diff -r ca4e9eb8d576 -r 7c89d3f6923e thys/Sulzmann.thy
--- a/thys/Sulzmann.thy	Tue Jun 27 08:59:11 2017 +0100
+++ b/thys/Sulzmann.thy	Tue Jun 27 13:15:55 2017 +0100
@@ -1,1865 +1,11 @@
    
 theory Sulzmann
-  imports "Lexer" "~~/src/HOL/Library/Multiset"
+  imports "Positions" 
 begin
 
 
 section {* Sulzmann's "Ordering" of Values *}
 
-fun 
-  size :: "val \<Rightarrow> nat"
-where
-  "size (Void) = 0"
-| "size (Char c) = 0"
-| "size (Left v) = 1 + size v"
-| "size (Right v) = 1 + size v"
-| "size (Seq v1 v2) = 1 + (size v1) + (size v2)"
-| "size (Stars []) = 0"
-| "size (Stars (v#vs)) = 1 + (size v) + (size (Stars vs))" 
-
-lemma Star_size [simp]:
-  "\<lbrakk>n < length vs; 0 < length vs\<rbrakk> \<Longrightarrow> size (nth vs n) < size (Stars vs)"
-apply(induct vs arbitrary: n)
-apply(simp)
-apply(auto)
-by (metis One_nat_def Suc_pred less_Suc0 less_Suc_eq list.size(3) not_add_less1 not_less_eq nth_Cons' trans_less_add2)
-
-lemma Star_size0 [simp]:
-  "0 < length vs \<Longrightarrow> 0 < size (Stars vs)"
-apply(induct vs)
-apply(auto)
-done
-
-
-fun 
-  at :: "val \<Rightarrow> nat list \<Rightarrow> val"
-where
-  "at v [] = v"
-| "at (Left v) (0#ps)= at v ps"
-| "at (Right v) (Suc 0#ps)= at v ps"
-| "at (Seq v1 v2) (0#ps)= at v1 ps"
-| "at (Seq v1 v2) (Suc 0#ps)= at v2 ps"
-| "at (Stars vs) (n#ps)= at (nth vs n) ps"
-
-fun 
-  ato :: "val \<Rightarrow> nat list \<Rightarrow> val option"
-where
-  "ato v [] = Some v"
-| "ato (Left v) (0#ps)= ato v ps"
-| "ato (Right v) (Suc 0#ps)= ato v ps"
-| "ato (Seq v1 v2) (0#ps)= ato v1 ps"
-| "ato (Seq v1 v2) (Suc 0#ps)= ato v2 ps"
-| "ato (Stars vs) (n#ps)= (if (n < length vs) then ato (nth vs n) ps else None)"
-| "ato v p = None"
-
-fun Pos :: "val \<Rightarrow> (nat list) set"
-where
-  "Pos (Void) = {[]}"
-| "Pos (Char c) = {[]}"
-| "Pos (Left v) = {[]} \<union> {0#ps | ps. ps \<in> Pos v}"
-| "Pos (Right v) = {[]} \<union> {1#ps | ps. ps \<in> Pos v}"
-| "Pos (Seq v1 v2) = {[]} \<union> {0#ps | ps. ps \<in> Pos v1} \<union> {1#ps | ps. ps \<in> Pos v2}" 
-| "Pos (Stars []) = {[]}"
-| "Pos (Stars (v#vs)) = {[]} \<union> {0#ps | ps. ps \<in> Pos v} \<union> {(Suc n)#ps | n ps. n#ps \<in> Pos (Stars vs)}"
-
-lemma Pos_empty:
-  shows "[] \<in> Pos v"
-apply(induct v rule: Pos.induct)
-apply(auto)
-done
-
-lemma Pos_finite_aux:
-  assumes "\<forall>v \<in> set vs. finite (Pos v)"
-  shows "finite (Pos (Stars vs))"
-using assms
-apply(induct vs)
-apply(simp)
-apply(simp)
-apply(subgoal_tac "finite (Pos (Stars vs) - {[]})")
-apply(rule_tac f="\<lambda>l. Suc (hd l) # tl l" in finite_surj)
-apply(assumption)
-back
-apply(auto simp add: image_def)
-apply(rule_tac x="n#ps" in bexI)
-apply(simp)
-apply(simp)
-done
-
-lemma Pos_finite:
-  shows "finite (Pos v)"
-apply(induct v rule: val.induct)
-apply(auto)
-apply(simp add: Pos_finite_aux)
-done
-
-
-lemma ato_test:
-  assumes "p \<in> Pos v"
-  shows "\<exists>v'. ato v p = Some v'"
-using assms
-apply(induct v arbitrary: p rule: Pos.induct)
-apply(auto)
-apply force
-by (metis ato.simps(6) option.distinct(1))
-
-definition pflat :: "val \<Rightarrow> nat list => string option"
-where
-  "pflat v p \<equiv> (if p \<in> Pos v then Some (flat (at v p)) else None)"
-
-fun intlen :: "'a list \<Rightarrow> int"
-where
-  "intlen [] = 0"
-| "intlen (x#xs) = 1 + intlen xs"
-
-lemma inlen_bigger:
-  shows "0 \<le> intlen xs"
-apply(induct xs)
-apply(auto)
-done 
-
-lemma intlen_append:
-  shows "intlen (xs @ ys) = intlen xs + intlen ys"
-apply(induct xs arbitrary: ys)
-apply(auto)
-done 
-
-lemma intlen_length:
-  assumes "length xs < length ys"
-  shows "intlen xs < intlen ys"
-using assms
-apply(induct xs arbitrary: ys)
-apply(auto)
-apply(case_tac ys)
-apply(simp_all)
-apply (smt inlen_bigger)
-by (smt Suc_lessE intlen.simps(2) length_Suc_conv)
-
-
-definition pflat_len :: "val \<Rightarrow> nat list => int"
-where
-  "pflat_len v p \<equiv> (if p \<in> Pos v then intlen (flat (at v p)) else -1)"
-
-lemma pflat_len_simps:
-  shows "pflat_len (Seq v1 v2) (0#p) = pflat_len v1 p"
-  and   "pflat_len (Seq v1 v2) (Suc 0#p) = pflat_len v2 p"
-  and   "pflat_len (Left v) (0#p) = pflat_len v p"
-  and   "pflat_len (Left v) (Suc 0#p) = -1"
-  and   "pflat_len (Right v) (Suc 0#p) = pflat_len v p"
-  and   "pflat_len (Right v) (0#p) = -1"
-  and   "pflat_len v [] = intlen (flat v)"
-apply(auto simp add: pflat_len_def Pos_empty)
-done
-
-lemma pflat_len_Stars_simps:
-  assumes "n < length vs"
-  shows "pflat_len (Stars vs) (n#p) = pflat_len (vs!n) p"
-using assms 
-apply(induct vs arbitrary: n p)
-apply(simp)
-apply(simp)
-apply(simp add: pflat_len_def)
-apply(auto)[1]
-apply (metis at.simps(6))
-apply (metis Suc_less_eq Suc_pred)
-by (metis diff_Suc_1 less_Suc_eq_0_disj nth_Cons')
-
-
-lemma pflat_len_Stars_simps2:
-  shows "pflat_len (Stars (v#vs)) (Suc n # p) = pflat_len (Stars vs) (n#p)"
-  and   "pflat_len (Stars (v#vs)) (0 # p) = pflat_len v p"
-using assms 
-apply(simp_all add: pflat_len_def)
-done
-
-lemma Two_to_Three_aux:
-  assumes "p \<in> Pos v1 \<union> Pos v2" "pflat_len v1 p = pflat_len v2 p"
-  shows "p \<in> Pos v1 \<inter> Pos v2"
-using assms
-apply(simp add: pflat_len_def)
-apply(auto split: if_splits)
-apply (smt inlen_bigger)+
-done
-
-lemma Two_to_Three:
-  assumes "\<forall>p \<in> Pos v1 \<union> Pos v2. pflat v1 p = pflat v2 p"
-  shows "Pos v1 = Pos v2"
-using assms
-by (metis Un_iff option.distinct(1) pflat_def subsetI subset_antisym)
-
-lemma Two_to_Three_orig:
-  assumes "\<forall>p \<in> Pos v1 \<union> Pos v2. pflat_len v1 p = pflat_len v2 p"
-  shows "Pos v1 = Pos v2"
-using assms
-by (metis Un_iff inlen_bigger neg_0_le_iff_le not_one_le_zero pflat_len_def subsetI subset_antisym)
-
-lemma set_eq1:
-  assumes "insert [] A = insert [] B" "[] \<notin> A"  "[] \<notin> B"
-  shows "A = B"
-using assms
-by (simp add: insert_ident)
-
-lemma set_eq2:
-  assumes "A \<union> B = A \<union> C"
-  and "A \<inter> B = {}" "A \<inter> C = {}"  
-  shows "B = C"
-using assms
-using Un_Int_distrib sup_bot.left_neutral sup_commute by blast
-
-
-
-lemma Three_to_One:
-  assumes "\<turnstile> v1 : r" "\<turnstile> v2 : r" 
-  and "Pos v1 = Pos v2" 
-  shows "v1 = v2"
-using assms
-apply(induct v1 arbitrary: r v2 rule: Pos.induct)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(clarify)
-apply(simp add: insert_ident)
-apply(drule_tac x="r1a" in meta_spec)
-apply(drule_tac x="v1a" in meta_spec)
-apply(simp)
-apply(drule_tac meta_mp)
-thm subset_antisym
-apply(rule subset_antisym)
-apply(auto)[3]
-apply(clarify)
-apply(simp add: insert_ident)
-using Pos_empty apply blast
-apply(erule Prf.cases)
-apply(simp_all)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(clarify)
-apply(simp add: insert_ident)
-using Pos_empty apply blast
-apply(simp add: insert_ident)
-apply(drule_tac x="r2a" in meta_spec)
-apply(drule_tac x="v2b" in meta_spec)
-apply(simp)
-apply(drule_tac meta_mp)
-apply(rule subset_antisym)
-apply(auto)[3]
-apply(erule Prf.cases)
-apply(simp_all)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(simp add: insert_ident)
-apply(clarify)
-apply(drule_tac x="r1a" in meta_spec)
-apply(drule_tac x="r2a" in meta_spec)
-apply(drule_tac x="v1b" in meta_spec)
