theory BlexerSimp

  imports Blexer 

begin

fun distinctWith :: "'a list \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'a list"

  where

  "distinctWith [] eq acc = []"

| "distinctWith (x # xs) eq acc = 

     (if (\<exists> y \<in> acc. eq x y) then distinctWith xs eq acc 

      else x # (distinctWith xs eq ({x} \<union> acc)))"

fun eq1 ("_ ~1 _" [80, 80] 80) where  

  "AZERO ~1 AZERO = True"

| "(AONE bs1) ~1 (AONE bs2) = True"

| "(ACHAR bs1 c) ~1 (ACHAR bs2 d) = (if c = d then True else False)"

| "(ASEQ bs1 ra1 ra2) ~1 (ASEQ bs2 rb1 rb2) = (ra1 ~1 rb1 \<and> ra2 ~1 rb2)"

| "(AALTs bs1 []) ~1 (AALTs bs2 []) = True"

| "(AALTs bs1 (r1 # rs1)) ~1 (AALTs bs2 (r2 # rs2)) = (r1 ~1 r2 \<and> (AALTs bs1 rs1) ~1 (AALTs bs2 rs2))"

| "(ASTAR bs1 r1) ~1 (ASTAR bs2 r2) = r1 ~1 r2"

| "(ANTIMES bs1 r1 n1) ~1 (ANTIMES bs2 r2 n2) = (r1 ~1 r2 \<and> n1 = n2)"

| "_ ~1 _ = False"

lemma eq1_L:

  assumes "x ~1 y"

  shows "L (erase x) = L (erase y)"

  using assms

  apply(induct rule: eq1.induct)

  apply(auto elim: eq1.elims)

  apply presburger

  done

fun flts :: "arexp list \<Rightarrow> arexp list"

  where 

  "flts [] = []"

| "flts (AZERO # rs) = flts rs"

| "flts ((AALTs bs  rs1) # rs) = (map (fuse bs) rs1) @ flts rs"

| "flts (r1 # rs) = r1 # flts rs"

fun bsimp_ASEQ :: "bit list \<Rightarrow> arexp \<Rightarrow> arexp \<Rightarrow> arexp"

  where

  "bsimp_ASEQ _ AZERO _ = AZERO"

| "bsimp_ASEQ _ _ AZERO = AZERO"

| "bsimp_ASEQ bs1 (AONE bs2) r2 = fuse (bs1 @ bs2) r2"

| "bsimp_ASEQ bs1 r1 r2 = ASEQ  bs1 r1 r2"

lemma bsimp_ASEQ0[simp]:

  shows "bsimp_ASEQ bs r1 AZERO = AZERO"

  by (case_tac r1)(simp_all)

lemma bsimp_ASEQ1:

  assumes "r1 \<noteq> AZERO" "r2 \<noteq> AZERO" "\<nexists>bs. r1 = AONE bs"

  shows "bsimp_ASEQ bs r1 r2 = ASEQ bs r1 r2"

  using assms

  apply(induct bs r1 r2 rule: bsimp_ASEQ.induct)

  apply(auto)

  done

lemma bsimp_ASEQ2[simp]:

  shows "bsimp_ASEQ bs1 (AONE bs2) r2 = fuse (bs1 @ bs2) r2"

  by (case_tac r2) (simp_all)

fun bsimp_AALTs :: "bit list \<Rightarrow> arexp list \<Rightarrow> arexp"

  where

  "bsimp_AALTs _ [] = AZERO"

| "bsimp_AALTs bs1 [r] = fuse bs1 r"

| "bsimp_AALTs bs1 rs = AALTs bs1 rs"

fun bsimp :: "arexp \<Rightarrow> arexp" 

  where

  "bsimp (ASEQ bs1 r1 r2) = bsimp_ASEQ bs1 (bsimp r1) (bsimp r2)"

| "bsimp (AALTs bs1 rs) = bsimp_AALTs bs1 (distinctWith (flts (map bsimp rs)) eq1 {}) "

| "bsimp r = r"

fun 

  bders_simp :: "arexp \<Rightarrow> string \<Rightarrow> arexp"

where

  "bders_simp r [] = r"

| "bders_simp r (c # s) = bders_simp (bsimp (bder c r)) s"

definition blexer_simp where

 "blexer_simp r s \<equiv> if bnullable (bders_simp (intern r) s) then 

                    decode (bmkeps (bders_simp (intern r) s)) r else None"

lemma bders_simp_append:

