--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/thys2/Re.thy	Sun Oct 10 18:35:21 2021 +0100
@@ -0,0 +1,3515 @@
+   
+theory Re
+  imports "Main" 
+begin
+
+
+section {* Sequential Composition of Sets *}
+
+definition
+  Sequ :: "string set \<Rightarrow> string set \<Rightarrow> string set" ("_ ;; _" [100,100] 100)
+where 
+  "A ;; B = {s1 @ s2 | s1 s2. s1 \<in> A \<and> s2 \<in> B}"
+
+text {* Two Simple Properties about Sequential Composition *}
+
+lemma seq_empty [simp]:
+  shows "A ;; {[]} = A"
+  and   "{[]} ;; A = A"
+by (simp_all add: Sequ_def)
+
+lemma seq_null [simp]:
+  shows "A ;; {} = {}"
+  and   "{} ;; A = {}"
+by (simp_all add: Sequ_def)
+
+section {* Regular Expressions *}
+
+datatype rexp =
+  NULL
+| EMPTY
+| CHAR char
+| SEQ rexp rexp
+| ALT rexp rexp
+
+section {* Semantics of Regular Expressions *}
+ 
+fun
+  L :: "rexp \<Rightarrow> string set"
+where
+  "L (NULL) = {}"
+| "L (EMPTY) = {[]}"
+| "L (CHAR c) = {[c]}"
+| "L (SEQ r1 r2) = (L r1) ;; (L r2)"
+| "L (ALT r1 r2) = (L r1) \<union> (L r2)"
+
+fun
+ nullable :: "rexp \<Rightarrow> bool"
+where
+  "nullable (NULL) = False"
+| "nullable (EMPTY) = True"
+| "nullable (CHAR c) = False"
+| "nullable (ALT r1 r2) = (nullable r1 \<or> nullable r2)"
+| "nullable (SEQ r1 r2) = (nullable r1 \<and> nullable r2)"
+
+lemma nullable_correctness:
+  shows "nullable r  \<longleftrightarrow> [] \<in> (L r)"
+apply (induct r) 
+apply(auto simp add: Sequ_def) 
+done
+
+section {* Values *}
+
+datatype val = 
+  Void
+| Char char
+| Seq val val
+| Right val
+| Left val
+
+section {* The string behind a value *}
+
+fun 
+  flat :: "val \<Rightarrow> string"
+where
+  "flat(Void) = []"
+| "flat(Char c) = [c]"
+| "flat(Left v) = flat(v)"
+| "flat(Right v) = flat(v)"
+| "flat(Seq v1 v2) = flat(v1) @ flat(v2)"
+
+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 : EMPTY"
+| "\<turnstile> Char c : CHAR c"
+
+lemma not_nullable_flat:
+  assumes "\<turnstile> v : r" "\<not>nullable r"
+  shows "flat v \<noteq> []"
+using assms
+apply(induct)
+apply(auto)
+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)
+done
+
+lemma L_flat_Prf:
+  "L(r) = {flat v | v. \<turnstile> v : r}"
+apply(induct r)
+apply(auto dest: Prf_flat_L simp add: Sequ_def)
+apply (metis Prf.intros(4) flat.simps(1))
+apply (metis Prf.intros(5) flat.simps(2))
+apply (metis Prf.intros(1) flat.simps(5))
+apply (metis Prf.intros(2) flat.simps(3))
+apply (metis Prf.intros(3) flat.simps(4))
+apply(erule Prf.cases)
+apply(auto)
+done
+
+section {* Greedy Ordering according to Frisch/Cardelli *}
+
+inductive 
+  GrOrd :: "val \<Rightarrow> val \<Rightarrow> bool" ("_ gr\<succ> _")
+where 
+  "v1 gr\<succ> v1' \<Longrightarrow> (Seq v1 v2) gr\<succ> (Seq v1' v2')"
+| "v2 gr\<succ> v2' \<Longrightarrow> (Seq v1 v2) gr\<succ> (Seq v1 v2')"
+| "v1 gr\<succ> v2 \<Longrightarrow> (Left v1) gr\<succ> (Left v2)"
+| "v1 gr\<succ> v2 \<Longrightarrow> (Right v1) gr\<succ> (Right v2)"
+| "(Left v2) gr\<succ>(Right v1)"
+| "(Char c) gr\<succ> (Char c)"
+| "(Void) gr\<succ> (Void)"
+
+lemma Gr_refl:
+  assumes "\<turnstile> v : r"
+  shows "v gr\<succ> v"
+using assms
+apply(induct)
+apply(auto intro: GrOrd.intros)
+done
+
+lemma Gr_total:
+  assumes "\<turnstile> v1 : r" "\<turnstile> v2 : r"
+  shows "v1 gr\<succ> v2 \<or> v2 gr\<succ> v1"
+using assms
+apply(induct v1 r arbitrary: v2 rule: Prf.induct)
+apply(rotate_tac 4)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(clarify)
+apply (metis GrOrd.intros(1) GrOrd.intros(2))
+apply(rotate_tac 2)
+apply(erule Prf.cases)
+apply(simp_all)
+apply(clarify)
+apply (metis GrOrd.intros(3))
+apply(clarify)
+apply (metis GrOrd.intros(5))
+apply(rotate_tac 2)
+apply(erule Prf.cases)
+apply(simp_all)
+apply(clarify)
+apply (metis GrOrd.intros(5))
+apply(clarify)
+apply (metis GrOrd.intros(4))
+apply(erule Prf.cases)
+apply(simp_all)
+apply (metis GrOrd.intros(7))
+apply(erule Prf.cases)
+apply(simp_all)
+apply (metis GrOrd.intros(6))
+done
+
+lemma Gr_trans: 
+  assumes "v1 gr\<succ> v2" "v2 gr\<succ> v3" 
+  and     "\<turnstile> v1 : r" "\<turnstile> v2 : r" "\<turnstile> v3 : r"
+  shows "v1 gr\<succ> v3"
+using assms
+apply(induct r arbitrary: v1 v2 v3)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+defer
+(* ALT case *)
+apply(erule Prf.cases)
+apply(simp_all (no_asm_use))[5]
+apply(erule Prf.cases)
+apply(simp_all (no_asm_use))[5]
+apply(erule Prf.cases)
+apply(simp_all (no_asm_use))[5]
+apply(clarify)
+apply(erule GrOrd.cases)
+apply(simp_all (no_asm_use))[7]
+apply(erule GrOrd.cases)
+apply(simp_all (no_asm_use))[7]
+apply (metis GrOrd.intros(3))
+apply(clarify)
+apply(erule GrOrd.cases)
+apply(simp_all (no_asm_use))[7]
+apply(erule GrOrd.cases)
+apply(simp_all (no_asm_use))[7]
+apply (metis GrOrd.intros(5))
+apply(erule Prf.cases)
+apply(simp_all (no_asm_use))[5]
+apply(clarify)
+apply(erule GrOrd.cases)
+apply(simp_all (no_asm_use))[7]
+apply(erule GrOrd.cases)
+apply(simp_all (no_asm_use))[7]
+apply (metis GrOrd.intros(5))
+apply(erule Prf.cases)
+apply(simp_all (no_asm_use))[5]
+apply(erule Prf.cases)
+apply(simp_all (no_asm_use))[5]
+apply(clarify)
+apply(erule GrOrd.cases)
+apply(simp_all (no_asm_use))[7]
+apply(clarify)
+apply(erule GrOrd.cases)
+apply(simp_all (no_asm_use))[7]
+apply(erule Prf.cases)
+apply(simp_all (no_asm_use))[5]
+apply(clarify)
+apply(erule GrOrd.cases)
+apply(simp_all (no_asm_use))[7]
+apply(erule GrOrd.cases)
+apply(simp_all (no_asm_use))[7]
+apply(clarify)
+apply(erule GrOrd.cases)
+apply(simp_all (no_asm_use))[7]
+apply(erule GrOrd.cases)
+apply(simp_all (no_asm_use))[7]
+apply (metis GrOrd.intros(4))
+(* SEQ case *)
+apply(erule Prf.cases)
+apply(simp_all (no_asm_use))[5]
+apply(erule Prf.cases)
+apply(simp_all (no_asm_use))[5]
+apply(erule Prf.cases)
+apply(simp_all (no_asm_use))[5]
+apply(clarify)
+apply(erule GrOrd.cases)
+apply(simp_all (no_asm_use))[7]
+apply(erule GrOrd.cases)
+apply(simp_all (no_asm_use))[7]
+apply(clarify)
+apply (metis GrOrd.intros(1))
+apply (metis GrOrd.intros(1))
+apply(erule GrOrd.cases)
+apply(simp_all (no_asm_use))[7]
+apply (metis GrOrd.intros(1))
+by (metis GrOrd.intros(1) Gr_refl)
+
+
+section {* Values Sets *}
+
+definition prefix :: "string \<Rightarrow> string \<Rightarrow> bool" ("_ \<sqsubseteq> _" [100, 100] 100)
+where
+  "s1 \<sqsubseteq> s2 \<equiv> \<exists>s3. s1 @ s3 = s2"
+
+definition sprefix :: "string \<Rightarrow> string \<Rightarrow> bool" ("_ \<sqsubset> _" [100, 100] 100)
+where
+  "s1 \<sqsubset> s2 \<equiv> (s1 \<sqsubseteq> s2 \<and> s1 \<noteq> s2)"
+
+lemma length_sprefix:
+  "s1 \<sqsubset> s2 \<Longrightarrow> length s1 < length s2"
+unfolding sprefix_def prefix_def
+by (auto)
+
+definition Prefixes :: "string \<Rightarrow> string set" where
+  "Prefixes s \<equiv> {sp. sp \<sqsubseteq> s}"
+
+definition Suffixes :: "string \<Rightarrow> string set" where
+  "Suffixes s \<equiv> rev ` (Prefixes (rev s))"
+
+lemma Suffixes_in: 
+  "\<exists>s1. s1 @ s2 = s3 \<Longrightarrow> s2 \<in> Suffixes s3"
+unfolding Suffixes_def Prefixes_def prefix_def image_def
+apply(auto)
+by (metis rev_rev_ident)
+
+lemma Prefixes_Cons:
+  "Prefixes (c # s) = {[]} \<union> {c # sp | sp. sp \<in> Prefixes s}"
+unfolding Prefixes_def prefix_def
+apply(auto simp add: append_eq_Cons_conv) 
+done
+
+lemma finite_Prefixes:
+  "finite (Prefixes s)"
+apply(induct s)
+apply(auto simp add: Prefixes_def prefix_def)[1]
+apply(simp add: Prefixes_Cons)
+done
+
+lemma finite_Suffixes:
+  "finite (Suffixes s)"
+unfolding Suffixes_def
+apply(rule finite_imageI)
+apply(rule finite_Prefixes)
+done
+
+lemma prefix_Cons:
+  "((c # s1) \<sqsubseteq> (c # s2)) = (s1 \<sqsubseteq> s2)"
+apply(auto simp add: prefix_def)
+done
+
+lemma prefix_append:
+  "((s @ s1) \<sqsubseteq> (s @ s2)) = (s1 \<sqsubseteq> s2)"
+apply(induct s)
+apply(simp)
+apply(simp add: prefix_Cons)
+done
+
+
+definition Values :: "rexp \<Rightarrow> string \<Rightarrow> val set" where
+  "Values r s \<equiv> {v. \<turnstile> v : r \<and> flat v \<sqsubseteq> s}"
+
+definition rest :: "val \<Rightarrow> string \<Rightarrow> string" where
+  "rest v s \<equiv> drop (length (flat v)) s"
+
+lemma rest_flat:
+  assumes "flat v1 \<sqsubseteq> s"
+  shows "flat v1 @ rest v1 s = s"
+using assms
+apply(auto simp add: rest_def prefix_def)
+done
+
+
+lemma rest_Suffixes:
+  "rest v s \<in> Suffixes s"
+unfolding rest_def
+by (metis Suffixes_in append_take_drop_id)
+
+
+lemma Values_recs:
+  "Values (NULL) s = {}"
+  "Values (EMPTY) s = {Void}"
+  "Values (CHAR c) s = (if [c] \<sqsubseteq> s then {Char c} else {})" 
+  "Values (ALT r1 r2) s = {Left v | v. v \<in> Values r1 s} \<union> {Right v | v. v \<in> Values r2 s}"
+  "Values (SEQ r1 r2) s = {Seq v1 v2 | v1 v2. v1 \<in> Values r1 s \<and> v2 \<in> Values r2 (rest v1 s)}"
+unfolding Values_def
+apply(auto)
+(*NULL*)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+(*EMPTY*)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(rule Prf.intros)
+apply (metis append_Nil prefix_def)
+(*CHAR*)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(rule Prf.intros)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+(*ALT*)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply (metis Prf.intros(2))
+apply (metis Prf.intros(3))
+(*SEQ*)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply (simp add: append_eq_conv_conj prefix_def rest_def)
+apply (metis Prf.intros(1))
+apply (simp add: append_eq_conv_conj prefix_def rest_def)
+done
+
+lemma Values_finite:
+  "finite (Values r s)"
+apply(induct r arbitrary: s)
+apply(simp_all add: Values_recs)
+thm finite_surj
+apply(rule_tac f="\<lambda>(x, y). Seq x y" and 
