
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