-apply(drule_tac x="v2c" in meta_spec)
-apply(simp)
-apply(drule_tac meta_mp)
-apply(rule subset_antisym)
-apply(rule subsetI)
-apply(subgoal_tac "0 # x \<in> {0 # ps |ps. ps \<in> Pos v1a}")
-prefer 2
-apply(auto)[1]
-apply(subgoal_tac "0 # x \<in> {0 # ps |ps. ps \<in> Pos v1b}  \<union> {Suc 0 # ps |ps. ps \<in> Pos v2c}")
-prefer 2
-apply (metis (no_types, lifting) Un_iff)
-apply(simp)
-apply(rule subsetI)
-apply(subgoal_tac "0 # x \<in> {0 # ps |ps. ps \<in> Pos v1b}")
-prefer 2
-apply(auto)[1]
-apply(subgoal_tac "0 # x \<in> {0 # ps |ps. ps \<in> Pos v1a}  \<union> {Suc 0 # ps |ps. ps \<in> Pos v2b}")
-prefer 2
-apply (metis (no_types, lifting) Un_iff)
-apply(simp (no_asm_use))
-apply(simp)
-apply(drule_tac meta_mp)
-apply(rule subset_antisym)
-apply(rule subsetI)
-apply(subgoal_tac "Suc 0 # x \<in> {Suc 0 # ps |ps. ps \<in> Pos v2b}")
-prefer 2
-apply(auto)[1]
-apply(subgoal_tac "Suc 0 # x \<in> {0 # ps |ps. ps \<in> Pos v1b}  \<union> {Suc 0 # ps |ps. ps \<in> Pos v2c}")
-prefer 2
-apply (metis (no_types, lifting) Un_iff)
-apply(simp)
-apply(rule subsetI)
-apply(subgoal_tac "Suc 0 # x \<in> {Suc 0 # ps |ps. ps \<in> Pos v2c}")
-prefer 2
-apply(auto)[1]
-apply(subgoal_tac "Suc 0 # x \<in> {0 # ps |ps. ps \<in> Pos v1b}  \<union> {Suc 0 # ps |ps. ps \<in> Pos v2b}")
-prefer 2
-apply (metis (no_types, lifting) Un_iff)
-apply(simp (no_asm_use))
-apply(simp)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(auto)[1]
-using Pos_empty apply fastforce
-apply(erule Prf.cases)
-apply(simp_all)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(auto)[1]
-using Pos_empty apply fastforce
-apply(clarify)
-apply(simp add: insert_ident)
-apply(drule_tac x="rb" in meta_spec)
-apply(drule_tac x="STAR rb" in meta_spec)
-apply(drule_tac x="vb" in meta_spec)
-apply(drule_tac x="Stars vsb" in meta_spec)
-apply(simp)
-apply(drule_tac meta_mp)
-apply(rule subset_antisym)
-apply(rule subsetI)
-apply(subgoal_tac "0 # x \<in> {0 # ps |ps. ps \<in> Pos va}")
-prefer 2
-apply(auto)[1]
-apply(subgoal_tac "0 # x \<in> {0 # ps |ps. ps \<in> Pos vb} \<union> {Suc n # ps |n ps. n # ps \<in> Pos (Stars vsb)}")
-prefer 2
-apply (metis (no_types, lifting) Un_iff)
-apply(simp)
-apply(rule subsetI)
-apply(subgoal_tac "0 # x \<in> {0 # ps |ps. ps \<in> Pos vb}")
-prefer 2
-apply(auto)[1]
-apply(subgoal_tac "0 # x \<in> {0 # ps |ps. ps \<in> Pos va} \<union> {Suc n # ps |n ps. n # ps \<in> Pos (Stars vsa)}")
-prefer 2
-apply (metis (no_types, lifting) Un_iff)
-apply(simp (no_asm_use))
-apply(simp)
-apply(drule_tac meta_mp)
-apply(rule subset_antisym)
-apply(rule subsetI)
-apply(case_tac vsa)
-apply(simp)
-apply (simp add: Pos_empty)
-apply(simp)
-apply(clarify)
-apply(erule disjE)
-apply (simp add: Pos_empty)
-apply(erule disjE)
-apply(clarify)
-apply(subgoal_tac 
-  "Suc 0 # ps \<in> {Suc n # ps |n ps. n = 0 \<and> ps \<in> Pos a \<or> (\<exists>na. n = Suc na \<and> na # ps \<in> Pos (Stars list))}")
-prefer 2
-apply blast
-apply(subgoal_tac "Suc 0 # ps \<in> {0 # ps |ps. ps \<in> Pos vb} \<union>  {Suc n # ps |n ps. n # ps \<in> Pos (Stars vsb)}")
-prefer 2
-apply (metis (no_types, lifting) Un_iff)
-apply(simp)
-apply(clarify)
-apply(subgoal_tac 
-  "Suc (Suc n) # ps \<in> {Suc n # ps |n ps. n = 0 \<and> ps \<in> Pos a \<or> (\<exists>na. n = Suc na \<and> na # ps \<in> Pos (Stars list))}")
-prefer 2
-apply blast
-apply(subgoal_tac "Suc (Suc n) # ps \<in> {0 # ps |ps. ps \<in> Pos vb} \<union> {Suc n # ps |n ps. n # ps \<in> Pos (Stars vsb)}")
-prefer 2
-apply (metis (no_types, lifting) Un_iff)
-apply(simp)
-apply(rule subsetI)
-apply(case_tac vsb)
-apply(simp)
-apply (simp add: Pos_empty)
-apply(simp)
-apply(clarify)
-apply(erule disjE)
-apply (simp add: Pos_empty)
-apply(erule disjE)
-apply(clarify)
-apply(subgoal_tac 
-  "Suc 0 # ps \<in> {Suc n # ps |n ps. n = 0 \<and> ps \<in> Pos a \<or> (\<exists>na. n = Suc na \<and> na # ps \<in> Pos (Stars list))}")
-prefer 2
-apply(simp)
-apply(subgoal_tac "Suc 0 # ps \<in> {Suc n # ps |n ps. n # ps \<in> Pos (Stars vsa)}")
-apply blast
-using list.inject apply blast
-apply(clarify)
-apply(subgoal_tac 
-  "Suc (Suc n) # ps \<in> {Suc n # ps |n ps. n = 0 \<and> ps \<in> Pos a \<or> (\<exists>na. n = Suc na \<and> na # ps \<in> Pos (Stars list))}")
-prefer 2
-apply(simp)
-apply(subgoal_tac "Suc (Suc n) # ps \<in> {0 # ps |ps. ps \<in> Pos vb} \<union> {Suc n # ps |n ps. n # ps \<in> Pos (Stars vsa)}")
-prefer 2
-apply (metis (no_types, lifting) Un_iff)
-apply(simp (no_asm_use))
-apply(simp)
-done
-
-definition prefix_list:: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" ("_ \<sqsubseteq>pre _")
-where
-  "ps1 \<sqsubseteq>pre ps2 \<equiv> (\<exists>ps'. ps1 @ps' = ps2)"
-
-definition sprefix_list:: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" ("_ \<sqsubset>spre _")
-where
-  "ps1 \<sqsubset>spre ps2 \<equiv> (ps1 \<sqsubseteq>pre ps2 \<and> ps1 \<noteq> ps2)"
-
-inductive lex_lists :: "nat list \<Rightarrow> nat list \<Rightarrow> bool" ("_ \<sqsubset>lex _")
-where
-  "[] \<sqsubset>lex p#ps"
-| "ps1 \<sqsubset>lex ps2 \<Longrightarrow> (p#ps1) \<sqsubset>lex (p#ps2)"
-| "p1 < p2 \<Longrightarrow> (p1#ps1) \<sqsubset>lex (p2#ps2)"
-
-lemma lex_irrfl:
-  fixes ps1 ps2 :: "nat list"
-  assumes "ps1 \<sqsubset>lex ps2"
-  shows "ps1 \<noteq> ps2"
-using assms
-apply(induct rule: lex_lists.induct)
-apply(auto)
-done
-
-lemma lex_append:
-  assumes "ps2 \<noteq> []"  
-  shows "ps \<sqsubset>lex ps @ ps2"
-using assms
-apply(induct ps)
-apply(auto intro: lex_lists.intros)
-apply(case_tac ps2)
-apply(simp)
-apply(simp)
-apply(auto intro: lex_lists.intros)
-done
-
-lemma lexordp_simps [simp]:
-  fixes xs ys :: "nat list"
-  shows "[] \<sqsubset>lex ys = (ys \<noteq> [])"
-  and   "xs \<sqsubset>lex [] = False"
-  and   "(x # xs) \<sqsubset>lex (y # ys) \<longleftrightarrow> (x < y \<or> (\<not> y < x \<and> xs \<sqsubset>lex ys))"
-apply -
-apply (metis lex_append lex_lists.simps list.simps(3))
-using lex_lists.cases apply blast
-using lex_lists.cases lex_lists.intros(2) lex_lists.intros(3) not_less_iff_gr_or_eq by fastforce
-
-lemma lex_append_cancel [simp]:
-  fixes ps ps1 ps2 :: "nat list"
-  shows "ps @ ps1 \<sqsubset>lex ps @ ps2 \<longleftrightarrow> ps1 \<sqsubset>lex ps2"
-apply(induct ps)
-apply(auto)
-done
-
-lemma lex_trans:
-  fixes ps1 ps2 ps3 :: "nat list"
-  assumes "ps1 \<sqsubset>lex ps2" "ps2 \<sqsubset>lex ps3"
-  shows "ps1 \<sqsubset>lex ps3"
-using assms
-apply(induct arbitrary: ps3 rule: lex_lists.induct)
-apply(erule lex_lists.cases)
-apply(simp_all)
-apply(rotate_tac 2)
-apply(erule lex_lists.cases)
-apply(simp_all)
-apply(erule lex_lists.cases)
-apply(simp_all)
-done
-
-lemma trichotomous_aux:
-  fixes p q :: "nat list"
-  assumes "p \<sqsubset>lex q" "p \<noteq> q"
-  shows "\<not>(q \<sqsubset>lex p)"
-using assms
-apply(induct rule: lex_lists.induct)
-apply(auto)
-done
-
-lemma trichotomous_aux2:
-  fixes p q :: "nat list"
-  assumes "p \<sqsubset>lex q" "q \<sqsubset>lex p"
-  shows "False"
-using assms
-apply(induct rule: lex_lists.induct)
-apply(auto)
-done
-
-lemma trichotomous:
-  fixes p q :: "nat list"
-  shows "p = q \<or> p \<sqsubset>lex q \<or> q \<sqsubset>lex p"
