
diff -r 232aa2f19a75 -r ec5e4fe4cc70 thys2/SizeBound.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/thys2/SizeBound.thy	Sun Oct 10 18:35:21 2021 +0100
@@ -0,0 +1,1717 @@
+
+theory SizeBound
+  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 good :: "arexp \<Rightarrow> bool" where
+  "good AZERO = False"
+| "good (AONE cs) = True" 
+| "good (ACHAR cs c) = True"
+| "good (AALTs cs []) = False"
+| "good (AALTs cs [r]) = False"
+| "good (AALTs cs (r1#r2#rs)) = (\<forall>r' \<in> set (r1#r2#rs). good r' \<and> nonalt r')"
+| "good (ASEQ _ AZERO _) = False"
+| "good (ASEQ _ (AONE _) _) = False"
+| "good (ASEQ _ _ AZERO) = False"
+| "good (ASEQ cs r1 r2) = (good r1 \<and> good r2)"
+| "good (ASTAR cs r) = True"
+
+
+
+
+fun fuse :: "bit list \<Rightarrow> arexp \<Rightarrow> arexp" where
+  "fuse bs AZERO = AZERO"
+| "fuse bs (AONE cs) = AONE (bs @ cs)" 
+| "fuse bs (ACHAR cs c) = ACHAR (bs @ cs) c"
+| "fuse bs (AALTs cs rs) = AALTs (bs @ cs) rs"
+| "fuse bs (ASEQ cs r1 r2) = ASEQ (bs @ cs) r1 r2"
+| "fuse bs (ASTAR cs r) = ASTAR (bs @ cs) r"
+
+lemma fuse_append:
+  shows "fuse (bs1 @ bs2) r = fuse bs1 (fuse bs2 r)"
+  apply(induct r)
+  apply(auto)
+  done
+
+
+fun intern :: "rexp \<Rightarrow> arexp" where
+  "intern ZERO = AZERO"
+| "intern ONE = AONE []"
+| "intern (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
+
+thm Posix.induct
+
+lemma erase_intern [simp]:
+  shows "erase (intern r) = r"
+  apply(induct r)
+  apply(simp_all add: erase_fuse)
+  done
+
+lemma erase_bder [simp]:
+  shows "erase (bder a r) = der a (erase r)"
+  apply(induct r rule: erase.induct)
+  apply(simp_all add: erase_fuse bnullable_correctness)
+  done
+
+lemma erase_bders [simp]:
+  shows "erase (bders r s) = ders s (erase r)"
+  apply(induct s arbitrary: r )
+  apply(simp_all)
+  done
+
+lemma retrieve_encode_STARS:
+  assumes "\<forall>v\<in>set vs. \<Turnstile> v : r \<and> code v = retrieve (intern r) v"
+  shows "code (Stars vs) = retrieve (ASTAR [] (intern r)) (Stars vs)"
+  using assms
+  apply(induct vs)
+  apply(simp_all)
+  done
+
+
+lemma retrieve_fuse2:
+  assumes "\<Turnstile> v : (erase r)"
+  shows "retrieve (fuse bs r) v = bs @ retrieve r v"
+  using assms
+  apply(induct r arbitrary: v bs)
+         apply(auto elim: Prf_elims)[4]
+   defer
+  using retrieve_encode_STARS
+   apply(auto elim!: Prf_elims)[1]
+   apply(case_tac vs)
+    apply(simp)
+   apply(simp)
+  (* AALTs  case *)
+  apply(simp)
+  apply(case_tac x2a)
+   apply(simp)
+   apply(auto elim!: Prf_elims)[1]
+  apply(simp)
+   apply(case_tac list)
+   apply(simp)
+  apply(auto)
+  apply(auto elim!: Prf_elims)[1]
+  done
+
+lemma retrieve_fuse:
+  assumes "\<Turnstile> v : r"
+  shows "retrieve (fuse bs (intern r)) v = bs @ retrieve (intern r) v"
+  using assms 
+  by (simp_all add: retrieve_fuse2)
+
+
+lemma retrieve_code:
+  assumes "\<Turnstile> v : r"
+  shows "code v = retrieve (intern r) v"
+  using assms
