
diff -r 232aa2f19a75 -r ec5e4fe4cc70 thys2/BitCoded.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/thys2/BitCoded.thy	Sun Oct 10 18:35:21 2021 +0100
@@ -0,0 +1,3369 @@
+
+theory BitCodedCT
+  imports "Lexer" 
+begin
+
+section \<open>Bit-Encodings\<close>
+
+datatype bit = Z | S
+
+fun 
+  code :: "val \<Rightarrow> bit list"
+where
+  "code Void = []"
+| "code (Char c) = []"
+| "code (Left v) = Z # (code v)"
+| "code (Right v) = S # (code v)"
+| "code (Seq v1 v2) = (code v1) @ (code v2)"
+| "code (Stars []) = [S]"
+| "code (Stars (v # vs)) =  (Z # code v) @ code (Stars vs)"
+
+
+fun 
+  Stars_add :: "val \<Rightarrow> val \<Rightarrow> val"
+where
+  "Stars_add v (Stars vs) = Stars (v # vs)"
+
+function
+  decode' :: "bit list \<Rightarrow> rexp \<Rightarrow> (val * bit list)"
+where
+  "decode' ds ZERO = (Void, [])"
+| "decode' ds ONE = (Void, ds)"
+| "decode' ds (CHAR d) = (Char d, ds)"
+| "decode' [] (ALT r1 r2) = (Void, [])"
+| "decode' (Z # ds) (ALT r1 r2) = (let (v, ds') = decode' ds r1 in (Left v, ds'))"
+| "decode' (S # ds) (ALT r1 r2) = (let (v, ds') = decode' ds r2 in (Right v, ds'))"
+| "decode' ds (SEQ r1 r2) = (let (v1, ds') = decode' ds r1 in
+                             let (v2, ds'') = decode' ds' r2 in (Seq v1 v2, ds''))"
+| "decode' [] (STAR r) = (Void, [])"
+| "decode' (S # ds) (STAR r) = (Stars [], ds)"
+| "decode' (Z # ds) (STAR r) = (let (v, ds') = decode' ds r in
+                                    let (vs, ds'') = decode' ds' (STAR r) 
+                                    in (Stars_add v vs, ds''))"
+by pat_completeness auto
+
+lemma decode'_smaller:
+  assumes "decode'_dom (ds, r)"
+  shows "length (snd (decode' ds r)) \<le> length ds"
+using assms
+apply(induct ds r)
+apply(auto simp add: decode'.psimps split: prod.split)
+using dual_order.trans apply blast
+by (meson dual_order.trans le_SucI)
+
+termination "decode'"  
+apply(relation "inv_image (measure(%cs. size cs) <*lex*> measure(%s. size s)) (%(ds,r). (r,ds))") 
+apply(auto dest!: decode'_smaller)
+by (metis less_Suc_eq_le snd_conv)
+
+definition
+  decode :: "bit list \<Rightarrow> rexp \<Rightarrow> val option"
+where
+  "decode ds r \<equiv> (let (v, ds') = decode' ds r 
+                  in (if ds' = [] then Some v else None))"
+
+lemma decode'_code_Stars:
+  assumes "\<forall>v\<in>set vs. \<Turnstile> v : r \<and> (\<forall>x. decode' (code v @ x) r = (v, x)) \<and> flat v \<noteq> []" 
+  shows "decode' (code (Stars vs) @ ds) (STAR r) = (Stars vs, ds)"
+  using assms
+  apply(induct vs)
+  apply(auto)
+  done
+
+lemma decode'_code:
+  assumes "\<Turnstile> v : r"
+  shows "decode' ((code v) @ ds) r = (v, ds)"
+using assms
+  apply(induct v r arbitrary: ds) 
+  apply(auto)
+  using decode'_code_Stars by blast
+
+lemma decode_code:
+  assumes "\<Turnstile> v : r"
+  shows "decode (code v) r = Some v"
+  using assms unfolding decode_def
+  by (smt append_Nil2 decode'_code old.prod.case)
+
+
+section {* Annotated Regular Expressions *}
+
+datatype arexp = 
+  AZERO
+| AONE "bit list"
+| ACHAR "bit list" char
+| ASEQ "bit list" arexp arexp
+| AALTs "bit list" "arexp list"
+| ASTAR "bit list" arexp
+
+abbreviation
+  "AALT bs r1 r2 \<equiv> AALTs bs [r1, r2]"
+
+fun asize :: "arexp \<Rightarrow> nat" where
+  "asize AZERO = 1"
+| "asize (AONE cs) = 1" 
+| "asize (ACHAR cs c) = 1"
+| "asize (AALTs cs rs) = Suc (sum_list (map asize rs))"
+| "asize (ASEQ cs r1 r2) = Suc (asize r1 + asize r2)"
+| "asize (ASTAR cs r) = Suc (asize r)"
+
+fun 
+  erase :: "arexp \<Rightarrow> rexp"
+where
+  "erase AZERO = ZERO"
+| "erase (AONE _) = ONE"
+| "erase (ACHAR _ c) = CHAR c"
+| "erase (AALTs _ []) = ZERO"
+| "erase (AALTs _ [r]) = (erase r)"
+| "erase (AALTs bs (r#rs)) = ALT (erase r) (erase (AALTs bs rs))"
+| "erase (ASEQ _ r1 r2) = SEQ (erase r1) (erase r2)"
+| "erase (ASTAR _ r) = STAR (erase r)"
+
+lemma decode_code_erase:
+  assumes "\<Turnstile> v : (erase  a)"
+  shows "decode (code v) (erase a) = Some v"
+  using assms
+  by (simp add: decode_code) 
+
+
+fun nonalt :: "arexp \<Rightarrow> bool"
+  where
+  "nonalt (AALTs bs2 rs) = False"
+| "nonalt r = True"
+
+
+fun good :: "arexp \<Rightarrow> bool" where
+  "good AZERO = False"
+| "good (AONE cs) = True" 
+| "good (ACHAR cs c) = True"
+| "good (AALTs cs []) = False"
+| "good (AALTs cs [r]) = False"
+| "good (AALTs cs (r1#r2#rs)) = (\<forall>r' \<in> set (r1#r2#rs). good r' \<and> nonalt r')"
+| "good (ASEQ _ AZERO _) = False"
+| "good (ASEQ _ (AONE _) _) = False"
+| "good (ASEQ _ _ AZERO) = False"
+| "good (ASEQ cs r1 r2) = (good r1 \<and> good r2)"
+| "good (ASTAR cs r) = True"
+
+
+
+
+fun fuse :: "bit list \<Rightarrow> arexp \<Rightarrow> arexp" where
+  "fuse bs AZERO = AZERO"
+| "fuse bs (AONE cs) = AONE (bs @ cs)" 
+| "fuse bs (ACHAR cs c) = ACHAR (bs @ cs) c"
+| "fuse bs (AALTs cs rs) = AALTs (bs @ cs) rs"
+| "fuse bs (ASEQ cs r1 r2) = ASEQ (bs @ cs) r1 r2"
+| "fuse bs (ASTAR cs r) = ASTAR (bs @ cs) r"
+
+lemma fuse_append:
+  shows "fuse (bs1 @ bs2) r = fuse bs1 (fuse bs2 r)"
+  apply(induct r)
+  apply(auto)
+  done
+
+
+fun intern :: "rexp \<Rightarrow> arexp" where
+  "intern ZERO = AZERO"
+| "intern ONE = AONE []"
+| "intern (CHAR c) = ACHAR [] c"
+| "intern (ALT r1 r2) = AALT [] (fuse [Z] (intern r1)) 
+                                (fuse [S]  (intern r2))"
+| "intern (SEQ r1 r2) = ASEQ [] (intern r1) (intern r2)"
+| "intern (STAR r) = ASTAR [] (intern r)"
+
+
+fun retrieve :: "arexp \<Rightarrow> val \<Rightarrow> bit list" where
+  "retrieve (AONE bs) Void = bs"
+| "retrieve (ACHAR bs c) (Char d) = bs"
+| "retrieve (AALTs bs [r]) v = bs @ retrieve r v" 
+| "retrieve (AALTs bs (r#rs)) (Left v) = bs @ retrieve r v"
+| "retrieve (AALTs bs (r#rs)) (Right v) = bs @ retrieve (AALTs [] rs) v"
+| "retrieve (ASEQ bs r1 r2) (Seq v1 v2) = bs @ retrieve r1 v1 @ retrieve r2 v2"
+| "retrieve (ASTAR bs r) (Stars []) = bs @ [S]"
+| "retrieve (ASTAR bs r) (Stars (v#vs)) = 
+     bs @ [Z] @ retrieve r v @ retrieve (ASTAR [] r) (Stars vs)"
+
+
+
+fun
+ bnullable :: "arexp \<Rightarrow> bool"
+where
+  "bnullable (AZERO) = False"
+| "bnullable (AONE bs) = True"
+| "bnullable (ACHAR bs c) = False"
+| "bnullable (AALTs bs rs) = (\<exists>r \<in> set rs. bnullable r)"
+| "bnullable (ASEQ bs r1 r2) = (bnullable r1 \<and> bnullable r2)"
+| "bnullable (ASTAR bs r) = True"
+
+fun 
+  bmkeps :: "arexp \<Rightarrow> bit list"
+where
+  "bmkeps(AONE bs) = bs"
+| "bmkeps(ASEQ bs r1 r2) = bs @ (bmkeps r1) @ (bmkeps r2)"
+| "bmkeps(AALTs bs [r]) = bs @ (bmkeps r)"
+| "bmkeps(AALTs bs (r#rs)) = (if bnullable(r) then bs @ (bmkeps r) else (bmkeps (AALTs bs rs)))"
+| "bmkeps(ASTAR bs r) = bs @ [S]"
+
+
+fun
+ bder :: "char \<Rightarrow> arexp \<Rightarrow> arexp"
+where
+  "bder c (AZERO) = AZERO"
+| "bder c (AONE bs) = AZERO"
+| "bder c (ACHAR bs d) = (if c = d then AONE bs else AZERO)"
+| "bder c (AALTs bs rs) = AALTs bs (map (bder c) rs)"
+| "bder c (ASEQ bs r1 r2) = 
+     (if bnullable r1
+      then AALT bs (ASEQ [] (bder c r1) r2) (fuse (bmkeps r1) (bder c r2))
+      else ASEQ bs (bder c r1) r2)"
+| "bder c (ASTAR bs r) = ASEQ bs (fuse [Z] (bder c r)) (ASTAR [] r)"
+
+
+fun 
+  bders :: "arexp \<Rightarrow> string \<Rightarrow> arexp"
+where
+  "bders r [] = r"
+| "bders r (c#s) = bders (bder c r) s"
+
+lemma bders_append:
+  "bders r (s1 @ s2) = bders (bders r s1) s2"
+  apply(induct s1 arbitrary: r s2)
+  apply(simp_all)
+  done
+
+lemma bnullable_correctness:
+  shows "nullable (erase r) = bnullable r"
+  apply(induct r rule: erase.induct)
+  apply(simp_all)
+  done
+
+lemma erase_fuse:
+  shows "erase (fuse bs r) = erase r"
+  apply(induct r rule: erase.induct)
+  apply(simp_all)
+  done
+
+lemma erase_intern [simp]:
+  shows "erase (intern r) = r"
+  apply(induct r)
+  apply(simp_all add: erase_fuse)
+  done
+
+lemma erase_bder [simp]:
+  shows "erase (bder a r) = der a (erase r)"
+  apply(induct r rule: erase.induct)
+  apply(simp_all add: erase_fuse bnullable_correctness)
+  done
+
+lemma erase_bders [simp]:
+  shows "erase (bders r s) = ders s (erase r)"
+  apply(induct s arbitrary: r )
+  apply(simp_all)
+  done
+
+lemma retrieve_encode_STARS:
+  assumes "\<forall>v\<in>set vs. \<Turnstile> v : r \<and> code v = retrieve (intern r) v"
+  shows "code (Stars vs) = retrieve (ASTAR [] (intern r)) (Stars vs)"
+  using assms
+  apply(induct vs)
+  apply(simp_all)
+  done
+
+lemma retrieve_fuse2:
+  assumes "\<Turnstile> v : (erase r)"
+  shows "retrieve (fuse bs r) v = bs @ retrieve r v"
+  using assms
+  apply(induct r arbitrary: v bs)
+         apply(auto elim: Prf_elims)[4]
+   defer
+  using retrieve_encode_STARS
+   apply(auto elim!: Prf_elims)[1]
+   apply(case_tac vs)
+    apply(simp)
+   apply(simp)
+  (* AALTs  case *)
+  apply(simp)
+  apply(case_tac x2a)
+   apply(simp)
+   apply(auto elim!: Prf_elims)[1]
+  apply(simp)
+   apply(case_tac list)
+   apply(simp)
+  apply(auto)
+  apply(auto elim!: Prf_elims)[1]
+  done
+
+lemma retrieve_fuse:
+  assumes "\<Turnstile> v : r"
+  shows "retrieve (fuse bs (intern r)) v = bs @ retrieve (intern r) v"
+  using assms 
+  by (simp_all add: retrieve_fuse2)
+
+
+lemma retrieve_code:
+  assumes "\<Turnstile> v : r"
+  shows "code v = retrieve (intern r) v"
+  using assms
+  apply(induct v r )
+  apply(simp_all add: retrieve_fuse retrieve_encode_STARS)
+  done
+
+lemma r:
+  assumes "bnullable (AALTs bs (a # rs))"
+  shows "bnullable a \<or> (\<not> bnullable a \<and> bnullable (AALTs bs rs))"
+  using assms
+  apply(induct rs)
+   apply(auto)
+  done
+
+lemma r0:
+  assumes "bnullable a" 
+  shows  "bmkeps (AALTs bs (a # rs)) = bs @ (bmkeps a)"
+  using assms
+  by (metis bmkeps.simps(3) bmkeps.simps(4) list.exhaust)
+
+lemma r1:
+  assumes "\<not> bnullable a" "bnullable (AALTs bs rs)"
+  shows  "bmkeps (AALTs bs (a # rs)) = bmkeps (AALTs bs rs)"
+  using assms
+  apply(induct rs)
+   apply(auto)
+  done
+
+lemma r2:
+  assumes "x \<in> set rs" "bnullable x"
+  shows "bnullable (AALTs bs rs)"
+  using assms
+  apply(induct rs)
+   apply(auto)
+  done
+
+lemma  r3:
+  assumes "\<not> bnullable r" 
+          " \<exists> x \<in> set rs. bnullable x"
+  shows "retrieve (AALTs bs rs) (mkeps (erase (AALTs bs rs))) =
+         retrieve (AALTs bs (r # rs)) (mkeps (erase (AALTs bs (r # rs))))"
+  using assms
+  apply(induct rs arbitrary: r bs)
+   apply(auto)[1]
+  apply(auto)
+  using bnullable_correctness apply blast
+    apply(auto simp add: bnullable_correctness mkeps_nullable retrieve_fuse2)
+   apply(subst retrieve_fuse2[symmetric])
+  apply (smt bnullable.simps(4) bnullable_correctness erase.simps(5) erase.simps(6) insert_iff list.exhaust list.set(2) mkeps.simps(3) mkeps_nullable)
+   apply(simp)
+  apply(case_tac "bnullable a")
+  apply (smt append_Nil2 bnullable.simps(4) bnullable_correctness erase.simps(5) erase.simps(6) fuse.simps(4) insert_iff list.exhaust list.set(2) mkeps.simps(3) mkeps_nullable retrieve_fuse2)
+  apply(drule_tac x="a" in meta_spec)
+  apply(drule_tac x="bs" in meta_spec)
+  apply(drule meta_mp)
+   apply(simp)
+  apply(drule meta_mp)
+   apply(auto)
+  apply(subst retrieve_fuse2[symmetric])
+  apply(case_tac rs)
+    apply(simp)
+   apply(auto)[1]
+      apply (simp add: bnullable_correctness)
+  apply (metis append_Nil2 bnullable_correctness erase_fuse fuse.simps(4) list.set_intros(1) mkeps.simps(3) mkeps_nullable nullable.simps(4) r2)
+    apply (simp add: bnullable_correctness)
+  apply (metis append_Nil2 bnullable_correctness erase.simps(6) erase_fuse fuse.simps(4) list.set_intros(2) mkeps.simps(3) mkeps_nullable r2)
+  apply(simp)
+  done
+
+
+lemma t: 
+  assumes "\<forall>r \<in> set rs. nullable (erase r) \<longrightarrow> bmkeps r = retrieve r (mkeps (erase r))" 
+          "nullable (erase (AALTs bs rs))"
+  shows " bmkeps (AALTs bs rs) = retrieve (AALTs bs rs) (mkeps (erase (AALTs bs rs)))"
+  using assms
+  apply(induct rs arbitrary: bs)
+   apply(simp)
+  apply(auto simp add: bnullable_correctness)
+   apply(case_tac rs)
+     apply(auto simp add: bnullable_correctness)[2]
+   apply(subst r1)
+     apply(simp)
+    apply(rule r2)
+     apply(assumption)
+    apply(simp)
+   apply(drule_tac x="bs" in meta_spec)
+   apply(drule meta_mp)
+    apply(auto)[1]
+   prefer 2
+  apply(case_tac "bnullable a")
+    apply(subst r0)
+     apply blast
+    apply(subgoal_tac "nullable (erase a)")
+  prefer 2
+  using bnullable_correctness apply blast