-apply(induct p arbitrary: q)
-apply(auto)
-apply(case_tac q)
-apply(auto)
-done
-
-definition dpos :: "val \<Rightarrow> val \<Rightarrow> nat list \<Rightarrow> bool" 
-  where
-  "dpos v1 v2 p \<equiv> (p \<in> Pos v1 \<union> Pos v2) \<and> (p \<notin> Pos v1 \<inter> Pos v2)"
-
-definition
-  "DPos v1 v2 \<equiv> {p. dpos v1 v2 p}"
-
-lemma outside_lemma:
-  assumes "p \<notin> Pos v1 \<union> Pos v2"
-  shows "pflat_len v1 p = pflat_len v2 p"
-using assms
-apply(auto simp add: pflat_len_def)
-done
-
-lemma dpos_lemma_aux:
-  assumes "p \<in> Pos v1 \<union> Pos v2"
-  and "pflat_len v1 p = pflat_len v2 p"
-  shows "p \<in> Pos v1 \<inter> Pos v2"
-using assms
-apply(auto simp add: pflat_len_def)
-apply (smt inlen_bigger)
-apply (smt inlen_bigger)
-done
-
-lemma dpos_lemma:
-  assumes "p \<in> Pos v1 \<union> Pos v2"
-  and "pflat_len v1 p = pflat_len v2 p"
-  shows "\<not>dpos v1 v2 p"
-using assms
-apply(auto simp add: dpos_def dpos_lemma_aux)
-using dpos_lemma_aux apply auto[1]
-using dpos_lemma_aux apply auto[1]
-done
-
-lemma dpos_lemma2:
-  assumes "p \<in> Pos v1 \<union> Pos v2"
-  and "dpos v1 v2 p"
-  shows "pflat_len v1 p \<noteq> pflat_len v2 p"
-using assms
-using dpos_lemma by blast
-
-lemma DPos_lemma:
-  assumes "p \<in> DPos v1 v2"
-  shows "pflat_len v1 p \<noteq> pflat_len v2 p"
-using assms
-unfolding DPos_def 
-apply(auto simp add: pflat_len_def dpos_def)
-apply (smt inlen_bigger)
-by (smt inlen_bigger)
-
-
-definition val_ord:: "val \<Rightarrow> nat list \<Rightarrow> val \<Rightarrow> bool" ("_ \<sqsubset>val _ _")
-where
-  "v1 \<sqsubset>val p v2 \<equiv> (p \<in> Pos v1 \<and> pflat_len v1 p > pflat_len v2 p \<and>
-                   (\<forall>q \<in> Pos v1 \<union> Pos v2. q \<sqsubset>lex p \<longrightarrow> pflat_len v1 q = pflat_len v2 q))"
-
-
-definition val_ord_ex:: "val \<Rightarrow> val \<Rightarrow> bool" ("_ :\<sqsubset>val _")
-where
-  "v1 :\<sqsubset>val v2 \<equiv> (\<exists>p. v1 \<sqsubset>val p v2)"
-
-definition val_ord_ex1:: "val \<Rightarrow> val \<Rightarrow> bool" ("_ :\<sqsubseteq>val _")
-where
-  "v1 :\<sqsubseteq>val v2 \<equiv> v1 :\<sqsubset>val v2 \<or> v1 = v2"
-
-lemma val_ord_shorterI:
-  assumes "length (flat v') < length (flat v)"
-  shows "v :\<sqsubset>val v'" 
-using assms(1)
-apply(subst val_ord_ex_def)
-apply(rule_tac x="[]" in exI)
-apply(subst val_ord_def)
-apply(rule conjI)
-apply (simp add: Pos_empty)
-apply(rule conjI)
-apply(simp add: pflat_len_simps)
-apply (simp add: intlen_length)
-apply(simp)
-done
-
-lemma val_ord_spre:
-  assumes "(flat v') \<sqsubset>spre (flat v)"
-  shows "v :\<sqsubset>val v'" 
-using assms(1)
-apply(rule_tac val_ord_shorterI)
-apply(simp add: sprefix_list_def prefix_list_def)
-apply(auto)
-apply(case_tac ps')
-apply(auto)
-by (metis append_eq_conv_conj drop_all le_less_linear neq_Nil_conv)
-
-
-lemma val_ord_ALTI:
-  assumes "v \<sqsubset>val p v'" "flat v = flat v'"
-  shows "(Left v) \<sqsubset>val (0#p) (Left v')" 
-using assms(1)
-apply(subst (asm) val_ord_def)
-apply(erule conjE)
-apply(subst val_ord_def)
-apply(rule conjI)
-apply(simp)
-apply(rule conjI)
-apply(simp add: pflat_len_simps)
-apply(rule ballI)
-apply(rule impI)
-apply(simp only: Pos.simps)
-apply(auto)[1]
-using assms(2)
-apply(simp add: pflat_len_simps)
-apply(auto simp add: pflat_len_simps)[2]
-done
-
-lemma val_ord_ALTI2:
-  assumes "v \<sqsubset>val p v'" "flat v = flat v'"
-  shows "(Right v) \<sqsubset>val (1#p) (Right v')" 
-using assms(1)
-apply(subst (asm) val_ord_def)
-apply(erule conjE)
-apply(subst val_ord_def)
-apply(rule conjI)
-apply(simp)
-apply(rule conjI)
-apply(simp add: pflat_len_simps)
-apply(rule ballI)
-apply(rule impI)
-apply(simp only: Pos.simps)
-apply(auto)[1]
-using assms(2)
-apply(simp add: pflat_len_simps)
-apply(auto simp add: pflat_len_simps)[2]
-done
-
-lemma val_ord_ALTE:
-  assumes "(Left v1) \<sqsubset>val (p # ps) (Left v2)"
-  shows "p = 0 \<and> v1 \<sqsubset>val ps v2"
-using assms(1)
-apply(simp add: val_ord_def)
-apply(auto simp add: pflat_len_simps)
-apply (metis (no_types, hide_lams) assms lex_lists.intros(2) outside_lemma pflat_len_simps(3) val_ord_def)
-by (metis (no_types, hide_lams) assms lex_lists.intros(2) outside_lemma pflat_len_simps(3) val_ord_def)
-
-lemma val_ord_ALTE2:
-  assumes "(Right v1) \<sqsubset>val (p # ps) (Right v2)"
-  shows "p = 1 \<and> v1 \<sqsubset>val ps v2"
-using assms(1)
-apply(simp add: val_ord_def)
-apply(auto simp add: pflat_len_simps)
-apply (metis (no_types, hide_lams) assms lex_lists.intros(2) outside_lemma pflat_len_simps(5) val_ord_def)
-by (metis (no_types, hide_lams) assms lex_lists.intros(2) outside_lemma pflat_len_simps(5) val_ord_def)
-
-lemma val_ord_STARI:
-  assumes "v1 \<sqsubset>val p v2" "flat (Stars (v1#vs1)) = flat (Stars (v2#vs2))"
-  shows "(Stars (v1#vs1)) \<sqsubset>val (0#p) (Stars (v2#vs2))" 
-using assms(1)
-apply(subst (asm) val_ord_def)
-apply(erule conjE)
-apply(subst val_ord_def)
-apply(rule conjI)
-apply(simp)
-apply(rule conjI)
-apply(subst pflat_len_Stars_simps)
-apply(simp)
-apply(subst pflat_len_Stars_simps)
-apply(simp)
-apply(simp)
-apply(rule ballI)
-apply(rule impI)
-apply(simp)
-apply(auto)
-using assms(2)
-apply(simp add: pflat_len_simps)
-apply(auto simp add: pflat_len_Stars_simps)
-done
-
-lemma val_ord_STARI2:
-  assumes "(Stars vs1) \<sqsubset>val n#p (Stars vs2)" "flat (Stars vs1) = flat (Stars vs2)"
-  shows "(Stars (v#vs1)) \<sqsubset>val (Suc n#p) (Stars (v#vs2))" 
-using assms(1)
-apply(subst (asm) val_ord_def)
-apply(erule conjE)+
-apply(subst val_ord_def)
-apply(rule conjI)
-apply(simp)
-apply(rule conjI)
-apply(case_tac vs1)
-apply(simp)
-apply(simp)
-apply(auto)[1]
-apply(case_tac vs2)
-apply(simp)
-apply (simp add: pflat_len_def)
-apply(simp)
-apply(auto)[1]
-apply (simp add: pflat_len_Stars_simps)
-using pflat_len_def apply auto[1]
-apply(rule ballI)
-apply(rule impI)
-apply(simp)
-using assms(2)
-apply(auto)
-apply (simp add: pflat_len_simps(7))
-apply (simp add: pflat_len_Stars_simps)
-using assms(2)
-apply(auto simp add: pflat_len_def)[1]
-apply force
-apply force
-apply(auto simp add: pflat_len_def)[1]
-apply force
-apply force
-apply(auto simp add: pflat_len_def)[1]
-apply(auto simp add: pflat_len_def)[1]
-apply force
-apply force
-apply(auto simp add: pflat_len_def)[1]
-apply force
-apply force
-done
-
-
-lemma val_ord_SEQI:
-  assumes "v1 \<sqsubset>val p v1'" "flat (Seq v1 v2) = flat (Seq v1' v2')"
-  shows "(Seq v1 v2) \<sqsubset>val (0#p) (Seq v1' v2')" 
-using assms(1)
-apply(subst (asm) val_ord_def)
-apply(erule conjE)
-apply(subst val_ord_def)
-apply(rule conjI)
-apply(simp)
-apply(rule conjI)
-apply(simp add: pflat_len_simps)
-apply(rule ballI)
-apply(rule impI)
-apply(simp only: Pos.simps)
-apply(auto)[1]
-apply(simp add: pflat_len_simps)
-using assms(2)
-apply(simp)
-apply(auto simp add: pflat_len_simps)[2]
-done
-
-
-lemma val_ord_SEQI2:
-  assumes "v2 \<sqsubset>val p v2'" "flat v2 = flat v2'"
-  shows "(Seq v v2) \<sqsubset>val (1#p) (Seq v v2')" 
-using assms(1)
-apply(subst (asm) val_ord_def)
-apply(erule conjE)+
-apply(subst val_ord_def)
-apply(rule conjI)
-apply(simp)
-apply(rule conjI)