+  apply(induct v r )
+  apply(simp_all add: retrieve_fuse retrieve_encode_STARS)
+  done
+
+
+lemma 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)))"
+
+
+
+
+fun flts :: "arexp list \<Rightarrow> arexp list"
+  where 
+  "flts [] = []"
+| "flts (AZERO # rs) = flts rs"
+| "flts ((AALTs bs  rs1) # rs) = (map (fuse bs) rs1) @ flts rs"
+| "flts (r1 # rs) = r1 # flts rs"
+
+
+
+
+fun li :: "bit list \<Rightarrow> arexp list \<Rightarrow> arexp"
+  where
+  "li _ [] = AZERO"
+| "li bs [a] = fuse bs a"
+| "li bs as = AALTs bs as"
+
+
+
+
+fun bsimp_ASEQ :: "bit list \<Rightarrow> arexp \<Rightarrow> arexp \<Rightarrow> arexp"
+  where
+  "bsimp_ASEQ _ AZERO _ = AZERO"
+| "bsimp_ASEQ _ _ AZERO = AZERO"
+| "bsimp_ASEQ bs1 (AONE bs2) r2 = fuse (bs1 @ bs2) r2"
+| "bsimp_ASEQ bs1 r1 r2 = ASEQ  bs1 r1 r2"
+
+
+fun bsimp_AALTs :: "bit list \<Rightarrow> arexp list \<Rightarrow> arexp"
+  where
+  "bsimp_AALTs _ [] = AZERO"
+| "bsimp_AALTs bs1 [r] = fuse bs1 r"
+| "bsimp_AALTs bs1 rs = AALTs bs1 rs"
+
+
+fun bsimp :: "arexp \<Rightarrow> arexp" 
+  where
+  "bsimp (ASEQ bs1 r1 r2) = bsimp_ASEQ bs1 (bsimp r1) (bsimp r2)"
+| "bsimp (AALTs bs1 rs) = bsimp_AALTs bs1 (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)
+  apply(simp)
+  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:
+  shows "bnullable (AALTs bs rs) = (\<exists>r \<in> set rs. bnullable r)"
+  apply(induct rs arbitrary: bs)
+  apply(simp)
+  apply(simp)
+  done
+
+
+
+
+
+fun nonnested :: "arexp \<Rightarrow> bool"
+  where
+  "nonnested (AALTs bs2 []) = True"
+| "nonnested (AALTs bs2 ((AALTs bs1 rs1) # rs2)) = False"
+| "nonnested (AALTs bs2 (r # rs2)) = nonnested (AALTs bs2 rs2)"
+| "nonnested r = True"
+
+
+lemma  k0:
+  shows "flts (r # rs1) = flts [r] @ flts rs1"
+  apply(induct r arbitrary: rs1)
+   apply(auto)
+  done
+
+lemma  k00:
+  shows "flts (rs1 @ rs2) = flts rs1 @ flts rs2"
+  apply(induct rs1 arbitrary: rs2)
+   apply(auto)
+  by (metis append.assoc k0)
+
+lemma  k0a:
+  shows "flts [AALTs bs rs] = map (fuse bs)  rs"
+  apply(simp)
+  done
+
+
+
+
+
+
+
+
+lemma bsimp_AALTs_qq:
+  assumes "1 < length rs"
+  shows "bsimp_AALTs bs rs = AALTs bs  rs"
+  using  assms
+  apply(case_tac rs)
+   apply(simp)
+  apply(case_tac list)
+   apply(simp_all)
+  done
+
+
+
+lemma bbbbs1:
+  shows "nonalt r \<or> (\<exists>bs rs. r  = AALTs bs rs)"
+  using nonalt.elims(3) by auto
+  
+
+
+
+
+lemma flts_append:
+  "flts (xs1 @ xs2) = flts xs1 @ flts xs2"
+  apply(induct xs1  arbitrary: xs2  rule: rev_induct)
+   apply(auto)
+  apply(case_tac xs)
+   apply(auto)
+   apply(case_tac x)
+        apply(auto)
+  apply(case_tac x)
+        apply(auto)
+  done
+
+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
+
+
+fun flts2 :: "char \<Rightarrow> arexp list \<Rightarrow> arexp list"
+  where 
+  "flts2 _ [] = []"
+| "flts2 c (AZERO # rs) = flts2 c rs"
+| "flts2 c (AONE _ # rs) = flts2 c rs"
