
diff -r fff2e1b40dfc -r 32b222d77fa0 thys/Positions.thy
--- a/thys/Positions.thy	Wed Jul 19 14:55:46 2017 +0100
+++ b/thys/Positions.thy	Fri Aug 11 20:29:01 2017 +0100
@@ -31,10 +31,8 @@
 lemma Pos_stars:
   "Pos (Stars vs) = {[]} \<union> (\<Union>n < length vs. {n#ps | ps. ps \<in> Pos (vs ! n)})"
 apply(induct vs)
-apply(simp) 
-apply(simp add: insert_ident)
-apply(rule subset_antisym)
-using less_Suc_eq_0_disj by auto
+apply(auto simp add: insert_ident less_Suc_eq_0_disj)
+done
 
 lemma Pos_empty:
   shows "[] \<in> Pos v"
@@ -45,31 +43,25 @@
   "intlen [] = 0"
 | "intlen (x # xs) = 1 + intlen xs"
 
+lemma intlen_int:
+  shows "intlen xs = int (length xs)"
+by (induct xs)(simp_all)
+
 lemma intlen_bigger:
   shows "0 \<le> intlen xs"
 by (induct xs)(auto)
 
 lemma intlen_append:
   shows "intlen (xs @ ys) = intlen xs + intlen ys"
-by (induct xs arbitrary: ys) (auto)
+by (simp add: intlen_int)
 
 lemma intlen_length:
   shows "intlen xs < intlen ys \<longleftrightarrow> length xs < length ys"
-apply(induct xs arbitrary: ys)
-apply (auto simp add: intlen_bigger not_less)
-apply (metis intlen.elims intlen_bigger le_imp_0_less)
-apply (smt Suc_lessI intlen.simps(2) length_Suc_conv nat_neq_iff)
-by (smt Suc_lessE intlen.simps(2) length_Suc_conv)
+by (simp add: intlen_int)
 
 lemma intlen_length_eq:
   shows "intlen xs = intlen ys \<longleftrightarrow> length xs = length ys"
-apply(induct xs arbitrary: ys)
-apply (auto simp add: intlen_bigger not_less)
-apply(case_tac ys)
-apply(simp_all)
-apply (smt intlen_bigger)
-apply (smt intlen.elims intlen_bigger length_Suc_conv)
-by (metis intlen.simps(2) length_Suc_conv)
+by (simp add: intlen_int)
 
 definition pflat_len :: "val \<Rightarrow> nat list => int"
 where
@@ -90,7 +82,7 @@
 lemma pflat_len_Stars_simps:
   assumes "n < length vs"
   shows "pflat_len (Stars vs) (n#p) = pflat_len (vs!n) p"
-using assms 
+using assms
 apply(induct vs arbitrary: n p)
 apply(auto simp add: less_Suc_eq_0_disj pflat_len_simps)
 done
@@ -98,7 +90,8 @@
 lemma pflat_len_outside:
   assumes "p \<notin> Pos v1"
   shows "pflat_len v1 p = -1 "
-using assms by (auto simp add: pflat_len_def)
+using assms by (simp add: pflat_len_def)
+
 
 
 section {* Orderings *}
@@ -175,15 +168,10 @@
 lemma PosOrd_shorterE:
   assumes "v1 :\<sqsubset>val v2" 
   shows "length (flat v2) \<le> length (flat v1)"
-using assms
-apply(auto simp add: pflat_len_simps PosOrd_ex_def PosOrd_def)
-apply(case_tac p)
-apply(simp add: pflat_len_simps intlen_length)
-apply(simp)
-apply(drule_tac x="[]" in bspec)
-apply(simp add: Pos_empty)
-apply(simp add: pflat_len_simps le_less intlen_length_eq)
-done
+using assms unfolding PosOrd_ex_def PosOrd_def
+apply(auto simp add: pflat_len_def intlen_int split: if_splits)
+apply (metis Pos_empty Un_iff at.simps(1) eq_iff lex_simps(1) nat_less_le)
+by (metis Pos_empty UnI2 at.simps(1) lex_simps(2) lex_trichotomous linear)
 
 lemma PosOrd_shorterI:
   assumes "length (flat v2) < length (flat v1)"
@@ -206,8 +194,7 @@
 unfolding PosOrd_ex_def
 apply(rule_tac x="[0]" in exI)
 using assms 
-apply(auto simp add: PosOrd_def pflat_len_simps)
-apply(smt intlen_bigger)
+apply(auto simp add: PosOrd_def pflat_len_simps intlen_int)
 done
 
 lemma PosOrd_Left_eq:
@@ -547,34 +534,35 @@
 by (metis PosOrd_SeqI1 PosOrd_shorterI WW1 antisym_conv3 append_eq_append_conv assms(2))
 
