate_tac 4)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(rotate_tac 2)
-apply(erule Prf.cases)
-apply(simp_all)
+apply(auto elim!: Prf_elims intro: mkeps_flat split: if_splits)
+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(auto intro!: Prf.intros mkeps_nullable elim!: Prf_elims split: if_splits)[6]
+    apply(simp add: Prf_injval_flat)
+   apply(simp)
+  apply(case_tac x2)
+    apply(simp)
+  apply(erule Prf_elims)
+   apply(simp)
+   apply(erule Prf_elims)
+   apply(simp)
+  apply(erule Prf_elims)
+   apply(simp)
+  using Prf.intros(7) Prf_injval_flat apply auto[1]
+    apply(simp)
+    apply(case_tac x2)
+     apply(simp)
+    apply(erule Prf_elims)
+    apply(simp)
+    apply(erule Prf_elims)
+   apply(simp)
+  apply(erule Prf_elims)
+   apply(simp)
+    apply(subst append.simps(2)[symmetric])
+    apply(rule Prf.intros)
+      apply(simp add: Prf_injval_flat)
+     apply(simp)
+    apply(simp)
+    prefer 2
+   apply(simp)
+   apply(case_tac "x3a < x2")
+    apply(simp)
+    apply(erule Prf_elims)
+   apply(simp)
+    apply(case_tac x2)
+    apply(simp)
+    apply(case_tac x3a)
+     apply(simp)
+    apply(erule Prf_elims)
+    apply(simp)
+    apply(erule Prf_elims)
+    apply(simp)
+    apply(erule Prf_elims)
+    apply(simp)
+  using Prf.intros(12) Prf_injval_flat apply auto[1]
+   apply(simp)
+    apply(erule Prf_elims)
+   apply(simp)
+    apply(erule Prf_elims)
+    apply(simp)
+    apply(subst append.simps(2)[symmetric])
+    apply(rule Prf.intros)
+     apply(simp add: Prf_injval_flat)
+     apply(simp)
+     apply(simp)
+    apply(simp)
+   apply(simp)
+  using Prf.intros(12) Prf_injval_flat apply auto[1]
+    apply(case_tac x2)
+   apply(simp)
+    apply(erule Prf_elims)
+   apply(simp)
+    apply(erule Prf_elims)
+    apply(simp_all)
+    apply (simp add: Prf.intros(10) Prf_injval_flat)
+  using Prf.intros(10) Prf_injval_flat apply auto[1]
+  apply(erule Prf_elims)
+  apply(simp)
+    apply(erule Prf_elims)
+    apply(simp_all)
+    apply(subst append.simps(2)[symmetric])
+    apply(rule Prf.intros)
+     apply(simp add: Prf_injval_flat)
+     apply(simp)
+   apply(simp)
+  using Prf.intros(10) Prf_injval_flat apply auto
 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 []"
-| Posix_UPNTIMES1: "\<lbrakk>s1 \<in> r \<rightarrow> v; s2 \<in> UPNTIMES r n \<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 (UPNTIMES r n))\<rbrakk>
-    \<Longrightarrow> (s1 @ s2) \<in> UPNTIMES r (Suc n) \<rightarrow> Stars (v # vs)"
-| Posix_UPNTIMES2: "[] \<in> UPNTIMES r n \<rightarrow> Stars []"
-| Posix_NTIMES1: "\<lbrakk>s1 \<in> r \<rightarrow> v; s2 \<in> NTIMES r n \<rightarrow> Stars vs; flat v = [] \<Longrightarrow> flat (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 r \<and> s\<^sub>4 \<in> L (NTIMES r n))\<rbrakk>
-    \<Longrightarrow> (s1 @ s2) \<in> NTIMES r (Suc n) \<rightarrow> Stars (v # vs)"
-| Posix_NTIMES2: "[] \<in> NTIMES r 0 \<rightarrow> Stars []"
-| Posix_FROMNTIMES1: "\<lbrakk>s1 \<in> r \<rightarrow> v; s2 \<in> FROMNTIMES r n \<rightarrow> Stars vs; flat v = [] \<Longrightarrow> flat (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 r \<and> s\<^sub>4 \<in> L (FROMNTIMES r n))\<rbrakk>
-    \<Longrightarrow> (s1 @ s2) \<in> FROMNTIMES r (Suc n) \<rightarrow> Stars (v # vs)"
-| Posix_FROMNTIMES2: "s \<in> STAR r \<rightarrow> Stars vs \<Longrightarrow>  s \<in> FROMNTIMES r 0 \<rightarrow> Stars vs"
-| Posix_NMTIMES1: "\<lbrakk>s1 \<in> r \<rightarrow> v; s2 \<in> NMTIMES r n m \<rightarrow> Stars vs; flat v = [] \<Longrightarrow> flat (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 r \<and> s\<^sub>4 \<in> L (NMTIMES r n m))\<rbrakk>
-    \<Longrightarrow> (s1 @ s2) \<in> NMTIMES r (Suc n) (Suc m) \<rightarrow> Stars (v # vs)"
-| Posix_NMTIMES2: "s \<in> UPNTIMES r m \<rightarrow> Stars vs \<Longrightarrow>  s \<in> NMTIMES r 0 m \<rightarrow> Stars vs"
-| Posix_PLUS1: "\<lbrakk>s1 \<in> r \<rightarrow> v; s2 \<in> STAR r \<rightarrow> Stars vs; flat v = [] \<Longrightarrow> flat (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 r \<and> s\<^sub>4 \<in> L (STAR r))\<rbrakk>
-    \<Longrightarrow> (s1 @ s2) \<in> PLUS r \<rightarrow> Stars (v # vs)"
-
-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
-apply (induct s r v rule: Posix.induct)
-apply(auto simp add: Sequ_def)
-apply(rule_tac x="Suc x" in bexI)
-apply(auto simp add: Sequ_def)
-apply(rule_tac x="Suc x" in bexI)
-using Sequ_def apply auto[2]
-apply(simp add: Star_def)
-apply(rule_tac x="Suc x" in bexI)
-apply(auto simp add: Sequ_def)
-apply(simp add: Star_def)
-apply(clarify)
-apply(rule_tac x="Suc x" in bexI)
-apply(auto simp add: Sequ_def)
-done
-
-
-lemma 
-  "([] @ []) \<in> PLUS (ONE) \<rightarrow> Stars ([Void])"
-apply(rule Posix_PLUS1)
-apply(rule Posix.intros)
-apply(rule Posix.intros)
-apply(simp)
-apply(simp)
-done
-
-lemma 
-  assumes "s \<in> r \<rightarrow> v" "flat v \<noteq> []" "\<forall>s' \<in> L r. length s' < length s"
-  shows "([] @ (s @ [])) \<in> PLUS (ALT ONE r) \<rightarrow> Stars ([Left Void, Right v])"
-using assms
-oops
-
-lemma 
-  "([] @ ([] @ [])) \<in> FROMNTIMES (ONE) (Suc (Suc 0)) \<rightarrow> Stars ([Void, Void])"
-apply(rule Posix.intros)
-apply(rule Posix.intros)
-apply(rule Posix.intros)
-apply(rule Posix.intros)
-apply(rule Posix.intros)
-apply(rule Posix.intros)
-apply(auto)
-done
-
-
-lemma 
-  assumes "s \<in> r \<rightarrow> v" "flat v \<noteq> []"
-  "s \<in> PLUS (ALT ONE r) \<rightarrow> Stars ([Left(Void), Right(v)])"
-  shows "False"
-using assms
-apply(rotate_tac 2)
-apply(erule_tac Posix.cases)
-apply(simp_all)
-apply(auto)
-oops
-
-
-
-
-
-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)
-apply(rule Prf.intros)
-apply(auto)[1]
-apply(rotate_tac 3)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(rule Prf.intros)
-apply(auto)[1]
-apply(rotate_tac 3)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(rotate_tac 3)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(rule Prf.intros)
-apply(auto)[1]
-apply(rotate_tac 3)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(rotate_tac 3)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(rule Prf.intros)
-apply(auto)[1]
-apply(rotate_tac 3)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(rotate_tac 3)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(rule Prf.intros)
-apply(auto)[2]
-apply(rotate_tac 3)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(rule Prf.intros)
-apply(auto)[3]
-apply(rotate_tac 3)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(rotate_tac 3)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(rotate_tac 3)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(rule Prf.intros)
-apply(auto)[3]
-apply(rotate_tac 3)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(rotate_tac 3)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(rule Prf.intros)
-apply(auto)
-done
-
-
-lemma  Posix_NTIMES_mkeps:
-  assumes "[] \<in> r \<rightarrow> mkeps r"
-  shows "[] \<in> NTIMES r n \<rightarrow> Stars (replicate n (mkeps r))"
-apply(induct n)
-apply(simp)
-apply (rule Posix_NTIMES2)
-apply(simp)
-apply(subst append_Nil[symmetric])
-apply (rule Posix_NTIMES1)
-apply(auto)
-apply(rule assms)
-done
-
-lemma  Posix_FROMNTIMES_mkeps:
-  assumes "[] \<in> r \<rightarrow> mkeps r"
-  shows "[] \<in> FROMNTIMES r n \<rightarrow> Stars (replicate n (mkeps r))"
-apply(induct n)
-apply(simp)
-apply (rule Posix_FROMNTIMES2)
-apply (rule Posix.intros)
-apply(simp)
-apply(subst append_Nil[symmetric])
-apply (rule Posix_FROMNTIMES1)
-apply(auto)
-apply(rule assms)
-done
-
-lemma  Posix_NMTIMES_mkeps:
-  assumes "[] \<in> r \<rightarrow> mkeps r" "n \<le> m"
-  shows "[] \<in> NMTIMES r n m \<rightarrow> Stars (replicate n (mkeps r))"
-using assms(2)
-apply(induct n arbitrary: m)
-apply(simp)
-apply(rule Posix.intros)
-apply(rule Posix.intros)
-apply(case_tac m)
-apply(simp)
-apply(simp)
-apply(subst append_Nil[symmetric])
-apply(rule Posix.intros)
-apply(auto)
-apply(rule assms)
-done
-
-
+text {*
+  Mkeps and injval produce, or preserve, Posix values.
+*}
 
 lemma Posix_mkeps:
   assumes "nullable r"
@@ -984,261 +187,21 @@
 apply(auto intro: Posix.intros simp add: nullable_correctness Sequ_def)
 apply(subst append.simps(1)[symmetric])
 apply(rule Posix.intros)
-apply(auto)
-apply(case_tac n)
-apply(simp)
-apply (simp add: Posix_NTIMES2)
-apply(simp)
-apply(subst append.simps(1)[symmetric])
-apply(rule Posix.intros)
-apply(auto)
-apply(rule Posix_NTIMES_mkeps)
-apply(simp)
-apply(rule Posix.intros)
-apply(rule Posix.intros)
-apply(case_tac n)
-apply(simp)
-apply(rule Posix.intros)
-apply(rule Posix.intros)
-apply(simp)
-apply(subst append.simps(1)[symmetric])
-apply(rule Posix.intros)
-apply(auto)
-apply(rule Posix_FROMNTIMES_mkeps)
-apply(simp)
-apply(rule Posix.intros)
-apply(rule Posix.intros)
-apply(case_tac n)
-apply(simp)
-apply(rule Posix.intros)
-apply(rule Posix.intros)
-apply(simp)
-apply(case_tac m)
-apply(simp)
-apply(simp)
-apply(subst append_Nil[symmetric])
-apply(rule Posix.intros)
-apply(auto)
-apply(rule Posix_NMTIMES_mkeps)
-apply(auto)
-apply(subst append_Nil[symmetric])
-apply(rule Posix.intros)
-apply(auto)
-apply(rule Posix.intros)
-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)
-next
-  case (Posix_UPNTIMES1 s1 r v s2 n vs v2)
-  have "(s1 @ s2) \<in> UPNTIMES r (Suc n) \<rightarrow> v2" 
-       "s1 \<in> r \<rightarrow> v" "s2 \<in> (UPNTIMES r n) \<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 (UPNTIMES r n))" by fact+
-  then obtain v' vs' where "v2 = Stars (v' # vs')" "s1 \<in> r \<rightarrow> v'" "s2 \<in> (UPNTIMES r n) \<rightarrow> (Stars vs')"
-  apply(cases) apply (auto simp add: append_eq_append_conv2)
-  using Posix1(1) apply fastforce
-  apply (metis Posix1(1) Posix_UPNTIMES1.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> UPNTIMES r n \<rightarrow> v2 \<Longrightarrow> Stars vs = v2" by fact+
-  ultimately show "Stars (v # vs) = v2" by auto
-next
-  case (Posix_UPNTIMES2 r n v2)
-  have "[] \<in> UPNTIMES r n \<rightarrow> v2" by fact
-  then show "Stars [] = v2" by cases (auto simp add: Posix1)
-next
-  case (Posix_NTIMES2 r v2)
-  have "[] \<in> NTIMES r 0 \<rightarrow> v2" by fact
-  then show "Stars [] = v2" by cases (auto simp add: Posix1)
-next
-  case (Posix_NTIMES1 s1 r v s2 n vs v2)
-  have "(s1 @ s2) \<in> NTIMES r (Suc n) \<rightarrow> v2"  "flat v = [] \<Longrightarrow> flat (Stars vs) = []"
-       "s1 \<in> r \<rightarrow> v" "s2 \<in> (NTIMES r n) \<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 r \<and> s\<^sub>4 \<in> L (NTIMES r n))" by fact+
-  then obtain v' vs' where "v2 = Stars (v' # vs')" "s1 \<in> r \<rightarrow> v'" "s2 \<in> (NTIMES r n) \<rightarrow> (Stars vs')"
-  apply(cases) apply (auto simp add: append_eq_append_conv2)
-  using Posix1(1) apply fastforce
-  by (metis Posix1(1) Posix_NTIMES1.hyps(6) append_Nil append_Nil2)
-  moreover
-  have IHs: "\<And>v2. s1 \<in> r \<rightarrow> v2 \<Longrightarrow> v = v2"
-            "\<And>v2. s2 \<in> NTIMES r n \<rightarrow> v2 \<Longrightarrow> Stars vs = v2" by fact+
-  ultimately show "Stars (v # vs) = v2" by auto
-next
-  case (Posix_FROMNTIMES2 s r v1 v2)
-  have "s \<in> FROMNTIMES r 0 \<rightarrow> v2" "s \<in> STAR r \<rightarrow> Stars v1" 
-       "\<And>v3. s \<in> STAR r \<rightarrow> v3 \<Longrightarrow> Stars v1 = v3" by fact+
-  then show ?case 
-  apply(cases)
+      apply(auto)
+  done
+    
+lemma test:
+  assumes "s \<in> der c (FROMNTIMES r 0) \<rightarrow> v"
+  shows "XXX"
+using assms    
+  apply(simp)
+  apply(erule Posix_elims)
+  apply(drule FROMNTIMES_0)
   apply(auto)
-  done
-next
-  case (Posix_FROMNTIMES1 s1 r v s2 n vs v2)
-  have "(s1 @ s2) \<in> FROMNTIMES r (Suc n) \<rightarrow> v2" 
-       "s1 \<in> r \<rightarrow> v" "s2 \<in> (FROMNTIMES r n) \<rightarrow> Stars vs" "flat v = [] \<Longrightarrow> flat (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 r \<and> s\<^sub>4 \<in> L (FROMNTIMES r n))" by fact+
-  then obtain v' vs' where "v2 = Stars (v' # vs')" "s1 \<in> r \<rightarrow> v'" "s2 \<in> (FROMNTIMES r n) \<rightarrow> (Stars vs')"
-  apply(cases) apply (auto simp add: append_eq_append_conv2)
-  using Posix1(1) apply fastforce
-  by (metis Posix1(1) Posix_FROMNTIMES1.hyps(6) append_Nil append_Nil2)
-  moreover
-  have IHs: "\<And>v2. s1 \<in> r \<rightarrow> v2 \<Longrightarrow> v = v2"
-            "\<And>v2. s2 \<in> FROMNTIMES r n \<rightarrow> v2 \<Longrightarrow> Stars vs = v2" by fact+
-  ultimately show "Stars (v # vs) = v2" by auto
-next
-  case (Posix_NMTIMES2 s r m v1 v2)
-  have "s \<in> NMTIMES r 0 m \<rightarrow> v2" "s \<in> UPNTIMES r m \<rightarrow> Stars v1" 
-       "\<And>v3. s \<in> UPNTIMES r m \<rightarrow> v3 \<Longrightarrow> Stars v1 = v3" by fact+
-  then show ?case 
-  apply(cases)
-  apply(auto)
-  done
-next
-  case (Posix_NMTIMES1 s1 r v s2 n m vs v2)
-  have "(s1 @ s2) \<in> NMTIMES r (Suc n) (Suc m) \<rightarrow> v2" 
-       "s1 \<in> r \<rightarrow> v" "s2 \<in> (NMTIMES r n m) \<rightarrow> Stars vs" "flat v = [] \<Longrightarrow> flat (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 r \<and> s\<^sub>4 \<in> L (NMTIMES r n m))" by fact+
-  then obtain v' vs' where "v2 = Stars (v' # vs')" "s1 \<in> r \<rightarrow> v'" "s2 \<in> (NMTIMES r n m) \<rightarrow> (Stars vs')"
-  apply(cases) apply (auto simp add: append_eq_append_conv2)
-  using Posix1(1) apply fastforce
-  by (metis Posix1(1) Posix_NMTIMES1.hyps(6) self_append_conv self_append_conv2)
-  moreover
-  have IHs: "\<And>v2. s1 \<in> r \<rightarrow> v2 \<Longrightarrow> v = v2"
-            "\<And>v2. s2 \<in> NMTIMES r n m \<rightarrow> v2 \<Longrightarrow> Stars vs = v2" by fact+
-  ultimately show "Stars (v # vs) = v2" by auto
-next
-  case (Posix_PLUS1 s1 r v s2 vs v2)
-  have "(s1 @ s2) \<in> PLUS r \<rightarrow> v2" 
-       "s1 \<in> r \<rightarrow> v" "s2 \<in> (STAR r) \<rightarrow> Stars vs" (*"flat v = [] \<Longrightarrow> flat (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 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
-  by (metis Posix1(1) Posix_PLUS1.hyps(6) append_self_conv append_self_conv2)
-  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
-qed
-
-lemma NTIMES_Stars:
- assumes "\<turnstile> v : NTIMES r n"
- shows "\<exists>vs. v = Stars vs \<and> (\<forall>v \<in> set vs. \<turnstile> v : r) \<and> length vs = n"
-using assms
-apply(induct n arbitrary: v)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(erule Prf.cases)
-apply(simp_all)
-done
-
-lemma UPNTIMES_Stars:
- assumes "\<turnstile> v : UPNTIMES r n"
- shows "\<exists>vs. v = Stars vs \<and> (\<forall>v \<in> set vs. \<turnstile> v : r) \<and> length vs \<le> n"
-using assms
-apply(induct n arbitrary: v)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(erule Prf.cases)
-apply(simp_all)
-done
-
-lemma FROMNTIMES_Stars:
- assumes "\<turnstile> v : FROMNTIMES r n"
- shows "\<exists>vs. v = Stars vs \<and> (\<forall>v \<in> set vs. \<turnstile> v : r) \<and> n \<le> length vs"
-using assms
-apply(induct n arbitrary: v)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(erule Prf.cases)
-apply(simp_all)
-done
-
-lemma PLUS_Stars:
- assumes "\<turnstile> v : PLUS r"
- shows "\<exists>vs. v = Stars vs \<and> (\<forall>v \<in> set vs. \<turnstile> v : r) \<and> 1 \<le> length vs"
-using assms
-apply(erule_tac Prf.cases)
-apply(simp_all)
-done
-
-lemma NMTIMES_Stars:
- assumes "\<turnstile> v : NMTIMES r n m"
- shows "\<exists>vs. v = Stars vs \<and> (\<forall>v \<in> set vs. \<turnstile> v : r) \<and> n \<le> length vs \<and> length vs \<le> m"
-using assms
-apply(induct n arbitrary: m v)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(erule Prf.cases)
-apply(simp_all)
-done
-
+oops    
 
 lemma Posix_injval:
-  assumes "s \<in> (der c r) \<rightarrow> v"
+  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)
@@ -1352,7 +315,57 @@
       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)
+case (UPNTIMES r n s v)
+  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 (UPNTIMES r n) \<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> (UPNTIMES r (n - 1)) \<rightarrow> (Stars vs)" "0 < n"
+        "\<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 (UPNTIMES r (n - 1)))" 
+    (* here *)
+    apply(auto elim: Posix_elims simp add: der_correctness Der_def intro: Posix.intros split: if_splits)
+    apply(erule Posix_elims)
+    apply(simp)
+    apply(subgoal_tac "\<exists>vss. v2 = Stars vss")
+    apply(clarify)
+    apply(drule_tac x="v1" in meta_spec)
+    apply(drule_tac x="vss" in meta_spec)
+    apply(drule_tac x="s1" in meta_spec)
+    apply(drule_tac x="s2" in meta_spec)
+     apply(simp add: der_correctness Der_def)
+    apply(erule Posix_elims)
+     apply(auto)
+      done
+    then show "(c # s) \<in> (UPNTIMES r n) \<rightarrow> injval (UPNTIMES r n) 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> (UPNTIMES r (n - 1)) \<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 (UPNTIMES r (n - 1)))" 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 (UPNTIMES r (n - 1)))" 
+            by (simp add: der_correctness Der_def)
+        ultimately 
+        have "((c # s1) @ s2) \<in> UPNTIMES r n \<rightarrow> Stars (injval r c v1 # vs)" 
+           thm Posix.intros
+           apply (rule_tac Posix.intros)
+               apply(simp_all)
+              apply(case_tac n)
+            apply(simp)
+           using Posix_elims(1) UPNTIMES.prems apply auto[1]
+             apply(simp)
+             done
+        then show "(c # s) \<in> UPNTIMES r n \<rightarrow> injval (UPNTIMES r n) c v" using cons 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
@@ -1384,242 +397,146 @@
         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
-next 
-  case (UPNTIMES r n)
-  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 (UPNTIMES r n) \<rightarrow> v" by fact
+  next
+    case (NTIMES r n s v)
+     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 (NTIMES r n) \<rightarrow> v" by fact
   then consider
-      (cons) m v1 vs s1 s2 where 
-        "n = Suc m"
+      (cons) v1 vs s1 s2 where 
         "v = Seq v1 (Stars vs)" "s = s1 @ s2" 
-        "s1 \<in> der c r \<rightarrow> v1" "s2 \<in> (UPNTIMES r m) \<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 (UPNTIMES r m))" 
-        apply(case_tac n)
-        apply(simp)
-        using Posix_elims(1) apply blast
-        apply(simp)
-        apply(auto elim!: Posix_elims(1-5) simp add: der_correctness Der_def intro: Posix.intros)        
-        by (metis Posix1a UPNTIMES_Stars)
-    then show "(c # s) \<in> UPNTIMES r n \<rightarrow> injval (UPNTIMES r n) c v" 
+        "s1 \<in> der c r \<rightarrow> v1" "s2 \<in> (NTIMES r (n - 1)) \<rightarrow> (Stars vs)" "0 < n"
+        "\<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 (NTIMES r (n - 1)))" 