+| "flts2 c (ACHAR bs d # rs) = (if c = d then (ACHAR bs d # flts2 c rs) else flts2 c rs)"
+| "flts2 c ((AALTs bs rs1) # rs) = (map (fuse bs) rs1) @ flts2 c rs"
+| "flts2 c (ASEQ bs r1 r2 # rs) = (if (bnullable(r1) \<and> r2 = AZERO) then 
+    flts2 c rs
+    else ASEQ bs r1 r2 # flts2 c rs)"
+| "flts2 c (r1 # rs) = r1 # flts2 c rs"
+
+
+
+
+
+
+
+
+ 
+
+
+
+
+lemma WQ1:
+  assumes "s \<in> L (der c r)"
+  shows "s \<in> der c r \<rightarrow> mkeps (ders s (der c r))"
+  using assms
+  oops
+
+
+
+lemma bder_bsimp_AALTs:
+  shows "bder c (bsimp_AALTs bs rs) = bsimp_AALTs bs (map (bder c) rs)"
+  apply(induct bs rs rule: bsimp_AALTs.induct)  
+    apply(simp)
+   apply(simp)
+   apply (simp add: bder_fuse)
+  apply(simp)
+  done
+
+
+
+lemma
+  assumes "asize (bsimp a) = asize a"  "a = AALTs bs [AALTs bs2 [], AZERO, AONE bs3]"
+  shows "bsimp a = a"
+  using assms
+  apply(simp)
+  oops
+
+
+
+
+
+
+
+
+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 bs (AONE bs1) r \<leadsto> fuse (bs@bs1) 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)"
+(*the below rule for extracting common prefixes between a list of rexp's bitcodes*)
+| "AALTs bs (map (fuse bs1) rs) \<leadsto> AALTs (bs@bs1) rs"
+(*opposite direction also allowed, which means bits  are free to be moved around
+as long as they are on the right path*)
+| "AALTs (bs@bs1) rs \<leadsto> AALTs bs (map (fuse bs1) rs)"
+| "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')"
+(*rs1 = [r1, r2, ..., rn] rs2 = [r1', r2', ..., rn']
+[r1, r2, ..., rn] \<leadsto>* [r1', r2, ..., rn] \<leadsto>* [...r2',...] \<leadsto>* [r1', r2',... rn']
+*)
+
+
+
+lemma r_in_rstar : "r1 \<leadsto> r2 \<Longrightarrow> r1 \<leadsto>* r2"
+  using rrewrites.intros(1) rrewrites.intros(2) by blast
+ 
+lemma real_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: "rs1 s\<leadsto>* rs2 \<Longrightarrow> AALTs bs rs1 \<leadsto>* AALTs bs rs2"
+  by (metis append.left_neutral srewrites_alt)
+
+
+lemma star_seq:  "r1 \<leadsto>* r2 \<Longrightarrow> ASEQ bs r1 r3 \<leadsto>* ASEQ bs r2 r3"
+  apply(induct r1 r2 arbitrary: r3 rule: rrewrites.induct)
+   apply(rule rs1)
+  apply(erule rrewrites.cases)
+   apply(simp)
+   apply(rule r_in_rstar)
+   apply(rule rrewrite.intros(4))
+   apply simp
+  apply(rule rs2)
+   apply(assumption)
+  apply(rule rrewrite.intros(4))
+  by assumption
+
+lemma star_seq2:  "r3 \<leadsto>* r4 \<Longrightarrow> ASEQ bs r1 r3 \<leadsto>* ASEQ bs r1 r4"
+  apply(induct r3 r4 arbitrary: r1 rule: rrewrites.induct)
+   apply auto
+  using rrewrite.intros(5) by blast
+
+
+lemma continuous_rewrite: "\<lbrakk>r1 \<leadsto>* AZERO\<rbrakk> \<Longrightarrow> ASEQ bs1 r1 r2 \<leadsto>* AZERO"
+  apply(induction ra\<equiv>"r1" rb\<equiv>"AZERO" arbitrary: bs1 r1 r2 rule: rrewrites.induct)
+   apply (simp add: r_in_rstar rrewrite.intros(1))
+
+  by (meson rrewrite.intros(1) rrewrites.intros(2) star_seq)
+  
+
+