 
+
 section {* The Posix Value is smaller than any other Value *}
 
 
 lemma Posix_PosOrd:
-  assumes "s \<in> r \<rightarrow> v1" "v2 \<in> CPT r s" 
+  assumes "s \<in> r \<rightarrow> v1" "v2 \<in> CV r s" 
   shows "v1 :\<sqsubseteq>val v2"
 using assms
 proof (induct arbitrary: v2 rule: Posix.induct)
   case (Posix_ONE v)
-  have "v \<in> CPT ONE []" by fact
+  have "v \<in> CV ONE []" by fact
   then have "v = Void"
-    by (simp add: CPT_simps)
+    by (simp add: CV_simps)
   then show "Void :\<sqsubseteq>val v"
     by (simp add: PosOrd_ex_eq_def)
 next
   case (Posix_CHAR c v)
-  have "v \<in> CPT (CHAR c) [c]" by fact
+  have "v \<in> CV (CHAR c) [c]" by fact
   then have "v = Char c"
-    by (simp add: CPT_simps)
+    by (simp add: CV_simps)
   then show "Char c :\<sqsubseteq>val v"
     by (simp add: PosOrd_ex_eq_def)
 next
   case (Posix_ALT1 s r1 v r2 v2)
   have as1: "s \<in> r1 \<rightarrow> v" by fact
-  have IH: "\<And>v2. v2 \<in> CPT r1 s \<Longrightarrow> v :\<sqsubseteq>val v2" by fact
-  have "v2 \<in> CPT (ALT r1 r2) s" by fact
+  have IH: "\<And>v2. v2 \<in> CV r1 s \<Longrightarrow> v :\<sqsubseteq>val v2" by fact
+  have "v2 \<in> CV (ALT r1 r2) s" by fact
   then have "\<Turnstile> v2 : ALT r1 r2" "flat v2 = s"
-    by(auto simp add: CPT_def prefix_list_def)
+    by(auto simp add: CV_def prefix_list_def)
   then consider
     (Left) v3 where "v2 = Left v3" "\<Turnstile> v3 : r1" "flat v3 = s" 
   | (Right) v3 where "v2 = Right v3" "\<Turnstile> v3 : r2" "flat v3 = s"
@@ -582,8 +570,8 @@
   then show "Left v :\<sqsubseteq>val v2"
   proof(cases)
      case (Left v3)
-     have "v3 \<in> CPT r1 s" using Left(2,3) 
-       by (auto simp add: CPT_def prefix_list_def)
+     have "v3 \<in> CV r1 s" using Left(2,3) 
+       by (auto simp add: CV_def prefix_list_def)
      with IH have "v :\<sqsubseteq>val v3" by simp
      moreover
      have "flat v3 = flat v" using as1 Left(3)
@@ -603,10 +591,10 @@
   case (Posix_ALT2 s r2 v r1 v2)
   have as1: "s \<in> r2 \<rightarrow> v" by fact
   have as2: "s \<notin> L r1" by fact
-  have IH: "\<And>v2. v2 \<in> CPT r2 s \<Longrightarrow> v :\<sqsubseteq>val v2" by fact
-  have "v2 \<in> CPT (ALT r1 r2) s" by fact
+  have IH: "\<And>v2. v2 \<in> CV r2 s \<Longrightarrow> v :\<sqsubseteq>val v2" by fact
+  have "v2 \<in> CV (ALT r1 r2) s" by fact
   then have "\<Turnstile> v2 : ALT r1 r2" "flat v2 = s"
-    by(auto simp add: CPT_def prefix_list_def)
+    by(auto simp add: CV_def prefix_list_def)
   then consider
     (Left) v3 where "v2 = Left v3" "\<Turnstile> v3 : r1" "flat v3 = s" 
   | (Right) v3 where "v2 = Right v3" "\<Turnstile> v3 : r2" "flat v3 = s"
@@ -614,8 +602,8 @@
   then show "Right v :\<sqsubseteq>val v2"
   proof (cases)
     case (Right v3)
-     have "v3 \<in> CPT r2 s" using Right(2,3) 
-       by (auto simp add: CPT_def prefix_list_def)
+     have "v3 \<in> CV r2 s" using Right(2,3) 
+       by (auto simp add: CV_def prefix_list_def)
      with IH have "v :\<sqsubseteq>val v3" by simp
      moreover
      have "flat v3 = flat v" using as1 Right(3)
@@ -625,10 +613,10 @@
      then show "Right v :\<sqsubseteq>val v2" unfolding Right .
   next
      case (Left v3)
-     have "v3 \<in> CPT r1 s" using Left(2,3) as2  
-       by (auto simp add: CPT_def prefix_list_def)
+     have "v3 \<in> CV r1 s" using Left(2,3) as2  
+       by (auto simp add: CV_def prefix_list_def)
      then have "flat v3 = flat v \<and> \<Turnstile> v3 : r1" using as1 Left(3)
-       by (simp add: Posix1(2) CPT_def) 
+       by (simp add: Posix1(2) CV_def) 
      then have "False" using as1 as2 Left
        by (auto simp add: Posix1(2) L_flat_Prf1 Prf_CPrf)
      then show "Right v :\<sqsubseteq>val v2" by simp
@@ -637,22 +625,22 @@
   case (Posix_SEQ s1 r1 v1 s2 r2 v2 v3)
   have "s1 \<in> r1 \<rightarrow> v1" "s2 \<in> r2 \<rightarrow> v2" by fact+
   then have as1: "s1 = flat v1" "s2 = flat v2" by (simp_all add: Posix1(2))
-  have IH1: "\<And>v3. v3 \<in> CPT r1 s1 \<Longrightarrow> v1 :\<sqsubseteq>val v3" by fact
-  have IH2: "\<And>v3. v3 \<in> CPT r2 s2 \<Longrightarrow> v2 :\<sqsubseteq>val v3" by fact
