
diff -r 47c75ba5ad28 -r 994403dbbed5 thys2/SizeBoundStrong.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/thys2/SizeBoundStrong.thy	Wed Feb 16 17:20:40 2022 +0000
@@ -0,0 +1,1644 @@
+
+theory SizeBoundStrong
+  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 (CH 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) = 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 nonalt :: "arexp \<Rightarrow> bool"
+  where
+  "nonalt (AALTs bs2 rs) = False"
+| "nonalt 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
+
+lemma fuse_Nil:
+  shows "fuse [] r = r"
+  by (induct r)(simp_all)
+
+lemma map_fuse_Nil:
+  shows "map (fuse []) rs = rs"
+  by (induct rs)(simp_all add: fuse_Nil)
+
+
+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"
+
+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 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]
+   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 bnullable_Hdbmkeps_Hd:
+  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 bnullable_Hdbmkeps_Hd)
+     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)))"
+
+(*filter  (\<lambda>rt. case rt of 
+                                      SEQ r1p r2p \<Rightarrow> r2p = (erase r2)
+                                      r           \<Rightarrow> False ) allowableTerms*)
+
+
+lemma dB_single_step: 
+  shows "distinctBy (a#rs) f {} = a # distinctBy rs f {f a}"
+  by simp 
+
+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"
+
+
+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_ASEQ1 :: "bit list \<Rightarrow> arexp \<Rightarrow> arexp \<Rightarrow> arexp"
+  where
+  "bsimp_ASEQ1 _ AZERO _ = AZERO"
+| "bsimp_ASEQ1 bs (AONE bs1) r2 = fuse (bs @ bs1) r2"
+| "bsimp_ASEQ1 bs r1 r2 = ASEQ  bs r1 r2"
+
+
+fun collect where
+  \<open>collect _ [] = []\<close>
+|  \<open>collect erasedR2  ((SEQ r1 r2) # rs) = (if r2 = erasedR2 then r1 # (collect erasedR2 rs)
+                                        else collect erasedR2 rs)\<close>
+| \<open>collect erasedR2 (r # rs) = collect erasedR2 rs\<close>
+
+
+fun pruneRexp where
+  \<open>pruneRexp (ASEQ bs r1 r2) allowableTerms = 
+( let termsTruncated = (collect (erase r2) allowableTerms) in (let pruned = pruneRexp r1 termsTruncated in (bsimp_ASEQ1 bs pruned r2))    )\<close>
+| \<open>pruneRexp (AALTs bs rs) allowableTerms = (let rsp = (filter (\<lambda>r. r \<noteq> AZERO)  (map (\<lambda>r. pruneRexp r allowableTerms) rs) ) in bsimp_AALTs bs rsp )
+\<close>
+| \<open>pruneRexp r allowableTerms = (if (erase r) \<in> (set allowableTerms) then r else AZERO)\<close>
+
+
+fun oneSimp :: \<open>rexp \<Rightarrow> rexp\<close> where
+  \<open> oneSimp (SEQ ONE r) = r \<close>
+| \<open> oneSimp (SEQ r1 r2) = SEQ (oneSimp r1) r2 \<close>
+| \<open> oneSimp r = r \<close>
+
+fun breakIntoTerms where
+  \<open>breakIntoTerms (SEQ r1 r2) = map (\<lambda>r1p. SEQ r1p r2) (breakIntoTerms r1)\<close>
+| \<open>breakIntoTerms (ALT r1 r2) = (breakIntoTerms r1) @ (breakIntoTerms r2)\<close>
+| \<open>breakIntoTerms r = r # [] \<close>
+
+fun addToAcc :: "arexp \<Rightarrow> rexp list \<Rightarrow> rexp list"
+  where
+\<open>addToAcc r acc = filter (\<lambda>r1. oneSimp r1 \<notin> set acc) (breakIntoTerms (erase r)) \<close>
+
+fun dBStrong :: "arexp list \<Rightarrow> rexp list \<Rightarrow> arexp list"
+  where
+"dBStrong [] acc = []"
+| "dBStrong (r # rs) acc = (if (erase r) \<in> (set acc) then dBStrong rs acc
+                            else (case (pruneRexp r (addToAcc r acc)) of  