  shows "bders_simp r (s1 @ s2) = bders_simp (bders_simp r s1) s2"

  apply(induct s1 arbitrary: r s2)

  apply(simp_all)

  done

lemma bmkeps_fuse:

  assumes "bnullable r"

  shows "bmkeps (fuse bs r) = bs @ bmkeps r"

  using assms

  apply(induct r rule: bnullable.induct) 

        apply(auto)

  apply (metis append.assoc bmkeps_retrieve bmkeps_retrieve_AALTs bnullable.simps(4))  

   apply(case_tac n)

  apply(auto)

  done

lemma bmkepss_fuse: 

  assumes "bnullables rs"

  shows "bmkepss (map (fuse bs) rs) = bs @ bmkepss rs"

  using assms

  apply(induct rs arbitrary: bs)

  apply(auto simp add: bmkeps_fuse bnullable_fuse)

  done

lemma bder_fuse:

  shows "bder c (fuse bs a) = fuse bs  (bder c a)"

  apply(induct a arbitrary: bs c)

  apply(simp_all)

  done

inductive 

  rrewrite:: "arexp \<Rightarrow> arexp \<Rightarrow> bool" ("_ \<leadsto> _" [99, 99] 99)

and 

  srewrite:: "arexp list \<Rightarrow> arexp list \<Rightarrow> bool" (" _ s\<leadsto> _" [100, 100] 100)

where

  bs1: "ASEQ bs AZERO r2 \<leadsto> AZERO"

| bs2: "ASEQ bs r1 AZERO \<leadsto> AZERO"

| bs3: "ASEQ bs1 (AONE bs2) r \<leadsto> fuse (bs1@bs2) r"

| bs4: "r1 \<leadsto> r2 \<Longrightarrow> ASEQ bs r1 r3 \<leadsto> ASEQ bs r2 r3"

| bs5: "r3 \<leadsto> r4 \<Longrightarrow> ASEQ bs r1 r3 \<leadsto> ASEQ bs r1 r4"

| bs6: "AALTs bs [] \<leadsto> AZERO"

| bs7: "AALTs bs [r] \<leadsto> fuse bs r"

| bs10: "rs1 s\<leadsto> rs2 \<Longrightarrow> AALTs bs rs1 \<leadsto> AALTs bs rs2"

| ss1:  "[] s\<leadsto> []"

| ss2:  "rs1 s\<leadsto> rs2 \<Longrightarrow> (r # rs1) s\<leadsto> (r # rs2)"

| ss3:  "r1 \<leadsto> r2 \<Longrightarrow> (r1 # rs) s\<leadsto> (r2 # rs)"

| ss4:  "(AZERO # rs) s\<leadsto> rs"

| ss5:  "(AALTs bs1 rs1 # rsb) s\<leadsto> ((map (fuse bs1) rs1) @ rsb)"

| ss6:  "L (erase a2) \<subseteq> L (erase a1) \<Longrightarrow> (rsa@[a1]@rsb@[a2]@rsc) s\<leadsto> (rsa@[a1]@rsb@rsc)"

inductive 

  rrewrites:: "arexp \<Rightarrow> arexp \<Rightarrow> bool" ("_ \<leadsto>* _" [100, 100] 100)

where 

  rs1[intro, simp]:"r \<leadsto>* r"

| rs2[intro]: "\<lbrakk>r1 \<leadsto>* r2; r2 \<leadsto> r3\<rbrakk> \<Longrightarrow> r1 \<leadsto>* r3"

inductive 

  srewrites:: "arexp list \<Rightarrow> arexp list \<Rightarrow> bool" ("_ s\<leadsto>* _" [100, 100] 100)

where 

  sss1[intro, simp]:"rs s\<leadsto>* rs"

| sss2[intro]: "\<lbrakk>rs1 s\<leadsto> rs2; rs2 s\<leadsto>* rs3\<rbrakk> \<Longrightarrow> rs1 s\<leadsto>* rs3"

lemma r_in_rstar : "r1 \<leadsto> r2 \<Longrightarrow> r1 \<leadsto>* r2"

  using rrewrites.intros(1) rrewrites.intros(2) by blast

lemma rs_in_rstar: 

  shows "r1 s\<leadsto> r2 \<Longrightarrow> r1 s\<leadsto>* r2"

  using rrewrites.intros(1) rrewrites.intros(2) by blast

lemma rrewrites_trans[trans]: 

  assumes a1: "r1 \<leadsto>* r2"  and a2: "r2 \<leadsto>* r3"

  shows "r1 \<leadsto>* r3"

  using a2 a1

  apply(induct r2 r3 arbitrary: r1 rule: rrewrites.induct) 

  apply(auto)

  done

lemma srewrites_trans[trans]: 

  assumes a1: "r1 s\<leadsto>* r2"  and a2: "r2 s\<leadsto>* r3"

  shows "r1 s\<leadsto>* r3"

  using a1 a2

  apply(induct r1 r2 arbitrary: r3 rule: srewrites.induct) 

   apply(auto)

  done

lemma contextrewrites0: 