+  have IH1: "\<And>v3. v3 \<in> CV r1 s1 \<Longrightarrow> v1 :\<sqsubseteq>val v3" by fact
+  have IH2: "\<And>v3. v3 \<in> CV r2 s2 \<Longrightarrow> v2 :\<sqsubseteq>val v3" by fact
   have cond: "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L r1 \<and> s\<^sub>4 \<in> L r2)" by fact
-  have "v3 \<in> CPT (SEQ r1 r2) (s1 @ s2)" by fact
+  have "v3 \<in> CV (SEQ r1 r2) (s1 @ s2)" by fact
   then obtain v3a v3b where eqs:
     "v3 = Seq v3a v3b" "\<Turnstile> v3a : r1" "\<Turnstile> v3b : r2"
     "flat v3a @ flat v3b = s1 @ s2" 
-    by (force simp add: prefix_list_def CPT_def elim: CPrf.cases)
+    by (force simp add: prefix_list_def CV_def elim: CPrf.cases)
   with cond have "flat v3a \<sqsubseteq>pre s1" unfolding prefix_list_def
     by (smt L_flat_Prf1 Prf_CPrf append_eq_append_conv2 append_self_conv)
   then have "flat v3a \<sqsubset>spre s1 \<or> (flat v3a = s1 \<and> flat v3b = s2)" using eqs
     by (simp add: sprefix_list_def append_eq_conv_conj)
   then have q2: "v1 :\<sqsubset>val v3a \<or> (flat v3a = s1 \<and> flat v3b = s2)" 
     using PosOrd_spreI as1(1) eqs by blast
-  then have "v1 :\<sqsubset>val v3a \<or> (v3a \<in> CPT r1 s1 \<and> v3b \<in> CPT r2 s2)" using eqs(2,3)
-    by (auto simp add: CPT_def)
+  then have "v1 :\<sqsubset>val v3a \<or> (v3a \<in> CV r1 s1 \<and> v3b \<in> CV r2 s2)" using eqs(2,3)
+    by (auto simp add: CV_def)
   then have "v1 :\<sqsubset>val v3a \<or> (v1 :\<sqsubseteq>val v3a \<and> v2 :\<sqsubseteq>val v3b)" using IH1 IH2 by blast         
   then have "Seq v1 v2 :\<sqsubseteq>val Seq v3a v3b" using eqs q2 as1
     unfolding  PosOrd_ex_eq_def by (auto simp add: PosOrd_SeqI1 PosOrd_SeqI2) 
@@ -661,17 +649,17 @@
   case (Posix_STAR1 s1 r v s2 vs v3) 
   have "s1 \<in> r \<rightarrow> v" "s2 \<in> STAR r \<rightarrow> Stars vs" by fact+
   then have as1: "s1 = flat v" "s2 = flat (Stars vs)" by (auto dest: Posix1(2))
-  have IH1: "\<And>v3. v3 \<in> CPT r s1 \<Longrightarrow> v :\<sqsubseteq>val v3" by fact
-  have IH2: "\<And>v3. v3 \<in> CPT (STAR r) s2 \<Longrightarrow> Stars vs :\<sqsubseteq>val v3" by fact
+  have IH1: "\<And>v3. v3 \<in> CV r s1 \<Longrightarrow> v :\<sqsubseteq>val v3" by fact
+  have IH2: "\<And>v3. v3 \<in> CV (STAR r) s2 \<Longrightarrow> Stars vs :\<sqsubseteq>val v3" by fact
   have cond: "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L r \<and> s\<^sub>4 \<in> L (STAR r))" by fact
   have cond2: "flat v \<noteq> []" by fact
-  have "v3 \<in> CPT (STAR r) (s1 @ s2)" by fact
+  have "v3 \<in> CV (STAR r) (s1 @ s2)" by fact
   then consider 
     (NonEmpty) v3a vs3 where "v3 = Stars (v3a # vs3)" 
     "\<Turnstile> v3a : r" "\<Turnstile> Stars vs3 : STAR r"
     "flat (Stars (v3a # vs3)) = s1 @ s2"
   | (Empty) "v3 = Stars []"
-  unfolding CPT_def
+  unfolding CV_def
   apply(auto)
   apply(erule CPrf.cases)
   apply(simp_all)
@@ -690,8 +678,8 @@
         by (simp add: sprefix_list_def append_eq_conv_conj)
       then have q2: "v :\<sqsubset>val v3a \<or> (flat v3a = s1 \<and> flat (Stars vs3) = s2)" 
         using PosOrd_spreI as1(1) NonEmpty(4) by blast
-      then have "v :\<sqsubset>val v3a \<or> (v3a \<in> CPT r s1 \<and> Stars vs3 \<in> CPT (STAR r) s2)" 
-        using NonEmpty(2,3) by (auto simp add: CPT_def)
+      then have "v :\<sqsubset>val v3a \<or> (v3a \<in> CV r s1 \<and> Stars vs3 \<in> CV (STAR r) s2)" 
+        using NonEmpty(2,3) by (auto simp add: CV_def)
       then have "v :\<sqsubset>val v3a \<or> (v :\<sqsubseteq>val v3a \<and> Stars vs :\<sqsubseteq>val Stars vs3)" using IH1 IH2 by blast
       then have "v :\<sqsubset>val v3a \<or> (v = v3a \<and> Stars vs :\<sqsubseteq>val Stars vs3)" 
          unfolding PosOrd_ex_eq_def by auto     
@@ -708,58 +696,26 @@
     qed      
 next 
   case (Posix_STAR2 r v2)
-  have "v2 \<in> CPT (STAR r) []" by fact
+  have "v2 \<in> CV (STAR r) []" by fact
   then have "v2 = Stars []" 
-    unfolding CPT_def by (auto elim: CPrf.cases) 
+    unfolding CV_def by (auto elim: CPrf.cases) 
   then show "Stars [] :\<sqsubseteq>val v2"
   by (simp add: PosOrd_ex_eq_def)
 qed
 