+  apply (metis (no_types, lifting) erase.simps(5) erase.simps(6) list.exhaust mkeps.simps(3) retrieve.simps(3) retrieve.simps(4))
+  apply(subst r1)
+     apply(simp)
+  using r2 apply blast
+  apply(drule_tac x="bs" in meta_spec)
+   apply(drule meta_mp)
+    apply(auto)[1]
+   apply(simp)
+  using r3 apply blast
+  apply(auto)
+  using r3 by blast
+
+lemma bmkeps_retrieve:
+  assumes "nullable (erase r)"
+  shows "bmkeps r = retrieve r (mkeps (erase r))"
+  using assms
+  apply(induct r)
+         apply(simp)
+        apply(simp)
+       apply(simp)
+    apply(simp)
+   defer
+   apply(simp)
+  apply(rule t)
+   apply(auto)
+  done
+
+lemma bder_retrieve:
+  assumes "\<Turnstile> v : der c (erase r)"
+  shows "retrieve (bder c r) v = retrieve r (injval (erase r) c v)"
+  using assms
+  apply(induct r arbitrary: v rule: erase.induct)
+         apply(simp)
+         apply(erule Prf_elims)
+        apply(simp)
+        apply(erule Prf_elims) 
+        apply(simp)
+      apply(case_tac "c = ca")
+       apply(simp)
+       apply(erule Prf_elims)
+       apply(simp)
+      apply(simp)
+       apply(erule Prf_elims)
+  apply(simp)
+      apply(erule Prf_elims)
+     apply(simp)
+    apply(simp)
+  apply(rename_tac "r\<^sub>1" "r\<^sub>2" rs v)
+    apply(erule Prf_elims)
+     apply(simp)
+    apply(simp)
+    apply(case_tac rs)
+     apply(simp)
+    apply(simp)
+  apply (smt Prf_elims(3) injval.simps(2) injval.simps(3) retrieve.simps(4) retrieve.simps(5) same_append_eq)
+   apply(simp)
+   apply(case_tac "nullable (erase r1)")
+    apply(simp)
+  apply(erule Prf_elims)
+     apply(subgoal_tac "bnullable r1")
+  prefer 2
+  using bnullable_correctness apply blast
+    apply(simp)
+     apply(erule Prf_elims)
+     apply(simp)
+   apply(subgoal_tac "bnullable r1")
+  prefer 2
+  using bnullable_correctness apply blast
+    apply(simp)
+    apply(simp add: retrieve_fuse2)
+    apply(simp add: bmkeps_retrieve)
+   apply(simp)
+   apply(erule Prf_elims)
+   apply(simp)
+  using bnullable_correctness apply blast
+  apply(rename_tac bs r v)
+  apply(simp)
+  apply(erule Prf_elims)
+     apply(clarify)
+  apply(erule Prf_elims)
+  apply(clarify)
+  apply(subst injval.simps)
+  apply(simp del: retrieve.simps)
+  apply(subst retrieve.simps)
+  apply(subst retrieve.simps)
+  apply(simp)
+  apply(simp add: retrieve_fuse2)
+  done
+  
+
+
+lemma MAIN_decode:
+  assumes "\<Turnstile> v : ders s r"
+  shows "Some (flex r id s v) = decode (retrieve (bders (intern r) s) v) r"
+  using assms
+proof (induct s arbitrary: v rule: rev_induct)
+  case Nil
+  have "\<Turnstile> v : ders [] r" by fact
+  then have "\<Turnstile> v : r" by simp
+  then have "Some v = decode (retrieve (intern r) v) r"
+    using decode_code retrieve_code by auto
+  then show "Some (flex r id [] v) = decode (retrieve (bders (intern r) []) v) r"
+    by simp
+next
+  case (snoc c s v)
+  have IH: "\<And>v. \<Turnstile> v : ders s r \<Longrightarrow> 
+     Some (flex r id s v) = decode (retrieve (bders (intern r) s) v) r" by fact
+  have asm: "\<Turnstile> v : ders (s @ [c]) r" by fact
+  then have asm2: "\<Turnstile> injval (ders s r) c v : ders s r" 
+    by (simp add: Prf_injval ders_append)
+  have "Some (flex r id (s @ [c]) v) = Some (flex r id s (injval (ders s r) c v))"
+    by (simp add: flex_append)
+  also have "... = decode (retrieve (bders (intern r) s) (injval (ders s r) c v)) r"
+    using asm2 IH by simp
+  also have "... = decode (retrieve (bder c (bders (intern r) s)) v) r"
+    using asm by (simp_all add: bder_retrieve ders_append)
+  finally show "Some (flex r id (s @ [c]) v) = 
+                 decode (retrieve (bders (intern r) (s @ [c])) v) r" by (simp add: bders_append)
+qed
+
+
+definition blex where
+ "blex a s \<equiv> if bnullable (bders a s) then Some (bmkeps (bders a s)) else None"
+
+
+
+definition blexer where
+ "blexer r s \<equiv> if bnullable (bders (intern r) s) then 
+                decode (bmkeps (bders (intern r) s)) r else None"
+
+lemma blexer_correctness:
+  shows "blexer r s = lexer r s"
+proof -
+  { define bds where "bds \<equiv> bders (intern r) s"
+    define ds  where "ds \<equiv> ders s r"
+    assume asm: "nullable ds"
+    have era: "erase bds = ds" 
+      unfolding ds_def bds_def by simp
+    have mke: "\<Turnstile> mkeps ds : ds"
+      using asm by (simp add: mkeps_nullable)
+    have "decode (bmkeps bds) r = decode (retrieve bds (mkeps ds)) r"
+      using bmkeps_retrieve
+      using asm era by (simp add: bmkeps_retrieve)
+    also have "... =  Some (flex r id s (mkeps ds))"
+      using mke by (simp_all add: MAIN_decode ds_def bds_def)
+    finally have "decode (bmkeps bds) r = Some (flex r id s (mkeps ds))" 
+      unfolding bds_def ds_def .
+  }
+  then show "blexer r s = lexer r s"
+    unfolding blexer_def lexer_flex
+    apply(subst bnullable_correctness[symmetric])
+    apply(simp)
+    done
+qed
+
+
+fun distinctBy :: "'a list \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b set \<Rightarrow> 'a list"
+  where
+  "distinctBy [] f acc = []"
+| "distinctBy (x#xs) f acc = 
+     (if (f x) \<in> acc then distinctBy xs f acc 
+      else x # (distinctBy xs f ({f x} \<union> acc)))"
+
+fun flts :: "arexp list \<Rightarrow> arexp list"
+  where 
+  "flts [] = []"
+| "flts (AZERO # rs) = flts rs"
+| "flts ((AALTs bs  rs1) # rs) = (map (fuse bs) rs1) @ flts rs"
+| "flts (r1 # rs) = r1 # flts rs"
+
+
+
+
+fun li :: "bit list \<Rightarrow> arexp list \<Rightarrow> arexp"
+  where
+  "li _ [] = AZERO"
+| "li bs [a] = fuse bs a"
+| "li bs as = AALTs bs as"
+
+
+
+
+fun bsimp_ASEQ :: "bit list \<Rightarrow> arexp \<Rightarrow> arexp \<Rightarrow> arexp"
+  where
+  "bsimp_ASEQ _ AZERO _ = AZERO"
+| "bsimp_ASEQ _ _ AZERO = AZERO"
+| "bsimp_ASEQ bs1 (AONE bs2) r2 = fuse (bs1 @ bs2) r2"
+| "bsimp_ASEQ bs1 r1 r2 = ASEQ  bs1 r1 r2"
+
+
+fun bsimp_AALTs :: "bit list \<Rightarrow> arexp list \<Rightarrow> arexp"
+  where
+  "bsimp_AALTs _ [] = AZERO"
+| "bsimp_AALTs bs1 [r] = fuse bs1 r"
+| "bsimp_AALTs bs1 rs = AALTs bs1 rs"
+
+
+fun bsimp :: "arexp \<Rightarrow> arexp" 
+  where
+  "bsimp (ASEQ bs1 r1 r2) = bsimp_ASEQ bs1 (bsimp r1) (bsimp r2)"
+| "bsimp (AALTs bs1 rs) = bsimp_AALTs bs1 (flts (map bsimp rs))"
+| "bsimp r = r"
+
+
+
+
+fun 
+  bders_simp :: "arexp \<Rightarrow> string \<Rightarrow> arexp"
+where
+  "bders_simp r [] = r"
+| "bders_simp r (c # s) = bders_simp (bsimp (bder c r)) s"
+
+definition blexer_simp where
+ "blexer_simp r s \<equiv> if bnullable (bders_simp (intern r) s) then 
+                decode (bmkeps (bders_simp (intern r) s)) r else None"
+
+
+lemma asize0:
+  shows "0 < asize r"
+  apply(induct  r)
+       apply(auto)
+  done
+
+
+lemma bders_simp_append:
+  shows "bders_simp r (s1 @ s2) = bders_simp (bders_simp r s1) s2"
+  apply(induct s1 arbitrary: r s2)
+   apply(simp)
+  apply(simp)
+  done
+
+lemma bsimp_ASEQ_size:
+  shows "asize (bsimp_ASEQ bs r1 r2) \<le> Suc (asize r1 + asize r2)"
+  apply(induct bs r1 r2 rule: bsimp_ASEQ.induct)
+  apply(auto)
+  done
+
+lemma fuse_size:
+  shows "asize (fuse bs r) = asize r"
+  apply(induct r)
+  apply(auto)
+  done
+
+lemma flts_size:
+  shows "sum_list (map asize (flts rs)) \<le> sum_list (map asize rs)"
+  apply(induct rs rule: flts.induct)
+        apply(simp_all)
+  by (metis (mono_tags, lifting) add_mono comp_apply eq_imp_le fuse_size le_SucI map_eq_conv)
+  
+
+lemma bsimp_AALTs_size:
+  shows "asize (bsimp_AALTs bs rs) \<le> Suc (sum_list (map asize rs))"
+  apply(induct rs rule: bsimp_AALTs.induct)
+  apply(auto simp add: fuse_size)
+  done
+
+
+lemma bsimp_size:
+  shows "asize (bsimp r) \<le> asize r"
+  apply(induct r)
+       apply(simp_all)
+   apply (meson Suc_le_mono add_mono_thms_linordered_semiring(1) bsimp_ASEQ_size le_trans)
+  apply(rule le_trans)
+   apply(rule bsimp_AALTs_size)
+  apply(simp)
+   apply(rule le_trans)
+   apply(rule flts_size)
+  by (simp add: sum_list_mono)
+
+lemma bsimp_asize0:
+  shows "(\<Sum>x\<leftarrow>rs. asize (bsimp x)) \<le> sum_list (map asize rs)"
+  apply(induct rs)
+   apply(auto)
+  by (simp add: add_mono bsimp_size)
+
+lemma bsimp_AALTs_size2:
+  assumes "\<forall>r \<in> set  rs. nonalt r"
+  shows "asize (bsimp_AALTs bs rs) \<ge> sum_list (map asize rs)"
+  using assms
+  apply(induct rs rule: bsimp_AALTs.induct)
+    apply(simp_all add: fuse_size)
+  done
+
+
+lemma qq:
+  shows "map (asize \<circ> fuse bs) rs = map asize rs"
+  apply(induct rs)
+   apply(auto simp add: fuse_size)
+  done
+
+lemma flts_size2:
+  assumes "\<exists>bs rs'. AALTs bs  rs' \<in> set rs"
+  shows "sum_list (map asize (flts rs)) < sum_list (map asize rs)"
+  using assms
+  apply(induct rs)
+   apply(auto simp add: qq)
+   apply (simp add: flts_size less_Suc_eq_le)
+  apply(case_tac a)
+       apply(auto simp add: qq)
+   prefer 2
+   apply (simp add: flts_size le_imp_less_Suc)
+  using less_Suc_eq by auto
+
+lemma bsimp_AALTs_size3:
+  assumes "\<exists>r \<in> set  (map bsimp rs). \<not>nonalt r"
+  shows "asize (bsimp (AALTs bs rs)) < asize (AALTs bs rs)"
+  using assms flts_size2
+  apply  -
+  apply(clarify)
+  apply(simp)
+  apply(drule_tac x="map bsimp rs" in meta_spec)
+  apply(drule meta_mp)
+  apply (metis list.set_map nonalt.elims(3))
+  apply(simp)
+  apply(rule order_class.order.strict_trans1)
+   apply(rule bsimp_AALTs_size)
+  apply(simp)
+  by (smt Suc_leI bsimp_asize0 comp_def le_imp_less_Suc le_trans map_eq_conv not_less_eq)
+
+
+
+
+lemma L_bsimp_ASEQ:
+  "L (SEQ (erase r1) (erase r2)) = L (erase (bsimp_ASEQ bs r1 r2))"
+  apply(induct bs r1 r2 rule: bsimp_ASEQ.induct)
+  apply(simp_all)
+  by (metis erase_fuse fuse.simps(4))
+
+lemma L_bsimp_AALTs:
+  "L (erase (AALTs bs rs)) = L (erase (bsimp_AALTs bs rs))"
+  apply(induct bs rs rule: bsimp_AALTs.induct)
+  apply(simp_all add: erase_fuse)
+  done
+
+lemma L_erase_AALTs:
+  shows "L (erase (AALTs bs rs)) = \<Union> (L ` erase ` (set rs))"
+  apply(induct rs)
+   apply(simp)
+  apply(simp)
+  apply(case_tac rs)
+   apply(simp)
+  apply(simp)
+  done
+
+lemma L_erase_flts:
+  shows "\<Union> (L ` erase ` (set (flts rs))) = \<Union> (L ` erase ` (set rs))"
+  apply(induct rs rule: flts.induct)
+        apply(simp_all)
+  apply(auto)
+  using L_erase_AALTs erase_fuse apply auto[1]
+  by (simp add: L_erase_AALTs erase_fuse)
+
+
+lemma L_bsimp_erase:
+  shows "L (erase r) = L (erase (bsimp r))"
+  apply(induct r)
+  apply(simp)
+  apply(simp)
+  apply(simp)
+  apply(auto simp add: Sequ_def)[1]
+  apply(subst L_bsimp_ASEQ[symmetric])
+  apply(auto simp add: Sequ_def)[1]
+  apply(subst (asm)  L_bsimp_ASEQ[symmetric])
+  apply(auto simp add: Sequ_def)[1]
+   apply(simp)
+   apply(subst L_bsimp_AALTs[symmetric])
+   defer
+   apply(simp)
+  apply(subst (2)L_erase_AALTs)
+  apply(subst L_erase_flts)
+  apply(auto)
+   apply (simp add: L_erase_AALTs)
+  using L_erase_AALTs by blast
+
+lemma bsimp_ASEQ0:
+  shows "bsimp_ASEQ bs r1 AZERO = AZERO"
+  apply(induct r1)
+  apply(auto)
+  done
+
+
+
+lemma bsimp_ASEQ1:
+  assumes "r1 \<noteq> AZERO" "r2 \<noteq> AZERO" "\<forall>bs. r1 \<noteq> AONE bs"
+  shows "bsimp_ASEQ bs r1 r2 = ASEQ bs r1 r2"
+  using assms
+  apply(induct bs r1 r2 rule: bsimp_ASEQ.induct)
+  apply(auto)
+  done
+
+lemma bsimp_ASEQ2:
+  shows "bsimp_ASEQ bs (AONE bs1) r2 = fuse (bs @ bs1) r2"
+  apply(induct r2)
+  apply(auto)
+  done
+
+
+lemma L_bders_simp:
+  shows "L (erase (bders_simp r s)) = L (erase (bders r s))"
+  apply(induct s arbitrary: r rule: rev_induct)
+   apply(simp)
+  apply(simp)
+  apply(simp add: ders_append)
+  apply(simp add: bders_simp_append)
+  apply(simp add: L_bsimp_erase[symmetric])
+  by (simp add: der_correctness)
+
+lemma b1:
+  "bsimp_ASEQ bs1 (AONE bs) r =  fuse (bs1 @ bs) r" 
+  apply(induct r)
+       apply(auto)
+  done
+
+lemma b2:
+  assumes "bnullable r"
+  shows "bmkeps (fuse bs r) = bs @ bmkeps r"
+  by (simp add: assms bmkeps_retrieve bnullable_correctness erase_fuse mkeps_nullable retrieve_fuse2)
+
+lemma b3:
+  shows "bnullable r = bnullable (bsimp r)"
+  using L_bsimp_erase bnullable_correctness nullable_correctness by auto
+
+
+lemma b4:
+  shows "bnullable (bders_simp r s) = bnullable (bders r s)"
+  by (metis L_bders_simp bnullable_correctness lexer.simps(1) lexer_correct_None option.distinct(1))
+
+lemma q1:
+  assumes "\<forall>r \<in> set rs. bmkeps(bsimp r) = bmkeps r"