+lemma bsimp_aalts_simpcases: "AONE bs \<leadsto>* (bsimp (AONE bs))"  "AZERO \<leadsto>* bsimp AZERO" "ACHAR bs c \<leadsto>* (bsimp (ACHAR bs c))"
+  apply (simp add: rrewrites.intros(1))
+  apply (simp add: rrewrites.intros(1))
+  by (simp add: rrewrites.intros(1))
+
+lemma trivialbsimpsrewrites: "\<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_AALTsrewrites: "AALTs bs1 rs \<leadsto>* bsimp_AALTs bs1 rs"
+  apply(induction rs)
+  apply simp
+   apply(rule r_in_rstar)
+   apply(simp add:  rrewrite.intros(11))
+  apply(case_tac "rs = Nil")
+   apply(simp)
+  using rrewrite.intros(12) apply auto[1]
+  apply(subgoal_tac "length (a#rs) > 1")
+   apply(simp add: bsimp_AALTs_qq)
+  apply(simp)
+  done 
+
+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 flts_prepend: "\<lbrakk>nonalt a; nonazero a\<rbrakk> \<Longrightarrow> flts (a#rs) = a # (flts rs)"
+  by (metis append_Cons append_Nil flts_single1 k00)
+
+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 add: nonalt.simps)+
+   
+   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_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 r_in_rstar rrewrite.intros(7) apply presburger
+    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 presburger
+    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 fltsrewrites: "  AALTs bs1 rs \<leadsto>* AALTs bs1 (flts rs)"
+  apply(induction rs)
+   apply simp
+  apply(case_tac "a = AZERO")
+  apply (metis append_Nil flts.simps(2) many_steps_later rrewrite.intros(7))
+
+
+
+  apply(case_tac "\<exists>bs2 rs2. a = AALTs bs2 rs2")
+   apply(erule exE)+
+   apply(simp add: flts.simps)
+   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 k00 k0a)
+
+lemma alts_simpalts: "\<And>bs1 rs. (\<And>x. x \<in> set rs \<Longrightarrow> x \<leadsto>* bsimp x) \<Longrightarrow> 
+AALTs bs1 rs \<leadsto>* AALTs bs1 (map bsimp rs)"
+  apply(subgoal_tac " rs s\<leadsto>*  (map bsimp rs)")
+   prefer 2
+  using trivialbsimpsrewrites apply auto[1]
+  using srewrites_alt1 by auto
+
+
+lemma threelistsappend: "rsa@a#rsb = (rsa@[a])@rsb"
+  apply auto
+  done
+
+fun distinctByAcc :: "'a list \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b set \<Rightarrow> 'b set"
+  where
+  "distinctByAcc [] f acc = acc"
+| "distinctByAcc (x#xs) f acc = 
+     (if (f x) \<in> acc then distinctByAcc xs f acc 
+      else  (distinctByAcc xs f ({f x} \<union> acc)))"
+
+lemma dB_single_step: "distinctBy (a#rs) f {} = a # distinctBy rs f {f a}"
+  apply simp
+  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(13) 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(13) 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: " (rrewrites r ( bsimp r))"
+  apply(induction r rule: bsimp.induct)
+       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)
+      defer
+  using bsimp_aalts_simpcases(2) apply blast
+  apply simp
+  apply simp
+  apply simp
+
+  apply auto
+
+
+  apply(subgoal_tac "AALTs bs1 rs \<leadsto>* AALTs bs1 (map bsimp rs)")