  "rs1 s\<leadsto>* rs2 \<Longrightarrow> AALTs bs rs1 \<leadsto>* AALTs bs rs2"

  apply(induct rs1 rs2 rule: srewrites.inducts)

   apply simp

  using bs10 r_in_rstar rrewrites_trans by blast

lemma contextrewrites1: 

  "r \<leadsto>* r' \<Longrightarrow> AALTs bs (r # rs) \<leadsto>* AALTs bs (r' # rs)"

  apply(induct r r' rule: rrewrites.induct)

   apply simp

  using bs10 ss3 by blast

lemma srewrite1: 

  shows "rs1 s\<leadsto> rs2 \<Longrightarrow> (rs @ rs1) s\<leadsto> (rs @ rs2)"

  apply(induct rs)

   apply(auto)

  using ss2 by auto

lemma srewrites1: 

  shows "rs1 s\<leadsto>* rs2 \<Longrightarrow> (rs @ rs1) s\<leadsto>* (rs @ rs2)"

  apply(induct rs1 rs2 rule: srewrites.induct)

   apply(auto)

  using srewrite1 by blast

lemma srewrite2: 

  shows  "r1 \<leadsto> r2 \<Longrightarrow> True"

  and "rs1 s\<leadsto> rs2 \<Longrightarrow> (rs1 @ rs) s\<leadsto>* (rs2 @ rs)"

  apply(induct rule: rrewrite_srewrite.inducts)

  apply(auto)

  apply (metis append_Cons append_Nil srewrites1)

  apply(meson srewrites.simps ss3)

  apply (meson srewrites.simps ss4)

  apply (meson srewrites.simps ss5)

  by (metis append_Cons append_Nil srewrites.simps ss6)

lemma srewrites3: 

  shows "rs1 s\<leadsto>* rs2 \<Longrightarrow> (rs1 @ rs) s\<leadsto>* (rs2 @ rs)"

  apply(induct rs1 rs2 arbitrary: rs rule: srewrites.induct)

   apply(auto)

  by (meson srewrite2(2) srewrites_trans)

(*

lemma srewrites4:

  assumes "rs3 s\<leadsto>* rs4" "rs1 s\<leadsto>* rs2" 

  shows "(rs1 @ rs3) s\<leadsto>* (rs2 @ rs4)"

  using assms

  apply(induct rs3 rs4 arbitrary: rs1 rs2 rule: srewrites.induct)

  apply (simp add: srewrites3)

  using srewrite1 by blast

*)

lemma srewrites6:

  assumes "r1 \<leadsto>* r2" 

  shows "[r1] s\<leadsto>* [r2]"

  using assms

  apply(induct r1 r2 rule: rrewrites.induct)

   apply(auto)

  by (meson srewrites.simps srewrites_trans ss3)

lemma srewrites7:

  assumes "rs3 s\<leadsto>* rs4" "r1 \<leadsto>* r2" 

  shows "(r1 # rs3) s\<leadsto>* (r2 # rs4)"

  using assms

  by (smt (verit, best) append_Cons append_Nil srewrites1 srewrites3 srewrites6 srewrites_trans)

lemma ss6_stronger_aux:

  shows "(rs1 @ rs2) s\<leadsto>* (rs1 @ distinctWith rs2 eq1 (set rs1))"

  apply(induct rs2 arbitrary: rs1)

  apply(auto)

  prefer 2

  apply(drule_tac x="rs1 @ [a]" in meta_spec)

  apply(simp)

  apply(drule_tac x="rs1" in meta_spec)

  apply(subgoal_tac "(rs1 @ a # rs2) s\<leadsto>* (rs1 @ rs2)")

  using srewrites_trans apply blast

  apply(subgoal_tac "\<exists>rs1a rs1b. rs1 = rs1a @ [x] @ rs1b")

  prefer 2

  apply (simp add: split_list)