+  shows "map (\<lambda>r. bmkeps(bsimp r)) rs = map bmkeps rs"
+  using assms
+  apply(induct rs)
+  apply(simp)
+  apply(simp)
+  done
+
+lemma q3:
+  assumes "\<exists>r \<in> set rs. bnullable r"
+  shows "bmkeps (AALTs bs rs) = bmkeps (bsimp_AALTs bs rs)"
+  using assms
+  apply(induct bs rs rule: bsimp_AALTs.induct)
+    apply(simp)
+   apply(simp)
+  apply (simp add: b2)
+  apply(simp)
+  done
+
+lemma qq1:
+  assumes "\<exists>r \<in> set rs. bnullable r"
+  shows "bmkeps (AALTs bs (rs @ rs1)) = bmkeps (AALTs bs rs)"
+  using assms
+  apply(induct rs arbitrary: rs1 bs)
+  apply(simp)
+  apply(simp)
+  by (metis Nil_is_append_conv bmkeps.simps(4) neq_Nil_conv r0 split_list_last)
+
+lemma qq2:
+  assumes "\<forall>r \<in> set rs. \<not> bnullable r" "\<exists>r \<in> set rs1. bnullable r"
+  shows "bmkeps (AALTs bs (rs @ rs1)) = bmkeps (AALTs bs rs1)"
+  using assms
+  apply(induct rs arbitrary: rs1 bs)
+  apply(simp)
+  apply(simp)
+  by (metis append_assoc in_set_conv_decomp r1 r2)
+  
+lemma qq3:
+  shows "bnullable (AALTs bs rs) = (\<exists>r \<in> set rs. bnullable r)"
+  apply(induct rs arbitrary: bs)
+  apply(simp)
+  apply(simp)
+  done
+
+lemma fuse_empty:
+  shows "fuse [] r = r"
+  apply(induct r)
+       apply(auto)
+  done
+
+lemma flts_fuse:
+  shows "map (fuse bs) (flts rs) = flts (map (fuse bs) rs)"
+  apply(induct rs arbitrary: bs rule: flts.induct)
+        apply(auto simp add: fuse_append)
+  done
+
+lemma bsimp_ASEQ_fuse:
+  shows "fuse bs1 (bsimp_ASEQ bs2 r1 r2) = bsimp_ASEQ (bs1 @ bs2) r1 r2"
+  apply(induct r1 r2 arbitrary: bs1 bs2 rule: bsimp_ASEQ.induct)
+  apply(auto)
+  done
+
+lemma bsimp_AALTs_fuse:
+  assumes "\<forall>r \<in> set rs. fuse bs1 (fuse bs2 r) = fuse (bs1 @ bs2) r"
+  shows "fuse bs1 (bsimp_AALTs bs2 rs) = bsimp_AALTs (bs1 @ bs2) rs"
+  using assms
+  apply(induct bs2 rs arbitrary: bs1 rule: bsimp_AALTs.induct)
+  apply(auto)
+  done
+
+
+
+lemma bsimp_fuse:
+  shows "fuse bs (bsimp r) = bsimp (fuse bs r)"
+apply(induct r arbitrary: bs)
+       apply(simp)
+      apply(simp)
+     apply(simp)
+    prefer 3
+    apply(simp)
+   apply(simp)
+   apply (simp add: bsimp_ASEQ_fuse)
+  apply(simp)
+  by (simp add: bsimp_AALTs_fuse fuse_append)
+
+lemma bsimp_fuse_AALTs:
+  shows "fuse bs (bsimp (AALTs [] rs)) = bsimp (AALTs bs rs)"
+  apply(subst bsimp_fuse) 
+  apply(simp)
+  done
+
+lemma bsimp_fuse_AALTs2:
+  shows "fuse bs (bsimp_AALTs [] rs) = bsimp_AALTs bs rs"
+  using bsimp_AALTs_fuse fuse_append by auto
+  
+
+lemma bsimp_ASEQ_idem:
+  assumes "bsimp (bsimp r1) = bsimp r1" "bsimp (bsimp r2) = bsimp r2"
+  shows "bsimp (bsimp_ASEQ x1 (bsimp r1) (bsimp r2)) = bsimp_ASEQ x1 (bsimp r1) (bsimp r2)"
+  using assms
+  apply(case_tac "bsimp r1 = AZERO")
+    apply(simp)
+ apply(case_tac "bsimp r2 = AZERO")
+    apply(simp)
+  apply (metis bnullable.elims(2) bnullable.elims(3) bsimp.simps(3) bsimp_ASEQ.simps(2) bsimp_ASEQ.simps(3) bsimp_ASEQ.simps(4) bsimp_ASEQ.simps(5) bsimp_ASEQ.simps(6))  
+  apply(case_tac "\<exists>bs. bsimp r1 = AONE bs")
+    apply(auto)[1]
+    apply(subst bsimp_ASEQ2)
+   apply(subst bsimp_ASEQ2)
+  apply (metis assms(2) bsimp_fuse)
+      apply(subst bsimp_ASEQ1)
+      apply(auto)
+  done
+
+
+fun nonnested :: "arexp \<Rightarrow> bool"
+  where
+  "nonnested (AALTs bs2 []) = True"
+| "nonnested (AALTs bs2 ((AALTs bs1 rs1) # rs2)) = False"
+| "nonnested (AALTs bs2 (r # rs2)) = nonnested (AALTs bs2 rs2)"
+| "nonnested r = True"
+
+
+lemma  k0:
+  shows "flts (r # rs1) = flts [r] @ flts rs1"
+  apply(induct r arbitrary: rs1)
+   apply(auto)
+  done
+
+lemma  k00:
+  shows "flts (rs1 @ rs2) = flts rs1 @ flts rs2"
+  apply(induct rs1 arbitrary: rs2)
+   apply(auto)
+  by (metis append.assoc k0)
+
+lemma  k0a:
+  shows "flts [AALTs bs rs] = map (fuse bs)  rs"
+  apply(simp)
+  done
+
+
+lemma  k0b:
+  assumes "nonalt r" "r \<noteq> AZERO"
+  shows "flts [r] = [r]"
+  using assms
+  apply(case_tac  r)
+  apply(simp_all)
+  done
+
+lemma nn1:
+  assumes "nonnested (AALTs bs rs)"
+  shows "\<nexists>bs1 rs1. flts rs = [AALTs bs1 rs1]"
+  using assms
+  apply(induct rs rule: flts.induct)
+  apply(auto)
+  done
+
+lemma nn1q:
+  assumes "nonnested (AALTs bs rs)"
+  shows "\<nexists>bs1 rs1. AALTs bs1 rs1 \<in> set (flts rs)"
+  using assms
+  apply(induct rs rule: flts.induct)
+  apply(auto)
+  done
+
+lemma nn1qq:
+  assumes "nonnested (AALTs bs rs)"
+  shows "\<nexists>bs1 rs1. AALTs bs1 rs1 \<in> set rs"
+  using assms
+  apply(induct rs rule: flts.induct)
+  apply(auto)
+  done
+
+lemma nn10:
+  assumes "nonnested (AALTs cs rs)" 
+  shows "nonnested (AALTs (bs @ cs) rs)"
+  using assms
+  apply(induct rs arbitrary: cs bs)
+   apply(simp_all)
+  apply(case_tac a)
+       apply(simp_all)
+  done
+
+lemma nn11a:
+  assumes "nonalt r"
+  shows "nonalt (fuse bs r)"
+  using assms
+  apply(induct r)
+       apply(auto)
+  done
+
+
+lemma nn1a:
+  assumes "nonnested r"
+  shows "nonnested (fuse bs r)"
+  using assms
+  apply(induct bs r arbitrary: rule: fuse.induct)
+       apply(simp_all add: nn10)
+  done  
+
+lemma n0:
+  shows "nonnested (AALTs bs rs) \<longleftrightarrow> (\<forall>r \<in> set rs. nonalt r)"
+  apply(induct rs  arbitrary: bs)
+   apply(auto)
+    apply (metis list.set_intros(1) nn1qq nonalt.elims(3))
+   apply (metis list.set_intros(2) nn1qq nonalt.elims(3))
+  by (metis nonalt.elims(2) nonnested.simps(3) nonnested.simps(4) nonnested.simps(5) nonnested.simps(6) nonnested.simps(7))
+
+  
+  
+
+lemma nn1c:
+  assumes "\<forall>r \<in> set rs. nonnested r"
+  shows "\<forall>r \<in> set (flts rs). nonalt r"
+  using assms
+  apply(induct rs rule: flts.induct)
+        apply(auto)
+  apply(rule nn11a)
+  by (metis nn1qq nonalt.elims(3))
+
+lemma nn1bb:
+  assumes "\<forall>r \<in> set rs. nonalt r"
+  shows "nonnested (bsimp_AALTs bs rs)"
+  using assms
+  apply(induct bs rs rule: bsimp_AALTs.induct)
+    apply(auto)
+   apply (metis nn11a nonalt.simps(1) nonnested.elims(3))
+  using n0 by auto
+    
+lemma nn1b:
+  shows "nonnested (bsimp r)"
+  apply(induct r)
+       apply(simp_all)
+  apply(case_tac "bsimp r1 = AZERO")
+    apply(simp)
+ apply(case_tac "bsimp r2 = AZERO")
+   apply(simp)
+    apply(subst bsimp_ASEQ0)
+  apply(simp)
+  apply(case_tac "\<exists>bs. bsimp r1 = AONE bs")
+    apply(auto)[1]
+    apply(subst bsimp_ASEQ2)
+  apply (simp add: nn1a)    
+   apply(subst bsimp_ASEQ1)
+      apply(auto)
+  apply(rule nn1bb)
+  apply(auto)
+  by (metis (mono_tags, hide_lams) imageE nn1c set_map)
+
+lemma nn1d:
+  assumes "bsimp r = AALTs bs rs"
+  shows "\<forall>r1 \<in> set rs. \<forall>  bs. r1 \<noteq> AALTs bs  rs2"
+  using nn1b assms
+  by (metis nn1qq)
+
+lemma nn_flts:
+  assumes "nonnested (AALTs bs rs)"
+  shows "\<forall>r \<in>  set (flts rs). nonalt r"
+  using assms
+  apply(induct rs arbitrary: bs rule: flts.induct)
+        apply(auto)
+  done
+
+
+
+lemma rt:
+  shows "sum_list (map asize (flts (map bsimp rs))) \<le> sum_list (map asize rs)"
+  apply(induct rs)
+   apply(simp)
+  apply(simp)
+  apply(subst  k0)
+  apply(simp)
+  by (smt add_le_cancel_right add_mono bsimp_size flts.simps(1) flts_size k0 le_iff_add list.simps(9) map_append sum_list.Cons sum_list.append trans_le_add1)
+
+lemma bsimp_AALTs_qq:
+  assumes "1 < length rs"
+  shows "bsimp_AALTs bs rs = AALTs bs  rs"
+  using  assms
+  apply(case_tac rs)
+   apply(simp)
+  apply(case_tac list)
+   apply(simp_all)
+  done
+
+
+lemma bsimp_AALTs1:
+  assumes "nonalt r"
+  shows "bsimp_AALTs bs (flts [r]) = fuse bs r"
+  using  assms
+  apply(case_tac r)
+   apply(simp_all)
+  done
+
+lemma bbbbs:
+  assumes "good r" "r = AALTs bs1 rs"
+  shows "bsimp_AALTs bs (flts [r]) = AALTs bs (map (fuse bs1) rs)"
+  using  assms
+  by (metis (no_types, lifting) Nil_is_map_conv append.left_neutral append_butlast_last_id bsimp_AALTs.elims butlast.simps(2) good.simps(4) good.simps(5) k0a map_butlast)
+
+lemma bbbbs1:
+  shows "nonalt r \<or> (\<exists>bs rs. r  = AALTs bs rs)"
+  using nonalt.elims(3) by auto
+  
+
+lemma good_fuse:
+  shows "good (fuse bs r) = good r"
+  apply(induct r arbitrary: bs)
+       apply(auto)
+     apply(case_tac r1)
+          apply(simp_all)
+  apply(case_tac r2)
+          apply(simp_all)
+  apply(case_tac r2)
+            apply(simp_all)
+  apply(case_tac r2)
+           apply(simp_all)
+  apply(case_tac r2)
+          apply(simp_all)
+  apply(case_tac r1)
+          apply(simp_all)
+  apply(case_tac r2)
+           apply(simp_all)
+  apply(case_tac r2)
+           apply(simp_all)
+  apply(case_tac r2)
+           apply(simp_all)
+  apply(case_tac r2)
+         apply(simp_all)
+  apply(case_tac x2a)
+    apply(simp_all)
+  apply(case_tac list)
+    apply(simp_all)
+  apply(case_tac x2a)
+    apply(simp_all)
+  apply(case_tac list)
+    apply(simp_all)
+  done
+
+lemma good0:
+  assumes "rs \<noteq> Nil" "\<forall>r \<in> set rs. nonalt r"
+  shows "good (bsimp_AALTs bs rs) \<longleftrightarrow> (\<forall>r \<in> set rs. good r)"
+  using  assms
+  apply(induct bs rs rule: bsimp_AALTs.induct)
+  apply(auto simp add: good_fuse)
+  done
+
+lemma good0a:
+  assumes "flts (map bsimp rs) \<noteq> Nil" "\<forall>r \<in> set (flts (map bsimp rs)). nonalt r"
+  shows "good (bsimp (AALTs bs rs)) \<longleftrightarrow> (\<forall>r \<in> set (flts (map bsimp rs)). good r)"
+  using  assms
+  apply(simp)
+  apply(auto)
+  apply(subst (asm) good0)
+   apply(simp)
+    apply(auto)
+   apply(subst good0)
+   apply(simp)
+    apply(auto)
+  done
+
+lemma flts0:
+  assumes "r \<noteq> AZERO" "nonalt r"
+  shows "flts [r] \<noteq> []"
+  using  assms
+  apply(induct r)
+       apply(simp_all)
+  done
+
+lemma flts1:
+  assumes "good r" 
+  shows "flts [r] \<noteq> []"
+  using  assms
+  apply(induct r)
+       apply(simp_all)
+  apply(case_tac x2a)
+   apply(simp)
+  apply(simp)
+  done
+
+lemma flts2:
+  assumes "good r" 
+  shows "\<forall>r' \<in> set (flts [r]). good r' \<and> nonalt r'"
+  using  assms
+  apply(induct r)
+       apply(simp)
+      apply(simp)
+     apply(simp)
+    prefer 2
+    apply(simp)
+    apply(auto)[1]
+     apply (metis bsimp_AALTs.elims good.simps(4) good.simps(5) good.simps(6) good_fuse)
+  apply (metis bsimp_AALTs.elims good.simps(4) good.simps(5) good.simps(6) nn11a)
+   apply fastforce
+  apply(simp)
+  done  
+
+
+lemma flts3:
+  assumes "\<forall>r \<in> set rs. good r \<or> r = AZERO" 
+  shows "\<forall>r \<in> set (flts rs). good r"
+  using  assms
+  apply(induct rs arbitrary: rule: flts.induct)
+        apply(simp_all)
+  by (metis UnE flts2 k0a set_map)
+
+lemma flts3b:
+  assumes "\<exists>r\<in>set rs. good r"
+  shows "flts rs \<noteq> []"
+  using  assms
+  apply(induct rs arbitrary: rule: flts.induct)
+        apply(simp)
+       apply(simp)
+      apply(simp)
+      apply(auto)
+  done
+
+lemma flts4:
+  assumes "bsimp_AALTs bs (flts rs) = AZERO"
+  shows "\<forall>r \<in> set rs. \<not> good r"
+  using assms
+  apply(induct rs arbitrary: bs rule: flts.induct)
+        apply(auto)
+        defer
+  apply (metis (no_types, lifting) Nil_is_append_conv append_self_conv2 bsimp_AALTs.elims butlast.simps(2) butlast_append flts3b nonalt.simps(1) nonalt.simps(2))
+  apply (metis arexp.distinct(7) bsimp_AALTs.elims flts2 good.simps(1) good.simps(2) good0 k0b list.distinct(1) list.inject nonalt.simps(3))
+  apply (metis arexp.distinct(3) arexp.distinct(7) bsimp_AALTs.elims fuse.simps(3) list.distinct(1) list.inject)
+  apply (metis arexp.distinct(7) bsimp_AALTs.elims good.simps(1) good_fuse list.distinct(1) list.inject)
+    apply (metis arexp.distinct(7) bsimp_AALTs.elims list.distinct(1) list.inject)
+  apply (metis arexp.distinct(7) bsimp_AALTs.elims flts2 good.simps(1) good.simps(33) good0 k0b list.distinct(1) list.inject nonalt.simps(6))
+  by (metis (no_types, lifting) Nil_is_append_conv append_Nil2 arexp.distinct(7) bsimp_AALTs.elims butlast.simps(2) butlast_append flts1 flts2 good.simps(1) good0 k0a)
+
+
+lemma flts_nil:
+  assumes "\<forall>y. asize y < Suc (sum_list (map asize rs)) \<longrightarrow>
+            good (bsimp y) \<or> bsimp y = AZERO"
+  and "\<forall>r\<in>set rs. \<not> good (bsimp r)"
+  shows "flts (map bsimp rs) = []"
+  using assms
+  apply(induct rs)
+   apply(simp)
+  apply(simp)
+  apply(subst k0)
+  apply(simp)
+  by force
+
+lemma flts_nil2:
+  assumes "\<forall>y. asize y < Suc (sum_list (map asize rs)) \<longrightarrow>
+            good (bsimp y) \<or> bsimp y = AZERO"