+                                    AZERO \<Rightarrow> dBStrong rs ((addToAcc r acc) @ acc) |
+                                    r1    \<Rightarrow> r1 # (dBStrong rs ((addToAcc r acc) @ acc))
+                                 )
+                           )
+                    "
+fun bsimpStrong :: "arexp \<Rightarrow> arexp "
+  where
+  "bsimpStrong (ASEQ bs1 r1 r2) = bsimp_ASEQ bs1 (bsimpStrong r1) (bsimpStrong r2)"
+| "bsimpStrong (AALTs bs1 rs) = bsimp_AALTs bs1 (dBStrong (flts (map bsimpStrong rs)) []) "
+| "bsimpStrong r = r"
+
+
+fun bdersStrong :: "arexp \<Rightarrow> string \<Rightarrow> arexp"
+where
+  "bdersStrong r [] = r"
+| "bdersStrong r (c # s) = bdersStrong (bsimpStrong (bder c r)) s"
+
+
+definition blexerStrong where
+ "blexerStrong r s \<equiv> if bnullable (bdersStrong (intern r) s) then 
+                    decode (bmkeps (bdersStrong (intern r) s)) r else None"
+
+
+
+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"
+
+export_code bders_simp in Scala module_name Example
+
+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 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_erase_dB_acc:
+  shows "( \<Union>(L ` acc) \<union> ( \<Union> (L ` erase ` (set (distinctBy rs erase acc) ) ) )) = \<Union>(L ` acc) \<union>  \<Union> (L ` erase ` (set rs))"
+  apply(induction rs arbitrary: acc)
+   apply simp
+  apply simp
+  by (smt (z3) SUP_absorb UN_insert sup_assoc sup_commute)
+
+lemma L_erase_dB:
+  shows " ( \<Union> (L ` erase ` (set (distinctBy rs erase {}) ) ) ) = \<Union> (L ` erase ` (set rs))"
+  by (metis L_erase_dB_acc Un_commute Union_image_empty)
+
+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_dB)
+  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 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 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 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 bnullable_Hdbmkeps_Hd 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:
+  assumes "bnullable (AALTs bs (rs @ rs1))"
+          "bnullable (AALTs bs (rs @ rs2))"
+          "\<lbrakk>bnullable (AALTs bs rs1); bnullable (AALTs bs rs2); \<forall>r\<in>set rs. \<not>bnullable r\<rbrakk> \<Longrightarrow> 
+               bmkeps (AALTs bs rs1) = bmkeps (AALTs bs rs2)"
+  shows "bmkeps (AALTs bs (rs @ rs1)) = bmkeps (AALTs bs (rs @ rs2))"
+  using assms
+  apply(case_tac "\<exists>r \<in> set rs. bnullable r")
+  using qq1 apply auto[1]
+  by (metis UnE bnullable.simps(4) qq2 set_append)
+  
+
+lemma flts_append:
+  shows "flts (xs1 @ xs2) = flts xs1 @ flts xs2"
+by (induct xs1  arbitrary: xs2  rule: flts.induct)(auto)
+
+lemma  k0a:
+  shows "flts [AALTs bs rs] = map (fuse bs)  rs"
+  apply(simp)
+  done
+
+
+lemma bbbbs1:
+  shows "nonalt r \<or> (\<exists>bs rs. r  = AALTs bs rs)"
+  using nonalt.elims(3) by auto
+  
+
+
+fun nonazero :: "arexp \<Rightarrow> bool"
+  where
+  "nonazero AZERO = False"
+| "nonazero r = True"
+
+
+lemma flts_single1:
+  assumes "nonalt r" "nonazero r"
+  shows "flts [r] = [r]"
+  using assms
+  apply(induct r)
+  apply(auto)
+  done
+
+
+
+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 bnullable_Hdbmkeps_Hd)
+  apply(case_tac "bnullable a")
+   apply (metis append.assoc b2 bnullable_correctness erase_fuse bnullable_Hdbmkeps_Hd)
+  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: bnullable_Hdbmkeps_Hd)
+     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: bnullable_Hdbmkeps_Hd)