+               A="{(v1, v2) | v1 v2. v1 \<in> Values r1 s \<and> v2 \<in> Values r2 (rest v1 s)}" in finite_surj)
+prefer 2
+apply(auto)[1]
+apply(rule_tac B="\<Union>sp \<in> Suffixes s. {(v1, v2). v1 \<in> Values r1 s \<and> v2 \<in> Values r2 sp}" in finite_subset)
+apply(auto)[1]
+apply (metis rest_Suffixes)
+apply(rule finite_UN_I)
+apply(rule finite_Suffixes)
+apply(simp)
+done
+
+section {* Sulzmann functions *}
+
+fun 
+  mkeps :: "rexp \<Rightarrow> val"
+where
+  "mkeps(EMPTY) = 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))"
+
+section {* Derivatives *}
+
+fun
+ der :: "char \<Rightarrow> rexp \<Rightarrow> rexp"
+where
+  "der c (NULL) = NULL"
+| "der c (EMPTY) = NULL"
+| "der c (CHAR c') = (if c = c' then EMPTY else NULL)"
+| "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)"
+
+fun 
+ ders :: "string \<Rightarrow> rexp \<Rightarrow> rexp"
+where
+  "ders [] r = r"
+| "ders (c # s) r = ders s (der c r)"
+
+
+section {* Injection function *}
+
+fun injval :: "rexp \<Rightarrow> char \<Rightarrow> val \<Rightarrow> val"
+where
+  "injval (EMPTY) c Void = Char c"
+| "injval (CHAR d) c Void = Char d"
+| "injval (CHAR d) c (Char c') = Seq (Char d) (Char c')"
+| "injval (ALT r1 r2) c (Left v1) = Left(injval r1 c v1)"
+| "injval (ALT r1 r2) c (Right v2) = Right(injval r2 c v2)"
+| "injval (SEQ r1 r2) c (Char c') = Seq (Char c) (Char c')"
+| "injval (SEQ r1 r2) c (Seq v1 v2) = Seq (injval r1 c v1) v2"
+| "injval (SEQ r1 r2) c (Left (Seq v1 v2)) = Seq (injval r1 c v1) v2"
+| "injval (SEQ r1 r2) c (Right v2) = Seq (mkeps r1) (injval r2 c v2)"
+
+fun 
+  lex :: "rexp \<Rightarrow> string \<Rightarrow> val option"
+where
+  "lex r [] = (if nullable r then Some(mkeps r) else None)"
+| "lex r (c#s) = (case (lex (der c r) s) of  
+                    None \<Rightarrow> None
+                  | Some(v) \<Rightarrow> Some(injval r c v))"
+
+fun 
+  lex2 :: "rexp \<Rightarrow> string \<Rightarrow> val"
+where
+  "lex2 r [] = mkeps r"
+| "lex2 r (c#s) = injval r c (lex2 (der c r) s)"
+
+
+section {* Projection function *}
+
+fun projval :: "rexp \<Rightarrow> char \<Rightarrow> val \<Rightarrow> val"
+where
+  "projval (CHAR d) c _ = Void"
+| "projval (ALT r1 r2) c (Left v1) = Left (projval r1 c v1)"
+| "projval (ALT r1 r2) c (Right v2) = Right (projval r2 c v2)"
+| "projval (SEQ r1 r2) c (Seq v1 v2) = 
+     (if flat v1 = [] then Right(projval r2 c v2) 
+      else if nullable r1 then Left (Seq (projval r1 c v1) v2)
+                          else Seq (projval r1 c v1) v2)"
+
+
+
+lemma mkeps_nullable:
+  assumes "nullable(r)" 
+  shows "\<turnstile> mkeps r : r"
+using assms
+apply(induct rule: nullable.induct)
+apply(auto intro: Prf.intros)
+done
+
+lemma mkeps_flat:
+  assumes "nullable(r)" 
+  shows "flat (mkeps r) = []"
+using assms
+apply(induct rule: nullable.induct)
+apply(auto)
+done
+
+lemma v3:
+  assumes "\<turnstile> v : der c r" 
+  shows "\<turnstile> (injval r c v) : r"
+using assms
+apply(induct arbitrary: v rule: der.induct)
+apply(simp)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(simp)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(case_tac "c = c'")
+apply(simp)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply (metis Prf.intros(5))
+apply(simp)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(simp)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply (metis Prf.intros(2))
+apply (metis Prf.intros(3))
+apply(simp)
+apply(case_tac "nullable r1")
+apply(simp)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(auto)[1]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(auto)[1]
+apply (metis Prf.intros(1))
+apply(auto)[1]
+apply (metis Prf.intros(1) mkeps_nullable)
+apply(simp)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(auto)[1]
+apply(rule Prf.intros)
+apply(auto)[2]
+done
+
+lemma v3_proj:
+  assumes "\<turnstile> v : r" and "\<exists>s. (flat v) = c # s"
+  shows "\<turnstile> (projval r c v) : der c r"
+using assms
+apply(induct rule: Prf.induct)
+prefer 4
+apply(simp)
+prefer 4
+apply(simp)
+apply (metis Prf.intros(4))
+prefer 2
+apply(simp)
+apply (metis Prf.intros(2))
+prefer 2
+apply(simp)
+apply (metis Prf.intros(3))
+apply(auto)
+apply(rule Prf.intros)
+apply(simp)
+apply (metis Prf_flat_L nullable_correctness)
+apply(rule Prf.intros)
+apply(rule Prf.intros)
+apply (metis Cons_eq_append_conv)
+apply(simp)
+apply(rule Prf.intros)
+apply (metis Cons_eq_append_conv)
+apply(simp)
+done
+
+lemma v4:
+  assumes "\<turnstile> v : der c r" 
+  shows "flat (injval r c v) = c # (flat v)"
+using assms
+apply(induct arbitrary: v rule: der.induct)
+apply(simp)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(simp)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(simp)
+apply(case_tac "c = c'")
+apply(simp)
+apply(auto)[1]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(simp)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(simp)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(simp)
+apply(case_tac "nullable r1")
+apply(simp)
+apply(erule Prf.cases)
+apply(simp_all (no_asm_use))[5]
+apply(auto)[1]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(clarify)
+apply(simp only: injval.simps flat.simps)
+apply(auto)[1]
+apply (metis mkeps_flat)
+apply(simp)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+done
+
+lemma v4_proj:
+  assumes "\<turnstile> v : r" and "\<exists>s. (flat v) = c # s"
+  shows "c # flat (projval r c v) = flat v"
+using assms
+apply(induct rule: Prf.induct)
+prefer 4
+apply(simp)
+prefer 4
+apply(simp)
+prefer 2
+apply(simp)
+prefer 2
+apply(simp)
+apply(auto)
+by (metis Cons_eq_append_conv)
+
+lemma v4_proj2:
+  assumes "\<turnstile> v : r" and "(flat v) = c # s"
+  shows "flat (projval r c v) = s"
+using assms
+by (metis list.inject v4_proj)
+
+
+section {* Alternative Posix definition *}
+
+inductive 
+  PMatch :: "string \<Rightarrow> rexp \<Rightarrow> val \<Rightarrow> bool" ("_ \<in> _ \<rightarrow> _" [100, 100, 100] 100)
+where
+  "[] \<in> EMPTY \<rightarrow> Void"
+| "[c] \<in> (CHAR c) \<rightarrow> (Char c)"
+| "s \<in> r1 \<rightarrow> v \<Longrightarrow> s \<in> (ALT r1 r2) \<rightarrow> (Left v)"
+| "\<lbrakk>s \<in> r2 \<rightarrow> v; s \<notin> L(r1)\<rbrakk> \<Longrightarrow> s \<in> (ALT r1 r2) \<rightarrow> (Right v)"
+| "\<lbrakk>s1 \<in> r1 \<rightarrow> v1; s2 \<in> r2 \<rightarrow> v2;
+    \<not>(\<exists>s3 s4. s3 \<noteq> [] \<and> s3 @ s4 = s2 \<and> (s1 @ s3) \<in> L r1 \<and> s4 \<in> L r2)\<rbrakk> \<Longrightarrow> 
+    (s1 @ s2) \<in> (SEQ r1 r2) \<rightarrow> (Seq v1 v2)"
+
+
+lemma PMatch_mkeps:
+  assumes "nullable r"
+  shows "[] \<in> r \<rightarrow> mkeps r"
+using assms
+apply(induct r)
+apply(auto)
+apply (metis PMatch.intros(1))
+apply(subst append.simps(1)[symmetric])
+apply (rule PMatch.intros)
+apply(simp)
+apply(simp)
+apply(auto)[1]
+apply (rule PMatch.intros)
+apply(simp)
+apply (rule PMatch.intros)
+apply(simp)
+apply (rule PMatch.intros)
+apply(simp)
+by (metis nullable_correctness)
+
+
+lemma PMatch1:
+  assumes "s \<in> r \<rightarrow> v"
+  shows "\<turnstile> v : r" "flat v = s"
+using assms
+apply(induct s r v rule: PMatch.induct)
+apply(auto)
+apply (metis Prf.intros(4))
+apply (metis Prf.intros(5))
+apply (metis Prf.intros(2))
+apply (metis Prf.intros(3))
+apply (metis Prf.intros(1))
+done
+
+lemma PMatch_Values:
+  assumes "s \<in> r \<rightarrow> v"
+  shows "v \<in> Values r s"
+using assms
+apply(simp add: Values_def PMatch1)
+by (metis append_Nil2 prefix_def)
+
+lemma PMatch2:
+  assumes "s \<in> (der c r) \<rightarrow> v"
+  shows "(c#s) \<in> r \<rightarrow> (injval r c v)"
+using assms
+apply(induct c r arbitrary: s v rule: der.induct)
+apply(auto)
+apply(erule PMatch.cases)
+apply(simp_all)[5]
+apply(erule PMatch.cases)
+apply(simp_all)[5]
+apply(case_tac "c = c'")
+apply(simp)
+apply(erule PMatch.cases)
+apply(simp_all)[5]
+apply (metis PMatch.intros(2))
+apply(simp)
+apply(erule PMatch.cases)
+apply(simp_all)[5]
+apply(erule PMatch.cases)
+apply(simp_all)[5]
+apply (metis PMatch.intros(3))
+apply(clarify)
+apply(rule PMatch.intros)
+apply metis
+apply(simp add: L_flat_Prf)
+apply(auto)[1]
+thm v3_proj
+apply(frule_tac c="c" in v3_proj)
+apply metis
+apply(drule_tac x="projval r1 c v" in spec)
+apply(drule mp)
+apply (metis v4_proj2)
+apply(simp)
+apply(case_tac "nullable r1")
+apply(simp)
+defer
+apply(simp)
+apply(erule PMatch.cases)
+apply(simp_all)[5]
+apply(clarify)
+apply(subst append.simps(2)[symmetric])
+apply(rule PMatch.intros)
+apply metis
+apply metis
+apply(auto)[1]
+apply(simp add: L_flat_Prf)
+apply(auto)[1]
+apply(frule_tac c="c" in v3_proj)
+apply metis
+apply(drule_tac x="s3" in spec)
+apply(drule mp)
+apply(rule_tac x="projval r1 c v" in exI)
+apply(rule conjI)
+apply (metis v4_proj2)
+apply(simp)
+apply metis
+(* nullable case *)
+apply(erule PMatch.cases)
+apply(simp_all)[5]
+apply(clarify)
+apply(erule PMatch.cases)
+apply(simp_all)[5]
+apply(clarify)
+apply(subst append.simps(2)[symmetric])
+apply(rule PMatch.intros)
+apply metis
+apply metis
+apply(auto)[1]
+apply(simp add: L_flat_Prf)
+apply(auto)[1]
+apply(frule_tac c="c" in v3_proj)
+apply metis
+apply(drule_tac x="s3" in spec)
+apply(drule mp)
+apply (metis v4_proj2)
+apply(simp)
+(* interesting case *)
+apply(clarify)
+apply(simp)
+thm L.simps
+apply(subst (asm) L.simps(4)[symmetric])
+apply(simp only: L_flat_Prf)
+apply(simp)
+apply(subst append.simps(1)[symmetric])
+apply(rule PMatch.intros)
+apply (metis PMatch_mkeps)
+apply metis
+apply(auto)
+apply(simp only: L_flat_Prf)
+apply(simp)
+apply(auto)
+apply(drule_tac x="Seq (projval r1 c v) vb" in spec)
+apply(drule mp)
+apply(simp)