-apply(simp add: pflat_len_simps)
-apply(rule ballI)
-apply(rule impI)
-apply(simp only: Pos.simps)
-apply(auto)
-apply(auto simp add: pflat_len_def intlen_append)
-apply(auto simp add: assms(2))
-done
-
-lemma val_ord_SEQE_0:
-  assumes "(Seq v1 v2) \<sqsubset>val 0#p (Seq v1' v2')" 
-  shows "v1 \<sqsubset>val p v1'"
-using assms(1)
-apply(simp add: val_ord_def val_ord_ex_def)
-apply(auto)[1]
-apply(simp add: pflat_len_simps)
-apply(simp add: val_ord_def pflat_len_def)
-apply(auto)[1]
-apply(drule_tac x="0#q" in bspec)
-apply(simp)
-apply(simp)
-apply(drule_tac x="0#q" in bspec)
-apply(simp)
-apply(simp)
-apply(drule_tac x="0#q" in bspec)
-apply(simp)
-apply(simp)
-apply(simp add: val_ord_def pflat_len_def)
-apply(auto)[1]
-done
-
-lemma val_ord_SEQE_1:
-  assumes "(Seq v1 v2) \<sqsubset>val (Suc 0)#p (Seq v1' v2')" 
-  shows "v2 \<sqsubset>val p v2'"
-using assms(1)
-apply(simp add: val_ord_def pflat_len_def)
-apply(auto)[1]
-apply(drule_tac x="1#q" in bspec)
-apply(simp)
-apply(simp)
-apply(drule_tac x="1#q" in bspec)
-apply(simp)
-apply(simp)
-apply(drule_tac x="1#q" in bspec)
-apply(simp)
-apply(auto)[1]
-apply(drule_tac x="1#q" in bspec)
-apply(simp)
-apply(auto)
-apply(simp add: intlen_append)
-apply force
-apply(simp add: intlen_append)
-apply force
-apply(simp add: intlen_append)
-apply force
-apply(simp add: intlen_append)
-apply force
-done
-
-lemma val_ord_SEQE_2:
-  assumes "(Seq v1 v2) \<sqsubset>val (Suc 0)#p (Seq v1' v2')" 
-  and "\<turnstile> v1 : r" "\<turnstile> v1' : r" 
-  shows "v1 = v1'"
-proof -
-  have "\<forall>q \<in> Pos v1 \<union> Pos v1'. 0 # q \<sqsubset>lex 1#p \<longrightarrow> pflat_len v1 q = pflat_len v1' q"
-  using assms(1) 
-  apply(simp add: val_ord_def)
-  apply(rule ballI)
-  apply(clarify)
-  apply(drule_tac x="0#q" in bspec)
-  apply(auto)[1]
-  apply(simp add: pflat_len_simps)
-  done
-  then have "Pos v1 = Pos v1'"
-  apply(rule_tac Two_to_Three_orig)
-  apply(rule ballI)
-  apply(drule_tac x="pa" in bspec)
-  apply(simp)
-  apply(simp)
-  done
-  then show "v1 = v1'" 
-  apply(rule_tac Three_to_One)
-  apply(rule assms)
-  apply(rule assms)
-  apply(simp)
-  done
-qed
-
-lemma val_ord_SEQ:
-  assumes "(Seq v1 v2) :\<sqsubset>val (Seq v1' v2')" 
-  and "flat (Seq v1 v2) = flat (Seq v1' v2')"
-  and "\<turnstile> v1 : r" "\<turnstile> v1' : r" 
-  shows "(v1 :\<sqsubset>val v1') \<or> (v1 = v1' \<and> (v2 :\<sqsubset>val v2'))"
-using assms(1)
-apply(subst (asm) val_ord_ex_def)
-apply(erule exE)
-apply(simp only: val_ord_def)
-apply(simp)
-apply(erule conjE)+
-apply(erule disjE)
-prefer 2
-apply(erule disjE)
-apply(erule exE)
-apply(rule disjI1)
-apply(simp)
-apply(subst val_ord_ex_def)
-apply(rule_tac x="ps" in exI)
-apply(rule val_ord_SEQE_0)
-apply(simp add: val_ord_def)
-apply(erule exE)
-apply(rule disjI2)
-apply(rule conjI)
-thm val_ord_SEQE_1
-apply(rule_tac val_ord_SEQE_2)
-apply(auto simp add: val_ord_def)[3]
-apply(rule assms(3))
-apply(rule assms(4))
-apply(subst val_ord_ex_def)
-apply(rule_tac x="ps" in exI)
-apply(rule_tac val_ord_SEQE_1)
-apply(auto simp add: val_ord_def)[1]
-apply(simp)
-using assms(2)
-apply(simp add: pflat_len_simps)
-done
-
-
-lemma val_ord_ex_trans:
-  assumes "v1 :\<sqsubset>val v2" "v2 :\<sqsubset>val v3"
-  shows "v1 :\<sqsubset>val v3"
-using assms
-unfolding val_ord_ex_def
-apply(clarify)
-apply(subgoal_tac "p = pa \<or> p \<sqsubset>lex pa \<or> pa \<sqsubset>lex p")
-prefer 2
-apply(rule trichotomous)
-apply(erule disjE)
-apply(simp)
-apply(rule_tac x="pa" in exI)
-apply(subst val_ord_def)
-apply(rule conjI)
-apply(simp add: val_ord_def)
-apply(auto)[1]
-apply(simp add: val_ord_def)
-apply(simp add: val_ord_def)
-apply(auto)[1]
-using outside_lemma apply blast
-apply(simp add: val_ord_def)
-apply(auto)[1]
-using outside_lemma apply force
-apply auto[1]
-apply(simp add: val_ord_def)
-apply(auto)[1]
-apply (metis (no_types, hide_lams) lex_trans outside_lemma)
-apply(simp add: val_ord_def)
-apply(auto)[1]
-by (metis IntE Two_to_Three_aux UnCI lex_trans outside_lemma)
-
-
-definition fdpos :: "val \<Rightarrow> val \<Rightarrow> nat list \<Rightarrow> bool" 
-where
-  "fdpos v1 v2 p \<equiv>  ({q. q \<sqsubset>lex p} \<inter> DPos v1 v2 = {})"
-
-
-lemma pos_append:
-  assumes "p @ q \<in> Pos v"
-  shows "q \<in> Pos (at v p)"
-using assms
-apply(induct arbitrary: p q rule: Pos.induct) 
-apply(simp_all)
-apply(auto)[1]
-apply(simp add: append_eq_Cons_conv)
-apply(auto)[1]
-apply(auto)[1]
-apply(simp add: append_eq_Cons_conv)
-apply(auto)[1]
-apply(auto)[1]
-apply(simp add: append_eq_Cons_conv)
-apply(auto)[1]
-apply(simp add: append_eq_Cons_conv)
-apply(auto)[1]
-apply(auto)[1]
-apply(simp add: append_eq_Cons_conv)
-apply(auto)[1]
-apply(simp add: append_eq_Cons_conv)
-apply(auto)[1]
-by (metis append_Cons at.simps(6))
-
-
-lemma Pos_pre:
-  assumes "p \<in> Pos v" "q \<sqsubseteq>pre p"
-  shows "q \<in> Pos v"
-using assms
-apply(induct v arbitrary: p q rule: Pos.induct)
-apply(simp_all add: prefix_list_def)
-apply (meson append_eq_Cons_conv append_is_Nil_conv)
-apply (meson append_eq_Cons_conv append_is_Nil_conv)
-apply (metis (no_types, lifting) Cons_eq_append_conv append_is_Nil_conv)
-apply(auto)
-apply (meson append_eq_Cons_conv)
-apply(simp add: append_eq_Cons_conv)
-apply(auto)
-done
-
-lemma lex_lists_order:
-  assumes "q' \<sqsubset>lex q" "\<not>(q' \<sqsubseteq>pre q)"
-  shows "\<not>(q \<sqsubset>lex q')"
-using assms
-apply(induct rule: lex_lists.induct)
-apply(simp add: prefix_list_def)
-apply(auto)
-using trichotomous_aux2 by auto
-
-lemma lex_appendL:
-  assumes "q \<sqsubset>lex p" 
-  shows "q \<sqsubset>lex p @ q'"
-using assms
-apply(induct arbitrary: q' rule: lex_lists.induct)
-apply(auto)
-done
-
-
-inductive 
-  CPrf :: "val \<Rightarrow> rexp \<Rightarrow> bool" ("\<Turnstile> _ : _" [100, 100] 100)
-where
- "\<lbrakk>\<Turnstile> v1 : r1; \<Turnstile> v2 : r2\<rbrakk> \<Longrightarrow> \<Turnstile>  Seq v1 v2 : SEQ r1 r2"
-| "\<Turnstile> v1 : r1 \<Longrightarrow> \<Turnstile> Left v1 : ALT r1 r2"
-| "\<Turnstile> v2 : r2 \<Longrightarrow> \<Turnstile> Right v2 : ALT r1 r2"
-| "\<Turnstile> Void : ONE"
-| "\<Turnstile> Char c : CHAR c"
-| "\<Turnstile> Stars [] : STAR r"
-| "\<lbrakk>\<Turnstile> v : r; flat v \<noteq> []; \<Turnstile> Stars vs : STAR r\<rbrakk> \<Longrightarrow> \<Turnstile> Stars (v # vs) : STAR r"
-
-lemma Prf_CPrf:
-  assumes "\<Turnstile> v : r"
-  shows "\<turnstile> v : r"
-using assms
-apply(induct)
-apply(auto intro: Prf.intros)
-done
-
-definition
-  "CPT r s = {v. flat v = s \<and> \<Turnstile> v : r}"
-
-definition
-  "CPTpre r s = {v. \<exists>s'. flat v @ s' = s \<and> \<Turnstile> v : r}"
-
-lemma CPT_CPTpre_subset:
-  shows "CPT r s \<subseteq> CPTpre r s"
-apply(auto simp add: CPT_def CPTpre_def)
-done
-
-
-lemma CPTpre_subsets:
-  "CPTpre ZERO s = {}"
-  "CPTpre ONE s \<subseteq> {Void}"
-  "CPTpre (CHAR c) s \<subseteq> {Char c}"
-  "CPTpre (ALT r1 r2) s \<subseteq> Left ` CPTpre r1 s \<union> Right ` CPTpre r2 s"
-  "CPTpre (SEQ r1 r2) s \<subseteq> {Seq v1 v2 | v1 v2. v1 \<in> CPTpre r1 s \<and> v2 \<in> CPTpre r2 (drop (length (flat v1)) s)}"
-  "CPTpre (STAR r) s \<subseteq> {Stars []} \<union>