+    apply(auto elim: Posix_elims simp add: der_correctness Der_def intro: Posix.intros split: if_splits)
+    apply(erule Posix_elims)
+    apply(simp)
+    apply(subgoal_tac "\<exists>vss. v2 = Stars vss")
+    apply(clarify)
+    apply(drule_tac x="v1" in meta_spec)
+    apply(drule_tac x="vss" in meta_spec)
+    apply(drule_tac x="s1" in meta_spec)
+    apply(drule_tac x="s2" in meta_spec)
+     apply(simp add: der_correctness Der_def)
+    apply(erule Posix_elims)
+     apply(auto)
+      done
+    then show "(c # s) \<in> (NTIMES r n) \<rightarrow> injval (NTIMES r n) c v" 
     proof (cases)
       case cons
-        have "n = Suc m" by fact
-        moreover
           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> UPNTIMES r m \<rightarrow> Stars vs" by fact
+          have "s2 \<in> (NTIMES r (n - 1)) \<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 (UPNTIMES r m))" 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 (UPNTIMES r m))" 
+          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 (NTIMES r (n - 1)))" 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 (NTIMES r (n - 1)))" 
             by (simp add: der_correctness Der_def)
         ultimately 
-        have "((c # s1) @ s2) \<in> UPNTIMES r (Suc m) \<rightarrow> Stars (injval r c v1 # vs)" 
-          apply(rule_tac Posix.intros(8))
-          apply(simp_all)
-          done
-        then show "(c # s) \<in> UPNTIMES r n \<rightarrow> injval (UPNTIMES r n) c v" using cons by(simp)
-    qed
-next 
-  case (NTIMES r n)
-  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 (NTIMES r n) \<rightarrow> v" by fact
-  then consider
-      (cons) m v1 vs s1 s2 where 
-        "n = Suc m"
-        "v = Seq v1 (Stars vs)" "s = s1 @ s2" "flat v = [] \<Longrightarrow> flat (Stars vs) = []"
-        "s1 \<in> der c r \<rightarrow> v1" "s2 \<in> (NTIMES r m) \<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 (NTIMES r m))" 
-        apply(case_tac n)
-        apply(simp)
-        using Posix_elims(1) apply blast
-        apply(simp)
-        apply(auto elim!: Posix_elims(1-5) simp add: der_correctness Der_def intro: Posix.intros)
-        by (metis NTIMES_Stars Posix1a)
-    then show "(c # s) \<in> NTIMES r n \<rightarrow> injval (NTIMES r n) c v" 
-    proof (cases)
-      case cons
-        have "n = Suc m" by fact
-        moreover
-          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 "flat v = [] \<Longrightarrow> flat (Stars vs) = []" by fact
-        moreover
-          have "s2 \<in> NTIMES r m \<rightarrow> Stars vs" by fact
-        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 (NTIMES r m))" 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 (NTIMES r m))" 
-            by (simp add: der_correctness Der_def)
-        ultimately 
-        have "((c # s1) @ s2) \<in> NTIMES r (Suc m) \<rightarrow> Stars (injval r c v1 # vs)" 
-          apply(rule_tac Posix.intros(10))
-          apply(simp_all)
-          by (simp add: Posix1(2))
+        have "((c # s1) @ s2) \<in> NTIMES r n \<rightarrow> Stars (injval r c v1 # vs)" 
+           apply (rule_tac Posix.intros)
+               apply(simp_all)
+              apply(case_tac n)
+            apply(simp)
+           using Posix_elims(1) NTIMES.prems apply auto[1]
+             apply(simp)
+             done
         then show "(c # s) \<in> NTIMES r n \<rightarrow> injval (NTIMES r n) c v" using cons by(simp)
-    qed
-next 
-  case (FROMNTIMES r n)
+      qed  
+  next
+    case (FROMNTIMES r n s v)
   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 (FROMNTIMES r n) \<rightarrow> v" by fact
   then consider
-        (null) v1 vs s1 s2 where 
-        "n = 0"
+      (cons) v1 vs s1 s2 where 
         "v = Seq v1 (Stars vs)" "s = s1 @ s2" 
-        "s1 \<in> der c r \<rightarrow> v1" "s2 \<in> (FROMNTIMES r 0) \<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 (FROMNTIMES r 0))" 
-      | (cons) m v1 vs s1 s2 where 
-        "n = Suc m"
-        "v = Seq v1 (Stars vs)" "s = s1 @ s2" "flat v = [] \<Longrightarrow> flat (Stars vs) = []"
-        "s1 \<in> der c r \<rightarrow> v1" "s2 \<in> (FROMNTIMES r m) \<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 (FROMNTIMES r m))" 
-        apply(case_tac n)
-        apply(simp)
-        apply(auto elim!: Posix_elims(1-5) simp add: der_correctness Der_def intro: Posix.intros)
-        defer
-        apply (metis FROMNTIMES_Stars Posix1a)
-        apply(rotate_tac 5)
-        apply(erule Posix.cases)
-        apply(simp_all)
-        apply(clarify)
-        by (simp add: Posix_FROMNTIMES2)
-    then show "(c # s) \<in> FROMNTIMES r n \<rightarrow> injval (FROMNTIMES r n) c v" 
+        "s1 \<in> der c r \<rightarrow> v1" "s2 \<in> (FROMNTIMES r (n - 1)) \<rightarrow> (Stars vs)" "0 < n"
+        "\<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 (FROMNTIMES r (n - 1)))"
+     | (null)  v1 where 
+        "v = Seq v1 (Stars [])"  
+        "s \<in> der c r \<rightarrow> v1" "[] \<in> (FROMNTIMES r 0) \<rightarrow> (Stars [])" "n = 0"
+    (* here *)    
+    apply(auto elim: Posix_elims simp add: der_correctness Der_def intro: Posix.intros split: if_splits)
+    apply(erule Posix_elims)
+    apply(simp)
+    apply(case_tac "n = 0")
+     apply(simp)
+     apply(drule FROMNTIMES_0)
+     apply(simp)
+    using FROMNTIMES_0 Posix_mkeps apply force
+    apply(subgoal_tac "\<exists>vss. v2 = Stars vss")
+    apply(clarify)
+    apply(drule_tac x="v1" in meta_spec)
+    apply(drule_tac x="vss" in meta_spec)
+    apply(drule_tac x="s1" in meta_spec)
+    apply(drule_tac x="s2" in meta_spec)
+     apply(simp add: der_correctness Der_def)
+     apply(case_tac "n = 0")
+      apply(simp)
+      apply(simp)
+      apply(rotate_tac 4)
+    apply(erule Posix_elims)
+      apply(auto)[2]
+      done
+    then show "(c # s) \<in> (FROMNTIMES r n) \<rightarrow> injval (FROMNTIMES r n) c v" 
     proof (cases)
       case cons
-        have "n = Suc m" by fact
-        moreover
           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> FROMNTIMES r m \<rightarrow> Stars vs" by fact
-        moreover
-          have "flat v = [] \<Longrightarrow> flat (Stars vs) = []" by fact
+          have "s2 \<in> (FROMNTIMES r (n - 1)) \<rightarrow> Stars vs" by fact
         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 (FROMNTIMES r m))" 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 (FROMNTIMES r m))" 
+          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 (FROMNTIMES r (n - 1)))" 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 (FROMNTIMES r (n - 1)))" 
             by (simp add: der_correctness Der_def)
         ultimately 
-        have "((c # s1) @ s2) \<in> FROMNTIMES r (Suc m) \<rightarrow> Stars (injval r c v1 # vs)" 
-          apply(rule_tac Posix.intros(12))
-          apply(simp_all) 
-          by (simp add: Posix1(2))
+        have "((c # s1) @ s2) \<in> FROMNTIMES r n \<rightarrow> Stars (injval r c v1 # vs)" 
+           apply (rule_tac Posix.intros)
+               apply(simp_all)
+              apply(case_tac n)
+            apply(simp)
+          using Posix_elims(1) FROMNTIMES.prems apply auto[1]
+          using cons(5) apply blast
+             apply(simp)
+             done
         then show "(c # s) \<in> FROMNTIMES r n \<rightarrow> injval (FROMNTIMES r n) c v" using cons by(simp)
-      next
-      case null
-      then have "((c # s1) @ s2) \<in> FROMNTIMES r 0 \<rightarrow> Stars (injval r c v1 # vs)"
-      apply(rule_tac Posix.intros)
-      apply(rule_tac Posix.intros)
-      apply(rule IH)
-      apply(simp)
-      apply(rotate_tac 4)
-      apply(erule Posix.cases)
-      apply(simp_all)
-      apply (simp add: Posix1a Prf_injval_flat)
-      apply(simp only: Star_def)
-      apply(simp only: der_correctness Der_def)
-      apply(simp)
-      done
-      then show "(c # s) \<in> FROMNTIMES r n \<rightarrow> injval (FROMNTIMES r n) c v" using null by simp
-    qed
-next 
-  case (NMTIMES r n m)
-  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 (NMTIMES r n m) \<rightarrow> v" by fact
-  then show "(c # s) \<in> (NMTIMES r n m) \<rightarrow> injval (NMTIMES r n m) c v"
-        apply(case_tac "m < n")
-        apply(simp)
-        using Posix_elims(1) apply blast
-        apply(case_tac n)
-        apply(simp_all)
-        apply(case_tac m)
-        apply(simp)
-        apply(erule Posix_elims(1))
-        apply(simp)
-        apply(erule Posix.cases)
-        apply(simp_all)
-        apply(clarify)
-        apply(rotate_tac 5)
-        apply(frule Posix1a)
-        apply(drule UPNTIMES_Stars)
-        apply(clarify)
-        apply(simp)
-        apply(subst append_Cons[symmetric])
-        apply(rule Posix.intros)
-        apply(rule Posix.intros)
-        apply(auto)
-        apply(rule IH)
-        apply blast
-        using Posix1a Prf_injval_flat apply auto[1]
-        apply(simp add: der_correctness Der_def)
-        apply blast
-        apply(case_tac m)
-        apply(simp)
-        apply(simp)
-        apply(erule Posix.cases)
-        apply(simp_all)
-        apply(clarify)
-        apply(rotate_tac 6)
-        apply(frule Posix1a)
-        apply(drule NMTIMES_Stars)
-        apply(clarify)
-        apply(simp)
-        apply(subst append_Cons[symmetric])
-        apply(rule Posix.intros)
-        apply(rule IH)
-        apply(blast)
-        apply(simp)
-        apply(simp add: der_correctness Der_def)
-        using Posix1a Prf_injval_flat list.distinct(1) apply auto[1]
-        apply(simp add: der_correctness Der_def)
-        done
-next 
-  case (PLUS 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 (PLUS r) \<rightarrow> v" by fact
-  then show "(c # s) \<in> PLUS r \<rightarrow> injval (PLUS r) c v"
-  apply -
-  apply(erule Posix.cases)
-  apply(simp_all)
-  apply(auto)
-  apply(rotate_tac 3)
-  apply(erule Posix.cases)
-  apply(simp_all)
-  apply(subst append_Cons[symmetric])
-  apply(rule Posix.intros)
-  using PLUS.hyps apply auto[1]
-  apply(rule Posix.intros)
-  apply(simp)
-  apply(simp)
-  apply(simp)
-  apply(simp)
-  apply(simp)
-  using Posix1a Prf_injval_flat apply auto[1]
-  apply(simp add: der_correctness Der_def)
-  apply(subst append_Nil2[symmetric])
-  apply(rule Posix.intros)
-  using PLUS.hyps apply auto[1]
-  apply(rule Posix.intros)
-  apply(simp)
-  apply(simp)
-  done  
+      next 
+       case null
+          have "s \<in> der c r \<rightarrow> v1" by fact
+          then have "(c # s) \<in> r \<rightarrow> injval r c v1" using IH by simp
+        moreover
+          have "[] \<in> (FROMNTIMES r 0) \<rightarrow> Stars []" by fact
+        moreover 
+          have "(c # s) \<in> r \<rightarrow> injval r c v1" by fact 
+          then have "flat (injval r c v1) = (c # s)" by (rule Posix1)
+          then have "flat (injval r c v1) \<noteq> []" by simp
+        ultimately 
+        have "((c # s) @ []) \<in> FROMNTIMES r 1 \<rightarrow> Stars [injval r c v1]" 
+           apply (rule_tac Posix.intros)
+              apply(simp_all)
+           done
+        then show "(c # s) \<in> FROMNTIMES r n \<rightarrow> injval (FROMNTIMES r n) c v" using null 
+          apply(simp)
+          apply(erule Posix_elims)
+           apply(auto)
+           apply(rotate_tac 6) 
+          apply(drule FROMNTIMES_0)
+          apply(simp)  
+            sorry
+      qed  
+  next
+    case (NMTIMES x1 x2 m s v)
+    then show ?case sorry      
 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:
   shows "s \<notin> L r \<longleftrightarrow> lexer r s = None"
@@ -1645,8 +562,4 @@
 using Posix1(1) Posix_determ lexer_correct_None lexer_correct_Some apply fastforce
 using Posix1(1) lexer_correct_None lexer_correct_Some by blast
 
-value "lexer (PLUS (ALT ONE (SEQ (CHAR (CHR ''a'')) (CHAR (CHR ''b''))))) [CHR ''a'', CHR ''b'']"
-
-
-unused_thms
 end
\ No newline at end of file
diff -r deea42c83c9e -r a3134f7de065 thys/PositionsExt.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/thys/PositionsExt.thy	Sat Oct 07 22:16:16 2017 +0100
@@ -0,0 +1,990 @@
+   
+theory PositionsExt
+  imports "SpecExt" "LexerExt" 
+begin
+
+section {* Positions in Values *}
+
+fun 
+  at :: "val \<Rightarrow> nat list \<Rightarrow> val"
+where
+  "at v [] = v"
+| "at (Left v) (0#ps)= at v ps"
+| "at (Right v) (Suc 0#ps)= at v ps"
+| "at (Seq v1 v2) (0#ps)= at v1 ps"
+| "at (Seq v1 v2) (Suc 0#ps)= at v2 ps"
+| "at (Stars vs) (n#ps)= at (nth vs n) ps"
+
+
+
+fun Pos :: "val \<Rightarrow> (nat list) set"
+where
+  "Pos (Void) = {[]}"
+| "Pos (Char c) = {[]}"
+| "Pos (Left v) = {[]} \<union> {0#ps | ps. ps \<in> Pos v}"
+| "Pos (Right v) = {[]} \<union> {1#ps | ps. ps \<in> Pos v}"
+| "Pos (Seq v1 v2) = {[]} \<union> {0#ps | ps. ps \<in> Pos v1} \<union> {1#ps | ps. ps \<in> Pos v2}" 
+| "Pos (Stars []) = {[]}"
+| "Pos (Stars (v#vs)) = {[]} \<union> {0#ps | ps. ps \<in> Pos v} \<union> {Suc n#ps | n ps. n#ps \<in> Pos (Stars vs)}"
+
+
+lemma Pos_stars:
+  "Pos (Stars vs) = {[]} \<union> (\<Union>n < length vs. {n#ps | ps. ps \<in> Pos (vs ! n)})"
+apply(induct vs)
+apply(auto simp add: insert_ident less_Suc_eq_0_disj)
+done
+
+lemma Pos_empty:
+  shows "[] \<in> Pos v"
+by (induct v rule: Pos.induct)(auto)
+
+
+abbreviation
+  "intlen vs \<equiv> int (length vs)"
+
+
+definition pflat_len :: "val \<Rightarrow> nat list => int"
+where
+  "pflat_len v p \<equiv> (if p \<in> Pos v then intlen (flat (at v p)) else -1)"
+
+lemma pflat_len_simps:
+  shows "pflat_len (Seq v1 v2) (0#p) = pflat_len v1 p"
+  and   "pflat_len (Seq v1 v2) (Suc 0#p) = pflat_len v2 p"
+  and   "pflat_len (Left v) (0#p) = pflat_len v p"
+  and   "pflat_len (Left v) (Suc 0#p) = -1"
+  and   "pflat_len (Right v) (Suc 0#p) = pflat_len v p"
+  and   "pflat_len (Right v) (0#p) = -1"
+  and   "pflat_len (Stars (v#vs)) (Suc n#p) = pflat_len (Stars vs) (n#p)"
+  and   "pflat_len (Stars (v#vs)) (0#p) = pflat_len v p"
+  and   "pflat_len v [] = intlen (flat v)"
+by (auto simp add: pflat_len_def Pos_empty)
+
+lemma pflat_len_Stars_simps:
+  assumes "n < length vs"
+  shows "pflat_len (Stars vs) (n#p) = pflat_len (vs!n) p"
+using assms
+apply(induct vs arbitrary: n p)
+apply(auto simp add: less_Suc_eq_0_disj pflat_len_simps)
+done
+
+lemma pflat_len_outside:
+  assumes "p \<notin> Pos v1"
+  shows "pflat_len v1 p = -1 "
+using assms by (simp add: pflat_len_def)
+
+
+
+section {* Orderings *}
+
+
+definition prefix_list:: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" ("_ \<sqsubseteq>pre _" [60,59] 60)
+where
+  "ps1 \<sqsubseteq>pre ps2 \<equiv> \<exists>ps'. ps1 @ps' = ps2"
+
+definition sprefix_list:: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" ("_ \<sqsubset>spre _" [60,59] 60)
+where
+  "ps1 \<sqsubset>spre ps2 \<equiv> ps1 \<sqsubseteq>pre ps2 \<and> ps1 \<noteq> ps2"
+
+inductive lex_list :: "nat list \<Rightarrow> nat list \<Rightarrow> bool" ("_ \<sqsubset>lex _" [60,59] 60)
+where
+  "[] \<sqsubset>lex (p#ps)"
+| "ps1 \<sqsubset>lex ps2 \<Longrightarrow> (p#ps1) \<sqsubset>lex (p#ps2)"
+| "p1 < p2 \<Longrightarrow> (p1#ps1) \<sqsubset>lex (p2#ps2)"
+
+lemma lex_irrfl:
+  fixes ps1 ps2 :: "nat list"
+  assumes "ps1 \<sqsubset>lex ps2"
+  shows "ps1 \<noteq> ps2"
+using assms
+by(induct rule: lex_list.induct)(auto)
+
+lemma lex_simps [simp]:
+  fixes xs ys :: "nat list"
+  shows "[] \<sqsubset>lex ys \<longleftrightarrow> ys \<noteq> []"
+  and   "xs \<sqsubset>lex [] \<longleftrightarrow> False"
+  and   "(x # xs) \<sqsubset>lex (y # ys) \<longleftrightarrow> (x < y \<or> (x = y \<and> xs \<sqsubset>lex ys))"
+by (auto simp add: neq_Nil_conv elim: lex_list.cases intro: lex_list.intros)
+
+lemma lex_trans:
+  fixes ps1 ps2 ps3 :: "nat list"
+  assumes "ps1 \<sqsubset>lex ps2" "ps2 \<sqsubset>lex ps3"
+  shows "ps1 \<sqsubset>lex ps3"
+using assms
+by (induct arbitrary: ps3 rule: lex_list.induct)
+   (auto elim: lex_list.cases)
+
+
+lemma lex_trichotomous:
+  fixes p q :: "nat list"
+  shows "p = q \<or> p \<sqsubset>lex q \<or> q \<sqsubset>lex p"
+apply(induct p arbitrary: q)
+apply(auto elim: lex_list.cases)
+apply(case_tac q)
+apply(auto)
+done
+
+
+
+
+section {* POSIX Ordering of Values According to Okui & Suzuki *}
+
+
+definition PosOrd:: "val \<Rightarrow> nat list \<Rightarrow> val \<Rightarrow> bool" ("_ \<sqsubset>val _ _" [60, 60, 59] 60)
+where
+  "v1 \<sqsubset>val p v2 \<equiv> pflat_len v1 p > pflat_len v2 p \<and>
+                   (\<forall>q \<in> Pos v1 \<union> Pos v2. q \<sqsubset>lex p \<longrightarrow> pflat_len v1 q = pflat_len v2 q)"
+
+lemma PosOrd_def2:
+  shows "v1 \<sqsubset>val p v2 \<longleftrightarrow> 
+         pflat_len v1 p > pflat_len v2 p \<and>
+         (\<forall>q \<in> Pos v1. q \<sqsubset>lex p \<longrightarrow> pflat_len v1 q = pflat_len v2 q) \<and>
+         (\<forall>q \<in> Pos v2. q \<sqsubset>lex p \<longrightarrow> pflat_len v1 q = pflat_len v2 q)"
+unfolding PosOrd_def
+apply(auto)
+done
+
+
+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"
+
+definition PosOrd_ex_eq:: "val \<Rightarrow> val \<Rightarrow> bool" ("_ :\<sqsubseteq>val _" [60, 59] 60)
+where
+  "v1 :\<sqsubseteq>val v2 \<equiv> v1 :\<sqsubset>val v2 \<or> v1 = v2"
+
+
+lemma PosOrd_trans:
+  assumes "v1 :\<sqsubset>val v2" "v2 :\<sqsubset>val v3"
+  shows "v1 :\<sqsubset>val v3"
+proof -
+  from assms obtain p p'
+    where as: "v1 \<sqsubset>val p v2" "v2 \<sqsubset>val p' v3" unfolding PosOrd_ex_def by blast
+  then have pos: "p \<in> Pos v1" "p' \<in> Pos v2" unfolding PosOrd_def pflat_len_def
+    by (smt not_int_zless_negative)+
+  have "p = p' \<or> p \<sqsubset>lex p' \<or> p' \<sqsubset>lex p"
+    by (rule lex_trichotomous)
+  moreover
+    { assume "p = p'"
+      with as have "v1 \<sqsubset>val p v3" unfolding PosOrd_def pflat_len_def
+      by (smt Un_iff)
+      then have " v1 :\<sqsubset>val v3" unfolding PosOrd_ex_def by blast
+    }   
+  moreover
+    { assume "p \<sqsubset>lex p'"
+      with as have "v1 \<sqsubset>val p v3" unfolding PosOrd_def pflat_len_def
+      by (smt Un_iff lex_trans)
+      then have " v1 :\<sqsubset>val v3" unfolding PosOrd_ex_def by blast
+    }
+  moreover
+    { assume "p' \<sqsubset>lex p"
+      with as have "v1 \<sqsubset>val p' v3" unfolding PosOrd_def
+      by (smt Un_iff lex_trans pflat_len_def)
+      then have "v1 :\<sqsubset>val v3" unfolding PosOrd_ex_def by blast
+    }
+  ultimately show "v1 :\<sqsubset>val v3" by blast
+qed
+
+lemma PosOrd_irrefl:
+  assumes "v :\<sqsubset>val v"
+  shows "False"
+using assms unfolding PosOrd_ex_def PosOrd_def
+by auto
+
+lemma PosOrd_assym:
+  assumes "v1 :\<sqsubset>val v2" 
+  shows "\<not>(v2 :\<sqsubset>val v1)"
+using assms
+using PosOrd_irrefl PosOrd_trans by blast 
+
+text {*
+  :\<sqsubseteq>val and :\<sqsubset>val are partial orders.