+apply (metis append_Cons butlast_snoc last_snoc neq_Nil_conv rotate1.simps(2) v4_proj2)
+apply(subgoal_tac "\<turnstile> projval r1 c v : der c r1")
+apply (metis Prf.intros(1))
+apply(rule v3_proj)
+apply(simp)
+by (metis Cons_eq_append_conv)
+
+lemma lex_correct1:
+  assumes "s \<notin> L r"
+  shows "lex r s = None"
+using assms
+apply(induct s arbitrary: r)
+apply(simp)
+apply (metis nullable_correctness)
+apply(auto)
+apply(drule_tac x="der a r" in meta_spec)
+apply(drule meta_mp)
+apply(auto)
+apply(simp add: L_flat_Prf)
+by (metis v3 v4)
+
+
+lemma lex_correct2:
+  assumes "s \<in> L r"
+  shows "\<exists>v. lex r s = Some(v) \<and> \<turnstile> v : r \<and> flat v = s"
+using assms
+apply(induct s arbitrary: r)
+apply(simp)
+apply (metis mkeps_flat mkeps_nullable nullable_correctness)
+apply(drule_tac x="der a r" in meta_spec)
+apply(drule meta_mp)
+apply(simp add: L_flat_Prf)
+apply(auto)
+apply (metis v3_proj v4_proj2)
+apply (metis v3)
+apply(rule v4)
+by metis
+
+lemma lex_correct3:
+  assumes "s \<in> L r"
+  shows "\<exists>v. lex r s = Some(v) \<and> s \<in> r \<rightarrow> v"
+using assms
+apply(induct s arbitrary: r)
+apply(simp)
+apply (metis PMatch_mkeps nullable_correctness)
+apply(drule_tac x="der a r" in meta_spec)
+apply(drule meta_mp)
+apply(simp add: L_flat_Prf)
+apply(auto)
+apply (metis v3_proj v4_proj2)
+apply(rule PMatch2)
+apply(simp)
+done
+
+lemma lex_correct4:
+  assumes "s \<in> L r"
+  shows "s \<in> r \<rightarrow> (lex2 r s)"
+using assms
+apply(induct s arbitrary: r)
+apply(simp)
+apply (metis PMatch_mkeps nullable_correctness)
+apply(simp)
+apply(rule PMatch2)
+apply(drule_tac x="der a r" in meta_spec)
+apply(drule meta_mp)
+apply(simp add: L_flat_Prf)
+apply(auto)
+apply (metis v3_proj v4_proj2)
+done
+
+lemma 
+  "lex2 (ALT (CHAR a) (ALT (CHAR b) (SEQ (CHAR a) (CHAR b)))) [a,b] = Right (Right (Seq (Char a) (Char b)))"
+apply(simp)
+done
+
+
+section {* Sulzmann's Ordering of values *}
+
+
+inductive ValOrd :: "val \<Rightarrow> rexp \<Rightarrow> val \<Rightarrow> bool" ("_ \<succ>_ _" [100, 100, 100] 100)
+where
+  "v2 \<succ>r2 v2' \<Longrightarrow> (Seq v1 v2) \<succ>(SEQ r1 r2) (Seq v1 v2')" 
+| "\<lbrakk>v1 \<succ>r1 v1'; v1 \<noteq> v1'\<rbrakk> \<Longrightarrow> (Seq v1 v2) \<succ>(SEQ r1 r2) (Seq v1' v2')" 
+| "length (flat v1) \<ge> length (flat v2) \<Longrightarrow> (Left v1) \<succ>(ALT r1 r2) (Right v2)"
+| "length (flat v2) > length (flat v1) \<Longrightarrow> (Right v2) \<succ>(ALT r1 r2) (Left v1)"
+| "v2 \<succ>r2 v2' \<Longrightarrow> (Right v2) \<succ>(ALT r1 r2) (Right v2')"
+| "v1 \<succ>r1 v1' \<Longrightarrow> (Left v1) \<succ>(ALT r1 r2) (Left v1')"
+| "Void \<succ>EMPTY Void"
+| "(Char c) \<succ>(CHAR c) (Char c)"
+
+inductive ValOrd2 :: "val \<Rightarrow> val \<Rightarrow> bool" ("_ 2\<succ> _" [100, 100] 100)
+where
+  "v2 2\<succ> v2' \<Longrightarrow> (Seq v1 v2) 2\<succ> (Seq v1 v2')" 
+| "\<lbrakk>v1 2\<succ> v1'; v1 \<noteq> v1'\<rbrakk> \<Longrightarrow> (Seq v1 v2) 2\<succ> (Seq v1' v2')" 
+| "length (flat v1) \<ge> length (flat v2) \<Longrightarrow> (Left v1) 2\<succ> (Right v2)"
+| "length (flat v2) > length (flat v1) \<Longrightarrow> (Right v2) 2\<succ> (Left v1)"
+| "v2 2\<succ> v2' \<Longrightarrow> (Right v2) 2\<succ> (Right v2')"
+| "v1 2\<succ> v1' \<Longrightarrow> (Left v1) 2\<succ> (Left v1')"
+| "Void 2\<succ> Void"
+| "(Char c) 2\<succ> (Char c)"
+
+lemma Ord1:
+  "v1 \<succ>r v2 \<Longrightarrow> v1 2\<succ> v2"
+apply(induct rule: ValOrd.induct)
+apply(auto intro: ValOrd2.intros)
+done
+
+lemma Ord2:
+  "v1 2\<succ> v2 \<Longrightarrow> \<exists>r. v1 \<succ>r v2"
+apply(induct v1 v2 rule: ValOrd2.induct)
+apply(auto intro: ValOrd.intros)
+done
+
+lemma Ord3:
+  "\<lbrakk>v1 2\<succ> v2; \<turnstile> v1 : r\<rbrakk> \<Longrightarrow> v1 \<succ>r v2"
+apply(induct v1 v2 arbitrary: r rule: ValOrd2.induct)
+apply(auto intro: ValOrd.intros elim: Prf.cases)
+done
+
+section {* Posix definition *}
+
+definition POSIX :: "val \<Rightarrow> rexp \<Rightarrow> bool" 
+where
+  "POSIX v r \<equiv> (\<turnstile> v : r \<and> (\<forall>v'. (\<turnstile> v' : r \<and> flat v' \<sqsubseteq> flat v) \<longrightarrow> v \<succ>r v'))"
+
+lemma ValOrd_refl:
+  assumes "\<turnstile> v : r"
+  shows "v \<succ>r v"
+using assms
+apply(induct)
+apply(auto intro: ValOrd.intros)
+done
+
+lemma ValOrd_total:
+  shows "\<lbrakk>\<turnstile> v1 : r; \<turnstile> v2 : r\<rbrakk>  \<Longrightarrow> v1 \<succ>r v2 \<or> v2 \<succ>r v1"
+apply(induct r arbitrary: v1 v2)
+apply(auto)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply (metis ValOrd.intros(7))
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply (metis ValOrd.intros(8))
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(clarify)
+apply(case_tac "v1a = v1b")
+apply(simp)
+apply(rule ValOrd.intros(1))
+apply (metis ValOrd.intros(1))
+apply(rule ValOrd.intros(2))
+apply(auto)[2]
+apply(erule contrapos_np)
+apply(rule ValOrd.intros(2))
+apply(auto)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply (metis Ord1 Ord3 Prf.intros(2) ValOrd2.intros(6))
+apply(rule ValOrd.intros)
+apply(erule contrapos_np)
+apply(rule ValOrd.intros)
+apply (metis le_eq_less_or_eq neq_iff)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(rule ValOrd.intros)
+apply(erule contrapos_np)
+apply(rule ValOrd.intros)
+apply (metis le_eq_less_or_eq neq_iff)
+apply(rule ValOrd.intros)
+apply(erule contrapos_np)
+apply(rule ValOrd.intros)
+by metis
+
+lemma ValOrd_anti:
+  shows "\<lbrakk>\<turnstile> v1 : r; \<turnstile> v2 : r; v1 \<succ>r v2; v2 \<succ>r v1\<rbrakk> \<Longrightarrow> v1 = v2"
+apply(induct r arbitrary: v1 v2)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(erule ValOrd.cases)
+apply(simp_all)[8]
+apply(erule ValOrd.cases)
+apply(simp_all)[8]
+apply(erule ValOrd.cases)
+apply(simp_all)[8]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(erule ValOrd.cases)
+apply(simp_all)[8]
+apply(erule ValOrd.cases)
+apply(simp_all)[8]
+apply(erule ValOrd.cases)
+apply(simp_all)[8]
+apply(erule ValOrd.cases)
+apply(simp_all)[8]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(erule ValOrd.cases)
+apply(simp_all)[8]
+apply(erule ValOrd.cases)
+apply(simp_all)[8]
+apply(erule ValOrd.cases)
+apply(simp_all)[8]
+apply(erule ValOrd.cases)
+apply(simp_all)[8]
+done
+
+lemma POSIX_ALT_I1:
+  assumes "POSIX v1 r1" 
+  shows "POSIX (Left v1) (ALT r1 r2)"
+using assms
+unfolding POSIX_def
+apply(auto)
+apply (metis Prf.intros(2))
+apply(rotate_tac 2)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(auto)
+apply(rule ValOrd.intros)
+apply(auto)
+apply(rule ValOrd.intros)
+by (metis le_eq_less_or_eq length_sprefix sprefix_def)
+
+lemma POSIX_ALT_I2:
+  assumes "POSIX v2 r2" "\<forall>v'. \<turnstile> v' : r1 \<longrightarrow> length (flat v2) > length (flat v')"
+  shows "POSIX (Right v2) (ALT r1 r2)"
+using assms
+unfolding POSIX_def
+apply(auto)
+apply (metis Prf.intros)
+apply(rotate_tac 3)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(auto)
+apply(rule ValOrd.intros)
+apply metis
+apply(rule ValOrd.intros)
+apply metis
+done
+
+section {* tryout with all-mkeps *}
+
+fun 
+  alleps :: "rexp \<Rightarrow> val set"
+where
+  "alleps(NULL) = {}"
+| "alleps(EMPTY) = {Void}"
+| "alleps(CHAR c) = {}"
+| "alleps(SEQ r1 r2) = {Seq v1 v2 | v1 v2. v1 \<in> alleps r1 \<and> v2 \<in> alleps r2}"
+| "alleps(ALT r1 r2) = {Left v1 | v1. v1 \<in> alleps r1} \<union> {Right v2 | v2. v2 \<in> alleps r2}"
+
+fun injall :: "rexp \<Rightarrow> char \<Rightarrow> val \<Rightarrow> val set"
+where
+  "injall (EMPTY) c Void = {}"
+| "injall (CHAR d) c Void = (if c = d then {Char d} else {})"
+| "injall (ALT r1 r2) c (Left v1) = {Left v | v. v \<in> injall r1 c v1}"
+| "injall (ALT r1 r2) c (Right v2) = {Right v | v. v \<in> injall r2 c v2}"
+| "injall (SEQ r1 r2) c (Seq v1 v2) = {Seq v v2 | v. v \<in> injall r1 c v1}"
+| "injall (SEQ r1 r2) c (Left (Seq v1 v2)) = {Seq v v2 | v. v \<in> injall r1 c v1}"
+| "injall (SEQ r1 r2) c (Right v2) = {Seq v v' | v v'. v \<in> alleps r1 \<and> v' \<in> injall r2 c v2}"
+
+fun 
+  allvals :: "rexp \<Rightarrow> string \<Rightarrow> val set"
+where
+  "allvals r [] = alleps r"
+| "allvals r (c#s) = {v | v v'. v \<in> injall r c v' \<and> v' \<in> allvals (der c r) s}"
+
+lemma q1: 
+  assumes "v \<in> alleps r"
+  shows "\<turnstile> v : r \<and> flat v = []"
+using assms
+apply(induct r arbitrary: v)
+apply(auto intro: Prf.intros)
+done
+
+
+lemma allvals_NULL:
+  shows "allvals (NULL) s = {}"
+apply(induct_tac s)
+apply(simp)
+apply(simp)
+done
+
+lemma allvals_EMPTY:
+  shows "allvals (EMPTY) [] = {Void}"
+  and   "s \<noteq> [] \<Longrightarrow> allvals (EMPTY) s = {}"
+apply(simp)
+apply(case_tac s)
+apply(simp)
+apply(simp add: allvals_NULL)
+done
+
+lemma allvals_CHAR:
+  shows "allvals (CHAR c) [c] = {Char c}"
+  and   "s \<noteq> [c] \<Longrightarrow> allvals (CHAR c) s = {}"
+apply(simp)
+apply(case_tac s)
+apply(simp)
+apply(case_tac "c = a")
+apply(simp add: allvals_EMPTY)
+apply(simp add: allvals_NULL)
+done
+
+lemma allvals_ALT:
+  shows "allvals (ALT r1 r2) s = {Left v1 | v1. v1 \<in> allvals r1 s} \<union>
+                                 {Right v2 | v2. v2 \<in> allvals r2 s}"
+apply(induct s arbitrary: r1 r2)
+apply(simp)
+apply(simp)
+apply(auto)
+apply blast
+apply(rule_tac x="Left v'" in exI)
+apply(simp)
+apply(rule_tac x="Right v'" in exI)
+apply(simp)
+done
+
+lemma allvals_SEQ_Nil:
+  "allvals (SEQ r1 r2) [] = {Seq v1 v2 | v1 v2. v1 \<in> allvals r1 [] \<and> v2 \<in> allvals r2 []}"
+by simp
+
+lemma allvals_SEQ:
+  shows "allvals (SEQ r1 r2) s = {Seq v1 v2 | v1 v2 s1 s2. 
