theory BitCodedlotschris
  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 (CR 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) = CR c"
| "erase (AALTs _ []) = ZERO"
| "erase (AALTs _ [r]) = (erase r)"
| "erase (AALTs bs (r#rs)) = ALT (erase r) (erase (AALTs bs rs))"
| "erase (ASEQ _ r1 r2) = SEQ (erase r1) (erase r2)"
| "erase (ASTAR _ r) = STAR (erase r)"

lemma decode_code_erase:
  assumes "\<Turnstile> v : (erase  a)"
  shows "decode (code v) (erase a) = Some v"
  using assms
  by (simp add: decode_code)

fun nonalt :: "arexp \<Rightarrow> bool"
  where
  "nonalt (AALTs bs2 rs) = False"
| "nonalt r = True"

fun good :: "arexp \<Rightarrow> bool" where
  "good AZERO = False"
| "good (AONE cs) = True"
| "good (ACHAR cs c) = True"
| "good (AALTs cs []) = False"
| "good (AALTs cs [r]) = False"
| "good (AALTs cs (r1#r2#rs)) = (\<forall>r' \<in> set (r1#r2#rs). good r' \<and> nonalt r')"
| "good (ASEQ _ AZERO _) = False"
| "good (ASEQ _ (AONE _) _) = False"
| "good (ASEQ _ _ AZERO) = False"
| "good (ASEQ cs r1 r2) = (good r1 \<and> good r2)"
| "good (ASTAR cs r) = True"

fun fuse :: "bit list \<Rightarrow> arexp \<Rightarrow> arexp" where
  "fuse bs AZERO = AZERO"
| "fuse bs (AONE cs) = AONE (bs @ cs)"
| "fuse bs (ACHAR cs c) = ACHAR (bs @ cs) c"
| "fuse bs (AALTs cs rs) = AALTs (bs @ cs) rs"
| "fuse bs (ASEQ cs r1 r2) = ASEQ (bs @ cs) r1 r2"
| "fuse bs (ASTAR cs r) = ASTAR (bs @ cs) r"

lemma fuse_append:
  shows "fuse (bs1 @ bs2) r = fuse bs1 (fuse bs2 r)"
  apply(induct r)
  apply(auto)
  done

fun intern :: "rexp \<Rightarrow> arexp" where
  "intern ZERO = AZERO"
| "intern ONE = AONE []"
| "intern (CR 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 r:
  assumes "bnullable (AALTs bs (a # rs))"
  shows "bnullable a \<or> (\<not> bnullable a \<and> bnullable (AALTs bs rs))"
  using assms
  apply(induct rs)
   apply(auto)
  done

lemma r0:
  assumes "bnullable a"
  shows  "bmkeps (AALTs bs (a # rs)) = bs @ (bmkeps a)"
  using assms
  by (metis bmkeps.simps(3) bmkeps.simps(4) list.exhaust)

lemma r1:
  assumes "\<not> bnullable a" "bnullable (AALTs bs rs)"
  shows  "bmkeps (AALTs bs (a # rs)) = bmkeps (AALTs bs rs)"
  using assms
  apply(induct rs)
   apply(auto)
  done

lemma r2:
  assumes "x \<in> set rs" "bnullable x"
  shows "bnullable (AALTs bs rs)"
  using assms
  apply(induct rs)
   apply(auto)
  done

lemma  r3:
  assumes "\<not> bnullable r"
          " \<exists> x \<in> set rs. bnullable x"
  shows "retrieve (AALTs bs rs) (mkeps (erase (AALTs bs rs))) =
         retrieve (AALTs bs (r # rs)) (mkeps (erase (AALTs bs (r # rs))))"
  using assms
  apply(induct rs arbitrary: r bs)
   apply(auto)[1]
  apply(auto)
  using bnullable_correctness apply blast
    apply(auto simp add: bnullable_correctness mkeps_nullable retrieve_fuse2)
   apply(subst retrieve_fuse2[symmetric])
  apply (smt bnullable.simps(4) bnullable_correctness erase.simps(5) erase.simps(6) insert_iff list.exhaust list.set(2) mkeps.simps(3) mkeps_nullable)
   apply(simp)
  apply(case_tac "bnullable a")
  apply (smt append_Nil2 bnullable.simps(4) bnullable_correctness erase.simps(5) erase.simps(6) fuse.simps(4) insert_iff list.exhaust list.set(2) mkeps.simps(3) mkeps_nullable retrieve_fuse2)
  apply(drule_tac x="a" in meta_spec)
  apply(drule_tac x="bs" in meta_spec)
  apply(drule meta_mp)
   apply(simp)
  apply(drule meta_mp)
   apply(auto)
  apply(subst retrieve_fuse2[symmetric])
  apply(case_tac rs)
    apply(simp)
   apply(auto)[1]
      apply (simp add: bnullable_correctness)
  apply (metis append_Nil2 bnullable_correctness erase_fuse fuse.simps(4) list.set_intros(1) mkeps.simps(3) mkeps_nullable nullable.simps(4) r2)
    apply (simp add: bnullable_correctness)
  apply (metis append_Nil2 bnullable_correctness erase.simps(6) erase_fuse fuse.simps(4) list.set_intros(2) mkeps.simps(3) mkeps_nullable r2)
  apply(simp)
  done

lemma t:
  assumes "\<forall>r \<in> set rs. nullable (erase r) \<longrightarrow> bmkeps r = retrieve r (mkeps (erase r))"
          "nullable (erase (AALTs bs rs))"
  shows " bmkeps (AALTs bs rs) = retrieve (AALTs bs rs) (mkeps (erase (AALTs bs rs)))"
  using assms
  apply(induct rs arbitrary: bs)
   apply(simp)
  apply(auto simp add: bnullable_correctness)
   apply(case_tac rs)
     apply(auto simp add: bnullable_correctness)[2]
   apply(subst r1)
     apply(simp)
    apply(rule r2)
     apply(assumption)
    apply(simp)
   apply(drule_tac x="bs" in meta_spec)
   apply(drule meta_mp)
    apply(auto)[1]
   prefer 2
  apply(case_tac "bnullable a")
    apply(subst r0)
     apply blast
    apply(subgoal_tac "nullable (erase a)")
  prefer 2
  using bnullable_correctness apply blast
  apply (metis (no_types, lifting) erase.simps(5) erase.simps(6) list.exhaust mkeps.simps(3) retrieve.simps(3) retrieve.simps(4))
  apply(subst r1)
     apply(simp)
  using r2 apply blast
  apply(drule_tac x="bs" in meta_spec)
   apply(drule meta_mp)
    apply(auto)[1]
   apply(simp)
  using r3 apply blast
  apply(auto)
  using r3 by blast

lemma bmkeps_retrieve:
  assumes "nullable (erase r)"
  shows "bmkeps r = retrieve r (mkeps (erase r))"
  using assms
  apply(induct r)
         apply(simp)
        apply(simp)
       apply(simp)
    apply(simp)
   defer
   apply(simp)
  apply(rule t)
   apply(auto)
  done

lemma bder_retrieve:
  assumes "\<Turnstile> v : der c (erase r)"
  shows "retrieve (bder c r) v = retrieve r (injval (erase r) c v)"
  using assms
  apply(induct r arbitrary: v rule: erase.induct)
         apply(simp)
         apply(erule Prf_elims)
        apply(simp)
        apply(erule Prf_elims)
        apply(simp)
      apply(case_tac "c = ca")
       apply(simp)
       apply(erule Prf_elims)
       apply(simp)
      apply(simp)
       apply(erule Prf_elims)
  apply(simp)
      apply(erule Prf_elims)
     apply(simp)
    apply(simp)
  apply(rename_tac "r\<^sub>1" "r\<^sub>2" rs v)
    apply(erule Prf_elims)
     apply(simp)
    apply(simp)
    apply(case_tac rs)
     apply(simp)
    apply(simp)
  apply (smt Prf_elims(3) injval.simps(2) injval.simps(3) retrieve.simps(4) retrieve.simps(5) same_append_eq)
   apply(simp)
   apply(case_tac "nullable (erase r1)")
    apply(simp)
  apply(erule Prf_elims)
     apply(subgoal_tac "bnullable r1")
  prefer 2
  using bnullable_correctness apply blast
    apply(simp)
     apply(erule Prf_elims)
     apply(simp)
   apply(subgoal_tac "bnullable r1")
  prefer 2
  using bnullable_correctness apply blast
    apply(simp)
    apply(simp add: retrieve_fuse2)
    apply(simp add: bmkeps_retrieve)
   apply(simp)
   apply(erule Prf_elims)
   apply(simp)
  using bnullable_correctness apply blast
  apply(rename_tac bs r v)
  apply(simp)
  apply(erule Prf_elims)
     apply(clarify)
  apply(erule Prf_elims)
  apply(clarify)
  apply(subst injval.simps)
  apply(simp del: retrieve.simps)
  apply(subst retrieve.simps)
  apply(subst retrieve.simps)
  apply(simp)
  apply(simp add: retrieve_fuse2)
  done

lemma MAIN_decode:
  assumes "\<Turnstile> v : ders s r"
  shows "Some (flex r id s v) = decode (retrieve (bders (intern r) s) v) r"
  using assms
proof (induct s arbitrary: v rule: rev_induct)
  case Nil
  have "\<Turnstile> v : ders [] r" by fact
  then have "\<Turnstile> v : r" by simp
  then have "Some v = decode (retrieve (intern r) v) r"
    using decode_code retrieve_code by auto
  then show "Some (flex r id [] v) = decode (retrieve (bders (intern r) []) v) r"
    by simp
next
  case (snoc c s v)
  have IH: "\<And>v. \<Turnstile> v : ders s r \<Longrightarrow>
     Some (flex r id s v) = decode (retrieve (bders (intern r) s) v) r" by fact
  have asm: "\<Turnstile> v : ders (s @ [c]) r" by fact
  then have asm2: "\<Turnstile> injval (ders s r) c v : ders s r"
    by (simp add: Prf_injval ders_append)
  have "Some (flex r id (s @ [c]) v) = Some (flex r id s (injval (ders s r) c v))"
    by (simp add: flex_append)
  also have "... = decode (retrieve (bders (intern r) s) (injval (ders s r) c v)) r"
    using asm2 IH by simp
  also have "... = decode (retrieve (bder c (bders (intern r) s)) v) r"
    using asm by (simp_all add: bder_retrieve ders_append)
  finally show "Some (flex r id (s @ [c]) v) =
                 decode (retrieve (bders (intern r) (s @ [c])) v) r" by (simp add: bders_append)
qed

definition blex where
 "blex a s \<equiv> if bnullable (bders a s) then Some (bmkeps (bders a s)) else None"

definition blexer where
 "blexer r s \<equiv> if bnullable (bders (intern r) s) then
                decode (bmkeps (bders (intern r) s)) r else None"

lemma blexer_correctness:
  shows "blexer r s = lexer r s"
proof -
  { define bds where "bds \<equiv> bders (intern r) s"
    define ds  where "ds \<equiv> ders s r"
    assume asm: "nullable ds"
    have era: "erase bds = ds"
      unfolding ds_def bds_def by simp
    have mke: "\<Turnstile> mkeps ds : ds"
      using asm by (simp add: mkeps_nullable)
    have "decode (bmkeps bds) r = decode (retrieve bds (mkeps ds)) r"
      using bmkeps_retrieve
      using asm era by (simp add: bmkeps_retrieve)
    also have "... =  Some (flex r id s (mkeps ds))"
      using mke by (simp_all add: MAIN_decode ds_def bds_def)
    finally have "decode (bmkeps bds) r = Some (flex r id s (mkeps ds))"
      unfolding bds_def ds_def .
  }
  then show "blexer r s = lexer r s"
    unfolding blexer_def lexer_flex
    apply(subst bnullable_correctness[symmetric])
    apply(simp)
    done
qed

fun distinctBy :: "'a list \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b set \<Rightarrow> 'a list"
  where
  "distinctBy [] f acc = []"
| "distinctBy (x#xs) f acc =
     (if (f x) \<in> acc then distinctBy xs f acc
      else x # (distinctBy xs f ({f x} \<union> acc)))"

fun flts :: "arexp list \<Rightarrow> arexp list"
  where
  "flts [] = []"
| "flts (AZERO # rs) = flts rs"
| "flts ((AALTs bs  rs1) # rs) = (map (fuse bs) rs1) @ flts rs"
| "flts (r1 # rs) = r1 # flts rs"

fun li :: "bit list \<Rightarrow> arexp list \<Rightarrow> arexp"
  where
  "li _ [] = AZERO"
| "li bs [a] = fuse bs a"
| "li bs as = AALTs bs as"

fun bsimp_ASEQ :: "bit list \<Rightarrow> arexp \<Rightarrow> arexp \<Rightarrow> arexp"
  where
  "bsimp_ASEQ _ AZERO _ = AZERO"
| "bsimp_ASEQ _ _ AZERO = AZERO"
| "bsimp_ASEQ bs1 (AONE bs2) r2 = fuse (bs1 @ bs2) r2"
| "bsimp_ASEQ bs1 r1 r2 = ASEQ  bs1 r1 r2"