+   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: bnullable_Hdbmkeps_Hd)
+  apply(case_tac "bnullable (ASEQ x41 x42 x43)")
+   apply(case_tac list)
+    apply(simp)
+   apply(simp)
+   apply (simp add: bnullable_Hdbmkeps_Hd)
+  apply(simp)
+  using qq4 r1 r2 by auto
+
+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)
+where
+  "ASEQ bs AZERO r2 \<leadsto> AZERO"
+| "ASEQ bs r1 AZERO \<leadsto> AZERO"
+| "ASEQ bs1 (AONE bs2) r \<leadsto> fuse (bs1@bs2) r"
+| "r1 \<leadsto> r2 \<Longrightarrow> ASEQ bs r1 r3 \<leadsto> ASEQ bs r2 r3"
+| "r3 \<leadsto> r4 \<Longrightarrow> ASEQ bs r1 r3 \<leadsto> ASEQ bs r1 r4"
+| "r \<leadsto> r' \<Longrightarrow> (AALTs bs (rs1 @ [r] @ rs2)) \<leadsto> (AALTs bs (rs1 @ [r'] @ rs2))"
+(*context rule for eliminating 0, alts--corresponds to the recursive call flts r::rs = r::(flts rs)*)
+| "AALTs bs (rsa@ [AZERO] @ rsb) \<leadsto> AALTs bs (rsa @ rsb)"
+| "AALTs bs (rsa@ [AALTs bs1 rs1] @ rsb) \<leadsto> AALTs bs (rsa@(map (fuse bs1) rs1)@rsb)"
+| "AALTs bs [] \<leadsto> AZERO"
+| "AALTs bs [r] \<leadsto> fuse bs r"
+| "erase a1 = erase a2 \<Longrightarrow> AALTs bs (rsa@[a1]@rsb@[a2]@rsc) \<leadsto> AALTs bs (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
+  ss1: "[] s\<leadsto>* []"
+| ss2: "\<lbrakk>r \<leadsto>* r'; rs s\<leadsto>* rs'\<rbrakk> \<Longrightarrow> (r#rs) s\<leadsto>* (r'#rs')"
+
+
+(* rewrites for lists *)
+inductive 
+  frewrites:: "arexp list \<Rightarrow> arexp list \<Rightarrow> bool" (" _ f\<leadsto>* _" [100, 100] 100)
+where
+  fs1: "[] f\<leadsto>* []"
+| fs2: "\<lbrakk>rs f\<leadsto>* rs'\<rbrakk> \<Longrightarrow> (AZERO#rs) f\<leadsto>* rs'"
+| fs3: "\<lbrakk>rs f\<leadsto>* rs'\<rbrakk> \<Longrightarrow> ((AALTs bs rs1) # rs) f\<leadsto>* ((map (fuse bs) rs1) @ rs')"
+| fs4: "\<lbrakk>rs f\<leadsto>* rs'; nonalt r; nonazero r\<rbrakk> \<Longrightarrow> (r#rs) f\<leadsto>* (r#rs')"
+
+
+lemma r_in_rstar : "r1 \<leadsto> r2 \<Longrightarrow> r1 \<leadsto>* r2"
+  using rrewrites.intros(1) rrewrites.intros(2) by blast
+ 
+lemma real_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  many_steps_later: "\<lbrakk>r1 \<leadsto> r2; r2 \<leadsto>* r3 \<rbrakk> \<Longrightarrow> r1 \<leadsto>* r3"
+  by (meson r_in_rstar real_trans)
+
+
+lemma contextrewrites1: "r \<leadsto>* r' \<Longrightarrow> (AALTs bs (r#rs)) \<leadsto>* (AALTs bs (r'#rs))"
+  apply(induct r r' rule: rrewrites.induct)
+   apply simp
+  by (metis append_Cons append_Nil rrewrite.intros(6) rs2)
+
+
+lemma contextrewrites2: "r \<leadsto>* r' \<Longrightarrow> (AALTs bs (rs1@[r]@rs)) \<leadsto>* (AALTs bs (rs1@[r']@rs))"
+  apply(induct r r' rule: rrewrites.induct)
+   apply simp
+  using rrewrite.intros(6) by blast
+
+
+
+lemma srewrites_alt: "rs1 s\<leadsto>* rs2 \<Longrightarrow> (AALTs bs (rs@rs1)) \<leadsto>* (AALTs bs (rs@rs2))"
+
+  apply(induct rs1 rs2 arbitrary: bs rs rule: srewrites.induct)
+   apply(rule rs1)
+  apply(drule_tac x = "bs" in meta_spec)
+  apply(drule_tac x = "rsa@[r']" in meta_spec)
+  apply simp
+  apply(rule real_trans)
+   prefer 2
+   apply(assumption)
+  apply(drule contextrewrites2)
+  apply auto
+  done
+
+corollary srewrites_alt1: 
+  assumes "rs1 s\<leadsto>* rs2"
+  shows "AALTs bs rs1 \<leadsto>* AALTs bs rs2"
+using assms
+  by (metis append.left_neutral srewrites_alt)
+
+
+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.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.intros)
+done
+
+lemma continuous_rewrite: 
+  assumes "r1 \<leadsto>* AZERO"