+      s = s1 @ s2 \<and> v1 \<in> allvals r1 s1 \<and> v2 \<in> allvals r2 s2}"
+using assms
+apply(induct s arbitrary: r1 r2)
+apply(simp)
+apply(simp)
+apply(auto)
+apply(simp_all add: allvals_ALT)
+apply(auto)
+apply (metis (mono_tags, lifting) allvals.simps(2) append_Cons mem_Collect_eq)
+apply fastforce
+prefer 2
+apply(rule_tac x="a#s1" in exI)
+apply(rule_tac x="s2" in exI)
+apply(simp)
+apply(fastforce)
+prefer 2
+apply(subst (asm) Cons_eq_append_conv)
+apply(auto)[1]
+using Prf_flat_L nullable_correctness q1 apply fastforce
+apply(rule_tac x="Seq v' v2" in exI)
+apply(simp)
+apply(rule_tac x="ys'" in exI)
+apply(rule_tac x="s2" in exI)
+apply(simp)
+apply(subst (asm) Cons_eq_append_conv)
+apply(auto)[1]
+apply(rule_tac x="Right v'" in exI)
+apply(simp)
+apply(rule_tac x="Left (Seq v' v2)" in exI)
+apply(simp)
+apply(auto)[1]
+done
+
+lemma q11:
+  assumes "nullable r" "\<turnstile> v : r" "flat v = []"
+  shows "v \<in> alleps r"
+using assms
+apply(induct r arbitrary: v)
+apply(auto)
+apply(erule Prf.cases)
+apply(simp_all)
+apply(erule Prf.cases)
+apply(simp_all)
+apply(erule Prf.cases)
+apply(simp_all)
+apply(auto)
+apply(subgoal_tac "nullable r2a")
+apply(auto)
+using not_nullable_flat apply auto[1]
+ apply(erule Prf.cases)
+apply(simp_all)
+apply(auto)
+apply(subgoal_tac "nullable r1a")
+apply(auto)
+using not_nullable_flat apply auto[1]
+done
+
+lemma q33:
+  assumes "nullable r"
+  shows "alleps r = {v. \<turnstile> v : r \<and> flat v = []}"
+using assms
+apply(auto) 
+apply (simp_all add: q1)
+by (simp add: q11)
+
+
+lemma k0:
+  assumes "\<turnstile> v : der a r" "v' \<in> injall r a v"
+  shows "flat v' = a # flat v"
+using assms
+apply(induct a r arbitrary: v v' rule: der.induct)
+apply(simp_all)
+apply(erule Prf.cases)
+apply(simp_all)
+apply(erule Prf.cases)
+apply(simp_all)
+apply(case_tac "c = c'")
+apply(erule Prf.cases)
+apply(simp_all)
+apply(erule Prf.cases)
+apply(simp_all)
+apply(erule Prf.cases)
+apply(simp_all)
+apply(auto)[1]
+apply(auto)[1]
+apply(case_tac "nullable r1")
+apply(auto)
+apply(erule Prf.cases)
+apply(simp_all)
+apply(auto)
+apply(erule Prf.cases)
+apply(simp_all)
+apply(auto)
+using q1 apply blast
+apply(erule Prf.cases)
+apply(simp_all)
+apply(auto)
+done
+
+lemma k:
+  assumes "\<turnstile> v' : der a r" "v \<in> injall r a v'"
+  shows "\<turnstile> v : r"
+using assms
+apply(induct a r arbitrary: v v' rule: der.induct)
+apply(simp_all)
+apply(erule Prf.cases)
+apply(simp_all)
+apply(erule Prf.cases)
+apply(simp_all)
+apply(case_tac "c = c'")
+apply(erule Prf.cases)
+apply(simp_all)
+apply(rule Prf.intros)
+apply(erule Prf.cases)
+apply(simp_all)
+apply(erule Prf.cases)
+apply(simp_all)
+apply(auto intro: Prf.intros)[1]
+apply(auto intro: Prf.intros)[1]
+apply(case_tac "nullable r1")
+apply(auto)
+apply(erule Prf.cases)
+apply(simp_all)
+apply(auto)
+apply(erule Prf.cases)
+apply(simp_all)
+apply(auto)
+apply(auto intro: Prf.intros)[1]
+using Prf.intros(1) q1 apply blast
+apply(erule Prf.cases)
+apply(simp_all)
+apply(auto)
+using Prf.intros(1) by auto
+
+
+
+lemma q22: 
+  assumes "v \<in> allvals r s"
+  shows "\<turnstile> v : r \<and> s \<in> L (r) \<and> flat v = s"
+using assms
+apply(induct s arbitrary: v r)
+apply(auto)
+apply(simp_all add: q1)
+using Prf_flat_L q1 apply fastforce
+apply(drule_tac x="v'" in meta_spec)
+apply(drule_tac x="der a r" in meta_spec)
+apply(simp)
+apply(clarify)
+apply(rule k)
+apply(assumption)
+apply(assumption)
+apply(drule_tac x="v'" in meta_spec)
+apply(drule_tac x="der a r" in meta_spec)
+apply(simp)
+apply(clarify)
+using Prf_flat_L v3 v4 apply fastforce
+apply(drule_tac x="v'" in meta_spec)
+apply(drule_tac x="der a r" in meta_spec)
+apply(simp)
+apply(clarify)
+using k0 by blast
+
+lemma ra:
+  assumes "v \<in> allvals r1 s"
+  shows "Left v \<in> allvals (ALT r1 r2) s"
+using assms
+apply(induct s arbitrary: r1 r2 v)
+apply(simp)
+apply(auto)
+apply(rule_tac x="Left v'" in exI)
+apply(simp)
+done
+
+lemma rb:
+  assumes "v \<in> allvals r2 s"
+  shows "Right v \<in> allvals (ALT r1 r2) s"
+using assms
+apply(induct s arbitrary: r1 r2 v)
+apply(simp)
+apply(auto)
+apply(rule_tac x="Right v'" in exI)
+apply(simp)
+done
+
+lemma r1:
+  assumes "v1 \<in> alleps r1" "v2 \<in> allvals r2 s2"
+  shows "Seq v1 v2 \<in> allvals (SEQ r1 r2) s2"
+using assms
+apply(induct s2 arbitrary: r1 r2 v1 v2)
+apply(simp)
+apply(simp)
+apply(auto)
+apply(rule_tac x="Right v'" in exI)
+apply(simp)
+apply(rule rb)
+apply(simp)
+using not_nullable_flat q1 by blast
+
+lemma r2:
+  assumes "v1 \<in> allvals r1 s1" "v2 \<in> allvals r2 s2"
+  shows "Seq v1 v2 \<in> allvals (SEQ r1 r2) (s1 @ s2)"
+using assms
+apply(induct s1 arbitrary: r1 r2 v1 v2 s2)
+apply(simp)
+apply(rule r1) 
+apply(simp)
+apply(simp)
+apply(simp)
+apply(auto)
+apply(drule_tac x="der a r1" in meta_spec)
+apply(drule_tac x="r2" in meta_spec)
+apply(drule_tac x="v'" in meta_spec)
+apply(drule_tac x="v2" in meta_spec)
+apply(drule_tac x="s2" in meta_spec)
+apply(simp)
+apply(rule_tac x="Left (Seq v' v2)" in exI)
+apply(simp)
+apply(rule ra)
+apply(simp)
+apply(drule_tac x="der a r1" in meta_spec)
+apply(drule_tac x="r2" in meta_spec)
+apply(drule_tac x="v'" in meta_spec)
+apply(drule_tac x="v2" in meta_spec)
+apply(drule_tac x="s2" in meta_spec)
+apply(simp)
+apply(rule_tac x="Seq v' v2" in exI)
+apply(simp)
+done
+
+lemma q22a: 
+  assumes " s \<in> L (r)"
+  shows "\<exists>v. v \<in> allvals r s \<and> flat v = s"
+using assms
+apply(induct r arbitrary: s)
+apply(auto)
+apply(simp add: Sequ_def)
+apply(auto)
+apply(drule_tac x="s1" in meta_spec) 
+apply(drule_tac x="s2" in meta_spec) 
+apply(auto)
+apply(rule_tac x="Seq v va" in exI)
+apply(simp)
+apply(rule r2)
+apply(simp)
+apply(simp)
+apply(drule_tac x="s" in meta_spec) 
+apply(simp)
+apply(auto)
+apply(rule_tac x="Left v" in exI)
+apply(simp)
+apply(rule ra)
+apply(simp)
+apply(drule_tac x="s" in meta_spec) 
+apply(drule_tac x="s" in meta_spec) 
+apply(simp)
+apply(auto)
+apply(rule_tac x="Right v" in exI)
+apply(simp)
+apply(rule rb)
+apply(simp)
+done
+
+lemma q22b: 
+  assumes " s \<in> L (r)" "\<turnstile> v : r" "flat v = s"
+  shows "v \<in> allvals r s"
+using assms
+apply(induct r arbitrary: s v)
+apply(auto)
+apply(erule Prf.cases)
+apply(simp_all)
+apply(erule Prf.cases)
+apply(simp_all)
+apply(simp add: Sequ_def)
+apply(auto)
+apply(erule Prf.cases)
+apply(simp_all)
+apply(simp add: append_eq_append_conv2)
+apply(auto)
+apply (metis Prf_flat_L append_assoc r2)
+apply (metis Prf_flat_L r2)
+apply(erule Prf.cases)
+apply(simp_all)
+apply(auto intro: ra rb)[2]
+apply(rule rb)
+apply (simp add: Prf_flat_L)
+apply(erule Prf.cases)
+apply(simp_all)
+apply(auto intro: ra rb)[2]
+apply(rule ra)
+by (simp add: Prf_flat_L)
+
+
+lemma q2:
+  assumes "s \<in> L(r)" 
+  shows "allvals r s = {v. \<turnstile> v : r \<and> s \<in> L (r) \<and> flat v = s}"
+using assms
+apply(auto)
+apply (simp add: q22)
+apply (simp add: q22)
+by (simp add: q22b)
+
+lemma r3a:
+  assumes "v' \<in> allvals (SEQ r1 r2) (s1 @ s2)" 
+          "(s1 @ s2) \<in> L (SEQ r1 r2)"
+  shows "\<exists>v1 v2. v' = Seq v1 v2 \<and> v1 \<in> allvals r1 s1 \<and> v2 \<in> allvals r2 s2" 
+using assms
+apply(subst (asm) q2)
+apply(auto)
+apply(erule_tac Prf.cases)
+apply(simp_all)
+apply(rule conjI)
+apply(simp add: append_eq_append_conv2)
+apply(auto)
+apply(subst q2)
+oops
+
+lemma r3:
+  assumes "Seq v1 v2 \<in> allvals (SEQ r1 r2) (s1 @ s2)" 
+          "flat v1 = s1" "flat v2 = s2"
+          "(s1 @ s2) \<in> L (SEQ r1 r2)"
+  shows "v1 \<in> allvals r1 s1"  "v2 \<in> allvals r2 s2" 
+using assms
+apply(subst (asm) q2)
+apply(auto)
+apply(erule_tac Prf.cases)
+apply(simp_all)
+apply(subst q2)
+apply(auto)
+using Prf_flat_L apply blast
+using Prf_flat_L apply blast
+using assms
+apply(subst (asm) q2)
+apply(auto)
+apply(erule_tac Prf.cases)
+apply(simp_all)
+apply(subst q2)
+apply(auto)
+using Prf_flat_L apply blast
+using Prf_flat_L apply blast
+done
+
+
+definition POSIX2 :: "val \<Rightarrow> rexp \<Rightarrow> string \<Rightarrow> bool" 
+where
+  "POSIX2 v r s \<equiv> (\<turnstile> v : r \<and> flat v = s \<and> (\<forall>v'\<in>allvals r s. v \<succ>r v'))"
+
+
+
+
+lemma k1:
+  assumes "nullable r"
+  shows "POSIX2 v r [] \<Longrightarrow> \<forall>v' \<in> alleps r. v \<succ>r v'"
+using assms
+apply(induct r arbitrary: v)
+apply(simp_all)
+apply(simp add: POSIX2_def)
+apply(auto)
+apply(simp add: POSIX2_def)
+apply(simp add: POSIX2_def)
+apply(simp add: POSIX2_def)
+apply(simp add: POSIX2_def)
+apply(simp add: POSIX2_def)
+done
+
+lemma k2:
+  assumes "s \<in> L r"
+  shows "POSIX2 v r s \<Longrightarrow> \<forall>v' \<in> allvals r s. v \<succ>r v'"
+using assms
+apply(induct s arbitrary: r v)
+apply(simp)
+apply(rule k1)
+apply (simp add: nullable_correctness)
+apply(simp)
+apply(simp)
+apply(auto)
+apply(simp only: POSIX2_def)
+apply(simp)
+apply(clarify)
+apply(drule_tac x="x" in spec)
+apply(drule mp)
+apply(auto)
+done
+
+lemma pos:
+  assumes "s \<in> r \<rightarrow> v" 
+  shows "v \<in> allvals r s"
+using assms
+apply(subst q2)
+using PMatch1(1) PMatch1(2) Prf_flat_L apply blast
+apply(simp)
+using PMatch1(1) PMatch1(2) Prf_flat_L by blast
+
+lemma mkeps_val:
+  assumes "nullable r"
+  shows "mkeps r \<in> alleps r"
+using assms
+apply(induct r)
+apply(auto)
+done
+
+lemma injval_injall:
+  assumes "\<turnstile> v : der a r"
+  shows "injval r a v \<in> injall r a v"
+using assms
+apply(induct a r arbitrary: v rule: der.induct)
+apply(simp)
+apply(erule Prf.cases)
+apply(simp_all)
+apply(erule Prf.cases)
+apply(simp_all)
+apply(case_tac "c = c'")
+apply(simp)
+apply(erule Prf.cases)
+apply(simp_all)
+apply(erule Prf.cases)
+apply(simp_all)
+apply(erule Prf.cases)
+apply(simp_all)
+apply(case_tac "nullable r1")
+apply(simp)
+apply(erule Prf.cases)
+apply(simp_all)
+apply(auto)
+apply(erule Prf.cases)
+apply(simp_all)
+using mkeps_val apply blast
+apply(erule Prf.cases)
+apply(simp_all)
+done
+
+
+lemma mkeps_val1:
+  assumes "nullable r" "v \<succ>r mkeps r" "flat v = []" "\<turnstile> v : r"
+  shows "v = mkeps r"
+using assms
+apply(induct r arbitrary: v)
+apply(auto)
+apply(erule ValOrd.cases)
+apply(auto)
+apply(erule ValOrd.cases)
+apply(auto)
+apply(erule Prf.cases)
+apply(auto)
+apply(erule Prf.cases)
+apply(auto)
+apply(erule Prf.cases)
+apply(auto)
+apply(erule ValOrd.cases)
+apply(auto)
+apply(erule ValOrd.cases)
+apply(auto)
+apply(erule Prf.cases)
+apply(auto)
+apply(erule ValOrd.cases)
+apply(auto)
+apply(erule ValOrd.cases)
+apply(auto)
+apply(erule Prf.cases)
+apply(auto)
+apply(erule ValOrd.cases)
+apply(auto)
+apply (simp add: not_nullable_flat)
+apply(erule ValOrd.cases)
+apply(auto)
+done
+
+lemma sulzmann_our:
+  assumes "v \<in> alleps r" "nullable r"
+  shows "mkeps r \<succ>r v"
+using assms
+apply(induct r arbitrary: v)
+apply(simp_all)
+apply(rule ValOrd.intros)
+apply(auto)[1]
+apply(case_tac "mkeps r1 = v1")
+apply(simp)
+apply(rule ValOrd.intros)
+apply(blast)
+apply(rule ValOrd.intros)
+apply(blast)
+apply(simp)
+apply(auto)
+apply(rule ValOrd.intros)
+apply(blast)
+apply(rule ValOrd.intros)
+apply(blast)
+apply(rule ValOrd.intros)
+using not_nullable_flat q1 apply blast
+apply(rule ValOrd.intros)
+using q1 apply auto[1]