fun bsimp_AALTs :: "bit list \<Rightarrow> arexp list \<Rightarrow> arexp"
  where
  "bsimp_AALTs _ [] = AZERO"
| "bsimp_AALTs bs1 [r] = fuse bs1 r"
| "bsimp_AALTs bs1 rs = AALTs bs1 rs"

fun bsimp :: "arexp \<Rightarrow> arexp"
  where
  "bsimp (ASEQ bs1 r1 r2) = bsimp_ASEQ bs1 (bsimp r1) (bsimp r2)"
| "bsimp (AALTs bs1 rs) = bsimp_AALTs bs1 (flts (map bsimp rs))"
| "bsimp r = r"

fun
  bders_simp :: "arexp \<Rightarrow> string \<Rightarrow> arexp"
where
  "bders_simp r [] = r"
| "bders_simp r (c # s) = bders_simp (bsimp (bder c r)) s"

definition blexer_simp where
 "blexer_simp r s \<equiv> if bnullable (bders_simp (intern r) s) then
                decode (bmkeps (bders_simp (intern r) s)) r else None"

lemma asize0:
  shows "0 < asize r"
  apply(induct  r)
       apply(auto)
  done

lemma bders_simp_append:
  shows "bders_simp r (s1 @ s2) = bders_simp (bders_simp r s1) s2"
  apply(induct s1 arbitrary: r s2)
   apply(simp)
  apply(simp)
  done

lemma bsimp_ASEQ_size:
  shows "asize (bsimp_ASEQ bs r1 r2) \<le> Suc (asize r1 + asize r2)"
  apply(induct bs r1 r2 rule: bsimp_ASEQ.induct)
  apply(auto)
  done

lemma fuse_size:
  shows "asize (fuse bs r) = asize r"
  apply(induct r)
  apply(auto)
  done

lemma flts_size:
  shows "sum_list (map asize (flts rs)) \<le> sum_list (map asize rs)"
  apply(induct rs rule: flts.induct)
        apply(simp_all)
  by (metis (mono_tags, lifting) add_mono comp_apply eq_imp_le fuse_size le_SucI map_eq_conv)

lemma bsimp_AALTs_size:
  shows "asize (bsimp_AALTs bs rs) \<le> Suc (sum_list (map asize rs))"
  apply(induct rs rule: bsimp_AALTs.induct)
  apply(auto simp add: fuse_size)
  done

lemma bsimp_size:
  shows "asize (bsimp r) \<le> asize r"
  apply(induct r)
       apply(simp_all)
   apply (meson Suc_le_mono add_mono_thms_linordered_semiring(1) bsimp_ASEQ_size le_trans)
  apply(rule le_trans)
   apply(rule bsimp_AALTs_size)
  apply(simp)
   apply(rule le_trans)
   apply(rule flts_size)
  by (simp add: sum_list_mono)

lemma bsimp_asize0:
  shows "(\<Sum>x\<leftarrow>rs. asize (bsimp x)) \<le> sum_list (map asize rs)"
  apply(induct rs)
   apply(auto)
  by (simp add: add_mono bsimp_size)

lemma bsimp_AALTs_size2:
  assumes "\<forall>r \<in> set  rs. nonalt r"
  shows "asize (bsimp_AALTs bs rs) \<ge> sum_list (map asize rs)"
  using assms
  apply(induct rs rule: bsimp_AALTs.induct)
    apply(simp_all add: fuse_size)
  done

lemma qq:
  shows "map (asize \<circ> fuse bs) rs = map asize rs"
  apply(induct rs)
   apply(auto simp add: fuse_size)
  done

lemma flts_size2:
  assumes "\<exists>bs rs'. AALTs bs  rs' \<in> set rs"
  shows "sum_list (map asize (flts rs)) < sum_list (map asize rs)"
  using assms
  apply(induct rs)
   apply(auto simp add: qq)
   apply (simp add: flts_size less_Suc_eq_le)
  apply(case_tac a)
       apply(auto simp add: qq)
   prefer 2
   apply (simp add: flts_size le_imp_less_Suc)
  using less_Suc_eq by auto

lemma bsimp_AALTs_size3:
  assumes "\<exists>r \<in> set  (map bsimp rs). \<not>nonalt r"
  shows "asize (bsimp (AALTs bs rs)) < asize (AALTs bs rs)"
  using assms flts_size2
  apply  -
  apply(clarify)
  apply(simp)
  apply(drule_tac x="map bsimp rs" in meta_spec)
  apply(drule meta_mp)
  apply (metis list.set_map nonalt.elims(3))
  apply(simp)
  apply(rule order_class.order.strict_trans1)
   apply(rule bsimp_AALTs_size)
  apply(simp)
  by (smt Suc_leI bsimp_asize0 comp_def le_imp_less_Suc le_trans map_eq_conv not_less_eq)

lemma L_bsimp_ASEQ:
  "L (SEQ (erase r1) (erase r2)) = L (erase (bsimp_ASEQ bs r1 r2))"
  apply(induct bs r1 r2 rule: bsimp_ASEQ.induct)
  apply(simp_all)
  by (metis erase_fuse fuse.simps(4))

lemma L_bsimp_AALTs:
  "L (erase (AALTs bs rs)) = L (erase (bsimp_AALTs bs rs))"
  apply(induct bs rs rule: bsimp_AALTs.induct)
  apply(simp_all add: erase_fuse)
  done

lemma L_erase_AALTs:
  shows "L (erase (AALTs bs rs)) = \<Union> (L ` erase ` (set rs))"
  apply(induct rs)
   apply(simp)
  apply(simp)
  apply(case_tac rs)
   apply(simp)
  apply(simp)
  done

lemma L_erase_flts:
  shows "\<Union> (L ` erase ` (set (flts rs))) = \<Union> (L ` erase ` (set rs))"
  apply(induct rs rule: flts.induct)
        apply(simp_all)
  apply(auto)
  using L_erase_AALTs erase_fuse apply auto[1]
  by (simp add: L_erase_AALTs erase_fuse)

lemma L_bsimp_erase:
  shows "L (erase r) = L (erase (bsimp r))"
  apply(induct r)
  apply(simp)
  apply(simp)
  apply(simp)
  apply(auto simp add: Sequ_def)[1]
  apply(subst L_bsimp_ASEQ[symmetric])
  apply(auto simp add: Sequ_def)[1]
  apply(subst (asm)  L_bsimp_ASEQ[symmetric])
  apply(auto simp add: Sequ_def)[1]
   apply(simp)
   apply(subst L_bsimp_AALTs[symmetric])
   defer
   apply(simp)
  apply(subst (2)L_erase_AALTs)
  apply(subst L_erase_flts)
  apply(auto)
   apply (simp add: L_erase_AALTs)
  using L_erase_AALTs by blast

lemma bsimp_ASEQ0:
  shows "bsimp_ASEQ bs r1 AZERO = AZERO"
  apply(induct r1)
  apply(auto)
  done

lemma bsimp_ASEQ1:
  assumes "r1 \<noteq> AZERO" "r2 \<noteq> AZERO" "\<forall>bs. r1 \<noteq> AONE bs"
  shows "bsimp_ASEQ bs r1 r2 = ASEQ bs r1 r2"
  using assms
  apply(induct bs r1 r2 rule: bsimp_ASEQ.induct)
  apply(auto)
  done

lemma bsimp_ASEQ2:
  shows "bsimp_ASEQ bs (AONE bs1) r2 = fuse (bs @ bs1) r2"
  apply(induct r2)
  apply(auto)
  done

lemma L_bders_simp:
  shows "L (erase (bders_simp r s)) = L (erase (bders r s))"
  apply(induct s arbitrary: r rule: rev_induct)
   apply(simp)
  apply(simp)
  apply(simp add: ders_append)
  apply(simp add: bders_simp_append)
  apply(simp add: L_bsimp_erase[symmetric])
  by (simp add: der_correctness)

lemma b1:
  "bsimp_ASEQ bs1 (AONE bs) r =  fuse (bs1 @ bs) r"
  apply(induct r)
       apply(auto)
  done

lemma b2:
  assumes "bnullable r"
  shows "bmkeps (fuse bs r) = bs @ bmkeps r"
  by (simp add: assms bmkeps_retrieve bnullable_correctness erase_fuse mkeps_nullable retrieve_fuse2)

lemma b3:
  shows "bnullable r = bnullable (bsimp r)"
  using L_bsimp_erase bnullable_correctness nullable_correctness by auto

lemma b4:
  shows "bnullable (bders_simp r s) = bnullable (bders r s)"
  by (metis L_bders_simp bnullable_correctness lexer.simps(1) lexer_correct_None option.distinct(1))

lemma q1:
  assumes "\<forall>r \<in> set rs. bmkeps(bsimp r) = bmkeps r"
  shows "map (\<lambda>r. bmkeps(bsimp r)) rs = map bmkeps rs"
  using assms
  apply(induct rs)
  apply(simp)
  apply(simp)
  done

lemma q3:
  assumes "\<exists>r \<in> set rs. bnullable r"
  shows "bmkeps (AALTs bs rs) = bmkeps (bsimp_AALTs bs rs)"
  using assms
  apply(induct bs rs rule: bsimp_AALTs.induct)
    apply(simp)
   apply(simp)
  apply (simp add: b2)
  apply(simp)
  done

lemma qq1:
  assumes "\<exists>r \<in> set rs. bnullable r"
  shows "bmkeps (AALTs bs (rs @ rs1)) = bmkeps (AALTs bs rs)"
  using assms
  apply(induct rs arbitrary: rs1 bs)
  apply(simp)
  apply(simp)
  by (metis Nil_is_append_conv bmkeps.simps(4) neq_Nil_conv r0 split_list_last)

lemma qq2:
  assumes "\<forall>r \<in> set rs. \<not> bnullable r" "\<exists>r \<in> set rs1. bnullable r"
  shows "bmkeps (AALTs bs (rs @ rs1)) = bmkeps (AALTs bs rs1)"
  using assms
  apply(induct rs arbitrary: rs1 bs)
  apply(simp)
  apply(simp)
  by (metis append_assoc in_set_conv_decomp r1 r2)

lemma qq3:
  shows "bnullable (AALTs bs rs) = (\<exists>r \<in> set rs. bnullable r)"
  apply(induct rs arbitrary: bs)
  apply(simp)
  apply(simp)
  done

lemma fuse_empty:
  shows "fuse [] r = r"
  apply(induct r)
       apply(auto)
  done

lemma flts_fuse:
  shows "map (fuse bs) (flts rs) = flts (map (fuse bs) rs)"
  apply(induct rs arbitrary: bs rule: flts.induct)
        apply(auto simp add: fuse_append)
  done

lemma bsimp_ASEQ_fuse:
  shows "fuse bs1 (bsimp_ASEQ bs2 r1 r2) = bsimp_ASEQ (bs1 @ bs2) r1 r2"
  apply(induct r1 r2 arbitrary: bs1 bs2 rule: bsimp_ASEQ.induct)
  apply(auto)
  done

lemma bsimp_AALTs_fuse:
  assumes "\<forall>r \<in> set rs. fuse bs1 (fuse bs2 r) = fuse (bs1 @ bs2) r"
  shows "fuse bs1 (bsimp_AALTs bs2 rs) = bsimp_AALTs (bs1 @ bs2) rs"
  using assms
  apply(induct bs2 rs arbitrary: bs1 rule: bsimp_AALTs.induct)
  apply(auto)
  done

lemma bsimp_fuse:
  shows "fuse bs (bsimp r) = bsimp (fuse bs r)"
apply(induct r arbitrary: bs)
       apply(simp)
      apply(simp)
     apply(simp)
    prefer 3
    apply(simp)
   apply(simp)
   apply (simp add: bsimp_ASEQ_fuse)
  apply(simp)
  by (simp add: bsimp_AALTs_fuse fuse_append)

lemma bsimp_fuse_AALTs:
  shows "fuse bs (bsimp (AALTs [] rs)) = bsimp (AALTs bs rs)"
  apply(subst bsimp_fuse)
  apply(simp)
  done

lemma bsimp_fuse_AALTs2:
  shows "fuse bs (bsimp_AALTs [] rs) = bsimp_AALTs bs rs"
  using bsimp_AALTs_fuse fuse_append by auto

lemma bsimp_ASEQ_idem:
  assumes "bsimp (bsimp r1) = bsimp r1" "bsimp (bsimp r2) = bsimp r2"
  shows "bsimp (bsimp_ASEQ x1 (bsimp r1) (bsimp r2)) = bsimp_ASEQ x1 (bsimp r1) (bsimp r2)"
  using assms
  apply(case_tac "bsimp r1 = AZERO")
    apply(simp)
 apply(case_tac "bsimp r2 = AZERO")
    apply(simp)
  apply (metis bnullable.elims(2) bnullable.elims(3) bsimp.simps(3) bsimp_ASEQ.simps(2) bsimp_ASEQ.simps(3) bsimp_ASEQ.simps(4) bsimp_ASEQ.simps(5) bsimp_ASEQ.simps(6))
  apply(case_tac "\<exists>bs. bsimp r1 = AONE bs")
    apply(auto)[1]
    apply(subst bsimp_ASEQ2)
   apply(subst bsimp_ASEQ2)
  apply (metis assms(2) bsimp_fuse)
      apply(subst bsimp_ASEQ1)
      apply(auto)
  done

