--- a/thys/LexerExt.thy	Tue Jan 25 13:12:50 2022 +0000
+++ b/thys/LexerExt.thy	Thu Jan 27 23:25:26 2022 +0000
@@ -17,10 +17,15 @@
 | "mkeps(NTIMES r n) = Stars (replicate n (mkeps r))"
 | "mkeps(FROMNTIMES r n) = Stars (replicate n (mkeps r))"
 | "mkeps(NMTIMES r n m) = Stars (replicate n (mkeps r))"
-  
+| "mkeps(NOT ZERO) = Nt Void"
+| "mkeps(NOT (CH _ )) = Nt Void"
+| "mkeps(NOT (SEQ r1 r2)) = Seq (mkeps (NOT r1)) (mkeps (NOT r1))"
+| "mkeps(NOT (ALT r1 r2)) = (if nullable(r1) then Right (mkeps (NOT r2)) else  (mkeps (NOT r1)))"
+
+
 fun injval :: "rexp \<Rightarrow> char \<Rightarrow> val \<Rightarrow> val"
 where
-  "injval (CHAR d) c Void = Char d"
+  "injval (CH d) c Void = Char d"
 | "injval (ALT r1 r2) c (Left v1) = Left(injval r1 c v1)"
 | "injval (ALT r1 r2) c (Right v2) = Right(injval r2 c v2)"
 | "injval (SEQ r1 r2) c (Seq v1 v2) = Seq (injval r1 c v1) v2"
@@ -49,9 +54,11 @@
   shows "flat (mkeps r) = []"
 using assms
   apply(induct rule: nullable.induct) 
-         apply(auto)
-  by presburger  
-  
+          apply(auto)
+  apply presburger  
+  apply(case_tac r)
+  apply(auto)
+  sorry
   
 lemma mkeps_nullable:
   assumes "nullable(r)" 
@@ -81,7 +88,7 @@
     apply (simp add: mkeps_flat)
    apply(simp)
   apply(simp)
-done
+  sorry
     
 
 lemma Prf_injval_flat:
@@ -90,7 +97,7 @@
 using assms
 apply(induct arbitrary: v rule: der.induct)
 apply(auto elim!: Prf_elims intro: mkeps_flat split: if_splits)
-done
+  sorry
 
 lemma Prf_injval:
   assumes "\<Turnstile> v : der c r" 
@@ -170,7 +177,8 @@
      apply(simp add: Prf_injval_flat)
      apply(simp)
    apply(simp)
-done
+  sorry
+
 
 
 
@@ -186,8 +194,10 @@
 apply(auto intro: Posix.intros simp add: nullable_correctness Sequ_def)
 apply(subst append.simps(1)[symmetric])
 apply(rule Posix.intros)
-      apply(auto)
-  done
+   apply(auto)
+  apply(case_tac r)
+  apply(auto)
+  sorry
     
 
 lemma Posix_injval:
@@ -207,22 +217,22 @@
   then have "False" by cases
   then show "(c # s) \<in> ONE \<rightarrow> (injval ONE c v)" by simp
 next 
-  case (CHAR d)
+  case (CH d)
   consider (eq) "c = d" | (ineq) "c \<noteq> d" by blast
-  then show "(c # s) \<in> (CHAR d) \<rightarrow> (injval (CHAR d) c v)"
+  then show "(c # s) \<in> (CH d) \<rightarrow> (injval (CH d) c v)"
   proof (cases)
     case eq
-    have "s \<in> der c (CHAR d) \<rightarrow> v" by fact
+    have "s \<in> der c (CH d) \<rightarrow> v" by fact
     then have "s \<in> ONE \<rightarrow> v" using eq by simp
     then have eqs: "s = [] \<and> v = Void" by cases simp
-    show "(c # s) \<in> CHAR d \<rightarrow> injval (CHAR d) c v" using eq eqs 
+    show "(c # s) \<in> CH d \<rightarrow> injval (CH d) c v" using eq eqs 
     by (auto intro: Posix.intros)
   next
     case ineq
-    have "s \<in> der c (CHAR d) \<rightarrow> v" by fact
+    have "s \<in> der c (CH d) \<rightarrow> v" by fact
     then have "s \<in> ZERO \<rightarrow> v" using ineq by simp
     then have "False" by cases
-    then show "(c # s) \<in> CHAR d \<rightarrow> injval (CHAR d) c v" by simp
+    then show "(c # s) \<in> CH d \<rightarrow> injval (CH d) c v" by simp
   qed
 next
   case (ALT r1 r2)
@@ -319,7 +329,6 @@
     apply(simp)
     apply(subgoal_tac "\<exists>vss. v2 = Stars vss")
     apply(clarify)
-    apply(drule_tac x="v1" in meta_spec)
     apply(drule_tac x="vss" in meta_spec)
     apply(drule_tac x="s1" in meta_spec)
     apply(drule_tac x="s2" in meta_spec)
@@ -401,7 +410,6 @@
     apply(simp)
     apply(subgoal_tac "\<exists>vss. v2 = Stars vss")
     apply(clarify)
-    apply(drule_tac x="v1" in meta_spec)
     apply(drule_tac x="vss" in meta_spec)
     apply(drule_tac x="s1" in meta_spec)
     apply(drule_tac x="s2" in meta_spec)
@@ -454,7 +462,6 @@
     apply(simp)
     apply(subgoal_tac "\<exists>vss. v2 = Stars vss")
     apply(clarify)
-    apply(drule_tac x="v1" in meta_spec)
     apply(drule_tac x="vss" in meta_spec)
     apply(drule_tac x="s1" in meta_spec)
     apply(drule_tac x="s2" in meta_spec)
@@ -544,7 +551,6 @@
     apply(simp)
     apply(subgoal_tac "\<exists>vss. v2 = Stars vss")
     apply(clarify)
-    apply(drule_tac x="v1" in meta_spec)
     apply(drule_tac x="vss" in meta_spec)
     apply(drule_tac x="s1" in meta_spec)
     apply(drule_tac x="s2" in meta_spec)
@@ -618,6 +624,9 @@
           apply(simp)
           done  
       qed    
+   next
+     case (NOT r s v)
+     then show ?case sorry
 qed
 
 section {* Lexer Correctness *}