--- a/thys/Sulzmann.thy	Wed May 17 12:10:27 2017 +0100
+++ b/thys/Sulzmann.thy	Fri Jun 23 00:27:53 2017 +0100
@@ -6,19 +6,1626 @@
 
 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))" 
 
-inductive ValOrd :: "val \<Rightarrow> rexp \<Rightarrow> val \<Rightarrow> bool" ("_ >_ _" [100, 100, 100] 100)
+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 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
-  C2: "v1 >r1 v1' \<Longrightarrow> (Seq v1 v2) >(SEQ r1 r2) (Seq v1' v2')" 
-| C1: "v2 >r2 v2' \<Longrightarrow> (Seq v1 v2) >(SEQ r1 r2) (Seq v1 v2')" 
-| A1: "length (flat v2) > length (flat v1) \<Longrightarrow> (Right v2) >(ALT r1 r2) (Left v1)"
-| A2: "length (flat v1) \<ge> length (flat v2) \<Longrightarrow> (Left v1) >(ALT r1 r2) (Right v2)"
-| A3: "v2 >r2 v2' \<Longrightarrow> (Right v2) >(ALT r1 r2) (Right v2')"
-| A4: "v1 >r1 v1' \<Longrightarrow> (Left v1) >(ALT r1 r2) (Left v1')"
-| K1: "flat (Stars (v # vs)) = [] \<Longrightarrow> (Stars []) >(STAR r) (Stars (v # vs))"
-| K2: "flat (Stars (v # vs)) \<noteq> [] \<Longrightarrow> (Stars (v # vs)) >(STAR r) (Stars [])"
-| K3: "v1 >r v2 \<Longrightarrow> (Stars (v1 # vs1)) >(STAR r) (Stars (v2 # vs2))"
-| K4: "(Stars vs1) >(STAR r) (Stars vs2) \<Longrightarrow> (Stars (v # vs1)) >(STAR r) (Stars (v # vs2))"
+  "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_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_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 CPTpre_test:
+  assumes "s \<in> r \<rightarrow> v"
+  shows "\<not>(\<exists>v' \<in> CPT r s. v :\<sqsubset>val v')"
+using assms
+apply(induct r arbitrary: s v rule: rexp.induct)
+apply(erule Posix.cases)
+apply(simp_all)
+apply(erule Posix.cases)
+apply(simp_all)
+apply(simp add: CPT_simps)
+apply(simp add: val_ord_def val_ord_ex_def)
+apply(erule Posix.cases)
+apply(simp_all)
+apply(simp add: CPT_simps)
+apply (simp add: val_ord_def val_ord_ex_def)
+(* SEQ *)
+apply(rule ballI)
+apply(erule Posix.cases)
+apply(simp_all)
+apply(clarify)
+apply(subst (asm) CPT_simps)
+apply(simp)
+apply(clarify)
+thm val_ord_SEQ
+apply(drule_tac ?r="r1" in val_ord_SEQ)
+apply(simp)
+apply (simp add: CPT_def Posix1(2))
+apply (simp add: Posix1a)
+apply (simp add: CPT_def Posix1a)
+using Prf_CPrf apply auto[1]
+apply(erule disjE)
+apply(drule_tac x="s1" in meta_spec)
+apply(drule_tac x="v1" in meta_spec)
+apply(simp)
+apply(drule_tac x="v1a" in bspec)
+apply(subgoal_tac "s1 = s1a")
+apply(simp)
+apply(auto simp add: append_eq_append_conv2)[1]
+apply (metis (mono_tags, lifting) CPT_def L_flat_Prf1 Prf_CPrf append_Nil append_Nil2 mem_Collect_eq)
+apply(simp add: CPT_def)
+apply(auto)[1]
+oops
+
+
+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