-lemma Posix_PosOrd_stronger:
-  assumes "s \<in> r \<rightarrow> v1" "v2 \<in> CPTpre r s" 
-  shows "v1 :\<sqsubseteq>val v2"
-proof -
-  from assms(2) have "v2 \<in> CPT r s \<or> flat v2 \<sqsubset>spre s"
-  unfolding CPTpre_def CPT_def sprefix_list_def prefix_list_def by auto
-  moreover
-    { assume "v2 \<in> CPT r s" 
-      with assms(1) 
-      have "v1 :\<sqsubseteq>val v2" by (rule Posix_PosOrd)
-    }
-  moreover
-    { assume "flat v2 \<sqsubset>spre s"
-      then have "flat v2 \<sqsubset>spre flat v1" using assms(1)
-        using Posix1(2) by blast
-      then have "v1 :\<sqsubseteq>val v2"
-        by (simp add: PosOrd_ex_eq_def PosOrd_spreI) 
-    }
-  ultimately show "v1 :\<sqsubseteq>val v2" by blast
-qed
 
 lemma Posix_PosOrd_reverse:
   assumes "s \<in> r \<rightarrow> v1" 
-  shows "\<not>(\<exists>v2 \<in> CPTpre r s. v2 :\<sqsubset>val v1)"
+  shows "\<not>(\<exists>v2 \<in> CV r s. v2 :\<sqsubset>val v1)"
 using assms
-by (metis Posix_PosOrd_stronger less_irrefl PosOrd_def 
+by (metis Posix_PosOrd less_irrefl PosOrd_def 
     PosOrd_ex_eq_def PosOrd_ex_def PosOrd_trans)
 
 
-lemma test2: 
-  assumes "\<forall>v \<in> set vs. flat v \<in> r \<rightarrow> v \<and> flat v \<noteq> []"
-  shows "(Stars vs) \<in> CPT (STAR r) (flat (Stars vs))" 
-using assms
-apply(induct vs)
-apply(auto simp add: CPT_def)
-apply(rule CPrf.intros)
-apply(simp)
-apply(rule CPrf.intros)
-apply(simp_all)
-by (metis (no_types, lifting) CPT_def Posix_CPT mem_Collect_eq)
-
 