+   apply(subgoal_tac "AALTs bs1 (map bsimp rs) \<leadsto>* AALTs bs1 (flts (map bsimp rs))")
+  apply(subgoal_tac "AALTs bs1 (flts (map bsimp rs)) \<leadsto>* AALTs bs1 (distinctBy (flts (map bsimp rs)) erase {})")
+    apply(subgoal_tac "AALTs bs1 (distinctBy (flts (map bsimp rs)) erase {}) \<leadsto>* bsimp_AALTs bs1 (distinctBy (flts (map bsimp rs)) erase {} )")
+
+  
+      apply (meson real_trans)
+
+   apply (meson bsimp_AALTsrewrites)
+
+  apply (meson alts_dBrewrites)
+
+  using fltsrewrites apply auto[1]
+
+  using alts_simpalts by force
+
+
+lemma rewritenullable: "\<lbrakk>r1 \<leadsto> r2; bnullable r1 \<rbrakk> \<Longrightarrow> bnullable r2"
+  apply(induction r1 r2 rule: rrewrite.induct)
+             apply(simp)+
+  apply (metis bnullable_correctness erase_fuse)
+          apply simp
+         apply simp
+        apply auto[1]
+       apply auto[1]
+      apply auto[4]
+     apply (metis UnCI bnullable_correctness erase_fuse imageI)
+    apply (metis bnullable_correctness erase_fuse)
+    apply (metis bnullable_correctness erase_fuse)
+  
+   apply (metis bnullable_correctness erase.simps(5) erase_fuse)
+  
+
+  by (smt (z3) Un_iff bnullable_correctness insert_iff list.set(2) qq3 set_append)
+
+lemma rewrite_non_nullable: "\<lbrakk>r1 \<leadsto> r2; \<not>bnullable r1 \<rbrakk> \<Longrightarrow> \<not>bnullable r2"
+  apply(induction r1 r2 rule: rrewrite.induct)
+             apply auto 
+      apply (metis bnullable_correctness erase_fuse)+
+  done
+
+
+lemma rewritesnullable: "\<lbrakk> r1 \<leadsto>* r2; bnullable r1 \<rbrakk> \<Longrightarrow> bnullable r2"
+  apply(induction r1 r2 rule: rrewrites.induct)
+   apply simp
+  apply(rule rewritenullable)
+   apply simp
+  apply simp
+  done
+
+lemma nonbnullable_lists_concat: " \<lbrakk> \<not> (\<exists>r0\<in>set rs1. bnullable r0); \<not> bnullable r; \<not> (\<exists>r0\<in>set rs2. bnullable r0)\<rbrakk> \<Longrightarrow> 
+\<not>(\<exists>r0 \<in> (set (rs1@[r]@rs2)). bnullable r0 ) "
+  apply simp
+  apply blast
+  done
+
+
+
+lemma nomember_bnullable: "\<lbrakk> \<not> (\<exists>r0\<in>set rs1. bnullable r0); \<not> bnullable r; \<not> (\<exists>r0\<in>set rs2. bnullable r0)\<rbrakk>
+ \<Longrightarrow> \<not>bnullable (AALTs bs (rs1 @ [r] @ rs2))"
+  using nonbnullable_lists_concat qq3 by presburger
+
+lemma bnullable_segment: " bnullable (AALTs bs (rs1@[r]@rs2)) \<Longrightarrow> bnullable (AALTs bs rs1) \<or> bnullable (AALTs bs rs2) \<or> bnullable r"
+  apply(case_tac "\<exists>r0\<in>set rs1.  bnullable r0")
+
+  using qq3 apply blast
+  apply(case_tac "bnullable r")
+
+  apply blast
+  apply(case_tac "\<exists>r0\<in>set rs2.  bnullable r0")
+
+  using bnullable.simps(4) apply presburger
+  apply(subgoal_tac "False")
+
+  apply blast
+
+  using nomember_bnullable by blast
+
+  
+
+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 rrewrite_nbnullable: "\<lbrakk> r1 \<leadsto> r2 ; \<not> bnullable r1 \<rbrakk> \<Longrightarrow> \<not>bnullable r2"