  apply(erule exE)+

  apply(simp)

  using eq1_L rs_in_rstar ss6 by force

lemma ss6_stronger:

  shows "rs1 s\<leadsto>* distinctWith rs1 eq1 {}"

  by (metis append_Nil list.set(1) ss6_stronger_aux)

lemma rewrite_preserves_fuse: 

  shows "r2 \<leadsto> r3 \<Longrightarrow> fuse bs r2 \<leadsto> fuse bs r3"

  and   "rs2 s\<leadsto> rs3 \<Longrightarrow> map (fuse bs) rs2 s\<leadsto>* map (fuse bs) rs3"

proof(induct rule: rrewrite_srewrite.inducts)

  case (bs3 bs1 bs2 r)

  then show ?case

    by (metis fuse.simps(5) fuse_append rrewrite_srewrite.bs3) 

next

  case (bs7 bs r)

  then show ?case

    by (metis fuse.simps(4) fuse_append rrewrite_srewrite.bs7) 

next

  case (ss2 rs1 rs2 r)

  then show ?case

    using srewrites7 by force 

next

  case (ss3 r1 r2 rs)

  then show ?case by (simp add: r_in_rstar srewrites7)

next

  case (ss5 bs1 rs1 rsb)

  then show ?case 

    apply(simp)

    by (metis (mono_tags, lifting) comp_def fuse_append map_eq_conv rrewrite_srewrite.ss5 srewrites.simps)

next

  case (ss6 a1 a2 rsa rsb rsc)

  then show ?case 

    apply(simp only: map_append)

    by (smt (verit, best) erase_fuse list.simps(8) list.simps(9) rrewrite_srewrite.ss6 srewrites.simps)

qed (auto intro: rrewrite_srewrite.intros)

lemma rewrites_fuse:  

  assumes "r1 \<leadsto>* r2"

  shows "fuse bs r1 \<leadsto>* fuse bs r2"

using assms

apply(induction r1 r2 arbitrary: bs rule: rrewrites.induct)

apply(auto intro: rewrite_preserves_fuse rrewrites_trans)

done

lemma star_seq:  

  assumes "r1 \<leadsto>* r2"

  shows "ASEQ bs r1 r3 \<leadsto>* ASEQ bs r2 r3"

using assms

apply(induct r1 r2 arbitrary: r3 rule: rrewrites.induct)

apply(auto intro: rrewrite_srewrite.intros)

done

lemma star_seq2:  

  assumes "r3 \<leadsto>* r4"

  shows "ASEQ bs r1 r3 \<leadsto>* ASEQ bs r1 r4"

  using assms

apply(induct r3 r4 arbitrary: r1 rule: rrewrites.induct)

apply(auto intro: rrewrite_srewrite.intros)

done

lemma continuous_rewrite: 

  assumes "r1 \<leadsto>* AZERO"

  shows "ASEQ bs1 r1 r2 \<leadsto>* AZERO"

using assms bs1 star_seq by blast

(*

lemma continuous_rewrite2: 

  assumes "r1 \<leadsto>* AONE bs"

  shows "ASEQ bs1 r1 r2 \<leadsto>* (fuse (bs1 @ bs) r2)"

  using assms  by (meson bs3 rrewrites.simps star_seq)

*)

lemma bsimp_aalts_simpcases: 

  shows "AONE bs \<leadsto>* bsimp (AONE bs)"  

  and   "AZERO \<leadsto>* bsimp AZERO" 

  and   "ACHAR bs c \<leadsto>* bsimp (ACHAR bs c)"

  by (simp_all)

lemma bsimp_AALTs_rewrites: 

  shows "AALTs bs1 rs \<leadsto>* bsimp_AALTs bs1 rs"

  by (smt (verit) bs6 bs7 bsimp_AALTs.elims rrewrites.simps)

lemma trivialbsimp_srewrites: 