+  shows "ASEQ bs1 r1 r2 \<leadsto>* AZERO"
+using assms
+  apply(induction ra\<equiv>"r1" rb\<equiv>"AZERO" arbitrary: bs1 r1 r2 rule: rrewrites.induct)
+  apply(auto intro: rrewrite.intros r_in_rstar star_seq)
+  by (meson rrewrite.intros(1) rs2 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 trivialbsimp_srewrites: 
+  "\<lbrakk>\<And>x. x \<in> set rs \<Longrightarrow> x \<leadsto>* f x \<rbrakk> \<Longrightarrow> rs s\<leadsto>* (map f rs)"
+
+  apply(induction rs)
+   apply simp
+   apply(rule ss1)
+  by (metis insert_iff list.simps(15) list.simps(9) srewrites.simps)
+
+
+lemma bsimp_AALTs_rewrites: 
+  "AALTs bs1 rs \<leadsto>* bsimp_AALTs bs1 rs"
+  apply(induction rs)
+  apply simp
+   apply(rule r_in_rstar)
+  using rrewrite.intros(9) apply blast
+  by (metis bsimp_AALTs.elims list.discI rrewrite.intros(10) rrewrites.simps)
+
+
+
+lemma flts_prepend: "\<lbrakk>nonalt a; nonazero a\<rbrakk> \<Longrightarrow> flts (a#rs) = a # (flts rs)"
+  by (metis append_Cons append_Nil flts_single1 flts_append)
+
+lemma fltsfrewrites: "rs f\<leadsto>* (flts rs)"
+  apply(induction rs)
+  apply simp
+   apply(rule fs1)
+
+  apply(case_tac "a = AZERO")
+
+   
+  using fs2 apply auto[1]
+  apply(case_tac "\<exists>bs rs. a = AALTs bs rs")
+   apply(erule exE)+
+   
+   apply (simp add: fs3)
+  apply(subst flts_prepend)
+    apply(rule nonalt.elims(2))
+  prefer 2
+  thm nonalt.elims
+   
+         apply blast
+   
+  using bbbbs1 apply blast
+       apply(simp)+
+   
+   apply (meson nonazero.elims(3))
+   
+  by (meson fs4 nonalt.elims(3) nonazero.elims(3))
+
+
+lemma rrewrite0away: "AALTs bs (AZERO # rsb) \<leadsto> AALTs bs rsb"
+  by (metis append_Cons append_Nil rrewrite.intros(7))
+
+
+lemma frewritesaalts:"rs f\<leadsto>* rs' \<Longrightarrow> (AALTs bs (rs1@rs)) \<leadsto>* (AALTs bs (rs1@rs'))"
+  apply(induct rs rs' arbitrary: bs rs1 rule:frewrites.induct)
+    apply(rule rs1)
+    apply(drule_tac x = "bs" in meta_spec)
+  apply(drule_tac x = "rs1 @ [AZERO]" in meta_spec)
+    apply(rule real_trans)
+     apply simp
+  using rrewrite.intros(7) apply auto[1]
+    apply(drule_tac x = "bsa" in meta_spec)
+  apply(drule_tac x = "rs1a @ [AALTs bs rs1]" in meta_spec)
+   apply(rule real_trans)
+    apply simp
+  using r_in_rstar rrewrite.intros(8) apply auto[1]
+    apply(drule_tac x = "bs" in meta_spec)
+  apply(drule_tac x = "rs1@[r]" in meta_spec)
+    apply(rule real_trans)
+   apply simp
+  apply auto
+  done
+
+lemma flts_rewrites: "  AALTs bs1 rs \<leadsto>* AALTs bs1 (flts rs)"
+  apply(induction rs)
+   apply simp
+  apply(case_tac "a = AZERO")
+  apply (metis flts.simps(2) many_steps_later rrewrite0away)
+
+  apply(case_tac "\<exists>bs2 rs2. a = AALTs bs2 rs2")
+   apply(erule exE)+
+   apply(simp)
+   prefer 2
+
+  apply(subst flts_prepend)
+   
+     apply (meson nonalt.elims(3))
+   
+    apply (meson nonazero.elims(3))
+   apply(subgoal_tac "(a#rs) f\<leadsto>* (a#flts rs)")
+  apply (metis append_Nil frewritesaalts)
+  apply (meson fltsfrewrites fs4 nonalt.elims(3) nonazero.elims(3))
+  by (metis append_Cons append_Nil fltsfrewrites frewritesaalts flts_append k0a)
+
+(* TEST *)
+lemma r: 
+  assumes "AALTs bs rs1 \<leadsto> AALTs bs rs2"
+  shows "AALTs bs (x # rs1) \<leadsto>* AALTs bs (x # rs2)"
+  using assms
+  apply(erule_tac rrewrite.cases)
+            apply(auto)
+  apply (metis append_Cons append_Nil rrewrite.intros(6) r_in_rstar)
+  apply (metis append_Cons append_self_conv2 rrewrite.intros(7) r_in_rstar)