 lemma PosOrd_Posix_Stars:
   assumes "\<forall>v \<in> set vs. flat v \<in> r \<rightarrow> v \<and> flat v \<noteq> []"
-  and "\<not>(\<exists>vs2 \<in> PT (STAR r) (flat (Stars vs)). vs2 :\<sqsubset>val (Stars vs))"
+  and "\<not>(\<exists>vs2 \<in> LV (STAR r) (flat (Stars vs)). vs2 :\<sqsubset>val (Stars vs))"
   shows "(flat (Stars vs)) \<in> (STAR r) \<rightarrow> Stars vs" 
 using assms
 proof(induct vs)
@@ -769,24 +725,24 @@
 next
   case (Cons v vs)
   have IH: "\<lbrakk>\<forall>v\<in>set vs. flat v \<in> r \<rightarrow> v \<and> flat v \<noteq> []; 
-             \<not> (\<exists>vs2\<in>PT (STAR r) (flat (Stars vs)). vs2 :\<sqsubset>val Stars vs)\<rbrakk>
+             \<not> (\<exists>vs2\<in>LV (STAR r) (flat (Stars vs)). vs2 :\<sqsubset>val Stars vs)\<rbrakk>
              \<Longrightarrow> flat (Stars vs) \<in> STAR r \<rightarrow> Stars vs" by fact
   have as2: "\<forall>v\<in>set (v # vs). flat v \<in> r \<rightarrow> v \<and> flat v \<noteq> []" by fact
-  have as3: "\<not> (\<exists>vs2\<in>PT (STAR r) (flat (Stars (v # vs))). vs2 :\<sqsubset>val Stars (v # vs))" by fact
+  have as3: "\<not> (\<exists>vs2\<in>LV (STAR r) (flat (Stars (v # vs))). vs2 :\<sqsubset>val Stars (v # vs))" by fact
   have "flat v \<in> r \<rightarrow> v" using as2 by simp
   moreover
   have  "flat (Stars vs) \<in> STAR r \<rightarrow> Stars vs" 
     proof (rule IH)
       show "\<forall>v\<in>set vs. flat v \<in> r \<rightarrow> v \<and> flat v \<noteq> []" using as2 by simp
     next 
-      show "\<not> (\<exists>vs2\<in>PT (STAR r) (flat (Stars vs)). vs2 :\<sqsubset>val Stars vs)" using as3
+      show "\<not> (\<exists>vs2\<in>LV (STAR r) (flat (Stars vs)). vs2 :\<sqsubset>val Stars vs)" using as3
         apply(auto)
-        apply(subst (asm) (2) PT_def)
+        apply(subst (asm) (2) LV_def)
         apply(auto)
         apply(erule Prf.cases)
         apply(simp_all)
         apply(drule_tac x="Stars (v # vs)" in bspec)
-        apply(simp add: PT_def CPT_def)
+        apply(simp add: LV_def CV_def)
         using Posix_Prf Prf.intros(6) calculation
         apply(rule_tac Prf.intros)
         apply(simp add:)
@@ -810,7 +766,7 @@
    apply(simp_all)
    apply(clarify)
    apply(drule_tac x="Stars (va#vs)" in bspec)
-   apply(auto simp add: PT_def)[1]   
+   apply(auto simp add: LV_def)[1]   
    apply(rule Prf.intros)
    apply(simp)
    by (simp add: PosOrd_StarsI PosOrd_shorterI)
@@ -823,69 +779,69 @@
 section {* The Smallest Value is indeed the Posix Value *}
 
 text {*
-  The next lemma seems to require PT instead of CPT in the Star-case.
+  The next lemma seems to require LV instead of CV in the Star-case.
 *}
 