+  and "bsimp_AALTs bs (flts (map bsimp rs)) = AZERO"
+  shows "flts (map bsimp rs) = []"
+  using assms
+  apply(induct rs arbitrary: bs)
+   apply(simp)
+  apply(simp)
+  apply(subst k0)
+  apply(simp)
+  apply(subst (asm) k0)
+  apply(auto)
+  apply (metis flts.simps(1) flts.simps(2) flts4 k0 less_add_Suc1 list.set_intros(1))
+  by (metis flts.simps(2) flts4 k0 less_add_Suc1 list.set_intros(1))
+  
+  
+
+lemma good_SEQ:
+  assumes "r1 \<noteq> AZERO" "r2 \<noteq> AZERO" "\<forall>bs. r1 \<noteq> AONE bs"
+  shows "good (ASEQ bs r1 r2) = (good r1 \<and> good r2)"
+  using assms
+  apply(case_tac r1)
+       apply(simp_all)
+  apply(case_tac r2)
+          apply(simp_all)
+  apply(case_tac r2)
+         apply(simp_all)
+  apply(case_tac r2)
+        apply(simp_all)
+  apply(case_tac r2)
+       apply(simp_all)
+  done
+
+lemma good1:
+  shows "good (bsimp a) \<or> bsimp a = AZERO"
+  apply(induct a taking: asize rule: measure_induct)
+  apply(case_tac x)
+  apply(simp)
+  apply(simp)
+  apply(simp)
+  prefer 3
+    apply(simp)
+   prefer 2
+  (*  AALTs case  *)
+  apply(simp only:)
+   apply(case_tac "x52")
+    apply(simp)
+  thm good0a
+   (*  AALTs list at least one - case *)
+   apply(simp only: )
+  apply(frule_tac x="a" in spec)
+   apply(drule mp)
+    apply(simp)
+   (* either first element is good, or AZERO *)
+    apply(erule disjE)
+     prefer 2
+    apply(simp)
+   (* in  the AZERO case, the size  is smaller *)
+   apply(drule_tac x="AALTs x51 list" in spec)
+   apply(drule mp)
+     apply(simp add: asize0)
+    apply(subst (asm) bsimp.simps)
+  apply(subst (asm) bsimp.simps)
+    apply(assumption)
+   (* in the good case *)
+  apply(frule_tac x="AALTs x51 list" in spec)
+   apply(drule mp)
+    apply(simp add: asize0)
+   apply(erule disjE)
+    apply(rule disjI1)
+  apply(simp add: good0)
+    apply(subst good0)
+      apply (metis Nil_is_append_conv flts1 k0)
+  apply (metis ex_map_conv list.simps(9) nn1b nn1c)
+  apply(simp)
+    apply(subst k0)
+    apply(simp)
+    apply(auto)[1]
+  using flts2 apply blast
+    apply(subst  (asm) good0)
+      prefer 3
+      apply(auto)[1]
+     apply auto[1]
+    apply (metis ex_map_conv nn1b nn1c)
+  (* in  the AZERO case *)
+   apply(simp)
+   apply(frule_tac x="a" in spec)
+   apply(drule mp)
+  apply(simp)
+   apply(erule disjE)
+    apply(rule disjI1)
+    apply(subst good0)
+  apply(subst k0)
+  using flts1 apply blast
+     apply(auto)[1]
+  apply (metis (no_types, hide_lams) ex_map_conv list.simps(9) nn1b nn1c)
+    apply(auto)[1]
+  apply(subst (asm) k0)
+  apply(auto)[1]
+  using flts2 apply blast
+  apply(frule_tac x="AALTs x51 list" in spec)
+   apply(drule mp)
+     apply(simp add: asize0)
+    apply(erule disjE)
+     apply(simp)
+    apply(simp)
+  apply (metis add.left_commute flts_nil2 less_add_Suc1 less_imp_Suc_add list.distinct(1) list.set_cases nat.inject)
+   apply(subst (2) k0)
+  apply(simp)
+  (* SEQ case *)
+  apply(simp)
+  apply(case_tac "bsimp x42 = AZERO")
+    apply(simp)
+ apply(case_tac "bsimp x43 = AZERO")
+   apply(simp)
+    apply(subst (2) bsimp_ASEQ0)
+  apply(simp)
+  apply(case_tac "\<exists>bs. bsimp x42 = AONE bs")
+    apply(auto)[1]
+   apply(subst bsimp_ASEQ2)
+  using good_fuse apply force
+   apply(subst bsimp_ASEQ1)
+     apply(auto)
+  apply(subst  good_SEQ)
+  apply(simp)
+    apply(simp)
+   apply(simp)
+  using less_add_Suc1 less_add_Suc2 by blast
+
+lemma good1a:
+  assumes "L(erase a) \<noteq> {}"
+  shows "good (bsimp a)"
+  using good1 assms
+  using L_bsimp_erase by force
+  
+
+
+lemma flts_append:
+  "flts (xs1 @ xs2) = flts xs1 @ flts xs2"
+  apply(induct xs1  arbitrary: xs2  rule: rev_induct)
+   apply(auto)
+  apply(case_tac xs)
+   apply(auto)
+   apply(case_tac x)
+        apply(auto)
+  apply(case_tac x)
+        apply(auto)
+  done
+
+lemma g1:
+  assumes "good (bsimp_AALTs bs rs)"
+  shows "bsimp_AALTs bs rs = AALTs bs rs \<or> (\<exists>r. rs = [r] \<and> bsimp_AALTs bs [r] = fuse bs r)"
+using assms
+    apply(induct rs arbitrary: bs)
+  apply(simp)
+  apply(case_tac rs)
+  apply(simp only:)
+  apply(simp)
+  apply(case_tac  list)
+  apply(simp)
+  by simp
+
+lemma flts_0:
+  assumes "nonnested (AALTs bs  rs)"
+  shows "\<forall>r \<in> set (flts rs). r \<noteq> AZERO"
+  using assms
+  apply(induct rs arbitrary: bs rule: flts.induct)
+        apply(simp) 
+       apply(simp) 
+      defer
+      apply(simp) 
+     apply(simp) 
+    apply(simp) 
+apply(simp) 
+  apply(rule ballI)
+  apply(simp)
+  done
+
+lemma flts_0a:
+  assumes "nonnested (AALTs bs  rs)"
+  shows "AZERO \<notin> set (flts rs)"
+  using assms
+  using flts_0 by blast 
+  
+lemma qqq1:
+  shows "AZERO \<notin> set (flts (map bsimp rs))"
+  by (metis ex_map_conv flts3 good.simps(1) good1)
+
+
+fun nonazero :: "arexp \<Rightarrow> bool"
+  where
+  "nonazero AZERO = False"
+| "nonazero r = True"
+
+lemma flts_concat:
+  shows "flts rs = concat (map (\<lambda>r. flts [r]) rs)"
+  apply(induct rs)
+   apply(auto)
+  apply(subst k0)
+  apply(simp)
+  done
+
+lemma flts_single1:
+  assumes "nonalt r" "nonazero r"
+  shows "flts [r] = [r]"
+  using assms
+  apply(induct r)
+  apply(auto)
+  done
+
+lemma flts_qq:
+  assumes "\<forall>y. asize y < Suc (sum_list (map asize rs)) \<longrightarrow> good y \<longrightarrow> bsimp y = y" 
+          "\<forall>r'\<in>set rs. good r' \<and> nonalt r'"
+  shows "flts (map bsimp rs) = rs"
+  using assms
+  apply(induct rs)
+   apply(simp)
+  apply(simp)
+  apply(subst k0)
+  apply(subgoal_tac "flts [bsimp a] =  [a]")
+   prefer 2
+   apply(drule_tac x="a" in spec)
+   apply(drule mp)
+    apply(simp)
+   apply(auto)[1]
+  using good.simps(1) k0b apply blast
+  apply(auto)[1]  
+  done
+  
+lemma test:
+  assumes "good r"
+  shows "bsimp r = r"
+  using assms
+  apply(induct r taking: "asize" rule: measure_induct)
+  apply(erule good.elims)
+  apply(simp_all)
+  apply(subst k0)
+  apply(subst (2) k0)
+                apply(subst flts_qq)
+                  apply(auto)[1]
+                 apply(auto)[1]
+                apply (metis append_Cons append_Nil bsimp_AALTs.simps(3) good.simps(1) k0b)
+               apply force+
+  apply (metis (no_types, lifting) add_Suc add_Suc_right asize.simps(5) bsimp.simps(1) bsimp_ASEQ.simps(19) less_add_Suc1 less_add_Suc2)
+  apply (metis add_Suc add_Suc_right arexp.distinct(5) arexp.distinct(7) asize.simps(4) asize.simps(5) bsimp.simps(1) bsimp.simps(2) bsimp_ASEQ1 good.simps(21) good.simps(8) less_add_Suc1 less_add_Suc2)
+         apply force+
+  apply (metis (no_types, lifting) add_Suc add_Suc_right arexp.distinct(5) arexp.distinct(7) asize.simps(4) asize.simps(5) bsimp.simps(1) bsimp.simps(2) bsimp_ASEQ1 good.simps(25) good.simps(8) less_add_Suc1 less_add_Suc2)
+  apply (metis add_Suc add_Suc_right arexp.distinct(7) asize.simps(4) bsimp.simps(2) bsimp_ASEQ1 good.simps(26) good.simps(8) less_add_Suc1 less_add_Suc2)
+    apply force+
+  done
+
+lemma test2:
+  assumes "good r"
+  shows "bsimp r = r"
+  using assms
+  apply(induct r taking: "asize" rule: measure_induct)
+  apply(case_tac x)
+       apply(simp_all)
+   defer  
+  (* AALT case *)
+   apply(subgoal_tac "1 < length x52")
+    prefer 2
+    apply(case_tac x52)
+     apply(simp)
+    apply(simp)
+    apply(case_tac list)
+     apply(simp)
+  apply(simp)
+    apply(subst bsimp_AALTs_qq)
+    prefer 2
+    apply(subst flts_qq)
+      apply(auto)[1]
+     apply(auto)[1]
+   apply(case_tac x52)
+     apply(simp)
+    apply(simp)
+    apply(case_tac list)
+     apply(simp)
+      apply(simp)
+      apply(auto)[1]
+  apply (metis (no_types, lifting) bsimp_AALTs.elims good.simps(6) length_Cons length_pos_if_in_set list.size(3) nat_neq_iff)
+  apply(simp)  
+  apply(case_tac x52)
+     apply(simp)
+    apply(simp)
+    apply(case_tac list)
+     apply(simp)
+   apply(simp)
+   apply(subst k0)
+   apply(simp)
+   apply(subst (2) k0)
+   apply(simp)
+  apply (simp add: Suc_lessI flts1 one_is_add)
+  (* SEQ case *)
+  apply(case_tac "bsimp x42 = AZERO")
+   apply simp
+  apply (metis asize.elims good.simps(10) good.simps(11) good.simps(12) good.simps(2) good.simps(7) good.simps(9) good_SEQ less_add_Suc1)  
+   apply(case_tac "\<exists>bs'. bsimp x42 = AONE bs'")
+   apply(auto)[1]
+  defer
+  apply(case_tac "bsimp x43 = AZERO")
+    apply(simp)
+  apply (metis bsimp.elims bsimp.simps(3) good.simps(10) good.simps(11) good.simps(12) good.simps(8) good.simps(9) good_SEQ less_add_Suc2)
+  apply(auto)  
+   apply (subst bsimp_ASEQ1)
+      apply(auto)[3]
+   apply(auto)[1]
+    apply (metis bsimp.simps(3) good.simps(2) good_SEQ less_add_Suc1)
+   apply (metis bsimp.simps(3) good.simps(2) good_SEQ less_add_Suc1 less_add_Suc2)
+  apply (subst bsimp_ASEQ2)
+  apply(drule_tac x="x42" in spec)
+  apply(drule mp)
+   apply(simp)
+  apply(drule mp)
+   apply (metis bsimp.elims bsimp.simps(3) good.simps(10) good.simps(11) good.simps(2) good_SEQ)
+  apply(simp)
+  done
+
+
+lemma bsimp_idem:
+  shows "bsimp (bsimp r) = bsimp r"
+  using test good1
+  by force
+
+
+lemma q3a:
+  assumes "\<exists>r \<in> set rs. bnullable r"
+  shows "bmkeps (AALTs bs (map (fuse bs1) rs)) = bmkeps (AALTs (bs@bs1) rs)"
+  using assms
+  apply(induct rs arbitrary: bs bs1)
+   apply(simp)
+  apply(simp)
+  apply(auto)
+   apply (metis append_assoc b2 bnullable_correctness erase_fuse r0)
+  apply(case_tac "bnullable a")
+   apply (metis append.assoc b2 bnullable_correctness erase_fuse r0)
+  apply(case_tac rs)
+  apply(simp)
+  apply(simp)
+  apply(auto)[1]
+   apply (metis bnullable_correctness erase_fuse)+
+  done
+
+lemma qq4:
+  assumes "\<exists>x\<in>set list. bnullable x"
+  shows "\<exists>x\<in>set (flts list). bnullable x"
+  using assms
+  apply(induct list rule: flts.induct)
+        apply(auto)
+  by (metis UnCI bnullable_correctness erase_fuse imageI)
+  
+
+lemma qs3:
+  assumes "\<exists>r \<in> set rs. bnullable r"
+  shows "bmkeps (AALTs bs rs) = bmkeps (AALTs bs (flts rs))"
+  using assms
+  apply(induct rs arbitrary: bs taking: size rule: measure_induct)
+  apply(case_tac x)
+  apply(simp)
+  apply(simp)
+  apply(case_tac a)
+       apply(simp)
+       apply (simp add: r1)
+      apply(simp)
+      apply (simp add: r0)
+     apply(simp)
+     apply(case_tac "flts list")
+      apply(simp)
+  apply (metis L_erase_AALTs L_erase_flts L_flat_Prf1 L_flat_Prf2 Prf_elims(1) bnullable_correctness erase.simps(4) mkeps_nullable r2)
+     apply(simp)
+     apply (simp add: r1)
+    prefer 3
+    apply(simp)
+    apply (simp add: r0)
+   prefer 2
+   apply(simp)
+  apply(case_tac "\<exists>x\<in>set x52. bnullable x")
+  apply(case_tac "list")
+    apply(simp)
+    apply (metis b2 fuse.simps(4) q3a r2)
+   apply(erule disjE)
+    apply(subst qq1)
+     apply(auto)[1]
+     apply (metis bnullable_correctness erase_fuse)
+    apply(simp)
+     apply (metis b2 fuse.simps(4) q3a r2)
+    apply(simp)
+    apply(auto)[1]
+     apply(subst qq1)
+      apply (metis bnullable_correctness erase_fuse image_eqI set_map)
+     apply (metis b2 fuse.simps(4) q3a r2)
+  apply(subst qq1)
+      apply (metis bnullable_correctness erase_fuse image_eqI set_map)
+    apply (metis b2 fuse.simps(4) q3a r2)
+   apply(simp)
+   apply(subst qq2)
+     apply (metis bnullable_correctness erase_fuse imageE set_map)
+  prefer 2
+  apply(case_tac "list")
+     apply(simp)
+    apply(simp)
+   apply (simp add: qq4)
+  apply(simp)
+  apply(auto)
+   apply(case_tac list)
+    apply(simp)
+   apply(simp)
+   apply (simp add: r0)
+  apply(case_tac "bnullable (ASEQ x41 x42 x43)")
+   apply(case_tac list)
+    apply(simp)
+   apply(simp)
+   apply (simp add: r0)
+  apply(simp)
+  using qq4 r1 r2 by auto
+
+
+
+lemma k1:
+  assumes "\<And>x2aa. \<lbrakk>x2aa \<in> set x2a; bnullable x2aa\<rbrakk> \<Longrightarrow> bmkeps x2aa = bmkeps (bsimp x2aa)"
+          "\<exists>x\<in>set x2a. bnullable x"
+        shows "bmkeps (AALTs x1 (flts x2a)) = bmkeps (AALTs x1 (flts (map bsimp x2a)))"
+  using assms
+  apply(induct x2a)
+  apply fastforce
+  apply(simp)
+  apply(subst k0)
+  apply(subst (2) k0)
+  apply(auto)[1]
+  apply (metis b3 k0 list.set_intros(1) qs3 r0)
+  by (smt b3 imageI insert_iff k0 list.set(2) qq3 qs3 r0 r1 set_map)
+  
+  
+  
+lemma bmkeps_simp:
+  assumes "bnullable r"
+  shows "bmkeps r = bmkeps (bsimp r)"
+  using  assms
+  apply(induct r)
+       apply(simp)
+      apply(simp)
+     apply(simp)
+    apply(simp)
+    prefer 3
+  apply(simp)
+   apply(case_tac "bsimp r1 = AZERO")
+    apply(simp)
+    apply(auto)[1]
+  apply (metis L_bsimp_erase L_flat_Prf1 L_flat_Prf2 Prf_elims(1) bnullable_correctness erase.simps(1) mkeps_nullable)
+ apply(case_tac "bsimp r2 = AZERO")
+    apply(simp)  
+    apply(auto)[1]
+  apply (metis L_bsimp_erase L_flat_Prf1 L_flat_Prf2 Prf_elims(1) bnullable_correctness erase.simps(1) mkeps_nullable)
+  apply(case_tac "\<exists>bs. bsimp r1 = AONE bs")
+    apply(auto)[1]
+    apply(subst b1)
+    apply(subst b2)
+  apply(simp add: b3[symmetric])
+    apply(simp)
+   apply(subgoal_tac "bsimp_ASEQ x1 (bsimp r1) (bsimp r2) = ASEQ x1 (bsimp r1) (bsimp r2)")
+    prefer 2
+  apply (smt b3 bnullable.elims(2) bsimp_ASEQ.simps(17) bsimp_ASEQ.simps(19) bsimp_ASEQ.simps(20) bsimp_ASEQ.simps(21) bsimp_ASEQ.simps(22) bsimp_ASEQ.simps(24) bsimp_ASEQ.simps(25) bsimp_ASEQ.simps(26) bsimp_ASEQ.simps(27) bsimp_ASEQ.simps(29) bsimp_ASEQ.simps(30) bsimp_ASEQ.simps(31))
+   apply(simp)
+  apply(simp)
+  thm q3
+  apply(subst q3[symmetric])
+   apply simp
+  using b3 qq4 apply auto[1]
+  apply(subst qs3)
+   apply simp
+  using k1 by blast
+
+thm bmkeps_retrieve bmkeps_simp bder_retrieve
+
+lemma bmkeps_bder_AALTs:
+  assumes "\<exists>r \<in> set rs. bnullable (bder c r)" 
+  shows "bmkeps (bder c (bsimp_AALTs bs rs)) = bmkeps (bsimp_AALTs bs (map (bder c) rs))"
+  using assms
+  apply(induct rs)
+   apply(simp)
+  apply(simp)
+  apply(auto)
+  apply(case_tac rs)
+    apply(simp)
+  apply (metis (full_types) Prf_injval bder_retrieve bmkeps_retrieve bnullable_correctness erase_bder erase_fuse mkeps_nullable retrieve_fuse2)
+   apply(simp)
+  apply(case_tac  rs)
+   apply(simp_all)
+  done
+
+lemma bbs0:
+  shows "blexer_simp r [] = blexer r []"
+  apply(simp add: blexer_def blexer_simp_def)
+  done
+
+lemma bbs1:
+  shows "blexer_simp r [c] = blexer r [c]"
+  apply(simp add: blexer_def blexer_simp_def)
+  apply(auto)
+    defer
+  using b3 apply auto[1]
+  using b3 apply auto[1]  
+  apply(subst bmkeps_simp[symmetric])
+   apply(simp)
+  apply(simp)
+  done
+
+lemma oo:
+  shows "(case (blexer (der c r) s) of None \<Rightarrow> None | Some v \<Rightarrow> Some (injval r c v)) = blexer r (c # s)"
+  apply(simp add: blexer_correctness)
+  done
+
+
+lemma bder_fuse:
+  shows "bder c (fuse bs a) = fuse bs  (bder c a)"
+  apply(induct a arbitrary: bs c)
+       apply(simp_all)
+  done
+
+
+fun flts2 :: "char \<Rightarrow> arexp list \<Rightarrow> arexp list"
+  where 
+  "flts2 _ [] = []"
+| "flts2 c (AZERO # rs) = flts2 c rs"
+| "flts2 c (AONE _ # rs) = flts2 c rs"
+| "flts2 c (ACHAR bs d # rs) = (if c = d then (ACHAR bs d # flts2 c rs) else flts2 c rs)"
+| "flts2 c ((AALTs bs rs1) # rs) = (map (fuse bs) rs1) @ flts2 c rs"
+| "flts2 c (ASEQ bs r1 r2 # rs) = (if (bnullable(r1) \<and> r2 = AZERO) then 
+    flts2 c rs
+    else ASEQ bs r1 r2 # flts2 c rs)"
+| "flts2 c (r1 # rs) = r1 # flts2 c rs"
+
+lemma  flts2_k0:
+  shows "flts2 c (r # rs1) = flts2 c [r] @ flts2 c rs1"
+  apply(induct r arbitrary: c rs1)
+   apply(auto)
+  done
+
+lemma  flts2_k00:
+  shows "flts2 c (rs1 @ rs2) = flts2 c rs1 @ flts2 c rs2"
+  apply(induct rs1 arbitrary: rs2 c)
+   apply(auto)
+  by (metis append.assoc flts2_k0)
+
+
+lemma
+  shows "flts (map (bder c) rs) = (map (bder c) (flts2 c rs))"
+  apply(induct c rs rule: flts2.induct)
+        apply(simp)
+       apply(simp)
+      apply(simp)
+     apply(simp)
+  apply(simp)
+    apply(auto simp add: bder_fuse)[1]
+  defer
+   apply(simp)
+  apply(simp del: flts2.simps)
+  apply(rule conjI)
+   prefer 2
+   apply(auto)[1]
+  apply(rule impI)
+  apply(subst flts2_k0)
+  apply(subst map_append)
+  apply(subst flts2.simps)
+  apply(simp only: flts2.simps)
+  apply(auto)
+
+
+
+lemma XXX2_helper:
+  assumes "\<forall>y. asize y < Suc (sum_list (map asize rs)) \<longrightarrow> good y \<longrightarrow> bsimp y = y" 
+          "\<forall>r'\<in>set rs. good r' \<and> nonalt r'"
+  shows "flts (map (bsimp \<circ> bder c) (flts (map bsimp rs))) = flts (map (bsimp \<circ> bder c) rs)"
+  using assms
+  apply(induct rs arbitrary: c)
+   apply(simp)
+  apply(simp)
+  apply(subst k0)
+  apply(simp add: flts_append)
+  apply(subst (2) k0)
+  apply(simp add: flts_append)
+  apply(subgoal_tac "flts [a] =  [a]")
+   prefer 2
+  using good.simps(1) k0b apply blast
+  apply(simp)
+  done
+
+lemma bmkeps_good:
+  assumes "good a"
+  shows "bmkeps (bsimp a) = bmkeps a"
+  using assms
+  using test2 by auto
+
+
+lemma xxx_bder:
+  assumes "good r"
+  shows "L (erase r) \<noteq> {}"
+  using assms
+  apply(induct r rule: good.induct)
+  apply(auto simp add: Sequ_def)
+  done
+
+lemma xxx_bder2:
+  assumes "L (erase (bsimp r)) = {}"
+  shows "bsimp r = AZERO"
+  using assms xxx_bder test2 good1
+  by blast
+
+lemma XXX2aa:
+  assumes "good a"
+  shows "bsimp (bder c (bsimp a)) = bsimp (bder c a)"
+  using  assms
+  by (simp add: test2)
+
+lemma XXX2aa_ders:
+  assumes "good a"
+  shows "bsimp (bders (bsimp a) s) = bsimp (bders a s)"
+  using  assms
+  by (simp add: test2)
+
+lemma XXX4a:
+  shows "good (bders_simp (bsimp r) s)  \<or> bders_simp (bsimp r) s = AZERO"
+  apply(induct s arbitrary: r rule:  rev_induct)
+   apply(simp)
+  apply (simp add: good1)
+  apply(simp add: bders_simp_append)
+  apply (simp add: good1)
+  done
+
+lemma XXX4a_good:
+  assumes "good a"
+  shows "good (bders_simp a s) \<or> bders_simp a s = AZERO"
+  using assms
+  apply(induct s arbitrary: a rule:  rev_induct)
+   apply(simp)
+  apply(simp add: bders_simp_append)
+  apply (simp add: good1)
+  done
+
+lemma XXX4a_good_cons:
+  assumes "s \<noteq> []"
+  shows "good (bders_simp a s) \<or> bders_simp a s = AZERO"
+  using assms
+  apply(case_tac s)
+   apply(auto)
+  using XXX4a by blast
+
+lemma XXX4b:
+  assumes "good a" "L (erase (bders_simp a s)) \<noteq> {}"
+  shows "good (bders_simp a s)"
+  using assms
+  apply(induct s arbitrary: a)
+   apply(simp)
+  apply(simp)
+  apply(subgoal_tac "L (erase (bder a aa)) = {} \<or> L (erase (bder a aa)) \<noteq> {}")
+   prefer 2
+   apply(auto)[1]
+  apply(erule disjE)
+   apply(subgoal_tac "bsimp (bder a aa) = AZERO")
+    prefer 2
+  using L_bsimp_erase xxx_bder2 apply auto[1]
+   apply(simp)
+  apply (metis L.simps(1) XXX4a erase.simps(1))  
+  apply(drule_tac x="bsimp (bder a aa)" in meta_spec)
+  apply(drule meta_mp)
+  apply simp
+  apply(rule good1a)
+  apply(auto)
+  done
+
+lemma bders_AZERO:
+  shows "bders AZERO s = AZERO"
+  and   "bders_simp AZERO s = AZERO"
+   apply (induct s)
+     apply(auto)
+  done
+
+lemma LA:
+  assumes "\<Turnstile> v : ders s (erase r)"
+  shows "retrieve (bders r s) v = retrieve r (flex (erase r) id s v)"
+  using assms
+  apply(induct s arbitrary: r v rule: rev_induct)
+   apply(simp)
+  apply(simp add: bders_append ders_append)
+  apply(subst bder_retrieve)
+   apply(simp)
+  apply(drule Prf_injval)
+  by (simp add: flex_append)
+
+
+lemma LB:
+  assumes "s \<in> (erase r) \<rightarrow> v" 
+  shows "retrieve r v = retrieve r (flex (erase r) id s (mkeps (ders s (erase r))))"
+  using assms
+  apply(induct s arbitrary: r v rule: rev_induct)
+   apply(simp)
+   apply(subgoal_tac "v = mkeps (erase r)")
+    prefer 2
+  apply (simp add: Posix1(1) Posix_determ Posix_mkeps nullable_correctness)
+   apply(simp)
+  apply(simp add: flex_append ders_append)
+  by (metis Posix_determ Posix_flex Posix_injval Posix_mkeps ders_snoc lexer_correctness(2) lexer_flex)
+
+lemma LB_sym:
+  assumes "s \<in> (erase r) \<rightarrow> v" 
+  shows "retrieve r v = retrieve r (flex (erase r) id s (mkeps (erase (bders r s))))"
+  using assms
+  by (simp add: LB)
+
+
+lemma LC:
+  assumes "s \<in> (erase r) \<rightarrow> v" 
+  shows "retrieve r v = retrieve (bders r s) (mkeps (erase (bders r s)))"
+  apply(simp)
+  by (metis LA LB Posix1(1) assms lexer_correct_None lexer_flex mkeps_nullable)
+
+
+lemma L0:
+  assumes "bnullable a"
+  shows "retrieve (bsimp a) (mkeps (erase (bsimp a))) = retrieve a (mkeps (erase a))"
+  using assms
+  by (metis b3 bmkeps_retrieve bmkeps_simp bnullable_correctness)
+
+thm bmkeps_retrieve
+
+lemma L0a:
+  assumes "s \<in> L(erase a)"
+  shows "retrieve (bsimp (bders a s)) (mkeps (erase (bsimp (bders a s)))) = 
+         retrieve (bders a s) (mkeps (erase (bders a s)))"
+  using assms
+  by (metis L0 bnullable_correctness erase_bders lexer_correct_None lexer_flex)
+  
+lemma L0aa:
+  assumes "s \<in> L (erase a)"
+  shows "[] \<in> erase (bsimp (bders a s)) \<rightarrow> mkeps (erase (bsimp (bders a s)))"
+  using assms
+  by (metis Posix_mkeps b3 bnullable_correctness erase_bders lexer_correct_None lexer_flex)
+
+lemma L0aaa:
+  assumes "[c] \<in> L (erase a)"
+  shows "[c] \<in> (erase a) \<rightarrow> flex (erase a) id [c] (mkeps (erase (bder c a)))"
+  using assms
+  by (metis bders.simps(1) bders.simps(2) erase_bders lexer_correct_None lexer_correct_Some lexer_flex option.inject)
+
+lemma L0aaaa:
+  assumes "[c] \<in> L (erase a)"
+  shows "[c] \<in> (erase a) \<rightarrow> flex (erase a) id [c] (mkeps (erase (bders a [c])))"
+  using assms
+  using L0aaa by auto
+    
+
+lemma L02:
+  assumes "bnullable (bder c a)"
+  shows "retrieve (bsimp a) (flex (erase (bsimp a)) id [c] (mkeps (erase (bder c (bsimp a))))) = 
+         retrieve (bder c (bsimp a)) (mkeps (erase (bder c (bsimp a))))"
+  using assms
+  apply(simp)
+  using bder_retrieve L0 bmkeps_simp bmkeps_retrieve L0  LA LB
+  apply(subst bder_retrieve[symmetric])
+  apply (metis L_bsimp_erase bnullable_correctness der_correctness erase_bder mkeps_nullable nullable_correctness)
+  apply(simp)
+  done
+
+lemma L02_bders:
+  assumes "bnullable (bders a s)"
+  shows "retrieve (bsimp a) (flex (erase (bsimp a)) id s (mkeps (erase (bders (bsimp a) s)))) = 
+         retrieve (bders (bsimp a) s) (mkeps (erase (bders (bsimp a) s)))"
+  using assms
+  by (metis LA L_bsimp_erase bnullable_correctness ders_correctness erase_bders mkeps_nullable nullable_correctness)
+
+
+  
+
+lemma L03:
+  assumes "bnullable (bder c a)"
+  shows "retrieve (bder c (bsimp a)) (mkeps (erase (bder c (bsimp a)))) =
+         bmkeps (bsimp (bder c (bsimp a)))"
+  using assms
+  by (metis L0 L_bsimp_erase bmkeps_retrieve bnullable_correctness der_correctness erase_bder nullable_correctness)
+
+lemma L04:
+  assumes "bnullable (bder c a)"
+  shows "retrieve (bder c (bsimp a)) (mkeps (erase (bder c (bsimp a)))) =
+         retrieve (bsimp (bder c (bsimp a))) (mkeps (erase (bsimp (bder c (bsimp a)))))"     
+  using assms
+  by (metis L0 L_bsimp_erase bnullable_correctness der_correctness erase_bder nullable_correctness)
+    
+lemma L05:
+  assumes "bnullable (bder c a)"
+  shows "retrieve (bder c (bsimp a)) (mkeps (erase (bder c (bsimp a)))) =
+         retrieve (bsimp (bder c (bsimp a))) (mkeps (erase (bsimp (bder c (bsimp a)))))" 
+  using assms
+  using L04 by auto 
+
+lemma L06:
+  assumes "bnullable (bder c a)"
+  shows "bmkeps (bder c (bsimp a)) = bmkeps (bsimp (bder c (bsimp a)))"
+  using assms
+  by (metis L03 L_bsimp_erase bmkeps_retrieve bnullable_correctness der_correctness erase_bder nullable_correctness) 
+
+lemma L07:
+  assumes "s \<in> L (erase r)"
+  shows "retrieve r (flex (erase r) id s (mkeps (ders s (erase r)))) 
+            = retrieve (bders r s) (mkeps (erase (bders r s)))"
+  using assms
+  using LB LC lexer_correct_Some by auto
+
+lemma LXXX:
+  assumes "s \<in> (erase r) \<rightarrow> v" "s \<in> (erase (bsimp r)) \<rightarrow> v'"
+  shows "retrieve r v = retrieve (bsimp r) v'"
+  using  assms
+  apply -
+  thm LC
+  apply(subst LC)
+   apply(assumption)
+  apply(subst  L0[symmetric])
+  using bnullable_correctness lexer_correctness(2) lexer_flex apply fastforce
+  apply(subst (2) LC)
+   apply(assumption)
+  apply(subst (2)  L0[symmetric])
+  using bnullable_correctness lexer_correctness(2) lexer_flex apply fastforce
+   
+  oops  
+
+
+lemma L07a:
+  assumes "s \<in> L (erase r)"
+  shows "retrieve (bsimp r) (flex (erase (bsimp r)) id s (mkeps (ders s (erase (bsimp r))))) 
+         = retrieve r (flex (erase r) id s (mkeps (ders s (erase r))))"
+  using assms
+  apply(induct s arbitrary: r)
+   apply(simp)
+  using L0a apply force
+  apply(drule_tac x="(bder a r)" in meta_spec)
+  apply(drule meta_mp)
+  apply (metis L_bsimp_erase erase_bder lexer.simps(2) lexer_correct_None option.case(1))
+  apply(drule sym)
+  apply(simp)
+  apply(subst (asm) bder_retrieve)
+   apply (metis Posix_Prf Posix_flex Posix_mkeps ders.simps(2) lexer_correct_None lexer_flex)
+  apply(simp only: flex_fun_apply)
+  apply(simp)
+  using L0[no_vars] bder_retrieve[no_vars] LA[no_vars] LC[no_vars] L07[no_vars]
+  oops
+
+lemma L08:
+  assumes "s \<in> L (erase r)"
+  shows "retrieve (bders (bsimp r) s) (mkeps (erase (bders (bsimp r) s)))
+         = retrieve (bders r s) (mkeps (erase (bders r s)))"
+  using assms
+  apply(induct s arbitrary: r)
+   apply(simp)
+  using L0 bnullable_correctness nullable_correctness apply blast
+  apply(simp add: bders_append)
+  apply(drule_tac x="(bder a (bsimp r))" in meta_spec)
+  apply(drule meta_mp)
+  apply (metis L_bsimp_erase erase_bder lexer.simps(2) lexer_correct_None option.case(1))
+  apply(drule sym)
+  apply(simp)
+  apply(subst LA)
+  apply (metis L0aa L_bsimp_erase Posix1(1) ders.simps(2) ders_correctness erase_bder erase_bders mkeps_nullable nullable_correctness)
+  apply(subst LA)
+  using lexer_correct_None lexer_flex mkeps_nullable apply force
+  
+  using L0[no_vars] bder_retrieve[no_vars] LA[no_vars] LC[no_vars] L07[no_vars]
+
+thm L0[no_vars] bder_retrieve[no_vars] LA[no_vars] LC[no_vars] L07[no_vars]
+  oops
+
+lemma test:
+  assumes "s = [c]"
+  shows "retrieve (bders r s) v = XXX" and "YYY = retrieve r (flex (erase r) id s v)"
+  using assms
+   apply(simp only: bders.simps)
+   defer
+  using assms
+   apply(simp only: flex.simps id_simps)
+  using  L0[no_vars] bder_retrieve[no_vars] LA[no_vars] LC[no_vars] 
+  find_theorems "retrieve (bders _ _) _"
+  find_theorems "retrieve _ (mkeps _)"
+  oops
+
+lemma L06X:
+  assumes "bnullable (bder c a)"
+  shows "bmkeps (bder c (bsimp a)) = bmkeps (bder c a)"
+  using assms
+  apply(induct a arbitrary: c)
+       apply(simp)
+      apply(simp)
+     apply(simp)
+    prefer 3
+    apply(simp)
+   prefer 2
+   apply(simp)
+  
+   defer
+  oops
+
+lemma L06_2:
+  assumes "bnullable (bders a [c,d])"
+  shows "bmkeps (bders (bsimp a) [c,d]) = bmkeps (bsimp (bders (bsimp a) [c,d]))"
+  using assms
+  apply(simp)
+  by (metis L_bsimp_erase bmkeps_simp bnullable_correctness der_correctness erase_bder nullable_correctness)
+  
+lemma L06_bders:
+  assumes "bnullable (bders a s)"
+  shows "bmkeps (bders (bsimp a) s) = bmkeps (bsimp (bders (bsimp a) s))"
+  using assms
+  by (metis L_bsimp_erase bmkeps_simp bnullable_correctness ders_correctness erase_bders nullable_correctness)
+
+lemma LLLL:
+  shows "L (erase a) =  L (erase (bsimp a))"
+  and "L (erase a) = {flat v | v. \<Turnstile> v: (erase a)}"
+  and "L (erase a) = {flat v | v. \<Turnstile> v: (erase (bsimp a))}"
+  using L_bsimp_erase apply(blast)
+  apply (simp add: L_flat_Prf)
+  using L_bsimp_erase L_flat_Prf apply(auto)[1]
+  done  
+    
+
+
+lemma L07XX:
+  assumes "s \<in> L (erase a)"
+  shows "s \<in> erase a \<rightarrow> flex (erase a) id s (mkeps (ders s (erase a)))"
+  using assms
+  by (meson lexer_correct_None lexer_correctness(1) lexer_flex)
+
+lemma LX0:
+  assumes "s \<in> L r"
+  shows "decode (bmkeps (bders (intern r) s)) r = Some(flex r id s (mkeps (ders s r)))"
+  by (metis assms blexer_correctness blexer_def lexer_correct_None lexer_flex)
+
+
+lemma L02_bders2:
+  assumes "bnullable (bders a s)" "s = [c]"
+  shows "retrieve (bders (bsimp a) s) (mkeps (erase (bders (bsimp a) s)))  =
+         retrieve (bders a s) (mkeps (erase (bders a s)))"
+  using assms
+  apply(simp)
+  
+  apply(induct s arbitrary: a)
+   apply(simp)
+  using L0 apply auto[1]
+  oops
+
+thm bmkeps_retrieve bmkeps_simp Posix_mkeps
+
+lemma WQ1:
+  assumes "s \<in> L (der c r)"
+  shows "s \<in> der c r \<rightarrow> mkeps (ders s (der c r))"
+  using assms
+  oops
+
+lemma L02_bsimp:
+  assumes "bnullable (bders a s)"
+  shows "retrieve (bsimp a) (flex (erase (bsimp a)) id s (mkeps (erase (bders (bsimp a) s)))) =
+         retrieve a (flex (erase a) id s (mkeps (erase (bders a s))))"
+  using assms
+  apply(induct s arbitrary: a)
+   apply(simp)
+   apply (simp add: L0)
+  apply(simp)
+  apply(drule_tac x="bder a aa" in meta_spec)
+  apply(simp)
+  apply(subst (asm) bder_retrieve)
+  using Posix_Prf Posix_flex Posix_mkeps bnullable_correctness apply fastforce
+  apply(simp add: flex_fun_apply)
+  apply(drule sym)
+  apply(simp)
+  apply(subst flex_injval)
+  apply(subst bder_retrieve[symmetric])
+  apply (metis L_bsimp_erase Posix_Prf Posix_flex Posix_mkeps bders.simps(2) bnullable_correctness ders.simps(2) erase_bders lexer_correct_None lexer_flex option.distinct(1))
+  apply(simp only: erase_bder[symmetric] erase_bders[symmetric])  
+  apply(subst LB_sym[symmetric])
+   apply(simp)
+  oops
+
+lemma L1:
+  assumes "s \<in> r \<rightarrow> v" 
+  shows "decode (bmkeps (bders (intern r) s)) r = Some v"
+  using assms
+  by (metis blexer_correctness blexer_def lexer_correctness(1) option.distinct(1))
+
+lemma L2:
+  assumes "s \<in> (der c r) \<rightarrow> v" 
+  shows "decode (bmkeps (bders (intern r) (c # s))) r = Some (injval r c v)"
+  using assms
+  apply(subst bmkeps_retrieve)
+  using Posix1(1) lexer_correct_None lexer_flex apply fastforce
+  using MAIN_decode
+  apply(subst MAIN_decode[symmetric])
+   apply(simp)
+   apply (meson Posix1(1) lexer_correct_None lexer_flex mkeps_nullable)
+  apply(simp)
+  apply(subgoal_tac "v = flex (der c r) id s (mkeps (ders s (der c r)))")
+   prefer 2
+   apply (metis Posix_determ lexer_correctness(1) lexer_flex option.distinct(1))
+  apply(simp)
+  apply(subgoal_tac "injval r c (flex (der c r) id s (mkeps (ders s (der c r)))) =
+    (flex (der c r) ((\<lambda>v. injval r c v) o id) s (mkeps (ders s (der c r))))")
+   apply(simp)
+  using flex_fun_apply by blast
+  
+lemma L3:
+  assumes "s2 \<in> (ders s1 r) \<rightarrow> v" 
+  shows "decode (bmkeps (bders (intern r) (s1 @ s2))) r = Some (flex r id s1 v)"
+  using assms
+  apply(induct s1 arbitrary: r s2 v rule: rev_induct)
+   apply(simp)
+  using L1 apply blast
+  apply(simp add: ders_append)
+  apply(drule_tac x="r" in meta_spec)
+  apply(drule_tac x="x # s2" in meta_spec)
+  apply(drule_tac x="injval (ders xs r) x v" in meta_spec)
+  apply(drule meta_mp)
+   defer
+   apply(simp)
+   apply(simp add:  flex_append)
+  by (simp add: Posix_injval)
+
+
+
+lemma bders_snoc:
+  "bder c (bders a s) = bders a (s @ [c])"
+  apply(simp add: bders_append)
+  done
+
+
+lemma QQ1:
+  shows "bsimp (bders (bsimp a) []) = bders_simp (bsimp a) []"
+  apply(simp)
+  apply(simp add: bsimp_idem)
+  done
+
+lemma QQ2:
+  shows "bsimp (bders (bsimp a) [c]) = bders_simp (bsimp a) [c]"
+  apply(simp)
+  done
+
+lemma XXX2a_long:
+  assumes "good a"
+  shows "bsimp (bder c (bsimp a)) = bsimp (bder c a)"
+  using  assms
+  apply(induct a arbitrary: c taking: asize rule: measure_induct)
+  apply(case_tac x)
+       apply(simp)
+      apply(simp)
+     apply(simp)
+  prefer 3
+    apply(simp)
+   apply(simp)
+   apply(auto)[1]
+apply(case_tac "x42 = AZERO")
+     apply(simp)
+   apply(case_tac "x43 = AZERO")
+     apply(simp)
+  using test2 apply force  
+  apply(case_tac "\<exists>bs. x42 = AONE bs")
+     apply(clarify)
+     apply(simp)
+    apply(subst bsimp_ASEQ1)
+       apply(simp)
+  using b3 apply force
+  using bsimp_ASEQ0 test2 apply force
+  thm good_SEQ test2
+     apply (simp add: good_SEQ test2)
+    apply (simp add: good_SEQ test2)
+  apply(case_tac "x42 = AZERO")
+     apply(simp)
+   apply(case_tac "x43 = AZERO")
+    apply(simp)
+  apply (simp add: bsimp_ASEQ0)
+  apply(case_tac "\<exists>bs. x42 = AONE bs")
+     apply(clarify)
+     apply(simp)
+    apply(subst bsimp_ASEQ1)
+      apply(simp)
+  using bsimp_ASEQ0 test2 apply force
+     apply (simp add: good_SEQ test2)
+    apply (simp add: good_SEQ test2)
+  apply (simp add: good_SEQ test2)
+  (* AALTs case *)
+  apply(simp)
+  using test2 by fastforce
+
+lemma XXX2a_long_without_good:
+  assumes "a = AALTs bs0  [AALTs bs1 [AALTs bs2 [ASTAR [] (AONE bs7), AONE bs6, ASEQ bs3 (ACHAR bs4 d) (AONE bs5)]]]" 
+  shows "bsimp (bder c (bsimp a)) = bsimp (bder c a)"
+        "bsimp (bder c (bsimp a)) = XXX"
+        "bsimp (bder c a) = YYY"
+  using  assms
+    apply(simp)
+  using  assms
+   apply(simp)
+   prefer 2
+  using  assms
+   apply(simp)
+  oops
+
+lemma bder_bsimp_AALTs:
+  shows "bder c (bsimp_AALTs bs rs) = bsimp_AALTs bs (map (bder c) rs)"
+  apply(induct bs rs rule: bsimp_AALTs.induct)  
+    apply(simp)
+   apply(simp)
+   apply (simp add: bder_fuse)
+  apply(simp)
+  done
+
+lemma flts_nothing:
+  assumes "\<forall>r \<in> set rs. r \<noteq> AZERO" "\<forall>r \<in> set rs. nonalt r"
+  shows "flts rs = rs"
+  using assms
+  apply(induct rs rule: flts.induct)
+        apply(auto)
+  done
+
+lemma flts_flts:
+  assumes "\<forall>r \<in> set rs. good r"
+  shows "flts (flts rs) = flts rs"
+  using assms
+  apply(induct rs taking: "\<lambda>rs. sum_list  (map asize rs)" rule: measure_induct)
+  apply(case_tac x)
+   apply(simp)
+  apply(simp)
+  apply(case_tac a)
+       apply(simp_all  add: bder_fuse flts_append)
+  apply(subgoal_tac "\<forall>r \<in> set x52. r \<noteq> AZERO")
+   prefer 2
+  apply (metis Nil_is_append_conv bsimp_AALTs.elims good.simps(1) good.simps(5) good0 list.distinct(1) n0 nn1b split_list_last test2)
+  apply(subgoal_tac "\<forall>r \<in> set x52. nonalt r")
+   prefer 2
+   apply (metis n0 nn1b test2)
+  by (metis flts_fuse flts_nothing)
+
+
+lemma PP:
+  assumes "bnullable (bders r s)" 
+  shows "bmkeps (bders (bsimp r) s) = bmkeps (bders r s)"
+  using assms
+  apply(induct s arbitrary: r)
+   apply(simp)
+  using bmkeps_simp apply auto[1]
+  apply(simp add: bders_append bders_simp_append)
+  oops
+
+lemma PP:
+  assumes "bnullable (bders r s)"
+  shows "bmkeps (bders_simp (bsimp r) s) = bmkeps (bders r s)"
+  using assms
+  apply(induct s arbitrary: r rule: rev_induct)
+   apply(simp)
+  using bmkeps_simp apply auto[1]
+  apply(simp add: bders_append bders_simp_append)
+  apply(drule_tac x="bder a (bsimp r)" in meta_spec)
+  apply(drule_tac meta_mp)
+   defer
+  oops
+
+
+lemma
+  assumes "asize (bsimp a) = asize a"  "a = AALTs bs [AALTs bs2 [], AZERO, AONE bs3]"
+  shows "bsimp a = a"
+  using assms
+  apply(simp)
+  oops
+
+
+lemma iii:
+  assumes "bsimp_AALTs bs rs \<noteq> AZERO"
+  shows "rs \<noteq> []"
+  using assms
+  apply(induct bs  rs rule: bsimp_AALTs.induct)
+    apply(auto)
+  done
+
+lemma CT1_SEQ:
+  shows "bsimp (ASEQ bs a1 a2) = bsimp (ASEQ bs (bsimp a1) (bsimp a2))"
+  apply(simp add: bsimp_idem)
+  done
+
+lemma CT1:
+  shows "bsimp (AALTs bs as) = bsimp (AALTs bs (map  bsimp as))"
+  apply(induct as arbitrary: bs)
+   apply(simp)
+  apply(simp)
+  by (simp add: bsimp_idem comp_def)
+  
+lemma CT1a:
+  shows "bsimp (AALT bs a1 a2) = bsimp(AALT bs (bsimp a1) (bsimp a2))"
+  by (metis CT1 list.simps(8) list.simps(9))
+
+lemma WWW2:
+  shows "bsimp (bsimp_AALTs bs1 (flts (map bsimp as1))) =
+         bsimp_AALTs bs1 (flts (map bsimp as1))"
+  by (metis bsimp.simps(2) bsimp_idem)
+
+lemma CT1b:
+  shows "bsimp (bsimp_AALTs bs as) = bsimp (bsimp_AALTs bs (map bsimp as))"
+  apply(induct bs as rule: bsimp_AALTs.induct)
+    apply(auto simp add: bsimp_idem)
+  apply (simp add: bsimp_fuse bsimp_idem)
+  by (metis bsimp_idem comp_apply)
+  
+  
+
+
+(* CT *)
+
+lemma CTU:
+  shows "bsimp_AALTs bs as = li bs as"
+  apply(induct bs as rule: li.induct)
+    apply(auto)
+  done
+
+find_theorems "bder _ (bsimp_AALTs _ _)"
+
+lemma CTa:
+  assumes "\<forall>r \<in> set as. nonalt r \<and> r \<noteq> AZERO"
+  shows  "flts as = as"
+  using assms
+  apply(induct as)
+   apply(simp)
+  apply(case_tac as)
+   apply(simp)
+  apply (simp add: k0b)
+  using flts_nothing by auto
+
+lemma CT0:
+  assumes "\<forall>r \<in> set as1. nonalt r \<and> r \<noteq> AZERO" 
+  shows "flts [bsimp_AALTs bs1 as1] =  flts (map (fuse bs1) as1)"
+  using assms CTa
+  apply(induct as1 arbitrary: bs1)
+    apply(simp)
+   apply(simp)
+  apply(case_tac as1)
+   apply(simp)
+  apply(simp)
+proof -
+fix a :: arexp and as1a :: "arexp list" and bs1a :: "bit list" and aa :: arexp and list :: "arexp list"
+  assume a1: "nonalt a \<and> a \<noteq> AZERO \<and> nonalt aa \<and> aa \<noteq> AZERO \<and> (\<forall>r\<in>set list. nonalt r \<and> r \<noteq> AZERO)"
+  assume a2: "\<And>as. \<forall>r\<in>set as. nonalt r \<and> r \<noteq> AZERO \<Longrightarrow> flts as = as"
+  assume a3: "as1a = aa # list"
+  have "flts [a] = [a]"
+using a1 k0b by blast
+then show "fuse bs1a a # fuse bs1a aa # map (fuse bs1a) list = flts (fuse bs1a a # fuse bs1a aa # map (fuse bs1a) list)"
+  using a3 a2 a1 by (metis (no_types) append.left_neutral append_Cons flts_fuse k00 k0b list.simps(9))
+qed
+  
+  
+lemma CT01:
+  assumes "\<forall>r \<in> set as1. nonalt r \<and> r \<noteq> AZERO" "\<forall>r \<in> set as2. nonalt r \<and> r \<noteq> AZERO" 
+  shows "flts [bsimp_AALTs bs1 as1, bsimp_AALTs bs2 as2] =  flts ((map (fuse bs1) as1) @ (map (fuse bs2) as2))"
+  using assms CT0
+  by (metis k0 k00)
+  
+
+
+lemma CT_exp:
+  assumes "\<forall>a \<in> set as. bsimp (bder c (bsimp a)) = bsimp (bder c a)"
+  shows "map bsimp (map (bder c) as) = map bsimp (map (bder c) (map bsimp as))"
+  using assms
+  apply(induct as)
+   apply(auto)
+  done
+
+lemma asize_set:
+  assumes "a \<in> set as"
+  shows "asize a < Suc (sum_list (map asize as))"
+  using assms
+  apply(induct as arbitrary: a)
+   apply(auto)
+  using le_add2 le_less_trans not_less_eq by blast
+  
+
+lemma XXX2a_long_without_good:
+  shows "bsimp (bder c (bsimp a)) = bsimp (bder c a)"
+  apply(induct a arbitrary: c taking: "\<lambda>a. asize a" rule: measure_induct)
+  apply(case_tac x)
+       apply(simp)
+      apply(simp)
+     apply(simp)
+  prefer 3
+    apply(simp)
+  (* AALT case *)
+   prefer 2
+   apply(simp del: bsimp.simps)
+   apply(subst (2) CT1)
+   apply(subst CT_exp)
+    apply(auto)[1]
+  using asize_set apply blast
+   apply(subst CT1[symmetric])
+  apply(simp)
+  oops
+
+lemma YY:
+  assumes "flts (map bsimp as1) = xs"
+  shows "flts (map bsimp (map (fuse bs1) as1)) = map (fuse bs1) xs"
+  using assms
+  apply(induct as1 arbitrary: bs1 xs)
+   apply(simp)
+  apply(auto)
+  by (metis bsimp_fuse flts_fuse k0 list.simps(9))
+  
+
+lemma flts_nonalt:
+  assumes "flts (map bsimp xs) = ys"
+  shows "\<forall>y \<in> set ys. nonalt y"
+  using assms
+  apply(induct xs arbitrary: ys)
+   apply(auto)
+  apply(case_tac xs)
+   apply(auto)
+  using flts2 good1 apply fastforce
+  by (smt ex_map_conv list.simps(9) nn1b nn1c)
+
+
+lemma WWW3:
+  shows "flts [bsimp_AALTs bs1 (flts (map bsimp as1))] =
+         flts (map bsimp (map (fuse bs1) as1))"
+  by (metis CT0 YY flts_nonalt flts_nothing qqq1)
+
+lemma WWW4:
+  shows "map (bder c \<circ> fuse bs1) as1 = map (fuse bs1) (map (bder c) as1)"
+  apply(induct as1)
+   apply(auto)
+  using bder_fuse by blast
+
+lemma WWW5:
+  shows "map (bsimp \<circ> bder c) as1 = map bsimp (map (bder c) as1)"
+  apply(induct as1)
+   apply(auto)
+  done
+
+lemma WWW6:
+  shows "bsimp (bder c (bsimp_AALTs x51 (flts [bsimp a1, bsimp a2]) ) )  = 
+ bsimp(bsimp_AALTs x51 (map (bder c) (flts [bsimp a1, bsimp a2]))) "
+  using bder_bsimp_AALTs by auto
+
+lemma WWW7:
+  shows "bsimp (bsimp_AALTs x51 (map (bder c) (flts [bsimp a1, bsimp a2]))) =
+  bsimp(bsimp_AALTs x51 (flts (map (bder c) [bsimp a1, bsimp a2])))"
+  sorry
+
+
+lemma stupid:
+  assumes "a = b"
+  shows "bsimp(a) = bsimp(b)"
+  using assms
+  apply(auto)
+  done
+(*
+proving idea:
+bsimp_AALTs x51  (map (bder c) (flts [a1, a2])) = bsimp_AALTs x51 (map (bder c) (flts [a1]++[a2]))
+= bsimp_AALTs x51  (map (bder c) ((flts [a1])++(flts [a2]))) =  
+bsimp_AALTs x51 (map (bder c) (flts [a1]))++(map (bder c) (flts [a2])) = A
+and then want to prove that
+map (bder c) (flts [a]) = flts [bder c a] under the condition 
+that a is either a seq with the first elem being not nullable, or a character equal to c,
+or an AALTs, or a star
+Then, A = bsimp_AALTs x51 (flts [bder c a]) ++ (map (bder c) (flts [a2])) = A1
+Using the same condition for a2, we get
+A1 = bsimp_AALTs x51 (flts [bder c a1]) ++ (flts [bder c a2])
+=bsimp_AALTs x51 flts ([bder c a1] ++ [bder c a2])
+=bsimp_AALTs x51 flts ([bder c a1, bder c a2])
+ *)
+lemma manipulate_flts:
+  shows "bsimp_AALTs x51  (map (bder c) (flts [a1, a2])) = 
+bsimp_AALTs x51 ((map (bder c) (flts [a1])) @ (map (bder c) (flts [a2])))"
+  by (metis k0 map_append)
+  
+lemma go_inside_flts:
+  assumes " (bder c a1 \<noteq> AZERO) "
+ "\<not>(\<exists> a01 a02 x02. (  (a1 = ASEQ x02 a01 a02) \<and> bnullable(a01) )      )"
+shows "map (bder c) (flts [a1]) = flts [bder c a1]"
+  using assms
+  apply -
+  apply(case_tac a1)
+  apply(simp)
+  apply(simp)
+     apply(case_tac "x32 = c")
+  prefer 2
+      apply(simp)
+     apply(simp)
+    apply(simp)
+  apply (simp add: WWW4)
+   apply(simp add: bder_fuse)
+  done
+
+lemma medium010:
+  assumes " (bder c a1 = AZERO) "
+  shows "map (bder c) (flts [a1]) = [AZERO] \<or> map (bder c) (flts [a1]) = []"
+  using assms
+  apply -
+  apply(case_tac a1)
+       apply(simp)
+      apply(simp)
+  apply(simp)
+    apply(simp)
+  apply(simp)
+  apply(simp)
+  done
+
+lemma medium011:
+  assumes " (bder c a1 = AZERO) "
+  shows "flts (map (bder c)  [a1, a2]) = flts [bder c a2]"
+  using assms
+  apply -
+  apply(simp)
+  done
+
+lemma medium01central:
+  shows "bsimp(bsimp_AALTs x51 (map (bder c) (flts [a2])) ) = bsimp(bsimp_AALTs x51 (flts [bder c a2]))"
+  sorry
+
+
+lemma plus_bsimp:
+  assumes "bsimp( bsimp a) = bsimp (bsimp b)"
+  shows "bsimp a = bsimp b"
+  using assms
+  apply -
+  by (simp add: bsimp_idem)
+lemma patience_good5:
+  assumes "bsimp r = AALTs x y"
+  shows " \<exists> a aa list. y = a#aa#list"
+  by (metis Nil_is_map_conv arexp.simps(13) assms bsimp_AALTs.elims flts1 good.simps(5) good1 k0a)
+
+(*SAD*)
+(*this does not hold actually
+lemma bsimp_equiv0:
+  shows "bsimp(bsimp r) = bsimp(bsimp (AALTs []  [r]))"
+  apply(simp)
+  apply(case_tac "bsimp r")
+       apply(simp)
+      apply(simp)
+     apply(simp)
+    apply(simp)
+ thm good1
+  using good1
+   apply -
+   apply(drule_tac x="r" in meta_spec)
+   apply(erule disjE)
+
+    apply(simp only: bsimp_AALTs.simps)
+    apply(simp only:flts.simps)
+    apply(drule patience_good5)
+    apply(clarify)
+    apply(subst  bsimp_AALTs_qq)
+     apply simp
+    prefer 2
+  sorry*)
+
+(*exercise: try multiple ways of proving this*)
+(*this lemma does not hold.........
+lemma bsimp_equiv1:
+  shows "bsimp r = bsimp (AALTs []  [r])"
+  using plus_bsimp
+  apply -
+  using bsimp_equiv0 by blast
+  (*apply(simp)
+  apply(case_tac "bsimp r")
+       apply(simp)
+      apply(simp)
+     apply(simp)
+    apply(simp)
+(*use lemma good1*)
+  thm good1
+  using good1
+   apply -
+   apply(drule_tac x="r" in meta_spec)
+   apply(erule disjE)
+  
+  apply(subst flts_single1)
+  apply(simp only: bsimp_AALTs.simps)
+    prefer 2
+  
+  thm flts_single1
+
+  find_theorems "flts _ = _"*)
+*)
+lemma bsimp_equiv2:
+  shows "bsimp (AALTs x51 [r])  =  bsimp (AALT x51 AZERO r)"
+  sorry
+
+lemma medium_stupid_isabelle:
+  assumes "rs = a # list"
+  shows  "bsimp_AALTs x51 (AZERO # rs) = AALTs x51 (AZERO#rs)"
+  using assms
+  apply -
+  apply(simp)
+  done 
+(*
+lemma mediumlittle:
+  shows "bsimp(bsimp_AALTs x51 rs) = bsimp(bsimp_AALTs x51 (AZERO # rs))"
+  apply(case_tac rs)
+   apply(simp)
+  apply(case_tac list)
+   apply(subst medium_stupid_isabelle)
+    apply(simp)
+   prefer 2
+   apply simp
+  apply(rule_tac s="a#list" and t="rs" in subst)
+   apply(simp)
+  apply(rule_tac t="list" and s= "[]" in subst)
+   apply(simp)
+ (*dunno what is the rule for x#nil = x*)
+   apply(case_tac a)
+        apply(simp)
+       apply(simp)
+     apply(simp)
+    prefer 3
+    apply simp
+   apply(simp only:bsimp_AALTs.simps)
+
+  apply simp
+     apply(case_tac "bsimp x42")
+        apply(simp)
+       apply simp
+       apply(case_tac "bsimp x43")
+            apply simp
+           apply simp
+  apply simp
+         apply simp
+        apply(simp only:bsimp_ASEQ.simps)
+  using good1
+        apply -
+        apply(drule_tac x="x43" in meta_spec)
+  apply(erule disjE)
+        apply(subst bsimp_AALTs_qq)
+  using patience_good5 apply force
+         apply(simp only:bsimp_AALTs.simps)
+  apply(simp only:fuse.simps)
+         apply(simp only:flts.simps)
+(*OK from here you actually realize this lemma doesnt hold*)
+  apply(simp)
+        apply(simp)
+       apply(rule_tac t="rs" and s="a#list" in subst)
+        apply(simp)
+   apply(rule_tac t="list" and s="[]" in subst)
+        apply(simp)
+       (*apply(simp only:bsimp_AALTs.simps)*)
+       (*apply(simp only:fuse.simps)*)
+  sorry
+*)
+lemma singleton_list_map:
+  shows"map f [a] = [f a]"
+  apply simp
+  done
+lemma map_application2:
+  shows"map f [a,b] = [f a, f b]"
+  apply simp
+  done
+(*SAD*)
+(* bsimp (bder c (bsimp_AALTs x51 (flts [bsimp a1, bsimp a2]))) =
+       bsimp (AALT x51 (bder c (bsimp a1)) (bder c (bsimp a2)))*)
+(*This equality does not hold*)
+lemma medium01:
+  assumes " (bder c a1 = AZERO) "
+  shows "bsimp(bsimp_AALTs x51 (map (bder c) (flts [ a1, a2]))) =
+         bsimp(bsimp_AALTs x51 (flts (map (bder c) [ a1, a2])))"
+  apply(subst manipulate_flts)
+  using assms
+  apply -
+  apply(subst medium011)
+   apply(simp)
+  apply(case_tac "map (bder c) (flts [a1]) = []")
+   apply(simp)
+  using medium01central apply blast
+apply(frule medium010)
+  apply(erule disjE)
+  prefer 2
+   apply(simp)
+  apply(simp)
+  apply(case_tac a2)
+       apply simp
+      apply simp
+     apply simp
+    apply(simp only:flts.simps)
+(*HOW do i say here to replace ASEQ ..... back into a2*)
+(*how do i say here to use the definition of map function
+without lemma, of course*)
+(*how do i say here that AZERO#map (bder c) [ASEQ x41 x42 x43]'s list.len >1
+without a lemma, of course*)
+    apply(subst singleton_list_map)
+    apply(simp only: bsimp_AALTs.simps)
+    apply(case_tac "bder c (ASEQ x41 x42 x43)")
+         apply simp
+        apply simp
+       apply simp
+      prefer 3
+      apply simp
+     apply(rule_tac t="bder c (ASEQ x41 x42 x43)" 
+and s="ASEQ x41a x42a x43a" in subst)
+      apply simp
+     apply(simp only: flts.simps)
+     apply(simp only: bsimp_AALTs.simps)
+     apply(simp only: fuse.simps)
+     apply(subst (2) bsimp_idem[symmetric])
+     apply(subst (1) bsimp_idem[symmetric])
+     apply(simp only:bsimp.simps)
+     apply(subst map_application2)
+     apply(simp only: bsimp.simps)
+     apply(simp only:flts.simps)
+(*want to happily change between a2 and ASEQ x41 42 43, and eliminate now 
+redundant conditions such as  map (bder c) (flts [a1]) = [AZERO] *)
+     apply(case_tac "bsimp x42a")
+          apply(simp)
+         apply(case_tac "bsimp x43a")
+              apply(simp)
+             apply(simp)
+            apply(simp)
+           apply(simp)
+          prefer 2
+          apply(simp)
+     apply(rule_tac t="bsimp x43a" 
+and s="AALTs x51a x52" in subst)
+          apply simp
+         apply(simp only:bsimp_ASEQ.simps)
+         apply(simp only:fuse.simps)
+         apply(simp only:flts.simps)
+         
+  using medium01central mediumlittle by auto
+ 
+  
+
+lemma medium1:
+  assumes " (bder c a1 \<noteq> AZERO) "
+ "\<not>(\<exists> a01 a02 x02. (  (a1 = ASEQ x02 a01 a02) \<and> bnullable(a01) )      )"
+" (bder c a2 \<noteq> AZERO)"
+ "\<not>(\<exists> a11 a12 x12. (  (a2 = ASEQ x12 a11 a12) \<and> bnullable(a11) )      )"
+  shows "bsimp_AALTs x51 (map (bder c) (flts [ a1, a2])) =
+         bsimp_AALTs x51 (flts (map (bder c) [ a1, a2]))"
+  using assms
+  apply -
+  apply(subst manipulate_flts)
+  apply(case_tac "a1")
+       apply(simp)
+      apply(simp)
+     apply(case_tac "x32 = c")
+      prefer 2
+  apply(simp)
+     prefer 2
+     apply(case_tac "bnullable x42")
+      apply(simp)
+       apply(simp)
+
+  apply(case_tac "a2")
+            apply(simp)
+         apply(simp)
+        apply(case_tac "x32 = c")
+         prefer 2 
+  apply(simp)
+        apply(simp)
+       apply(case_tac "bnullable x42a")
+        apply(simp)
+       apply(subst go_inside_flts)
+  apply(simp)
+        apply(simp)
+       apply(simp)
+      apply(simp)
+      apply (simp add: WWW4)
+      apply(simp)
+      apply (simp add: WWW4)
+  apply (simp add: go_inside_flts)
+  apply (metis (no_types, lifting) go_inside_flts k0 list.simps(8) list.simps(9))
+  by (smt bder.simps(6) flts.simps(1) flts.simps(6) flts.simps(7) go_inside_flts k0 list.inject list.simps(9))
+  
+lemma big0:
+  shows "bsimp (AALT x51 (AALTs bs1 as1) (AALTs bs2 as2)) =
+         bsimp (AALTs x51 ((map (fuse bs1) as1) @ (map (fuse bs2) as2)))"
+  by (smt WWW3 bsimp.simps(2) k0 k00 list.simps(8) list.simps(9) map_append)
+
+lemma bignA:
+  shows "bsimp (AALTs x51 (AALTs bs1 as1 # as2)) =
+         bsimp (AALTs x51 ((map (fuse bs1) as1) @ as2))"
+  apply(simp)
+  apply(subst k0)
+  apply(subst WWW3)
+  apply(simp add: flts_append)
+  done
+
+lemma XXX2a_long_without_good:
+  shows "bsimp (bder c (bsimp a)) = bsimp (bder c a)"
+  apply(induct a arbitrary: c taking: "\<lambda>a. asize a" rule: measure_induct)
+  apply(case_tac x)
+       apply(simp)
+      apply(simp)
+     apply(simp)
+    prefer 3
+    apply(simp)
+  (* SEQ case *)
+   apply(simp only:)
+   apply(subst CT1_SEQ)
+  apply(simp only: bsimp.simps)
+
+  (* AALT case *)
+   prefer 2
+   apply(simp only:)
+   apply(case_tac "\<exists>a1 a2. x52 = [a1, a2]")
+    apply(clarify)
+    apply(simp del: bsimp.simps)
+  apply(subst (2) CT1) 
+    apply(simp del: bsimp.simps)
+  apply(rule_tac t="bsimp (bder c a1)" and  s="bsimp (bder c (bsimp a1))" in subst)
+  apply(simp del: bsimp.simps)
+  apply(rule_tac t="bsimp (bder c a2)" and  s="bsimp (bder c (bsimp a2))" in subst)
+     apply(simp del: bsimp.simps)
+    apply(subst  CT1a[symmetric])
+  (* \<rightarrow> *)
+  apply(rule_tac t="AALT x51 (bder c (bsimp a1)) (bder c (bsimp a2))"
+            and  s="bder c (AALT x51 (bsimp a1) (bsimp a2))" in subst)
+     apply(simp)
+     apply(subst bsimp.simps)
+    apply(simp del: bsimp.simps bder.simps)
+
+    apply(subst bder_bsimp_AALTs)
+    apply(subst bsimp.simps)
+    apply(subst WWW2[symmetric])
+    apply(subst bsimp_AALTs_qq)
+  defer 
+    apply(subst bsimp.simps)
+    
+  (* <- *)
+    apply(subst bsimp.simps)
+    apply(simp del: bsimp.simps)
+(*bsimp_AALTs x51 (map (bder c) (flts [a1, a2])) =
+    bsimp_AALTs x51 (flts (map (bder c) [a1, a2]))*)
+  apply(case_tac "\<exists>bs1 as1. bsimp a1 = AALTs bs1 as1")
+  apply(case_tac "\<exists>bs2 as2. bsimp a2 = AALTs bs2 as2")
+      apply(clarify)
+  apply(simp only:)
+      apply(simp del: bsimp.simps bder.simps)
+      apply(subst bsimp_AALTs_qq)
+       prefer 2
+       apply(simp del: bsimp.simps)
+       apply(subst big0)
+       apply(simp add: WWW4)
+  apply (m etis One_nat_def Suc_eq_plus1 Suc_lessI arexp.distinct(7) bsimp.simps(2) bsimp_AALTs.simps(1) bsimp_idem flts.simps(1) length_append length_greater_0_conv length_map not_add_less2 not_less_eq)
+  oops
+
+lemma XXX2a_long_without_good:
+  shows "bsimp (bder c (bsimp a)) = bsimp (bder c a)"
+  apply(induct a arbitrary: c taking: "\<lambda>a. asize a" rule: measure_induct)
+  apply(case_tac x)
+       apply(simp)
+      apply(simp)
+     apply(simp)
+  prefer 3
+    apply(simp)
+  (* AALT case *)
+   prefer 2
+   apply(subgoal_tac "nonnested (bsimp x)")
+    prefer 2
+  using nn1b apply blast
+   apply(simp only:)
+  apply(drule_tac x="AALTs x51 (flts x52)" in spec)
+   apply(drule mp)
+    defer
+    apply(drule_tac x="c" in spec)
+    apply(simp)
+    apply(rotate_tac 2)
+  
+    apply(drule sym)
+  apply(simp)
+
+   apply(simp only: bder.simps)
+   apply(simp only: bsimp.simps)
+   apply(subst bder_bsimp_AALTs)
+   apply(case_tac x52)
+    apply(simp)
+   apply(simp)
+  apply(case_tac list)
+    apply(simp)
+    apply(case_tac a)
+         apply(simp)
+        apply(simp)
+       apply(simp)
+      defer
+      apply(simp)
+  
+
+   (* case AALTs list is not empty *)
+   apply(simp)
+   apply(subst k0)
+   apply(subst (2) k0)
+   apply(simp)
+   apply(case_tac "bsimp a = AZERO")
+    apply(subgoal_tac "bsimp (bder c a) = AZERO")
+     prefer 2
+  using less_iff_Suc_add apply auto[1]
+    apply(simp)
+  apply(drule_tac x="AALTs x51 list" in  spec)
+   apply(drule mp)
+    apply(simp add: asize0)
+   apply(drule_tac x="c" in spec)
+    apply(simp add: bder_bsimp_AALTs)
+   apply(case_tac  "nonalt (bsimp a)")
+    prefer 2
+  apply(drule_tac x="bsimp (AALTs x51 (a#list))" in  spec)
+    apply(drule mp)
+     apply(rule order_class.order.strict_trans2)
+      apply(rule bsimp_AALTs_size3)
+      apply(auto)[1]
+     apply(simp)
+    apply(subst (asm) bsimp_idem)
+  apply(drule_tac x="c" in spec)
+  apply(simp)  
+  find_theorems "_ < _ \<Longrightarrow> _ \<le> _ \<Longrightarrow>_ < _"
+  apply(rule le_trans)
+  apply(subgoal_tac "flts [bsimp a] = [bsimp a]")
+     prefer 2
+  using k0b apply blast
+    apply(simp)
+  find_theorems "asize _ < asize _"
+  
+  using bder_bsimp_AALTs
+   apply(case_tac list)
+    apply(simp)
+   sledgeha mmer [timeout=6000]  
+
+   apply(case_tac "\<exists>r \<in> set (map bsimp x52). \<not>nonalt r")
+    apply(drule_tac x="bsimp (AALTs x51 x52)" in spec)
+    apply(drule mp)
+  using bsimp_AALTs_size3 apply blast
+    apply(drule_tac x="c" in spec)
+  apply(subst (asm) (2) test)
+  
+   apply(case_tac x52)
+    apply(simp)
+   apply(simp)
+  apply(case_tac "bsimp a = AZERO")
+     apply(simp)
+     apply(subgoal_tac "bsimp (bder c a) = AZERO")
+      prefer 2
+     apply auto[1]
+  apply (metis L.simps(1) L_bsimp_erase der.simps(1) der_correctness erase.simps(1) erase_bder xxx_bder2)
+    apply(simp)
+    apply(drule_tac x="AALTs x51 list" in spec)
+    apply(drule mp)
+     apply(simp add: asize0)
+  apply(simp)
+   apply(case_tac list)
+    prefer 2
+    apply(simp)
+  apply(case_tac "bsimp aa = AZERO")
+     apply(simp)
+     apply(subgoal_tac "bsimp (bder c aa) = AZERO")
+      prefer 2
+      apply auto[1]
+      apply (metis add.left_commute bder.simps(1) bsimp.simps(3) less_add_Suc1)
+     apply(simp)
+  apply(drule_tac x="AALTs x51 (a#lista)" in spec)
+    apply(drule mp)
+     apply(simp  add: asize0)
+     apply(simp)
+     apply (metis flts.simps(2) k0)
+    apply(subst k0)
+  apply(subst (2) k0)
+  
+  
+  using less_add_Suc1 apply fastforce
+    apply(subst k0)
+  
+
+    apply(simp)
+    apply(case_tac "bsimp a = AZERO")
+     apply(simp)
+     apply(subgoal_tac "bsimp (bder c a) = AZERO")
+      prefer 2
+  apply auto[1]
+     apply(simp)
+  apply(case_tac "nonalt (bsimp a)")
+     apply(subst bsimp_AALTs1)
+      apply(simp)
+  using less_add_Suc1 apply fastforce
+  
+     apply(subst bsimp_AALTs1)
+  
+  using nn11a apply b last
+
+  (* SEQ case *)
+   apply(clarify)
+  apply(subst  bsimp.simps)
+   apply(simp del: bsimp.simps)
+   apply(auto simp del: bsimp.simps)[1]
+    apply(subgoal_tac "bsimp x42 \<noteq> AZERO")
+  prefer 2
+  using b3 apply force
+  apply(case_tac "bsimp x43 = AZERO")
+     apply(simp)
+     apply (simp add: bsimp_ASEQ0)
+  apply (metis bder.simps(1) bsimp.simps(3) bsimp_AALTs.simps(1) bsimp_fuse flts.simps(1) flts.simps(2) fuse.simps(1) less_add_Suc2)
+    apply(case_tac "\<exists>bs. bsimp x42 = AONE bs")
+     apply(clarify)
+     apply(simp)
+     apply(subst bsimp_ASEQ2)
+     apply(subgoal_tac "bsimp (bder c x42) = AZERO")
+      prefer 2
+  using less_add_Suc1 apply fastforce
+     apply(simp)
+     apply(frule_tac x="x43" in spec)
+  apply(drule mp)
+     apply(simp)
+  apply(drule_tac x="c" in spec)
+     apply(subst bder_fuse)
+  apply(subst bsimp_fuse[symmetric])
+     apply(simp)
+  apply(subgoal_tac "bmkeps x42 = bs")
+      prefer 2
+      apply (simp add: bmkeps_simp)
+     apply(simp)
+     apply(subst bsimp_fuse[symmetric])
+  apply(case_tac "nonalt (bsimp (bder c x43))")
+      apply(subst bsimp_AALTs1)
+  using nn11a apply blast
+  using fuse_append apply blast
+     apply(subgoal_tac "\<exists>bs rs. bsimp (bder c x43) = AALTs bs rs")
+      prefer 2
+  using bbbbs1 apply blast
+  apply(clarify)
+     apply(simp)
+     apply(case_tac rs)
+      apply(simp)
+      apply (metis arexp.distinct(7) good.simps(4) good1)
+     apply(simp)
+     apply(case_tac list)
+      apply(simp)
+      apply (metis arexp.distinct(7) good.simps(5) good1)
+  apply(simp del: bsimp_AALTs.simps)
+  apply(simp only: bsimp_AALTs.simps)
+     apply(simp)
+  
+  
+
+
+(* HERE *)
+apply(case_tac "x42 = AZERO")
+     apply(simp)
+   apply(case_tac "bsimp x43 = AZERO")
+     apply(simp)
+     apply (simp add: bsimp_ASEQ0)
+     apply(subgoal_tac "bsimp (fuse (bmkeps x42) (bder c x43)) = AZERO")
+      apply(simp)
+  apply (met is bder.simps(1) bsimp.simps(3) bsimp_fuse fuse.simps(1) less_add_Suc2)
+  apply(case_tac "\<exists>bs. bsimp x42 = AONE bs")
+     apply(clarify)
+     apply(simp)
+     apply(subst bsimp_ASEQ2)
+     apply(subgoal_tac "bsimp (bder c x42) = AZERO")
+      apply(simp)
+  prefer 2
+  using less_add_Suc1 apply fastforce
+     apply(subgoal_tac "bmkeps x42 = bs")
+      prefer 2
+      apply (simp add: bmkeps_simp)
+     apply(simp)
+     apply(case_tac "nonalt (bsimp (bder c x43))")
+  apply (metis bder_fuse bsimp_AALTs.simps(1) bsimp_AALTs.simps(2) bsimp_fuse flts.simps(1) flts.simps(2) fuse.simps(1) fuse_append k0b less_add_Suc2 nn11a)
+     apply(subgoal_tac "nonnested (bsimp (bder c x43))")
+      prefer 2
+  using nn1b apply blast
+     apply(case_tac x43)
+          apply(simp)
+         apply(simp)
+        apply(simp)
+       prefer 3
+       apply(simp)
+       apply (metis arexp.distinct(25) arexp.distinct(7) arexp.distinct(9) bsimp_ASEQ.simps(1) bsimp_ASEQ.simps(11) bsimp_ASEQ1 nn11a nonalt.elims(3) nonalt.simps(6)) 
+      apply(simp)
+      apply(auto)[1]
+       apply(case_tac "(bsimp (bder c x42a)) = AZERO")
+        apply(simp)
+  
+       apply(simp)
+  
+  
+  
+     apply(subgoal_tac "(\<exists>bs1 rs1. 1 < length rs1 \<and> bsimp (bder c x43) =  AALTs bs1 rs1 ) \<or>
+                        (\<exists>bs1 r. bsimp (bder c x43) =  fuse bs1 r)")
+      prefer 2
+  apply (metis fuse_empty)
+     apply(erule disjE)
+  prefer 2
+     apply(clarify)
+     apply(simp only:)
+     apply(simp)
+     apply(simp add: fuse_append)
+     apply(subst bder_fuse)
+     apply(subst bsimp_fuse[symmetric])
+     apply(subst bder_fuse)
+     apply(subst bsimp_fuse[symmetric])
+     apply(subgoal_tac "bsimp (bder c (bsimp x43)) = bsimp (bder c x43)")
+      prefer 2
+  using less_add_Suc2 apply bl ast
+     apply(simp only: )
+     apply(subst bsimp_fuse[symmetric])
+      apply(simp only: )
+  
+     apply(simp only: fuse.simps)
+  apply(simp)
+      apply(case_tac rs1)
+      apply(simp)
+      apply (me tis arexp.distinct(7) fuse.simps(1) good.simps(4) good1 good_fuse)
+  apply(simp)
+  apply(case_tac list)
+      apply(simp)
+      apply (me tis arexp.distinct(7) fuse.simps(1) good.simps(5) good1 good_fuse)
+     apply(simp only: bsimp_AALTs.simps map_cons.simps)
+     apply(auto)[1]
+  
+  
+      
+      apply(subst bsimp_fuse[symmetric])
+  apply(subgoal_tac "bmkeps x42 = bs")
+      prefer 2
+      apply (simp add: bmkeps_simp)
+  
+  
+        apply(simp)
+  
+  using b3 apply force
+  using bsimp_ASEQ0 test2 apply fo rce
+  thm good_SEQ test2
+     apply (simp add: good_SEQ test2)
+    apply (simp add: good_SEQ test2)
+  apply(case_tac "x42 = AZERO")
+     apply(simp)
+   apply(case_tac "x43 = AZERO")
+    apply(simp)
+  apply (simp add: bsimp_ASEQ0)
+  apply(case_tac "\<exists>bs. x42 = AONE bs")
+     apply(clarify)
+     apply(simp)
+    apply(subst bsimp_ASEQ1)
+      apply(simp)
+  using bsimp_ASEQ0 test2 apply fo rce
+     apply (simp add: good_SEQ test2)
+    apply (simp add: good_SEQ test2)
+  apply (simp add: good_SEQ test2)
+  (* AALTs case *)
+  apply(simp)
+  using test2 by fa st force
+
+
+lemma XXX4ab:
+  shows "good (bders_simp (bsimp r) s)  \<or> bders_simp (bsimp r) s = AZERO"
+  apply(induct s arbitrary: r rule:  rev_induct)
+   apply(simp)
+  apply (simp add: good1)
+  apply(simp add: bders_simp_append)
+  apply (simp add: good1)
+  done
+
+lemma XXX4:
+  assumes "good a"
+  shows "bders_simp a s = bsimp (bders a s)"
+  using  assms
+  apply(induct s arbitrary: a rule: rev_induct)
+   apply(simp)
+   apply (simp add: test2)
+  apply(simp add: bders_append bders_simp_append)
+  oops
+
+
+lemma MAINMAIN:
+  "blexer r s = blexer_simp r s"
+  apply(induct s arbitrary: r)
+  apply(simp add: blexer_def blexer_simp_def)
+  apply(simp add: blexer_def blexer_simp_def del: bders.simps bders_simp.simps)
+  apply(auto simp del: bders.simps bders_simp.simps)
+    prefer 2
+  apply (metis b4 bders.simps(2) bders_simp.simps(2))
+   prefer 2
+  apply (metis b4 bders.simps(2))
+  apply(subst bmkeps_simp)
+   apply(simp)
+  apply(case_tac s)
+   apply(simp only: bders.simps)
+   apply(subst bders_simp.simps)
+  apply(simp)
+  oops   
+
+
+lemma
+  fixes n :: nat
+  shows "(\<Sum>i \<in> {0..n}. i) = n * (n + 1) div 2"
+  apply(induct n)
+  apply(simp)
+  apply(simp)
+  done
+
+
+
+
+
+end
\ No newline at end of file