+  apply (metis Cons_eq_appendI append_eq_append_conv2 rrewrite.intros(8) self_append_conv r_in_rstar)
+   apply(case_tac rs2)
+    apply(auto)
+  apply(case_tac r)
+         apply(auto)
+  apply (metis append_Nil2 append_butlast_last_id butlast.simps(2) last.simps list.distinct(1) list.map_disc_iff r_in_rstar rrewrite.intros(8))
+     apply(case_tac r)
+        apply(auto)
+   defer
+   apply(rule r_in_rstar)
+  apply (metis append_Cons append_Nil rrewrite.intros(11))
+  apply(rule real_trans)
+   apply(rule r_in_rstar)
+  using rrewrite.intros(8)[where ?rsb = "[]", of "bs" "[x]" "[]" , simplified]
+   apply(rule_tac rrewrite.intros(8)[where ?rsb = "[]", of "bs" "[x]" "[]" , simplified])
+  apply(simp add: map_fuse_Nil fuse_Nil)
+  done 
+
+lemma alts_simpalts: 
+  "(\<And>x. x \<in> set rs \<Longrightarrow> x \<leadsto>* bsimp x) \<Longrightarrow> 
+  AALTs bs1 rs \<leadsto>* AALTs bs1 (map bsimp rs)"
+  apply(induct rs)
+   apply(auto)[1]
+  using trivialbsimp_srewrites apply auto[1]
+  by (simp add: srewrites_alt1 ss2)
+
+lemma threelistsappend: "rsa@a#rsb = (rsa@[a])@rsb"
+  apply auto
+  done
+
+
+lemma somewhereInside: "r \<in> set rs \<Longrightarrow> \<exists>rs1 rs2. rs = rs1@[r]@rs2"
+  using split_list by fastforce
+
+lemma somewhereMapInside: "f r \<in> f ` set rs \<Longrightarrow> \<exists>rs1 rs2 a. rs = rs1@[a]@rs2 \<and> f a = f r"
+  apply auto
+  by (metis split_list)
+
+lemma alts_dBrewrites_withFront: 
+  "AALTs bs (rsa @ rs) \<leadsto>* AALTs bs (rsa @ distinctBy rs erase (erase ` set rsa))"
+  apply(induction rs arbitrary: rsa)
+   apply simp
+  apply(drule_tac x = "rsa@[a]" in meta_spec)
+  apply(subst threelistsappend)
+  apply(rule real_trans)
+  apply simp
+  apply(case_tac "a \<in> set rsa")
+   apply simp
+   apply(drule somewhereInside)
+   apply(erule exE)+
+   apply simp
+  apply(subgoal_tac " AALTs bs
+            (rs1 @
+             a #
+             rs2 @
+             a #
+             distinctBy rs erase
+              (insert (erase a)
+                (erase `
+                 (set rs1 \<union> set rs2)))) \<leadsto> AALTs bs (rs1@ a # rs2 @  distinctBy rs erase
+              (insert (erase a)
+                (erase `
+                 (set rs1 \<union> set rs2)))) ")
+  prefer 2
+  using rrewrite.intros(11) apply force
+  using r_in_rstar apply force
+  apply(subgoal_tac "erase ` set (rsa @ [a]) = insert (erase a) (erase ` set rsa)")
+  prefer 2
+    
+   apply auto[1]
+  apply(case_tac "erase a \<in> erase `set rsa")
+
+   apply simp
+  apply(subgoal_tac "AALTs bs (rsa @ a # distinctBy rs erase (insert (erase a) (erase ` set rsa))) \<leadsto>
+                     AALTs bs (rsa @ distinctBy rs erase (insert (erase a) (erase ` set rsa)))")
+  apply force
+  apply (smt (verit, ccfv_threshold) append_Cons append_assoc append_self_conv2 r_in_rstar rrewrite.intros(11) same_append_eq somewhereMapInside)
+  by force
+
+ 
+
+lemma alts_dBrewrites: "AALTs bs rs \<leadsto>* AALTs bs (distinctBy rs erase {})"
+  apply(induction rs)
+   apply simp
+  apply simp
+  using alts_dBrewrites_withFront
+  by (metis append_Nil dB_single_step empty_set image_empty)
+
+lemma bsimp_rewrite: 
+  shows "r \<leadsto>* bsimp r"
+proof (induction r rule: bsimp.induct)
+  case (1 bs1 r1 r2)
+  then show "ASEQ bs1 r1 r2 \<leadsto>* bsimp (ASEQ bs1 r1 r2)"
+    apply(simp)
+    apply(case_tac "bsimp r1 = AZERO")
+        apply simp
+  using continuous_rewrite apply blast