 lemma PosOrd_Posix:
-  assumes "v1 \<in> CPT r s" "\<forall>v\<^sub>2 \<in> PT r s. \<not> v\<^sub>2 :\<sqsubset>val v1"
+  assumes "v1 \<in> CV r s" "\<forall>v\<^sub>2 \<in> LV r s. \<not> v\<^sub>2 :\<sqsubset>val v1"
   shows "s \<in> r \<rightarrow> v1" 
 using assms
 proof(induct r arbitrary: s v1)
   case (ZERO s v1)
-  have "v1 \<in> CPT ZERO s" by fact
-  then show "s \<in> ZERO \<rightarrow> v1" unfolding CPT_def
+  have "v1 \<in> CV ZERO s" by fact
+  then show "s \<in> ZERO \<rightarrow> v1" unfolding CV_def
     by (auto elim: CPrf.cases)
 next 
   case (ONE s v1)
-  have "v1 \<in> CPT ONE s" by fact
-  then show "s \<in> ONE \<rightarrow> v1" unfolding CPT_def
+  have "v1 \<in> CV ONE s" by fact
+  then show "s \<in> ONE \<rightarrow> v1" unfolding CV_def
     by(auto elim!: CPrf.cases intro: Posix.intros)
 next 
   case (CHAR c s v1)
-  have "v1 \<in> CPT (CHAR c) s" by fact
-  then show "s \<in> CHAR c \<rightarrow> v1" unfolding CPT_def
+  have "v1 \<in> CV (CHAR c) s" by fact
+  then show "s \<in> CHAR c \<rightarrow> v1" unfolding CV_def
     by (auto elim!: CPrf.cases intro: Posix.intros)
 next
   case (ALT r1 r2 s v1)
-  have IH1: "\<And>s v1. \<lbrakk>v1 \<in> CPT r1 s; \<forall>v2 \<in> PT r1 s. \<not> v2 :\<sqsubset>val v1\<rbrakk> \<Longrightarrow> s \<in> r1 \<rightarrow> v1" by fact
-  have IH2: "\<And>s v1. \<lbrakk>v1 \<in> CPT r2 s; \<forall>v2 \<in> PT r2 s. \<not> v2 :\<sqsubset>val v1\<rbrakk> \<Longrightarrow> s \<in> r2 \<rightarrow> v1" by fact
-  have as1: "\<forall>v2\<in>PT (ALT r1 r2) s. \<not> v2 :\<sqsubset>val v1" by fact
-  have as2: "v1 \<in> CPT (ALT r1 r2) s" by fact
+  have IH1: "\<And>s v1. \<lbrakk>v1 \<in> CV r1 s; \<forall>v2 \<in> LV r1 s. \<not> v2 :\<sqsubset>val v1\<rbrakk> \<Longrightarrow> s \<in> r1 \<rightarrow> v1" by fact
+  have IH2: "\<And>s v1. \<lbrakk>v1 \<in> CV r2 s; \<forall>v2 \<in> LV r2 s. \<not> v2 :\<sqsubset>val v1\<rbrakk> \<Longrightarrow> s \<in> r2 \<rightarrow> v1" by fact
+  have as1: "\<forall>v2\<in>LV (ALT r1 r2) s. \<not> v2 :\<sqsubset>val v1" by fact
+  have as2: "v1 \<in> CV (ALT r1 r2) s" by fact
   then consider 
      (Left) v1' where
         "v1 = Left v1'" "s = flat v1'"
-        "v1' \<in> CPT r1 s"
+        "v1' \<in> CV r1 s"
   |  (Right) v1' where
         "v1 = Right v1'" "s = flat v1'"
-        "v1' \<in> CPT r2 s"
-  unfolding CPT_def by (auto elim: CPrf.cases)
+        "v1' \<in> CV r2 s"
+  unfolding CV_def by (auto elim: CPrf.cases)
   then show "s \<in> ALT r1 r2 \<rightarrow> v1"
    proof (cases)
      case (Left v1')
-     have "v1' \<in> CPT r1 s" using as2
-       unfolding CPT_def Left by (auto elim: CPrf.cases)
+     have "v1' \<in> CV r1 s" using as2
+       unfolding CV_def Left by (auto elim: CPrf.cases)
      moreover
-     have "\<forall>v2 \<in> PT r1 s. \<not> v2 :\<sqsubset>val v1'" using as1
-       unfolding PT_def Left using Prf.intros(2) PosOrd_Left_eq by force  
+     have "\<forall>v2 \<in> LV r1 s. \<not> v2 :\<sqsubset>val v1'" using as1
+       unfolding LV_def Left using Prf.intros(2) PosOrd_Left_eq by force  
      ultimately have "s \<in> r1 \<rightarrow> v1'" using IH1 by simp
      then have "s \<in> ALT r1 r2 \<rightarrow> Left v1'" by (rule Posix.intros)
      then show "s \<in> ALT r1 r2 \<rightarrow> v1" using Left by simp
    next
      case (Right v1')
-     have "v1' \<in> CPT r2 s" using as2
-       unfolding CPT_def Right by (auto elim: CPrf.cases)
+     have "v1' \<in> CV r2 s" using as2
+       unfolding CV_def Right by (auto elim: CPrf.cases)
      moreover
-     have "\<forall>v2 \<in> PT r2 s. \<not> v2 :\<sqsubset>val v1'" using as1
-       unfolding PT_def Right using Prf.intros(3) PosOrd_RightI by force   
+     have "\<forall>v2 \<in> LV r2 s. \<not> v2 :\<sqsubset>val v1'" using as1
+       unfolding LV_def Right using Prf.intros(3) PosOrd_RightI by force   
      ultimately have "s \<in> r2 \<rightarrow> v1'" using IH2 by simp
      moreover 
        { assume "s \<in> L r1"
-         then obtain v' where "v' \<in>  PT r1 s"
-            unfolding PT_def using L_flat_Prf2 by blast
-         then have "Left v' \<in>  PT (ALT r1 r2) s" 
-            unfolding PT_def by (auto intro: Prf.intros)
+         then obtain v' where "v' \<in>  LV r1 s"
+            unfolding LV_def using L_flat_Prf2 by blast
+         then have "Left v' \<in>  LV (ALT r1 r2) s" 