fun nonnested :: "arexp \<Rightarrow> bool"
  where
  "nonnested (AALTs bs2 []) = True"
| "nonnested (AALTs bs2 ((AALTs bs1 rs1) # rs2)) = False"
| "nonnested (AALTs bs2 (r # rs2)) = nonnested (AALTs bs2 rs2)"
| "nonnested r = True"

lemma  k0:
  shows "flts (r # rs1) = flts [r] @ flts rs1"
  apply(induct r arbitrary: rs1)
   apply(auto)
  done

lemma  k00:
  shows "flts (rs1 @ rs2) = flts rs1 @ flts rs2"
  apply(induct rs1 arbitrary: rs2)
   apply(auto)
  by (metis append.assoc k0)

lemma  k0a:
  shows "flts [AALTs bs rs] = map (fuse bs)  rs"
  apply(simp)
  done

lemma  k0b:
  assumes "nonalt r" "r \<noteq> AZERO"
  shows "flts [r] = [r]"
  using assms
  apply(case_tac  r)
  apply(simp_all)
  done

lemma nn1:
  assumes "nonnested (AALTs bs rs)"
  shows "\<nexists>bs1 rs1. flts rs = [AALTs bs1 rs1]"
  using assms
  apply(induct rs rule: flts.induct)
  apply(auto)
  done

lemma nn1q:
  assumes "nonnested (AALTs bs rs)"
  shows "\<nexists>bs1 rs1. AALTs bs1 rs1 \<in> set (flts rs)"
  using assms
  apply(induct rs rule: flts.induct)
  apply(auto)
  done

lemma nn1qq:
  assumes "nonnested (AALTs bs rs)"
  shows "\<nexists>bs1 rs1. AALTs bs1 rs1 \<in> set rs"
  using assms
  apply(induct rs rule: flts.induct)
  apply(auto)
  done

lemma nn10:
  assumes "nonnested (AALTs cs rs)"
  shows "nonnested (AALTs (bs @ cs) rs)"
  using assms
  apply(induct rs arbitrary: cs bs)
   apply(simp_all)
  apply(case_tac a)
       apply(simp_all)
  done

lemma nn11a:
  assumes "nonalt r"
  shows "nonalt (fuse bs r)"
  using assms
  apply(induct r)
       apply(auto)
  done

lemma nn1a:
  assumes "nonnested r"
  shows "nonnested (fuse bs r)"
  using assms
  apply(induct bs r arbitrary: rule: fuse.induct)
       apply(simp_all add: nn10)
  done

lemma n0:
  shows "nonnested (AALTs bs rs) \<longleftrightarrow> (\<forall>r \<in> set rs. nonalt r)"
  apply(induct rs  arbitrary: bs)
   apply(auto)
    apply (metis list.set_intros(1) nn1qq nonalt.elims(3))
   apply (metis list.set_intros(2) nn1qq nonalt.elims(3))
  by (metis nonalt.elims(2) nonnested.simps(3) nonnested.simps(4) nonnested.simps(5) nonnested.simps(6) nonnested.simps(7))

lemma nn1c:
  assumes "\<forall>r \<in> set rs. nonnested r"
  shows "\<forall>r \<in> set (flts rs). nonalt r"
  using assms
  apply(induct rs rule: flts.induct)
        apply(auto)
  apply(rule nn11a)
  by (metis nn1qq nonalt.elims(3))

lemma nn1bb:
  assumes "\<forall>r \<in> set rs. nonalt r"
  shows "nonnested (bsimp_AALTs bs rs)"
  using assms
  apply(induct bs rs rule: bsimp_AALTs.induct)
    apply(auto)
   apply (metis nn11a nonalt.simps(1) nonnested.elims(3))
  using n0 by auto

lemma nn1b:
  shows "nonnested (bsimp r)"
  apply(induct r)
       apply(simp_all)
  apply(case_tac "bsimp r1 = AZERO")
    apply(simp)
 apply(case_tac "bsimp r2 = AZERO")
   apply(simp)
    apply(subst bsimp_ASEQ0)
  apply(simp)
  apply(case_tac "\<exists>bs. bsimp r1 = AONE bs")
    apply(auto)[1]
    apply(subst bsimp_ASEQ2)
  apply (simp add: nn1a)
   apply(subst bsimp_ASEQ1)
      apply(auto)
  apply(rule nn1bb)
  apply(auto)
  by (metis (mono_tags, hide_lams) imageE nn1c set_map)

lemma nn1d:
  assumes "bsimp r = AALTs bs rs"
  shows "\<forall>r1 \<in> set rs. \<forall>  bs. r1 \<noteq> AALTs bs  rs2"
  using nn1b assms
  by (metis nn1qq)

lemma nn_flts:
  assumes "nonnested (AALTs bs rs)"
  shows "\<forall>r \<in>  set (flts rs). nonalt r"
  using assms
  apply(induct rs arbitrary: bs rule: flts.induct)
        apply(auto)
  done

lemma rt:
  shows "sum_list (map asize (flts (map bsimp rs))) \<le> sum_list (map asize rs)"
  apply(induct rs)
   apply(simp)
  apply(simp)
  apply(subst  k0)
  apply(simp)
  by (smt add_le_cancel_right add_mono bsimp_size flts.simps(1) flts_size k0 le_iff_add list.simps(9) map_append sum_list.Cons sum_list.append trans_le_add1)

lemma bsimp_AALTs_qq:
  assumes "1 < length rs"
  shows "bsimp_AALTs bs rs = AALTs bs  rs"
  using  assms
  apply(case_tac rs)
   apply(simp)
  apply(case_tac list)
   apply(simp_all)
  done

lemma bsimp_AALTs1:
  assumes "nonalt r"
  shows "bsimp_AALTs bs (flts [r]) = fuse bs r"
  using  assms
  apply(case_tac r)
   apply(simp_all)
  done

lemma bbbbs:
  assumes "good r" "r = AALTs bs1 rs"
  shows "bsimp_AALTs bs (flts [r]) = AALTs bs (map (fuse bs1) rs)"
  using  assms
  by (metis (no_types, lifting) Nil_is_map_conv append.left_neutral append_butlast_last_id bsimp_AALTs.elims butlast.simps(2) good.simps(4) good.simps(5) k0a map_butlast)

lemma bbbbs1:
  shows "nonalt r \<or> (\<exists>bs rs. r  = AALTs bs rs)"
  using nonalt.elims(3) by auto

lemma good_fuse:
  shows "good (fuse bs r) = good r"
  apply(induct r arbitrary: bs)
       apply(auto)
     apply(case_tac r1)
          apply(simp_all)
  apply(case_tac r2)
          apply(simp_all)
  apply(case_tac r2)
            apply(simp_all)
  apply(case_tac r2)
           apply(simp_all)
  apply(case_tac r2)
          apply(simp_all)
  apply(case_tac r1)
          apply(simp_all)
  apply(case_tac r2)
           apply(simp_all)
  apply(case_tac r2)
           apply(simp_all)
  apply(case_tac r2)
           apply(simp_all)
  apply(case_tac r2)
         apply(simp_all)
  apply(case_tac x2a)
    apply(simp_all)
  apply(case_tac list)
    apply(simp_all)
  apply(case_tac x2a)
    apply(simp_all)
  apply(case_tac list)
    apply(simp_all)
  done

lemma good0:
  assumes "rs \<noteq> Nil" "\<forall>r \<in> set rs. nonalt r"
  shows "good (bsimp_AALTs bs rs) \<longleftrightarrow> (\<forall>r \<in> set rs. good r)"
  using  assms
  apply(induct bs rs rule: bsimp_AALTs.induct)
  apply(auto simp add: good_fuse)
  done

lemma good0a:
  assumes "flts (map bsimp rs) \<noteq> Nil" "\<forall>r \<in> set (flts (map bsimp rs)). nonalt r"
  shows "good (bsimp (AALTs bs rs)) \<longleftrightarrow> (\<forall>r \<in> set (flts (map bsimp rs)). good r)"
  using  assms
  apply(simp)
  apply(auto)
  apply(subst (asm) good0)
   apply(simp)
    apply(auto)
   apply(subst good0)
   apply(simp)
    apply(auto)
  done

lemma flts0:
  assumes "r \<noteq> AZERO" "nonalt r"
  shows "flts [r] \<noteq> []"
  using  assms
  apply(induct r)
       apply(simp_all)
  done

lemma flts1:
  assumes "good r"
  shows "flts [r] \<noteq> []"
  using  assms
  apply(induct r)
       apply(simp_all)
  apply(case_tac x2a)
   apply(simp)
  apply(simp)
  done

lemma flts2:
  assumes "good r"
  shows "\<forall>r' \<in> set (flts [r]). good r' \<and> nonalt r'"
  using  assms
  apply(induct r)
       apply(simp)
      apply(simp)
     apply(simp)
    prefer 2
    apply(simp)
    apply(auto)[1]
     apply (metis bsimp_AALTs.elims good.simps(4) good.simps(5) good.simps(6) good_fuse)
  apply (metis bsimp_AALTs.elims good.simps(4) good.simps(5) good.simps(6) nn11a)
   apply fastforce
  apply(simp)
  done

lemma flts3:
  assumes "\<forall>r \<in> set rs. good r \<or> r = AZERO"
  shows "\<forall>r \<in> set (flts rs). good r"
  using  assms
  apply(induct rs arbitrary: rule: flts.induct)
        apply(simp_all)
  by (metis UnE flts2 k0a set_map)

lemma flts3b:
  assumes "\<exists>r\<in>set rs. good r"
  shows "flts rs \<noteq> []"
  using  assms
  apply(induct rs arbitrary: rule: flts.induct)
        apply(simp)
       apply(simp)
      apply(simp)
      apply(auto)
  done

lemma flts4:
  assumes "bsimp_AALTs bs (flts rs) = AZERO"
  shows "\<forall>r \<in> set rs. \<not> good r"
  using assms
  apply(induct rs arbitrary: bs rule: flts.induct)
        apply(auto)
        defer
  apply (metis (no_types, lifting) Nil_is_append_conv append_self_conv2 bsimp_AALTs.elims butlast.simps(2) butlast_append flts3b nonalt.simps(1) nonalt.simps(2))
  apply (metis arexp.distinct(7) bsimp_AALTs.elims flts2 good.simps(1) good.simps(2) good0 k0b list.distinct(1) list.inject nonalt.simps(3))
  apply (metis arexp.distinct(3) arexp.distinct(7) bsimp_AALTs.elims fuse.simps(3) list.distinct(1) list.inject)
  apply (metis arexp.distinct(7) bsimp_AALTs.elims good.simps(1) good_fuse list.distinct(1) list.inject)
    apply (metis arexp.distinct(7) bsimp_AALTs.elims list.distinct(1) list.inject)
  apply (metis arexp.distinct(7) bsimp_AALTs.elims flts2 good.simps(1) good.simps(33) good0 k0b list.distinct(1) list.inject nonalt.simps(6))
  by (metis (no_types, lifting) Nil_is_append_conv append_Nil2 arexp.distinct(7) bsimp_AALTs.elims butlast.simps(2) butlast_append flts1 flts2 good.simps(1) good0 k0a)

lemma flts_nil:
  assumes "\<forall>y. asize y < Suc (sum_list (map asize rs)) \<longrightarrow>
            good (bsimp y) \<or> bsimp y = AZERO"
  and "\<forall>r\<in>set rs. \<not> good (bsimp r)"
  shows "flts (map bsimp rs) = []"
  using assms
  apply(induct rs)
   apply(simp)
  apply(simp)
  apply(subst k0)
  apply(simp)
  by force

lemma flts_nil2:
  assumes "\<forall>y. asize y < Suc (sum_list (map asize rs)) \<longrightarrow>
            good (bsimp y) \<or> bsimp y = AZERO"
  and "bsimp_AALTs bs (flts (map bsimp rs)) = AZERO"
  shows "flts (map bsimp rs) = []"
  using assms
  apply(induct rs arbitrary: bs)
   apply(simp)
  apply(simp)
  apply(subst k0)
  apply(simp)
  apply(subst (asm) k0)
  apply(auto)
  apply (metis flts.simps(1) flts.simps(2) flts4 k0 less_add_Suc1 list.set_intros(1))
  by (metis flts.simps(2) flts4 k0 less_add_Suc1 list.set_intros(1))

lemma good_SEQ:
  assumes "r1 \<noteq> AZERO" "r2 \<noteq> AZERO" "\<forall>bs. r1 \<noteq> AONE bs"
  shows "good (ASEQ bs r1 r2) = (good r1 \<and> good r2)"
  using assms
  apply(case_tac r1)
       apply(simp_all)
  apply(case_tac r2)
          apply(simp_all)
  apply(case_tac r2)
         apply(simp_all)
  apply(case_tac r2)
        apply(simp_all)
  apply(case_tac r2)
       apply(simp_all)
  done