-    {Stars (v#vs) | v vs. v \<in> CPTpre r s \<and> flat v \<noteq> [] \<and> Stars vs \<in> CPTpre (STAR r) (drop (length (flat v)) s)}"
-  "CPTpre (STAR r) [] = {Stars []}"
-apply(auto simp add: CPTpre_def)
-apply(erule CPrf.cases)
-apply(simp_all)
-apply(erule CPrf.cases)
-apply(simp_all)
-apply(erule CPrf.cases)
-apply(simp_all)
-apply(erule CPrf.cases)
-apply(simp_all)
-apply(erule CPrf.cases)
-apply(simp_all)
-apply(erule CPrf.cases)
-apply(simp_all)
-apply(erule CPrf.cases)
-apply(simp_all)
-apply(rule CPrf.intros)
-done
-
-
-lemma CPTpre_simps:
-  shows "CPTpre ONE s = {Void}"
-  and   "CPTpre (CHAR c) (d#s) = (if c = d then {Char c} else {})"
-  and   "CPTpre (ALT r1 r2) s = Left ` CPTpre r1 s \<union> Right ` CPTpre r2 s"
-  and   "CPTpre (SEQ r1 r2) s = 
-          {Seq v1 v2 | v1 v2. v1 \<in> CPTpre r1 s \<and> v2 \<in> CPTpre r2 (drop (length (flat v1)) s)}"
-apply -
-apply(rule subset_antisym)
-apply(rule CPTpre_subsets)
-apply(auto simp add: CPTpre_def intro: "CPrf.intros")[1]
-apply(case_tac "c = d")
-apply(simp)
-apply(rule subset_antisym)
-apply(rule CPTpre_subsets)
-apply(auto simp add: CPTpre_def intro: CPrf.intros)[1]
-apply(simp)
-apply(auto simp add: CPTpre_def intro: CPrf.intros)[1]
-apply(erule CPrf.cases)
-apply(simp_all)
-apply(rule subset_antisym)
-apply(rule CPTpre_subsets)
-apply(auto simp add: CPTpre_def intro: CPrf.intros)[1]
-apply(rule subset_antisym)
-apply(rule CPTpre_subsets)
-apply(auto simp add: CPTpre_def intro: CPrf.intros)[1]
-done
-
-lemma CPT_simps:
-  shows "CPT ONE s = (if s = [] then {Void} else {})"
-  and   "CPT (CHAR c) [d] = (if c = d then {Char c} else {})"
-  and   "CPT (ALT r1 r2) s = Left ` CPT r1 s \<union> Right ` CPT r2 s"
-  and   "CPT (SEQ r1 r2) s = 
-          {Seq v1 v2 | v1 v2 s1 s2. s1 @ s2 = s \<and> v1 \<in> CPT r1 s1 \<and> v2 \<in> CPT r2 s2}"
-apply -
-apply(rule subset_antisym)
-apply(auto simp add: CPT_def)[1]
-apply(erule CPrf.cases)
-apply(simp_all)[7]
-apply(erule CPrf.cases)
-apply(simp_all)[7]
-apply(auto simp add: CPT_def intro: CPrf.intros)[1]
-apply(auto simp add: CPT_def intro: CPrf.intros)[1]
-apply(erule CPrf.cases)
-apply(simp_all)[7]
-apply(erule CPrf.cases)
-apply(simp_all)[7]
-apply(auto simp add: CPT_def image_def intro: CPrf.intros)[1]
-apply(erule CPrf.cases)
-apply(simp_all)[7]
-apply(clarify)
-apply blast
-apply(auto simp add: CPT_def image_def intro: CPrf.intros)[1]
-apply(erule CPrf.cases)
-apply(simp_all)[7]
-done
-
-lemma CPTpre_SEQ:
-  assumes "v \<in> CPTpre (SEQ r1 r2) s"
-  shows "\<exists>s'. flat v = s' \<and> (s' \<sqsubseteq>pre s) \<and> s' \<in> L (SEQ r1 r2)"
-using assms
-apply(simp add: CPTpre_simps)
-apply(auto simp add: CPTpre_def)
-apply (simp add: prefix_list_def)
-by (metis L.simps(4) L_flat_Prf1 Prf.intros(1) Prf_CPrf flat.simps(5))
-
-lemma Cond_prefix:
-  assumes "\<forall>s\<^sub>3. s1 @ s\<^sub>3 \<in> L r1 \<longrightarrow> s\<^sub>3 = [] \<or> (\<forall>s\<^sub>4. s1 @ s\<^sub>3 @ s\<^sub>4 \<sqsubseteq>pre s1 @ s2 \<longrightarrow> s\<^sub>4 \<notin> L r2)"
-  and "t1 \<in> L r1" "t2 \<in> L r2" "t1 @ t2 \<sqsubseteq>pre s1 @ s2"
-  shows "t1 \<sqsubseteq>pre s1"
-using assms
-apply(auto simp add: Sequ_def prefix_list_def append_eq_append_conv2)
-done
-
-
-
-lemma test: 
-  assumes "finite A"
-  shows "finite {vs. Stars vs \<in> A}"
-using assms
-apply(induct A)
-apply(simp)
-apply(auto)
-apply(case_tac x)
-apply(simp_all)
-done
-
-lemma CPTpre_STAR_finite:
-  assumes "\<And>s. finite (CPTpre r s)"
-  shows "finite (CPTpre (STAR r) s)"
-apply(induct s rule: length_induct)
-apply(case_tac xs)
-apply(simp)
-apply(simp add: CPTpre_subsets)
-apply(rule finite_subset)
-apply(rule CPTpre_subsets)
-apply(simp)
-apply(rule_tac B="(\<lambda>(v, vs). Stars (v#vs)) ` {(v, vs). v \<in> CPTpre r (a#list) \<and> flat v \<noteq> [] \<and> Stars vs \<in> CPTpre (STAR r) (drop (length (flat v)) (a#list))}" in finite_subset)
-apply(auto)[1]
-apply(rule finite_imageI)
-apply(simp add: Collect_case_prod_Sigma)
-apply(rule finite_SigmaI)
-apply(rule assms)
-apply(case_tac "flat v = []")
-apply(simp)
-apply(drule_tac x="drop (length (flat v)) (a # list)" in spec)
-apply(simp)
-apply(auto)[1]
-apply(rule test)
-apply(simp)
-done
-
-lemma CPTpre_finite:
-  shows "finite (CPTpre r s)"
-apply(induct r arbitrary: s)
-apply(simp add: CPTpre_subsets)
-apply(rule finite_subset)
-apply(rule CPTpre_subsets)
-apply(simp)
-apply(rule finite_subset)
-apply(rule CPTpre_subsets)
-apply(simp)
-sorry
-
-
-lemma CPT_finite:
-  shows "finite (CPT r s)"
-apply(rule finite_subset)
-apply(rule CPT_CPTpre_subset)
-apply(rule CPTpre_finite)
-done
-
-lemma Posix_CPT:
-  assumes "s \<in> r \<rightarrow> v"
-  shows "v \<in> CPT r s"
-using assms
-apply(induct rule: Posix.induct)
-apply(simp add: CPT_def)
-apply(rule CPrf.intros)
-apply(simp add: CPT_def)
-apply(rule CPrf.intros)
-apply(simp add: CPT_def)
-apply(rule CPrf.intros)
-apply(simp)
-apply(simp add: CPT_def)
-apply(rule CPrf.intros)
-apply(simp)
-apply(simp add: CPT_def)
-apply(rule CPrf.intros)
-apply(simp)
-apply(simp)
-apply(simp add: CPT_def)
-apply(rule CPrf.intros)
-apply(simp)
-apply(simp)
-apply(simp)
-apply(simp add: CPT_def)
-apply(rule CPrf.intros)
-done
-
-lemma Posix_val_ord:
-  assumes "s \<in> r \<rightarrow> v1" "v2 \<in> CPTpre r s" 
-  shows "v1 :\<sqsubseteq>val v2"
-using assms
-apply(induct arbitrary: v2 rule: Posix.induct)
-apply(simp add: CPTpre_def)
-apply(clarify)
-apply(erule CPrf.cases)
-apply(simp_all)
-apply(simp add: val_ord_ex1_def)
-apply(simp add: CPTpre_def)
-apply(clarify)
-apply(erule CPrf.cases)
-apply(simp_all)
-apply(simp add: val_ord_ex1_def)
-(* ALT1 *)
-prefer 3
-(* SEQ case *)
-apply(subst (asm) (3) CPTpre_def)
-apply(clarify)
-apply(erule CPrf.cases)
-apply(simp_all)
-apply(case_tac "s' = []")
-apply(simp)
-prefer 2
-apply(simp add: val_ord_ex1_def)
-apply(clarify)
-apply(simp)
-apply(simp add: val_ord_ex_def)
-apply(simp (no_asm) add: val_ord_def)
-apply(rule_tac x="[]" in exI)
-apply(simp add: pflat_len_simps)
-apply(rule intlen_length)
-apply (metis Posix1(2) append_assoc append_eq_conv_conj drop_eq_Nil not_le)
-apply(subgoal_tac "length (flat v1a) \<le> length s1")
-prefer 2
-apply (metis L_flat_Prf1 Prf_CPrf append_eq_append_conv_if append_take_drop_id drop_eq_Nil)
-apply(subst (asm) append_eq_append_conv_if)
-apply(simp)
-apply(clarify)
-apply(drule_tac x="v1a" in meta_spec)
-apply(drule meta_mp)
-apply(auto simp add: CPTpre_def)[1]
-using append_eq_conv_conj apply blast
-apply(subst (asm) (2)val_ord_ex1_def)
-apply(erule disjE)
-apply(subst (asm) val_ord_ex_def)
-apply(erule exE)
-apply(subst val_ord_ex1_def)
-apply(rule disjI1)
-apply(subst val_ord_ex_def)
-apply(rule_tac x="0#p" in exI)
-apply(rule val_ord_SEQI)
-apply(simp)
-apply(simp)
-apply (metis Posix1(2) append_assoc append_take_drop_id)
-apply(simp)
-apply(drule_tac x="v2b" in meta_spec)
-apply(drule meta_mp)
-apply(auto simp add: CPTpre_def)[1]
-apply (simp add: Posix1(2))
-apply(subst (asm) val_ord_ex1_def)
-apply(erule disjE)
-apply(subst (asm) val_ord_ex_def)
-apply(erule exE)
-apply(subst val_ord_ex1_def)
-apply(rule disjI1)
-apply(subst val_ord_ex_def)
-apply(rule_tac x="1#p" in exI)
-apply(rule val_ord_SEQI2)