+*}
+
+lemma PosOrd_ordering:
+  shows "ordering (\<lambda>v1 v2. v1 :\<sqsubseteq>val v2) (\<lambda> v1 v2. v1 :\<sqsubset>val v2)"
+unfolding ordering_def PosOrd_ex_eq_def
+apply(auto)
+using PosOrd_irrefl apply blast
+using PosOrd_assym apply blast
+using PosOrd_trans by blast
+
+lemma PosOrd_order:
+  shows "class.order (\<lambda>v1 v2. v1 :\<sqsubseteq>val v2) (\<lambda> v1 v2. v1 :\<sqsubset>val v2)"
+using PosOrd_ordering
+apply(simp add: class.order_def class.preorder_def class.order_axioms_def)
+unfolding ordering_def
+by blast
+
+
+lemma PosOrd_ex_eq2:
+  shows "v1 :\<sqsubset>val v2 \<longleftrightarrow> (v1 :\<sqsubseteq>val v2 \<and> v1 \<noteq> v2)"
+using PosOrd_ordering 
+unfolding ordering_def
+by auto
+
+lemma PosOrdeq_trans:
+  assumes "v1 :\<sqsubseteq>val v2" "v2 :\<sqsubseteq>val v3"
+  shows "v1 :\<sqsubseteq>val v3"
+using assms PosOrd_ordering 
+unfolding ordering_def
+by blast
+
+lemma PosOrdeq_antisym:
+  assumes "v1 :\<sqsubseteq>val v2" "v2 :\<sqsubseteq>val v1"
+  shows "v1 = v2"
+using assms PosOrd_ordering 
+unfolding ordering_def
+by blast
+
+lemma PosOrdeq_refl:
+  shows "v :\<sqsubseteq>val v" 
+unfolding PosOrd_ex_eq_def
+by auto
+
+
+lemma PosOrd_shorterE:
+  assumes "v1 :\<sqsubset>val v2" 
+  shows "length (flat v2) \<le> length (flat v1)"
+using assms unfolding PosOrd_ex_def PosOrd_def
+apply(auto)
+apply(case_tac p)
+apply(simp add: pflat_len_simps)
+apply(drule_tac x="[]" in bspec)
+apply(simp add: Pos_empty)
+apply(simp add: pflat_len_simps)
+done
+
+lemma PosOrd_shorterI:
+  assumes "length (flat v2) < length (flat v1)"
+  shows "v1 :\<sqsubset>val v2"
+unfolding PosOrd_ex_def PosOrd_def pflat_len_def 
+using assms Pos_empty by force
+
+lemma PosOrd_spreI:
+  assumes "flat v' \<sqsubset>spre flat v"
+  shows "v :\<sqsubset>val v'" 
+using assms
+apply(rule_tac PosOrd_shorterI)
+unfolding prefix_list_def sprefix_list_def
+by (metis append_Nil2 append_eq_conv_conj drop_all le_less_linear)
+
+lemma pflat_len_inside:
+  assumes "pflat_len v2 p < pflat_len v1 p"
+  shows "p \<in> Pos v1"
+using assms 
+unfolding pflat_len_def
+by (auto split: if_splits)
+
+
+lemma PosOrd_Left_Right:
+  assumes "flat v1 = flat v2"
+  shows "Left v1 :\<sqsubset>val Right v2" 
+unfolding PosOrd_ex_def
+apply(rule_tac x="[0]" in exI)
+apply(auto simp add: PosOrd_def pflat_len_simps assms)
+done
+
+lemma PosOrd_LeftE:
+  assumes "Left v1 :\<sqsubset>val Left v2" "flat v1 = flat v2"
+  shows "v1 :\<sqsubset>val v2"
+using assms
+unfolding PosOrd_ex_def PosOrd_def2
+apply(auto simp add: pflat_len_simps)
+apply(frule pflat_len_inside)
+apply(auto simp add: pflat_len_simps)
+by (metis lex_simps(3) pflat_len_simps(3))
+
+lemma PosOrd_LeftI:
+  assumes "v1 :\<sqsubset>val v2" "flat v1 = flat v2"
+  shows  "Left v1 :\<sqsubset>val Left v2"
+using assms
+unfolding PosOrd_ex_def PosOrd_def2
+apply(auto simp add: pflat_len_simps)
+by (metis less_numeral_extra(3) lex_simps(3) pflat_len_simps(3))
+
+lemma PosOrd_Left_eq:
+  assumes "flat v1 = flat v2"
+  shows "Left v1 :\<sqsubset>val Left v2 \<longleftrightarrow> v1 :\<sqsubset>val v2" 
+using assms PosOrd_LeftE PosOrd_LeftI
+by blast
+
+
+lemma PosOrd_RightE:
+  assumes "Right v1 :\<sqsubset>val Right v2" "flat v1 = flat v2"
+  shows "v1 :\<sqsubset>val v2"
+using assms
+unfolding PosOrd_ex_def PosOrd_def2
+apply(auto simp add: pflat_len_simps)
+apply(frule pflat_len_inside)
+apply(auto simp add: pflat_len_simps)
+by (metis lex_simps(3) pflat_len_simps(5))
+
+lemma PosOrd_RightI:
+  assumes "v1 :\<sqsubset>val v2" "flat v1 = flat v2"
+  shows  "Right v1 :\<sqsubset>val Right v2"
+using assms
+unfolding PosOrd_ex_def PosOrd_def2
+apply(auto simp add: pflat_len_simps)
+by (metis lex_simps(3) nat_neq_iff pflat_len_simps(5))
+
+
+lemma PosOrd_Right_eq:
+  assumes "flat v1 = flat v2"
+  shows "Right v1 :\<sqsubset>val Right v2 \<longleftrightarrow> v1 :\<sqsubset>val v2" 
+using assms PosOrd_RightE PosOrd_RightI
+by blast
+
+
+lemma PosOrd_SeqI1:
+  assumes "v1 :\<sqsubset>val w1" "flat (Seq v1 v2) = flat (Seq w1 w2)"
+  shows "Seq v1 v2 :\<sqsubset>val Seq w1 w2" 
+using assms(1)
+apply(subst (asm) PosOrd_ex_def)
+apply(subst (asm) PosOrd_def)
+apply(clarify)
+apply(subst PosOrd_ex_def)
+apply(rule_tac x="0#p" in exI)
+apply(subst PosOrd_def)
+apply(rule conjI)
+apply(simp add: pflat_len_simps)
+apply(rule ballI)
+apply(rule impI)
+apply(simp only: Pos.simps)
+apply(auto)[1]
+apply(simp add: pflat_len_simps)
+apply(auto simp add: pflat_len_simps)
+using assms(2)
+apply(simp)
+apply(metis length_append of_nat_add)
+done
+
+lemma PosOrd_SeqI2:
+  assumes "v2 :\<sqsubset>val w2" "flat v2 = flat w2"
+  shows "Seq v v2 :\<sqsubset>val Seq v w2" 
+using assms(1)
+apply(subst (asm) PosOrd_ex_def)
+apply(subst (asm) PosOrd_def)
+apply(clarify)
+apply(subst PosOrd_ex_def)
+apply(rule_tac x="Suc 0#p" in exI)
+apply(subst PosOrd_def)
+apply(rule conjI)
+apply(simp add: pflat_len_simps)
+apply(rule ballI)
+apply(rule impI)
+apply(simp only: Pos.simps)
+apply(auto)[1]
+apply(simp add: pflat_len_simps)
+using assms(2)
+apply(simp)
+apply(auto simp add: pflat_len_simps)
+done
+
+lemma PosOrd_Seq_eq:
+  assumes "flat v2 = flat w2"
+  shows "(Seq v v2) :\<sqsubset>val (Seq v w2) \<longleftrightarrow> v2 :\<sqsubset>val w2"
+using assms 
+apply(auto)
+prefer 2
+apply(simp add: PosOrd_SeqI2)
+apply(simp add: PosOrd_ex_def)
+apply(auto)
+apply(case_tac p)
+apply(simp add: PosOrd_def pflat_len_simps)
+apply(case_tac a)
+apply(simp add: PosOrd_def pflat_len_simps)
+apply(clarify)
+apply(case_tac nat)
+prefer 2
+apply(simp add: PosOrd_def pflat_len_simps pflat_len_outside)
+apply(rule_tac x="list" in exI)
+apply(auto simp add: PosOrd_def2 pflat_len_simps)
+apply(smt Collect_disj_eq lex_list.intros(2) mem_Collect_eq pflat_len_simps(2))
+apply(smt Collect_disj_eq lex_list.intros(2) mem_Collect_eq pflat_len_simps(2))
+done
+
+
+
+lemma PosOrd_StarsI:
+  assumes "v1 :\<sqsubset>val v2" "flats (v1#vs1) = flats (v2#vs2)"
+  shows "Stars (v1#vs1) :\<sqsubset>val Stars (v2#vs2)" 
+using assms(1)
+apply(subst (asm) PosOrd_ex_def)
+apply(subst (asm) PosOrd_def)
+apply(clarify)
+apply(subst PosOrd_ex_def)
+apply(subst PosOrd_def)
+apply(rule_tac x="0#p" in exI)
+apply(simp add: pflat_len_Stars_simps pflat_len_simps)
+using assms(2)
+apply(simp add: pflat_len_simps)
+apply(auto simp add: pflat_len_Stars_simps pflat_len_simps)
+by (metis length_append of_nat_add)
+
+lemma PosOrd_StarsI2:
+  assumes "Stars vs1 :\<sqsubset>val Stars vs2" "flats vs1 = flats vs2"
+  shows "Stars (v#vs1) :\<sqsubset>val Stars (v#vs2)" 
+using assms(1)
+apply(subst (asm) PosOrd_ex_def)
+apply(subst (asm) PosOrd_def)
+apply(clarify)
+apply(subst PosOrd_ex_def)
+apply(subst PosOrd_def)
+apply(case_tac p)
+apply(simp add: pflat_len_simps)
+apply(rule_tac x="Suc a#list" in exI)
+apply(auto simp add: pflat_len_Stars_simps pflat_len_simps assms(2))
+done
+
+lemma PosOrd_Stars_appendI:
+  assumes "Stars vs1 :\<sqsubset>val Stars vs2" "flat (Stars vs1) = flat (Stars vs2)"
+  shows "Stars (vs @ vs1) :\<sqsubset>val Stars (vs @ vs2)"
+using assms
+apply(induct vs)
+apply(simp)
+apply(simp add: PosOrd_StarsI2)
+done
+
+lemma PosOrd_eq_Stars_zipI:
+  assumes "\<forall>(v1, v2) \<in> set (zip vs1 vs2). v1 :\<sqsubseteq>val v2" 
+     "length vs1 = length vs2" "flats vs1 = flats vs2"
+  shows "Stars vs1 :\<sqsubseteq>val Stars vs2"
+  using assms
+  apply(induct vs1 arbitrary: vs2)
+   apply(case_tac vs2)
+apply(simp add: PosOrd_ex_eq_def)    
+   apply(simp)
+  apply(case_tac vs2)
+   apply(simp)
+  apply(simp)
+  apply(auto)
+apply(subst (asm) (2)PosOrd_ex_eq_def)
+  apply(auto)
+   apply(subst PosOrd_ex_eq_def)
+   apply(rule disjI1)
+   apply(rule PosOrd_StarsI)
+    apply(simp)
+   apply(simp)
+  using PosOrd_StarsI2 PosOrd_ex_eq_def by fastforce
+  
+lemma PosOrd_StarsE2:
+  assumes "Stars (v # vs1) :\<sqsubset>val Stars (v # vs2)"
+  shows "Stars vs1 :\<sqsubset>val Stars vs2"
+using assms
+apply(subst (asm) PosOrd_ex_def)
+apply(erule exE)
+apply(case_tac p)
+apply(simp)
+apply(simp add: PosOrd_def pflat_len_simps)
+apply(subst PosOrd_ex_def)
+apply(rule_tac x="[]" in exI)
+apply(simp add: PosOrd_def pflat_len_simps Pos_empty)
+apply(simp)
+apply(case_tac a)
+apply(clarify)
+apply(auto simp add: pflat_len_simps PosOrd_def pflat_len_def split: if_splits)[1]
+apply(clarify)
+apply(simp add: PosOrd_ex_def)
+apply(rule_tac x="nat#list" in exI)
+apply(auto simp add: PosOrd_def pflat_len_simps)[1]
+apply(case_tac q)
+apply(simp add: PosOrd_def pflat_len_simps)
+apply(clarify)
+apply(drule_tac x="Suc a # lista" in bspec)
+apply(simp)
+apply(auto simp add: PosOrd_def pflat_len_simps)[1]
+apply(case_tac q)
+apply(simp add: PosOrd_def pflat_len_simps)
+apply(clarify)
+apply(drule_tac x="Suc a # lista" in bspec)
+apply(simp)
+apply(auto simp add: PosOrd_def pflat_len_simps)[1]
+done
+
+lemma PosOrd_Stars_appendE:
+  assumes "Stars (vs @ vs1) :\<sqsubset>val Stars (vs @ vs2)"
+  shows "Stars vs1 :\<sqsubset>val Stars vs2"
+using assms
+apply(induct vs)
+apply(simp)
+apply(simp add: PosOrd_StarsE2)
+done
+
+lemma PosOrd_Stars_append_eq:
+  assumes "flats vs1 = flats vs2"
+  shows "Stars (vs @ vs1) :\<sqsubset>val Stars (vs @ vs2) \<longleftrightarrow> Stars vs1 :\<sqsubset>val Stars vs2"
+using assms
+apply(rule_tac iffI)
+apply(erule PosOrd_Stars_appendE)
+apply(rule PosOrd_Stars_appendI)
+apply(auto)
+done  
+
+lemma PosOrd_almost_trichotomous:
+  shows "v1 :\<sqsubset>val v2 \<or> v2 :\<sqsubset>val v1 \<or> (length (flat v1) = length (flat v2))"
+apply(auto simp add: PosOrd_ex_def)
+apply(auto simp add: PosOrd_def)
+apply(rule_tac x="[]" in exI)
+apply(auto simp add: Pos_empty pflat_len_simps)
+apply(drule_tac x="[]" in spec)
+apply(auto simp add: Pos_empty pflat_len_simps)
+done
+
+
+
+section {* The Posix Value is smaller than any other Value *}
+
+
+lemma Posix_PosOrd:
+  assumes "s \<in> r \<rightarrow> v1" "v2 \<in> LV r s" 
+  shows "v1 :\<sqsubseteq>val v2"
+using assms
+proof (induct arbitrary: v2 rule: Posix.induct)
+  case (Posix_ONE v)
+  have "v \<in> LV ONE []" by fact
+  then have "v = Void"
+    by (simp add: LV_simps)
+  then show "Void :\<sqsubseteq>val v"
+    by (simp add: PosOrd_ex_eq_def)
+next
+  case (Posix_CHAR c v)
+  have "v \<in> LV (CHAR c) [c]" by fact
+  then have "v = Char c"
+    by (simp add: LV_simps)
+  then show "Char c :\<sqsubseteq>val v"
+    by (simp add: PosOrd_ex_eq_def)
+next
+  case (Posix_ALT1 s r1 v r2 v2)
+  have as1: "s \<in> r1 \<rightarrow> v" by fact
+  have IH: "\<And>v2. v2 \<in> LV r1 s \<Longrightarrow> v :\<sqsubseteq>val v2" by fact
+  have "v2 \<in> LV (ALT r1 r2) s" by fact
+  then have "\<Turnstile> v2 : ALT r1 r2" "flat v2 = s"
+    by(auto simp add: LV_def prefix_list_def)
+  then consider
+    (Left) v3 where "v2 = Left v3" "\<Turnstile> v3 : r1" "flat v3 = s" 
+  | (Right) v3 where "v2 = Right v3" "\<Turnstile> v3 : r2" "flat v3 = s"
+  by (auto elim: Prf.cases)
+  then show "Left v :\<sqsubseteq>val v2"
+  proof(cases)
+     case (Left v3)
+     have "v3 \<in> LV r1 s" using Left(2,3) 
+       by (auto simp add: LV_def prefix_list_def)
+     with IH have "v :\<sqsubseteq>val v3" by simp
+     moreover
+     have "flat v3 = flat v" using as1 Left(3)
+       by (simp add: Posix1(2)) 
+     ultimately have "Left v :\<sqsubseteq>val Left v3"
+       by (simp add: PosOrd_ex_eq_def PosOrd_Left_eq)
+     then show "Left v :\<sqsubseteq>val v2" unfolding Left .
+  next
+     case (Right v3)
+     have "flat v3 = flat v" using as1 Right(3)
+       by (simp add: Posix1(2)) 
+     then have "Left v :\<sqsubseteq>val Right v3" 
+       unfolding PosOrd_ex_eq_def
+       by (simp add: PosOrd_Left_Right)
+     then show "Left v :\<sqsubseteq>val v2" unfolding Right .
+  qed
+next
+  case (Posix_ALT2 s r2 v r1 v2)
+  have as1: "s \<in> r2 \<rightarrow> v" by fact
+  have as2: "s \<notin> L r1" by fact
+  have IH: "\<And>v2. v2 \<in> LV r2 s \<Longrightarrow> v :\<sqsubseteq>val v2" by fact
+  have "v2 \<in> LV (ALT r1 r2) s" by fact
+  then have "\<Turnstile> v2 : ALT r1 r2" "flat v2 = s"
+    by(auto simp add: LV_def prefix_list_def)
+  then consider
+    (Left) v3 where "v2 = Left v3" "\<Turnstile> v3 : r1" "flat v3 = s" 
+  | (Right) v3 where "v2 = Right v3" "\<Turnstile> v3 : r2" "flat v3 = s"
+  by (auto elim: Prf.cases)
+  then show "Right v :\<sqsubseteq>val v2"
+  proof (cases)
+    case (Right v3)
+     have "v3 \<in> LV r2 s" using Right(2,3) 
+       by (auto simp add: LV_def prefix_list_def)
+     with IH have "v :\<sqsubseteq>val v3" by simp
+     moreover
+     have "flat v3 = flat v" using as1 Right(3)
+       by (simp add: Posix1(2)) 
+     ultimately have "Right v :\<sqsubseteq>val Right v3" 
+        by (auto simp add: PosOrd_ex_eq_def PosOrd_RightI)
+     then show "Right v :\<sqsubseteq>val v2" unfolding Right .