lemma good1:
  shows "good (bsimp a) \<or> bsimp a = AZERO"
  apply(induct a taking: asize rule: measure_induct)
  apply(case_tac x)
  apply(simp)
  apply(simp)
  apply(simp)
  prefer 3
    apply(simp)
   prefer 2
  (*  AALTs case  *)
  apply(simp only:)
   apply(case_tac "x52")
    apply(simp)
  thm good0a
   (*  AALTs list at least one - case *)
   apply(simp only: )
  apply(frule_tac x="a" in spec)
   apply(drule mp)
    apply(simp)
   (* either first element is good, or AZERO *)
    apply(erule disjE)
     prefer 2
    apply(simp)
   (* in  the AZERO case, the size  is smaller *)
   apply(drule_tac x="AALTs x51 list" in spec)
   apply(drule mp)
     apply(simp add: asize0)
    apply(subst (asm) bsimp.simps)
  apply(subst (asm) bsimp.simps)
    apply(assumption)
   (* in the good case *)
  apply(frule_tac x="AALTs x51 list" in spec)
   apply(drule mp)
    apply(simp add: asize0)
   apply(erule disjE)
    apply(rule disjI1)
  apply(simp add: good0)
    apply(subst good0)
      apply (metis Nil_is_append_conv flts1 k0)
  apply (metis ex_map_conv list.simps(9) nn1b nn1c)
  apply(simp)
    apply(subst k0)
    apply(simp)
    apply(auto)[1]
  using flts2 apply blast
    apply(subst  (asm) good0)
      prefer 3
      apply(auto)[1]
     apply auto[1]
    apply (metis ex_map_conv nn1b nn1c)
  (* in  the AZERO case *)
   apply(simp)
   apply(frule_tac x="a" in spec)
   apply(drule mp)
  apply(simp)
   apply(erule disjE)
    apply(rule disjI1)
    apply(subst good0)
  apply(subst k0)
  using flts1 apply blast
     apply(auto)[1]
  apply (metis (no_types, hide_lams) ex_map_conv list.simps(9) nn1b nn1c)
    apply(auto)[1]
  apply(subst (asm) k0)
  apply(auto)[1]
  using flts2 apply blast
  apply(frule_tac x="AALTs x51 list" in spec)
   apply(drule mp)
     apply(simp add: asize0)
    apply(erule disjE)
     apply(simp)
    apply(simp)
  apply (metis add.left_commute flts_nil2 less_add_Suc1 less_imp_Suc_add list.distinct(1) list.set_cases nat.inject)
   apply(subst (2) k0)
  apply(simp)
  (* SEQ case *)
  apply(simp)
  apply(case_tac "bsimp x42 = AZERO")
    apply(simp)
 apply(case_tac "bsimp x43 = AZERO")
   apply(simp)
    apply(subst (2) bsimp_ASEQ0)
  apply(simp)
  apply(case_tac "\<exists>bs. bsimp x42 = AONE bs")
    apply(auto)[1]
   apply(subst bsimp_ASEQ2)
  using good_fuse apply force
   apply(subst bsimp_ASEQ1)
     apply(auto)
  apply(subst  good_SEQ)
  apply(simp)
    apply(simp)
   apply(simp)
  using less_add_Suc1 less_add_Suc2 by blast

lemma good1a:
  assumes "L(erase a) \<noteq> {}"
  shows "good (bsimp a)"
  using good1 assms
  using L_bsimp_erase by force

lemma flts_append:
  "flts (xs1 @ xs2) = flts xs1 @ flts xs2"
  apply(induct xs1  arbitrary: xs2  rule: rev_induct)
   apply(auto)
  apply(case_tac xs)
   apply(auto)
   apply(case_tac x)
        apply(auto)
  apply(case_tac x)
        apply(auto)
  done

lemma g1:
  assumes "good (bsimp_AALTs bs rs)"
  shows "bsimp_AALTs bs rs = AALTs bs rs \<or> (\<exists>r. rs = [r] \<and> bsimp_AALTs bs [r] = fuse bs r)"
using assms
    apply(induct rs arbitrary: bs)
  apply(simp)
  apply(case_tac rs)
  apply(simp only:)
  apply(simp)
  apply(case_tac  list)
  apply(simp)
  by simp

lemma flts_0:
  assumes "nonnested (AALTs bs  rs)"
  shows "\<forall>r \<in> set (flts rs). r \<noteq> AZERO"
  using assms
  apply(induct rs arbitrary: bs rule: flts.induct)
        apply(simp)
       apply(simp)
      defer
      apply(simp)
     apply(simp)
    apply(simp)
apply(simp)
  apply(rule ballI)
  apply(simp)
  done

lemma flts_0a:
  assumes "nonnested (AALTs bs  rs)"
  shows "AZERO \<notin> set (flts rs)"
  using assms
  using flts_0 by blast

lemma qqq1:
  shows "AZERO \<notin> set (flts (map bsimp rs))"
  by (metis ex_map_conv flts3 good.simps(1) good1)

fun nonazero :: "arexp \<Rightarrow> bool"
  where
  "nonazero AZERO = False"
| "nonazero r = True"

lemma flts_concat:
  shows "flts rs = concat (map (\<lambda>r. flts [r]) rs)"
  apply(induct rs)
   apply(auto)
  apply(subst k0)
  apply(simp)
  done

lemma flts_single1:
  assumes "nonalt r" "nonazero r"
  shows "flts [r] = [r]"
  using assms
  apply(induct r)
  apply(auto)
  done

lemma flts_qq:
  assumes "\<forall>y. asize y < Suc (sum_list (map asize rs)) \<longrightarrow> good y \<longrightarrow> bsimp y = y"
          "\<forall>r'\<in>set rs. good r' \<and> nonalt r'"
  shows "flts (map bsimp rs) = rs"
  using assms
  apply(induct rs)
   apply(simp)
  apply(simp)
  apply(subst k0)
  apply(subgoal_tac "flts [bsimp a] =  [a]")
   prefer 2
   apply(drule_tac x="a" in spec)
   apply(drule mp)
    apply(simp)
   apply(auto)[1]
  using good.simps(1) k0b apply blast
  apply(auto)[1]
  done

lemma test:
  assumes "good r"
  shows "bsimp r = r"
  using assms
  apply(induct r taking: "asize" rule: measure_induct)
  apply(erule good.elims)
  apply(simp_all)
  apply(subst k0)
  apply(subst (2) k0)
                apply(subst flts_qq)
                  apply(auto)[1]
                 apply(auto)[1]
                apply (metis append_Cons append_Nil bsimp_AALTs.simps(3) good.simps(1) k0b)
               apply force+
  apply (metis (no_types, lifting) add_Suc add_Suc_right asize.simps(5) bsimp.simps(1) bsimp_ASEQ.simps(19) less_add_Suc1 less_add_Suc2)
  apply (metis add_Suc add_Suc_right arexp.distinct(5) arexp.distinct(7) asize.simps(4) asize.simps(5) bsimp.simps(1) bsimp.simps(2) bsimp_ASEQ1 good.simps(21) good.simps(8) less_add_Suc1 less_add_Suc2)
         apply force+
  apply (metis (no_types, lifting) add_Suc add_Suc_right arexp.distinct(5) arexp.distinct(7) asize.simps(4) asize.simps(5) bsimp.simps(1) bsimp.simps(2) bsimp_ASEQ1 good.simps(25) good.simps(8) less_add_Suc1 less_add_Suc2)
  apply (metis add_Suc add_Suc_right arexp.distinct(7) asize.simps(4) bsimp.simps(2) bsimp_ASEQ1 good.simps(26) good.simps(8) less_add_Suc1 less_add_Suc2)
    apply force+
  done

lemma test2:
  assumes "good r"
  shows "bsimp r = r"
  using assms
  apply(induct r taking: "asize" rule: measure_induct)
  apply(case_tac x)
       apply(simp_all)
   defer
  (* AALT case *)
   apply(subgoal_tac "1 < length x52")
    prefer 2
    apply(case_tac x52)
     apply(simp)
    apply(simp)
    apply(case_tac list)
     apply(simp)
  apply(simp)
    apply(subst bsimp_AALTs_qq)
    prefer 2
    apply(subst flts_qq)
      apply(auto)[1]
     apply(auto)[1]
   apply(case_tac x52)
     apply(simp)
    apply(simp)
    apply(case_tac list)
     apply(simp)
      apply(simp)
      apply(auto)[1]
  apply (metis (no_types, lifting) bsimp_AALTs.elims good.simps(6) length_Cons length_pos_if_in_set list.size(3) nat_neq_iff)
  apply(simp)
  apply(case_tac x52)
     apply(simp)
    apply(simp)
    apply(case_tac list)
     apply(simp)
   apply(simp)
   apply(subst k0)
   apply(simp)
   apply(subst (2) k0)
   apply(simp)
  apply (simp add: Suc_lessI flts1 one_is_add)
  (* SEQ case *)
  apply(case_tac "bsimp x42 = AZERO")
   apply simp
  apply (metis asize.elims good.simps(10) good.simps(11) good.simps(12) good.simps(2) good.simps(7) good.simps(9) good_SEQ less_add_Suc1)
   apply(case_tac "\<exists>bs'. bsimp x42 = AONE bs'")
   apply(auto)[1]
  defer
  apply(case_tac "bsimp x43 = AZERO")
    apply(simp)
  apply (metis bsimp.elims bsimp.simps(3) good.simps(10) good.simps(11) good.simps(12) good.simps(8) good.simps(9) good_SEQ less_add_Suc2)
  apply(auto)
   apply (subst bsimp_ASEQ1)
      apply(auto)[3]
   apply(auto)[1]
    apply (metis bsimp.simps(3) good.simps(2) good_SEQ less_add_Suc1)
   apply (metis bsimp.simps(3) good.simps(2) good_SEQ less_add_Suc1 less_add_Suc2)
  apply (subst bsimp_ASEQ2)
  apply(drule_tac x="x42" in spec)
  apply(drule mp)
   apply(simp)
  apply(drule mp)
   apply (metis bsimp.elims bsimp.simps(3) good.simps(10) good.simps(11) good.simps(2) good_SEQ)
  apply(simp)
  done

lemma bsimp_idem:
  shows "bsimp (bsimp r) = bsimp r"
  using test good1
  by force

lemma q3a:
  assumes "\<exists>r \<in> set rs. bnullable r"
  shows "bmkeps (AALTs bs (map (fuse bs1) rs)) = bmkeps (AALTs (bs@bs1) rs)"
  using assms
  apply(induct rs arbitrary: bs bs1)
   apply(simp)
  apply(simp)
  apply(auto)
   apply (metis append_assoc b2 bnullable_correctness erase_fuse r0)
  apply(case_tac "bnullable a")
   apply (metis append.assoc b2 bnullable_correctness erase_fuse r0)
  apply(case_tac rs)
  apply(simp)
  apply(simp)
  apply(auto)[1]
   apply (metis bnullable_correctness erase_fuse)+
  done

lemma qq4:
  assumes "\<exists>x\<in>set list. bnullable x"
  shows "\<exists>x\<in>set (flts list). bnullable x"
  using assms
  apply(induct list rule: flts.induct)
        apply(auto)
  by (metis UnCI bnullable_correctness erase_fuse imageI)

lemma qs3:
  assumes "\<exists>r \<in> set rs. bnullable r"
  shows "bmkeps (AALTs bs rs) = bmkeps (AALTs bs (flts rs))"
  using assms
  apply(induct rs arbitrary: bs taking: size rule: measure_induct)
  apply(case_tac x)
  apply(simp)
  apply(simp)
  apply(case_tac a)
       apply(simp)
       apply (simp add: r1)
      apply(simp)
      apply (simp add: r0)
     apply(simp)
     apply(case_tac "flts list")
      apply(simp)
  apply (metis L_erase_AALTs L_erase_flts L_flat_Prf1 L_flat_Prf2 Prf_elims(1) bnullable_correctness erase.simps(4) mkeps_nullable r2)
     apply(simp)
     apply (simp add: r1)
    prefer 3
    apply(simp)
    apply (simp add: r0)
   prefer 2
   apply(simp)
  apply(case_tac "\<exists>x\<in>set x52. bnullable x")
  apply(case_tac "list")
    apply(simp)
    apply (metis b2 fuse.simps(4) q3a r2)
   apply(erule disjE)
    apply(subst qq1)
     apply(auto)[1]
     apply (metis bnullable_correctness erase_fuse)
    apply(simp)
     apply (metis b2 fuse.simps(4) q3a r2)
    apply(simp)
    apply(auto)[1]
     apply(subst qq1)
      apply (metis bnullable_correctness erase_fuse image_eqI set_map)
     apply (metis b2 fuse.simps(4) q3a r2)
  apply(subst qq1)
      apply (metis bnullable_correctness erase_fuse image_eqI set_map)
    apply (metis b2 fuse.simps(4) q3a r2)
   apply(simp)
   apply(subst qq2)
     apply (metis bnullable_correctness erase_fuse imageE set_map)
  prefer 2
  apply(case_tac "list")
     apply(simp)
    apply(simp)
   apply (simp add: qq4)
  apply(simp)
  apply(auto)
   apply(case_tac list)
    apply(simp)
   apply(simp)
   apply (simp add: r0)
  apply(case_tac "bnullable (ASEQ x41 x42 x43)")
   apply(case_tac list)
    apply(simp)
   apply(simp)
   apply (simp add: r0)
  apply(simp)
  using qq4 r1 r2 by auto