+  apply(induction rule: rrewrite.induct)
+             apply auto[1]
+            apply auto[1]
+           apply auto[1]
+           apply (metis bnullable_correctness erase_fuse)
+          apply auto[1]
+         apply auto[1]
+        apply auto[1]
+       apply auto[1]
+      apply auto[1]
+      apply (metis bnullable_correctness erase_fuse)
+     apply auto[1]
+     apply (metis bnullable_correctness erase_fuse)
+    apply auto[1]
+    apply (metis bnullable_correctness erase_fuse)
+   apply auto[1]
+   apply auto[1]
+
+  apply (metis bnullable_correctness erase_fuse)
+
+  by (meson rewrite_non_nullable rrewrite.intros(13))
+
+
+
+
+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)"
+  
+  by (metis append.left_neutral append_Cons bnullable.simps(1) bnullable_segment rrewrite.intros(7) rrewrite_nbnullable)
+
+
+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)"
+  apply(subgoal_tac "bnullable (AALTs bs rs3)")
+   apply(subgoal_tac "\<forall>r \<in> set (rs1@rs2). \<not>bnullable r")
+  apply(subgoal_tac "bmkeps (AALTs bs (rs1@rs2@rs3)) = bmkeps (AALTs bs ((rs1@rs2)@rs3) )")
+  apply (metis qq2 qq3)
+
+  apply (metis append.assoc)
+
+  apply (metis append.assoc in_set_conv_decomp r2 third_segment_bnullable)
+
+  using third_segment_bnullable by blast
+
+
+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(case_tac "bnullable (AALTs bs rsa)")
+  
+  using qq1 apply force
+  apply(case_tac "bnullable (AALTs bs1 rs1)")
+  apply(subst qq2)
+
+  
+  using r2 apply blast
+  
+    apply (metis list.set_intros(1))
+  apply (smt (verit, ccfv_threshold) append_eq_append_conv2 list.set_intros(1) qq2 qq3 rewritenullable rrewrite.intros(8) self_append_conv2 spillbmkepslistr)
+
+
+  thm qq1
+  apply(subgoal_tac "bmkeps  (AALTs bs (rsa @ AALTs bs1 rs1 # rsb)) = bmkeps (AALTs bs rsb) ")
+   prefer 2
+  
+  apply (metis append_Cons append_Nil bnullable.simps(1) bnullable_segment rewritenullable rrewrite.intros(11) third_segment_bmkeps)
+
+  by (metis bnullable.simps(4) rewrite_non_nullable rrewrite.intros(10) third_segment_bmkeps)
+
+
+
+lemma rewrite_bmkeps: "\<lbrakk> r1 \<leadsto> r2; (bnullable r1)\<rbrakk> \<Longrightarrow> bmkeps r1 = bmkeps r2"
+
+  apply(frule rewritenullable)
+  apply simp
+  apply(induction r1 r2 rule: rrewrite.induct)
+             apply simp
+  using bnullable.simps(1) bnullable.simps(5) apply blast
+         apply (simp add: b2)
+        apply simp
+         apply simp
+  apply(frule bnullable_segment)
+        apply(case_tac "bnullable (AALTs bs rs1)")
+  using qq1 apply force
+        apply(case_tac "bnullable r")
+  using bnullablewhichbmkeps rewritenullable apply presburger
+        apply(subgoal_tac "bnullable (AALTs bs rs2)")
+  apply(subgoal_tac "\<not> bnullable r'")
+  apply (simp add: qq2 r1)
+  
+  using rrewrite_nbnullable apply blast
+
+        apply blast
+       apply (simp add: flts_append qs3)