+       apply(case_tac "\<exists>bs. bsimp r1 = AONE bs")
+        apply(erule exE)
+        apply simp
+        apply(subst bsimp_ASEQ2)
+        apply (meson real_trans rrewrite.intros(3) rrewrites.intros(2) star_seq star_seq2)
+       apply (smt (verit, best) bsimp_ASEQ0 bsimp_ASEQ1 real_trans rrewrite.intros(2) rs2 star_seq star_seq2)
+  done
+next
+  case (2 bs1 rs)
+  then show "AALTs bs1 rs \<leadsto>* bsimp (AALTs bs1 rs)"
+    by (metis alts_dBrewrites alts_simpalts bsimp.simps(2) bsimp_AALTs_rewrites flts_rewrites real_trans)  
+next
+  case "3_1"
+  then show "AZERO \<leadsto>* bsimp AZERO"
+    by simp
+next
+  case ("3_2" v)
+  then show "AONE v \<leadsto>* bsimp (AONE v)" 
+    by simp
+next
+  case ("3_3" v va)
+  then show "ACHAR v va \<leadsto>* bsimp (ACHAR v va)" 
+    by simp
+next
+  case ("3_4" v va)
+  then show "ASTAR v va \<leadsto>* bsimp (ASTAR v va)" 
+    by simp
+qed
+
+lemma rewrite_non_nullable_strong: 
+  assumes "r1 \<leadsto> r2"
+  shows "bnullable r1 = bnullable r2"
+using assms
+apply(induction r1 r2 rule: rrewrite.induct)
+apply(auto)
+apply(metis bnullable_correctness erase_fuse)+
+apply(metis UnCI bnullable_correctness erase_fuse imageI)
+apply(metis bnullable_correctness erase_fuse)+
+done
+
+lemma rewrite_nullable: 
+  assumes "r1 \<leadsto> r2" "bnullable r1"
+  shows "bnullable r2"
+using assms rewrite_non_nullable_strong
+by auto
+
+lemma rewritesnullable: 
+  assumes "r1 \<leadsto>* r2" "bnullable r1"
+  shows "bnullable r2"
+using assms
+  apply(induction r1 r2 rule: rrewrites.induct)
+  apply simp
+  using rewrite_non_nullable_strong by blast
+
+
+lemma bnullable_segment: 
+  "bnullable (AALTs bs (rs1@[r]@rs2)) \<Longrightarrow> bnullable (AALTs bs rs1) \<or> bnullable (AALTs bs rs2) \<or> bnullable r"
+  by auto
+
+lemma bnullablewhichbmkeps: "\<lbrakk>bnullable  (AALTs bs (rs1@[r]@rs2)); \<not> bnullable (AALTs bs rs1); bnullable r \<rbrakk>
+ \<Longrightarrow> bmkeps (AALTs bs (rs1@[r]@rs2)) = bs @ (bmkeps r)"
+  
+  using qq2 bnullable_Hdbmkeps_Hd by force
+
+lemma spillbmkepslistr: "bnullable (AALTs bs1 rs1)
+    \<Longrightarrow> bmkeps (AALTs bs (AALTs bs1 rs1 # rsb)) = bmkeps (AALTs bs ( map (fuse bs1) rs1 @ rsb))"
+  apply(subst bnullable_Hdbmkeps_Hd)
+  
+   apply simp
+  by (metis bmkeps.simps(3) k0a list.set_intros(1) qq1 qq4 qs3)
+
+lemma third_segment_bnullable: 
+  "\<lbrakk>bnullable (AALTs bs (rs1@rs2@rs3)); \<not>bnullable (AALTs bs rs1); \<not>bnullable (AALTs bs rs2)\<rbrakk> \<Longrightarrow> 
+   bnullable (AALTs bs rs3)"
+  apply(auto)
+  done
+
+lemma third_segment_bmkeps:  
+  "\<lbrakk>bnullable (AALTs bs (rs1@rs2@rs3)); \<not>bnullable (AALTs bs rs1); \<not>bnullable (AALTs bs rs2)\<rbrakk> \<Longrightarrow> 
+   bmkeps (AALTs bs (rs1@rs2@rs3) ) = bmkeps (AALTs bs rs3)"
+  by (metis bnullable.simps(1) bnullable.simps(4) bsimp_AALTs.simps(1) bsimp_AALTs_rewrites qq2 rewritesnullable self_append_conv third_segment_bnullable)
+
+lemma rewrite_bmkepsalt: 
+  "\<lbrakk>bnullable (AALTs bs (rsa @ AALTs bs1 rs1 # rsb)); bnullable (AALTs bs (rsa @ map (fuse bs1) rs1 @ rsb))\<rbrakk>
+       \<Longrightarrow> bmkeps (AALTs bs (rsa @ AALTs bs1 rs1 # rsb)) = bmkeps (AALTs bs (rsa @ map (fuse bs1) rs1 @ rsb))"
+  apply(rule qq3)
+  apply(simp)
+  apply(simp)
+  apply(case_tac "bnullable (AALTs bs1 rs1)")
+  using spillbmkepslistr apply blast
+  apply(subst qq2)
+    apply(auto simp add: bnullable_fuse r1)
+  done
+
+lemma rewrite_bmkeps_aux: 
+  assumes "r1 \<leadsto> r2" "bnullable r1" "bnullable r2"