+apply(rule ValOrd.intros)
+apply (simp add: q1)
+apply(rule ValOrd.intros)
+apply(blast)
+done
+
+lemma destruct:
+  assumes "\<forall>s3. s1 @ s3 \<in> L r1 \<longrightarrow> s3 = [] \<or> (\<forall>s4. s3 @ s4 = s2 \<longrightarrow> s4 \<notin> L r2)"
+  and "s1 \<in> L r1" "s2 \<in> L r2" "(s1' @ s2') \<sqsubseteq> (s1 @ s2)"
+  and "s1'@ s2' \<in> L (SEQ r1 r2)"  "s1' \<in> L r1"
+  shows "s1' \<sqsubseteq> s1"
+using assms
+apply(simp add: prefix_def)
+apply(auto)
+apply(simp add: append_eq_append_conv2)
+apply(auto)
+apply(simp add: Sequ_def)
+apply(auto)
+apply(drule_tac x="us" in spec)
+apply(simp)
+apply(auto)
+apply(simp add: append_eq_append_conv2)
+apply(auto)
+oops
+
+lemma inj_ord:
+  assumes "v1 \<succ>(der a r) v2" "s \<in> (der a r) \<rightarrow> v1" "s' \<in> L (der a r)" 
+          "v1 \<in> allvals (der a r) s" "v2 \<in> allvals (der a r) s'" "s' \<sqsubseteq> s" 
+  shows "injval r a v1 \<succ>r injval r a v2"
+using assms
+apply(induct a r arbitrary: s s' v1 v2 rule: der.induct)
+apply(simp_all)
+(*apply(simp add: allvals_NULL)
+apply(simp add: allvals_NULL)*)
+apply(case_tac "c = c'")
+apply(simp)
+apply(case_tac "s = []")
+apply(subgoal_tac "s' = []")
+prefer 2
+using allvals_EMPTY(2) apply blast
+apply(simp add: allvals_EMPTY)
+apply(rule ValOrd.intros)
+apply(simp add: allvals_EMPTY)
+apply(simp)
+(*apply(simp add: allvals_NULL)*)
+(* ALT case *)
+apply(simp add: allvals_ALT)
+apply(auto)[1]
+apply(erule ValOrd.cases)
+apply(simp_all)
+apply(rule ValOrd.intros)
+apply(erule PMatch.cases)
+apply(simp_all)
+apply(erule ValOrd.cases)
+apply(simp_all)
+apply(rule ValOrd.intros)
+apply(subst v4)
+apply(simp)
+apply (simp add: q22)
+apply(subst v4)
+apply(simp)
+apply (simp add: q22)
+apply(simp)
+apply(erule ValOrd.cases)
+apply(simp_all)
+apply(rule ValOrd.intros)
+apply(subst v4)
+apply (simp add: q22)
+apply(subst v4)
+apply (simp add: q22)
+apply(simp)
+apply(erule ValOrd.cases)
+apply(simp_all)
+apply(rule ValOrd.intros)
+apply(erule PMatch.cases)
+apply(simp_all)
+using q22 apply auto[1]
+apply(erule PMatch.cases)
+apply(simp_all)
+apply(erule ValOrd.cases)
+apply(simp_all)
+apply(rule ValOrd.intros)
+using q22 apply auto[1]
+apply(erule PMatch.cases)
+apply(simp_all)
+apply(erule ValOrd.cases)
+apply(simp_all)
+apply(rule ValOrd.intros)
+apply(subst v4)
+apply (simp add: q22)
+apply(subst v4)
+apply (simp add: q22)
+apply(simp)
+apply(erule PMatch.cases)
+apply(simp_all)
+apply(erule ValOrd.cases)
+apply(simp_all)
+apply(rule ValOrd.intros)
+apply(subst v4)
+apply (simp add: q22)
+apply(subst v4)
+apply (simp add: q22)
+apply(simp)
+using q22 apply auto[1]
+apply(erule PMatch.cases)
+apply(simp_all)
+apply(erule ValOrd.cases)
+apply(simp_all)
+apply(rule ValOrd.intros)
+using q22 apply auto[1]
+(* seq case *)
+apply(case_tac "nullable r1")
+apply(simp)
+prefer 2
+apply(simp)
+apply(simp add: allvals_SEQ)
+apply(auto)[1]
+apply(erule ValOrd.cases)
+apply(simp_all)
+apply(clarify)
+apply(rule ValOrd.intros)
+apply(simp)
+apply(rule ValOrd.intros)
+apply(erule PMatch.cases)
+apply(simp_all)
+apply(clarify)
+apply(rotate_tac 1)
+apply(drule_tac x="s1b" in meta_spec)
+apply(rotate_tac 13)
+apply(drule_tac x="s1a" in meta_spec)
+apply(drule_tac x="v1c" in meta_spec)
+apply(drule_tac x="v1'" in meta_spec)
+apply(simp)
+apply(subgoal_tac "s1 = s1b")
+prefer 2
+apply (metis PMatch1(2) q22)
+apply(simp)
+apply(clarify)
+apply(drule destruct)
+apply (metis PMatch1(2) q22)
+apply (metis PMatch1(2) q22)
+apply(assumption)
+apply (metis PMatch1(2) q22)
+apply (metis PMatch1(2) q22)
+apply(subgoal_tac "s2a = s2b")
+prefer 2
+apply (metis PMatch1(2) q22)
+apply(drule destruct)
+apply (metis PMatch1(2) q22)
+apply (metis PMatch1(2) q22)
+apply(assumption)
+back
+apply (metis PMatch1(2) q22)
+apply (metis PMatch1(2) q22)
+
+
+
+apply(simp add: allvals_ALT)
+apply(auto)
+apply(erule ValOrd.cases)
+apply(simp_all)
+apply(clarify)
+apply(erule ValOrd.cases)
+apply(simp_all)
+apply(clarify)
+apply(rule ValOrd.intros)
+apply(blast)
+apply(rule ValOrd.intros)
+apply(clarify)
+apply(simp add: allvals_SEQ)
+apply(auto)[1]
+apply(erule PMatch.cases)
+apply(simp_all)
+apply(clarify)
+apply(erule PMatch.cases)
+apply(simp_all)
+apply(auto)
+apply(drule_tac x="s1b" in meta_spec)
+apply(drule_tac x="v1" in meta_spec)
+apply(drule_tac x="v1'a" in meta_spec)
+apply(simp)
+apply(drule_tac meta_mp)
+apply(subgoal_tac "s1 = s1b")
+apply(simp)
+apply (metis PMatch1(2) q22)
+apply(drule_tac meta_mp)
+apply(subgoal_tac "s1a = s1b")
+apply(simp)
+apply(simp add: append_eq_append_conv2)
+apply(auto)[1]
+apply(subgoal_tac "s2 = s2a")
+apply(simp)
+apply(clarify)
+prefer 2
+using q22 apply blast
+using q22 apply blast
+using q22 apply blast
+apply(subgoal_tac "usa = []")
+apply(simp)
+prefer 2
+using q22 apply blast
+prefer 3
+apply(simp)
+prefer 4
+apply(erule PMatch.cases)
+apply(simp_all)
+apply(clarify)
+apply(erule PMatch.cases)
+apply(simp_all)
+apply(auto)
+apply(erule ValOrd.cases)
+apply(simp_all)
+apply(clarify)
+apply(simp)
+prefer 5
+apply(erule PMatch.cases)
+apply(simp_all)
+apply(clarify)
+apply(erule ValOrd.cases)
+apply(simp_all)
+apply(clarify)
+apply(simp add: allvals_SEQ)
+apply(auto)[1]
+apply (simp add: q22)
+apply(simp add: allvals_SEQ)
+apply(auto)[1]
+apply(simp add: append_eq_append_conv2)
+apply(auto)[1]
+apply (simp add: q22)
+thm PMatch2
+apply(drule PMatch2)
+
+
+lemma sulzmann_our:
+  assumes "\<forall>v' \<in> allvals r s. v \<succ>r v'" "s \<in> L r" "\<turnstile> v : r" "flat v = s"
+  shows "s \<in> r \<rightarrow> v"
+using assms
+apply(induct s arbitrary: r v)
+apply(simp_all)
+apply(subst (asm) q33)
+apply (simp add: nullable_correctness)
+apply(simp)
+apply(drule_tac x="mkeps r" in spec)
+apply(drule mp)
+apply(rule conjI)
+using mkeps_val not_nullable_flat q1 apply blast
+using mkeps_flat not_nullable_flat apply blast
+apply(subgoal_tac "nullable r")
+apply(drule mkeps_val1)
+apply(assumption)
+apply(simp)
+apply(simp)
+apply(simp)
+using PMatch_mkeps not_nullable_flat apply blast
+using not_nullable_flat apply blast
+apply(drule_tac x="der a r" in meta_spec)
+apply(drule_tac x="projval r a v" in meta_spec)
+apply(drule meta_mp)
+defer
+apply(drule meta_mp)
+using Prf_flat_L v3_proj v4_proj2 apply blast
+apply(drule meta_mp)
+using v3_proj apply blast
+apply(drule meta_mp)
+apply (simp add: v4_proj2)
+defer
+apply(rule ballI)
+apply(drule_tac x="injval r a x" in spec)
+apply(auto)
+apply(drule_tac x="x" in spec)
+apply(drule mp)
+apply(rule injval_injall)
+using q22 apply blast
+apply(simp)
+(* HERE *)
+
+
+lemma our_sulzmann:
+  assumes "s \<in> r \<rightarrow> v" "v' \<in> allvals r s"
+  shows "v \<succ>r v'"
+using assms
+apply(induct r arbitrary: s v v')
+apply(erule PMatch.cases)
+apply(simp_all)[5]
+apply(erule PMatch.cases)
+apply(simp_all)[5]
+apply(rule ValOrd.intros)
+apply(erule PMatch.cases)
+apply(simp_all)[5]
+apply(rule ValOrd.intros)
+(* SEQ - case *)
+apply(erule PMatch.cases)
+apply(simp_all)[5]
+apply(clarify)
+apply(subst (asm) (3) q2)
+apply(simp add: Sequ_def)
+apply(rule_tac x="s1" in exI)
+apply(rule_tac x="s2" in exI)
+apply(simp)
+apply(rule conjI)
+using PMatch1(1) PMatch1(2) Prf_flat_L apply fastforce
+apply (metis PMatch1(1) PMatch1(2) Prf_flat_L)
+apply(simp)
+apply(clarify)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(case_tac "v1 = v1a")
+apply(simp)
+apply(rule ValOrd.intros)
+apply(rotate_tac 1)
+apply(drule_tac x="s2" in meta_spec)
+apply(drule_tac x="v2" in meta_spec)
+apply(drule_tac x="v2a" in meta_spec)
+apply(simp)
+apply(drule_tac meta_mp)
+apply(subst q2)
+using PMatch1(1) PMatch1(2) Prf_flat_L apply fastforce
+apply(simp)
+apply(rule conjI)
+using PMatch1(1) PMatch1(2) Prf_flat_L apply fastforce
+apply (simp add: PMatch1(2))
+apply(simp)
+apply(rule ValOrd.intros)
+prefer 2
+apply(simp)
+apply(drule_tac x="s1" in meta_spec)
+apply(drule_tac x="v1" in meta_spec)
+apply(drule_tac x="v1a" in meta_spec)
+apply(simp)
+apply(drule_tac meta_mp)
+apply(subst q2)
+apply (metis PMatch1(1) PMatch1(2) Prf_flat_L)
+apply(simp)
+apply(rule conjI)
+apply (metis PMatch1(1) PMatch1(2) Prf_flat_L)
+apply(subst (asm) append_eq_append_conv2)
+apply(auto)[1]
+using Prf_flat_L apply fastforce
+
+apply(drule_tac x="us" in spec)
+apply(auto)[1]
+
+using Prf_flat_L apply fastforce
+using Prf_flat_L apply blast
+apply(drule_tac meta_mp)
+apply(subst q2)
+using Prf_flat_L apply fastforce
+apply(simp)
+using Prf_flat_L apply fastforce
+apply(simp)
+apply(drule_tac x="flat v1a" in meta_spec)
+apply(drule_tac x="v1" in meta_spec)
+apply(drule_tac x="v1a" in meta_spec)
+apply(drule_tac meta_mp)
+apply(simp)
+apply(drule meta_mp)
+apply(subst q2)
+apply (metis PMatch1(1) PMatch1(2) Prf_flat_L)
+apply(simp)
+apply(rule conjI)
+apply (metis PMatch1(1) PMatch1(2) Prf_flat_L)
+apply(drule_tac x="[]" in spec)
+apply(auto)[1]
+
+using Prf_flat_L apply fast
+apply(drule_tac x="us" in spec)
+apply(simp)
+
+apply (simp add: Prf_flat_L)
+apply(simp)
+thm PMatch1
+qed
+done
+using PMatch1(1) PMatch1(2) Prf_flat_L apply fastforce
+apply(simp)
+apply(clarify)
+apply(erule Prf.cases)
+apply(simp_all)
+apply(rule ValOrd.intros)
+apply(drule_tac x="v1" in meta_spec)
+apply(drule meta_mp)
+apply(subst q2)
+apply (simp add: Prf_flat_L)
+apply(simp)
+apply (simp add: Prf_flat_L)
+apply(simp)
+apply(rule ValOrd.intros)
+apply(auto)[1]
+apply (simp add: PMatch1(2))
+apply (simp add: PMatch1(2))
+apply(subst (asm) (2) q2)
+using PMatch1(1) PMatch1(2) Prf_flat_L apply fastforce
+apply(simp)
+apply(clarify)
+apply(erule Prf.cases)
+apply(simp_all)
+prefer 2
+apply(rule ValOrd.intros)
+using q22b apply blast
+apply(rule ValOrd.intros)
+apply(auto)
+using Prf_flat_L apply blast
+apply(subst (asm) (3) q2)
+apply(simp add: Sequ_def)
+apply(rule_tac x="s1" in exI)
+apply(rule_tac x="s2" in exI)
+apply(simp)
+using PMatch1(1) PMatch1(2) Prf_flat_L apply fastforce
+apply(simp)
+apply(clarify)
+apply(erule Prf.cases)
+apply(simp_all)
+apply(auto simp add: Sequ_def)[1]
+apply(case_tac "v1 = v1a")
+apply(simp)
+apply(rule ValOrd.intros)
+apply(rotate_tac 3)
+apply(drule_tac x="v2a" in meta_spec)
+apply(drule_tac meta_mp)
+apply(subst q2)
+using PMatch1(1) PMatch1(2) Prf_flat_L apply fastforce
+apply(simp)
+apply(rule conjI)
+using PMatch1(1) PMatch1(2) Prf_flat_L apply fastforce
+apply (metis PMatch1(2) same_append_eq)
+apply(simp)
+apply(rule ValOrd.intros)
+apply(drule_tac x="v1a" in meta_spec)
+apply(drule_tac meta_mp)
+apply(subst q2)
+using PMatch1(1) PMatch1(2) Prf_flat_L apply fastforce
+apply(simp)
+apply(rule conjI)
+using PMatch1(1) PMatch1(2) Prf_flat_L apply fastforce
+prefer 2
+apply(simp)
+prefer 2
+apply(simp)
+apply(rotate_tac 7)
+apply(drule sym)
+apply(simp)
+apply(subst (asm) (2) append_eq_append_conv2)
+apply(auto)[1]
+apply(frule_tac x="us" in spec)
+apply(auto)[1]
+prefer 2
+apply(drule_tac x="flat v2a" in spec)
+apply(auto)[1]
+
+apply(subgoal_tac "flat v2a = s2")
+apply(simp)
+
+apply(simp add: append_eq_append_conv2)
+apply(auto)
+prefer 2
+apply blast
+prefer 2
+apply (metis Prf_flat_L append_self_conv2)
+prefer 4
+
+
+
+lemma our_sulzmann:
+  assumes "s \<in> r \<rightarrow> v"
+  shows "POSIX2 v r s"
+using assms
+apply(induct s r v)
+apply(auto)
+apply(simp add: POSIX2_def)
+using Prf.intros(4) ValOrd.intros(7) apply blast