lemma k1:
  assumes "\<And>x2aa. \<lbrakk>x2aa \<in> set x2a; bnullable x2aa\<rbrakk> \<Longrightarrow> bmkeps x2aa = bmkeps (bsimp x2aa)"
          "\<exists>x\<in>set x2a. bnullable x"
        shows "bmkeps (AALTs x1 (flts x2a)) = bmkeps (AALTs x1 (flts (map bsimp x2a)))"
  using assms
  apply(induct x2a)
  apply fastforce
  apply(simp)
  apply(subst k0)
  apply(subst (2) k0)
  apply(auto)[1]
  apply (metis b3 k0 list.set_intros(1) qs3 r0)
  by (smt b3 imageI insert_iff k0 list.set(2) qq3 qs3 r0 r1 set_map)

lemma bmkeps_simp:
  assumes "bnullable r"
  shows "bmkeps r = bmkeps (bsimp r)"
  using  assms
  apply(induct r)
       apply(simp)
      apply(simp)
     apply(simp)
    apply(simp)
    prefer 3
  apply(simp)
   apply(case_tac "bsimp r1 = AZERO")
    apply(simp)
    apply(auto)[1]
  apply (metis L_bsimp_erase L_flat_Prf1 L_flat_Prf2 Prf_elims(1) bnullable_correctness erase.simps(1) mkeps_nullable)
 apply(case_tac "bsimp r2 = AZERO")
    apply(simp)
    apply(auto)[1]
  apply (metis L_bsimp_erase L_flat_Prf1 L_flat_Prf2 Prf_elims(1) bnullable_correctness erase.simps(1) mkeps_nullable)
  apply(case_tac "\<exists>bs. bsimp r1 = AONE bs")
    apply(auto)[1]
    apply(subst b1)
    apply(subst b2)
  apply(simp add: b3[symmetric])
    apply(simp)
   apply(subgoal_tac "bsimp_ASEQ x1 (bsimp r1) (bsimp r2) = ASEQ x1 (bsimp r1) (bsimp r2)")
    prefer 2
  apply (smt b3 bnullable.elims(2) bsimp_ASEQ.simps(17) bsimp_ASEQ.simps(19) bsimp_ASEQ.simps(20) bsimp_ASEQ.simps(21) bsimp_ASEQ.simps(22) bsimp_ASEQ.simps(24) bsimp_ASEQ.simps(25) bsimp_ASEQ.simps(26) bsimp_ASEQ.simps(27) bsimp_ASEQ.simps(29) bsimp_ASEQ.simps(30) bsimp_ASEQ.simps(31))
   apply(simp)
  apply(simp)
  thm q3
  apply(subst q3[symmetric])
   apply simp
  using b3 qq4 apply auto[1]
  apply(subst qs3)
   apply simp
  using k1 by blast

thm bmkeps_retrieve bmkeps_simp bder_retrieve

lemma bmkeps_bder_AALTs:
  assumes "\<exists>r \<in> set rs. bnullable (bder c r)"
  shows "bmkeps (bder c (bsimp_AALTs bs rs)) = bmkeps (bsimp_AALTs bs (map (bder c) rs))"
  using assms
  apply(induct rs)
   apply(simp)
  apply(simp)
  apply(auto)
  apply(case_tac rs)
    apply(simp)
  apply (metis (full_types) Prf_injval bder_retrieve bmkeps_retrieve bnullable_correctness erase_bder erase_fuse mkeps_nullable retrieve_fuse2)
   apply(simp)
  apply(case_tac  rs)
   apply(simp_all)
  done

lemma bbs0:
  shows "blexer_simp r [] = blexer r []"
  apply(simp add: blexer_def blexer_simp_def)
  done

lemma bbs1:
  shows "blexer_simp r [c] = blexer r [c]"
  apply(simp add: blexer_def blexer_simp_def)
  apply(auto)
    defer
  using b3 apply auto[1]
  using b3 apply auto[1]
  apply(subst bmkeps_simp[symmetric])
   apply(simp)
  apply(simp)
  done

lemma oo:
  shows "(case (blexer (der c r) s) of None \<Rightarrow> None | Some v \<Rightarrow> Some (injval r c v)) = blexer r (c # s)"
  apply(simp add: blexer_correctness)
  done

lemma bder_fuse:
  shows "bder c (fuse bs a) = fuse bs  (bder c a)"
  apply(induct a arbitrary: bs c)
       apply(simp_all)
  done

fun flts2 :: "char \<Rightarrow> arexp list \<Rightarrow> arexp list"
  where
  "flts2 _ [] = []"
| "flts2 c (AZERO # rs) = flts2 c rs"
| "flts2 c (AONE _ # rs) = flts2 c rs"
| "flts2 c (ACHAR bs d # rs) = (if c = d then (ACHAR bs d # flts2 c rs) else flts2 c rs)"
| "flts2 c ((AALTs bs rs1) # rs) = (map (fuse bs) rs1) @ flts2 c rs"
| "flts2 c (ASEQ bs r1 r2 # rs) = (if (bnullable(r1) \<and> r2 = AZERO) then
    flts2 c rs
    else ASEQ bs r1 r2 # flts2 c rs)"
| "flts2 c (r1 # rs) = r1 # flts2 c rs"

lemma  flts2_k0:
  shows "flts2 c (r # rs1) = flts2 c [r] @ flts2 c rs1"
  apply(induct r arbitrary: c rs1)
   apply(auto)
  done

lemma  flts2_k00:
  shows "flts2 c (rs1 @ rs2) = flts2 c rs1 @ flts2 c rs2"
  apply(induct rs1 arbitrary: rs2 c)
   apply(auto)
  by (metis append.assoc flts2_k0)

lemma XXX2_helper:
  assumes "\<forall>y. asize y < Suc (sum_list (map asize rs)) \<longrightarrow> good y \<longrightarrow> bsimp y = y"
          "\<forall>r'\<in>set rs. good r' \<and> nonalt r'"
  shows "flts (map (bsimp \<circ> bder c) (flts (map bsimp rs))) = flts (map (bsimp \<circ> bder c) rs)"
  using assms
  apply(induct rs arbitrary: c)
   apply(simp)
  apply(simp)
  apply(subst k0)
  apply(simp add: flts_append)
  apply(subst (2) k0)
  apply(simp add: flts_append)
  apply(subgoal_tac "flts [a] =  [a]")
   prefer 2
  using good.simps(1) k0b apply blast
  apply(simp)
  done

lemma bmkeps_good:
  assumes "good a"
  shows "bmkeps (bsimp a) = bmkeps a"
  using assms
  using test2 by auto

lemma xxx_bder:
  assumes "good r"
  shows "L (erase r) \<noteq> {}"
  using assms
  apply(induct r rule: good.induct)
  apply(auto simp add: Sequ_def)
  done

lemma xxx_bder2:
  assumes "L (erase (bsimp r)) = {}"
  shows "bsimp r = AZERO"
  using assms xxx_bder test2 good1
  by blast

lemma XXX2aa:
  assumes "good a"
  shows "bsimp (bder c (bsimp a)) = bsimp (bder c a)"
  using  assms
  by (simp add: test2)

lemma XXX2aa_ders:
  assumes "good a"
  shows "bsimp (bders (bsimp a) s) = bsimp (bders a s)"
  using  assms
  by (simp add: test2)

lemma XXX4a:
  shows "good (bders_simp (bsimp r) s)  \<or> bders_simp (bsimp r) s = AZERO"
  apply(induct s arbitrary: r rule:  rev_induct)
   apply(simp)
  apply (simp add: good1)
  apply(simp add: bders_simp_append)
  apply (simp add: good1)
  done

lemma XXX4a_good:
  assumes "good a"
  shows "good (bders_simp a s) \<or> bders_simp a s = AZERO"
  using assms
  apply(induct s arbitrary: a rule:  rev_induct)
   apply(simp)
  apply(simp add: bders_simp_append)
  apply (simp add: good1)
  done

lemma XXX4a_good_cons:
  assumes "s \<noteq> []"
  shows "good (bders_simp a s) \<or> bders_simp a s = AZERO"
  using assms
  apply(case_tac s)
   apply(auto)
  using XXX4a by blast

lemma XXX4b:
  assumes "good a" "L (erase (bders_simp a s)) \<noteq> {}"
  shows "good (bders_simp a s)"
  using assms
  apply(induct s arbitrary: a)
   apply(simp)
  apply(simp)
  apply(subgoal_tac "L (erase (bder a aa)) = {} \<or> L (erase (bder a aa)) \<noteq> {}")
   prefer 2
   apply(auto)[1]
  apply(erule disjE)
   apply(subgoal_tac "bsimp (bder a aa) = AZERO")
    prefer 2
  using L_bsimp_erase xxx_bder2 apply auto[1]
   apply(simp)
  apply (metis L.simps(1) XXX4a erase.simps(1))
  apply(drule_tac x="bsimp (bder a aa)" in meta_spec)
  apply(drule meta_mp)
  apply simp
  apply(rule good1a)
  apply(auto)
  done

lemma bders_AZERO:
  shows "bders AZERO s = AZERO"
  and   "bders_simp AZERO s = AZERO"
   apply (induct s)
     apply(auto)
  done

lemma LA:
  assumes "\<Turnstile> v : ders s (erase r)"
  shows "retrieve (bders r s) v = retrieve r (flex (erase r) id s v)"
  using assms
  apply(induct s arbitrary: r v rule: rev_induct)
   apply(simp)
  apply(simp add: bders_append ders_append)
  apply(subst bder_retrieve)
   apply(simp)
  apply(drule Prf_injval)
  by (simp add: flex_append)

lemma LB:
  assumes "s \<in> (erase r) \<rightarrow> v"
  shows "retrieve r v = retrieve r (flex (erase r) id s (mkeps (ders s (erase r))))"
  using assms
  apply(induct s arbitrary: r v rule: rev_induct)
   apply(simp)
   apply(subgoal_tac "v = mkeps (erase r)")
    prefer 2
  apply (simp add: Posix1(1) Posix_determ Posix_mkeps nullable_correctness)
   apply(simp)
  apply(simp add: flex_append ders_append)
  sorry
(*  by (metis Posix_determ Posix_flex Posix_injval Posix_mkeps ders_snoc lexer_correctness(2) lexer_flex)
*)

lemma LB_sym:
  assumes "s \<in> (erase r) \<rightarrow> v"
  shows "retrieve r v = retrieve r (flex (erase r) id s (mkeps (erase (bders r s))))"
  using assms
  by (simp add: LB)

lemma LC:
  assumes "s \<in> (erase r) \<rightarrow> v"
  shows "retrieve r v = retrieve (bders r s) (mkeps (erase (bders r s)))"
  apply(simp)
  by (metis LA LB Posix1(1) assms lexer_correct_None lexer_flex mkeps_nullable)

lemma L0:
  assumes "bnullable a"
  shows "retrieve (bsimp a) (mkeps (erase (bsimp a))) = retrieve a (mkeps (erase a))"
  using assms
  by (metis b3 bmkeps_retrieve bmkeps_simp bnullable_correctness)

thm bmkeps_retrieve

lemma L0a:
  assumes "s \<in> L(erase a)"
  shows "retrieve (bsimp (bders a s)) (mkeps (erase (bsimp (bders a s)))) =
         retrieve (bders a s) (mkeps (erase (bders a s)))"
  using assms
  by (metis L0 bnullable_correctness erase_bders lexer_correct_None lexer_flex)

lemma L0aa:
  assumes "s \<in> L (erase a)"
  shows "[] \<in> erase (bsimp (bders a s)) \<rightarrow> mkeps (erase (bsimp (bders a s)))"
  using assms
  by (metis Posix_mkeps b3 bnullable_correctness erase_bders lexer_correct_None lexer_flex)

lemma L0aaa:
  assumes "[c] \<in> L (erase a)"
  shows "[c] \<in> (erase a) \<rightarrow> flex (erase a) id [c] (mkeps (erase (bder c a)))"
  using assms
  by (metis bders.simps(1) bders.simps(2) erase_bders lexer_correct_None lexer_correct_Some lexer_flex option.inject)

lemma L0aaaa:
  assumes "[c] \<in> L (erase a)"
  shows "[c] \<in> (erase a) \<rightarrow> flex (erase a) id [c] (mkeps (erase (bders a [c])))"
  using assms
  using L0aaa by auto

lemma L02:
  assumes "bnullable (bder c a)"
  shows "retrieve (bsimp a) (flex (erase (bsimp a)) id [c] (mkeps (erase (bder c (bsimp a))))) =
         retrieve (bder c (bsimp a)) (mkeps (erase (bder c (bsimp a))))"
  using assms
  apply(simp)
  using bder_retrieve L0 bmkeps_simp bmkeps_retrieve L0  LA LB
  apply(subst bder_retrieve[symmetric])
  apply (metis L_bsimp_erase bnullable_correctness der_correctness erase_bder mkeps_nullable nullable_correctness)
  apply(simp)
  done