-apply(simp)
-apply (simp add: Posix1(2))
-apply(subst val_ord_ex1_def)
-apply(simp)
-(* ALT *)
-apply(subst (asm) (2) CPTpre_def)
-apply(clarify)
-apply(erule CPrf.cases)
-apply(simp_all)
-apply(clarify)
-apply(case_tac "s' = []")
-apply(simp)
-apply(drule_tac x="v1" in meta_spec)
-apply(drule meta_mp)
-apply(auto simp add: CPTpre_def)[1]
-apply(subst (asm) val_ord_ex1_def)
-apply(erule disjE)
-apply(subst (asm) val_ord_ex_def)
-apply(erule exE)
-apply(subst val_ord_ex1_def)
-apply(rule disjI1)
-apply(subst val_ord_ex_def)
-apply(rule_tac x="0#p" in exI)
-apply(rule val_ord_ALTI)
-apply(simp)
-using Posix1(2) apply blast
-using val_ord_ex1_def apply blast
-apply(subst val_ord_ex1_def)
-apply(rule disjI1)
-apply (simp add: Posix1(2) val_ord_shorterI)
-apply(subst val_ord_ex1_def)
-apply(rule disjI1)
-apply(case_tac "s' = []")
-apply(simp)
-apply(subst val_ord_ex_def)
-apply(rule_tac x="[0]" in exI)
-apply(subst val_ord_def)
-apply(rule conjI)
-apply(simp add: Pos_empty)
-apply(rule conjI)
-apply(simp add: pflat_len_simps)
-apply (smt inlen_bigger)
-apply(simp)
-apply(rule conjI)
-apply(simp add: pflat_len_simps)
-using Posix1(2) apply auto[1]
-apply(rule ballI)
-apply(rule impI)
-apply(case_tac "q = []")
-using Posix1(2) apply auto[1]
-apply(auto)[1]
-apply(rule val_ord_shorterI)
-apply(simp)
-apply (simp add: Posix1(2))
-(* ALT RIGHT *)
-apply(subst (asm) (2) CPTpre_def)
-apply(clarify)
-apply(erule CPrf.cases)
-apply(simp_all)
-apply(clarify)
-apply(case_tac "s' = []")
-apply(simp)
-apply (simp add: L_flat_Prf1 Prf_CPrf)
-apply(subst val_ord_ex1_def)
-apply(rule disjI1)
-apply(rule val_ord_shorterI)
-apply(simp)
-apply (simp add: Posix1(2))
-apply(case_tac "s' = []")
-apply(simp)
-apply(drule_tac x="v2a" in meta_spec)
-apply(drule meta_mp)
-apply(auto simp add: CPTpre_def)[1]
-apply(subst (asm) val_ord_ex1_def)
-apply(erule disjE)
-apply(subst (asm) val_ord_ex_def)
-apply(erule exE)
-apply(subst val_ord_ex1_def)
-apply(rule disjI1)
-apply(subst val_ord_ex_def)
-apply(rule_tac x="1#p" in exI)
-apply(rule val_ord_ALTI2)
-apply(simp)
-using Posix1(2) apply blast
-apply (simp add: val_ord_ex1_def)
-apply(subst val_ord_ex1_def)
-apply(rule disjI1)
-apply(rule val_ord_shorterI)
-apply(simp)
-apply (simp add: Posix1(2))
-(* STAR empty case *)
-prefer 2
-apply(subst (asm) CPTpre_def)
-apply(clarify)
-apply(erule CPrf.cases)
-apply(simp_all)
-apply(clarify)
-apply (simp add: val_ord_ex1_def)
-(* STAR non-empty case *)
-apply(subst (asm) (3) CPTpre_def)
-apply(clarify)
-apply(erule CPrf.cases)
-apply(simp_all)
-apply(clarify)
-apply (simp add: val_ord_ex1_def)
-apply(rule val_ord_shorterI)
-apply(simp)
-apply(case_tac "s' = []")
-apply(simp)
-prefer 2
-apply (simp add: val_ord_ex1_def)
-apply(rule disjI1)
-apply(rule val_ord_shorterI)
-apply(simp)
-apply (metis Posix1(2) append_assoc append_eq_conv_conj drop_all flat.simps(7) flat_Stars length_append not_less)
-apply(drule_tac x="va" in meta_spec)
-apply(drule meta_mp)
-apply(auto simp add: CPTpre_def)[1]
-apply (smt L.simps(6) L_flat_Prf1 Prf_CPrf append_eq_append_conv2 flat_Stars self_append_conv)
-apply (subst (asm) (2) val_ord_ex1_def)
-apply(erule disjE)
-prefer 2
-apply(simp)
-apply(drule_tac x="Stars vsa" in meta_spec)
-apply(drule meta_mp)
-apply(auto simp add: CPTpre_def)[1]
-apply (simp add: Posix1(2))
-apply (subst (asm) val_ord_ex1_def)
-apply(erule disjE)
-apply (subst (asm) val_ord_ex_def)
-apply(erule exE)
-apply (subst val_ord_ex1_def)
-apply(rule disjI1)
-apply (subst val_ord_ex_def)
-apply(case_tac p)
-apply(simp)
-apply (metis Posix1(2) flat_Stars not_less_iff_gr_or_eq pflat_len_simps(7) same_append_eq val_ord_def)
-using Posix1(2) val_ord_STARI2 apply fastforce
-apply(simp add: val_ord_ex1_def)
-apply (subst (asm) val_ord_ex_def)
-apply(erule exE)
-apply (subst val_ord_ex1_def)
-apply(rule disjI1)
-apply (subst val_ord_ex_def)
-by (metis Posix1(2) flat.simps(7) flat_Stars val_ord_STARI)
-
-lemma Posix_val_ord_stronger:
-  assumes "s \<in> r \<rightarrow> v1" "v2 \<in> CPT r s" 
-  shows "v1 :\<sqsubseteq>val v2"
-using assms
-apply(rule_tac Posix_val_ord)
-apply(assumption)
-apply(simp add: CPTpre_def CPT_def)
-done
-
-
-lemma STAR_val_ord:
-  assumes "Stars (v1 # vs1) \<sqsubset>val (Suc p # ps) Stars (v2 # vs2)" "flat v1 = flat v2"
-  shows "(Stars vs1) \<sqsubset>val (p # ps) (Stars vs2)"
-using assms(1,2)
-apply -
-apply(simp(no_asm)  add: val_ord_def)
-apply(rule conjI)
-apply(simp add: val_ord_def)
-apply(rule conjI)
-apply(simp add: val_ord_def)
-apply(auto simp add: pflat_len_simps pflat_len_Stars_simps2)[1]
-apply(rule ballI)
-apply(rule impI)
-apply(simp add: val_ord_def pflat_len_simps pflat_len_Stars_simps2 intlen_append)
-apply(clarify)
-apply(case_tac q)
-apply(simp add: val_ord_def pflat_len_simps pflat_len_Stars_simps2 intlen_append)
-apply(clarify)
-apply(erule disjE)
-prefer 2
-apply(drule_tac x="Suc a # list" in bspec)
-apply(simp)
-apply(drule mp)
-apply(simp)
-apply(simp add: val_ord_def pflat_len_simps pflat_len_Stars_simps2 intlen_append)
-apply(drule_tac x="Suc a # list" in bspec)
-apply(simp)
-apply(drule mp)
-apply(simp)
-apply(simp add: val_ord_def pflat_len_simps pflat_len_Stars_simps2 intlen_append)
-done
-
-
-lemma Posix_val_ord_reverse:
-  assumes "s \<in> r \<rightarrow> v1" 
-  shows "\<not>(\<exists>v2 \<in> CPT r s. v2 :\<sqsubset>val v1)"
-using assms
-by (metis Posix_val_ord_stronger less_irrefl val_ord_def 
-    val_ord_ex1_def val_ord_ex_def val_ord_ex_trans)
-
-thm Posix.intros(6)
-
-inductive Prop :: "rexp \<Rightarrow> val list \<Rightarrow> bool"
-where
-  "Prop r []"
-| "\<lbrakk>Prop r vs; 
-    \<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = concat (map flat vs) \<and> flat v @ s\<^sub>3 \<in> L r \<and> s\<^sub>4 \<in> L (STAR r))\<rbrakk> 
-   \<Longrightarrow> Prop r (v # vs)"
-
-lemma STAR_val_ord_ex:
-  assumes "Stars (v # vs1) :\<sqsubset>val Stars (v # vs2)"
-  shows "Stars vs1 :\<sqsubset>val Stars vs2"
-using assms
-apply(subst (asm) val_ord_ex_def)
-apply(erule exE)
-apply(case_tac p)
-apply(simp)
-apply(simp add: val_ord_def pflat_len_simps intlen_append)
-apply(subst val_ord_ex_def)
-apply(rule_tac x="[]" in exI)
-apply(simp add: val_ord_def pflat_len_simps Pos_empty)
-apply(simp)
-apply(case_tac a)
-apply(clarify)
-prefer 2
-using STAR_val_ord val_ord_ex_def apply blast
-apply(auto simp add: pflat_len_Stars_simps2 val_ord_def pflat_len_def)[1]
-done
-
-lemma STAR_val_ord_exI:
-  assumes "Stars vs1 :\<sqsubset>val Stars vs2" "flat (Stars vs1) = flat (Stars vs2)"
-  shows "Stars (vs @ vs1) :\<sqsubset>val Stars (vs @ vs2)"
-using assms
-apply(induct vs)
-apply(simp)
-apply(simp)
-apply(simp add: val_ord_ex_def)
-apply(erule exE)
-apply(case_tac p)
-apply(simp)
-apply(rule_tac x="[]" in exI)
-apply(simp add: val_ord_def)
-apply(auto simp add: pflat_len_simps intlen_append)[1]
-apply(simp)
-apply(rule_tac x="Suc aa#list" in exI)
-apply(rule val_ord_STARI2)
-apply(simp)
-apply(simp)
-done
-
-lemma STAR_val_ord_ex_append:
-  assumes "Stars (vs @ vs1) :\<sqsubset>val Stars (vs @ vs2)"
-  shows "Stars vs1 :\<sqsubset>val Stars vs2"
-using assms
-apply(induct vs)
-apply(simp)
-apply(simp)
-apply(drule STAR_val_ord_ex)