+apply(simp add: POSIX2_def)
+apply (simp add: Prf.intros(5) ValOrd.intros(8))
+apply(simp add: POSIX2_def)
+apply(auto)
+apply(rule Prf.intros)
+apply(simp)
+apply(subgoal_tac "(\<exists>x1. x = Left x1) \<or> (\<exists>x1. x = Right x1)")
+apply(auto)[1]
+apply(rule ValOrd.intros)
+apply(drule_tac x="x1" in bspec)
+apply(subst (asm) q2)
+apply (simp add: Prf_flat_L)
+apply(simp)
+apply(subst q2)
+apply (simp add: Prf_flat_L)
+apply(auto)[1]
+apply(rotate_tac 2)
+apply(erule Prf.cases)
+apply(simp_all)
+apply(rotate_tac 2)
+apply(erule Prf.cases)
+apply(simp_all)
+apply (simp add: Prf_flat_L)
+apply(rule ValOrd.intros)
+apply(subst (asm) (2) q2)
+apply (simp add: Prf_flat_L)
+apply(auto)[1]
+defer
+apply(simp add: POSIX2_def)
+apply(auto)[1]
+apply(rule Prf.intros)
+apply (simp add: Prf_flat_L)
+apply(subgoal_tac "(\<exists>x1. x = Left x1) \<or> (\<exists>x1. x = Right x1)")
+apply(auto)[1]
+apply(rule ValOrd.intros)
+apply(subst (asm) (2) q2)
+apply (simp add: Prf_flat_L)
+apply(auto)[1]
+apply(rotate_tac 4)
+apply(erule Prf.cases)
+apply(simp_all)
+apply(auto)[1]
+using Prf_flat_L apply force
+apply(rule ValOrd.intros)
+apply(drule_tac x="x1" in bspec)
+apply(subst (asm) q2)
+apply (simp add: Prf_flat_L)
+apply(auto)[1]
+apply(subst q2)
+apply(rotate_tac 3)
+apply(erule Prf.cases)
+apply(simp_all)
+apply(rotate_tac 3)
+apply(erule Prf.cases)
+apply(simp_all)
+defer
+apply(auto)[1]
+apply(simp add: POSIX2_def)
+apply(auto intro: Prf.intros)[1]
+apply(subgoal_tac "(\<exists>x1 x2. x = Seq x1 x2 \<and> flat v1 @ flat v2 = flat x1 @ flat x2)")
+apply(auto)[1]
+apply(case_tac "v1 = x1")
+apply(simp)
+apply(rule ValOrd.intros)
+apply(rotate_tac 6)
+apply(drule_tac x="x2" in bspec)
+apply(subst (asm) q2)
+apply (simp add: Sequ_def Prf_flat_L)
+
+using Prf_flat_L apply blast
+apply(auto)[1]
+apply(rotate_tac 6)
+apply(erule Prf.cases)
+apply(simp_all)
+apply(subst q2)
+apply (simp add: Prf_flat_L)
+apply(auto)[1]
+apply(auto simp add: Sequ_def)
+using Prf_flat_L apply blast
+apply(rule ValOrd.intros)
+apply(rotate_tac 5)
+apply(drule_tac x="x1" in bspec)
+apply(rotate_tac 1)
+apply(subst (asm) q2)
+apply (simp add: Sequ_def Prf_flat_L)
+using Prf_flat_L apply blast
+apply(auto)[1]
+apply(subst q2)
+apply (simp add: Sequ_def Prf_flat_L)
+apply(auto)[1]
+apply(rotate_tac 7)
+apply(erule Prf.cases)
+apply(simp_all)
+apply (simp add: Sequ_def Prf_flat_L)
+apply(rotate_tac 7)
+apply(erule Prf.cases)
+apply(simp_all)
+apply(auto)[1]
+apply(simp add: append_eq_append_conv2)
+apply(auto simp add: Sequ_def)[1]
+using Prf_flat_L apply fastforce
+apply(simp add: append_eq_append_conv2)
+apply(auto simp add: Sequ_def)[1]
+
+apply(auto)[1]
+
+apply(simp add: POSIX2_def)
+apply(simp add: POSIX2_def)
+apply(simp add: POSIX2_def)
+apply(simp add: POSIX2_def)
+apply(simp add: POSIX2_def)
+
+lemma "s \<in> L(r) \<Longrightarrow> \<exists>v. POSIX2 v r s"
+apply(induct r arbitrary: s)
+apply(auto)
+apply(rule_tac x="Void" in exI)
+apply(simp add: POSIX2_def)
+apply (simp add: Prf.intros(4) ValOrd.intros(7))
+apply(rule_tac x="Char x" in exI)
+apply(simp add: POSIX2_def)
+apply (simp add: Prf.intros(5) ValOrd.intros(8))
+defer
+apply(drule_tac x="s" in meta_spec)
+apply(auto)[1]
+apply(rule_tac x="Left v" in exI)
+apply(simp add: POSIX2_def)
+apply(auto)[1]
+using Prf.intros(2) apply blast
+
+apply(case_tac s)
+apply(simp)
+apply(auto)[1]
+apply (simp add: ValOrd.intros(6))
+apply(rule ValOrd.intros)
+
+thm PMatch.intros[no_vars]
+
+lemma POSIX_PMatch:
+  assumes "s \<in> r \<rightarrow> v" "v' \<in> Values r s"
+  shows "v \<succ>r v' \<or> length (flat v') < length (flat v)" 
+using assms
+apply(induct r arbitrary: s v v' rule: rexp.induct)
+apply(simp_all add: Values_recs)
+apply(erule PMatch.cases)
+apply(simp_all)[5]
+apply (metis ValOrd.intros(7))
+apply(erule PMatch.cases)
+apply(simp_all)[5]
+apply(simp add: prefix_def)
+apply (metis ValOrd.intros(8))
+apply(auto)[1]
+defer
+apply(auto)[1]
+apply(erule PMatch.cases)
+apply(simp_all)[5]
+apply (metis ValOrd.intros(6))
+apply (metis (no_types, lifting) PMatch1(2) Prf_flat_L Values_def length_sprefix mem_Collect_eq sprefix_def)
+apply(erule PMatch.cases)
+apply(simp_all)[5]
+apply (metis (no_types, lifting) PMatch1(2) ValOrd.intros(3) Values_def length_sprefix mem_Collect_eq order_refl sprefix_def)
+apply (metis ValOrd.intros(5))
+apply(erule PMatch.cases)
+apply(simp_all)[5]
+apply(auto)
+apply(case_tac "v1a = v1")
+apply(simp)
+apply(rule ValOrd.intros(1))
+apply (metis PMatch1(2) append_Nil comm_monoid_diff_class.diff_cancel drop_0 drop_all drop_append order_refl rest_def)
+apply(rule ValOrd.intros(2))
+apply(auto)
+apply(drule_tac x="s1" in meta_spec)
+apply(drule_tac x="v1a" in meta_spec)
+apply(drule_tac x="v1" in meta_spec)
+apply(auto)
+apply(drule meta_mp)
+defer
+apply(auto)[1]
+apply(frule PMatch1)
+apply(drule PMatch1(2))
+apply(frule PMatch1)
+apply(drule PMatch1(2))
+apply(auto)[1]
+apply(simp add: Values_def)
+apply(simp add: prefix_def)
+apply(auto)[1]
+apply(simp add: append_eq_append_conv2)
+apply(auto)[1]
+apply(rotate_tac 10)
+apply(drule sym)
+apply(simp)
+apply(simp add: rest_def)
+apply(case_tac "s3a = []")
+apply(auto)[1]
+
+
+apply (metis ValOrd.intros(6))
+apply (metis (no_types, lifting) PMatch1(2) ValOrd.intros(3) Values_def length_sprefix mem_Collect_eq order_refl sprefix_def)
+apply(auto)[1]
+apply (metis (no_types, lifting) PMatch1(2) Prf_flat_L Values_def length_sprefix mem_Collect_eq sprefix_def)
+apply (metis ValOrd.intros(5))
+apply(auto)[1]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(erule PMatch.cases)
+apply(simp_all)[5]
+defer
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(erule PMatch.cases)
+apply(simp_all)[5]
+apply(clarify)
+apply(simp add: L_flat_Prf)
+
+apply(clarify)
+apply (metis ValOrd.intros(8))
+apply (metis POSIX_ALT_I1)
+apply(rule POSIX_ALT_I2)
+apply(simp)
+apply(auto)[1]
+apply(simp add: POSIX_def)
+apply(frule PMatch1(1))
+apply(frule PMatch1(2))
+apply(simp)
+
+
+lemma POSIX_PMatch:
+  assumes "s \<in> r \<rightarrow> v" 
+  shows "POSIX v r" 
+using assms
+apply(induct arbitrary: rule: PMatch.induct)
+apply(auto)
+apply(simp add: POSIX_def)
+apply(auto)[1]
+apply (metis Prf.intros(4))
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply (metis ValOrd.intros(7))
+apply(simp add: POSIX_def)
+apply(auto)[1]
+apply (metis Prf.intros(5))
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply (metis ValOrd.intros(8))
+apply (metis POSIX_ALT_I1)
+apply(rule POSIX_ALT_I2)
+apply(simp)
+apply(auto)[1]
+apply(simp add: POSIX_def)
+apply(frule PMatch1(1))
+apply(frule PMatch1(2))
+apply(simp)
+
+
+
+lemma ValOrd_PMatch:
+  assumes "s \<in> r \<rightarrow> v1" "\<turnstile> v2 : r" "flat v2 = s"
+  shows "v1 \<succ>r v2"
+using assms
+apply(induct arbitrary: v2 rule: PMatch.induct)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply (metis ValOrd.intros(7))
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply (metis ValOrd.intros(8))
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(clarify)
+apply (metis ValOrd.intros(6))
+apply(clarify)
+apply (metis PMatch1(2) ValOrd.intros(3) order_refl)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(clarify)
+apply (metis Prf_flat_L)
+apply (metis ValOrd.intros(5))
+(* Seq case *)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(auto)
+apply(case_tac "v1 = v1a")
+apply(auto)
+apply (metis PMatch1(2) ValOrd.intros(1) same_append_eq)
+apply(rule ValOrd.intros(2))
+apply(auto)
+apply(drule_tac x="v1a" in meta_spec)
+apply(drule_tac meta_mp)
+apply(simp)
+apply(drule_tac meta_mp)
+prefer 2
+apply(simp)
+apply(simp add: append_eq_append_conv2)
+apply(auto)
+apply (metis Prf_flat_L)
+apply(case_tac "us = []")
+apply(simp)
+apply(drule_tac x="us" in spec)
+apply(drule mp)
+
+thm L_flat_Prf
+apply(simp add: L_flat_Prf)
+thm append_eq_append_conv2
+apply(simp add: append_eq_append_conv2)
+apply(auto)
+apply(drule_tac x="us" in spec)
+apply(drule mp)
+apply metis
+apply (metis append_Nil2)
+apply(case_tac "us = []")
+apply(auto)
+apply(drule_tac x="s2" in spec)
+apply(drule mp)
+
+apply(auto)[1]
+apply(drule_tac x="v1a" in meta_spec)
+apply(simp)
+
+lemma refl_on_ValOrd:
+  "refl_on (Values r s) {(v1, v2). v1 \<succ>r v2 \<and> v1 \<in> Values r s \<and> v2 \<in> Values r s}"
+unfolding refl_on_def
+apply(auto)
+apply(rule ValOrd_refl)
+apply(simp add: Values_def)
+done
+
+
+section {* Posix definition *}
+
+definition POSIX :: "val \<Rightarrow> rexp \<Rightarrow> bool" 
+where
+  "POSIX v r \<equiv> (\<turnstile> v : r \<and> (\<forall>v'. (\<turnstile> v' : r \<and> flat v = flat v') \<longrightarrow> v \<succ>r v'))"
+
+definition POSIX2 :: "val \<Rightarrow> rexp \<Rightarrow> bool" 
+where
+  "POSIX2 v r \<equiv> (\<turnstile> v : r \<and> (\<forall>v'. (\<turnstile> v' : r \<and> flat v = flat v') \<longrightarrow> v 2\<succ> v'))"
+
+lemma "POSIX v r = POSIX2 v r"
+unfolding POSIX_def POSIX2_def
+apply(auto)
+apply(rule Ord1)
+apply(auto)
+apply(rule Ord3)
+apply(auto)
+done
+
+section {* POSIX for some constructors *}
+
+lemma POSIX_SEQ1:
+  assumes "POSIX (Seq v1 v2) (SEQ r1 r2)" "\<turnstile> v1 : r1" "\<turnstile> v2 : r2"
+  shows "POSIX v1 r1"
+using assms
+unfolding POSIX_def
+apply(auto)
+apply(drule_tac x="Seq v' v2" in spec)
+apply(simp)
+apply(erule impE)
+apply(rule Prf.intros)
+apply(simp)
+apply(simp)
+apply(erule ValOrd.cases)
+apply(simp_all)
+apply(clarify)
+by (metis ValOrd_refl)
+
+lemma POSIX_SEQ2:
+  assumes "POSIX (Seq v1 v2) (SEQ r1 r2)" "\<turnstile> v1 : r1" "\<turnstile> v2 : r2" 
+  shows "POSIX v2 r2"
+using assms
+unfolding POSIX_def
+apply(auto)
+apply(drule_tac x="Seq v1 v'" in spec)
+apply(simp)
+apply(erule impE)
+apply(rule Prf.intros)
+apply(simp)
+apply(simp)
+apply(erule ValOrd.cases)
+apply(simp_all)
+done
+
+lemma POSIX_ALT2:
+  assumes "POSIX (Left v1) (ALT r1 r2)"
+  shows "POSIX v1 r1"
+using assms
+unfolding POSIX_def
+apply(auto)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(drule_tac x="Left v'" in spec)
+apply(simp)
+apply(drule mp)
+apply(rule Prf.intros)
+apply(auto)
+apply(erule ValOrd.cases)
+apply(simp_all)
+done
+
+lemma POSIX_ALT1a:
+  assumes "POSIX (Right v2) (ALT r1 r2)"
+  shows "POSIX v2 r2"
+using assms
+unfolding POSIX_def
+apply(auto)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(drule_tac x="Right v'" in spec)
+apply(simp)
+apply(drule mp)
+apply(rule Prf.intros)
+apply(auto)
+apply(erule ValOrd.cases)
+apply(simp_all)
+done
+
+lemma POSIX_ALT1b:
+  assumes "POSIX (Right v2) (ALT r1 r2)"
+  shows "(\<forall>v'. (\<turnstile> v' : r2 \<and> flat v' = flat v2) \<longrightarrow> v2 \<succ>r2 v')"
+using assms
+apply(drule_tac POSIX_ALT1a)
+unfolding POSIX_def
+apply(auto)
+done
+
+lemma POSIX_ALT_I1:
+  assumes "POSIX v1 r1" 
+  shows "POSIX (Left v1) (ALT r1 r2)"
+using assms
+unfolding POSIX_def
+apply(auto)
+apply (metis Prf.intros(2))
+apply(rotate_tac 2)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(auto)
+apply(rule ValOrd.intros)
+apply(auto)
+apply(rule ValOrd.intros)
+by simp
+
+lemma POSIX_ALT_I2:
+  assumes "POSIX v2 r2" "\<forall>v'. \<turnstile> v' : r1 \<longrightarrow> length (flat v2) > length (flat v')"
+  shows "POSIX (Right v2) (ALT r1 r2)"
+using assms
+unfolding POSIX_def