lemma L02_bders:
  assumes "bnullable (bders a s)"
  shows "retrieve (bsimp a) (flex (erase (bsimp a)) id s (mkeps (erase (bders (bsimp a) s)))) =
         retrieve (bders (bsimp a) s) (mkeps (erase (bders (bsimp a) s)))"
  using assms
  by (metis LA L_bsimp_erase bnullable_correctness ders_correctness erase_bders mkeps_nullable nullable_correctness)

lemma L03:
  assumes "bnullable (bder c a)"
  shows "retrieve (bder c (bsimp a)) (mkeps (erase (bder c (bsimp a)))) =
         bmkeps (bsimp (bder c (bsimp a)))"
  using assms
  by (metis L0 L_bsimp_erase bmkeps_retrieve bnullable_correctness der_correctness erase_bder nullable_correctness)

lemma L04:
  assumes "bnullable (bder c a)"
  shows "retrieve (bder c (bsimp a)) (mkeps (erase (bder c (bsimp a)))) =
         retrieve (bsimp (bder c (bsimp a))) (mkeps (erase (bsimp (bder c (bsimp a)))))"
  using assms
  by (metis L0 L_bsimp_erase bnullable_correctness der_correctness erase_bder nullable_correctness)

lemma L05:
  assumes "bnullable (bder c a)"
  shows "retrieve (bder c (bsimp a)) (mkeps (erase (bder c (bsimp a)))) =
         retrieve (bsimp (bder c (bsimp a))) (mkeps (erase (bsimp (bder c (bsimp a)))))"
  using assms
  using L04 by auto

lemma L06:
  assumes "bnullable (bder c a)"
  shows "bmkeps (bder c (bsimp a)) = bmkeps (bsimp (bder c (bsimp a)))"
  using assms
  by (metis L03 L_bsimp_erase bmkeps_retrieve bnullable_correctness der_correctness erase_bder nullable_correctness)

lemma L07:
  assumes "s \<in> L (erase r)"
  shows "retrieve r (flex (erase r) id s (mkeps (ders s (erase r))))
            = retrieve (bders r s) (mkeps (erase (bders r s)))"
  using assms
  using LB LC lexer_correct_Some by auto

lemma LXXX:
  assumes "s \<in> (erase r) \<rightarrow> v" "s \<in> (erase (bsimp r)) \<rightarrow> v'"
  shows "retrieve r v = retrieve (bsimp r) v'"
  using  assms
  apply -
  thm LC
  apply(subst LC)
   apply(assumption)
  apply(subst  L0[symmetric])
  using bnullable_correctness lexer_correctness(2) lexer_flex apply fastforce
  apply(subst (2) LC)
   apply(assumption)
  apply(subst (2)  L0[symmetric])
  using bnullable_correctness lexer_correctness(2) lexer_flex apply fastforce

  oops

lemma L08:
  assumes "s \<in> L (erase r)"
  shows "retrieve (bders (bsimp r) s) (mkeps (erase (bders (bsimp r) s)))
         = retrieve (bders r s) (mkeps (erase (bders r s)))"
  using assms
  apply(induct s arbitrary: r)
   apply(simp)
  using L0 bnullable_correctness nullable_correctness apply blast
  apply(simp add: bders_append)
  apply(drule_tac x="(bder a (bsimp r))" in meta_spec)
  apply(drule meta_mp)
  apply (metis L_bsimp_erase erase_bder lexer.simps(2) lexer_correct_None option.case(1))
  apply(drule sym)
  apply(simp)
  apply(subst LA)
  apply (metis L0aa L_bsimp_erase Posix1(1) ders.simps(2) ders_correctness erase_bder erase_bders mkeps_nullable nullable_correctness)
  apply(subst LA)
  using lexer_correct_None lexer_flex mkeps_nullable apply force

  using L0[no_vars] bder_retrieve[no_vars] LA[no_vars] LC[no_vars] L07[no_vars]

thm L0[no_vars] bder_retrieve[no_vars] LA[no_vars] LC[no_vars] L07[no_vars]
  oops

lemma test:
  assumes "s = [c]"
  shows "retrieve (bders r s) v = XXX" and "YYY = retrieve r (flex (erase r) id s v)"
  using assms
   apply(simp only: bders.simps)
   defer
  using assms
   apply(simp only: flex.simps id_simps)
  using  L0[no_vars] bder_retrieve[no_vars] LA[no_vars] LC[no_vars]
  find_theorems "retrieve (bders _ _) _"
  find_theorems "retrieve _ (mkeps _)"
  oops

lemma L06X:
  assumes "bnullable (bder c a)"
  shows "bmkeps (bder c (bsimp a)) = bmkeps (bder c a)"
  using assms
  apply(induct a arbitrary: c)
       apply(simp)
      apply(simp)
     apply(simp)
    prefer 3
    apply(simp)
   prefer 2
   apply(simp)

   defer
  oops

lemma L06_2:
  assumes "bnullable (bders a [c,d])"
  shows "bmkeps (bders (bsimp a) [c,d]) = bmkeps (bsimp (bders (bsimp a) [c,d]))"
  using assms
  apply(simp)
  by (metis L_bsimp_erase bmkeps_simp bnullable_correctness der_correctness erase_bder nullable_correctness)

lemma L06_bders:
  assumes "bnullable (bders a s)"
  shows "bmkeps (bders (bsimp a) s) = bmkeps (bsimp (bders (bsimp a) s))"
  using assms
  by (metis L_bsimp_erase bmkeps_simp bnullable_correctness ders_correctness erase_bders nullable_correctness)

lemma LLLL:
  shows "L (erase a) =  L (erase (bsimp a))"
  and "L (erase a) = {flat v | v. \<Turnstile> v: (erase a)}"
  and "L (erase a) = {flat v | v. \<Turnstile> v: (erase (bsimp a))}"
  using L_bsimp_erase apply(blast)
  apply (simp add: L_flat_Prf)
  using L_bsimp_erase L_flat_Prf apply(auto)[1]
  done

lemma L07XX:
  assumes "s \<in> L (erase a)"
  shows "s \<in> erase a \<rightarrow> flex (erase a) id s (mkeps (ders s (erase a)))"
  using assms
  by (meson lexer_correct_None lexer_correctness(1) lexer_flex)

lemma LX0:
  assumes "s \<in> L r"
  shows "decode (bmkeps (bders (intern r) s)) r = Some(flex r id s (mkeps (ders s r)))"
  by (metis assms blexer_correctness blexer_def lexer_correct_None lexer_flex)

lemma L02_bders2:
  assumes "bnullable (bders a s)" "s = [c]"
  shows "retrieve (bders (bsimp a) s) (mkeps (erase (bders (bsimp a) s)))  =
         retrieve (bders a s) (mkeps (erase (bders a s)))"
  using assms
  apply(simp)

  apply(induct s arbitrary: a)
   apply(simp)
  using L0 apply auto[1]
  oops

thm bmkeps_retrieve bmkeps_simp Posix_mkeps

lemma WQ1:
  assumes "s \<in> L (der c r)"
  shows "s \<in> der c r \<rightarrow> mkeps (ders s (der c r))"
  using assms
  oops

lemma L1:
  assumes "s \<in> r \<rightarrow> v"
  shows "decode (bmkeps (bders (intern r) s)) r = Some v"
  using assms
  by (metis blexer_correctness blexer_def lexer_correctness(1) option.distinct(1))

lemma L2:
  assumes "s \<in> (der c r) \<rightarrow> v"
  shows "decode (bmkeps (bders (intern r) (c # s))) r = Some (injval r c v)"
  using assms
  apply(subst bmkeps_retrieve)
  using Posix1(1) lexer_correct_None lexer_flex apply fastforce
  using MAIN_decode
  apply(subst MAIN_decode[symmetric])
   apply(simp)
   apply (meson Posix1(1) lexer_correct_None lexer_flex mkeps_nullable)
  apply(simp)
  apply(subgoal_tac "v = flex (der c r) id s (mkeps (ders s (der c r)))")
   prefer 2
   apply (metis Posix_determ lexer_correctness(1) lexer_flex option.distinct(1))
  apply(simp)
  apply(subgoal_tac "injval r c (flex (der c r) id s (mkeps (ders s (der c r)))) =
    (flex (der c r) ((\<lambda>v. injval r c v) o id) s (mkeps (ders s (der c r))))")
   apply(simp)
  using flex_fun_apply by blast

lemma L3:
  assumes "s2 \<in> (ders s1 r) \<rightarrow> v"
  shows "decode (bmkeps (bders (intern r) (s1 @ s2))) r = Some (flex r id s1 v)"
  using assms
  apply(induct s1 arbitrary: r s2 v rule: rev_induct)
   apply(simp)
  using L1 apply blast
  apply(simp add: ders_append)
  apply(drule_tac x="r" in meta_spec)
  apply(drule_tac x="x # s2" in meta_spec)
  apply(drule_tac x="injval (ders xs r) x v" in meta_spec)
  apply(drule meta_mp)
   defer
   apply(simp)
   apply(simp add:  flex_append)
  by (simp add: Posix_injval)

lemma bders_snoc:
  "bder c (bders a s) = bders a (s @ [c])"
  apply(simp add: bders_append)
  done

lemma QQ1:
  shows "bsimp (bders (bsimp a) []) = bders_simp (bsimp a) []"
  apply(simp)
  apply(simp add: bsimp_idem)
  done

lemma QQ2:
  shows "bsimp (bders (bsimp a) [c]) = bders_simp (bsimp a) [c]"
  apply(simp)
  done

lemma XXX2a_long:
  assumes "good a"
  shows "bsimp (bder c (bsimp a)) = bsimp (bder c a)"
  using  assms
  apply(induct a arbitrary: c taking: asize rule: measure_induct)
  apply(case_tac x)
       apply(simp)
      apply(simp)
     apply(simp)
  prefer 3
    apply(simp)
   apply(simp)
   apply(auto)[1]
apply(case_tac "x42 = AZERO")
     apply(simp)
   apply(case_tac "x43 = AZERO")
     apply(simp)
  using test2 apply force
  apply(case_tac "\<exists>bs. x42 = AONE bs")
     apply(clarify)
     apply(simp)
    apply(subst bsimp_ASEQ1)
       apply(simp)
  using b3 apply force
  using bsimp_ASEQ0 test2 apply force
  thm good_SEQ test2
     apply (simp add: good_SEQ test2)
    apply (simp add: good_SEQ test2)
  apply(case_tac "x42 = AZERO")
     apply(simp)
   apply(case_tac "x43 = AZERO")
    apply(simp)
  apply (simp add: bsimp_ASEQ0)
  apply(case_tac "\<exists>bs. x42 = AONE bs")
     apply(clarify)
     apply(simp)
    apply(subst bsimp_ASEQ1)
      apply(simp)
  using bsimp_ASEQ0 test2 apply force
     apply (simp add: good_SEQ test2)
    apply (simp add: good_SEQ test2)
  apply (simp add: good_SEQ test2)
  (* AALTs case *)
  apply(simp)
  using test2 by fastforce

lemma XXX2a_long_without_good:
  assumes "a = AALTs bs0  [AALTs bs1 [AALTs bs2 [ASTAR [] (AONE bs7), AONE bs6, ASEQ bs3 (ACHAR bs4 d) (AONE bs5)]]]"
  shows "bsimp (bder c (bsimp a)) = bsimp (bder c a)"
        "bsimp (bder c (bsimp a)) = XXX"
        "bsimp (bder c a) = YYY"
  using  assms
    apply(simp)
  using  assms
   apply(simp)
   prefer 2
  using  assms
   apply(simp)
  oops

lemma bder_bsimp_AALTs:
  shows "bder c (bsimp_AALTs bs rs) = bsimp_AALTs bs (map (bder c) rs)"
  apply(induct bs rs rule: bsimp_AALTs.induct)
    apply(simp)
   apply(simp)
   apply (simp add: bder_fuse)
  apply(simp)
  done

lemma flts_nothing:
  assumes "\<forall>r \<in> set rs. r \<noteq> AZERO" "\<forall>r \<in> set rs. nonalt r"
  shows "flts rs = rs"
  using assms
  apply(induct rs rule: flts.induct)
        apply(auto)
  done

lemma flts_flts:
  assumes "\<forall>r \<in> set rs. good r"
  shows "flts (flts rs) = flts rs"
  using assms
  apply(induct rs taking: "\<lambda>rs. sum_list  (map asize rs)" rule: measure_induct)
  apply(case_tac x)
   apply(simp)
  apply(simp)
  apply(case_tac a)
       apply(simp_all  add: bder_fuse flts_append)
  apply(subgoal_tac "\<forall>r \<in> set x52. r \<noteq> AZERO")
   prefer 2
  apply (metis Nil_is_append_conv bsimp_AALTs.elims good.simps(1) good.simps(5) good0 list.distinct(1) n0 nn1b split_list_last test2)
  apply(subgoal_tac "\<forall>r \<in> set x52. nonalt r")
   prefer 2
   apply (metis n0 nn1b test2)
  by (metis flts_fuse flts_nothing)