-apply(simp)
-done
-
-lemma STAR_val_ord_ex_append_eq:
-  assumes "flat (Stars vs1) = flat (Stars vs2)"
-  shows "Stars (vs @ vs1) :\<sqsubset>val Stars (vs @ vs2) \<longleftrightarrow> Stars vs1 :\<sqsubset>val Stars vs2"
-using assms
-apply(rule_tac iffI)
-apply(erule STAR_val_ord_ex_append)
-apply(rule STAR_val_ord_exI)
-apply(auto)
-done
-
-lemma Posix_STARI:
-  assumes "\<forall>v \<in> set vs. flat v \<noteq> [] \<and> (flat v) \<in> r \<rightarrow> v" "Prop r vs"
-  shows "flat (Stars vs) \<in> STAR r \<rightarrow> Stars vs"
-using assms
-apply(induct vs arbitrary: r)
-apply(simp)
-apply(rule Posix.intros)
-apply(simp)
-apply(rule Posix.intros)
-apply(simp)
-apply(auto)[1]
-apply(erule Prop.cases)
-apply(simp)
-apply(simp)
-apply(simp)
-apply(erule Prop.cases)
-apply(simp)
-apply(auto)[1]
-done
-
-lemma CPrf_stars:
-  assumes "\<Turnstile> Stars vs : STAR r"
-  shows "\<forall>v \<in> set vs. flat v \<noteq> [] \<and> \<Turnstile> v : r"
-using assms
-apply(induct vs)
-apply(auto)
-apply(erule CPrf.cases)
-apply(simp_all)
-apply(erule CPrf.cases)
-apply(simp_all)
-apply(erule CPrf.cases)
-apply(simp_all)
-apply(erule CPrf.cases)
-apply(simp_all)
-done
-
-definition PT :: "rexp \<Rightarrow> string \<Rightarrow> val set"
-where "PT r s \<equiv> {v. flat v = s \<and> \<turnstile> v : r}"
-
-lemma exists:
-  assumes "s \<in> (L r)\<star>"
-  shows "\<exists>vs. concat (map flat vs) = s \<and> \<turnstile> Stars vs : STAR r"
-using assms
-apply(drule_tac Star_string)
-apply(auto)
-by (metis L_flat_Prf2 Prf_Stars Star_val)
-
-
-lemma val_ord_Posix_Stars:
-  assumes "(Stars vs) \<in> CPT (STAR r) (flat (Stars vs))" "\<forall>v \<in> set vs. flat v \<in> r \<rightarrow> v"
-  and "\<not>(\<exists>vs2 \<in> PT (STAR r) (flat (Stars vs)). vs2 :\<sqsubset>val (Stars vs))"
-  shows "(flat (Stars vs)) \<in> (STAR r) \<rightarrow> Stars vs" 
-using assms
-apply(induct vs)
-apply(simp)
-apply(rule Posix.intros)
-apply(simp (no_asm))
-apply(rule Posix.intros)
-apply(auto)[1]
-apply(auto simp add: CPT_def PT_def)[1]
-defer
-apply(simp)
-apply(drule meta_mp)
-apply(auto simp add: CPT_def PT_def)[1]
-apply(erule CPrf.cases)
-apply(simp_all)
-apply(drule meta_mp)
-apply(auto simp add: CPT_def PT_def)[1]
-apply(erule Prf.cases)
-apply(simp_all)
-apply (metis CPrf_stars Cons_eq_map_conv Posix_CPT Posix_STAR2 Posix_val_ord_reverse list.exhaust list.set_intros(2) map_idI val.distinct(25))
-apply(clarify)
-apply(drule_tac x="Stars (a#v#vsa)" in spec)
-apply(simp)
-apply(drule mp)
-apply (meson CPrf_stars Prf.intros(7) Prf_CPrf list.set_intros(1))
-apply(subst (asm) (2) val_ord_ex_def)
-apply(simp)
-apply (metis flat.simps(7) flat_Stars intlen.cases not_less_iff_gr_or_eq pflat_len_simps(7) val_ord_STARI2 val_ord_def val_ord_ex_def)
-apply(auto simp add: CPT_def PT_def)[1]
-using CPrf_stars apply auto[1]
-apply(auto)[1]
-apply(auto simp add: CPT_def PT_def)[1]
-apply(subgoal_tac "\<exists>vA. flat vA = flat a @ s\<^sub>3 \<and> \<turnstile> vA : r")
-prefer 2
-apply (meson L_flat_Prf2)
-apply(subgoal_tac "\<exists>vB. flat (Stars vB) = s\<^sub>4 \<and> \<turnstile> (Stars vB) : (STAR r)")
-apply(clarify)
-apply(drule_tac x="Stars (vA # vB)" in spec)
-apply(simp)
-apply(drule mp)
-using Prf.intros(7) apply blast
-apply(subst (asm) (2) val_ord_ex_def)
-apply(simp)
-prefer 2
-apply(simp)
-using exists apply blast
-prefer 2
-apply(drule meta_mp)
-apply(erule CPrf.cases)
-apply(simp_all)
-apply(drule meta_mp)
-apply(auto)[1]
-prefer 2
-apply(simp)
-apply(erule CPrf.cases)
-apply(simp_all)
-apply(clarify)
-apply(rotate_tac 3)
-apply(erule Prf.cases)
-apply(simp_all)
-apply (metis CPrf_stars intlen.cases less_irrefl list.set_intros(1) val_ord_def val_ord_ex_def)
-apply(drule_tac x="Stars (v#va#vsb)" in spec)
-apply(drule mp)
-apply (simp add: Posix1a Prf.intros(7))
-apply(simp)
-apply(subst (asm) (2) val_ord_ex_def)
-apply(simp)
-apply (metis flat.simps(7) flat_Stars intlen.cases not_less_iff_gr_or_eq pflat_len_simps(7) val_ord_STARI2 val_ord_def val_ord_ex_def)
-proof -
-  fix a :: val and vsa :: "val list" and s\<^sub>3 :: "char list" and vA :: val and vB :: "val list"
-  assume a1: "s\<^sub>3 \<noteq> []"
-  assume a2: "s\<^sub>3 @ concat (map flat vB) = concat (map flat vsa)"
-  assume a3: "flat vA = flat a @ s\<^sub>3"
-  assume a4: "\<forall>p. \<not> Stars (vA # vB) \<sqsubset>val p Stars (a # vsa)"
-  have f5: "\<And>n cs. drop n (cs::char list) = [] \<or> n < length cs"
-    by (meson drop_eq_Nil not_less)
-  have f6: "\<not> length (flat vA) \<le> length (flat a)"
-    using a3 a1 by simp
-  have "flat (Stars (a # vsa)) = flat (Stars (vA # vB))"
-    using a3 a2 by simp
-  then show False
-    using f6 f5 a4 by (metis (full_types) drop_eq_Nil val_ord_STARI val_ord_ex_def val_ord_shorterI)
-qed
-
-lemma Prf_Stars_append:
-  assumes "\<turnstile> Stars vs1 : STAR r" "\<turnstile> Stars vs2 : STAR r"
-  shows "\<turnstile> Stars (vs1 @ vs2) : STAR r"
-using assms
-apply(induct vs1 arbitrary: vs2)
-apply(auto intro: Prf.intros)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(auto intro: Prf.intros)
-done
-
-lemma CPrf_Stars_appendE:
-  assumes "\<Turnstile> Stars (vs1 @ vs2) : STAR r"
-  shows "\<Turnstile> Stars vs1 : STAR r \<and> \<Turnstile> Stars vs2 : STAR r" 
-using assms
-apply(induct vs1 arbitrary: vs2)
-apply(auto intro: CPrf.intros)[1]
-apply(erule CPrf.cases)
-apply(simp_all)
-apply(auto intro: CPrf.intros)
-done
-
-lemma val_ord_Posix:
-  assumes "v1 \<in> CPT r s" "\<not>(\<exists>v2 \<in> PT r s. v2 :\<sqsubset>val v1)"
-  shows "s \<in> r \<rightarrow> v1" 
-using assms
-apply(induct r arbitrary: s v1)
-apply(auto simp add: CPT_def PT_def)[1]
-apply(erule CPrf.cases)
-apply(simp_all)
-(* ONE *)
-apply(auto simp add: CPT_def)[1]
-apply(erule CPrf.cases)
-apply(simp_all)
-apply(rule Posix.intros)
-(* CHAR *)
-apply(auto simp add: CPT_def)[1]
-apply(erule CPrf.cases)
-apply(simp_all)
-apply(rule Posix.intros)
-prefer 2
-(* ALT *)
-apply(auto simp add: CPT_def PT_def)[1]
-apply(erule CPrf.cases)
-apply(simp_all)
-apply(rule Posix.intros)
-apply(drule_tac x="flat v1a" in meta_spec)
-apply(drule_tac x="v1a" in meta_spec)
-apply(drule meta_mp)
-apply(simp)
-apply(drule meta_mp)
-apply(auto)[1]
-apply(drule_tac x="Left v2" in spec)
-apply(simp)
-apply(drule mp)
-apply(rule Prf.intros)
-apply(simp)
-apply (meson val_ord_ALTI val_ord_ex_def)
-apply(assumption)
-(* ALT Right *)
-apply(auto simp add: CPT_def)[1]
-apply(rule Posix.intros)
-apply(rotate_tac 1)
-apply(drule_tac x="flat v2" in meta_spec)
-apply(drule_tac x="v2" in meta_spec)
-apply(drule meta_mp)
-apply(simp)
-apply(drule meta_mp)
-apply(auto)[1]
-apply(drule_tac x="Right v2a" in spec)
-apply(simp)
-apply(drule mp)
-apply(rule Prf.intros)
-apply(simp)
-apply(subst (asm) (2) val_ord_ex_def)
-apply(erule exE)
-apply(drule val_ord_ALTI2)
-apply(assumption)
-apply(auto simp add: val_ord_ex_def)[1]
-apply(assumption)
-apply(auto)[1]
-apply(subgoal_tac "\<exists>v2'. flat v2' = flat v2 \<and> \<turnstile> v2' : r1a")
-apply(clarify)
-apply(drule_tac x="Left v2'" in spec)
-apply(simp)
-apply(drule mp)
-apply(rule Prf.intros)
-apply(assumption)
-apply(simp add: val_ord_ex_def)