+  shows "bmkeps r1 = bmkeps r2"
+  using assms 
+proof (induction r1 r2 rule: rrewrite.induct)
+  case (1 bs r2)
+  then show ?case by simp 
+next
+  case (2 bs r1)
+  then show ?case by simp
+next
+  case (3 bs bs1 r)
+  then show ?case by (simp add: b2) 
+next
+  case (4 r1 r2 bs r3)
+  then show ?case by simp
+next
+  case (5 r3 r4 bs r1)
+  then show ?case by simp
+next
+  case (6 r r' bs rs1 rs2)
+  then show ?case
+    by (metis append_Cons append_Nil bnullable.simps(4) bnullable_segment bnullablewhichbmkeps qq3 r1 rewrite_non_nullable_strong)
+next
+  case (7 bs rsa rsb)
+  then show ?case
+    by (metis bnullable.simps(1) bnullable.simps(4) bnullable_segment qq1 qq2 rewrite_nullable rrewrite.intros(9) rrewrite0away third_segment_bmkeps) 
+next
+  case (8 bs rsa bs1 rs1 rsb)
+  then show ?case
+    by (simp add: rewrite_bmkepsalt) 
+next
+  case (9 bs)
+  then show ?case
+    by fastforce 
+next
+  case (10 bs r)
+  then show ?case
+    by (simp add: b2) 
+next
+  case (11 a1 a2 bs rsa rsb rsc)
+  then show ?case
+    by (smt (verit, ccfv_threshold) append_Cons append_eq_appendI append_self_conv2 bnullable_correctness list.set_intros(1) qq3 r1)
+qed
+
+
+lemma rewrite_bmkeps: 
+  assumes "r1 \<leadsto> r2" "bnullable r1"
+  shows "bmkeps r1 = bmkeps r2"
+  using assms(1) assms(2) rewrite_bmkeps_aux rewrite_nullable by blast
+  
+
+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: rewritesnullable) 
+  then have "bmkeps r2 = bmkeps r3" using rewrite_bmkeps a3 a4 by simp
+  then show "bmkeps r1 = bmkeps r3" using IH by simp
+qed
+
+lemma alts_rewrite_front: "r \<leadsto> r' \<Longrightarrow> AALTs bs (r # rs) \<leadsto> AALTs bs (r' # rs)"
+  by (metis append_Cons append_Nil rrewrite.intros(6))
+
+lemma to_zero_in_alt: " AALT bs (ASEQ [] AZERO r) r2 \<leadsto>  AALT bs AZERO r2"
+  by (simp add: alts_rewrite_front rrewrite.intros(1))
+
+lemma rewrite_fuse: 
+  assumes "r2 \<leadsto> r3"
+  shows "fuse bs r2 \<leadsto>* fuse bs r3"
+  using assms
+proof(induction r2 r3 arbitrary: bs rule: rrewrite.induct)
+  case (1 bs r2)
+  then show ?case
+    by (simp add: continuous_rewrite) 
+next
+  case (2 bs r1)
+  then show ?case
+    using rrewrite.intros(2) by force 
+next
+  case (3 bs bs1 r)
+  then show ?case
+    by (metis fuse.simps(5) fuse_append r_in_rstar rrewrite.intros(3)) 
+next
+  case (4 r1 r2 bs r3)
+  then show ?case
+    by (simp add: r_in_rstar star_seq) 
+next
+  case (5 r3 r4 bs r1)
+  then show ?case
+    using fuse.simps(5) r_in_rstar star_seq2 by auto  
+next
+  case (6 r r' bs rs1 rs2)
+  then show ?case
+    using contextrewrites2 r_in_rstar by force 
+next
+  case (7 bs rsa rsb)
+  then show ?case
+    using rrewrite.intros(7) by force  
+next
+  case (8 bs rsa bs1 rs1 rsb)
+  then show ?case
+    using rrewrite.intros(8) by force 
+next
+  case (9 bs)
+  then show ?case
+    by (simp add: r_in_rstar rrewrite.intros(9))
+next
+  case (10 bs r)
+  then show ?case
+    by (metis fuse.simps(4) fuse_append r_in_rstar rrewrite.intros(10)) 
+next
+  case (11 a1 a2 bs rsa rsb rsc)
+  then show ?case
+    using fuse.simps(4) r_in_rstar rrewrite.intros(11) by auto 
+qed
+
+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_fuse real_trans)
+done
+
+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_der_altmiddle: 
+  "bder c (AALTs bs (rsa @ AALTs bs1 rs1 # rsb)) \<leadsto>* bder c (AALTs bs (rsa @ map (fuse bs1) rs1 @ rsb))"
+   apply simp
+   apply(simp add: bder_fuse_list del: append.simps)