lemma PP:
  assumes "bnullable (bders r s)"
  shows "bmkeps (bders (bsimp r) s) = bmkeps (bders r s)"
  using assms
  apply(induct s arbitrary: r)
   apply(simp)
  using bmkeps_simp apply auto[1]
  apply(simp add: bders_append bders_simp_append)
  oops

lemma PP:
  assumes "bnullable (bders r s)"
  shows "bmkeps (bders_simp (bsimp r) s) = bmkeps (bders r s)"
  using assms
  apply(induct s arbitrary: r rule: rev_induct)
   apply(simp)
  using bmkeps_simp apply auto[1]
  apply(simp add: bders_append bders_simp_append)
  apply(drule_tac x="bder a (bsimp r)" in meta_spec)
  apply(drule_tac meta_mp)
   defer
  oops

lemma
  assumes "asize (bsimp a) = asize a"  "a = AALTs bs [AALTs bs2 [], AZERO, AONE bs3]"
  shows "bsimp a = a"
  using assms
  apply(simp)
  oops

lemma iii:
  assumes "bsimp_AALTs bs rs \<noteq> AZERO"
  shows "rs \<noteq> []"
  using assms
  apply(induct bs  rs rule: bsimp_AALTs.induct)
    apply(auto)
  done

lemma CT1_SEQ:
  shows "bsimp (ASEQ bs a1 a2) = bsimp (ASEQ bs (bsimp a1) (bsimp a2))"
  apply(simp add: bsimp_idem)
  done

lemma CT1:
  shows "bsimp (AALTs bs as) = bsimp (AALTs bs (map  bsimp as))"
  apply(induct as arbitrary: bs)
   apply(simp)
  apply(simp)
  by (simp add: bsimp_idem comp_def)

lemma CT1a:
  shows "bsimp (AALT bs a1 a2) = bsimp(AALT bs (bsimp a1) (bsimp a2))"
  by (metis CT1 list.simps(8) list.simps(9))

lemma WWW2:
  shows "bsimp (bsimp_AALTs bs1 (flts (map bsimp as1))) =
         bsimp_AALTs bs1 (flts (map bsimp as1))"
  by (metis bsimp.simps(2) bsimp_idem)

lemma CT1b:
  shows "bsimp (bsimp_AALTs bs as) = bsimp (bsimp_AALTs bs (map bsimp as))"
  apply(induct bs as rule: bsimp_AALTs.induct)
    apply(auto simp add: bsimp_idem)
  apply (simp add: bsimp_fuse bsimp_idem)
  by (metis bsimp_idem comp_apply)

lemma conj_rule: "\<lbrakk> P; Q\<rbrakk> \<Longrightarrow> P \<and> (Q \<and> P)"
  apply(rule conjI)
   apply assumption
  apply (rule conjI)
   apply assumption
  apply assumption
  done

  thm disjE

lemma disj_swap : "P \<or> Q \<Longrightarrow> Q \<or> P"
  apply(erule disjE)
   apply(rule disjI2)
   apply assumption
  apply (rule disjI1)
  apply assumption
  done

lemma disj_swapag: "P \<or> Q \<Longrightarrow> Q \<or> P"
  apply(rule disjE)
    apply assumption
   apply(rule disjI2)
   apply assumption
  apply (rule disjI1)
  apply assumption
  done

lemma conj_swap : "P \<and> Q \<Longrightarrow> Q \<and> P"
  apply(erule conjE)
  apply(rule conjI)
   apply assumption +
  done

lemma imp_uncurry : "P \<longrightarrow> (Q \<longrightarrow> R) \<Longrightarrow> P \<and> Q \<longrightarrow> R"
  apply(rule impI)
  apply(erule conjE)
  apply(drule mp)
  apply assumption
  apply(drule mp)
   apply assumption+
  done

thm conjunct2

lemma "\<lbrakk>\<not>(P \<longrightarrow> Q); \<not>(R \<longrightarrow> Q) \<rbrakk> \<Longrightarrow> R"
  apply (erule_tac Q = "R\<longrightarrow>Q" in contrapos_np)
  apply (intro impI)
  by (erule notE)

lemma "(P \<or> Q ) \<and> R \<Longrightarrow> P \<or> (Q \<and> R)"

thm disjI2

thm conjE

definition congruent
  where
"congruent r1 r2 \<equiv> (\<forall> v.
                    (
                      (\<Turnstile> v : (erase (bsimp r1)) \<longleftrightarrow> (\<Turnstile> v : erase (bsimp r2))) \<and>  (retrieve (bsimp r1) v = retrieve (bsimp r2) v)
                    )
                   )"

inductive sim :: "rexp \<Rightarrow> rexp \<Rightarrow> bool" ("_ ~~~ _" [100, 100] 100)
  where
  "CR c ~~~ CR c"
| "ONE ~~~ ONE"
| "\<lbrakk>r11 ~~~ r12; r21 ~~~ r22\<rbrakk> \<Longrightarrow> SEQ r11 r21 ~~~ SEQ r12 r22"
| "\<lbrakk>r11 ~~~ r12; r21 ~~~ r22\<rbrakk> \<Longrightarrow> ALT r11 r21 ~~~ ALT r12 r22"
| "r1 ~~~ r2 \<Longrightarrow> STAR r1 ~~~ STAR r2"

fun nzero :: "rexp \<Rightarrow> bool"
  where
  "nzero ZERO = False"
| "nzero ONE = True"
| "nzero (CR c) = True"
| "nzero (ALT r1 r2) = (nzero r1 \<and> nzero r2)"
| "nzero (SEQ r1 r2) = (nzero r1 \<and> nzero r2)"
| "nzero (STAR r1) = nzero r1"

lemma sim0:
  assumes "r1 ~~~ r2"
  shows "(\<forall>v.  \<Turnstile> v : r1  \<longleftrightarrow> \<Turnstile> v : r2)"
using assms
apply(induct r1 r2 rule: sim.induct)
apply(auto)
  apply (metis Prf.intros(1) Prf_elims(2))
  apply (metis Prf.intros(1) Prf_elims(2))
  apply (smt (verit, del_insts) Prf.intros(2) Prf.intros(3) Prf_elims(3))
  apply (smt (verit, del_insts) Prf.intros(2) Prf.intros(3) Prf_elims(3))
  apply (metis Prf.intros(6) Prf_elims(6))
  by (metis Prf.intros(6) Prf_elims(6))

lemma nzero_exists:
  assumes "nzero r"
  shows "\<exists>v. \<Turnstile> v : r"
using assms
apply(induct r rule: nzero.induct)
apply(auto intro: Prf.intros)
  using mkeps_nullable nullable.simps(6) by blast

thm Prf.intros mp

lemma sim_eq:
  assumes "r1 ~~~ r2"
  shows "r1 = r2"
using assms
apply(induct r1 r2)
apply(auto)
done

thm Prf.intros

lemma
  shows "\<forall>v. \<Turnstile> v : STAR ONE = \<Turnstile> v : STAR (ALT ONE ONE)"
apply(auto)
apply(erule Prf_elims)
apply(auto)
  apply (metis Prf.intros(6) Prf_elims(4) flat.simps(1))
apply(erule Prf_elims)
apply(auto)
  by (metis L.simps(2) L.simps(5) L_flat_Prf1 Prf.intros(6) Un_iff singletonD)

lemma purecontra: " (\<not>Q \<Longrightarrow> \<not>P) \<Longrightarrow> (P \<Longrightarrow> Q) "
  apply(simp add: contrapos_pp)
  apply auto
  done

lemma sim1starcp: "  \<exists>v. \<Turnstile> v : r1 \<noteq> \<Turnstile> v : x6 \<Longrightarrow>  \<not>(\<forall>v. \<Turnstile> v : STAR r1 = \<Turnstile> v : STAR x6 \<and> nzero r1 \<and> nzero x6)"
  apply(erule exE)
  apply simp
  sorry

lemma sim1star:" (\<forall>v. \<Turnstile> v : STAR r1 = \<Turnstile> v : STAR x6 \<and> nzero r1 \<and> nzero x6) \<Longrightarrow> \<forall>v. \<Turnstile> v : r1 = \<Turnstile> v : x6"
  using sim1starcp by auto

thm contrapos_nn

lemma sim1:
  assumes "(\<forall>v. \<Turnstile> v : r1  \<longleftrightarrow> \<Turnstile> v : r2)" "nzero r1" "nzero r2"
  shows "r1 ~~~ r2"
using assms
apply(induct r1 arbitrary: r2)
apply(auto)
  prefer 3
(* SEQ CASE*)
  apply(case_tac r2)
apply(auto)
  apply (meson Prf.intros(4) Prf_elims(2) val.distinct(3))
  apply (meson Prf.intros(5) Prf_elims(2) val.distinct(11))
apply(rule sim.intros)
  apply (metis Prf.intros(1) Prf_elims(2) nzero_exists val.inject(2))
  apply (metis Prf.intros(1) Prf_elims(2) nzero_exists val.inject(2))
  apply (metis Prf.intros(2) Prf_elims(2) nzero_exists val.distinct(21))
  apply (metis Prf_elims(2) mkeps.simps(4) mkeps_nullable nullable.simps(6) val.distinct(23))
  prefer 3
(* ALT CASE*)
  apply(case_tac r2)
  apply(auto)
  apply (metis Prf.intros(2) Prf_elims(4) nzero_exists val.distinct(7))
  apply (metis Prf.intros(2) Prf_elims(5) nzero_exists val.distinct(15))
  apply (metis Prf.intros(2) Prf_elims(2) nzero_exists val.distinct(21))
apply(rule sim.intros)
  apply (metis Prf.intros(2) Prf_elims(3) val.distinct(25) val.inject(4))
  apply (metis Prf.intros(3) Prf_elims(3) val.distinct(25) val.inject(3))
  apply(frule nzero_exists)
  apply(auto)
  apply (metis Prf_elims(3) Prf_elims(6) nzero.simps(4) nzero_exists val.distinct(27) val.distinct(29))
(* ONE CASE *)
  using Prf.cases Prf.intros(4) sim.intros(2) apply force
(* CR case *)
  apply (smt (verit, del_insts) Posix1(2) Posix_CHAR Posix_mkeps Prf.cases Prf.intros(5) Prf_elims(5) list.distinct(1) mkeps_nullable nullable.simps(6) sim.intros(1) val.distinct(1) val.distinct(11) val.distinct(13) val.distinct(15) val.inject(1))
(* STAR CASE *)
     apply(case_tac r2)
  apply(auto)
  apply (meson Prf.intros(4) Prf_elims(6) val.distinct(9))
  apply (meson Prf.intros(5) Prf_elims(6) val.distinct(17))
  apply (metis Prf_elims(2) mkeps.simps(4) mkeps_nullable nullable.simps(6) val.distinct(23))
  apply (meson Prf.intros(2) Prf_elims(6) nzero_exists val.distinct(29))
apply(rule sim.intros)
  apply(frule nzero_exists)
  apply(auto)
  apply(drule_tac x="x6" in meta_spec)
  apply(auto)
  apply(drule meta_mp)
  apply(auto)
  sorry
(* there might be a problem with the definition of |= for STARs *)

lemma symshape:
  shows "\<lbrakk>(\<forall>v.  \<Turnstile> v : r1  \<longleftrightarrow> \<Turnstile> v : r2); nzero r1; nzero r2\<rbrakk> \<Longrightarrow> r1 = r2"
  by (simp add: sim1 sim_eq)

lemma rsrcong: shows "congruent r (bsimp r)"
  by (simp add: congruent_def bsimp_idem)

thm congruent_def retrieve.induct

lemma congsymm: shows "congruent r1 r2 \<Longrightarrow> congruent r2 r1"
  by (simp add: congruent_def)

lemma
  shows "(\<forall>v.  \<Turnstile> v : ZERO  \<longleftrightarrow> \<Turnstile> v : SEQ ZERO ZERO)"
  apply(auto)
  apply(erule Prf_elims)
  apply(erule Prf_elims)
  apply(erule Prf_elims)
  done

lemma symshape0:
  shows "(\<forall>v.  \<Turnstile> v : r1  \<longleftrightarrow> \<Turnstile> v : r2) \<Longrightarrow> L r1 = L r2"
  using L_flat_Prf by auto

lemma congrueninduct: shows "\<lbrakk>congruent r1 r2; congruent r3 r4 \<rbrakk> \<Longrightarrow> congruent (ASEQ bs r1 r3) (ASEQ bs r2 r4)"
  apply(simp add: congruent_def)
  apply(case_tac "\<exists>bs1. bsimp r1 = AONE bs1")
  apply(erule exE)

   apply(simp only: bsimp_ASEQ.simps)
  apply(simp only: congshape)
  sorry

(* CT *)

lemma CTU:
  shows "bsimp_AALTs bs as = li bs as"
  apply(induct bs as rule: li.induct)
    apply(auto)
  done

find_theorems "bder _ (bsimp_AALTs _ _)"