+
+  apply (meson rewrite_bmkepsalt)
+  
+  using bnullable.simps(4) q3a apply blast
+
+  apply (simp add: q3a)
+
+  using bnullable.simps(1) apply blast
+
+  apply (simp add: b2)
+ 
+  by (smt (z3) Un_iff bnullable_correctness erase.simps(5) qq1 qq2 qq3 set_append)
+
+
+
+lemma rewrites_bmkeps: "\<lbrakk> (r1 \<leadsto>* r2); (bnullable r1)\<rbrakk> \<Longrightarrow> bmkeps r1 = bmkeps r2"
+  apply(induction r1 r2 rule: rrewrites.induct)
+   apply simp
+  apply(subgoal_tac "bnullable r2")
+  prefer 2
+   apply(metis rewritesnullable)
+  apply(subgoal_tac "bmkeps r1 = bmkeps r2")
+   prefer 2
+   apply fastforce
+  using rewrite_bmkeps by presburger
+
+
+thm rrewrite.intros(12)
+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 alt_rewrite_front: "r \<leadsto> r' \<Longrightarrow> AALT bs r r2 \<leadsto> AALT bs r' r2"
+  using alts_rewrite_front by blast
+
+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 alt_remove0_front: " AALT bs AZERO r \<leadsto> AALTs bs [r]"
+  by (simp add: rrewrite0away)
+
+lemma alt_rewrites_back: "r2 \<leadsto>* r2' \<Longrightarrow>AALT bs r1 r2 \<leadsto>* AALT bs r1 r2'"
+  apply(induction r2 r2' arbitrary: bs rule: rrewrites.induct)
+   apply simp
+  by (meson rs1 rs2 srewrites_alt1 ss1 ss2)
+
+lemma rewrite_fuse: " r2 \<leadsto> r3 \<Longrightarrow> fuse bs r2 \<leadsto>* fuse bs r3"
+  apply(induction r2 r3 arbitrary: bs rule: rrewrite.induct)
+             apply auto
+
+           apply (simp add: continuous_rewrite)
+
+          apply (simp add: r_in_rstar rrewrite.intros(2))
+
+         apply (metis fuse_append r_in_rstar rrewrite.intros(3))
+
+  using r_in_rstar star_seq apply blast
+
+  using r_in_rstar star_seq2 apply blast
+
+  using contextrewrites2 r_in_rstar apply auto[1]
+  
+       apply (simp add: r_in_rstar rrewrite.intros(7))
+
+  using rrewrite.intros(8) apply auto[1]
+
+   apply (metis append_assoc r_in_rstar rrewrite.intros(9))
+
+  apply (metis append_assoc r_in_rstar rrewrite.intros(10))
+
+  apply (simp add: r_in_rstar rrewrite.intros(11))
+
+  apply (metis fuse_append r_in_rstar rrewrite.intros(12))
+
+  using rrewrite.intros(13) by auto
+
+  
+
+lemma rewrites_fuse:  "r2 \<leadsto>* r2' \<Longrightarrow>  (fuse bs1 r2) \<leadsto>*  (fuse bs1 r2')"
+  apply(induction r2 r2' arbitrary: bs1 rule: rrewrites.induct)
+   apply simp
+  by (meson real_trans rewrite_fuse)
+
+lemma  bder_fuse_list: " map (bder c \<circ> fuse bs1) rs1 = map (fuse bs1 \<circ> bder c) rs1"
+  apply(induction rs1)
+  apply simp
+  by (simp add: bder_fuse)
+
+
+
+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)
+  apply(rule many_steps_later)
+   apply(subst rrewrite.intros(8))
+   apply simp
+
+  by fastforce
+
+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(13) by auto
+
+lemma rewrite_after_der: "r1 \<leadsto> r2 \<Longrightarrow> (bder c r1) \<leadsto>* (bder c r2)"
+  apply(induction r1 r2 arbitrary: c rule: rrewrite.induct)
+  
+              apply (simp add: r_in_rstar rrewrite.intros(1))
+  apply simp
+  
+  apply (meson contextrewrites1 r_in_rstar rrewrite.intros(11) rrewrite.intros(2) rrewrite0away rs2)
+           apply(simp)
+           apply(rule many_steps_later)
+            apply(rule to_zero_in_alt)
+           apply(rule many_steps_later)
+  apply(rule alt_remove0_front)
+           apply(rule many_steps_later)
+            apply(rule rrewrite.intros(12))
+  using bder_fuse fuse_append rs1 apply presburger
+          apply(case_tac "bnullable r1")
+  prefer 2
+           apply(subgoal_tac "\<not>bnullable r2")
+            prefer 2
+  using rewrite_non_nullable apply presburger
+           apply simp+
+  
+  using star_seq apply auto[1]
+          apply(subgoal_tac "bnullable r2")
+           apply simp+
+  apply(subgoal_tac "bmkeps r1 = bmkeps r2")
+  prefer 2
+  using rewrite_bmkeps apply auto[1]
+  using contextrewrites1 star_seq apply auto[1]
+  using rewritenullable apply auto[1]
+         apply(case_tac "bnullable r1")
+          apply simp
+          apply(subgoal_tac "ASEQ [] (bder c r1) r3 \<leadsto> ASEQ [] (bder c r1) r4") 
+           prefer 2
+  using rrewrite.intros(5) apply blast
+          apply(rule many_steps_later)
+           apply(rule alt_rewrite_front)
+           apply assumption
+  apply (meson alt_rewrites_back rewrites_fuse) 
+
+       apply (simp add: r_in_rstar rrewrite.intros(5))
+
+  using contextrewrites2 apply force
+
+  using rrewrite.intros(7) apply force
+  
+  using rewrite_der_altmiddle apply auto[1]
+  
+  apply (metis bder.simps(4) bder_fuse_list map_map r_in_rstar rrewrite.intros(9))
+
+  apply (metis List.map.compositionality bder.simps(4) bder_fuse_list r_in_rstar rrewrite.intros(10))
+
+  apply (simp add: r_in_rstar rrewrite.intros(11))
+
+   apply (metis bder.simps(4) bder_bsimp_AALTs bsimp_AALTs.simps(2) bsimp_AALTsrewrites)
+
+  
+  using lock_step_der_removal by auto
+
+
+
+lemma rewrites_after_der: "  r1 \<leadsto>* r2  \<Longrightarrow>  (bder c r1) \<leadsto>* (bder c r2)"
+  apply(induction r1 r2 rule: rrewrites.induct)
+   apply(rule rs1)
+  by (meson real_trans rewrite_after_der)
+  
+
+
+
+lemma central: " (bders r s) \<leadsto>*  (bders_simp r s)" 
+  apply(induct s arbitrary: r rule: rev_induct)
+
+   apply simp
+  apply(subst bders_append)
+  apply(subst bders_simp_append)
+  by (metis bders.simps(1) bders.simps(2) bders_simp.simps(1) bders_simp.simps(2) bsimp_rewrite real_trans rewrites_after_der)
+
+
+
+thm arexp.induct
+
+lemma quasi_main: "bnullable (bders r s) \<Longrightarrow> bmkeps (bders r s) = bmkeps (bders_simp r s)"
+  using central rewrites_bmkeps by blast
+
+theorem main_main: "blexer r s = blexer_simp r s"
+  by (simp add: b4 blexer_def blexer_simp_def quasi_main)
+
+
+theorem blexersimp_correctness: "blexer_simp r s= lexer r s"
+  using blexer_correctness main_main by auto
+
+
+unused_thms
+
+
+end