+  by (metis append.assoc map_map r_in_rstar rrewrite.intros(8) threelistsappend)
+
+lemma lock_step_der_removal: 
+  shows " erase a1 = erase a2 \<Longrightarrow> 
+                                  bder c (AALTs bs (rsa @ [a1] @ rsb @ [a2] @ rsc)) \<leadsto>* 
+                                  bder c (AALTs bs (rsa @ [a1] @ rsb @ rsc))"
+  apply(simp)
+  
+  using rrewrite.intros(11) by auto
+
+lemma rewrite_after_der: 
+  assumes "r1 \<leadsto> r2"
+  shows "(bder c r1) \<leadsto>* (bder c r2)"
+  using assms
+proof(induction r1 r2 rule: rrewrite.induct)
+  case (1 bs r2)
+  then show "bder c (ASEQ bs AZERO r2) \<leadsto>* bder c AZERO"
+    by (simp add: continuous_rewrite) 
+next
+  case (2 bs r1)
+  then show "bder c (ASEQ bs r1 AZERO) \<leadsto>* bder c AZERO"
+    apply(simp)
+    by (meson contextrewrites1 r_in_rstar real_trans rrewrite.intros(9) rrewrite.intros(2) rrewrite0away)
+next
+  case (3 bs bs1 r)
+  then show "bder c (ASEQ bs (AONE bs1) r) \<leadsto>* bder c (fuse (bs @ bs1) r)" 
+    apply(simp)
+    by (metis bder_fuse fuse_append rrewrite.intros(10) rrewrite0away rrewrites.simps to_zero_in_alt)
+next
+  case (4 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 (simp add: contextrewrites1 rewrite_bmkeps rewrite_non_nullable_strong star_seq) 
+next
+  case (5 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)"
+    using bder.simps(5) r_in_rstar rewrites_fuse srewrites_alt1 ss1 ss2 star_seq2 by presburger
+next
+  case (6 r r' bs rs1 rs2)
+  have as: "r \<leadsto> r'" by fact
+  have IH: "bder c r \<leadsto>* bder c r'" by fact
+  from as IH show "bder c (AALTs bs (rs1 @ [r] @ rs2)) \<leadsto>* bder c (AALTs bs (rs1 @ [r'] @ rs2))" 
+    apply(simp)
+    using contextrewrites2 by force
+next
+  case (7 bs rsa rsb)
+  then show "bder c (AALTs bs (rsa @ [AZERO] @ rsb)) \<leadsto>* bder c (AALTs bs (rsa @ rsb))" 
+    apply(simp)
+    using rrewrite.intros(7) by auto
+next
+  case (8 bs rsa bs1 rs1 rsb)
+  then show 
+    "bder c (AALTs bs (rsa @ [AALTs bs1 rs1] @ rsb)) \<leadsto>* bder c (AALTs bs (rsa @ map (fuse bs1) rs1 @ rsb))"
+    using rewrite_der_altmiddle by auto 
+next
+  case (9 bs)
+  then show "bder c (AALTs bs []) \<leadsto>* bder c AZERO"
+    by (simp add: r_in_rstar rrewrite.intros(9))
+next
+  case (10 bs r)
+  then show "bder c (AALTs bs [r]) \<leadsto>* bder c (fuse bs r)"
+    by (simp add: bder_fuse r_in_rstar rrewrite.intros(10)) 
+next
+  case (11 a1 a2 bs rsa rsb rsc)
+  have as: "erase a1 = erase a2" by fact
+  then show "bder c (AALTs bs (rsa @ [a1] @ rsb @ [a2] @ rsc)) \<leadsto>* bder c (AALTs bs (rsa @ [a1] @ rsb @ rsc))"
+    using lock_step_der_removal by force 
+qed
+
+
+lemma rewrites_after_der: 
+  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_after_der real_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_after_der)
+  also have "... \<leadsto>* bders_simp (bders_simp r xs) [x]" using IH
+    by (simp add: bsimp_rewrite)
+  finally show "bders r (xs @ [x]) \<leadsto>* bders_simp r (xs @ [x])" 
+    by (simp add: bders_simp_append)
+qed
+
+
+  
+
+
+lemma quasi_main: 
+  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_main: 
+  shows "blexer r s = blexer_simp r s"
+  unfolding blexer_def blexer_simp_def
+  using b4 quasi_main by simp
+
+
+theorem blexersimp_correctness: 
+  shows "lexer r s = blexer_simp r s"
+  using blexer_correctness main_main by simp
+
+
+
+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