  "\<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(simp)

  using srewrites7 by auto

lemma fltsfrewrites: "rs s\<leadsto>* flts rs"

  apply(induction rs rule: flts.induct)

  apply(auto intro: rrewrite_srewrite.intros)

  apply (meson srewrites.simps srewrites1 ss5)

  using rs1 srewrites7 apply presburger

  using srewrites7 apply force

  apply (simp add: srewrites7)

   apply(simp add: srewrites7)

  by (simp add: srewrites7)

lemma bnullable0:

shows "r1 \<leadsto> r2 \<Longrightarrow> bnullable r1 = bnullable r2" 

  and "rs1 s\<leadsto> rs2 \<Longrightarrow> bnullables rs1 = bnullables rs2" 

  apply(induct rule: rrewrite_srewrite.inducts)

  apply(auto simp add:  bnullable_fuse)

   apply (meson UnCI bnullable_fuse imageI)

  using bnullable_correctness nullable_correctness by blast 

lemma rewritesnullable: 

  assumes "r1 \<leadsto>* r2" 

  shows "bnullable r1 = bnullable r2"

using assms 

  apply(induction r1 r2 rule: rrewrites.induct)

  apply simp

  using bnullable0(1) by auto

lemma rewrite_bmkeps_aux: 

  shows "r1 \<leadsto> r2 \<Longrightarrow> (bnullable r1 \<and> bnullable r2 \<Longrightarrow> bmkeps r1 = bmkeps r2)"

  and   "rs1 s\<leadsto> rs2 \<Longrightarrow> (bnullables rs1 \<and> bnullables rs2 \<Longrightarrow> bmkepss rs1 = bmkepss rs2)" 

proof (induct rule: rrewrite_srewrite.inducts)

  case (bs3 bs1 bs2 r)

  then show ?case by (simp add: bmkeps_fuse) 

next

  case (bs7 bs r)

  then show ?case by (simp add: bmkeps_fuse) 

next

  case (ss3 r1 r2 rs)

  then show ?case

    using bmkepss.simps bnullable0(1) by presburger

next

  case (ss5 bs1 rs1 rsb)

  then show ?case

    apply (simp add: bmkepss1 bmkepss2 bmkepss_fuse bnullable_fuse)

    apply(case_tac rs1)

     apply(auto simp add: bnullable_fuse)

    apply (metis bmkeps_retrieve bmkeps_retrieve_AALTs bnullable.simps(4))

    apply (metis bmkeps_retrieve bmkeps_retrieve_AALTs bnullable.simps(4))

    apply (metis bmkeps_retrieve bmkeps_retrieve_AALTs bnullable.simps(4))

    by (metis bmkeps_retrieve bmkeps_retrieve_AALTs bnullable.simps(4))    

next

  case (ss6 a1 a2 rsa rsb rsc)

  then show ?case

    by (smt (verit, best) Nil_is_append_conv bmkepss1 bmkepss2 bnullable_correctness in_set_conv_decomp list.distinct(1) list.set_intros(1) nullable_correctness set_ConsD subsetD)

next

  case (bs10 rs1 rs2 bs)

  then show ?case

    by (metis bmkeps_retrieve bmkeps_retrieve_AALTs bnullable.simps(4)) 

qed (auto)

lemma rewrites_bmkeps: 

  assumes "r1 \<leadsto>* r2" "bnullable r1" 

  shows "bmkeps r1 = bmkeps r2"

  using assms

proof(induction r1 r2 rule: rrewrites.induct)

  case (rs1 r)

  then show "bmkeps r = bmkeps r" by simp

next

  case (rs2 r1 r2 r3)

  then have IH: "bmkeps r1 = bmkeps r2" by simp

  have a1: "bnullable r1" by fact

  have a2: "r1 \<leadsto>* r2" by fact

  have a3: "r2 \<leadsto> r3" by fact

  have a4: "bnullable r2" using a1 a2 by (simp add: rewritesnullable) 

  then have "bmkeps r2 = bmkeps r3"

    using a3 bnullable0(1) rewrite_bmkeps_aux(1) by blast 

  then show "bmkeps r1 = bmkeps r3" using IH by simp

qed

lemma rewrites_to_bsimp: 

  shows "r \<leadsto>* bsimp r"

proof (induction r rule: bsimp.induct)

  case (1 bs1 r1 r2)

  have IH1: "r1 \<leadsto>* bsimp r1" by fact

  have IH2: "r2 \<leadsto>* bsimp r2" by fact

  { assume as: "bsimp r1 = AZERO \<or> bsimp r2 = AZERO"

    with IH1 IH2 have "r1 \<leadsto>* AZERO \<or> r2 \<leadsto>* AZERO" by auto

    then have "ASEQ bs1 r1 r2 \<leadsto>* AZERO"

      by (metis bs2 continuous_rewrite rrewrites.simps star_seq2)  

    then have "ASEQ bs1 r1 r2 \<leadsto>* bsimp (ASEQ bs1 r1 r2)" using as by auto

  }

  moreover

  { assume "\<exists>bs. bsimp r1 = AONE bs"