+apply(auto)
+apply (metis Prf.intros)
+apply(rotate_tac 3)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(auto)
+apply(rule ValOrd.intros)
+apply metis
+done
+
+lemma mkeps_POSIX:
+  assumes "nullable r"
+  shows "POSIX (mkeps r) r"
+using assms
+apply(induct r)
+apply(auto)[1]
+apply(simp add: POSIX_def)
+apply(auto)[1]
+apply (metis Prf.intros(4))
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply (metis ValOrd.intros)
+apply(simp)
+apply(auto)[1]
+apply(simp add: POSIX_def)
+apply(auto)[1]
+apply (metis mkeps.simps(2) mkeps_nullable nullable.simps(5))
+apply(rotate_tac 6)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply (simp add: mkeps_flat)
+apply(case_tac "mkeps r1a = v1")
+apply(simp)
+apply (metis ValOrd.intros(1))
+apply (rule ValOrd.intros(2))
+apply metis
+apply(simp)
+(* ALT case *)
+thm mkeps.simps
+apply(simp)
+apply(erule disjE)
+apply(simp)
+apply (metis POSIX_ALT_I1)
+(* *)
+apply(auto)[1]
+thm  POSIX_ALT_I1
+apply (metis POSIX_ALT_I1)
+apply(simp (no_asm) add: POSIX_def)
+apply(auto)[1]
+apply(rule Prf.intros(3))
+apply(simp only: POSIX_def)
+apply(rotate_tac 4)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+thm mkeps_flat
+apply(simp add: mkeps_flat)
+apply(auto)[1]
+thm Prf_flat_L nullable_correctness
+apply (metis Prf_flat_L nullable_correctness)
+apply(rule ValOrd.intros)
+apply(subst (asm) POSIX_def)
+apply(clarify)
+apply(drule_tac x="v2" in spec)
+by simp
+
+
+
+text {*
+  Injection value is related to r
+*}
+
+
+
+text {*
+  The string behind the injection value is an added c
+*}
+
+
+lemma injval_inj: "inj_on (injval r c) {v. \<turnstile> v : der c r}"
+apply(induct c r rule: der.induct)
+unfolding inj_on_def
+apply(auto)[1]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(auto)[1]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(auto)[1]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(auto)[1]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(auto)[1]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(clarify)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(clarify)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(clarify)
+apply (metis list.distinct(1) mkeps_flat v4)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(clarify)
+apply(rotate_tac 6)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply (metis list.distinct(1) mkeps_flat v4)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+done
+
+lemma Values_nullable:
+  assumes "nullable r1"
+  shows "mkeps r1 \<in> Values r1 s"
+using assms
+apply(induct r1 arbitrary: s)
+apply(simp_all)
+apply(simp add: Values_recs)
+apply(simp add: Values_recs)
+apply(simp add: Values_recs)
+apply(auto)[1]
+done
+
+lemma Values_injval:
+  assumes "v \<in> Values (der c r) s"
+  shows "injval r c v \<in> Values r (c#s)"
+using assms
+apply(induct c r arbitrary: v s rule: der.induct)
+apply(simp add: Values_recs)
+apply(simp add: Values_recs)
+apply(case_tac "c = c'")
+apply(simp)
+apply(simp add: Values_recs)
+apply(simp add: prefix_def)
+apply(simp)
+apply(simp add: Values_recs)
+apply(simp)
+apply(simp add: Values_recs)
+apply(auto)[1]
+apply(case_tac "nullable r1")
+apply(simp)
+apply(simp add: Values_recs)
+apply(auto)[1]
+apply(simp add: rest_def)
+apply(subst v4)
+apply(simp add: Values_def)
+apply(simp add: Values_def)
+apply(rule Values_nullable)
+apply(assumption)
+apply(simp add: rest_def)
+apply(subst mkeps_flat)
+apply(assumption)
+apply(simp)
+apply(simp)
+apply(simp add: Values_recs)
+apply(auto)[1]
+apply(simp add: rest_def)
+apply(subst v4)
+apply(simp add: Values_def)
+apply(simp add: Values_def)
+done
+
+lemma Values_projval:
+  assumes "v \<in> Values r (c#s)" "\<exists>s. flat v = c # s"
+  shows "projval r c v \<in> Values (der c r) s"
+using assms
+apply(induct r arbitrary: v s c rule: rexp.induct)
+apply(simp add: Values_recs)
+apply(simp add: Values_recs)
+apply(case_tac "c = char")
+apply(simp)
+apply(simp add: Values_recs)
+apply(simp)
+apply(simp add: Values_recs)
+apply(simp add: prefix_def)
+apply(case_tac "nullable rexp1")
+apply(simp)
+apply(simp add: Values_recs)
+apply(auto)[1]
+apply(simp add: rest_def)
+apply (metis hd_Cons_tl hd_append2 list.sel(1))
+apply(simp add: rest_def)
+apply(simp add: append_eq_Cons_conv)
+apply(auto)[1]
+apply(subst v4_proj2)
+apply(simp add: Values_def)
+apply(assumption)
+apply(simp)
+apply(simp)
+apply(simp add: Values_recs)
+apply(auto)[1]
+apply(auto simp add: Values_def not_nullable_flat)[1]
+apply(simp add: append_eq_Cons_conv)
+apply(auto)[1]
+apply(simp add: append_eq_Cons_conv)
+apply(auto)[1]
+apply(simp add: rest_def)
+apply(subst v4_proj2)
+apply(simp add: Values_def)
+apply(assumption)
+apply(simp)
+apply(simp add: Values_recs)
+apply(auto)[1]
+done
+
+
+definition "MValue v r s \<equiv> (v \<in> Values r s \<and> (\<forall>v' \<in> Values r s. v 2\<succ> v'))"
+
+lemma MValue_ALTE:
+  assumes "MValue v (ALT r1 r2) s"
+  shows "(\<exists>vl. v = Left vl \<and> MValue vl r1 s \<and> (\<forall>vr \<in> Values r2 s. length (flat vr) \<le> length (flat vl))) \<or> 
+         (\<exists>vr. v = Right vr \<and> MValue vr r2 s \<and> (\<forall>vl \<in> Values r1 s. length (flat vl) < length (flat vr)))"
+using assms
+apply(simp add: MValue_def)
+apply(simp add: Values_recs)
+apply(auto)
+apply(drule_tac x="Left x" in bspec)
+apply(simp)
+apply(erule ValOrd2.cases)
+apply(simp_all)
+apply(drule_tac x="Right vr" in bspec)
+apply(simp)
+apply(erule ValOrd2.cases)
+apply(simp_all)
+apply(drule_tac x="Right x" in bspec)
+apply(simp)
+apply(erule ValOrd2.cases)
+apply(simp_all)
+apply(drule_tac x="Left vl" in bspec)
+apply(simp)
+apply(erule ValOrd2.cases)
+apply(simp_all)
+done
+
+lemma MValue_ALTI1:
+  assumes "MValue vl r1 s"  "\<forall>vr \<in> Values r2 s. length (flat vr) \<le> length (flat vl)"
+  shows "MValue (Left vl) (ALT r1 r2) s"
+using assms
+apply(simp add: MValue_def)
+apply(simp add: Values_recs)
+apply(auto)
+apply(rule ValOrd2.intros)
+apply metis
+apply(rule ValOrd2.intros)
+apply metis
+done
+
+lemma MValue_ALTI2:
+  assumes "MValue vr r2 s"  "\<forall>vl \<in> Values r1 s. length (flat vl) < length (flat vr)"
+  shows "MValue (Right vr) (ALT r1 r2) s"
+using assms
+apply(simp add: MValue_def)
+apply(simp add: Values_recs)
+apply(auto)
+apply(rule ValOrd2.intros)
+apply metis
+apply(rule ValOrd2.intros)
+apply metis
+done
+
+lemma t: "(c#xs = c#ys) \<Longrightarrow> xs = ys"
+by (metis list.sel(3))
+
+lemma t2: "(xs = ys) \<Longrightarrow> (c#xs) = (c#ys)"
+by (metis)
+
+lemma "\<not>(nullable r) \<Longrightarrow> \<not>(\<exists>v. \<turnstile> v : r \<and> flat v = [])"
+by (metis Prf_flat_L nullable_correctness)
+
+
+lemma LeftRight:
+  assumes "(Left v1) \<succ>(der c (ALT r1 r2)) (Right v2)"
+  and "\<turnstile> v1 : der c r1" "\<turnstile> v2 : der c r2" 
+  shows "(injval (ALT r1 r2) c (Left v1)) \<succ>(ALT r1 r2) (injval (ALT r1 r2) c (Right v2))"
+using assms
+apply(simp)
+apply(erule ValOrd.cases)
+apply(simp_all)[8]
+apply(rule ValOrd.intros)
+apply(clarify)
+apply(subst v4)
+apply(simp)
+apply(subst v4)
+apply(simp)
+apply(simp)
+done
+
+lemma RightLeft:
+  assumes "(Right v1) \<succ>(der c (ALT r1 r2)) (Left v2)"
+  and "\<turnstile> v1 : der c r2" "\<turnstile> v2 : der c r1" 
+  shows "(injval (ALT r1 r2) c (Right v1)) \<succ>(ALT r1 r2) (injval (ALT r1 r2) c (Left v2))"
+using assms
+apply(simp)
+apply(erule ValOrd.cases)
+apply(simp_all)[8]
+apply(rule ValOrd.intros)
+apply(clarify)
+apply(subst v4)
+apply(simp)
+apply(subst v4)
+apply(simp)
+apply(simp)
+done
+
+lemma h: 
+  assumes "nullable r1" "\<turnstile> v1 : der c r1"
+  shows "injval r1 c v1 \<succ>r1 mkeps r1"
+using assms
+apply(induct r1 arbitrary: v1 rule: der.induct)
+apply(simp)
+apply(simp)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(simp)
+apply(simp)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(clarify)
+apply(auto)[1]
+apply (metis ValOrd.intros(6))
+apply (metis ValOrd.intros(6))
+apply (metis ValOrd.intros(3) le_add2 list.size(3) mkeps_flat monoid_add_class.add.right_neutral)
+apply(auto)[1]
+apply (metis ValOrd.intros(4) length_greater_0_conv list.distinct(1) list.size(3) mkeps_flat v4)
+apply (metis ValOrd.intros(4) length_greater_0_conv list.distinct(1) list.size(3) mkeps_flat v4)
+apply (metis ValOrd.intros(5))
+apply(simp)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(clarify)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(clarify)
+apply (metis ValOrd.intros(2) list.distinct(1) mkeps_flat v4)
+apply(clarify)
+by (metis ValOrd.intros(1))
+
+lemma LeftRightSeq:
+  assumes "(Left (Seq v1 v2)) \<succ>(der c (SEQ r1 r2)) (Right v3)"
+  and "nullable r1" "\<turnstile> v1 : der c r1"
+  shows "(injval (SEQ r1 r2) c (Seq v1 v2)) \<succ>(SEQ r1 r2) (injval (SEQ r1 r2) c (Right v2))"
+using assms
+apply(simp)
+apply(erule ValOrd.cases)
+apply(simp_all)[8]
+apply(clarify)
+apply(simp)
+apply(rule ValOrd.intros(2))
+prefer 2
+apply (metis list.distinct(1) mkeps_flat v4)
+by (metis h)
+
+lemma rr1: 
+  assumes "\<turnstile> v : r" "\<not>nullable r" 
+  shows "flat v \<noteq> []"
+using assms
+by (metis Prf_flat_L nullable_correctness)
+
+(* HERE *)
+
+lemma Prf_inj_test:
+  assumes "v1 \<succ>(der c r) v2" 
+          "v1 \<in> Values (der c r) s"
+          "v2 \<in> Values (der c r) s"
+          "injval r c v1 \<in> Values r (c#s)"
+          "injval r c v2 \<in> Values r (c#s)"
+  shows "(injval r c v1) 2\<succ>  (injval r c v2)"
+using assms
+apply(induct c r arbitrary: v1 v2 s rule: der.induct)
+(* NULL case *)
+apply(simp add: Values_recs)
+(* EMPTY case *)
+apply(simp add: Values_recs)
+(* CHAR case *)
+apply(case_tac "c = c'")
+apply(simp)
+apply(simp add: Values_recs)
+apply (metis ValOrd2.intros(8))
+apply(simp add: Values_recs)
+(* ALT case *)
+apply(simp)
+apply(simp add: Values_recs)
+apply(auto)[1]
+apply(erule ValOrd.cases)
+apply(simp_all)[8]
+apply (metis ValOrd2.intros(6))
+apply(erule ValOrd.cases)
+apply(simp_all)[8]
+apply(rule ValOrd2.intros)
+apply(subst v4)
+apply(simp add: Values_def)
+apply(subst v4)
+apply(simp add: Values_def)
+apply(simp)
+apply(erule ValOrd.cases)
+apply(simp_all)[8]
+apply(rule ValOrd2.intros)
+apply(subst v4)
+apply(simp add: Values_def)
+apply(subst v4)
+apply(simp add: Values_def)
+apply(simp)
+apply(erule ValOrd.cases)
+apply(simp_all)[8]
+apply (metis ValOrd2.intros(5))
+(* SEQ case*)
+apply(simp)
+apply(case_tac "nullable r1")
+apply(simp)
+defer
+apply(simp)
+apply(simp add: Values_recs)
+apply(auto)[1]
+apply(erule ValOrd.cases)
+apply(simp_all)[8]
+apply(clarify)
+apply(rule ValOrd2.intros)
+apply(simp)
+apply (metis Ord1)
+apply(clarify)
+apply(rule ValOrd2.intros)
+apply(subgoal_tac "rest v1 (flat v1 @ flat v2) = flat v2")
+apply(simp)
+apply(subgoal_tac "rest (injval r1 c v1) (c # flat v1 @ flat v2) = flat v2")
+apply(simp)
+oops
+
+lemma Prf_inj_test:
+  assumes "v1 \<succ>(der c r) v2" 
+          "v1 \<in> Values (der c r) s"
+          "v2 \<in> Values (der c r) s"
+          "injval r c v1 \<in> Values r (c#s)"
+          "injval r c v2 \<in> Values r (c#s)"
+  shows "(injval r c v1) 2\<succ>  (injval r c v2)"
+using assms
+apply(induct c r arbitrary: v1 v2 s rule: der.induct)
+(* NULL case *)
+apply(simp add: Values_recs)
+(* EMPTY case *)
+apply(simp add: Values_recs)