+            unfolding LV_def by (auto intro: Prf.intros)
          with as1 have "\<not> (Left v' :\<sqsubset>val Right v1') \<and> (flat v' = s)" 
-            unfolding PT_def Right by (auto)
+            unfolding LV_def Right by (auto)
          then have False using PosOrd_Left_Right Right by blast  
        }
      then have "s \<notin> L r1" by rule 
@@ -894,36 +850,36 @@
   qed
 next 
   case (SEQ r1 r2 s v1)
-  have IH1: "\<And>s v1. \<lbrakk>v1 \<in> CPT r1 s; \<forall>v2 \<in> PT r1 s. \<not> v2 :\<sqsubset>val v1\<rbrakk> \<Longrightarrow> s \<in> r1 \<rightarrow> v1" by fact
-  have IH2: "\<And>s v1. \<lbrakk>v1 \<in> CPT r2 s; \<forall>v2 \<in> PT r2 s. \<not> v2 :\<sqsubset>val v1\<rbrakk> \<Longrightarrow> s \<in> r2 \<rightarrow> v1" by fact
-  have as1: "\<forall>v2\<in>PT (SEQ r1 r2) s. \<not> v2 :\<sqsubset>val v1" by fact
-  have as2: "v1 \<in> CPT (SEQ r1 r2) s" by fact
+  have IH1: "\<And>s v1. \<lbrakk>v1 \<in> CV r1 s; \<forall>v2 \<in> LV r1 s. \<not> v2 :\<sqsubset>val v1\<rbrakk> \<Longrightarrow> s \<in> r1 \<rightarrow> v1" by fact
+  have IH2: "\<And>s v1. \<lbrakk>v1 \<in> CV r2 s; \<forall>v2 \<in> LV r2 s. \<not> v2 :\<sqsubset>val v1\<rbrakk> \<Longrightarrow> s \<in> r2 \<rightarrow> v1" by fact
+  have as1: "\<forall>v2\<in>LV (SEQ r1 r2) s. \<not> v2 :\<sqsubset>val v1" by fact
+  have as2: "v1 \<in> CV (SEQ r1 r2) s" by fact
   then obtain 
     v1a v1b where eqs:
         "v1 = Seq v1a v1b" "s = flat v1a @ flat v1b"
-        "v1a \<in> CPT r1 (flat v1a)" "v1b \<in> CPT r2 (flat v1b)" 
-  unfolding CPT_def by(auto elim: CPrf.cases)
-  have "\<forall>v2 \<in> PT r1 (flat v1a). \<not> v2 :\<sqsubset>val v1a"
+        "v1a \<in> CV r1 (flat v1a)" "v1b \<in> CV r2 (flat v1b)" 
+  unfolding CV_def by(auto elim: CPrf.cases)
+  have "\<forall>v2 \<in> LV r1 (flat v1a). \<not> v2 :\<sqsubset>val v1a"
     proof
       fix v2
-      assume "v2 \<in> PT r1 (flat v1a)"
-      with eqs(2,4) have "Seq v2 v1b \<in> PT (SEQ r1 r2) s"
-         by (simp add: CPT_def PT_def Prf.intros(1) Prf_CPrf)
+      assume "v2 \<in> LV r1 (flat v1a)"
+      with eqs(2,4) have "Seq v2 v1b \<in> LV (SEQ r1 r2) s"
+         by (simp add: CV_def LV_def Prf.intros(1) Prf_CPrf)
       with as1 have "\<not> Seq v2 v1b :\<sqsubset>val Seq v1a v1b \<and> flat (Seq v2 v1b) = flat (Seq v1a v1b)" 
-         using eqs by (simp add: PT_def) 
+         using eqs by (simp add: LV_def) 
       then show "\<not> v2 :\<sqsubset>val v1a"
          using PosOrd_SeqI1 by blast
     qed     
   then have "flat v1a \<in> r1 \<rightarrow> v1a" using IH1 eqs by simp
   moreover
-  have "\<forall>v2 \<in> PT r2 (flat v1b). \<not> v2 :\<sqsubset>val v1b"
+  have "\<forall>v2 \<in> LV r2 (flat v1b). \<not> v2 :\<sqsubset>val v1b"
     proof 
       fix v2
-      assume "v2 \<in> PT r2 (flat v1b)"
-      with eqs(2,3,4) have "Seq v1a v2 \<in> PT (SEQ r1 r2) s"
-         by (simp add: CPT_def PT_def Prf.intros Prf_CPrf)
+      assume "v2 \<in> LV r2 (flat v1b)"
+      with eqs(2,3,4) have "Seq v1a v2 \<in> LV (SEQ r1 r2) s"
+         by (simp add: CV_def LV_def Prf.intros Prf_CPrf)
       with as1 have "\<not> Seq v1a v2 :\<sqsubset>val Seq v1a v1b \<and> flat v2 = flat v1b" 
-         using eqs by (simp add: PT_def) 
+         using eqs by (simp add: LV_def) 
       then show "\<not> v2 :\<sqsubset>val v1b"
          using PosOrd_SeqI2 by auto
     qed     
@@ -935,8 +891,8 @@
      then obtain s3 s4 where q1: "s3 \<noteq> [] \<and> s3 @ s4 = flat v1b \<and> flat v1a @ s3 \<in> L r1 \<and> s4 \<in> L r2" by blast
      then obtain vA vB where q2: "flat vA = flat v1a @ s3" "\<turnstile> vA : r1" "flat vB = s4" "\<turnstile> vB : r2"
         using L_flat_Prf2 by blast
-     then have "Seq vA vB \<in> PT (SEQ r1 r2) s" unfolding eqs using q1
-       by (auto simp add: PT_def intro: Prf.intros)
+     then have "Seq vA vB \<in> LV (SEQ r1 r2) s" unfolding eqs using q1
+       by (auto simp add: LV_def intro: Prf.intros)
      with as1 have "\<not> Seq vA vB :\<sqsubset>val Seq v1a v1b" unfolding eqs by auto
      then have "\<not> vA :\<sqsubset>val v1a \<and> length (flat vA) > length (flat v1a)" using q1 q2 PosOrd_SeqI1 by auto 
      then show "False"
@@ -947,57 +903,57 @@
     by (rule Posix.intros)
 next
    case (STAR r s v1)