+
+definition Minval :: "rexp \<Rightarrow> string \<Rightarrow> val \<Rightarrow> bool"
+  where
+  "Minval r s v \<equiv> \<Turnstile> v : r \<and> flat v = s \<and> (\<forall>v' \<in> CPT r s. v :\<sqsubset>val v' \<or> v = v')"   
+
+lemma 
+  assumes "s \<in> L(r)"
+  shows "\<exists>v. Minval r s v"
+using assms
+apply(induct r arbitrary: s)
+apply(simp)
+apply(simp)
+apply(rule_tac x="Void" in exI)
+apply(simp add: Minval_def)
+apply(rule conjI)
+apply (simp add: CPrf.intros(4))
+apply(clarify)
+apply(simp add: CPT_def)
+apply(auto)[1]
+apply(erule CPrf.cases)
+apply(simp_all)
+apply(rule_tac x="Char x" in exI)
+apply(simp add: Minval_def)
+apply(rule conjI)
+apply (simp add: CPrf.intros)
+apply(clarify)
+apply(simp add: CPT_def)
+apply(auto)[1]
+apply(erule CPrf.cases)
+apply(simp_all)
+prefer 2
+apply(auto)[1]
+apply(drule_tac x="s" in meta_spec) 
+apply(simp)
+apply(clarify)
+apply(rule_tac x="Left x" in exI)
+apply(simp (no_asm) add: Minval_def)
+apply(rule conjI)
+apply (simp add: CPrf.intros(2) Minval_def)
+apply(rule conjI)
+apply(simp add: Minval_def)
+apply(clarify)
+apply(simp add: CPT_def)
+apply(auto)[1]
+apply(erule CPrf.cases)
+apply(simp_all)
+apply(simp add: val_ord_ex_def)
+apply(simp only: val_ord_def)
+oops
+
+lemma
+  "wf {(v1, v2). v1 \<in> CPT r s \<and> v2 \<in> CPT r s \<and> (v1 :\<sqsubset>val v2)}"
+apply(rule wfI)
+oops
+
+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')" 
+| C1: "v2 \<preceq>r2 v2' \<Longrightarrow> (Seq v1 v2) \<preceq>(SEQ r1 r2) (Seq v1 v2')" 
+| A1: "length (flat v2) > length (flat v1) \<Longrightarrow> (Right v2) \<preceq>(ALT r1 r2) (Left v1)"
+| A2: "length (flat v1) \<ge> length (flat v2) \<Longrightarrow> (Left v1) \<preceq>(ALT r1 r2) (Right v2)"
+| A3: "v2 \<preceq>r2 v2' \<Longrightarrow> (Right v2) \<preceq>(ALT r1 r2) (Right v2')"
+| A4: "v1 \<preceq>r1 v1' \<Longrightarrow> (Left v1) \<preceq>(ALT r1 r2) (Left v1')"
+| K1: "flat (Stars (v # vs)) = [] \<Longrightarrow> (Stars []) \<preceq>(STAR r) (Stars (v # vs))"
+| K2: "flat (Stars (v # vs)) \<noteq> [] \<Longrightarrow> (Stars (v # vs)) \<preceq>(STAR r) (Stars [])"
+| K3: "v1 \<preceq>r v2 \<Longrightarrow> (Stars (v1 # vs1)) \<preceq>(STAR r) (Stars (v2 # vs2))"
+| K4: "(Stars vs1) \<preceq>(STAR r) (Stars vs2) \<Longrightarrow> (Stars (v # vs1)) \<preceq>(STAR r) (Stars (v # vs2))"
+| MY1: "Void \<preceq>ONE Void" 
+| MY2: "(Char c) \<preceq>(CHAR c) (Char c)" 
+| MY3: "(Stars []) \<preceq>(STAR r) (Stars [])" 
+
+lemma ValOrd_refl: 
+  assumes "\<turnstile> v : r" 
+  shows "v \<preceq>r v"
+using assms
+apply(induct r rule: Prf.induct)
+apply(rule ValOrd.intros)
+apply(simp)
+apply(rule ValOrd.intros)
+apply(simp)
+apply(rule ValOrd.intros)