    then obtain bs where as: "bsimp r1 = AONE bs" by blast

    with IH1 have "r1 \<leadsto>* AONE bs" by simp

    then have "ASEQ bs1 r1 r2 \<leadsto>* fuse (bs1 @ bs) r2" using bs3 star_seq by blast

    with IH2 have "ASEQ bs1 r1 r2 \<leadsto>* fuse (bs1 @ bs) (bsimp r2)"

      using rewrites_fuse by (meson rrewrites_trans) 

    then have "ASEQ bs1 r1 r2 \<leadsto>* bsimp (ASEQ bs1 (AONE bs) r2)" by simp

    then have "ASEQ bs1 r1 r2 \<leadsto>* bsimp (ASEQ bs1 r1 r2)" by (simp add: as) 

  } 

  moreover

  { assume as1: "bsimp r1 \<noteq> AZERO" "bsimp r2 \<noteq> AZERO" and as2: "(\<nexists>bs. bsimp r1 = AONE bs)" 

    then have "bsimp_ASEQ bs1 (bsimp r1) (bsimp r2) = ASEQ bs1 (bsimp r1) (bsimp r2)" 

      by (simp add: bsimp_ASEQ1) 

    then have "ASEQ bs1 r1 r2 \<leadsto>* bsimp_ASEQ bs1 (bsimp r1) (bsimp r2)" using as1 as2 IH1 IH2

      by (metis rrewrites_trans star_seq star_seq2) 

    then have "ASEQ bs1 r1 r2 \<leadsto>* bsimp (ASEQ bs1 r1 r2)" by simp

  } 

  ultimately show "ASEQ bs1 r1 r2 \<leadsto>* bsimp (ASEQ bs1 r1 r2)" by blast

next

  case (2 bs1 rs)

  have IH: "\<And>x. x \<in> set rs \<Longrightarrow> x \<leadsto>* bsimp x" by fact

  then have "rs s\<leadsto>* (map bsimp rs)" by (simp add: trivialbsimp_srewrites)

  also have "... s\<leadsto>* flts (map bsimp rs)" by (simp add: fltsfrewrites) 

  also have "... s\<leadsto>* distinctWith (flts (map bsimp rs)) eq1 {}" by (simp add: ss6_stronger)

  finally have "AALTs bs1 rs \<leadsto>* AALTs bs1 (distinctWith (flts (map bsimp rs)) eq1 {})"

    using contextrewrites0 by auto

  also have "... \<leadsto>* bsimp_AALTs  bs1 (distinctWith (flts (map bsimp rs)) eq1 {})"

    by (simp add: bsimp_AALTs_rewrites)     

  finally show "AALTs bs1 rs \<leadsto>* bsimp (AALTs bs1 rs)" by simp

qed (simp_all)

lemma to_zero_in_alt: 

  shows "AALT bs (ASEQ [] AZERO r) r2 \<leadsto> AALT bs AZERO r2"

  by (simp add: bs1 bs10 ss3)

lemma  bder_fuse_list: 

  shows "map (bder c \<circ> fuse bs1) rs1 = map (fuse bs1 \<circ> bder c) rs1"

  apply(induction rs1)

  apply(simp_all add: bder_fuse)

  done

lemma map1:

  shows "(map f [a]) = [f a]"

  by (simp)

lemma rewrite_preserves_bder: 

  shows "r1 \<leadsto> r2 \<Longrightarrow> (bder c r1) \<leadsto>* (bder c r2)"

  and   "rs1 s\<leadsto> rs2 \<Longrightarrow> map (bder c) rs1 s\<leadsto>* map (bder c) rs2"

proof(induction rule: rrewrite_srewrite.inducts)

  case (bs1 bs r2)

  then show ?case

    by (simp add: continuous_rewrite) 

next

  case (bs2 bs r1)

  then show ?case 

    apply(auto)

    apply (meson bs6 contextrewrites0 rrewrite_srewrite.bs2 rs2 ss3 ss4 sss1 sss2)

    by (simp add: r_in_rstar rrewrite_srewrite.bs2)

next

  case (bs3 bs1 bs2 r)

  then show ?case 

    apply(simp)

    by (metis (no_types, lifting) bder_fuse bs10 bs7 fuse_append rrewrites.simps ss4 to_zero_in_alt)

next

  case (bs4 r1 r2 bs r3)

  have as: "r1 \<leadsto> r2" by fact

  have IH: "bder c r1 \<leadsto>* bder c r2" by fact