-   have IH: "\<And>s v1. \<lbrakk>v1 \<in> CPT r s; \<forall>v2\<in>PT r s. \<not> v2 :\<sqsubset>val v1\<rbrakk> \<Longrightarrow> s \<in> r \<rightarrow> v1" by fact
-   have as1: "\<forall>v2\<in>PT (STAR r) s. \<not> v2 :\<sqsubset>val v1" by fact
-   have as2: "v1 \<in> CPT (STAR r) s" by fact
+   have IH: "\<And>s v1. \<lbrakk>v1 \<in> CV r s; \<forall>v2\<in>LV r s. \<not> v2 :\<sqsubset>val v1\<rbrakk> \<Longrightarrow> s \<in> r \<rightarrow> v1" by fact
+   have as1: "\<forall>v2\<in>LV (STAR r) s. \<not> v2 :\<sqsubset>val v1" by fact
+   have as2: "v1 \<in> CV (STAR r) s" by fact
    then obtain 
     vs where eqs:
         "v1 = Stars vs" "s = flat (Stars vs)"
-        "\<forall>v \<in> set vs. v \<in> CPT r (flat v)"
-        unfolding CPT_def by (auto elim: CPrf.cases dest!: CPrf_stars)
+        "\<forall>v \<in> set vs. v \<in> CV r (flat v)"
+        unfolding CV_def by (auto elim: CPrf.cases)
    have "\<forall>v\<in>set vs. flat v \<in> r \<rightarrow> v \<and> flat v \<noteq> []" 
      proof 
         fix v
         assume a: "v \<in> set vs"
         then obtain pre post where e: "vs = pre @ [v] @ post"
           by (metis append_Cons append_Nil in_set_conv_decomp_first)
-        then have q: "\<forall>v2\<in>PT (STAR r) s. \<not> v2 :\<sqsubset>val Stars (pre @ [v] @ post)" 
+        then have q: "\<forall>v2\<in>LV (STAR r) s. \<not> v2 :\<sqsubset>val Stars (pre @ [v] @ post)" 
           using as1 unfolding eqs by simp
-        have "\<forall>v2\<in>PT r (flat v). \<not> v2 :\<sqsubset>val v" unfolding eqs 
+        have "\<forall>v2\<in>LV r (flat v). \<not> v2 :\<sqsubset>val v" unfolding eqs 
           proof (rule ballI, rule notI) 
              fix v2
              assume w: "v2 :\<sqsubset>val v"
-             assume "v2 \<in> PT r (flat v)"
-             then have "Stars (pre @ [v2] @ post) \<in> PT (STAR r) s" 
+             assume "v2 \<in> LV r (flat v)"
+             then have "Stars (pre @ [v2] @ post) \<in> LV (STAR r) s" 
                  using as2 unfolding e eqs
-                 apply(auto simp add: CPT_def PT_def intro!: Prf.intros)[1]
-                 using CPrf_Stars_appendE CPrf_stars Prf_CPrf apply blast
-                 by (meson CPrf_Stars_appendE CPrf_stars Prf_CPrf list.set_intros(2))
+                 apply(auto simp add: CV_def LV_def intro!: Prf.intros)[1]
+                 using CPrf_Stars_appendE Prf_CPrf Prf_elims(6) list.set_intros apply blast
+                 by (metis CPrf_Stars_appendE Prf_CPrf Prf_elims(6) list.set_intros(2) val.inject(5))
              then have  "\<not> Stars (pre @ [v2] @ post) :\<sqsubset>val Stars (pre @ [v] @ post)"
                 using q by simp     
              with w show "False"
-                using PT_def \<open>v2 \<in> PT r (flat v)\<close> append_Cons flat.simps(7) mem_Collect_eq 
+                using LV_def \<open>v2 \<in> LV r (flat v)\<close> append_Cons flat.simps(7) mem_Collect_eq 
                 PosOrd_StarsI PosOrd_Stars_appendI by auto
           qed     
         with IH
-        show "flat v \<in> r \<rightarrow> v \<and> flat v \<noteq> []" using a as2 unfolding eqs
-          using eqs(3) by (smt CPT_def CPrf_stars mem_Collect_eq) 
+        show "flat v \<in> r \<rightarrow> v \<and> flat v \<noteq> []" using a as2 unfolding eqs CV_def
+        by (auto elim: CPrf.cases)
      qed
    moreover
-   have "\<not> (\<exists>vs2\<in>PT (STAR r) (flat (Stars vs)). vs2 :\<sqsubset>val Stars vs)" 
+   have "\<not> (\<exists>vs2\<in>LV (STAR r) (flat (Stars vs)). vs2 :\<sqsubset>val Stars vs)" 
      proof 
-       assume "\<exists>vs2 \<in> PT (STAR r) (flat (Stars vs)). vs2 :\<sqsubset>val Stars vs"
+       assume "\<exists>vs2 \<in> LV (STAR r) (flat (Stars vs)). vs2 :\<sqsubset>val Stars vs"
        then obtain vs2 where "\<turnstile> Stars vs2 : STAR r" "flat (Stars vs2) = flat (Stars vs)"
                              "Stars vs2 :\<sqsubset>val Stars vs" 
-         unfolding PT_def
-         apply(auto elim: Prf.cases)
+         unfolding LV_def
+         apply(auto)
          apply(erule Prf.cases)
          apply(auto intro: Prf.intros)
          done
        then show "False" using as1 unfolding eqs
          apply -
          apply(drule_tac x="Stars vs2" in bspec)
-         apply(auto simp add: PT_def)
+         apply(auto simp add: LV_def)
          done
      qed
    ultimately have "flat (Stars vs) \<in> STAR r \<rightarrow> Stars vs"