-apply(subst (asm) (3) val_ord_def)
-apply(simp)
-apply(simp add: pflat_len_simps)
-apply(drule_tac x="[0]" in spec)
-apply(simp add: pflat_len_simps Pos_empty)
-apply(drule mp)
-apply (smt inlen_bigger)
-apply(erule disjE)
-apply blast
-apply auto[1]
-apply (meson L_flat_Prf2)
-(* SEQ *)
-apply(auto simp add: CPT_def)[1]
-apply(erule CPrf.cases)
-apply(simp_all)
-apply(rule Posix.intros)
-apply(drule_tac x="flat v1a" in meta_spec)
-apply(drule_tac x="v1a" in meta_spec)
-apply(drule meta_mp)
-apply(simp)
-apply(drule meta_mp)
-apply(auto)[1]
-apply(auto simp add: PT_def)[1]
-apply(drule_tac x="Seq v2a v2" in spec)
-apply(simp)
-apply(drule mp)
-apply (simp add: Prf.intros(1) Prf_CPrf)
-using val_ord_SEQI val_ord_ex_def apply fastforce
-apply(assumption)
-apply(rotate_tac 1)
-apply(drule_tac x="flat v2" in meta_spec)
-apply(drule_tac x="v2" in meta_spec)
-apply(simp)
-apply(auto)[1]
-apply(drule meta_mp)
-apply(auto)[1]
-apply(auto simp add: PT_def)[1]
-apply(drule_tac x="Seq v1a v2a" in spec)
-apply(simp)
-apply(drule mp)
-apply (simp add: Prf.intros(1) Prf_CPrf)
-apply (meson val_ord_SEQI2 val_ord_ex_def)
-apply(assumption)
-(* SEQ side condition *)
-apply(auto simp add: PT_def)
-apply(subgoal_tac "\<exists>vA. flat vA = flat v1a @ s\<^sub>3 \<and> \<turnstile> vA : r1a")
-prefer 2
-apply (meson L_flat_Prf2)
-apply(subgoal_tac "\<exists>vB. flat vB = s\<^sub>4 \<and> \<turnstile> vB : r2a")
-prefer 2
-apply (meson L_flat_Prf2)
-apply(clarify)
-apply(drule_tac x="Seq vA vB" in spec)
-apply(simp)
-apply(drule mp)
-apply (simp add: Prf.intros(1))
-apply(subst (asm) (3) val_ord_ex_def)
-apply (metis append_Nil2 append_assoc append_eq_conv_conj flat.simps(5) length_append not_add_less1 not_less_iff_gr_or_eq val_ord_SEQI val_ord_ex_def val_ord_shorterI)
-(* STAR *)
-apply(auto simp add: CPT_def)[1]
-apply(erule CPrf.cases)
-apply(simp_all)[6]
-using Posix_STAR2 apply blast
-apply(clarify)
-apply(rule val_ord_Posix_Stars)
-apply(auto simp add: CPT_def)[1]
-apply (simp add: CPrf.intros(7))
-apply(auto)[1]
-apply(drule_tac x="flat v" in meta_spec)
-apply(drule_tac x="v" in meta_spec)
-apply(simp)
-apply(drule meta_mp)
-apply(auto)[1]
-apply(drule_tac x="Stars (v2#vs)" in spec)
-apply(simp)
-apply(drule mp)
-using Prf.intros(7) Prf_CPrf apply blast
-apply(subst (asm) (2) val_ord_ex_def)
-apply(simp)
-using val_ord_STARI val_ord_ex_def apply fastforce
-apply(assumption)
-apply(drule_tac x="flat va" in meta_spec)
-apply(drule_tac x="va" in meta_spec)
-apply(simp)
-apply(drule meta_mp)
-using CPrf_stars apply blast
-apply(drule meta_mp)
-apply(auto)[1]
-apply(subgoal_tac "\<exists>pre post. vs = pre @ [va] @ post")
-prefer 2
-apply (metis append_Cons append_Nil in_set_conv_decomp_first)
-apply(clarify)
-apply(drule_tac x="Stars (v#(pre @ [v2] @ post))" in spec)
-apply(simp)
-apply(drule mp)
-apply(rule Prf.intros)
-apply (simp add: Prf_CPrf)
-apply(rule Prf_Stars_append)
-apply(drule CPrf_Stars_appendE)
-apply(auto simp add: Prf_CPrf)[1]
-apply (metis CPrf_Stars_appendE CPrf_stars Prf_CPrf Prf_Stars list.set_intros(2) set_ConsD)
-apply(subgoal_tac "\<not> Stars ([v] @ pre @ v2 # post) :\<sqsubset>val Stars ([v] @ pre @ va # post)")
-apply(subst (asm) STAR_val_ord_ex_append_eq)
-apply(simp)
-apply(subst (asm) STAR_val_ord_ex_append_eq)
-apply(simp)
-prefer 2
-apply(simp)
-prefer 2
-apply(simp)
-apply(simp add: val_ord_ex_def)
-apply(erule exE)
-apply(rotate_tac 6)
-apply(drule_tac x="0#p" in spec)
-apply (simp add: val_ord_STARI)
-apply(auto simp add: PT_def)
-done
-
 inductive ValOrd :: "val \<Rightarrow> rexp \<Rightarrow> val \<Rightarrow> bool" ("_ \<preceq>_ _" [100, 100, 100] 100)
 where
   C2: "v1 \<preceq>r1 v1' \<Longrightarrow> (Seq v1 v2) \<preceq>(SEQ r1 r2) (Seq v1' v2')" 
@@ -1926,123 +72,25 @@
 
 
 lemma Posix_CPT2:
-  assumes "v1 \<preceq>r v2" "flat v2 \<sqsubseteq>pre flat v1"
+  assumes "v1 \<preceq>r v2" "flat v2 \<sqsubseteq>pre s" "flat v1 \<sqsubseteq>pre s"
   shows "v1 :\<sqsubset>val v2" 
 using assms
-apply(induct v1 r v2 arbitrary: rule: ValOrd.induct)
-prefer 3
-apply(simp)
-apply(auto simp add: prefix_sprefix)[1]
-apply(rule val_ord_spre)
-apply(simp)
-prefer 3
-apply(simp)
-apply(auto simp add: prefix_sprefix)[1]
-apply(auto simp add: val_ord_ex_def)[1]
-apply(rule_tac x="[0]" in exI)
-apply(auto simp add: val_ord_def Pos_empty pflat_len_simps)[1]
-apply (smt inlen_bigger)
-apply(rule val_ord_spre)
-apply(simp)
-prefer 3
-apply(simp)
-apply(auto simp add: prefix_sprefix)[1]
-using val_ord_ALTI2 val_ord_ex_def apply fastforce
-apply(rule val_ord_spre)
-apply(simp)
+apply(induct v1 r v2 arbitrary: s rule: ValOrd.induct)
 prefer 3
 apply(simp)
-apply(auto simp add: prefix_sprefix)[1]
-using val_ord_ALTI val_ord_ex_def apply fastforce
-apply(rule val_ord_spre)
-apply(simp)
-(* SEQ case *)
-apply(simp)
-apply(auto simp add: prefix_sprefix)[1]
-apply(auto simp add: append_eq_append_conv2)[1]
-apply(case_tac "us = []")
-apply(simp)
-apply(auto simp add: val_ord_ex1_def)[1]
-apply (metis flat.simps(5) val_ord_SEQI val_ord_ex_def)
-apply(simp)
-prefer 2
-apply(case_tac "us = []")
+apply(rule val_ord_shorterI)
 apply(simp)
-apply (metis flat.simps(5) val_ord_SEQI val_ord_ex_def)
-apply(drule meta_mp)
-apply(rule disjI2)
-apply (metis append_self_conv prefix_list_def sprefix_list_def)
-apply(auto simp add: val_ord_ex_def)[1]
-apply (metis append_assoc flat.simps(5) val_ord_SEQI)
-
-apply(sugoal_ tac "")
-thm val_ord_SEQI
-apply(rule val_ord_SEQI)
-thm val_ord_SEQI
-prefer 2
-apply(case_tac "us
-apply(case_tac "v1 = v1'")
-apply(simp)
-
-apply(auto simp add: val_ord_ex1_def)[1]
-apply(auto simp add: val_ord_ex_def)[1]
-apply(rule_tac x="[0]" in exI)
-
-prefer 2
-apply(rule val_ord_spre)
-apply(simp)
-prefer 3
-apply(simp)
-using val_ord_ex1_def val_ord_spre apply auto[1]
-
-apply(erule disjE)
-prefer 2
-apply(subst val_ord_ex1_def)
-apply(rule disjI1)
-apply(rule val_ord_spre)
-apply(simp)
+apply(subst (asm) (3) prefix_list_def)
+apply(subst (asm) (3) prefix_list_def)
+apply(clarify)
 apply(simp)
 apply(simp add: append_eq_append_conv2)
 apply(auto)[1]
-apply(case_tac "us = []")
+apply(drule_tac x="flat v1' @ flat v2' @ usa" in meta_spec)
+apply(simp add: prefix_list_def)
+apply(rule val_ord_SeqI1)
 apply(simp)
-apply(drule meta_mp)
-apply(auto simp add: prefix_sprefix)[1]
-apply(subst (asm) val_ord_ex1_def)
-apply(erule disjE)
-apply(subst val_ord_ex1_def)
-apply(rule disjI1)
-apply (metis flat.simps(5) val_ord_SEQI val_ord_ex_def)
-apply(clarify)
-prefer 2
-apply(subgoal_tac "flat v1' \<sqsubset>spre flat v1")
-prefer 2
-apply (simp add: prefix_list_def sprefix_list_def)
-apply(drule val_ord_spre)
-apply(auto)[1]
-apply(rule val_ord_sprefixI)
-apply(erule ValOrd.cases)
-apply(simp_all)
-apply(erule ValOrd.cases)
-apply(simp_all)
-apply(erule ValOrd.cases)
-apply(simp_all)
-apply(erule ValOrd.cases)
-apply(simp_all)
-apply(clarify)
-(* HERE *)
 apply(simp)
-apply(subst val_ord_ex_def)
-apply(simp)
-apply(drule_tac x="v2a" in  meta_spec)
-apply(rotate_tac 5)
-apply(drule_tac x="v2'" in  meta_spec)
-apply(rule_tac x="0#p" in exI)
-apply(rule val_ord_SEQI)
-
-apply(drule_tac r="r1a" in  val_ord_SEQ)
-apply(simp)
-apply(auto)[1]
 
 
 lemma Posix_CPT: