# HG changeset patch
# User Chengsong
# Date 1643933112 0
# Node ID f71df68776bb8969985a20a3439dfd50f04cb42e
# Parent  01d1285b08ed61a7107f80ead6a4633460a7dd42
5ct

diff -r 01d1285b08ed -r f71df68776bb thys2/SizeBound4.thy
--- a/thys2/SizeBound4.thy	Wed Feb 02 22:30:41 2022 +0000
+++ b/thys2/SizeBound4.thy	Fri Feb 04 00:05:12 2022 +0000
@@ -1444,6 +1444,40 @@
   using bsimp_ASEQ_idem apply presburger
   oops
 
+lemma neg:
+  shows " \<not>(\<exists>r2. r1 \<leadsto> r2 \<and>  (r2 \<leadsto>* bsimp r1) )"
+  apply(rule notI)
+  apply(erule exE)
+  apply(erule conjE)
+  oops
+
+
+
+
+lemma reduction_always_in_bsimp:
+  shows " \<lbrakk> r1 \<leadsto> r2 ; \<not>(r2 \<leadsto>* bsimp r1)\<rbrakk> \<Longrightarrow> False"
+  apply(erule rrewrite.cases)
+         apply simp
+        apply auto
+
+  oops
+
+(*
+AALTs [] [AZERO, AALTs(bs1, [a, b]) ] 
+rewrite seq 1: \<leadsto> AALTs [] [ AALTs(bs1, [a, b]) ] \<leadsto>
+fuse [] (AALTs bs1, [a, b])
+rewrite seq 2: \<leadsto> AALTs [] [AZERO, (fuse bs1 a), (fuse bs1 b)]) ]
+
+*)
+
+lemma normal_bsimp: 
+  shows "\<nexists>r'. bsimp r \<leadsto> r'"
+  oops
+
+  (*r' size bsimp r > size r' 
+    r' \<leadsto>* bsimp bsimp r
+size bsimp r > size r' \<ge> size bsimp bsimp r*)
+
 export_code blexer_simp blexer lexer bders bders_simp in Scala module_name VerifiedLexers
 
 
diff -r 01d1285b08ed -r f71df68776bb thys2/SizeBound4CT.thy
--- a/thys2/SizeBound4CT.thy	Wed Feb 02 22:30:41 2022 +0000
+++ b/thys2/SizeBound4CT.thy	Fri Feb 04 00:05:12 2022 +0000
@@ -1156,6 +1156,16 @@
   using bsimp_ASEQ_idem apply presburger
   oops
 
+
+lemma normal_form: 
+  shows "\<forall>r. \<nexists> r'. bsimp r \<leadsto> r'"
+
+  oops
+
+lemma another_normal:
+  shows "\<nexists>r'. bsimp r \<leadsto> r'"
+  oops
+
 export_code blexer_simp blexer lexer bders bders_simp in Scala module_name VerifiedLexers
 
 
diff -r 01d1285b08ed -r f71df68776bb thys2/SizeBound5CT.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/thys2/SizeBound5CT.thy	Fri Feb 04 00:05:12 2022 +0000
@@ -0,0 +1,1493 @@
+
+theory SizeBound5CT
+  imports "Lexer" "PDerivs"
+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' bs ZERO = (undefined, bs)"
+| "decode' bs ONE = (Void, bs)"
+| "decode' bs (CH d) = (Char d, bs)"
+| "decode' [] (ALT r1 r2) = (Void, [])"
+| "decode' (Z # bs) (ALT r1 r2) = (let (v, bs') = decode' bs r1 in (Left v, bs'))"
+| "decode' (S # bs) (ALT r1 r2) = (let (v, bs') = decode' bs r2 in (Right v, bs'))"
+| "decode' bs (SEQ r1 r2) = (let (v1, bs') = decode' bs r1 in
+                             let (v2, bs'') = decode' bs' r2 in (Seq v1 v2, bs''))"
+| "decode' [] (STAR r) = (Void, [])"
+| "decode' (S # bs) (STAR r) = (Stars [], bs)"
+| "decode' (Z # bs) (STAR r) = (let (v, bs') = decode' bs r in
+                                    let (vs, bs'') = decode' bs' (STAR r) 
+                                    in (Stars_add v vs, bs''))"
+by pat_completeness auto
+
+lemma decode'_smaller:
+  assumes "decode'_dom (bs, r)"
+  shows "length (snd (decode' bs r)) \<le> length bs"
+using assms
+apply(induct bs 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) = CH 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)"
+
+
+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 (CH 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"
+
+abbreviation
+  bnullables :: "arexp list \<Rightarrow> bool"
+where
+  "bnullables rs \<equiv> (\<exists>r \<in> set rs. bnullable r)"
+
+fun 
+  bmkeps :: "arexp \<Rightarrow> bit list" and
+  bmkepss :: "arexp list \<Rightarrow> bit list"
+where
+  "bmkeps(AONE bs) = bs"
+| "bmkeps(ASEQ bs r1 r2) = bs @ (bmkeps r1) @ (bmkeps r2)"
+| "bmkeps(AALTs bs rs) = bs @ (bmkepss rs)"
+| "bmkeps(ASTAR bs r) = bs @ [S]"
+| "bmkepss [] = []"
+| "bmkepss (r # rs) = (if bnullable(r) then (bmkeps r) else (bmkepss rs))"
+
+lemma bmkepss1:
+  assumes "\<not> bnullables rs1"
+  shows "bmkepss (rs1 @ rs2) = bmkepss rs2"
+  using assms
+  by (induct rs1) (auto)
+
+lemma bmkepss2:
+  assumes "bnullables rs1"
+  shows "bmkepss (rs1 @ rs2) = bmkepss rs1"
+  using assms
+  by (induct rs1) (auto)
+
+
+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 c (s1 @ s2) = bders (bders c s1) s2"
+  apply(induct s1 arbitrary: c 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 bnullable_fuse:
+  shows "bnullable (fuse bs r) = bnullable r"
+  apply(induct r arbitrary: bs)
+  apply(auto)
+  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]
+  apply(case_tac x2a)
+  apply(simp)
+  using Prf_elims(1) apply blast
+  apply(case_tac x2a)
+  apply(simp)
+  apply(simp)
+  apply(case_tac list)
+  apply(simp)
+  apply(simp)
+  apply (smt (verit, best) Prf_elims(3) append_assoc retrieve.simps(4) retrieve.simps(5))
+  apply(simp)
+  using retrieve_encode_STARS
+  apply(auto elim!: Prf_elims)[1]
+  apply(case_tac vs)
+  apply(simp)
+  apply(simp)
+  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 retrieve_AALTs_bnullable1:
+  assumes "bnullable r"
+  shows "retrieve (AALTs bs (r # rs)) (mkeps (erase (AALTs bs (r # rs))))
+         = bs @ retrieve r (mkeps (erase r))"
+  using assms
+  apply(case_tac rs)
+  apply(auto simp add: bnullable_correctness)
+  done
+
+lemma retrieve_AALTs_bnullable2:
+  assumes "\<not>bnullable r" "bnullables rs"
+  shows "retrieve (AALTs bs (r # rs)) (mkeps (erase (AALTs bs (r # rs))))
+         = retrieve (AALTs bs rs) (mkeps (erase (AALTs bs rs)))"
+  using assms
+  apply(induct rs arbitrary: r bs)
+  apply(auto)
+  using bnullable_correctness apply blast
+  apply(case_tac rs)
+  apply(auto)
+  using bnullable_correctness apply blast
+  apply(case_tac rs)
+  apply(auto)
+  done
+
+lemma bmkeps_retrieve_AALTs: 
+  assumes "\<forall>r \<in> set rs. bnullable r \<longrightarrow> bmkeps r = retrieve r (mkeps (erase r))" 
+          "bnullables rs"
+  shows "bs @ bmkepss rs = retrieve (AALTs bs rs) (mkeps (erase (AALTs bs rs)))"
+ using assms
+  apply(induct rs arbitrary: bs)
+  apply(auto)
+  using retrieve_AALTs_bnullable1 apply presburger
+  apply (metis retrieve_AALTs_bnullable2)
+  apply (simp add: retrieve_AALTs_bnullable1)
+  by (metis retrieve_AALTs_bnullable2)
+
+    
+lemma bmkeps_retrieve:
+  assumes "bnullable r"
+  shows "bmkeps r = retrieve r (mkeps (erase r))"
+  using assms
+  apply(induct r)
+  apply(auto)  
+  using bmkeps_retrieve_AALTs by auto
+
+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)
+  using Prf_elims(1) apply auto[1]
+  using Prf_elims(1) apply auto[1]
+  apply(auto)[1]
+  apply (metis Prf_elims(4) injval.simps(1) retrieve.simps(1) retrieve.simps(2))
+  using Prf_elims(1) apply blast
+  (* AALTs case *)
+  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)
+  using Prf_elims(3) apply fastforce
+  (* ASEQ case *) 
+  apply(simp)
+  apply(case_tac "nullable (erase r1)")
+  apply(simp)
+  apply(erule Prf_elims)
+  using Prf_elims(2) bnullable_correctness apply force
+  apply (simp add: bmkeps_retrieve bnullable_correctness retrieve_fuse2)
+  apply (simp add: bmkeps_retrieve bnullable_correctness retrieve_fuse2)
+  using Prf_elims(2) apply force
+  (* ASTAR case *)  
+  apply(rename_tac bs r v)
+  apply(simp)  
+  apply(erule Prf_elims)
+  apply(clarify)
+  apply(erule Prf_elims)
+  apply(clarify)
+  by (simp add: retrieve_fuse2)
+
+
+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 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
+      using bnullable_correctness by force 
+    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
+    by (auto simp add: bnullable_correctness[symmetric])
+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 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"
+
+lemma bsimp_ASEQ0[simp]:
+  shows "bsimp_ASEQ bs r1 AZERO = AZERO"
+  by (case_tac r1)(simp_all)
+
+lemma bsimp_ASEQ1:
+  assumes "r1 \<noteq> AZERO" "r2 \<noteq> AZERO" "\<nexists>bs. r1 = 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[simp]:
+  shows "bsimp_ASEQ bs1 (AONE bs2) r2 = fuse (bs1 @ bs2) r2"
+  by (case_tac r2) (simp_all)
+
+
+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 (distinctBy (flts (map bsimp rs)) erase {}) "
+| "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 bders_simp_append:
+  shows "bders_simp r (s1 @ s2) = bders_simp (bders_simp r s1) s2"
+  apply(induct s1 arbitrary: r s2)
+  apply(simp_all)
+  done
+
+
+lemma bmkeps_fuse:
+  assumes "bnullable r"
+  shows "bmkeps (fuse bs r) = bs @ bmkeps r"
+  using assms
+  by (metis bmkeps_retrieve bnullable_correctness erase_fuse mkeps_nullable retrieve_fuse2)
+
+lemma bmkepss_fuse: 
+  assumes "bnullables rs"
+  shows "bmkepss (map (fuse bs) rs) = bs @ bmkepss rs"
+  using assms
+  apply(induct rs arbitrary: bs)
+  apply(auto simp add: bmkeps_fuse bnullable_fuse)
+  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
+
+
+
+
+inductive 
+  rrewrite:: "arexp \<Rightarrow> arexp \<Rightarrow> bool" ("_ \<leadsto> _" [99, 99] 99)
+and 
+  srewrite:: "arexp list \<Rightarrow> arexp list \<Rightarrow> bool" (" _ s\<leadsto> _" [100, 100] 100)
+where
+  bs1: "ASEQ bs AZERO r2 \<leadsto> AZERO"
+| bs2: "ASEQ bs r1 AZERO \<leadsto> AZERO"
+| bs3: "ASEQ bs1 (AONE bs2) r \<leadsto> fuse (bs1@bs2) r"
+| bs4: "r1 \<leadsto> r2 \<Longrightarrow> ASEQ bs r1 r3 \<leadsto> ASEQ bs r2 r3"
+| bs5: "r3 \<leadsto> r4 \<Longrightarrow> ASEQ bs r1 r3 \<leadsto> ASEQ bs r1 r4"
+| bs6: "AALTs bs [] \<leadsto> AZERO"
+| bs7: "AALTs bs [r] \<leadsto> fuse bs r"
+| bs8: "rs1 s\<leadsto> rs2 \<Longrightarrow> AALTs bs rs1 \<leadsto> AALTs bs rs2"
+(*| ss1:  "[] s\<leadsto> []"*)
+| ss2:  "rs1 s\<leadsto> rs2 \<Longrightarrow> (r # rs1) s\<leadsto> (r # rs2)"
+| ss3:  "r1 \<leadsto> r2 \<Longrightarrow> (r1 # rs) s\<leadsto> (r2 # rs)"
+| ss4:  "(AZERO # rs) s\<leadsto> rs"
+| ss5:  "(AALTs bs1 rs1 # rsb) s\<leadsto> ((map (fuse bs1) rs1) @ rsb)"
+| ss6:  "erase a1 = erase a2 \<Longrightarrow> (rsa@[a1]@rsb@[a2]@rsc) s\<leadsto> (rsa@[a1]@rsb@rsc)"
+
+
+inductive 
+  rrewrites:: "arexp \<Rightarrow> arexp \<Rightarrow> bool" ("_ \<leadsto>* _" [100, 100] 100)
+where 
+  rs1[intro, simp]:"r \<leadsto>* r"
+| rs2[intro]: "\<lbrakk>r1 \<leadsto>* r2; r2 \<leadsto> r3\<rbrakk> \<Longrightarrow> r1 \<leadsto>* r3"
+
+inductive 
+  srewrites:: "arexp list \<Rightarrow> arexp list \<Rightarrow> bool" ("_ s\<leadsto>* _" [100, 100] 100)
+where 
+  sss1[intro, simp]:"rs s\<leadsto>* rs"
+| sss2[intro]: "\<lbrakk>rs1 s\<leadsto> rs2; rs2 s\<leadsto>* rs3\<rbrakk> \<Longrightarrow> rs1 s\<leadsto>* rs3"
+
+
+lemma r_in_rstar: 
+  shows "r1 \<leadsto> r2 \<Longrightarrow> r1 \<leadsto>* r2"
+  using rrewrites.intros(1) rrewrites.intros(2) by blast
+
+lemma rrewrites_trans[trans]: 
+  assumes a1: "r1 \<leadsto>* r2"  and a2: "r2 \<leadsto>* r3"
+  shows "r1 \<leadsto>* r3"
+  using a2 a1
+  apply(induct r2 r3 arbitrary: r1 rule: rrewrites.induct) 
+  apply(auto)
+  done
+
+lemma srewrites_trans[trans]: 
+  assumes a1: "r1 s\<leadsto>* r2"  and a2: "r2 s\<leadsto>* r3"
+  shows "r1 s\<leadsto>* r3"
+  using a1 a2
+  apply(induct r1 r2 arbitrary: r3 rule: srewrites.induct) 
+   apply(auto)
+  done
+
+
+
+lemma contextrewrites0: 
+  "rs1 s\<leadsto>* rs2 \<Longrightarrow> AALTs bs rs1 \<leadsto>* AALTs bs rs2"
+  apply(induct rs1 rs2 rule: srewrites.inducts)
+   apply simp
+  using bs8 r_in_rstar rrewrites_trans by blast
+
+lemma contextrewrites1: 
+  "r \<leadsto>* r' \<Longrightarrow> AALTs bs (r # rs) \<leadsto>* AALTs bs (r' # rs)"
+  apply(induct r r' rule: rrewrites.induct)
+   apply simp
+  using bs8 ss3 by blast
+
+lemma srewrite1: 
+  shows "rs1 s\<leadsto> rs2 \<Longrightarrow> (rs @ rs1) s\<leadsto> (rs @ rs2)"
+  apply(induct rs)
+   apply(auto)
+  using ss2 by auto
+
+lemma srewrites1: 
+  shows "rs1 s\<leadsto>* rs2 \<Longrightarrow> (rs @ rs1) s\<leadsto>* (rs @ rs2)"
+  apply(induct rs1 rs2 rule: srewrites.induct)
+   apply(auto)
+  using srewrite1 by blast
+
+lemma srewrite2: 
+  shows  "r1 \<leadsto> r2 \<Longrightarrow> True"
+  and "rs1 s\<leadsto> rs2 \<Longrightarrow> (rs1 @ rs) s\<leadsto>* (rs2 @ rs)"
+  apply(induct rule: rrewrite_srewrite.inducts)
+  apply(auto)
+  apply (metis append_Cons append_Nil srewrites1)
+  apply(meson srewrites.simps ss3)
+  apply (meson srewrites.simps ss4)
+  apply (meson srewrites.simps ss5)
+  by (metis append_Cons append_Nil srewrites.simps ss6)
+  
+
+lemma srewrites3: 
+  shows "rs1 s\<leadsto>* rs2 \<Longrightarrow> (rs1 @ rs) s\<leadsto>* (rs2 @ rs)"
+  apply(induct rs1 rs2 arbitrary: rs rule: srewrites.induct)
+   apply(auto)
+  by (meson srewrite2(2) srewrites_trans)
+
+(*
+lemma srewrites4:
+  assumes "rs3 s\<leadsto>* rs4" "rs1 s\<leadsto>* rs2" 
+  shows "(rs1 @ rs3) s\<leadsto>* (rs2 @ rs4)"
+  using assms
+  apply(induct rs3 rs4 arbitrary: rs1 rs2 rule: srewrites.induct)
+  apply (simp add: srewrites3)
+  using srewrite1 by blast
+*)
+
+lemma srewrites6:
+  assumes "r1 \<leadsto>* r2" 
+  shows "[r1] s\<leadsto>* [r2]"
+  using assms
+  apply(induct r1 r2 rule: rrewrites.induct)
+  apply(auto)
+  by (meson srewrites.simps srewrites_trans ss3)
+
+lemma srewrites7:
+  assumes "rs3 s\<leadsto>* rs4" "r1 \<leadsto>* r2" 
+  shows "(r1 # rs3) s\<leadsto>* (r2 # rs4)"
+  using assms
+  by (smt (verit, del_insts) append.simps srewrites1 srewrites3 srewrites6 srewrites_trans)
+  
+lemma ss6_stronger_aux:
+  shows "(rs1 @ rs2) s\<leadsto>* (rs1 @ distinctBy rs2 erase (set (map erase rs1)))"
+  apply(induct rs2 arbitrary: rs1)
+   apply(auto)
+  apply (smt (verit, best) append.assoc append.right_neutral append_Cons append_Nil split_list srewrite2(2) srewrites_trans ss6)
+  apply(drule_tac x="rs1 @ [a]" in meta_spec)
+  apply(simp)
+  done
+
+lemma ss6_stronger:
+  shows "rs1 s\<leadsto>* distinctBy rs1 erase {}"
+  using ss6_stronger_aux[of "[]" _] by auto
+
+lemma rewrite_preserves_fuse: 
+  shows "r2 \<leadsto> r3 \<Longrightarrow> fuse bs r2 \<leadsto> fuse bs r3"
+  and   "rs2 s\<leadsto> rs3 \<Longrightarrow> map (fuse bs) rs2 s\<leadsto> map (fuse bs) rs3"
+proof(induct rule: rrewrite_srewrite.inducts)
+  case (bs3 bs1 bs2 r)
+  then show "fuse bs (ASEQ bs1 (AONE bs2) r) \<leadsto> fuse bs (fuse (bs1 @ bs2) r)"
+    by (metis fuse.simps(5) fuse_append rrewrite_srewrite.bs3) 
+next
+  case (bs7 bs1 r)
+  then show "fuse bs (AALTs bs1 [r]) \<leadsto> fuse bs (fuse bs1 r)"
+    by (metis fuse.simps(4) fuse_append rrewrite_srewrite.bs7) 
+next
+  case (ss2 rs1 rs2 r)
+  then show "map (fuse bs) (r # rs1) s\<leadsto> map (fuse bs) (r # rs2)"
+    by (simp add: rrewrite_srewrite.ss2)
+next
+  case (ss3 r1 r2 rs)
+  then show "map (fuse bs) (r1 # rs) s\<leadsto> map (fuse bs) (r2 # rs)"
+    by (simp add: rrewrite_srewrite.ss3)
+next
+  case (ss5 bs1 rs1 rsb)
+  have "map (fuse bs) (AALTs bs1 rs1 # rsb) = AALTs (bs @ bs1) rs1 # (map (fuse bs) rsb)" by simp
+  also have "... s\<leadsto> ((map (fuse (bs @ bs1)) rs1) @ (map (fuse bs) rsb))"
+    by (simp add: rrewrite_srewrite.ss5)  
+  finally show "map (fuse bs) (AALTs bs1 rs1 # rsb) s\<leadsto> map (fuse bs) (map (fuse bs1) rs1 @ rsb)"
+    by (simp add: comp_def fuse_append)
+next
+  case (ss6 a1 a2 rsa rsb rsc)
+  then show "map (fuse bs) (rsa @ [a1] @ rsb @ [a2] @ rsc) s\<leadsto> map (fuse bs) (rsa @ [a1] @ rsb @ rsc)"
+    apply(simp)
+    apply(rule rrewrite_srewrite.ss6[simplified])
+    apply(simp add: erase_fuse)
+    done
+qed (auto intro: rrewrite_srewrite.intros)
+
+lemma rewrites_fuse:  
+  assumes "r1 \<leadsto>* r2"
+  shows "fuse bs r1 \<leadsto>* fuse bs r2"
+using assms
+apply(induction r1 r2 arbitrary: bs rule: rrewrites.induct)
+apply(auto intro: rewrite_preserves_fuse)
+done
+
+
+lemma star_seq:  
+  assumes "r1 \<leadsto>* r2"
+  shows "ASEQ bs r1 r3 \<leadsto>* ASEQ bs r2 r3"
+using assms
+apply(induct r1 r2 arbitrary: r3 rule: rrewrites.induct)
+apply(auto intro: rrewrite_srewrite.intros)
+done
+
+lemma star_seq2:  
+  assumes "r3 \<leadsto>* r4"
+  shows "ASEQ bs r1 r3 \<leadsto>* ASEQ bs r1 r4"
+  using assms
+apply(induct r3 r4 arbitrary: r1 rule: rrewrites.induct)
+apply(auto intro: rrewrite_srewrite.intros)
+done
+
+lemma continuous_rewrite: 
+  assumes "r1 \<leadsto>* AZERO"
+  shows "ASEQ bs1 r1 r2 \<leadsto>* AZERO"
+using assms bs1 star_seq by blast
+
+(*
+lemma continuous_rewrite2: 
+  assumes "r1 \<leadsto>* AONE bs"
+  shows "ASEQ bs1 r1 r2 \<leadsto>* (fuse (bs1 @ bs) r2)"
+  using assms  by (meson bs3 rrewrites.simps star_seq)
+*)
+
+lemma bsimp_aalts_simpcases: 
+  shows "AONE bs \<leadsto>* bsimp (AONE bs)"  
+  and   "AZERO \<leadsto>* bsimp AZERO" 
+  and   "ACHAR bs c \<leadsto>* bsimp (ACHAR bs c)"
+  by (simp_all)
+
+lemma bsimp_AALTs_rewrites: 
+  shows "AALTs bs1 rs \<leadsto>* bsimp_AALTs bs1 rs"
+  by (smt (verit) bs6 bs7 bsimp_AALTs.elims rrewrites.simps)
+
+lemma trivialbsimp_srewrites: 
+  assumes "\<And>x. x \<in> set rs \<Longrightarrow> x \<leadsto>* f x"
+  shows "rs s\<leadsto>* (map f rs)"
+using assms
+  apply(induction rs)
+  apply(simp_all add: srewrites7)
+  done
+
+lemma fltsfrewrites: "rs s\<leadsto>* flts rs"
+  apply(induction rs rule: flts.induct)
+  apply(auto intro: rrewrite_srewrite.intros)
+  apply (meson srewrites.simps srewrites1 ss5)
+  using rs1 srewrites7 apply presburger
+  using srewrites7 apply force
+  apply (simp add: srewrites7)
+  by (simp add: srewrites7)
+
+lemma bnullable0:
+shows "r1 \<leadsto> r2 \<Longrightarrow> bnullable r1 = bnullable r2" 
+  and "rs1 s\<leadsto> rs2 \<Longrightarrow> bnullables rs1 = bnullables rs2" 
+  apply(induct rule: rrewrite_srewrite.inducts)
+  apply(auto simp add:  bnullable_fuse)
+  apply (meson UnCI bnullable_fuse imageI)
+  by (metis bnullable_correctness)
+
+
+lemma rewrites_bnullable_eq: 
+  assumes "r1 \<leadsto>* r2" 
+  shows "bnullable r1 = bnullable r2"
+using assms 
+  apply(induction r1 r2 rule: rrewrites.induct)
+  apply simp
+  using bnullable0(1) by auto
+
+lemma rewrite_bmkeps_aux: 
+  shows "r1 \<leadsto> r2 \<Longrightarrow> bnullable r1 \<Longrightarrow> bmkeps r1 = bmkeps r2"
+  and   "rs1 s\<leadsto> rs2 \<Longrightarrow> bnullables rs1 \<Longrightarrow> bmkepss rs1 = bmkepss rs2" 
+proof (induct rule: rrewrite_srewrite.inducts)
+  case (bs3 bs1 bs2 r)
+  have IH2: "bnullable (ASEQ bs1 (AONE bs2) r)" by fact
+  then show "bmkeps (ASEQ bs1 (AONE bs2) r) = bmkeps (fuse (bs1 @ bs2) r)"
+    by (simp add: bmkeps_fuse)
+next
+  case (bs7 bs r)
+  have IH2: "bnullable (AALTs bs [r])" by fact
+  then show "bmkeps (AALTs bs [r]) = bmkeps (fuse bs r)" 
+    by (simp add: bmkeps_fuse)
+next
+  case (ss3 r1 r2 rs)
+  have IH1: "bnullable r1 \<Longrightarrow> bmkeps r1 = bmkeps r2" by fact
+  have as: "r1 \<leadsto> r2" by fact
+  from IH1 as show "bmkepss (r1 # rs) = bmkepss (r2 # rs)"
+    by (simp add: bnullable0)
+next
+  case (ss5 bs1 rs1 rsb)
+  have "bnullables (AALTs bs1 rs1 # rsb)" by fact
+  then show "bmkepss (AALTs bs1 rs1 # rsb) = bmkepss (map (fuse bs1) rs1 @ rsb)"
+    by (simp add: bmkepss1 bmkepss2 bmkepss_fuse bnullable_fuse)
+next
+  case (ss6 a1 a2 rsa rsb rsc)
+  have as1: "erase a1 = erase a2" by fact
+  have as3: "bnullables (rsa @ [a1] @ rsb @ [a2] @ rsc)" by fact
+  show "bmkepss (rsa @ [a1] @ rsb @ [a2] @ rsc) = bmkepss (rsa @ [a1] @ rsb @ rsc)" using as1 as3
+    by (smt (verit, best) append_Cons bmkeps.simps(3) bmkepss.simps(2) bmkepss1 bmkepss2 bnullable_correctness)
+qed (auto)
+
+lemma rewrites_bmkeps: 
+  assumes "r1 \<leadsto>* r2" "bnullable r1" 
+  shows "bmkeps r1 = bmkeps r2"
+  using assms
+proof(induction r1 r2 rule: rrewrites.induct)
+  case (rs1 r)
+  then show "bmkeps r = bmkeps r" by simp
+next
+  case (rs2 r1 r2 r3)
+  then have IH: "bmkeps r1 = bmkeps r2" by simp
+  have a1: "bnullable r1" by fact
+  have a2: "r1 \<leadsto>* r2" by fact
+  have a3: "r2 \<leadsto> r3" by fact
+  have a4: "bnullable r2" using a1 a2 by (simp add: rewrites_bnullable_eq) 
+  then have "bmkeps r2 = bmkeps r3"
+    using a3 bnullable0(1) rewrite_bmkeps_aux(1) by blast 
+  then show "bmkeps r1 = bmkeps r3" using IH by simp
+qed
+
+
+lemma rewrites_to_bsimp: 
+  shows "r \<leadsto>* bsimp r"
+proof (induction r rule: bsimp.induct)
+  case (1 bs1 r1 r2)
+  have IH1: "r1 \<leadsto>* bsimp r1" by fact
+  have IH2: "r2 \<leadsto>* bsimp r2" by fact
+  { assume as: "bsimp r1 = AZERO \<or> bsimp r2 = AZERO"
+    with IH1 IH2 have "r1 \<leadsto>* AZERO \<or> r2 \<leadsto>* AZERO" by auto
+    then have "ASEQ bs1 r1 r2 \<leadsto>* AZERO"
+      by (metis bs2 continuous_rewrite rrewrites.simps star_seq2)  
+    then have "ASEQ bs1 r1 r2 \<leadsto>* bsimp (ASEQ bs1 r1 r2)" using as by auto
+  }
+  moreover
+  { assume "\<exists>bs. bsimp r1 = AONE bs"
+    then obtain bs where as: "bsimp r1 = AONE bs" by blast
+    with IH1 have "r1 \<leadsto>* AONE bs" by simp
+    then have "ASEQ bs1 r1 r2 \<leadsto>* fuse (bs1 @ bs) r2" using bs3 star_seq by blast
+    with IH2 have "ASEQ bs1 r1 r2 \<leadsto>* fuse (bs1 @ bs) (bsimp r2)"
+      using rewrites_fuse by (meson rrewrites_trans) 
+    then have "ASEQ bs1 r1 r2 \<leadsto>* bsimp (ASEQ bs1 (AONE bs) r2)" by simp
+    then have "ASEQ bs1 r1 r2 \<leadsto>* bsimp (ASEQ bs1 r1 r2)" by (simp add: as) 
+  } 
+  moreover
+  { assume as1: "bsimp r1 \<noteq> AZERO" "bsimp r2 \<noteq> AZERO" and as2: "(\<nexists>bs. bsimp r1 = AONE bs)" 
+    then have "bsimp_ASEQ bs1 (bsimp r1) (bsimp r2) = ASEQ bs1 (bsimp r1) (bsimp r2)" 
+      by (simp add: bsimp_ASEQ1) 
+    then have "ASEQ bs1 r1 r2 \<leadsto>* bsimp_ASEQ bs1 (bsimp r1) (bsimp r2)" using as1 as2 IH1 IH2
+      by (metis rrewrites_trans star_seq star_seq2) 
+    then have "ASEQ bs1 r1 r2 \<leadsto>* bsimp (ASEQ bs1 r1 r2)" by simp
+  } 
+  ultimately show "ASEQ bs1 r1 r2 \<leadsto>* bsimp (ASEQ bs1 r1 r2)" by blast
+next
+  case (2 bs1 rs)
+  have IH: "\<And>x. x \<in> set rs \<Longrightarrow> x \<leadsto>* bsimp x" by fact
+  then have "rs s\<leadsto>* (map bsimp rs)" by (simp add: trivialbsimp_srewrites)
+  also have "... s\<leadsto>* flts (map bsimp rs)" by (simp add: fltsfrewrites) 
+  also have "... s\<leadsto>* distinctBy (flts (map bsimp rs)) erase {}" by (simp add: ss6_stronger) 
+  finally have "AALTs bs1 rs \<leadsto>* AALTs bs1 (distinctBy (flts (map bsimp rs)) erase {})"
+    using contextrewrites0 by blast
+  also have "... \<leadsto>* bsimp_AALTs  bs1 (distinctBy (flts (map bsimp rs)) erase {})"
+    by (simp add: bsimp_AALTs_rewrites)     
+  finally show "AALTs bs1 rs \<leadsto>* bsimp (AALTs bs1 rs)" by simp
+qed (simp_all)
+
+
+lemma to_zero_in_alt: 
+  shows "AALT bs (ASEQ [] AZERO r) r2 \<leadsto> AALT bs AZERO r2"
+  by (simp add: bs1 bs8 ss3)
+
+
+
+lemma  bder_fuse_list: 
+  shows "map (bder c \<circ> fuse bs1) rs1 = map (fuse bs1 \<circ> bder c) rs1"
+  apply(induction rs1)
+  apply(simp_all add: bder_fuse)
+  done
+
+lemma rewrite_preserves_bder: 
+  shows "r1 \<leadsto> r2 \<Longrightarrow> bder c r1 \<leadsto>* bder c r2"
+  and   "rs1 s\<leadsto> rs2 \<Longrightarrow> map (bder c) rs1 s\<leadsto>* map (bder c) rs2"
+proof(induction rule: rrewrite_srewrite.inducts)
+  case (bs1 bs r2)
+  show "bder c (ASEQ bs AZERO r2) \<leadsto>* bder c AZERO"
+    by (simp add: continuous_rewrite) 
+next
+  case (bs2 bs r1)
+  show "bder c (ASEQ bs r1 AZERO) \<leadsto>* bder c AZERO"
+    apply(auto)
+    apply (meson bs6 contextrewrites0 rrewrite_srewrite.bs2 rs2 ss3 ss4 sss1 sss2)
+    by (simp add: r_in_rstar rrewrite_srewrite.bs2)
+next
+  case (bs3 bs1 bs2 r)
+  show "bder c (ASEQ bs1 (AONE bs2) r) \<leadsto>* bder c (fuse (bs1 @ bs2) r)"
+    apply(simp)
+    by (metis (no_types, lifting) bder_fuse bs8 bs7 fuse_append rrewrites.simps ss4 to_zero_in_alt)
+next
+  case (bs4 r1 r2 bs r3)
+  have as: "r1 \<leadsto> r2" by fact
+  have IH: "bder c r1 \<leadsto>* bder c r2" by fact
+  from as IH show "bder c (ASEQ bs r1 r3) \<leadsto>* bder c (ASEQ bs r2 r3)"
+    by (metis bder.simps(5) bnullable0(1) contextrewrites1 rewrite_bmkeps_aux(1) star_seq)
+next
+  case (bs5 r3 r4 bs r1)
+  have as: "r3 \<leadsto> r4" by fact 
+  have IH: "bder c r3 \<leadsto>* bder c r4" by fact 
+  from as IH show "bder c (ASEQ bs r1 r3) \<leadsto>* bder c (ASEQ bs r1 r4)"
+    apply(simp)
+    apply(auto)
+    using contextrewrites0 r_in_rstar rewrites_fuse srewrites6 srewrites7 star_seq2 apply presburger
+    using star_seq2 by blast
+next
+  case (bs6 bs)
+  show "bder c (AALTs bs []) \<leadsto>* bder c AZERO"
+    using rrewrite_srewrite.bs6 by force 
+next
+  case (bs7 bs r)
+  show "bder c (AALTs bs [r]) \<leadsto>* bder c (fuse bs r)"
+    by (simp add: bder_fuse r_in_rstar rrewrite_srewrite.bs7) 
+next
+  case (bs8 rs1 rs2 bs)
+  have IH1: "map (bder c) rs1 s\<leadsto>* map (bder c) rs2" by fact
+  then show "bder c (AALTs bs rs1) \<leadsto>* bder c (AALTs bs rs2)" 
+    using contextrewrites0 by force    
+(*next
+  case ss1
+  show "map (bder c) [] s\<leadsto>* map (bder c) []" by simp*)
+next
+  case (ss2 rs1 rs2 r)
+  have IH1: "map (bder c) rs1 s\<leadsto>* map (bder c) rs2" by fact
+  then show "map (bder c) (r # rs1) s\<leadsto>* map (bder c) (r # rs2)"
+    by (simp add: srewrites7) 
+next
+  case (ss3 r1 r2 rs)
+  have IH: "bder c r1 \<leadsto>* bder c r2" by fact
+  then show "map (bder c) (r1 # rs) s\<leadsto>* map (bder c) (r2 # rs)"
+    by (simp add: srewrites7) 
+next
+  case (ss4 rs)
+  show "map (bder c) (AZERO # rs) s\<leadsto>* map (bder c) rs"
+    using rrewrite_srewrite.ss4 by fastforce 
+next
+  case (ss5 bs1 rs1 rsb)
+  show "map (bder c) (AALTs bs1 rs1 # rsb) s\<leadsto>* map (bder c) (map (fuse bs1) rs1 @ rsb)"
+    apply(simp)
+    using bder_fuse_list map_map rrewrite_srewrite.ss5 srewrites.simps by blast
+next
+  case (ss6 a1 a2 bs rsa rsb)
+  have as: "erase a1 = erase a2" by fact
+  show "map (bder c) (bs @ [a1] @ rsa @ [a2] @ rsb) s\<leadsto>* map (bder c) (bs @ [a1] @ rsa @ rsb)"
+    apply(simp only: map_append)
+    by (smt (verit, best) erase_bder list.simps(8) list.simps(9) as rrewrite_srewrite.ss6 srewrites.simps)
+qed
+
+lemma rewrites_preserves_bder: 
+  assumes "r1 \<leadsto>* r2"
+  shows "bder c r1 \<leadsto>* bder c r2"
+using assms  
+apply(induction r1 r2 rule: rrewrites.induct)
+apply(simp_all add: rewrite_preserves_bder rrewrites_trans)
+done
+
+
+lemma central:  
+  shows "bders r s \<leadsto>* bders_simp r s"
+proof(induct s arbitrary: r rule: rev_induct)
+  case Nil
+  then show "bders r [] \<leadsto>* bders_simp r []" by simp
+next
+  case (snoc x xs)
+  have IH: "\<And>r. bders r xs \<leadsto>* bders_simp r xs" by fact
+  have "bders r (xs @ [x]) = bders (bders r xs) [x]" by (simp add: bders_append)
+  also have "... \<leadsto>* bders (bders_simp r xs) [x]" using IH
+    by (simp add: rewrites_preserves_bder)
+  also have "... \<leadsto>* bders_simp (bders_simp r xs) [x]" using IH
+    by (simp add: rewrites_to_bsimp)
+  finally show "bders r (xs @ [x]) \<leadsto>* bders_simp r (xs @ [x])" 
+    by (simp add: bders_simp_append)
+qed
+
+lemma main_aux: 
+  assumes "bnullable (bders r s)"
+  shows "bmkeps (bders r s) = bmkeps (bders_simp r s)"
+proof -
+  have "bders r s \<leadsto>* bders_simp r s" by (rule central)
+  then 
+  show "bmkeps (bders r s) = bmkeps (bders_simp r s)" using assms
+    by (rule rewrites_bmkeps)
+qed  
+
+
+theorem main_blexer_simp: 
+  shows "blexer r s = blexer_simp r s"
+  unfolding blexer_def blexer_simp_def
+  by (metis central main_aux rewrites_bnullable_eq)
+
+
+theorem blexersimp_correctness: 
+  shows "lexer r s = blexer_simp r s"
+  using blexer_correctness main_blexer_simp by simp
+
+
+(* some tests *)
+
+lemma asize_fuse:
+  shows "asize (fuse bs r) = asize r"
+  apply(induct r arbitrary: bs)
+  apply(auto)
+  done
+
+lemma asize_rewrite2:
+  shows "r1 \<leadsto> r2 \<Longrightarrow> asize r1 \<ge> asize r2"
+  and   "rs1 s\<leadsto> rs2 \<Longrightarrow> (sum_list (map asize rs1)) \<ge> (sum_list (map asize rs2))"
+   apply(induct rule: rrewrite_srewrite.inducts)
+   apply(auto simp add: asize_fuse comp_def)
+  done
+
+lemma asize_rrewrites:
+  assumes "r1 \<leadsto>* r2"
+  shows "asize r1 \<ge> asize r2"
+  using assms
+  apply(induct rule: rrewrites.induct)
+   apply(auto)
+  using asize_rewrite2(1) le_trans by blast
+  
+
+
+fun asize2 :: "arexp \<Rightarrow> nat" where
+  "asize2 AZERO = 1"
+| "asize2 (AONE cs) = 1" 
+| "asize2 (ACHAR cs c) = 1"
+| "asize2 (AALTs cs rs) = Suc (Suc (sum_list (map asize2 rs)))"
+| "asize2 (ASEQ cs r1 r2) = Suc (asize2 r1 + asize2 r2)"
+| "asize2 (ASTAR cs r) = Suc (asize2 r)"
+
+
+lemma asize2_fuse:
+  shows "asize2 (fuse bs r) = asize2 r"
+  apply(induct r arbitrary: bs)
+  apply(auto)
+  done
+
+lemma asize2_not_zero:
+  shows "0 < asize2 r"
+  apply(induct r)
+  apply(auto)
+  done
+
+lemma asize_rewrite:
+  shows "r1 \<leadsto> r2 \<Longrightarrow> asize2 r1 > asize2 r2"
+  and   "rs1 s\<leadsto> rs2 \<Longrightarrow> (sum_list (map asize2 rs1)) > (sum_list (map asize2 rs2))"
+   apply(induct rule: rrewrite_srewrite.inducts)
+   apply(auto simp add: asize2_fuse comp_def)
+  apply(simp add: asize2_not_zero) 
+  done
+
+lemma asize2_bsimp_ASEQ:
+  shows "asize2 (bsimp_ASEQ bs r1 r2) \<le> Suc (asize2 r1 + asize2 r2)"
+  apply(induct bs r1 r2 rule: bsimp_ASEQ.induct)
+  apply(auto)
+  done
+
+lemma asize2_bsimp_AALTs:
+  shows "asize2 (bsimp_AALTs bs rs) \<le> Suc (Suc (sum_list (map asize2 rs)))"
+  apply(induct bs rs rule: bsimp_AALTs.induct)
+  apply(auto simp add: asize2_fuse)
+  done
+
+lemma distinctBy_asize2:
+  shows "sum_list (map asize2 (distinctBy rs f acc)) \<le> sum_list (map asize2 rs)"
+  apply(induct rs f acc rule: distinctBy.induct)
+  apply(auto)
+  done
+
+lemma flts_asize2:
+  shows "sum_list (map asize2 (flts rs)) \<le> sum_list (map asize2 rs)"
+  apply(induct rs rule: flts.induct)
+  apply(auto simp add: comp_def asize2_fuse)
+  done
+
+lemma sumlist_asize2:
+  assumes "\<And>x. x \<in> set rs \<Longrightarrow> asize2 (f x) \<le> asize2 x"
+  shows "sum_list (map asize2 (map f rs)) \<le> sum_list (map asize2 rs)"
+  using assms
+  apply(induct rs)
+   apply(auto simp add: comp_def)
+  by (simp add: add_le_mono)
+
+lemma test0:
+  assumes "r1 \<leadsto>* r2"
+  shows "r1 = r2 \<or> (\<exists>r3. r1 \<leadsto> r3 \<and> r3 \<leadsto>* r2)"
+  using assms
+  apply(induct r1 r2 rule: rrewrites.induct)
+   apply(auto)
+  done
+
+lemma test2:
+  assumes "r1 \<leadsto>* r2"
+  shows "asize2 r1 \<ge> asize2 r2"
+using assms
+  apply(induct r1 r2 rule: rrewrites.induct)
+  apply(auto)
+  using asize_rewrite(1) by fastforce
+  
+
+lemma test3:
+  shows "r = bsimp r \<or> (asize2 (bsimp r) < asize2 r)"
+proof -
+  have "r \<leadsto>* bsimp r"
+    by (simp add: rewrites_to_bsimp)
+  then have "r = bsimp r \<or> (\<exists>r3. r \<leadsto> r3 \<and> r3 \<leadsto>* bsimp r)"
+    using test0 by blast
+  then show ?thesis
+    by (meson asize_rewrite(1) dual_order.strict_trans2 test2)
+qed
+
+lemma test3Q:
+  shows "r = bsimp r \<or> (asize (bsimp r) \<le> asize r)"
+proof -
+  have "r \<leadsto>* bsimp r"
+    by (simp add: rewrites_to_bsimp)
+  then have "r = bsimp r \<or> (\<exists>r3. r \<leadsto> r3 \<and> r3 \<leadsto>* bsimp r)"
+    using test0 by blast
+  then show ?thesis
+    using asize_rewrite2(1) asize_rrewrites le_trans by blast
+qed
+
+lemma test4:
+  shows "asize2 (bsimp (bsimp r)) \<le> asize2 (bsimp r)"
+  apply(induct r rule: bsimp.induct)
+       apply(auto)
+  using rewrites_to_bsimp test2 apply fastforce
+  using rewrites_to_bsimp test2 by presburger
+
+lemma test4Q:
+  shows "asize (bsimp (bsimp r)) \<le> asize (bsimp r)"
+  apply(induct r rule: bsimp.induct)
+       apply(auto)
+  apply (metis order_refl test3Q)
+  by (metis le_refl test3Q)
+
+
+
+lemma testb0:
+  shows "fuse bs1 (bsimp_ASEQ bs r1 r2) =  bsimp_ASEQ (bs1 @ bs) r1 r2"
+  apply(induct bs r1 r2 arbitrary: bs1 rule: bsimp_ASEQ.induct)
+  apply(auto)
+  done
+
+lemma testb1:
+  shows "fuse bs1 (bsimp_AALTs bs rs) =  bsimp_AALTs (bs1 @ bs) rs"
+  apply(induct bs rs arbitrary: bs1 rule: bsimp_AALTs.induct)
+  apply(auto simp add: fuse_append)
+  done
+
+lemma testb2:
+  shows "bsimp (bsimp_ASEQ bs r1 r2) =  bsimp_ASEQ bs (bsimp r1) (bsimp r2)"
+  apply(induct bs r1 r2 rule: bsimp_ASEQ.induct)
+  apply(auto simp add: testb0 testb1)
+  done  
+
+lemma testb3:
+  shows "\<nexists>r'. (bsimp r \<leadsto> r') \<and> asize2 (bsimp r) > asize2 r'"
+apply(induct r rule: bsimp.induct)
+       apply(auto)
+       defer
+       defer
+  using rrewrite.cases apply blast
+  using rrewrite.cases apply blast
+  using rrewrite.cases apply blast
+  using rrewrite.cases apply blast
+  oops
+
+lemma testb4:
+  assumes "sum_list (map asize rs1) \<le> sum_list (map asize rs2)"
+  shows "asize (bsimp_AALTs bs1 rs1) \<le> Suc (asize (bsimp_AALTs bs1 rs2))"
+  using assms 
+apply(induct bs1 rs2 arbitrary: rs1 rule: bsimp_AALTs.induct)
+    apply(auto)
+    apply(case_tac rs1)
+     apply(auto)
+  using asize2.elims apply auto[1]
+  apply (metis One_nat_def Zero_not_Suc asize.elims)
+      apply(case_tac rs1)
+    apply(auto)
+   apply(case_tac list)
+    apply(auto)
+  using asize_fuse apply force
+  apply (simp add: asize_fuse)
+  by (smt (verit, ccfv_threshold) One_nat_def add.right_neutral asize.simps(1) asize.simps(4) asize_fuse bsimp_AALTs.elims le_Suc_eq list.map(1) list.map(2) not_less_eq_eq sum_list_simps(1) sum_list_simps(2))
+
+lemma flts_asize:
+  shows "sum_list (map asize (flts rs)) \<le> sum_list (map asize rs)"
+  apply(induct rs rule: flts.induct)
+  apply(auto simp add: comp_def asize_fuse)
+  done
+
+
+lemma test5:
+  shows "asize2 r \<ge> asize2 (bsimp r)"
+  apply(induct r rule: bsimp.induct)
+       apply(auto)
+  apply (meson Suc_le_mono add_le_mono asize2_bsimp_ASEQ order_trans)
+  apply(rule order_trans)
+   apply(rule asize2_bsimp_AALTs)
+  apply(simp)
+  apply(rule order_trans)
+   apply(rule distinctBy_asize2)
+  apply(rule order_trans)
+   apply(rule flts_asize2)
+  using sumlist_asize2 by force
+  
+
+fun awidth :: "arexp \<Rightarrow> nat" where
+  "awidth AZERO = 1"
+| "awidth (AONE cs) = 1" 
+| "awidth (ACHAR cs c) = 1"
+| "awidth (AALTs cs rs) = (sum_list (map awidth rs))"
+| "awidth (ASEQ cs r1 r2) = (awidth r1 + awidth r2)"
+| "awidth (ASTAR cs r) = (awidth r)"
+
+
+
+lemma 
+  shows "s \<notin> L r \<Longrightarrow> blexer_simp r s = None"
+  by (simp add: blexersimp_correctness lexer_correct_None)
+
+lemma g1:
+  "bders_simp AZERO s = AZERO"
+ apply(induct s)
+ apply(simp)
+ apply(simp)
+  done
+
+lemma g2:
+  "s \<noteq> Nil \<Longrightarrow> bders_simp (AONE bs) s = AZERO"
+ apply(induct s)
+ apply(simp)
+  apply(simp)
+  apply(case_tac s)
+  apply(simp)
+       apply(simp)
+  done
+
+lemma finite_pder:
+  shows "finite (pder c r)"
+  apply(induct c r rule: pder.induct)
+  apply(auto)
+  done
+
+
+
+lemma awidth_fuse:
+  shows "awidth (fuse bs r) = awidth r"
+  apply(induct r arbitrary: bs)
+  apply(auto)
+  done
+
+lemma pders_SEQs:
+  assumes "finite A"
+  shows "card (SEQs A (STAR r)) \<le> card A"
+  using assms
+  by (simp add: SEQs_eq_image card_image_le)
+
+lemma binullable_intern:
+  shows "bnullable (intern r) = nullable r"
+  apply(induct r)
+  apply(auto simp add: bnullable_fuse)
+  done
+
+lemma
+  "card (pder c r) \<le> awidth (bder c (intern r))"
+  apply(induct c r rule: pder.induct)
+  apply(simp)
+      apply(simp)
+     apply(simp)
+    apply(simp)
+  apply(rule order_trans)
+     apply(rule card_Un_le)
+  apply (simp add: awidth_fuse bder_fuse)
+   defer
+   apply(simp)
+  apply(rule order_trans)
+    apply(rule pders_SEQs)
+  using finite_pder apply presburger
+  apply (simp add: awidth_fuse)
+  apply(auto)
+     apply(rule order_trans)
+      apply(rule card_Un_le)
+     apply(simp add: awidth_fuse)
+     defer
+  using binullable_intern apply blast
+  using binullable_intern apply blast
+  apply (smt (verit, best) SEQs_eq_image add.commute add_Suc_right card_image_le dual_order.trans finite_pder trans_le_add2)
+  apply(subgoal_tac "card (SEQs (pder c r1) r2) \<le> card (pder c r1)")
+  apply(linarith)
+    by (simp add: UNION_singleton_eq_range card_image_le finite_pder)
+  
+lemma
+  "card (pder c r) \<le> asize (bder c (intern r))"
+  apply(induct c r rule: pder.induct)
+       apply(simp)
+  apply(simp)
+     apply(simp)
+    apply(simp)
+  apply (metis add_mono_thms_linordered_semiring(1) asize_fuse bder_fuse card_Un_le le_Suc_eq order_trans)
+  defer 
+   apply(simp)
+   apply(rule order_trans)
+    apply(rule pders_SEQs)
+  using finite_pder apply presburger
+  apply (simp add: asize_fuse)
+  apply(simp)
+  apply(auto)
+  apply(rule order_trans)
+      apply(rule card_Un_le)
+  apply (smt (z3) SEQs_eq_image add.commute add_Suc_right add_mono_thms_linordered_semiring(1) asize_fuse card_image_le dual_order.trans finite_pder le_add1)
+    apply(rule order_trans)
+     apply(rule card_Un_le)
+  using binullable_intern apply blast
+  using binullable_intern apply blast
+  by (smt (verit, best) SEQs_eq_image add.commute add_Suc_right card_image_le dual_order.trans finite_pder trans_le_add2)
+  
+lemma
+  "card (pder c r) \<le> asize (bsimp (bder c (intern r)))"
+  apply(induct c r rule: pder.induct)
+       apply(simp)
+      apply(simp)
+     apply(simp)
+    apply(simp)
+  apply(rule order_trans)
+     apply(rule card_Un_le)
+    prefer 3
+    apply(simp)
+  apply(rule order_trans)
+     apply(rule pders_SEQs)
+  using finite_pder apply blast
+  oops
+  
+
+(* below is the idempotency of bsimp *)
+
+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_all add: bsimp_ASEQ_fuse bsimp_AALTs_fuse fuse_append)
+  done
+
+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(case_tac "\<exists>bs. bsimp r1 = AONE bs")
+  apply(auto)[1]
+  apply (metis bsimp_fuse)
+  apply(simp add: bsimp_ASEQ1)
+  done  
+
+lemma bsimp_AALTs_idem:
+  assumes "\<forall>r \<in> set rs. bsimp (bsimp r) = bsimp r" 
+  shows "bsimp (bsimp_AALTs bs rs) = bsimp_AALTs bs (map bsimp rs)" 
+  using assms
+  apply(induct bs rs rule: bsimp_AALTs.induct)
+  apply(simp)
+   apply(simp)
+  using bsimp_fuse apply presburger
+  oops   
+  
+lemma bsimp_idem_rev:
+  shows "\<nexists>r2. bsimp r1 \<leadsto> r2"
+  apply(induct r1 rule: bsimp.induct)
+  apply(auto)
+  defer
+  defer
+  using rrewrite.simps apply blast
+  using rrewrite.cases apply blast
+  using rrewrite.simps apply blast
+  using rrewrite.cases apply blast
+  apply(case_tac "bsimp r1 = AZERO")
+  apply(simp)
+  apply(case_tac "bsimp r2 = AZERO")
+  apply(simp)
+  apply(case_tac "\<exists>bs. bsimp r1 = AONE bs")
+  apply(auto)[1]
+  prefer 2
+  apply (smt (verit, best) arexp.distinct(25) arexp.inject(3) bsimp_ASEQ1 rrewrite.simps)
+  defer
+  oops
+
+lemma bsimp_idem:
+  shows "bsimp (bsimp r) = bsimp r"
+  apply(induct r rule: bsimp.induct)
+  apply(auto)
+  using bsimp_ASEQ_idem apply presburger
+  oops
+
+lemma neg:
+  shows " \<not>(\<exists>r2. r1 \<leadsto> r2 \<and>  (r2 \<leadsto>* bsimp r1) )"
+  apply(rule notI)
+  apply(erule exE)
+  apply(erule conjE)
+  oops
+
+
+
+
+lemma reduction_always_in_bsimp:
+  shows " \<lbrakk> r1 \<leadsto> r2 ; \<not>(r2 \<leadsto>* bsimp r1)\<rbrakk> \<Longrightarrow> False"
+  apply(erule rrewrite.cases)
+         apply simp
+        apply auto
+
+  oops
+
+(*
+AALTs [] [AZERO, AALTs(bs1, [a, b]) ] 
+rewrite seq 1: \<leadsto> AALTs [] [ AALTs(bs1, [a, b]) ] \<leadsto>
+fuse [] (AALTs bs1, [a, b])
+rewrite seq 2: \<leadsto> AALTs [] [AZERO, (fuse bs1 a), (fuse bs1 b)]) ]
+
+*)
+
+lemma normal_bsimp: 
+  shows "\<nexists>r'. bsimp r \<leadsto> r'"
+  oops
+
+  (*r' size bsimp r > size r' 
+    r' \<leadsto>* bsimp bsimp r
+size bsimp r > size r' \<ge> size bsimp bsimp r*)
+
+export_code blexer_simp blexer lexer bders bders_simp in Scala module_name VerifiedLexers
+
+
+unused_thms
+
+
+inductive aggressive:: "arexp \<Rightarrow> arexp \<Rightarrow> bool" ("_ \<leadsto>? _" [99, 99] 99)
+  where
+ "ASEQ bs (AALTs bs1 rs) r \<leadsto>? AALTs (bs@bs1) (map (\<lambda>r'. ASEQ [] r' r) rs) "
+
+
+
+end
diff -r 01d1285b08ed -r f71df68776bb thys2/blexer1.sc
--- a/thys2/blexer1.sc	Wed Feb 02 22:30:41 2022 +0000
+++ b/thys2/blexer1.sc	Fri Feb 04 00:05:12 2022 +0000
@@ -361,7 +361,7 @@
       case AALTS(bs1, rs) => {
             val rs_simp = rs.map(bsimp(_))
             val flat_res = flats(rs_simp)
-            val dist_res = strongDistinctBy(flat_res)//distinctBy(flat_res, erase)
+            val dist_res = distinctBy(flat_res, erase)//strongDB(flat_res)//distinctBy(flat_res, erase)
             dist_res match {
               case Nil => AZERO
               case s :: Nil => fuse(bs1, s)
@@ -372,6 +372,30 @@
       case r => r
     }
   }
+  def strongBsimp(r: ARexp): ARexp =
+  {
+    r match {
+      case ASEQ(bs1, r1, r2) => (strongBsimp(r1), strongBsimp(r2)) match {
+          case (AZERO, _) => AZERO
+          case (_, AZERO) => AZERO
+          case (AONE(bs2), r2s) => fuse(bs1 ++ bs2, r2s)
+          case (r1s, r2s) => ASEQ(bs1, r1s, r2s)
+      }
+      case AALTS(bs1, rs) => {
+            val rs_simp = rs.map(strongBsimp(_))
+            val flat_res = flats(rs_simp)
+            val dist_res = strongDB(flat_res)//distinctBy(flat_res, erase)
+            dist_res match {
+              case Nil => AZERO
+              case s :: Nil => fuse(bs1, s)
+              case rs => AALTS(bs1, rs)  
+            }
+          
+      }
+      case r => r
+    }
+  }
+
   def bders (s: List[Char], r: ARexp) : ARexp = s match {
     case Nil => r
     case c::s => bders(s, bder(c, r))
@@ -487,13 +511,13 @@
   } 
 
 
-  def strongDistinctBy(xs: List[ARexp], 
+  def strongDB(xs: List[ARexp], 
                        acc1: List[Rexp] = Nil, 
                        acc2 : List[(List[Rexp], Rexp)] = Nil): List[ARexp] = xs match {
     case Nil => Nil
     case (x::xs) => 
         if(acc1.contains(erase(x)))
-            strongDistinctBy(xs, acc1, acc2)
+            strongDB(xs, acc1, acc2)
         else{
             x match {
                 case ASTAR(bs0, r0) => 
@@ -502,7 +526,7 @@
                         r2stl => {val (r2s, tl) = r2stl; tl == erase(r0) } 
                     )
                     if(i == -1){ 
-                        x::strongDistinctBy(
+                        x::strongDB(
                             xs, erase(x)::acc1, (ONE::Nil, erase(r0))::acc2
                         )
                     }
@@ -513,10 +537,10 @@
                         newHeads match{
                             case newHead::Nil =>
                                 ASTAR(bs0, r0) :: 
-                                strongDistinctBy(xs, erase(x)::acc1, 
+                                strongDB(xs, erase(x)::acc1, 
                                 acc2.updated(i, (oldHeadsUpdated, headListAlready._2)) )//TODO: acc2 already contains headListAlready
                             case Nil =>
-                                strongDistinctBy(xs, erase(x)::acc1, 
+                                strongDB(xs, erase(x)::acc1, 
                                 acc2)
                         }
                     }                
@@ -526,7 +550,7 @@
                         r2stl => {val (r2s, tl) = r2stl; tl == erase(r0) } 
                     )
                     if(i == -1){ 
-                        x::strongDistinctBy(
+                        x::strongDB(
                             xs, erase(x)::acc1, (headList.map(erase(_)), erase(r0))::acc2
                         )
                     }
@@ -537,18 +561,18 @@
                         newHeads match{
                             case newHead::Nil =>
                                 ASEQ(bs, newHead, ASTAR(bs0, r0)) :: 
-                                strongDistinctBy(xs, erase(x)::acc1, 
+                                strongDB(xs, erase(x)::acc1, 
                                 acc2.updated(i, (oldHeadsUpdated, headListAlready._2)) )//TODO: acc2 already contains headListAlready
                             case Nil =>
-                                strongDistinctBy(xs, erase(x)::acc1, 
+                                strongDB(xs, erase(x)::acc1, 
                                 acc2)
                             case hds => val AALTS(bsp, rsp) = r1
                                 ASEQ(bs, AALTS(bsp, hds), ASTAR(bs0, r0)) ::
-                                strongDistinctBy(xs, erase(x)::acc1,
+                                strongDB(xs, erase(x)::acc1,
                                 acc2.updated(i, (oldHeadsUpdated, headListAlready._2)))
                         }
                     }
-                case rPrime => x::strongDistinctBy(xs, erase(x)::acc1, acc2)    
+                case rPrime => x::strongDB(xs, erase(x)::acc1, acc2)    
             }
                 
         }
@@ -586,12 +610,12 @@
 
 
 // @arg(doc = "small tests")
-val STARREG = ("a" | "aa").%
+val STARREG = ((STAR("a") | STAR("aa") ).%).%
 
 @main
 def small() = {
 
-  val prog0 = """aaa"""
+  val prog0 = """aaaaaaaaa"""
   println(s"test: $prog0")
 //   println(lexing_simp(NOTREG, prog0))
 //   val v = lex_simp(NOTREG, prog0.toList)