+apply(simp)
+apply(rule ValOrd.intros)
+apply(rule ValOrd.intros)
+apply(rule ValOrd.intros)
+apply(rule ValOrd.intros)
+apply(simp)
+done
+
+lemma Posix_CPT2:
+  assumes "v1 \<preceq>r v2" "flat v1 = flat v2"
+  shows "v2 :\<sqsubset>val v1 \<or> v1 = v2" 
+using assms
+apply(induct r arbitrary: v1 v2 rule: rexp.induct)
+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:
+  assumes "v1 :\<sqsubset>val v2" "v1 \<in> CPT r s" "v2 \<in> CPT r s"
+  shows "v1 \<preceq>r v2" 
+using assms
+apply(induct r arbitrary: v1 v2 s rule: rexp.induct)
+apply(simp add: CPT_def)
+apply(clarify)
+apply(erule CPrf.cases)
+apply(simp_all)
+apply(simp add: CPT_def)
+apply(clarify)
+apply(erule CPrf.cases)
+apply(simp_all)
+apply(erule CPrf.cases)
+apply(simp_all)
+apply(rule ValOrd.intros)
+apply(simp add: CPT_def)
+apply(clarify)
+apply(erule CPrf.cases)
+apply(simp_all)
+apply(erule CPrf.cases)
+apply(simp_all)
+apply(rule ValOrd.intros)
+(*SEQ case *)
+apply(simp add: CPT_def)
+apply(clarify)
+apply(erule CPrf.cases)
+apply(simp_all)
+apply(clarify)
+apply(erule CPrf.cases)
+apply(simp_all)
+apply(clarify)
+thm val_ord_SEQ
+apply(drule_tac r="r1a" in  val_ord_SEQ)
+apply(simp)
+using Prf_CPrf apply blast
+using Prf_CPrf apply blast
+apply(erule disjE)
+apply(rule C2)
+prefer 2
+apply(simp)
+apply(rule C1)
+apply blast
+
+apply(simp add: append_eq_append_conv2)
+apply(clarify)
+apply(auto)[1]
+apply(drule_tac x="v1a" in meta_spec)
+apply(rotate_tac 8)
+apply(drule_tac x="v1b" in meta_spec)
+apply(rotate_tac 8)
+apply(simp)
+
+(* HERE *)
+apply(subst (asm) (3) val_ord_ex_def)
+apply(clarify)
+apply(subst (asm) val_ord_def)
+apply(clarify)
+apply(rule ValOrd.intros)
+
+
+apply(simp add: val_ord_ex_def)
+oops
+
+
+lemma ValOrd_trans:
+  assumes "x \<preceq>r y" "y \<preceq>r z"
+  and "x \<in> CPT r s" "y \<in> CPT r s" "z \<in> CPT r s"
+  shows "x \<preceq>r z"
+using assms
+apply(induct x r y arbitrary: s z rule: ValOrd.induct)
+apply(rotate_tac 2)
+apply(erule ValOrd.cases)
+apply(simp_all)[13]
+apply(rule ValOrd.intros)
+apply(drule_tac x="s" in meta_spec)
+apply(drule_tac x="v1'a" in meta_spec)
+apply(drule_tac meta_mp)
+apply(simp)
+apply(drule_tac meta_mp)
+apply(simp add: CPT_def)
+oops
+
+lemma ValOrd_preorder:
+  "preorder_on (CPT r s) {(v1, v2). v1 \<preceq>r v2 \<and> v1 \<in> (CPT r s) \<and> v2 \<in> (CPT r s)}"
+apply(simp add: preorder_on_def)
+apply(rule conjI)
+apply(simp add: refl_on_def)
+apply(auto)
+apply(rule ValOrd_refl)
+apply(simp add: CPT_def)
+apply(rule Prf_CPrf)
+apply(auto)[1]
+apply(simp add: trans_def)
+apply(auto)
 
 definition ValOrdEq :: "val \<Rightarrow> rexp \<Rightarrow> val \<Rightarrow> bool" ("_ \<ge>_ _" [100, 100, 100] 100)
 where