+  next
+     case (Left v3)
+     have "v3 \<in> LV r1 s" using Left(2,3) as2  
+       by (auto simp add: LV_def prefix_list_def)
+     then have "flat v3 = flat v \<and> \<Turnstile> v3 : r1" using as1 Left(3)
+       by (simp add: Posix1(2) LV_def) 
+     then have "False" using as1 as2 Left
+       by (auto simp add: Posix1(2) L_flat_Prf1)
+     then show "Right v :\<sqsubseteq>val v2" by simp
+  qed
+next 
+  case (Posix_SEQ s1 r1 v1 s2 r2 v2 v3)
+  have "s1 \<in> r1 \<rightarrow> v1" "s2 \<in> r2 \<rightarrow> v2" by fact+
+  then have as1: "s1 = flat v1" "s2 = flat v2" by (simp_all add: Posix1(2))
+  have IH1: "\<And>v3. v3 \<in> LV r1 s1 \<Longrightarrow> v1 :\<sqsubseteq>val v3" by fact
+  have IH2: "\<And>v3. v3 \<in> LV r2 s2 \<Longrightarrow> v2 :\<sqsubseteq>val v3" by fact
+  have cond: "\<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
+  have "v3 \<in> LV (SEQ r1 r2) (s1 @ s2)" by fact
+  then obtain v3a v3b where eqs:
+    "v3 = Seq v3a v3b" "\<Turnstile> v3a : r1" "\<Turnstile> v3b : r2"
+    "flat v3a @ flat v3b = s1 @ s2" 
+    by (force simp add: prefix_list_def LV_def elim: Prf.cases)
+  with cond have "flat v3a \<sqsubseteq>pre s1" unfolding prefix_list_def
+    by (smt L_flat_Prf1 append_eq_append_conv2 append_self_conv)
+  then have "flat v3a \<sqsubset>spre s1 \<or> (flat v3a = s1 \<and> flat v3b = s2)" using eqs
+    by (simp add: sprefix_list_def append_eq_conv_conj)
+  then have q2: "v1 :\<sqsubset>val v3a \<or> (flat v3a = s1 \<and> flat v3b = s2)" 
+    using PosOrd_spreI as1(1) eqs by blast
+  then have "v1 :\<sqsubset>val v3a \<or> (v3a \<in> LV r1 s1 \<and> v3b \<in> LV r2 s2)" using eqs(2,3)
+    by (auto simp add: LV_def)
+  then have "v1 :\<sqsubset>val v3a \<or> (v1 :\<sqsubseteq>val v3a \<and> v2 :\<sqsubseteq>val v3b)" using IH1 IH2 by blast         
+  then have "Seq v1 v2 :\<sqsubseteq>val Seq v3a v3b" using eqs q2 as1
+    unfolding  PosOrd_ex_eq_def by (auto simp add: PosOrd_SeqI1 PosOrd_Seq_eq) 
+  then show "Seq v1 v2 :\<sqsubseteq>val v3" unfolding eqs by blast
+next 
+  case (Posix_STAR1 s1 r v s2 vs v3) 
+  have "s1 \<in> r \<rightarrow> v" "s2 \<in> STAR r \<rightarrow> Stars vs" by fact+
+  then have as1: "s1 = flat v" "s2 = flat (Stars vs)" by (auto dest: Posix1(2))
+  have IH1: "\<And>v3. v3 \<in> LV r s1 \<Longrightarrow> v :\<sqsubseteq>val v3" by fact
+  have IH2: "\<And>v3. v3 \<in> LV (STAR r) s2 \<Longrightarrow> Stars vs :\<sqsubseteq>val v3" by fact
+  have cond: "\<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
+  have cond2: "flat v \<noteq> []" by fact
+  have "v3 \<in> LV (STAR r) (s1 @ s2)" by fact
+  then consider 
+    (NonEmpty) v3a vs3 where "v3 = Stars (v3a # vs3)" 
+    "\<Turnstile> v3a : r" "\<Turnstile> Stars vs3 : STAR r"
+    "flat (Stars (v3a # vs3)) = s1 @ s2"
+  | (Empty) "v3 = Stars []"
+  unfolding LV_def  
+  apply(auto)
+  apply(erule Prf.cases)
+  apply(auto)
+  apply(case_tac vs)
+  apply(auto intro: Prf.intros)
+  done
+  then show "Stars (v # vs) :\<sqsubseteq>val v3" 
+    proof (cases)
+      case (NonEmpty v3a vs3)
+      have "flat (Stars (v3a # vs3)) = s1 @ s2" using NonEmpty(4) . 
+      with cond have "flat v3a \<sqsubseteq>pre s1" using NonEmpty(2,3)
+        unfolding prefix_list_def
+        by (smt L_flat_Prf1 append_Nil2 append_eq_append_conv2 flat.simps(7)) 
+      then have "flat v3a \<sqsubset>spre s1 \<or> (flat v3a = s1 \<and> flat (Stars vs3) = s2)" using NonEmpty(4)
+        by (simp add: sprefix_list_def append_eq_conv_conj)
+      then have q2: "v :\<sqsubset>val v3a \<or> (flat v3a = s1 \<and> flat (Stars vs3) = s2)" 
+        using PosOrd_spreI as1(1) NonEmpty(4) by blast
+      then have "v :\<sqsubset>val v3a \<or> (v3a \<in> LV r s1 \<and> Stars vs3 \<in> LV (STAR r) s2)" 
+        using NonEmpty(2,3) by (auto simp add: LV_def)
+      then have "v :\<sqsubset>val v3a \<or> (v :\<sqsubseteq>val v3a \<and> Stars vs :\<sqsubseteq>val Stars vs3)" using IH1 IH2 by blast
+      then have "v :\<sqsubset>val v3a \<or> (v = v3a \<and> Stars vs :\<sqsubseteq>val Stars vs3)" 
+         unfolding PosOrd_ex_eq_def by auto     
+      then have "Stars (v # vs) :\<sqsubseteq>val Stars (v3a # vs3)" using NonEmpty(4) q2 as1
+        unfolding  PosOrd_ex_eq_def
+        using PosOrd_StarsI PosOrd_StarsI2 by auto 
+      then show "Stars (v # vs) :\<sqsubseteq>val v3" unfolding NonEmpty by blast
+    next 
+      case Empty
+      have "v3 = Stars []" by fact
+      then show "Stars (v # vs) :\<sqsubseteq>val v3"
+      unfolding PosOrd_ex_eq_def using cond2
+      by (simp add: PosOrd_shorterI)
+    qed      
+next 
+  case (Posix_STAR2 r v2)
+  have "v2 \<in> LV (STAR r) []" by fact
+  then have "v2 = Stars []" 
+    unfolding LV_def by (auto elim: Prf.cases) 
+  then show "Stars [] :\<sqsubseteq>val v2"
+    by (simp add: PosOrd_ex_eq_def)
+next 
+  case (Posix_NTIMES1 s1 r v s2 n vs v3) 
+  have "s1 \<in> r \<rightarrow> v" "s2 \<in> NTIMES r (n - 1) \<rightarrow> Stars vs" by fact+
+  then have as1: "s1 = flat v" "s2 = flats vs" by (auto dest: Posix1(2))
+  have IH1: "\<And>v3. v3 \<in> LV r s1 \<Longrightarrow> v :\<sqsubseteq>val v3" by fact
+  have IH2: "\<And>v3. v3 \<in> LV (NTIMES r (n - 1)) s2 \<Longrightarrow> Stars vs :\<sqsubseteq>val v3" by fact
+  have cond: "\<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 (NTIMES r (n - 1)))" by fact
+  have cond2: "flat v \<noteq> []" by fact
+  have "v3 \<in> LV (NTIMES r n) (s1 @ s2)" by fact
+  then consider 
+    (NonEmpty) v3a vs3 where "v3 = Stars (v3a # vs3)" 
+    "\<Turnstile> v3a : r" "\<Turnstile> Stars vs3 : NTIMES r (n - 1)"
+    "flats (v3a # vs3) = s1 @ s2"
+  | (Empty) "v3 = Stars []"
+  unfolding LV_def  
+  apply(auto)
+  apply(erule Prf.cases)
+             apply(auto)  
+  apply(case_tac vs1)
+   apply(auto intro: Prf.intros)
+   apply(case_tac vs2)
+    apply(auto intro: Prf.intros)
+  apply (simp add: as1(1) cond2 flats_empty)
+  by (simp add: Prf.intros(8))
+  then show "Stars (v # vs) :\<sqsubseteq>val v3" 
+    proof (cases)
+      case (NonEmpty v3a vs3)
+      have "flats (v3a # vs3) = s1 @ s2" using NonEmpty(4) . 
+      with cond have "flat v3a \<sqsubseteq>pre s1" using NonEmpty(2,3)
+        unfolding prefix_list_def
+        by (smt L_flat_Prf1 append_Nil2 append_eq_append_conv2 flat.simps(7) flat_Stars)
+      then have "flat v3a \<sqsubset>spre s1 \<or> (flat v3a = s1 \<and> flat (Stars vs3) = s2)" using NonEmpty(4)
+        by (simp add: sprefix_list_def append_eq_conv_conj)
+      then have q2: "v :\<sqsubset>val v3a \<or> (flat v3a = s1 \<and> flat (Stars vs3) = s2)" 
+        using PosOrd_spreI as1(1) NonEmpty(4) by blast
+      then have "v :\<sqsubset>val v3a \<or> (v3a \<in> LV r s1 \<and> Stars vs3 \<in> LV (NTIMES r (n - 1)) s2)" 
+        using NonEmpty(2,3) by (auto simp add: LV_def)
+      then have "v :\<sqsubset>val v3a \<or> (v :\<sqsubseteq>val v3a \<and> Stars vs :\<sqsubseteq>val Stars vs3)" using IH1 IH2 by blast
+      then have "v :\<sqsubset>val v3a \<or> (v = v3a \<and> Stars vs :\<sqsubseteq>val Stars vs3)" 
+         unfolding PosOrd_ex_eq_def by auto     
+      then have "Stars (v # vs) :\<sqsubseteq>val Stars (v3a # vs3)" using NonEmpty(4) q2 as1
+        unfolding  PosOrd_ex_eq_def
+        using PosOrd_StarsI PosOrd_StarsI2 by auto 
+      then show "Stars (v # vs) :\<sqsubseteq>val v3" unfolding NonEmpty by blast
+    next 
+      case Empty
+      have "v3 = Stars []" by fact
+      then show "Stars (v # vs) :\<sqsubseteq>val v3"
+      unfolding PosOrd_ex_eq_def using cond2
+      by (simp add: PosOrd_shorterI)
+  qed 
+next
+  case (Posix_NTIMES2 vs r n v2) 
+  then show "Stars vs :\<sqsubseteq>val v2"
+    apply(simp add: LV_def)
+    apply(auto)  
+    apply(erule Prf_elims)
+    apply(auto)
+    apply(rule PosOrd_eq_Stars_zipI) 
+      prefer 2
+      apply(simp)
+     prefer 2
+     apply (metis Posix1(2) flats_empty)
+    apply(auto)
+    by (meson in_set_zipE)
+next
+  case (Posix_UPNTIMES2 r n v2)
+    then show "Stars [] :\<sqsubseteq>val v2"
+    apply(simp add: LV_def)
+      apply(auto)  
+    apply(erule Prf_elims)
+      apply(auto)
+      unfolding PosOrd_ex_eq_def by simp
+next 
+  case (Posix_UPNTIMES1 s1 r v s2 n vs v3)
+  have "s1 \<in> r \<rightarrow> v" "s2 \<in> UPNTIMES r (n - 1) \<rightarrow> Stars vs" by fact+
+  then have as1: "s1 = flat v" "s2 = flat (Stars vs)" by (auto dest: Posix1(2))
+  have IH1: "\<And>v3. v3 \<in> LV r s1 \<Longrightarrow> v :\<sqsubseteq>val v3" by fact
+  have IH2: "\<And>v3. v3 \<in> LV (UPNTIMES r (n - 1)) s2 \<Longrightarrow> Stars vs :\<sqsubseteq>val v3" by fact
+  have cond: "\<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 (UPNTIMES r (n - 1)))" by fact
+  have cond2: "flat v \<noteq> []" by fact
+  have "v3 \<in> LV (UPNTIMES r n) (s1 @ s2)" by fact
+  then consider 
+    (NonEmpty) v3a vs3 where "v3 = Stars (v3a # vs3)" 
+    "\<Turnstile> v3a : r" "\<Turnstile> Stars vs3 : UPNTIMES r (n - 1)" 
+    "flats (v3a # vs3) = s1 @ s2"
+  | (Empty) "v3 = Stars []"
+  unfolding LV_def  
+  apply(auto)
+  apply(erule Prf.cases)
+  apply(auto)
+  apply(case_tac vs)
+   apply(auto intro: Prf.intros)
+  by (simp add: Prf.intros(7) as1(1) cond2)
+  then show "Stars (v # vs) :\<sqsubseteq>val v3" 
+    proof (cases)
+      case (NonEmpty v3a vs3)
+      have "flats (v3a # vs3) = s1 @ s2" using NonEmpty(4) . 
+      with cond have "flat v3a \<sqsubseteq>pre s1" using NonEmpty(2,3)
+        unfolding prefix_list_def
+        apply(simp)
+        apply(simp add: append_eq_append_conv2)
+        apply(auto)
+        by (metis L_flat_Prf1 One_nat_def cond flat_Stars)
+      then have "flat v3a \<sqsubset>spre s1 \<or> (flat v3a = s1 \<and> flat (Stars vs3) = s2)" using NonEmpty(4)
+        by (simp add: sprefix_list_def append_eq_conv_conj)
+      then have q2: "v :\<sqsubset>val v3a \<or> (flat v3a = s1 \<and> flat (Stars vs3) = s2)" 
+        using PosOrd_spreI as1(1) NonEmpty(4) by blast
+      then have "v :\<sqsubset>val v3a \<or> (v3a \<in> LV r s1 \<and> Stars vs3 \<in> LV (UPNTIMES r (n - 1)) s2)" 
+        using NonEmpty(2,3) by (auto simp add: LV_def)
+      then have "v :\<sqsubset>val v3a \<or> (v :\<sqsubseteq>val v3a \<and> Stars vs :\<sqsubseteq>val Stars vs3)" using IH1 IH2 by blast
+      then have "v :\<sqsubset>val v3a \<or> (v = v3a \<and> Stars vs :\<sqsubseteq>val Stars vs3)" 
+         unfolding PosOrd_ex_eq_def by auto     
+      then have "Stars (v # vs) :\<sqsubseteq>val Stars (v3a # vs3)" using NonEmpty(4) q2 as1
+        unfolding  PosOrd_ex_eq_def
+        using PosOrd_StarsI PosOrd_StarsI2 by auto 
+      then show "Stars (v # vs) :\<sqsubseteq>val v3" unfolding NonEmpty by blast
+    next 
+      case Empty
+      have "v3 = Stars []" by fact
+      then show "Stars (v # vs) :\<sqsubseteq>val v3"
+      unfolding PosOrd_ex_eq_def using cond2
+      by (simp add: PosOrd_shorterI)
+  qed        
+next
+  case (Posix_FROMNTIMES2 vs r n v2)
+    then show "Stars vs :\<sqsubseteq>val v2"
+    apply(simp add: LV_def)
+      apply(auto)  
+    apply(erule Prf_elims)
+       apply(auto)
+        apply(rule PosOrd_eq_Stars_zipI) 
+      prefer 2
+      apply(simp)
+     prefer 2
+     apply (metis Posix1(2) flats_empty)
+    apply(auto)
+      by (meson in_set_zipE)
+next 
+  case (Posix_FROMNTIMES1 s1 r v s2 n vs v3) 
+  have "s1 \<in> r \<rightarrow> v" "s2 \<in> FROMNTIMES r (n - 1) \<rightarrow> Stars vs" by fact+
+  then have as1: "s1 = flat v" "s2 = flats vs" by (auto dest: Posix1(2))
+  have IH1: "\<And>v3. v3 \<in> LV r s1 \<Longrightarrow> v :\<sqsubseteq>val v3" by fact
+  have IH2: "\<And>v3. v3 \<in> LV (FROMNTIMES r (n - 1)) s2 \<Longrightarrow> Stars vs :\<sqsubseteq>val v3" by fact
+  have cond: "\<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 (FROMNTIMES r (n - 1)))" by fact
+  have cond2: "flat v \<noteq> []" by fact
+  have "v3 \<in> LV (FROMNTIMES r n) (s1 @ s2)" by fact
+  then consider 
+    (NonEmpty) v3a vs3 where "v3 = Stars (v3a # vs3)" 
+    "\<Turnstile> v3a : r" "\<Turnstile> Stars vs3 : FROMNTIMES r (n - 1)"
+    "flats (v3a # vs3) = s1 @ s2"
+  | (Empty) "v3 = Stars []" 
+  unfolding LV_def  
+  apply(auto)
+  apply(erule Prf.cases)
+             apply(auto)  
+  apply(case_tac vs1)
+   apply(auto intro: Prf.intros)
+   apply(case_tac vs2)
+    apply(auto intro: Prf.intros)
+    apply (simp add: as1(1) cond2 flats_empty)
+  apply (simp add: Prf.intros)
+  apply(case_tac vs)
+   apply(auto)
+  using Posix_FROMNTIMES1.hyps(6) Prf.intros(10) by auto
+  then show "Stars (v # vs) :\<sqsubseteq>val v3" 
+    proof (cases)
+      case (NonEmpty v3a vs3)
+      have "flats (v3a # vs3) = s1 @ s2" using NonEmpty(4) . 
+      with cond have "flat v3a \<sqsubseteq>pre s1" using NonEmpty(2,3)
+        unfolding prefix_list_def
+        by (smt L_flat_Prf1 append_Nil2 append_eq_append_conv2 flat.simps(7) flat_Stars)
+      then have "flat v3a \<sqsubset>spre s1 \<or> (flat v3a = s1 \<and> flat (Stars vs3) = s2)" using NonEmpty(4)
+        by (simp add: sprefix_list_def append_eq_conv_conj)
+      then have q2: "v :\<sqsubset>val v3a \<or> (flat v3a = s1 \<and> flat (Stars vs3) = s2)" 
+        using PosOrd_spreI as1(1) NonEmpty(4) by blast
+      then have "v :\<sqsubset>val v3a \<or> (v3a \<in> LV r s1 \<and> Stars vs3 \<in> LV (FROMNTIMES r (n - 1)) s2)" 
+        using NonEmpty(2,3) by (auto simp add: LV_def)
+      then have "v :\<sqsubset>val v3a \<or> (v :\<sqsubseteq>val v3a \<and> Stars vs :\<sqsubseteq>val Stars vs3)" using IH1 IH2 by blast
+      then have "v :\<sqsubset>val v3a \<or> (v = v3a \<and> Stars vs :\<sqsubseteq>val Stars vs3)" 
+         unfolding PosOrd_ex_eq_def by auto     
+      then have "Stars (v # vs) :\<sqsubseteq>val Stars (v3a # vs3)" using NonEmpty(4) q2 as1
+        unfolding  PosOrd_ex_eq_def
+        using PosOrd_StarsI PosOrd_StarsI2 by auto 
+      then show "Stars (v # vs) :\<sqsubseteq>val v3" unfolding NonEmpty by blast
+    next 
+      case Empty
+      have "v3 = Stars []" by fact
+      then show "Stars (v # vs) :\<sqsubseteq>val v3"
+      unfolding PosOrd_ex_eq_def using cond2
+      by (simp add: PosOrd_shorterI)
+  qed         
+next
+  case (Posix_NMTIMES2 vs r n m v2) 
+  then show "Stars vs :\<sqsubseteq>val v2" sorry
+next
+  case (Posix_NMTIMES1 s1 r v s2 n m vs v2) 
+  then show "Stars (v # vs) :\<sqsubseteq>val v2" sorry      
+qed
+
+
+lemma Posix_PosOrd_reverse:
+  assumes "s \<in> r \<rightarrow> v1" 
+  shows "\<not>(\<exists>v2 \<in> LV r s. v2 :\<sqsubset>val v1)"
+using assms
+by (metis Posix_PosOrd less_irrefl PosOrd_def 
+    PosOrd_ex_eq_def PosOrd_ex_def PosOrd_trans)
+
+lemma PosOrd_Posix:
+  assumes "v1 \<in> LV r s" "\<forall>v\<^sub>2 \<in> LV r s. \<not> v\<^sub>2 :\<sqsubset>val v1"
+  shows "s \<in> r \<rightarrow> v1" 
+proof -
+  have "s \<in> L r" using assms(1) unfolding LV_def
+    using L_flat_Prf1 by blast 
+  then obtain vposix where vp: "s \<in> r \<rightarrow> vposix"
+    using lexer_correct_Some by blast 
+  with assms(1) have "vposix :\<sqsubseteq>val v1" by (simp add: Posix_PosOrd) 
+  then have "vposix = v1 \<or> vposix :\<sqsubset>val v1" unfolding PosOrd_ex_eq2 by auto
+  moreover
+    { assume "vposix :\<sqsubset>val v1"
+      moreover
+      have "vposix \<in> LV r s" using vp 
+         using Posix_LV by blast 
+      ultimately have "False" using assms(2) by blast
+    }
+  ultimately show "s \<in> r \<rightarrow> v1" using vp by blast
+qed
+
+lemma Least_existence:
+  assumes "LV r s \<noteq> {}"
+  shows " \<exists>vmin \<in> LV r s. \<forall>v \<in> LV r s. vmin :\<sqsubseteq>val v"
+proof -
+  from assms
+  obtain vposix where "s \<in> r \<rightarrow> vposix"
+  unfolding LV_def 
+  using L_flat_Prf1 lexer_correct_Some by blast
+  then have "\<forall>v \<in> LV r s. vposix :\<sqsubseteq>val v"
+    by (simp add: Posix_PosOrd)
+  then show "\<exists>vmin \<in> LV r s. \<forall>v \<in> LV r s. vmin :\<sqsubseteq>val v"
+    using Posix_LV \<open>s \<in> r \<rightarrow> vposix\<close> by blast
+qed 
+
+lemma Least_existence1:
+  assumes "LV r s \<noteq> {}"
+  shows " \<exists>!vmin \<in> LV r s. \<forall>v \<in> LV r s. vmin :\<sqsubseteq>val v"
+using Least_existence[OF assms] assms
+  using PosOrdeq_antisym by blast
+
+
+
+
+
+lemma Least_existence1_pre:
+  assumes "LV r s \<noteq> {}"
+  shows " \<exists>!vmin \<in> LV r s. \<forall>v \<in> (LV r s \<union> {v'. flat v' \<sqsubset>spre s}). vmin :\<sqsubseteq>val v"
+using Least_existence[OF assms] assms
+apply -
+apply(erule bexE)
+apply(rule_tac a="vmin" in ex1I)
+apply(auto)[1]
+apply (metis PosOrd_Posix PosOrd_ex_eq2 PosOrd_spreI PosOrdeq_antisym Posix1(2))
+apply(auto)[1]
+apply(simp add: PosOrdeq_antisym)
+done
+
+lemma
+  shows "partial_order_on UNIV {(v1, v2). v1 :\<sqsubseteq>val v2}"
+apply(simp add: partial_order_on_def)
+apply(simp add: preorder_on_def refl_on_def)
+apply(simp add: PosOrdeq_refl)
+apply(auto)
+apply(rule transI)
+apply(auto intro: PosOrdeq_trans)[1]
+apply(rule antisymI)
+apply(simp add: PosOrdeq_antisym)
+done
+
+lemma
+ "wf {(v1, v2). v1 :\<sqsubset>val v2 \<and> v1 \<in> LV r s \<and> v2 \<in> LV r s}"
+apply(rule finite_acyclic_wf)
+prefer 2
+apply(simp add: acyclic_def)
+apply(induct_tac rule: trancl.induct)
+apply(auto)[1]
+oops
+
+
+unused_thms
+
+end
\ No newline at end of file
diff -r deea42c83c9e -r a3134f7de065 thys/SpecExt.thy
--- a/thys/SpecExt.thy	Thu Oct 05 12:45:13 2017 +0100
+++ b/thys/SpecExt.thy	Sat Oct 07 22:16:16 2017 +0100
@@ -311,8 +311,9 @@
 apply(subst Pow_Sequ_Un)
 apply(simp)
 apply(simp only: Der_union Der_empty)
-apply(simp)
-apply(simp add: nullable_correctness del: Der_UNION)
+    apply(simp)
+(* FROMNTIMES *)    
+   apply(simp add: nullable_correctness del: Der_UNION)
 apply(subst Sequ_Union_in)
 apply(subst Der_Pow_Sequ[symmetric])
 apply(subst Pow.simps[symmetric])
@@ -326,7 +327,8 @@
 apply(auto simp add: Sequ_def)[1]
 apply(drule Pow_decomp)
 apply(auto)[1]
-apply (metis append_Cons)
+   apply (metis append_Cons)
+(* NMTIMES *)    
 apply(simp add: nullable_correctness del: Der_UNION)
 apply(rule impI)
 apply(rule conjI)
@@ -663,10 +665,10 @@
   case (FROMNTIMES r n)
   have IH: "\<And>s. s \<in> L r \<Longrightarrow> \<exists>v. \<Turnstile> v : r \<and> flat v = s" by fact
   have "s \<in> L (FROMNTIMES r n)" by fact 
-  then obtain ss1 ss2 m where "concat (ss1 @ ss2) = s" "length (ss1 @ ss2) = m"  "n \<le> m"
+  then obtain ss1 ss2 k where "concat (ss1 @ ss2) = s" "length (ss1 @ ss2) = k"  "n \<le> k"
     "\<forall>s \<in> set ss1. s \<in> L r \<and> s \<noteq> []" "\<forall>s \<in> set ss2. s \<in> L r \<and> s = []"
     using Pow_cstring by force 
-  then obtain vs1 vs2 where "flats (vs1 @ vs2) = s" "length (vs1 @ vs2) = m" "n \<le> m"
+  then obtain vs1 vs2 where "flats (vs1 @ vs2) = s" "length (vs1 @ vs2) = k" "n \<le> k"
       "\<forall>v\<in>set vs1. \<Turnstile> v : r \<and> flat v \<noteq> []" "\<forall>v\<in>set vs2. \<Turnstile> v : r \<and> flat v = []"
   using IH flats_cval 
   apply -
@@ -723,11 +725,26 @@
   apply(simp)
   done
   then show "\<exists>v. \<Turnstile> v : NMTIMES r n m \<and> flat v = s"
-  apply(rule_tac x="Stars (vs1 @ vs2)" in exI)
+    apply(case_tac "length vs1 \<le> n")
+  apply(rule_tac x="Stars (vs1 @ take (n - length vs1) vs2)" in exI)
   apply(simp)
-  apply(rule Prf.intros)
-       apply(auto) 
-  sorry
+  apply(subgoal_tac "flats (take (n - length vs1) vs2) = []")
+  prefer 2
+  apply (meson flats_empty in_set_takeD)
+  apply(clarify)
+    apply(rule conjI)
+      apply(rule Prf.intros)
+        apply(simp)
+       apply (meson in_set_takeD)
+      apply(simp)
+     apply(simp)
+     apply (simp add: flats_empty)
+      apply(rule_tac x="Stars vs1" in exI)
+  apply(simp)
+    apply(rule conjI)
+     apply(rule Prf.intros)
+      apply(auto)
+  done    
 next 
   case (UPNTIMES r n s)
   have IH: "\<And>s. s \<in> L r \<Longrightarrow> \<exists>v. \<Turnstile> v : r \<and> flat v = s" by fact
@@ -806,6 +823,58 @@
   ultimately show "finite (Prefixes s)" by simp
 qed
 
+definition
+  "Stars_Cons V Vs \<equiv> {Stars (v # vs) | v vs. v \<in> V \<and> Stars vs \<in> Vs}"
+  
+definition
+  "Stars_Append Vs1 Vs2 \<equiv> {Stars (vs1 @ vs2) | vs1 vs2. Stars vs1 \<in> Vs1 \<and> Stars vs2 \<in> Vs2}"
+
+fun Stars_Pow :: "val set \<Rightarrow> nat \<Rightarrow> val set"
+where  
+  "Stars_Pow Vs 0 = {Stars []}"
+| "Stars_Pow Vs (Suc n) = Stars_Cons Vs (Stars_Pow Vs n)"
+  
+lemma finite_Stars_Cons:
+  assumes "finite V" "finite Vs"
+  shows "finite (Stars_Cons V Vs)"
+  using assms  
+proof -
+  from assms(2) have "finite (Stars -` Vs)"
+    by(simp add: finite_vimageI inj_on_def) 
+  with assms(1) have "finite (V \<times> (Stars -` Vs))"
+    by(simp)
+  then have "finite ((\<lambda>(v, vs). Stars (v # vs)) ` (V \<times> (Stars -` Vs)))"
+    by simp
+  moreover have "Stars_Cons V Vs = (\<lambda>(v, vs). Stars (v # vs)) ` (V \<times> (Stars -` Vs))"
+    unfolding Stars_Cons_def by auto    
+  ultimately show "finite (Stars_Cons V Vs)"   
+    by simp
+qed
+
+lemma finite_Stars_Append:
+  assumes "finite Vs1" "finite Vs2"
+  shows "finite (Stars_Append Vs1 Vs2)"
+  using assms  
+proof -
+  define UVs1 where "UVs1 \<equiv> Stars -` Vs1"
+  define UVs2 where "UVs2 \<equiv> Stars -` Vs2"  
+  from assms have "finite UVs1" "finite UVs2"
+    unfolding UVs1_def UVs2_def
+    by(simp_all add: finite_vimageI inj_on_def) 
+  then have "finite ((\<lambda>(vs1, vs2). Stars (vs1 @ vs2)) ` (UVs1 \<times> UVs2))"
+    by simp
+  moreover 
+    have "Stars_Append Vs1 Vs2 = (\<lambda>(vs1, vs2). Stars (vs1 @ vs2)) ` (UVs1 \<times> UVs2)"
+    unfolding Stars_Append_def UVs1_def UVs2_def by auto    
+  ultimately show "finite (Stars_Append Vs1 Vs2)"   
+    by simp
+qed 
+ 
+lemma finite_Stars_Pow:
+  assumes "finite Vs"
+  shows "finite (Stars_Pow Vs n)"    
+by (induct n) (simp_all add: finite_Stars_Cons assms)
+    
 lemma LV_STAR_finite:
   assumes "\<forall>s. finite (LV r s)"
   shows "finite (LV (STAR r) s)"
@@ -816,18 +885,20 @@
     by (auto simp add: strict_suffix_def) 
   define f where "f \<equiv> \<lambda>(v, vs). Stars (v # vs)"
   define S1 where "S1 \<equiv> \<Union>s' \<in> Prefixes s. LV r s'"
-  define S2 where "S2 \<equiv> \<Union>s2 \<in> SSuffixes s. Stars -` (LV (STAR r) s2)"
+  define S2 where "S2 \<equiv> \<Union>s2 \<in> SSuffixes s. LV (STAR r) s2"
   have "finite S1" using assms
     unfolding S1_def by (simp_all add: finite_Prefixes)
   moreover 
   with IH have "finite S2" unfolding S2_def
-    by (auto simp add: finite_SSuffixes inj_on_def finite_vimageI)
+    by (auto simp add: finite_SSuffixes)
   ultimately 
-  have "finite ({Stars []} \<union> f ` (S1 \<times> S2))" by simp
+  have "finite ({Stars []} \<union> Stars_Cons S1 S2)" 
+    by (simp add: finite_Stars_Cons)
   moreover 
-  have "LV (STAR r) s \<subseteq> {Stars []} \<union> f ` (S1 \<times> S2)" 
-  unfolding S1_def S2_def f_def
-  unfolding LV_def image_def prefix_def strict_suffix_def
+  have "LV (STAR r) s \<subseteq> {Stars []} \<union> (Stars_Cons S1 S2)" 
+  unfolding S1_def S2_def f_def LV_def Stars_Cons_def
+  unfolding prefix_def strict_suffix_def 
+  unfolding image_def
   apply(auto)
   apply(case_tac x)
   apply(auto elim: Prf_elims)
@@ -837,12 +908,12 @@
   apply(auto intro: Prf.intros)  
   apply(rule exI)
   apply(rule conjI)
-  apply(rule_tac x="flat a" in exI)
+  apply(rule_tac x="flats list" in exI)
   apply(rule conjI)
-  apply(rule_tac x="flats list" in exI)
+  apply(rule_tac x="flat a" in exI)
   apply(simp)
   apply(blast)
-  using Prf.intros(6) by blast  
+  using Prf.intros(6) flat_Stars by blast  
   ultimately
   show "finite (LV (STAR r) s)" by (simp add: finite_subset)
 qed  
@@ -851,37 +922,6 @@
   "LV (UPNTIMES r n) s \<subseteq> LV (STAR r) s"
 by(auto simp add: LV_def intro: Prf.intros elim: Prf_elims)
 
-
-lemma LV_NTIMES_0:
-  shows "LV (NTIMES r 0) s \<subseteq> {Stars []}"
-unfolding LV_def
-apply(auto elim: Prf_elims)
-done
-
-lemma LV_NTIMES_1:
-  shows "LV (NTIMES r 1) s \<subseteq> (\<lambda>v. Stars [v]) ` (LV r s)"
-unfolding LV_def
-apply(auto elim!: Prf_elims)
-apply(case_tac vs1)
-apply(simp)
-apply(case_tac vs2)
-apply(simp)
-apply(simp)
-apply(simp)
-done
-
-lemma LV_NTIMES_2:
-  shows "LV (NTIMES r 2) [] \<subseteq> (\<lambda>(v1,v2). Stars [v1,v2]) ` (LV r [] \<times> LV r [])"
-unfolding LV_def
-apply(auto elim!: Prf_elims simp add: image_def)
-apply(case_tac vs1)
-apply(auto)
-apply(case_tac vs2)
-apply(auto)
-apply(case_tac list)
-apply(auto)
-done
-
 lemma LV_NTIMES_3:
   shows "LV (NTIMES r (Suc n)) [] = (\<lambda>(v,vs). Stars (v#vs)) ` (LV r [] \<times> (Stars -` (LV (NTIMES r n) [])))"
 unfolding LV_def
@@ -896,66 +936,147 @@
 apply(subst append.simps(1)[symmetric])
 apply(rule Prf.intros)
 apply(auto)
-done
-
-thm card_cartesian_product
-
-lemma finite_list:
-  assumes "finite A"
-  shows "finite {vs. \<forall>v\<in>set vs. v \<in> A \<and> length vs = n}"
-  apply(induct n)
-   apply(simp)
-   apply (smt Collect_cong empty_iff finite.emptyI finite.insertI 
-   in_listsI list.set(1) lists_empty mem_Collect_eq singleton_conv2)
-  apply(rule_tac B="{[]} \<union> (\<lambda>(v,vs). v # vs) `(A \<times> {vs. \<forall>v\<in>set vs. v \<in> A \<and> length vs = n})" in finite_subset)  
-   apply(auto simp add: image_def)[1]
-   apply(case_tac x)
-    apply(simp)
-   apply(simp)
-  apply(simp)
-  apply(rule finite_imageI)
-    using assms
-    apply(simp)
-    done
-      
-lemma test:
-  "LV (NTIMES r n) [] \<subseteq> Stars ` {vs. \<forall>v \<in> set vs. v \<in> LV r [] \<and> length vs = n}"
-apply(auto simp add: LV_def elim: Prf_elims)
-done
+  done 
     
-lemma test3:
-  "LV (FROMNTIMES r n) [] \<subseteq> Stars ` {vs. \<forall>v \<in> set vs. v \<in> LV r [] \<and> length vs = n}"
-  apply(auto simp add: image_def LV_def elim!: Prf_elims)
-   apply blast
-  apply(case_tac vs)
+lemma LV_FROMNTIMES_3:
+  shows "LV (FROMNTIMES r (Suc n)) [] = 
+    (\<lambda>(v,vs). Stars (v#vs)) ` (LV r [] \<times> (Stars -` (LV (FROMNTIMES r n) [])))"
+unfolding LV_def
+apply(auto elim!: Prf_elims simp add: image_def)
+apply(case_tac vs1)
+apply(auto)
+apply(case_tac vs2)
+apply(auto)
+apply(subst append.simps(1)[symmetric])
+apply(rule Prf.intros)
+     apply(auto)
+  apply (metis le_imp_less_Suc length_greater_0_conv less_antisym list.exhaust list.set_intros(1) not_less_eq zero_le)
+  prefer 2
+  using nth_mem apply blast
+  apply(case_tac vs1)
+  apply (smt Groups.add_ac(2) Prf.intros(9) add.right_neutral add_Suc_right append.simps(1) insert_iff length_append list.set(2) list.size(3) list.size(4))
     apply(auto)
-  done   
-    
-    
+done     
+  
+lemma LV_NTIMES_4:
+ "LV (NTIMES r n) [] = Stars_Pow (LV r []) n" 
+  apply(induct n)
+   apply(simp add: LV_def)    
+   apply(auto elim!: Prf_elims simp add: image_def)[1]
+   apply(subst append.simps[symmetric])
+    apply(rule Prf.intros)
+      apply(simp_all)
+    apply(simp add: LV_NTIMES_3 image_def Stars_Cons_def)
+  apply blast
+ done   
 
-lemma LV_NTIMES_empty_finite:
-  assumes "finite (LV r [])"
-  shows "finite (LV (NTIMES r n) [])"
-  using assms
-  apply -  
+lemma LV_NTIMES_5:
+  "LV (NTIMES r n) s \<subseteq> Stars_Append (LV (STAR r) s) (\<Union>i\<le>n. LV (NTIMES r i) [])"
+apply(auto simp add: LV_def)
+apply(auto elim!: Prf_elims)
+  apply(auto simp add: Stars_Append_def)
+  apply(rule_tac x="vs1" in exI)
+  apply(rule_tac x="vs2" in exI)  
+  apply(auto)
+    using Prf.intros(6) apply(auto)
+      apply(rule_tac x="length vs2" in bexI)
+    thm Prf.intros
+      apply(subst append.simps(1)[symmetric])
+    apply(rule Prf.intros)
+      apply(auto)[1]
+      apply(auto)[1]
+     apply(simp)
+    apply(simp)
+      done
+      
+lemma ttty:
+ "LV (FROMNTIMES r n) [] = Stars_Pow (LV r []) n" 
+  apply(induct n)
+   apply(simp add: LV_def)    
+   apply(auto elim: Prf_elims simp add: image_def)[1]
+   prefer 2
+    apply(subst append.simps[symmetric])
+    apply(rule Prf.intros)
+      apply(simp_all)
+   apply(erule Prf_elims) 
+    apply(case_tac vs1)
+     apply(simp)
+    apply(simp)
+   apply(case_tac x)
+    apply(simp_all)
+    apply(simp add: LV_FROMNTIMES_3 image_def Stars_Cons_def)
+  apply blast
+ done     
+
+lemma LV_FROMNTIMES_5:
+  "LV (FROMNTIMES r n) s \<subseteq> Stars_Append (LV (STAR r) s) (\<Union>i\<le>n. LV (FROMNTIMES r i) [])"
+apply(auto simp add: LV_def)
+apply(auto elim!: Prf_elims)
+  apply(auto simp add: Stars_Append_def)
+  apply(rule_tac x="vs1" in exI)
+  apply(rule_tac x="vs2" in exI)  
+  apply(auto)
+    using Prf.intros(6) apply(auto)
+      apply(rule_tac x="length vs2" in bexI)
+    thm Prf.intros
+      apply(subst append.simps(1)[symmetric])
+    apply(rule Prf.intros)
+      apply(auto)[1]
+      apply(auto)[1]
+     apply(simp)
+     apply(simp)
+      apply(rule_tac x="vs" in exI)
+    apply(rule_tac x="[]" in exI) 
+    apply(auto)
+    by (metis Prf.intros(9) append_Nil atMost_iff empty_iff le_imp_less_Suc less_antisym list.set(1) nth_mem zero_le)
+
+lemma LV_FROMNTIMES_6:
+  assumes "\<forall>s. finite (LV r s)"
+  shows "finite (LV (FROMNTIMES r n) s)"
   apply(rule finite_subset)
-   apply(rule test)
-  apply(rule finite_imageI)
-    apply(rule finite_list)
-   apply(simp)
-  done
+   apply(rule LV_FROMNTIMES_5)
+  apply(rule finite_Stars_Append)
+    apply(rule LV_STAR_finite)
+   apply(rule assms)
+  apply(rule finite_UN_I)
+   apply(auto)
+  by (simp add: assms finite_Stars_Pow ttty)
     
-lemma LV_NTIMES_STAR:
-  "LV (NTIMES r n) s \<subseteq> LV (STAR r) s"
-apply(auto simp add: LV_def intro: Prf.intros elim!: Prf_elims)
-apply(rule Prf.intros)
-oops
+lemma LV_NMTIMES_5:
+  "LV (NMTIMES r n m) s \<subseteq> Stars_Append (LV (STAR r) s) (\<Union>i\<le>n. LV (FROMNTIMES r i) [])"
+apply(auto simp add: LV_def)
+apply(auto elim!: Prf_elims)
+  apply(auto simp add: Stars_Append_def)
+  apply(rule_tac x="vs1" in exI)
+  apply(rule_tac x="vs2" in exI)  
+  apply(auto)
+    using Prf.intros(6) apply(auto)
+      apply(rule_tac x="length vs2" in bexI)
+    thm Prf.intros
+      apply(subst append.simps(1)[symmetric])
+    apply(rule Prf.intros)
+      apply(auto)[1]
+      apply(auto)[1]
+     apply(simp)
+     apply(simp)
+      apply(rule_tac x="vs" in exI)
+    apply(rule_tac x="[]" in exI) 
+    apply(auto)
+    by (metis Prf.intros(9) append_Nil atMost_iff empty_iff le_imp_less_Suc less_antisym list.set(1) nth_mem zero_le)
 
-lemma LV_FROMNTIMES_STAR:
-  "LV (FROMNTIMES r n) s \<subseteq> LV (STAR r) s"
-apply(auto simp add: LV_def intro: Prf.intros elim!: Prf_elims)
-oops
-
+lemma LV_NMTIMES_6:
+  assumes "\<forall>s. finite (LV r s)"
+  shows "finite (LV (NMTIMES r n m) s)"
+  apply(rule finite_subset)
+   apply(rule LV_NMTIMES_5)
+  apply(rule finite_Stars_Append)
+    apply(rule LV_STAR_finite)
+   apply(rule assms)
+  apply(rule finite_UN_I)
+   apply(auto)
+  by (simp add: assms finite_Stars_Pow ttty)
+        
+    
 lemma LV_finite:
   shows "finite (LV r s)"
 proof(induct r arbitrary: s)
@@ -999,7 +1120,17 @@
   case (FROMNTIMES r n s)
   have "\<And>s. finite (LV r s)" by fact
   then show "finite (LV (FROMNTIMES r n) s)"
-  
+    by (simp add: LV_FROMNTIMES_6)
+next 
+  case (NTIMES r n s)
+  have "\<And>s. finite (LV r s)" by fact
+  then show "finite (LV (NTIMES r n) s)"
+    by (metis (no_types, lifting) LV_NTIMES_4 LV_NTIMES_5 LV_STAR_finite finite_Stars_Append finite_Stars_Pow finite_UN_I finite_atMost finite_subset)
+next
+  case (NMTIMES r n m s)
+  have "\<And>s. finite (LV r s)" by fact
+  then show "finite (LV (NMTIMES r n m) s)"
+    by (simp add: LV_NMTIMES_6)         
 qed
 
 
@@ -1020,7 +1151,26 @@
     \<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 []"
-
+| Posix_NTIMES1: "\<lbrakk>s1 \<in> r \<rightarrow> v; s2 \<in> NTIMES r (n - 1) \<rightarrow> Stars vs; flat v \<noteq> []; 0 < n;
+    \<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 (NTIMES r (n - 1)))\<rbrakk>
+    \<Longrightarrow> (s1 @ s2) \<in> NTIMES r n \<rightarrow> Stars (v # vs)"
+| Posix_NTIMES2: "\<lbrakk>\<forall>v \<in> set vs. [] \<in> r \<rightarrow> v; length vs = n\<rbrakk>
+    \<Longrightarrow> [] \<in> NTIMES r n \<rightarrow> Stars vs"  
+| Posix_UPNTIMES1: "\<lbrakk>s1 \<in> r \<rightarrow> v; s2 \<in> UPNTIMES r (n - 1) \<rightarrow> Stars vs; flat v \<noteq> []; 0 < n;
+    \<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 (UPNTIMES r (n - 1)))\<rbrakk>
+    \<Longrightarrow> (s1 @ s2) \<in> UPNTIMES r n \<rightarrow> Stars (v # vs)"
+| Posix_UPNTIMES2: "[] \<in> UPNTIMES r n \<rightarrow> Stars []"
+| Posix_FROMNTIMES2: "\<lbrakk>\<forall>v \<in> set vs. [] \<in> r \<rightarrow> v; length vs = n\<rbrakk>
+    \<Longrightarrow> [] \<in> FROMNTIMES r n \<rightarrow> Stars vs"
+| Posix_FROMNTIMES1: "\<lbrakk>s1 \<in> r \<rightarrow> v; s2 \<in> FROMNTIMES r (n - 1) \<rightarrow> Stars vs; flat v \<noteq> []; 0 < n;
+    \<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 (FROMNTIMES r (n - 1)))\<rbrakk>
+    \<Longrightarrow> (s1 @ s2) \<in> FROMNTIMES r n \<rightarrow> Stars (v # vs)"  
+| Posix_NMTIMES2: "\<lbrakk>\<forall>v \<in> set vs. [] \<in> r \<rightarrow> v; length vs = n; n \<le> m\<rbrakk>
+    \<Longrightarrow> [] \<in> NMTIMES r n m \<rightarrow> Stars vs"  
+| Posix_NMTIMES1: "\<lbrakk>s1 \<in> r \<rightarrow> v; s2 \<in> NMTIMES r n m \<rightarrow> Stars vs; flat v \<noteq> []; n \<le> m;
+    \<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 (NMTIMES r n m))\<rbrakk>
+    \<Longrightarrow> (s1 @ s2) \<in> NMTIMES r (Suc n) (Suc m) \<rightarrow> Stars (v # vs)"  
+  
 inductive_cases Posix_elims:
   "s \<in> ZERO \<rightarrow> v"
   "s \<in> ONE \<rightarrow> v"
@@ -1028,17 +1178,60 @@
   "s \<in> ALT r1 r2 \<rightarrow> v"
   "s \<in> SEQ r1 r2 \<rightarrow> v"
   "s \<in> STAR r \<rightarrow> v"
-
+  "s \<in> NTIMES r n \<rightarrow> v"
+  "s \<in> UPNTIMES r n \<rightarrow> v"
+  "s \<in> FROMNTIMES r n \<rightarrow> v"
+  "s \<in> NMTIMES r n m \<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)
-
+  apply(induct s r v rule: Posix.induct)
+                    apply(auto simp add: Sequ_def)[18]
+            apply(case_tac n)
+             apply(simp)
+  apply(simp add: Sequ_def)
+            apply(auto)[1]
+           apply(simp)
+  apply(clarify)
+  apply(rule_tac x="Suc x" in bexI)
+  apply(simp add: Sequ_def)
+            apply(auto)[5]
+  using nth_mem nullable.simps(9) nullable_correctness apply auto[1]
+  apply simp
+       apply(simp)
+       apply(clarify)
+       apply(rule_tac x="Suc x" in bexI)
+        apply(simp add: Sequ_def)
+          apply(auto)[3]
+     apply(simp)
+  apply fastforce
+    apply(simp)
+   apply(simp)
+    apply(clarify)
+   apply(rule_tac x="Suc x" in bexI)
+    apply(auto simp add: Sequ_def)[2]
+   apply(simp)
+done  
+  
 text {*
   Our Posix definition determines a unique value.
 *}
+  
+lemma List_eq_zipI:
+  assumes "\<forall>(v1, v2) \<in> set (zip vs1 vs2). v1 = v2" 
+  and "length vs1 = length vs2"
+  shows "vs1 = vs2"  
+ using assms
+  apply(induct vs1 arbitrary: vs2)
+   apply(case_tac vs2)
+   apply(simp)    
+   apply(simp)
+   apply(case_tac vs2)
+   apply(simp)
+  apply(simp)
+done    
 
 lemma Posix_determ:
   assumes "s \<in> r \<rightarrow> v1" "s \<in> r \<rightarrow> v2"
@@ -1104,6 +1297,89 @@
   case (Posix_STAR2 r v2)
   have "[] \<in> STAR r \<rightarrow> v2" by fact
   then show "Stars [] = v2" by cases (auto simp add: Posix1)
+next
+  case (Posix_NTIMES2 vs r n v2)
+  then show "Stars vs = v2"
+    apply(erule_tac Posix_elims)
+     apply(auto)
+     apply (simp add: Posix1(2))
+    apply(rule List_eq_zipI)
+     apply(auto)
+    by (meson in_set_zipE)
+next
+  case (Posix_NTIMES1 s1 r v s2 n vs v2)
+  have "(s1 @ s2) \<in> NTIMES r n \<rightarrow> v2" 
+       "s1 \<in> r \<rightarrow> v" "s2 \<in> NTIMES r (n - 1) \<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 (NTIMES r (n - 1 )))" by fact+
+  then obtain v' vs' where "v2 = Stars (v' # vs')" "s1 \<in> r \<rightarrow> v'" "s2 \<in> (NTIMES r (n - 1)) \<rightarrow> (Stars vs')"
+  apply(cases) apply (auto simp add: append_eq_append_conv2)
+    using Posix1(1) apply fastforce
+    apply (metis One_nat_def Posix1(1) Posix_NTIMES1.hyps(7) append.right_neutral append_self_conv2)
+  using Posix1(2) by blast
+  moreover
+  have IHs: "\<And>v2. s1 \<in> r \<rightarrow> v2 \<Longrightarrow> v = v2"
+            "\<And>v2. s2 \<in> NTIMES r (n - 1) \<rightarrow> v2 \<Longrightarrow> Stars vs = v2" by fact+
+  ultimately show "Stars (v # vs) = v2" by auto
+next
+  case (Posix_UPNTIMES1 s1 r v s2 n vs v2)
+  have "(s1 @ s2) \<in> UPNTIMES r n \<rightarrow> v2" 
+       "s1 \<in> r \<rightarrow> v" "s2 \<in> UPNTIMES r (n - 1) \<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 (UPNTIMES r (n - 1 )))" by fact+
+  then obtain v' vs' where "v2 = Stars (v' # vs')" "s1 \<in> r \<rightarrow> v'" "s2 \<in> (UPNTIMES r (n - 1)) \<rightarrow> (Stars vs')"
+  apply(cases) apply (auto simp add: append_eq_append_conv2)
+    using Posix1(1) apply fastforce
+    apply (metis One_nat_def Posix1(1) Posix_UPNTIMES1.hyps(7) append.right_neutral append_self_conv2)
+  using Posix1(2) by blast
+  moreover
+  have IHs: "\<And>v2. s1 \<in> r \<rightarrow> v2 \<Longrightarrow> v = v2"
+            "\<And>v2. s2 \<in> UPNTIMES r (n - 1) \<rightarrow> v2 \<Longrightarrow> Stars vs = v2" by fact+
+  ultimately show "Stars (v # vs) = v2" by auto
+next
+  case (Posix_UPNTIMES2 r n v2)
+  then show "Stars [] = v2"
+    apply(erule_tac Posix_elims)
+     apply(auto)
+    by (simp add: Posix1(2))
+next
+  case (Posix_FROMNTIMES1 s1 r v s2 n vs v2)
+  have "(s1 @ s2) \<in> FROMNTIMES r n \<rightarrow> v2" 
+       "s1 \<in> r \<rightarrow> v" "s2 \<in> FROMNTIMES r (n - 1) \<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 (FROMNTIMES r (n - 1 )))" by fact+
+  then obtain v' vs' where "v2 = Stars (v' # vs')" "s1 \<in> r \<rightarrow> v'" "s2 \<in> (FROMNTIMES r (n - 1)) \<rightarrow> (Stars vs')"
+  apply(cases) apply (auto simp add: append_eq_append_conv2)
+    using Posix1(1) Posix1(2) apply blast 
+    apply(drule_tac x="v" in meta_spec)
+    apply(drule_tac x="vs" in meta_spec)
+    apply(simp)
+    apply(drule meta_mp)
+     apply(case_tac n)
+      apply(simp)
+      apply(rule conjI)      
+     apply (smt L.simps(9) One_nat_def Posix1(1) Posix_FROMNTIMES1.hyps Suc_mono Suc_pred UN_E append.right_neutral append_Nil lessI less_antisym list.size(3) nat.simps(3) neq0_conv not_less_eq val.inject(5))
+       apply(rule List_eq_zipI)
+        apply(auto)[1]
+    sorry  
+  moreover
+  have IHs: "\<And>v2. s1 \<in> r \<rightarrow> v2 \<Longrightarrow> v = v2"
+            "\<And>v2. s2 \<in> FROMNTIMES r (n - 1) \<rightarrow> v2 \<Longrightarrow> Stars vs = v2" by fact+
+  ultimately show "Stars (v # vs) = v2" by auto    
+next
+  case (Posix_FROMNTIMES2 vs r n v2)  
+  then show "Stars vs = v2"
+    apply(erule_tac Posix_elims)
+     apply(auto)
+    apply(rule List_eq_zipI)
+     apply(auto)
+     apply(meson in_set_zipE)
+    by (simp add: Posix1(2))
+next
+  case (Posix_NMTIMES1 s1 r v s2 n m vs v2)
+  then show "Stars (v # vs) = v2"
+    sorry
+next
+  case (Posix_NMTIMES2 vs r n m v2)
+  then show "Stars vs = v2"
+    sorry
 qed
 
 
@@ -1116,7 +1392,60 @@
   shows "v \<in> LV r s"
 using assms unfolding LV_def
 apply(induct rule: Posix.induct)
-apply(auto simp add: intro!: Prf.intros elim!: Prf_elims)
-done
-
+            apply(auto simp add: intro!: Prf.intros elim!: Prf_elims)[7]
+     defer
+  defer
+     apply(auto simp add: intro!: Prf.intros elim!: Prf_elims)[2]
+  apply (metis (mono_tags, lifting) Prf.intros(9) append_Nil empty_iff flat_Stars flats_empty list.set(1) mem_Collect_eq)
+  prefer 4
+  apply(auto simp add: intro!: Prf.intros elim!: Prf_elims)[1]
+   apply(subst append.simps(2)[symmetric])
+  apply(rule Prf.intros)
+    apply(auto)
+  prefer 4
+  apply (metis Prf.intros(8) append_Nil empty_iff list.set(1))
+  apply(erule Posix_elims)
+  apply(auto)
+  apply(rule_tac t="v # vsa" and s = "[v] @ vsa" in subst)
+  apply(simp)
+  apply(rule Prf.intros)
+   apply(simp)
+  apply(auto)[1]
+  apply(auto simp add: Sequ_def intro: Prf.intros elim: Prf_elims)[1]
+  apply(simp)
+  apply(rotate_tac 4)
+  apply(erule Prf_elims)
+    apply(auto)
+   apply(case_tac n)
+  apply(simp)
+  
+  apply(subst append.simps(2)[symmetric])
+  apply(rule Prf.intros)
+   apply(auto)
+  apply(rule Prf.intros(10))
+   apply(auto)
+  apply (metis Prf.intros(11) append_Nil empty_iff list.set(1))
+  apply(rotate_tac 6)
+  apply(erule Prf_elims)
+  apply(simp)
+  apply(subst append.simps(2)[symmetric])
+  apply(rule Prf.intros)
+    apply(auto)
+  apply(rule_tac t="v # vsa" and s = "(v # vsa) @ []" in subst)
+  apply(simp)
+  apply(simp add: Prf.intros(12))
+  done
+    
+lemma FROMNTIMES_0:
+  assumes "s \<in> FROMNTIMES r 0 \<rightarrow> v"
+  shows "s = [] \<and> v = Stars []"
+  using assms
+  apply(erule_tac Posix_elims)
+   apply(auto)
+  done    
+    
+lemma FROMNTIMES_der_0:
+  shows "der c (FROMNTIMES r 0) = SEQ (der c r) (FROMNTIMES r 0)"
+by(simp)    
+  
 end
\ No newline at end of file
diff -r deea42c83c9e -r a3134f7de065 thys/journal/llncs.cls
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/thys/journal/llncs.cls	Sat Oct 07 22:16:16 2017 +0100
@@ -0,0 +1,1189 @@
+% LLNCS DOCUMENT CLASS -- version 2.13 (28-Jan-2002)
+% Springer Verlag LaTeX2e support for Lecture Notes in Computer Science
+%
+%%
+%% \CharacterTable
+%%  {Upper-case    \A\B\C\D\E\F\G\H\I\J\K\L\M\N\O\P\Q\R\S\T\U\V\W\X\Y\Z
+%%   Lower-case    \a\b\c\d\e\f\g\h\i\j\k\l\m\n\o\p\q\r\s\t\u\v\w\x\y\z
+%%   Digits        \0\1\2\3\4\5\6\7\8\9
+%%   Exclamation   \!     Double quote  \"     Hash (number) \#
+%%   Dollar        \$     Percent       \%     Ampersand     \&
+%%   Acute accent  \'     Left paren    \(     Right paren   \)
+%%   Asterisk      \*     Plus          \+     Comma         \,
+%%   Minus         \-     Point         \.     Solidus       \/
+%%   Colon         \:     Semicolon     \;     Less than     \<
+%%   Equals        \=     Greater than  \>     Question mark \?
+%%   Commercial at \@     Left bracket  \[     Backslash     \\
+%%   Right bracket \]     Circumflex    \^     Underscore    \_
+%%   Grave accent  \`     Left brace    \{     Vertical bar  \|
+%%   Right brace   \}     Tilde         \~}
+%%
+\NeedsTeXFormat{LaTeX2e}[1995/12/01]
+\ProvidesClass{llncs}[2002/01/28 v2.13
+^^J LaTeX document class for Lecture Notes in Computer Science]
+% Options
+\let\if@envcntreset\iffalse
+\DeclareOption{envcountreset}{\let\if@envcntreset\iftrue}
+\DeclareOption{citeauthoryear}{\let\citeauthoryear=Y}
+\DeclareOption{oribibl}{\let\oribibl=Y}
+\let\if@custvec\iftrue
+\DeclareOption{orivec}{\let\if@custvec\iffalse}
+\let\if@envcntsame\iffalse
+\DeclareOption{envcountsame}{\let\if@envcntsame\iftrue}
+\let\if@envcntsect\iffalse
+\DeclareOption{envcountsect}{\let\if@envcntsect\iftrue}
+\let\if@runhead\iffalse
+\DeclareOption{runningheads}{\let\if@runhead\iftrue}
+
+\let\if@openbib\iffalse
+\DeclareOption{openbib}{\let\if@openbib\iftrue}
+
+% languages
+\let\switcht@@therlang\relax
+\def\ds@deutsch{\def\switcht@@therlang{\switcht@deutsch}}
+\def\ds@francais{\def\switcht@@therlang{\switcht@francais}}
+
+\DeclareOption*{\PassOptionsToClass{\CurrentOption}{article}}
+
+\ProcessOptions
+
+\LoadClass[twoside]{article}
+\RequirePackage{multicol} % needed for the list of participants, index
+
+\setlength{\textwidth}{12.2cm}
+\setlength{\textheight}{19.3cm}
+\renewcommand\@pnumwidth{2em}
+\renewcommand\@tocrmarg{3.5em}
+%
+\def\@dottedtocline#1#2#3#4#5{%
+  \ifnum #1>\c@tocdepth \else
+    \vskip \z@ \@plus.2\p@
+    {\leftskip #2\relax \rightskip \@tocrmarg \advance\rightskip by 0pt plus 2cm
+               \parfillskip -\rightskip \pretolerance=10000
+     \parindent #2\relax\@afterindenttrue
+     \interlinepenalty\@M
+     \leavevmode
+     \@tempdima #3\relax
+     \advance\leftskip \@tempdima \null\nobreak\hskip -\leftskip
+     {#4}\nobreak
+     \leaders\hbox{$\m@th
+        \mkern \@dotsep mu\hbox{.}\mkern \@dotsep
+        mu$}\hfill
+     \nobreak
+     \hb@xt@\@pnumwidth{\hfil\normalfont \normalcolor #5}%
+     \par}%
+  \fi}
+%
+\def\switcht@albion{%
+\def\abstractname{Abstract.}
+\def\ackname{Acknowledgement.}
+\def\andname{and}
+\def\lastandname{\unskip, and}
+\def\appendixname{Appendix}
+\def\chaptername{Chapter}
+\def\claimname{Claim}
+\def\conjecturename{Conjecture}
+\def\contentsname{Table of Contents}
+\def\corollaryname{Corollary}
+\def\definitionname{Definition}
+\def\examplename{Example}
+\def\exercisename{Exercise}
+\def\figurename{Fig.}
+\def\keywordname{{\bf Key words:}}
+\def\indexname{Index}
+\def\lemmaname{Lemma}
+\def\contriblistname{List of Contributors}
+\def\listfigurename{List of Figures}
+\def\listtablename{List of Tables}
+\def\mailname{{\it Correspondence to\/}:}
+\def\noteaddname{Note added in proof}
+\def\notename{Note}
+\def\partname{Part}
+\def\problemname{Problem}
+\def\proofname{Proof}
+\def\propertyname{Property}
+\def\propositionname{Proposition}
+\def\questionname{Question}
+\def\remarkname{Remark}
+\def\seename{see}
+\def\solutionname{Solution}
+\def\subclassname{{\it Subject Classifications\/}:}
+\def\tablename{Table}
+\def\theoremname{Theorem}}
+\switcht@albion
+% Names of theorem like environments are already defined
+% but must be translated if another language is chosen
+%
+% French section
+\def\switcht@francais{%\typeout{On parle francais.}%
+ \def\abstractname{R\'esum\'e.}%
+ \def\ackname{Remerciements.}%
+ \def\andname{et}%
+ \def\lastandname{ et}%
+ \def\appendixname{Appendice}
+ \def\chaptername{Chapitre}%
+ \def\claimname{Pr\'etention}%
+ \def\conjecturename{Hypoth\`ese}%
+ \def\contentsname{Table des mati\`eres}%
+ \def\corollaryname{Corollaire}%
+ \def\definitionname{D\'efinition}%
+ \def\examplename{Exemple}%
+ \def\exercisename{Exercice}%
+ \def\figurename{Fig.}%
+ \def\keywordname{{\bf Mots-cl\'e:}}
+ \def\indexname{Index}
+ \def\lemmaname{Lemme}%
+ \def\contriblistname{Liste des contributeurs}
+ \def\listfigurename{Liste des figures}%
+ \def\listtablename{Liste des tables}%
+ \def\mailname{{\it Correspondence to\/}:}
+ \def\noteaddname{Note ajout\'ee \`a l'\'epreuve}%
+ \def\notename{Remarque}%
+ \def\partname{Partie}%
+ \def\problemname{Probl\`eme}%
+ \def\proofname{Preuve}%
+ \def\propertyname{Caract\'eristique}%
+%\def\propositionname{Proposition}%
+ \def\questionname{Question}%
+ \def\remarkname{Remarque}%
+ \def\seename{voir}
+ \def\solutionname{Solution}%
+ \def\subclassname{{\it Subject Classifications\/}:}
+ \def\tablename{Tableau}%
+ \def\theoremname{Th\'eor\`eme}%
+}
+%
+% German section
+\def\switcht@deutsch{%\typeout{Man spricht deutsch.}%
+ \def\abstractname{Zusammenfassung.}%
+ \def\ackname{Danksagung.}%
+ \def\andname{und}%
+ \def\lastandname{ und}%
+ \def\appendixname{Anhang}%
+ \def\chaptername{Kapitel}%
+ \def\claimname{Behauptung}%
+ \def\conjecturename{Hypothese}%
+ \def\contentsname{Inhaltsverzeichnis}%
+ \def\corollaryname{Korollar}%
+%\def\definitionname{Definition}%
+ \def\examplename{Beispiel}%
+ \def\exercisename{\"Ubung}%
+ \def\figurename{Abb.}%
+ \def\keywordname{{\bf Schl\"usselw\"orter:}}
+ \def\indexname{Index}
+%\def\lemmaname{Lemma}%
+ \def\contriblistname{Mitarbeiter}
+ \def\listfigurename{Abbildungsverzeichnis}%
+ \def\listtablename{Tabellenverzeichnis}%
+ \def\mailname{{\it Correspondence to\/}:}
+ \def\noteaddname{Nachtrag}%
+ \def\notename{Anmerkung}%
+ \def\partname{Teil}%
+%\def\problemname{Problem}%
+ \def\proofname{Beweis}%
+ \def\propertyname{Eigenschaft}%
+%\def\propositionname{Proposition}%
+ \def\questionname{Frage}%
+ \def\remarkname{Anmerkung}%
+ \def\seename{siehe}
+ \def\solutionname{L\"osung}%
+ \def\subclassname{{\it Subject Classifications\/}:}
+ \def\tablename{Tabelle}%
+%\def\theoremname{Theorem}%
+}
+
+% Ragged bottom for the actual page
+\def\thisbottomragged{\def\@textbottom{\vskip\z@ plus.0001fil
+\global\let\@textbottom\relax}}
+
+\renewcommand\small{%
+   \@setfontsize\small\@ixpt{11}%
+   \abovedisplayskip 8.5\p@ \@plus3\p@ \@minus4\p@
+   \abovedisplayshortskip \z@ \@plus2\p@
+   \belowdisplayshortskip 4\p@ \@plus2\p@ \@minus2\p@
+   \def\@listi{\leftmargin\leftmargini
+               \parsep 0\p@ \@plus1\p@ \@minus\p@
+               \topsep 8\p@ \@plus2\p@ \@minus4\p@
+               \itemsep0\p@}%
+   \belowdisplayskip \abovedisplayskip
+}
+
+\frenchspacing
+\widowpenalty=10000
+\clubpenalty=10000
+
+\setlength\oddsidemargin   {63\p@}
+\setlength\evensidemargin  {63\p@}
+\setlength\marginparwidth  {90\p@}
+
+\setlength\headsep   {16\p@}
+
+\setlength\footnotesep{7.7\p@}
+\setlength\textfloatsep{8mm\@plus 2\p@ \@minus 4\p@}
+\setlength\intextsep   {8mm\@plus 2\p@ \@minus 2\p@}
+
+\setcounter{secnumdepth}{2}
+
+\newcounter {chapter}
+\renewcommand\thechapter      {\@arabic\c@chapter}
+
+\newif\if@mainmatter \@mainmattertrue
+\newcommand\frontmatter{\cleardoublepage
+            \@mainmatterfalse\pagenumbering{Roman}}
+\newcommand\mainmatter{\cleardoublepage
+       \@mainmattertrue\pagenumbering{arabic}}
+\newcommand\backmatter{\if@openright\cleardoublepage\else\clearpage\fi
+      \@mainmatterfalse}
+
+\renewcommand\part{\cleardoublepage
+                 \thispagestyle{empty}%
+                 \if@twocolumn
+                     \onecolumn
+                     \@tempswatrue
+                   \else
+                     \@tempswafalse
+                 \fi
+                 \null\vfil
+                 \secdef\@part\@spart}
+
+\def\@part[#1]#2{%
+    \ifnum \c@secnumdepth >-2\relax
+      \refstepcounter{part}%
+      \addcontentsline{toc}{part}{\thepart\hspace{1em}#1}%
+    \else
+      \addcontentsline{toc}{part}{#1}%
+    \fi
+    \markboth{}{}%
+    {\centering
+     \interlinepenalty \@M
+     \normalfont
+     \ifnum \c@secnumdepth >-2\relax
+       \huge\bfseries \partname~\thepart
+       \par
+       \vskip 20\p@
+     \fi
+     \Huge \bfseries #2\par}%
+    \@endpart}
+\def\@spart#1{%
+    {\centering
+     \interlinepenalty \@M
+     \normalfont
+     \Huge \bfseries #1\par}%
+    \@endpart}
+\def\@endpart{\vfil\newpage
+              \if@twoside
+                \null
+                \thispagestyle{empty}%
+                \newpage
+              \fi
+              \if@tempswa
+                \twocolumn
+              \fi}
+
+\newcommand\chapter{\clearpage
+                    \thispagestyle{empty}%
+                    \global\@topnum\z@
+                    \@afterindentfalse
+                    \secdef\@chapter\@schapter}
+\def\@chapter[#1]#2{\ifnum \c@secnumdepth >\m@ne
+                       \if@mainmatter
+                         \refstepcounter{chapter}%
+                         \typeout{\@chapapp\space\thechapter.}%
+                         \addcontentsline{toc}{chapter}%
+                                  {\protect\numberline{\thechapter}#1}%
+                       \else
+                         \addcontentsline{toc}{chapter}{#1}%
+                       \fi
+                    \else
+                      \addcontentsline{toc}{chapter}{#1}%
+                    \fi
+                    \chaptermark{#1}%
+                    \addtocontents{lof}{\protect\addvspace{10\p@}}%
+                    \addtocontents{lot}{\protect\addvspace{10\p@}}%
+                    \if@twocolumn
+                      \@topnewpage[\@makechapterhead{#2}]%
+                    \else
+                      \@makechapterhead{#2}%
+                      \@afterheading
+                    \fi}
+\def\@makechapterhead#1{%
+% \vspace*{50\p@}%
+  {\centering
+    \ifnum \c@secnumdepth >\m@ne
+      \if@mainmatter
+        \large\bfseries \@chapapp{} \thechapter
+        \par\nobreak
+        \vskip 20\p@
+      \fi
+    \fi
+    \interlinepenalty\@M
+    \Large \bfseries #1\par\nobreak
+    \vskip 40\p@
+  }}
+\def\@schapter#1{\if@twocolumn
+                   \@topnewpage[\@makeschapterhead{#1}]%
+                 \else
+                   \@makeschapterhead{#1}%
+                   \@afterheading
+                 \fi}
+\def\@makeschapterhead#1{%
+% \vspace*{50\p@}%
+  {\centering
+    \normalfont
+    \interlinepenalty\@M
+    \Large \bfseries  #1\par\nobreak
+    \vskip 40\p@
+  }}
+
+\renewcommand\section{\@startsection{section}{1}{\z@}%
+                       {-18\p@ \@plus -4\p@ \@minus -4\p@}%
+                       {12\p@ \@plus 4\p@ \@minus 4\p@}%
+                       {\normalfont\large\bfseries\boldmath
+                        \rightskip=\z@ \@plus 8em\pretolerance=10000 }}
+\renewcommand\subsection{\@startsection{subsection}{2}{\z@}%
+                       {-18\p@ \@plus -4\p@ \@minus -4\p@}%
+                       {8\p@ \@plus 4\p@ \@minus 4\p@}%
+                       {\normalfont\normalsize\bfseries\boldmath
+                        \rightskip=\z@ \@plus 8em\pretolerance=10000 }}
+\renewcommand\subsubsection{\@startsection{subsubsection}{3}{\z@}%
+                       {-18\p@ \@plus -4\p@ \@minus -4\p@}%
+                       {-0.5em \@plus -0.22em \@minus -0.1em}%
+                       {\normalfont\normalsize\bfseries\boldmath}}
+\renewcommand\paragraph{\@startsection{paragraph}{4}{\z@}%
+                       {-12\p@ \@plus -4\p@ \@minus -4\p@}%
+                       {-0.5em \@plus -0.22em \@minus -0.1em}%
+                       {\normalfont\normalsize\itshape}}
+\renewcommand\subparagraph[1]{\typeout{LLNCS warning: You should not use
+                  \string\subparagraph\space with this class}\vskip0.5cm
+You should not use \verb|\subparagraph| with this class.\vskip0.5cm}
+
+\DeclareMathSymbol{\Gamma}{\mathalpha}{letters}{"00}
+\DeclareMathSymbol{\Delta}{\mathalpha}{letters}{"01}
+\DeclareMathSymbol{\Theta}{\mathalpha}{letters}{"02}
+\DeclareMathSymbol{\Lambda}{\mathalpha}{letters}{"03}
+\DeclareMathSymbol{\Xi}{\mathalpha}{letters}{"04}
+\DeclareMathSymbol{\Pi}{\mathalpha}{letters}{"05}
+\DeclareMathSymbol{\Sigma}{\mathalpha}{letters}{"06}
+\DeclareMathSymbol{\Upsilon}{\mathalpha}{letters}{"07}
+\DeclareMathSymbol{\Phi}{\mathalpha}{letters}{"08}
+\DeclareMathSymbol{\Psi}{\mathalpha}{letters}{"09}
+\DeclareMathSymbol{\Omega}{\mathalpha}{letters}{"0A}
+
+\let\footnotesize\small
+
+\if@custvec
+\def\vec#1{\mathchoice{\mbox{\boldmath$\displaystyle#1$}}
+{\mbox{\boldmath$\textstyle#1$}}
+{\mbox{\boldmath$\scriptstyle#1$}}
+{\mbox{\boldmath$\scriptscriptstyle#1$}}}
+\fi
+
+\def\squareforqed{\hbox{\rlap{$\sqcap$}$\sqcup$}}
+\def\qed{\ifmmode\squareforqed\else{\unskip\nobreak\hfil
+\penalty50\hskip1em\null\nobreak\hfil\squareforqed
+\parfillskip=0pt\finalhyphendemerits=0\endgraf}\fi}
+
+\def\getsto{\mathrel{\mathchoice {\vcenter{\offinterlineskip
+\halign{\hfil
+$\displaystyle##$\hfil\cr\gets\cr\to\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\textstyle##$\hfil\cr\gets
+\cr\to\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\scriptstyle##$\hfil\cr\gets
+\cr\to\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\scriptscriptstyle##$\hfil\cr
+\gets\cr\to\cr}}}}}
+\def\lid{\mathrel{\mathchoice {\vcenter{\offinterlineskip\halign{\hfil
+$\displaystyle##$\hfil\cr<\cr\noalign{\vskip1.2pt}=\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\textstyle##$\hfil\cr<\cr
+\noalign{\vskip1.2pt}=\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\scriptstyle##$\hfil\cr<\cr
+\noalign{\vskip1pt}=\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\scriptscriptstyle##$\hfil\cr
+<\cr
+\noalign{\vskip0.9pt}=\cr}}}}}
+\def\gid{\mathrel{\mathchoice {\vcenter{\offinterlineskip\halign{\hfil
+$\displaystyle##$\hfil\cr>\cr\noalign{\vskip1.2pt}=\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\textstyle##$\hfil\cr>\cr
+\noalign{\vskip1.2pt}=\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\scriptstyle##$\hfil\cr>\cr
+\noalign{\vskip1pt}=\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\scriptscriptstyle##$\hfil\cr
+>\cr
+\noalign{\vskip0.9pt}=\cr}}}}}
+\def\grole{\mathrel{\mathchoice {\vcenter{\offinterlineskip
+\halign{\hfil
+$\displaystyle##$\hfil\cr>\cr\noalign{\vskip-1pt}<\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\textstyle##$\hfil\cr
+>\cr\noalign{\vskip-1pt}<\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\scriptstyle##$\hfil\cr
+>\cr\noalign{\vskip-0.8pt}<\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\scriptscriptstyle##$\hfil\cr
+>\cr\noalign{\vskip-0.3pt}<\cr}}}}}
+\def\bbbr{{\rm I\!R}} %reelle Zahlen
+\def\bbbm{{\rm I\!M}}
+\def\bbbn{{\rm I\!N}} %natuerliche Zahlen
+\def\bbbf{{\rm I\!F}}
+\def\bbbh{{\rm I\!H}}
+\def\bbbk{{\rm I\!K}}
+\def\bbbp{{\rm I\!P}}
+\def\bbbone{{\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l}
+{\rm 1\mskip-4.5mu l} {\rm 1\mskip-5mu l}}}
+\def\bbbc{{\mathchoice {\setbox0=\hbox{$\displaystyle\rm C$}\hbox{\hbox
+to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}
+{\setbox0=\hbox{$\textstyle\rm C$}\hbox{\hbox
+to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}
+{\setbox0=\hbox{$\scriptstyle\rm C$}\hbox{\hbox
+to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}
+{\setbox0=\hbox{$\scriptscriptstyle\rm C$}\hbox{\hbox
+to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}}}
+\def\bbbq{{\mathchoice {\setbox0=\hbox{$\displaystyle\rm
+Q$}\hbox{\raise
+0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.8\ht0\hss}\box0}}
+{\setbox0=\hbox{$\textstyle\rm Q$}\hbox{\raise
+0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.8\ht0\hss}\box0}}
+{\setbox0=\hbox{$\scriptstyle\rm Q$}\hbox{\raise
+0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.7\ht0\hss}\box0}}
+{\setbox0=\hbox{$\scriptscriptstyle\rm Q$}\hbox{\raise
+0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.7\ht0\hss}\box0}}}}
+\def\bbbt{{\mathchoice {\setbox0=\hbox{$\displaystyle\rm
+T$}\hbox{\hbox to0pt{\kern0.3\wd0\vrule height0.9\ht0\hss}\box0}}
+{\setbox0=\hbox{$\textstyle\rm T$}\hbox{\hbox
+to0pt{\kern0.3\wd0\vrule height0.9\ht0\hss}\box0}}
+{\setbox0=\hbox{$\scriptstyle\rm T$}\hbox{\hbox
+to0pt{\kern0.3\wd0\vrule height0.9\ht0\hss}\box0}}
+{\setbox0=\hbox{$\scriptscriptstyle\rm T$}\hbox{\hbox
+to0pt{\kern0.3\wd0\vrule height0.9\ht0\hss}\box0}}}}
+\def\bbbs{{\mathchoice
+{\setbox0=\hbox{$\displaystyle     \rm S$}\hbox{\raise0.5\ht0\hbox
+to0pt{\kern0.35\wd0\vrule height0.45\ht0\hss}\hbox
+to0pt{\kern0.55\wd0\vrule height0.5\ht0\hss}\box0}}
+{\setbox0=\hbox{$\textstyle        \rm S$}\hbox{\raise0.5\ht0\hbox
+to0pt{\kern0.35\wd0\vrule height0.45\ht0\hss}\hbox
+to0pt{\kern0.55\wd0\vrule height0.5\ht0\hss}\box0}}
+{\setbox0=\hbox{$\scriptstyle      \rm S$}\hbox{\raise0.5\ht0\hbox
+to0pt{\kern0.35\wd0\vrule height0.45\ht0\hss}\raise0.05\ht0\hbox
+to0pt{\kern0.5\wd0\vrule height0.45\ht0\hss}\box0}}
+{\setbox0=\hbox{$\scriptscriptstyle\rm S$}\hbox{\raise0.5\ht0\hbox
+to0pt{\kern0.4\wd0\vrule height0.45\ht0\hss}\raise0.05\ht0\hbox
+to0pt{\kern0.55\wd0\vrule height0.45\ht0\hss}\box0}}}}
+\def\bbbz{{\mathchoice {\hbox{$\mathsf\textstyle Z\kern-0.4em Z$}}
+{\hbox{$\mathsf\textstyle Z\kern-0.4em Z$}}
+{\hbox{$\mathsf\scriptstyle Z\kern-0.3em Z$}}
+{\hbox{$\mathsf\scriptscriptstyle Z\kern-0.2em Z$}}}}
+
+\let\ts\,
+
+\setlength\leftmargini  {17\p@}
+\setlength\leftmargin    {\leftmargini}
+\setlength\leftmarginii  {\leftmargini}
+\setlength\leftmarginiii {\leftmargini}
+\setlength\leftmarginiv  {\leftmargini}
+\setlength  \labelsep  {.5em}
+\setlength  \labelwidth{\leftmargini}
+\addtolength\labelwidth{-\labelsep}
+
+\def\@listI{\leftmargin\leftmargini
+            \parsep 0\p@ \@plus1\p@ \@minus\p@
+            \topsep 8\p@ \@plus2\p@ \@minus4\p@
+            \itemsep0\p@}
+\let\@listi\@listI
+\@listi
+\def\@listii {\leftmargin\leftmarginii
+              \labelwidth\leftmarginii
+              \advance\labelwidth-\labelsep
+              \topsep    0\p@ \@plus2\p@ \@minus\p@}
+\def\@listiii{\leftmargin\leftmarginiii
+              \labelwidth\leftmarginiii
+              \advance\labelwidth-\labelsep
+              \topsep    0\p@ \@plus\p@\@minus\p@
+              \parsep    \z@
+              \partopsep \p@ \@plus\z@ \@minus\p@}
+
+\renewcommand\labelitemi{\normalfont\bfseries --}
+\renewcommand\labelitemii{$\m@th\bullet$}
+
+\setlength\arraycolsep{1.4\p@}
+\setlength\tabcolsep{1.4\p@}
+
+\def\tableofcontents{\chapter*{\contentsname\@mkboth{{\contentsname}}%
+                                                    {{\contentsname}}}
+ \def\authcount##1{\setcounter{auco}{##1}\setcounter{@auth}{1}}
+ \def\lastand{\ifnum\value{auco}=2\relax
+                 \unskip{} \andname\
+              \else
+                 \unskip \lastandname\
+              \fi}%
+ \def\and{\stepcounter{@auth}\relax
+          \ifnum\value{@auth}=\value{auco}%
+             \lastand
+          \else
+             \unskip,
+          \fi}%
+ \@starttoc{toc}\if@restonecol\twocolumn\fi}
+
+\def\l@part#1#2{\addpenalty{\@secpenalty}%
+   \addvspace{2em plus\p@}%  % space above part line
+   \begingroup
+     \parindent \z@
+     \rightskip \z@ plus 5em
+     \hrule\vskip5pt
+     \large               % same size as for a contribution heading
+     \bfseries\boldmath   % set line in boldface
+     \leavevmode          % TeX command to enter horizontal mode.
+     #1\par
+     \vskip5pt
+     \hrule
+     \vskip1pt
+     \nobreak             % Never break after part entry
+   \endgroup}
+
+\def\@dotsep{2}
+
+\def\hyperhrefextend{\ifx\hyper@anchor\@undefined\else
+{chapter.\thechapter}\fi}
+
+\def\addnumcontentsmark#1#2#3{%
+\addtocontents{#1}{\protect\contentsline{#2}{\protect\numberline
+                     {\thechapter}#3}{\thepage}\hyperhrefextend}}
+\def\addcontentsmark#1#2#3{%
+\addtocontents{#1}{\protect\contentsline{#2}{#3}{\thepage}\hyperhrefextend}}
+\def\addcontentsmarkwop#1#2#3{%
+\addtocontents{#1}{\protect\contentsline{#2}{#3}{0}\hyperhrefextend}}
+
+\def\@adcmk[#1]{\ifcase #1 \or
+\def\@gtempa{\addnumcontentsmark}%
+  \or    \def\@gtempa{\addcontentsmark}%
+  \or    \def\@gtempa{\addcontentsmarkwop}%
+  \fi\@gtempa{toc}{chapter}}
+\def\addtocmark{\@ifnextchar[{\@adcmk}{\@adcmk[3]}}
+
+\def\l@chapter#1#2{\addpenalty{-\@highpenalty}
+ \vskip 1.0em plus 1pt \@tempdima 1.5em \begingroup
+ \parindent \z@ \rightskip \@tocrmarg
+ \advance\rightskip by 0pt plus 2cm
+ \parfillskip -\rightskip \pretolerance=10000
+ \leavevmode \advance\leftskip\@tempdima \hskip -\leftskip
+ {\large\bfseries\boldmath#1}\ifx0#2\hfil\null
+ \else
+      \nobreak
+      \leaders\hbox{$\m@th \mkern \@dotsep mu.\mkern
+      \@dotsep mu$}\hfill
+      \nobreak\hbox to\@pnumwidth{\hss #2}%
+ \fi\par
+ \penalty\@highpenalty \endgroup}
+
+\def\l@title#1#2{\addpenalty{-\@highpenalty}
+ \addvspace{8pt plus 1pt}
+ \@tempdima \z@
+ \begingroup
+ \parindent \z@ \rightskip \@tocrmarg
+ \advance\rightskip by 0pt plus 2cm
+ \parfillskip -\rightskip \pretolerance=10000
+ \leavevmode \advance\leftskip\@tempdima \hskip -\leftskip
+ #1\nobreak
+ \leaders\hbox{$\m@th \mkern \@dotsep mu.\mkern
+ \@dotsep mu$}\hfill
+ \nobreak\hbox to\@pnumwidth{\hss #2}\par
+ \penalty\@highpenalty \endgroup}
+
+\def\l@author#1#2{\addpenalty{\@highpenalty}
+ \@tempdima=\z@ %15\p@
+ \begingroup
+ \parindent \z@ \rightskip \@tocrmarg
+ \advance\rightskip by 0pt plus 2cm
+ \pretolerance=10000
+ \leavevmode \advance\leftskip\@tempdima %\hskip -\leftskip
+ \textit{#1}\par
+ \penalty\@highpenalty \endgroup}
+
+%\setcounter{tocdepth}{0}
+\newdimen\tocchpnum
+\newdimen\tocsecnum
+\newdimen\tocsectotal
+\newdimen\tocsubsecnum
+\newdimen\tocsubsectotal
+\newdimen\tocsubsubsecnum
+\newdimen\tocsubsubsectotal
+\newdimen\tocparanum
+\newdimen\tocparatotal
+\newdimen\tocsubparanum
+\tocchpnum=\z@            % no chapter numbers
+\tocsecnum=15\p@          % section 88. plus 2.222pt
+\tocsubsecnum=23\p@       % subsection 88.8 plus 2.222pt
+\tocsubsubsecnum=27\p@    % subsubsection 88.8.8 plus 1.444pt
+\tocparanum=35\p@         % paragraph 88.8.8.8 plus 1.666pt
+\tocsubparanum=43\p@      % subparagraph 88.8.8.8.8 plus 1.888pt
+\def\calctocindent{%
+\tocsectotal=\tocchpnum
+\advance\tocsectotal by\tocsecnum
+\tocsubsectotal=\tocsectotal
+\advance\tocsubsectotal by\tocsubsecnum
+\tocsubsubsectotal=\tocsubsectotal
+\advance\tocsubsubsectotal by\tocsubsubsecnum
+\tocparatotal=\tocsubsubsectotal
+\advance\tocparatotal by\tocparanum}
+\calctocindent
+
+\def\l@section{\@dottedtocline{1}{\tocchpnum}{\tocsecnum}}
+\def\l@subsection{\@dottedtocline{2}{\tocsectotal}{\tocsubsecnum}}
+\def\l@subsubsection{\@dottedtocline{3}{\tocsubsectotal}{\tocsubsubsecnum}}
+\def\l@paragraph{\@dottedtocline{4}{\tocsubsubsectotal}{\tocparanum}}
+\def\l@subparagraph{\@dottedtocline{5}{\tocparatotal}{\tocsubparanum}}
+
+\def\listoffigures{\@restonecolfalse\if@twocolumn\@restonecoltrue\onecolumn
+ \fi\section*{\listfigurename\@mkboth{{\listfigurename}}{{\listfigurename}}}
+ \@starttoc{lof}\if@restonecol\twocolumn\fi}
+\def\l@figure{\@dottedtocline{1}{0em}{1.5em}}
+
+\def\listoftables{\@restonecolfalse\if@twocolumn\@restonecoltrue\onecolumn
+ \fi\section*{\listtablename\@mkboth{{\listtablename}}{{\listtablename}}}
+ \@starttoc{lot}\if@restonecol\twocolumn\fi}
+\let\l@table\l@figure
+
+\renewcommand\listoffigures{%
+    \section*{\listfigurename
+      \@mkboth{\listfigurename}{\listfigurename}}%
+    \@starttoc{lof}%
+    }
+
+\renewcommand\listoftables{%
+    \section*{\listtablename
+      \@mkboth{\listtablename}{\listtablename}}%
+    \@starttoc{lot}%
+    }
+
+\ifx\oribibl\undefined
+\ifx\citeauthoryear\undefined
+\renewenvironment{thebibliography}[1]
+     {\section*{\refname}
+      \def\@biblabel##1{##1.}
+      \small
+      \list{\@biblabel{\@arabic\c@enumiv}}%
+           {\settowidth\labelwidth{\@biblabel{#1}}%
+            \leftmargin\labelwidth
+            \advance\leftmargin\labelsep
+            \if@openbib
+              \advance\leftmargin\bibindent
+              \itemindent -\bibindent
+              \listparindent \itemindent
+              \parsep \z@
+            \fi
+            \usecounter{enumiv}%
+            \let\p@enumiv\@empty
+            \renewcommand\theenumiv{\@arabic\c@enumiv}}%
+      \if@openbib
+        \renewcommand\newblock{\par}%
+      \else
+        \renewcommand\newblock{\hskip .11em \@plus.33em \@minus.07em}%
+      \fi
+      \sloppy\clubpenalty4000\widowpenalty4000%
+      \sfcode`\.=\@m}
+     {\def\@noitemerr
+       {\@latex@warning{Empty `thebibliography' environment}}%
+      \endlist}
+\def\@lbibitem[#1]#2{\item[{[#1]}\hfill]\if@filesw
+     {\let\protect\noexpand\immediate
+     \write\@auxout{\string\bibcite{#2}{#1}}}\fi\ignorespaces}
+\newcount\@tempcntc
+\def\@citex[#1]#2{\if@filesw\immediate\write\@auxout{\string\citation{#2}}\fi
+  \@tempcnta\z@\@tempcntb\m@ne\def\@citea{}\@cite{\@for\@citeb:=#2\do
+    {\@ifundefined
+       {b@\@citeb}{\@citeo\@tempcntb\m@ne\@citea\def\@citea{,}{\bfseries
+        ?}\@warning
+       {Citation `\@citeb' on page \thepage \space undefined}}%
+    {\setbox\z@\hbox{\global\@tempcntc0\csname b@\@citeb\endcsname\relax}%
+     \ifnum\@tempcntc=\z@ \@citeo\@tempcntb\m@ne
+       \@citea\def\@citea{,}\hbox{\csname b@\@citeb\endcsname}%
+     \else
+      \advance\@tempcntb\@ne
+      \ifnum\@tempcntb=\@tempcntc
+      \else\advance\@tempcntb\m@ne\@citeo
+      \@tempcnta\@tempcntc\@tempcntb\@tempcntc\fi\fi}}\@citeo}{#1}}
+\def\@citeo{\ifnum\@tempcnta>\@tempcntb\else
+               \@citea\def\@citea{,\,\hskip\z@skip}%
+               \ifnum\@tempcnta=\@tempcntb\the\@tempcnta\else
+               {\advance\@tempcnta\@ne\ifnum\@tempcnta=\@tempcntb \else
+                \def\@citea{--}\fi
+      \advance\@tempcnta\m@ne\the\@tempcnta\@citea\the\@tempcntb}\fi\fi}
+\else
+\renewenvironment{thebibliography}[1]
+     {\section*{\refname}
+      \small
+      \list{}%
+           {\settowidth\labelwidth{}%
+            \leftmargin\parindent
+            \itemindent=-\parindent
+            \labelsep=\z@
+            \if@openbib
+              \advance\leftmargin\bibindent
+              \itemindent -\bibindent
+              \listparindent \itemindent
+              \parsep \z@
+            \fi
+            \usecounter{enumiv}%
+            \let\p@enumiv\@empty
+            \renewcommand\theenumiv{}}%
+      \if@openbib
+        \renewcommand\newblock{\par}%
+      \else
+        \renewcommand\newblock{\hskip .11em \@plus.33em \@minus.07em}%
+      \fi
+      \sloppy\clubpenalty4000\widowpenalty4000%
+      \sfcode`\.=\@m}
+     {\def\@noitemerr
+       {\@latex@warning{Empty `thebibliography' environment}}%
+      \endlist}
+      \def\@cite#1{#1}%
+      \def\@lbibitem[#1]#2{\item[]\if@filesw
+        {\def\protect##1{\string ##1\space}\immediate
+      \write\@auxout{\string\bibcite{#2}{#1}}}\fi\ignorespaces}
+   \fi
+\else
+\@cons\@openbib@code{\noexpand\small}
+\fi
+
+\def\idxquad{\hskip 10\p@}% space that divides entry from number
+
+\def\@idxitem{\par\hangindent 10\p@}
+
+\def\subitem{\par\setbox0=\hbox{--\enspace}% second order
+                \noindent\hangindent\wd0\box0}% index entry
+
+\def\subsubitem{\par\setbox0=\hbox{--\,--\enspace}% third
+                \noindent\hangindent\wd0\box0}% order index entry
+
+\def\indexspace{\par \vskip 10\p@ plus5\p@ minus3\p@\relax}
+
+\renewenvironment{theindex}
+               {\@mkboth{\indexname}{\indexname}%
+                \thispagestyle{empty}\parindent\z@
+                \parskip\z@ \@plus .3\p@\relax
+                \let\item\par
+                \def\,{\relax\ifmmode\mskip\thinmuskip
+                             \else\hskip0.2em\ignorespaces\fi}%
+                \normalfont\small
+                \begin{multicols}{2}[\@makeschapterhead{\indexname}]%
+                }
+                {\end{multicols}}
+
+\renewcommand\footnoterule{%
+  \kern-3\p@
+  \hrule\@width 2truecm
+  \kern2.6\p@}
+  \newdimen\fnindent
+  \fnindent1em
+\long\def\@makefntext#1{%
+    \parindent \fnindent%
+    \leftskip \fnindent%
+    \noindent
+    \llap{\hb@xt@1em{\hss\@makefnmark\ }}\ignorespaces#1}
+
+\long\def\@makecaption#1#2{%
+  \vskip\abovecaptionskip
+  \sbox\@tempboxa{{\bfseries #1.} #2}%
+  \ifdim \wd\@tempboxa >\hsize
+    {\bfseries #1.} #2\par
+  \else
+    \global \@minipagefalse
+    \hb@xt@\hsize{\hfil\box\@tempboxa\hfil}%
+  \fi
+  \vskip\belowcaptionskip}
+
+\def\fps@figure{htbp}
+\def\fnum@figure{\figurename\thinspace\thefigure}
+\def \@floatboxreset {%
+        \reset@font
+        \small
+        \@setnobreak
+        \@setminipage
+}
+\def\fps@table{htbp}
+\def\fnum@table{\tablename~\thetable}
+\renewenvironment{table}
+               {\setlength\abovecaptionskip{0\p@}%
+                \setlength\belowcaptionskip{10\p@}%
+                \@float{table}}
+               {\end@float}
+\renewenvironment{table*}
+               {\setlength\abovecaptionskip{0\p@}%
+                \setlength\belowcaptionskip{10\p@}%
+                \@dblfloat{table}}
+               {\end@dblfloat}
+
+\long\def\@caption#1[#2]#3{\par\addcontentsline{\csname
+  ext@#1\endcsname}{#1}{\protect\numberline{\csname
+  the#1\endcsname}{\ignorespaces #2}}\begingroup
+    \@parboxrestore
+    \@makecaption{\csname fnum@#1\endcsname}{\ignorespaces #3}\par
+  \endgroup}
+
+% LaTeX does not provide a command to enter the authors institute
+% addresses. The \institute command is defined here.
+
+\newcounter{@inst}
+\newcounter{@auth}
+\newcounter{auco}
+\newdimen\instindent
+\newbox\authrun
+\newtoks\authorrunning
+\newtoks\tocauthor
+\newbox\titrun
+\newtoks\titlerunning
+\newtoks\toctitle
+
+\def\clearheadinfo{\gdef\@author{No Author Given}%
+                   \gdef\@title{No Title Given}%
+                   \gdef\@subtitle{}%
+                   \gdef\@institute{No Institute Given}%
+                   \gdef\@thanks{}%
+                   \global\titlerunning={}\global\authorrunning={}%
+                   \global\toctitle={}\global\tocauthor={}}
+
+\def\institute#1{\gdef\@institute{#1}}
+
+\def\institutename{\par
+ \begingroup
+ \parskip=\z@
+ \parindent=\z@
+ \setcounter{@inst}{1}%
+ \def\and{\par\stepcounter{@inst}%
+ \noindent$^{\the@inst}$\enspace\ignorespaces}%
+ \setbox0=\vbox{\def\thanks##1{}\@institute}%
+ \ifnum\c@@inst=1\relax
+   \gdef\fnnstart{0}%
+ \else
+   \xdef\fnnstart{\c@@inst}%
+   \setcounter{@inst}{1}%
+   \noindent$^{\the@inst}$\enspace
+ \fi
+ \ignorespaces
+ \@institute\par
+ \endgroup}
+
+\def\@fnsymbol#1{\ensuremath{\ifcase#1\or\star\or{\star\star}\or
+   {\star\star\star}\or \dagger\or \ddagger\or
+   \mathchar "278\or \mathchar "27B\or \|\or **\or \dagger\dagger
+   \or \ddagger\ddagger \else\@ctrerr\fi}}
+
+\def\inst#1{\unskip$^{#1}$}
+\def\fnmsep{\unskip$^,$}
+\def\email#1{{\tt#1}}
+\AtBeginDocument{\@ifundefined{url}{\def\url#1{#1}}{}%
+\@ifpackageloaded{babel}{%
+\@ifundefined{extrasenglish}{}{\addto\extrasenglish{\switcht@albion}}%
+\@ifundefined{extrasfrenchb}{}{\addto\extrasfrenchb{\switcht@francais}}%
+\@ifundefined{extrasgerman}{}{\addto\extrasgerman{\switcht@deutsch}}%
+}{\switcht@@therlang}%
+}
+\def\homedir{\~{ }}
+
+\def\subtitle#1{\gdef\@subtitle{#1}}
+\clearheadinfo
+
+\renewcommand\maketitle{\newpage
+  \refstepcounter{chapter}%
+  \stepcounter{section}%
+  \setcounter{section}{0}%
+  \setcounter{subsection}{0}%
+  \setcounter{figure}{0}
+  \setcounter{table}{0}
+  \setcounter{equation}{0}
+  \setcounter{footnote}{0}%
+  \begingroup
+    \parindent=\z@
+    \renewcommand\thefootnote{\@fnsymbol\c@footnote}%
+    \if@twocolumn
+      \ifnum \col@number=\@ne
+        \@maketitle
+      \else
+        \twocolumn[\@maketitle]%
+      \fi
+    \else
+      \newpage
+      \global\@topnum\z@   % Prevents figures from going at top of page.
+      \@maketitle
+    \fi
+    \thispagestyle{empty}\@thanks
+%
+    \def\\{\unskip\ \ignorespaces}\def\inst##1{\unskip{}}%
+    \def\thanks##1{\unskip{}}\def\fnmsep{\unskip}%
+    \instindent=\hsize
+    \advance\instindent by-\headlineindent
+%    \if!\the\toctitle!\addcontentsline{toc}{title}{\@title}\else
+%       \addcontentsline{toc}{title}{\the\toctitle}\fi
+    \if@runhead
+       \if!\the\titlerunning!\else
+         \edef\@title{\the\titlerunning}%
+       \fi
+       \global\setbox\titrun=\hbox{\small\rm\unboldmath\ignorespaces\@title}%
+       \ifdim\wd\titrun>\instindent
+          \typeout{Title too long for running head. Please supply}%
+          \typeout{a shorter form with \string\titlerunning\space prior to
+                   \string\maketitle}%
+          \global\setbox\titrun=\hbox{\small\rm
+          Title Suppressed Due to Excessive Length}%
+       \fi
+       \xdef\@title{\copy\titrun}%
+    \fi
+%
+    \if!\the\tocauthor!\relax
+      {\def\and{\noexpand\protect\noexpand\and}%
+      \protected@xdef\toc@uthor{\@author}}%
+    \else
+      \def\\{\noexpand\protect\noexpand\newline}%
+      \protected@xdef\scratch{\the\tocauthor}%
+      \protected@xdef\toc@uthor{\scratch}%
+    \fi
+%    \addcontentsline{toc}{author}{\toc@uthor}%
+    \if@runhead
+       \if!\the\authorrunning!
+         \value{@inst}=\value{@auth}%
+         \setcounter{@auth}{1}%
+       \else
+         \edef\@author{\the\authorrunning}%
+       \fi
+       \global\setbox\authrun=\hbox{\small\unboldmath\@author\unskip}%
+       \ifdim\wd\authrun>\instindent
+          \typeout{Names of authors too long for running head. Please supply}%
+          \typeout{a shorter form with \string\authorrunning\space prior to
+                   \string\maketitle}%
+          \global\setbox\authrun=\hbox{\small\rm
+          Authors Suppressed Due to Excessive Length}%
+       \fi
+       \xdef\@author{\copy\authrun}%
+       \markboth{\@author}{\@title}%
+     \fi
+  \endgroup
+  \setcounter{footnote}{\fnnstart}%
+  \clearheadinfo}
+%
+\def\@maketitle{\newpage
+ \markboth{}{}%
+ \def\lastand{\ifnum\value{@inst}=2\relax
+                 \unskip{} \andname\
+              \else
+                 \unskip \lastandname\
+              \fi}%
+ \def\and{\stepcounter{@auth}\relax
+          \ifnum\value{@auth}=\value{@inst}%
+             \lastand
+          \else
+             \unskip,
+          \fi}%
+ \begin{center}%
+ \let\newline\\
+ {\Large \bfseries\boldmath
+  \pretolerance=10000
+  \@title \par}\vskip .8cm
+\if!\@subtitle!\else {\large \bfseries\boldmath
+  \vskip -.65cm
+  \pretolerance=10000
+  \@subtitle \par}\vskip .8cm\fi
+ \setbox0=\vbox{\setcounter{@auth}{1}\def\and{\stepcounter{@auth}}%
+ \def\thanks##1{}\@author}%
+ \global\value{@inst}=\value{@auth}%
+ \global\value{auco}=\value{@auth}%
+ \setcounter{@auth}{1}%
+{\lineskip .5em
+\noindent\ignorespaces
+\@author\vskip.35cm}
+ {\small\institutename}
+ \end{center}%
+ }
+
+% definition of the "\spnewtheorem" command.
+%
+% Usage:
+%
+%     \spnewtheorem{env_nam}{caption}[within]{cap_font}{body_font}
+% or  \spnewtheorem{env_nam}[numbered_like]{caption}{cap_font}{body_font}
+% or  \spnewtheorem*{env_nam}{caption}{cap_font}{body_font}
+%
+% New is "cap_font" and "body_font". It stands for
+% fontdefinition of the caption and the text itself.
+%
+% "\spnewtheorem*" gives a theorem without number.
+%
+% A defined spnewthoerem environment is used as described
+% by Lamport.
+%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\def\@thmcountersep{}
+\def\@thmcounterend{.}
+
+\def\spnewtheorem{\@ifstar{\@sthm}{\@Sthm}}
+
+% definition of \spnewtheorem with number
+
+\def\@spnthm#1#2{%
+  \@ifnextchar[{\@spxnthm{#1}{#2}}{\@spynthm{#1}{#2}}}
+\def\@Sthm#1{\@ifnextchar[{\@spothm{#1}}{\@spnthm{#1}}}
+
+\def\@spxnthm#1#2[#3]#4#5{\expandafter\@ifdefinable\csname #1\endcsname
+   {\@definecounter{#1}\@addtoreset{#1}{#3}%
+   \expandafter\xdef\csname the#1\endcsname{\expandafter\noexpand
+     \csname the#3\endcsname \noexpand\@thmcountersep \@thmcounter{#1}}%
+   \expandafter\xdef\csname #1name\endcsname{#2}%
+   \global\@namedef{#1}{\@spthm{#1}{\csname #1name\endcsname}{#4}{#5}}%
+                              \global\@namedef{end#1}{\@endtheorem}}}
+
+\def\@spynthm#1#2#3#4{\expandafter\@ifdefinable\csname #1\endcsname
+   {\@definecounter{#1}%
+   \expandafter\xdef\csname the#1\endcsname{\@thmcounter{#1}}%
+   \expandafter\xdef\csname #1name\endcsname{#2}%
+   \global\@namedef{#1}{\@spthm{#1}{\csname #1name\endcsname}{#3}{#4}}%
+                               \global\@namedef{end#1}{\@endtheorem}}}
+
+\def\@spothm#1[#2]#3#4#5{%
+  \@ifundefined{c@#2}{\@latexerr{No theorem environment `#2' defined}\@eha}%
+  {\expandafter\@ifdefinable\csname #1\endcsname
+  {\global\@namedef{the#1}{\@nameuse{the#2}}%
+  \expandafter\xdef\csname #1name\endcsname{#3}%
+  \global\@namedef{#1}{\@spthm{#2}{\csname #1name\endcsname}{#4}{#5}}%
+  \global\@namedef{end#1}{\@endtheorem}}}}
+
+\def\@spthm#1#2#3#4{\topsep 7\p@ \@plus2\p@ \@minus4\p@
+\refstepcounter{#1}%
+\@ifnextchar[{\@spythm{#1}{#2}{#3}{#4}}{\@spxthm{#1}{#2}{#3}{#4}}}
+
+\def\@spxthm#1#2#3#4{\@spbegintheorem{#2}{\csname the#1\endcsname}{#3}{#4}%
+                    \ignorespaces}
+
+\def\@spythm#1#2#3#4[#5]{\@spopargbegintheorem{#2}{\csname
+       the#1\endcsname}{#5}{#3}{#4}\ignorespaces}
+
+\def\@spbegintheorem#1#2#3#4{\trivlist
+                 \item[\hskip\labelsep{#3#1\ #2\@thmcounterend}]#4}
+
+\def\@spopargbegintheorem#1#2#3#4#5{\trivlist
+      \item[\hskip\labelsep{#4#1\ #2}]{#4(#3)\@thmcounterend\ }#5}
+
+% definition of \spnewtheorem* without number
+
+\def\@sthm#1#2{\@Ynthm{#1}{#2}}
+
+\def\@Ynthm#1#2#3#4{\expandafter\@ifdefinable\csname #1\endcsname
+   {\global\@namedef{#1}{\@Thm{\csname #1name\endcsname}{#3}{#4}}%
+    \expandafter\xdef\csname #1name\endcsname{#2}%
+    \global\@namedef{end#1}{\@endtheorem}}}
+
+\def\@Thm#1#2#3{\topsep 7\p@ \@plus2\p@ \@minus4\p@
+\@ifnextchar[{\@Ythm{#1}{#2}{#3}}{\@Xthm{#1}{#2}{#3}}}
+
+\def\@Xthm#1#2#3{\@Begintheorem{#1}{#2}{#3}\ignorespaces}
+
+\def\@Ythm#1#2#3[#4]{\@Opargbegintheorem{#1}
+       {#4}{#2}{#3}\ignorespaces}
+
+\def\@Begintheorem#1#2#3{#3\trivlist
+                           \item[\hskip\labelsep{#2#1\@thmcounterend}]}
+
+\def\@Opargbegintheorem#1#2#3#4{#4\trivlist
+      \item[\hskip\labelsep{#3#1}]{#3(#2)\@thmcounterend\ }}
+
+\if@envcntsect
+   \def\@thmcountersep{.}
+   \spnewtheorem{theorem}{Theorem}[section]{\bfseries}{\itshape}
+\else
+   \spnewtheorem{theorem}{Theorem}{\bfseries}{\itshape}
+   \if@envcntreset
+      \@addtoreset{theorem}{section}
+   \else
+      \@addtoreset{theorem}{chapter}
+   \fi
+\fi
+
+%definition of divers theorem environments
+\spnewtheorem*{claim}{Claim}{\itshape}{\rmfamily}
+\spnewtheorem*{proof}{Proof}{\itshape}{\rmfamily}
+\if@envcntsame % alle Umgebungen wie Theorem.
+   \def\spn@wtheorem#1#2#3#4{\@spothm{#1}[theorem]{#2}{#3}{#4}}
+\else % alle Umgebungen mit eigenem Zaehler
+   \if@envcntsect % mit section numeriert
+      \def\spn@wtheorem#1#2#3#4{\@spxnthm{#1}{#2}[section]{#3}{#4}}
+   \else % nicht mit section numeriert
+      \if@envcntreset
+         \def\spn@wtheorem#1#2#3#4{\@spynthm{#1}{#2}{#3}{#4}
+                                   \@addtoreset{#1}{section}}
+      \else
+         \def\spn@wtheorem#1#2#3#4{\@spynthm{#1}{#2}{#3}{#4}
+                                   \@addtoreset{#1}{chapter}}%
+      \fi
+   \fi
+\fi
+\spn@wtheorem{case}{Case}{\itshape}{\rmfamily}
+\spn@wtheorem{conjecture}{Conjecture}{\itshape}{\rmfamily}
+\spn@wtheorem{corollary}{Corollary}{\bfseries}{\itshape}
+\spn@wtheorem{definition}{Definition}{\bfseries}{\itshape}
+\spn@wtheorem{example}{Example}{\itshape}{\rmfamily}
+\spn@wtheorem{exercise}{Exercise}{\itshape}{\rmfamily}
+\spn@wtheorem{lemma}{Lemma}{\bfseries}{\itshape}
+\spn@wtheorem{note}{Note}{\itshape}{\rmfamily}
+\spn@wtheorem{problem}{Problem}{\itshape}{\rmfamily}
+\spn@wtheorem{property}{Property}{\itshape}{\rmfamily}
+\spn@wtheorem{proposition}{Proposition}{\bfseries}{\itshape}
+\spn@wtheorem{question}{Question}{\itshape}{\rmfamily}
+\spn@wtheorem{solution}{Solution}{\itshape}{\rmfamily}
+\spn@wtheorem{remark}{Remark}{\itshape}{\rmfamily}
+
+\def\@takefromreset#1#2{%
+    \def\@tempa{#1}%
+    \let\@tempd\@elt
+    \def\@elt##1{%
+        \def\@tempb{##1}%
+        \ifx\@tempa\@tempb\else
+            \@addtoreset{##1}{#2}%
+        \fi}%
+    \expandafter\expandafter\let\expandafter\@tempc\csname cl@#2\endcsname
+    \expandafter\def\csname cl@#2\endcsname{}%
+    \@tempc
+    \let\@elt\@tempd}
+
+\def\theopargself{\def\@spopargbegintheorem##1##2##3##4##5{\trivlist
+      \item[\hskip\labelsep{##4##1\ ##2}]{##4##3\@thmcounterend\ }##5}
+                  \def\@Opargbegintheorem##1##2##3##4{##4\trivlist
+      \item[\hskip\labelsep{##3##1}]{##3##2\@thmcounterend\ }}
+      }
+
+\renewenvironment{abstract}{%
+      \list{}{\advance\topsep by0.35cm\relax\small
+      \leftmargin=1cm
+      \labelwidth=\z@
+      \listparindent=\z@
+      \itemindent\listparindent
+      \rightmargin\leftmargin}\item[\hskip\labelsep
+                                    \bfseries\abstractname]}
+    {\endlist}
+
+\newdimen\headlineindent             % dimension for space between
+\headlineindent=1.166cm              % number and text of headings.
+
+\def\ps@headings{\let\@mkboth\@gobbletwo
+   \let\@oddfoot\@empty\let\@evenfoot\@empty
+   \def\@evenhead{\normalfont\small\rlap{\thepage}\hspace{\headlineindent}%
+                  \leftmark\hfil}
+   \def\@oddhead{\normalfont\small\hfil\rightmark\hspace{\headlineindent}%
+                 \llap{\thepage}}
+   \def\chaptermark##1{}%
+   \def\sectionmark##1{}%
+   \def\subsectionmark##1{}}
+
+\def\ps@titlepage{\let\@mkboth\@gobbletwo
+   \let\@oddfoot\@empty\let\@evenfoot\@empty
+   \def\@evenhead{\normalfont\small\rlap{\thepage}\hspace{\headlineindent}%
+                  \hfil}
+   \def\@oddhead{\normalfont\small\hfil\hspace{\headlineindent}%
+                 \llap{\thepage}}
+   \def\chaptermark##1{}%
+   \def\sectionmark##1{}%
+   \def\subsectionmark##1{}}
+
+\if@runhead\ps@headings\else
+\ps@empty\fi
+
+\setlength\arraycolsep{1.4\p@}
+\setlength\tabcolsep{1.4\p@}
+
+\endinput
+%end of file llncs.cls