+(* CHAR case *)
+apply(case_tac "c = c'")
+apply(simp)
+apply(simp add: Values_recs)
+apply (metis ValOrd2.intros(8))
+apply(simp add: Values_recs)
+(* ALT case *)
+apply(simp)
+apply(simp add: Values_recs)
+apply(auto)[1]
+apply(erule ValOrd.cases)
+apply(simp_all)[8]
+apply (metis ValOrd2.intros(6))
+apply(erule ValOrd.cases)
+apply(simp_all)[8]
+apply(rule ValOrd2.intros)
+apply(subst v4)
+apply(simp add: Values_def)
+apply(subst v4)
+apply(simp add: Values_def)
+apply(simp)
+apply(erule ValOrd.cases)
+apply(simp_all)[8]
+apply(rule ValOrd2.intros)
+apply(subst v4)
+apply(simp add: Values_def)
+apply(subst v4)
+apply(simp add: Values_def)
+apply(simp)
+apply(erule ValOrd.cases)
+apply(simp_all)[8]
+apply (metis ValOrd2.intros(5))
+(* SEQ case*)
+apply(simp)
+apply(case_tac "nullable r1")
+apply(simp)
+defer
+apply(simp)
+apply(simp add: Values_recs)
+apply(auto)[1]
+apply(erule ValOrd.cases)
+apply(simp_all)[8]
+apply(clarify)
+apply(rule ValOrd2.intros)
+apply(simp)
+apply (metis Ord1)
+apply(clarify)
+apply(rule ValOrd2.intros)
+apply metis
+using injval_inj
+apply(simp add: Values_def inj_on_def)
+apply metis
+apply(simp add: Values_recs)
+apply(auto)[1]
+apply(erule ValOrd.cases)
+apply(simp_all)[8]
+apply(clarify)
+apply(erule ValOrd.cases)
+apply(simp_all)[8]
+apply(clarify)
+apply (metis Ord1 ValOrd2.intros(1))
+apply(clarify)
+apply(rule ValOrd2.intros(2))
+apply metis
+using injval_inj
+apply(simp add: Values_def inj_on_def)
+apply metis
+apply(erule ValOrd.cases)
+apply(simp_all)[8]
+apply(rule ValOrd2.intros(2))
+thm h
+apply(rule Ord1)
+apply(rule h)
+apply(simp)
+apply(simp add: Values_def)
+apply(simp add: Values_def)
+apply (metis list.distinct(1) mkeps_flat v4)
+apply(erule ValOrd.cases)
+apply(simp_all)[8]
+apply(clarify)
+apply(simp add: Values_def)
+defer
+apply(erule ValOrd.cases)
+apply(simp_all)[8]
+apply(clarify)
+apply(rule ValOrd2.intros(1))
+apply(rotate_tac 1)
+apply(drule_tac x="v2" in meta_spec)
+apply(rotate_tac 8)
+apply(drule_tac x="v2'" in meta_spec)
+apply(rotate_tac 8)
+oops
+
+lemma POSIX_der:
+  assumes "POSIX v (der c r)" "\<turnstile> v : der c r"
+  shows "POSIX (injval r c v) r"
+using assms
+unfolding POSIX_def
+apply(auto)
+thm v3
+apply (erule v3)
+thm v4
+apply(subst (asm) v4)
+apply(assumption)
+apply(drule_tac x="projval r c v'" in spec)
+apply(drule mp)
+apply(rule conjI)
+thm v3_proj
+apply(rule v3_proj)
+apply(simp)
+apply(rule_tac x="flat v" in exI)
+apply(simp)
+thm t
+apply(rule_tac c="c" in  t)
+apply(simp)
+thm v4_proj
+apply(subst v4_proj)
+apply(simp)
+apply(rule_tac x="flat v" in exI)
+apply(simp)
+apply(simp)
+oops
+
+lemma POSIX_der:
+  assumes "POSIX v (der c r)" "\<turnstile> v : der c r"
+  shows "POSIX (injval r c v) r"
+using assms
+apply(induct c r arbitrary: v rule: der.induct)
+(* null case*)
+apply(simp add: POSIX_def)
+apply(auto)[1]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+(* empty case *)
+apply(simp add: POSIX_def)
+apply(auto)[1]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+(* char case *)
+apply(simp add: POSIX_def)
+apply(case_tac "c = c'")
+apply(auto)[1]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply (metis Prf.intros(5))
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply (metis ValOrd.intros(8))
+apply(auto)[1]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+(* alt case *)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(clarify)
+apply(simp (no_asm) add: POSIX_def)
+apply(auto)[1]
+apply (metis Prf.intros(2) v3)
+apply(rotate_tac 4)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply (metis POSIX_ALT2 POSIX_def ValOrd.intros(6))
+apply (metis ValOrd.intros(3) order_refl)
+apply(simp (no_asm) add: POSIX_def)
+apply(auto)[1]
+apply (metis Prf.intros(3) v3)
+apply(rotate_tac 4)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+defer
+apply (metis POSIX_ALT1a POSIX_def ValOrd.intros(5))
+prefer 2
+apply(subst (asm) (5) POSIX_def)
+apply(auto)[1]
+apply(rotate_tac 5)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(rule ValOrd.intros)
+apply(subst (asm) v4)
+apply(simp)
+apply(drule_tac x="Left (projval r1a c v1)" in spec)
+apply(clarify)
+apply(drule mp)
+apply(rule conjI)
+apply (metis Prf.intros(2) v3_proj)
+apply(simp)
+apply (metis v4_proj2)
+apply(erule ValOrd.cases)
+apply(simp_all)[8]
+apply (metis less_not_refl v4_proj2)
+(* seq case *)
+apply(case_tac "nullable r1")
+defer
+apply(simp add: POSIX_def)
+apply(auto)[1]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply (metis Prf.intros(1) v3)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(clarify)
+apply(subst (asm) (3) v4)
+apply(simp)
+apply(simp)
+apply(subgoal_tac "flat v1a \<noteq> []")
+prefer 2
+apply (metis Prf_flat_L nullable_correctness)
+apply(subgoal_tac "\<exists>s. flat v1a = c # s")
+prefer 2
+apply (metis append_eq_Cons_conv)
+apply(auto)[1]
+oops
+
+
+lemma POSIX_ex: "\<turnstile> v : r \<Longrightarrow> \<exists>v. POSIX v r"
+apply(induct r arbitrary: v)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(rule_tac x="Void" in exI)
+apply(simp add: POSIX_def)
+apply(auto)[1]
+apply (metis Prf.intros(4))
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply (metis ValOrd.intros(7))
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(rule_tac x="Char c" in exI)
+apply(simp add: POSIX_def)
+apply(auto)[1]
+apply (metis Prf.intros(5))
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply (metis ValOrd.intros(8))
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(auto)[1]
+apply(drule_tac x="v1" in meta_spec)
+apply(drule_tac x="v2" in meta_spec)
+apply(auto)[1]
+defer
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(auto)[1]
+apply (metis POSIX_ALT_I1)
+apply (metis POSIX_ALT_I1 POSIX_ALT_I2)
+apply(case_tac "nullable r1a")
+apply(rule_tac x="Seq (mkeps r1a) va" in exI)
+apply(auto simp add: POSIX_def)[1]
+apply (metis Prf.intros(1) mkeps_nullable)
+apply(simp add: mkeps_flat)
+apply(rotate_tac 7)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(case_tac "mkeps r1 = v1a")
+apply(simp)
+apply (rule ValOrd.intros(1))
+apply (metis append_Nil mkeps_flat)
+apply (rule ValOrd.intros(2))
+apply(drule mkeps_POSIX)
+apply(simp add: POSIX_def)
+oops
+
+lemma POSIX_ex2: "\<turnstile> v : r \<Longrightarrow> \<exists>v. POSIX v r \<and> \<turnstile> v : r"
+apply(induct r arbitrary: v)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(rule_tac x="Void" in exI)
+apply(simp add: POSIX_def)
+apply(auto)[1]
+oops
+
+lemma POSIX_ALT_cases:
+  assumes "\<turnstile> v : (ALT r1 r2)" "POSIX v (ALT r1 r2)"
+  shows "(\<exists>v1. v = Left v1 \<and> POSIX v1 r1) \<or> (\<exists>v2. v = Right v2 \<and> POSIX v2 r2)"
+using assms
+apply(erule_tac Prf.cases)
+apply(simp_all)
+unfolding POSIX_def
+apply(auto)
+apply (metis POSIX_ALT2 POSIX_def assms(2))
+by (metis POSIX_ALT1b assms(2))
+
+lemma POSIX_ALT_cases2:
+  assumes "POSIX v (ALT r1 r2)" "\<turnstile> v : (ALT r1 r2)" 
+  shows "(\<exists>v1. v = Left v1 \<and> POSIX v1 r1) \<or> (\<exists>v2. v = Right v2 \<and> POSIX v2 r2)"
+using assms POSIX_ALT_cases by auto
+
+lemma Prf_flat_empty:
+  assumes "\<turnstile> v : r" "flat v = []"
+  shows "nullable r"
+using assms
+apply(induct)
+apply(auto)
+done
+
+lemma POSIX_proj:
+  assumes "POSIX v r" "\<turnstile> v : r" "\<exists>s. flat v = c#s"
+  shows "POSIX (projval r c v) (der c r)"
+using assms
+apply(induct r c v arbitrary: rule: projval.induct)
+defer
+defer
+defer
+defer
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(simp add: POSIX_def)
+apply(auto)[1]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+oops
+
+lemma POSIX_proj:
+  assumes "POSIX v r" "\<turnstile> v : r" "\<exists>s. flat v = c#s"
+  shows "POSIX (projval r c v) (der c r)"
+using assms
+apply(induct r arbitrary: c v rule: rexp.induct)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(simp add: POSIX_def)
+apply(auto)[1]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+oops
+
+lemma POSIX_proj:
+  assumes "POSIX v r" "\<turnstile> v : r" "\<exists>s. flat v = c#s"
+  shows "POSIX (projval r c v) (der c r)"
+using assms
+apply(induct r c v arbitrary: rule: projval.induct)
+defer
+defer
+defer
+defer
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(simp add: POSIX_def)
+apply(auto)[1]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+oops
+
+lemma Prf_inj:
+  assumes "v1 \<succ>(der c r) v2" "\<turnstile> v1 : der c r" "\<turnstile> v2 : der c r" "flat v1 = flat v2"
+  shows "(injval r c v1) \<succ>r (injval r c v2)"
+using assms
+apply(induct arbitrary: v1 v2 rule: der.induct)
+(* NULL case *)
+apply(simp)
+apply(erule ValOrd.cases)
+apply(simp_all)[8]
+(* EMPTY case *)
+apply(erule ValOrd.cases)
+apply(simp_all)[8]
+(* CHAR case *)
+apply(case_tac "c = c'")
+apply(simp)
+apply(erule ValOrd.cases)
+apply(simp_all)[8]
+apply(rule ValOrd.intros)
+apply(simp)
+apply(erule ValOrd.cases)
+apply(simp_all)[8]
+(* ALT case *)
+apply(simp)
+apply(erule ValOrd.cases)
+apply(simp_all)[8]
+apply(rule ValOrd.intros)
+apply(subst v4)
+apply(clarify)
+apply(rotate_tac 3)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(subst v4)
+apply(clarify)
+apply(rotate_tac 2)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(simp)
+apply(rule ValOrd.intros)
+apply(clarify)
+apply(rotate_tac 3)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(clarify)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(rule ValOrd.intros)
+apply(clarify)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+(* SEQ case*)
+apply(simp)
+apply(case_tac "nullable r1")
+defer
+apply(simp)
+apply(erule ValOrd.cases)
+apply(simp_all)[8]
+apply(clarify)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(clarify)
+apply(rule ValOrd.intros)
+apply(simp)
+oops
+
+
+text {*
+  Injection followed by projection is the identity.
+*}
+
+lemma proj_inj_id:
+  assumes "\<turnstile> v : der c r" 
+  shows "projval r c (injval r c v) = v"
+using assms
+apply(induct r arbitrary: c v rule: rexp.induct)
+apply(simp)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(simp)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(simp)
+apply(case_tac "c = char")
+apply(simp)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(simp)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+defer
+apply(simp)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(simp)
+apply(case_tac "nullable rexp1")
+apply(simp)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(auto)[1]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(auto)[1]
+apply (metis list.distinct(1) v4)
+apply(auto)[1]
+apply (metis mkeps_flat)
+apply(auto)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+apply(auto)[1]
+apply(simp add: v4)
+done
+
+text {* 
+
+  HERE: Crucial lemma that does not go through in the sequence case. 
+
+*}
+lemma v5:
+  assumes "\<turnstile> v : der c r" "POSIX v (der c r)"
+  shows "POSIX (injval r c v) r"
+using assms
+apply(induct arbitrary: v rule: der.induct)
+(* NULL case *)
+apply(simp)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+(* EMPTY case *)
+apply(simp)
+apply(erule Prf.cases)
+apply(simp_all)[5]
+(* CHAR case *)
+apply(simp)
+apply(case_tac "c = c'")
+apply(auto simp add: POSIX_def)[1]
+apply(erule Prf.cases)
+apply(simp_all)[5]
+oops
\ No newline at end of file