lemma CTa:
  assumes "\<forall>r \<in> set as. nonalt r \<and> r \<noteq> AZERO"
  shows  "flts as = as"
  using assms
  apply(induct as)
   apply(simp)
  apply(case_tac as)
   apply(simp)
  apply (simp add: k0b)
  using flts_nothing by auto

lemma CT0:
  assumes "\<forall>r \<in> set as1. nonalt r \<and> r \<noteq> AZERO"
  shows "flts [bsimp_AALTs bs1 as1] =  flts (map (fuse bs1) as1)"
  using assms CTa
  apply(induct as1 arbitrary: bs1)
    apply(simp)
   apply(simp)
  apply(case_tac as1)
   apply(simp)
  apply(simp)
proof -
fix a :: arexp and as1a :: "arexp list" and bs1a :: "bit list" and aa :: arexp and list :: "arexp list"
  assume a1: "nonalt a \<and> a \<noteq> AZERO \<and> nonalt aa \<and> aa \<noteq> AZERO \<and> (\<forall>r\<in>set list. nonalt r \<and> r \<noteq> AZERO)"
  assume a2: "\<And>as. \<forall>r\<in>set as. nonalt r \<and> r \<noteq> AZERO \<Longrightarrow> flts as = as"
  assume a3: "as1a = aa # list"
  have "flts [a] = [a]"
using a1 k0b by blast
then show "fuse bs1a a # fuse bs1a aa # map (fuse bs1a) list = flts (fuse bs1a a # fuse bs1a aa # map (fuse bs1a) list)"
  using a3 a2 a1 by (metis (no_types) append.left_neutral append_Cons flts_fuse k00 k0b list.simps(9))
qed

lemma CT01:
  assumes "\<forall>r \<in> set as1. nonalt r \<and> r \<noteq> AZERO" "\<forall>r \<in> set as2. nonalt r \<and> r \<noteq> AZERO"
  shows "flts [bsimp_AALTs bs1 as1, bsimp_AALTs bs2 as2] =  flts ((map (fuse bs1) as1) @ (map (fuse bs2) as2))"
  using assms CT0
  by (metis k0 k00)

lemma CT_exp:
  assumes "\<forall>a \<in> set as. bsimp (bder c (bsimp a)) = bsimp (bder c a)"
  shows "map bsimp (map (bder c) as) = map bsimp (map (bder c) (map bsimp as))"
  using assms
  apply(induct as)
   apply(auto)
  done

lemma asize_set:
  assumes "a \<in> set as"
  shows "asize a < Suc (sum_list (map asize as))"
  using assms
  apply(induct as arbitrary: a)
   apply(auto)
  using le_add2 le_less_trans not_less_eq by blast

lemma XXX2a_long_without_good:
  shows "bsimp (bder c (bsimp a)) = bsimp (bder c a)"
  apply(induct a arbitrary: c taking: "\<lambda>a. asize a" rule: measure_induct)
  apply(case_tac x)
       apply(simp)
      apply(simp)
     apply(simp)
  prefer 3
    apply(simp)
  (* AALT case *)
   prefer 2
   apply(simp del: bsimp.simps)
   apply(subst (2) CT1)
   apply(subst CT_exp)
    apply(auto)[1]
  using asize_set apply blast
   apply(subst CT1[symmetric])
  apply(simp)
  oops

lemma YY:
  assumes "flts (map bsimp as1) = xs"
  shows "flts (map bsimp (map (fuse bs1) as1)) = map (fuse bs1) xs"
  using assms
  apply(induct as1 arbitrary: bs1 xs)
   apply(simp)
  apply(auto)
  by (metis bsimp_fuse flts_fuse k0 list.simps(9))

lemma flts_nonalt:
  assumes "flts (map bsimp xs) = ys"
  shows "\<forall>y \<in> set ys. nonalt y"
  using assms
  apply(induct xs arbitrary: ys)
   apply(auto)
  apply(case_tac xs)
   apply(auto)
  using flts2 good1 apply fastforce
  by (smt ex_map_conv list.simps(9) nn1b nn1c)

lemma WWW3:
  shows "flts [bsimp_AALTs bs1 (flts (map bsimp as1))] =
         flts (map bsimp (map (fuse bs1) as1))"
  by (metis CT0 YY flts_nonalt flts_nothing qqq1)

lemma WWW4:
  shows "map (bder c \<circ> fuse bs1) as1 = map (fuse bs1) (map (bder c) as1)"
  apply(induct as1)
   apply(auto)
  using bder_fuse by blast

lemma WWW5:
  shows "map (bsimp \<circ> bder c) as1 = map bsimp (map (bder c) as1)"
  apply(induct as1)
   apply(auto)
  done

lemma WWW6:
  shows "bsimp (bder c (bsimp_AALTs x51 (flts [bsimp a1, bsimp a2]) ) )  =
 bsimp(bsimp_AALTs x51 (map (bder c) (flts [bsimp a1, bsimp a2]))) "
  using bder_bsimp_AALTs by auto

lemma WWW7:
  shows "bsimp (bsimp_AALTs x51 (map (bder c) (flts [bsimp a1, bsimp a2]))) =
  bsimp(bsimp_AALTs x51 (flts (map (bder c) [bsimp a1, bsimp a2])))"
  sorry

lemma stupid:
  assumes "a = b"
  shows "bsimp(a) = bsimp(b)"
  using assms
  apply(auto)
  done
(*
proving idea:
bsimp_AALTs x51  (map (bder c) (flts [a1, a2])) = bsimp_AALTs x51 (map (bder c) (flts [a1]++[a2]))
= bsimp_AALTs x51  (map (bder c) ((flts [a1])++(flts [a2]))) =
bsimp_AALTs x51 (map (bder c) (flts [a1]))++(map (bder c) (flts [a2])) = A
and then want to prove that
map (bder c) (flts [a]) = flts [bder c a] under the condition
that a is either a seq with the first elem being not nullable, or a character equal to c,
or an AALTs, or a star
Then, A = bsimp_AALTs x51 (flts [bder c a]) ++ (map (bder c) (flts [a2])) = A1
Using the same condition for a2, we get
A1 = bsimp_AALTs x51 (flts [bder c a1]) ++ (flts [bder c a2])
=bsimp_AALTs x51 flts ([bder c a1] ++ [bder c a2])
=bsimp_AALTs x51 flts ([bder c a1, bder c a2])
 *)
lemma manipulate_flts:
  shows "bsimp_AALTs x51  (map (bder c) (flts [a1, a2])) =
bsimp_AALTs x51 ((map (bder c) (flts [a1])) @ (map (bder c) (flts [a2])))"
  by (metis k0 map_append)

lemma go_inside_flts:
  assumes " (bder c a1 \<noteq> AZERO) "
 "\<not>(\<exists> a01 a02 x02. (  (a1 = ASEQ x02 a01 a02) \<and> bnullable(a01) )      )"
shows "map (bder c) (flts [a1]) = flts [bder c a1]"
  using assms
  apply -
  apply(case_tac a1)
  apply(simp)
  apply(simp)
     apply(case_tac "x32 = c")
  prefer 2
      apply(simp)
     apply(simp)
    apply(simp)
  apply (simp add: WWW4)
   apply(simp add: bder_fuse)
  done

lemma medium010:
  assumes " (bder c a1 = AZERO) "
  shows "map (bder c) (flts [a1]) = [AZERO] \<or> map (bder c) (flts [a1]) = []"
  using assms
  apply -
  apply(case_tac a1)
       apply(simp)
      apply(simp)
  apply(simp)
    apply(simp)
  apply(simp)
  apply(simp)
  done

lemma medium011:
  assumes " (bder c a1 = AZERO) "
  shows "flts (map (bder c)  [a1, a2]) = flts [bder c a2]"
  using assms
  apply -
  apply(simp)
  done

lemma medium01central:
  shows "bsimp(bsimp_AALTs x51 (map (bder c) (flts [a2])) ) = bsimp(bsimp_AALTs x51 (flts [bder c a2]))"
  sorry

lemma plus_bsimp:
  assumes "bsimp( bsimp a) = bsimp (bsimp b)"
  shows "bsimp a = bsimp b"
  using assms
  apply -
  by (simp add: bsimp_idem)
lemma patience_good5:
  assumes "bsimp r = AALTs x y"
  shows " \<exists> a aa list. y = a#aa#list"
  by (metis Nil_is_map_conv arexp.simps(13) assms bsimp_AALTs.elims flts1 good.simps(5) good1 k0a)

(*SAD*)
(*this does not hold actually
lemma bsimp_equiv0:
  shows "bsimp(bsimp r) = bsimp(bsimp (AALTs []  [r]))"
  apply(simp)
  apply(case_tac "bsimp r")
       apply(simp)
      apply(simp)
     apply(simp)
    apply(simp)
 thm good1
  using good1
   apply -
   apply(drule_tac x="r" in meta_spec)
   apply(erule disjE)

    apply(simp only: bsimp_AALTs.simps)
    apply(simp only:flts.simps)
    apply(drule patience_good5)
    apply(clarify)
    apply(subst  bsimp_AALTs_qq)
     apply simp
    prefer 2
  sorry*)

(*exercise: try multiple ways of proving this*)
(*this lemma does not hold.........
lemma bsimp_equiv1:
  shows "bsimp r = bsimp (AALTs []  [r])"
  using plus_bsimp
  apply -
  using bsimp_equiv0 by blast
  (*apply(simp)
  apply(case_tac "bsimp r")
       apply(simp)
      apply(simp)
     apply(simp)
    apply(simp)
(*use lemma good1*)
  thm good1
  using good1
   apply -
   apply(drule_tac x="r" in meta_spec)
   apply(erule disjE)

  apply(subst flts_single1)
  apply(simp only: bsimp_AALTs.simps)
    prefer 2

  thm flts_single1

  find_theorems "flts _ = _"*)
*)
lemma bsimp_equiv2:
  shows "bsimp (AALTs x51 [r])  =  bsimp (AALT x51 AZERO r)"
  sorry

lemma medium_stupid_isabelle:
  assumes "rs = a # list"
  shows  "bsimp_AALTs x51 (AZERO # rs) = AALTs x51 (AZERO#rs)"
  using assms
  apply -
  apply(simp)
  done

lemma singleton_list_map:
  shows"map f [a] = [f a]"
  apply simp
  done
lemma map_application2:
  shows"map f [a,b] = [f a, f b]"
  apply simp
  done

lemma medium01:
  assumes " (bder c a1 = AZERO) "
  shows "bsimp(bsimp_AALTs x51 (map (bder c) (flts [ a1, a2]))) =
         bsimp(bsimp_AALTs x51 (flts (map (bder c) [ a1, a2])))"
  apply(subst manipulate_flts)
  using assms
  apply -
  apply(subst medium011)
   apply(simp)
  apply(case_tac "map (bder c) (flts [a1]) = []")
   apply(simp)
  using medium01central apply blast
apply(frule medium010)
  apply(erule disjE)
  prefer 2
   apply(simp)
  apply(simp)
  apply(case_tac a2)
       apply simp
      apply simp
     apply simp
    apply(simp only:flts.simps)
(*HOW do i say here to replace ASEQ ..... back into a2*)
(*how do i say here to use the definition of map function
without lemma, of course*)
(*how do i say here that AZERO#map (bder c) [ASEQ x41 x42 x43]'s list.len >1
without a lemma, of course*)
    apply(subst singleton_list_map)
    apply(simp only: bsimp_AALTs.simps)
    apply(case_tac "bder c (ASEQ x41 x42 x43)")
         apply simp
        apply simp
       apply simp
      prefer 3
      apply simp
     apply(rule_tac t="bder c (ASEQ x41 x42 x43)"
and s="ASEQ x41a x42a x43a" in subst)
      apply simp
     apply(simp only: flts.simps)
     apply(simp only: bsimp_AALTs.simps)
     apply(simp only: fuse.simps)
     apply(subst (2) bsimp_idem[symmetric])
     apply(subst (1) bsimp_idem[symmetric])
     apply(simp only:bsimp.simps)
     apply(subst map_application2)
     apply(simp only: bsimp.simps)
     apply(simp only:flts.simps)
(*want to happily change between a2 and ASEQ x41 42 43, and eliminate now
redundant conditions such as  map (bder c) (flts [a1]) = [AZERO] *)
     apply(case_tac "bsimp x42a")
          apply(simp)
         apply(case_tac "bsimp x43a")
              apply(simp)
             apply(simp)
            apply(simp)
           apply(simp)
          prefer 2
          apply(simp)
     apply(rule_tac t="bsimp x43a"
and s="AALTs x51a x52" in subst)
          apply simp
         apply(simp only:bsimp_ASEQ.simps)
         apply(simp only:fuse.simps)
         apply(simp only:flts.simps)

  using medium01central mediumlittle by auto

lemma medium1:
  assumes " (bder c a1 \<noteq> AZERO) "
 "\<not>(\<exists> a01 a02 x02. (  (a1 = ASEQ x02 a01 a02) \<and> bnullable(a01) )      )"
" (bder c a2 \<noteq> AZERO)"
 "\<not>(\<exists> a11 a12 x12. (  (a2 = ASEQ x12 a11 a12) \<and> bnullable(a11) )      )"
  shows "bsimp_AALTs x51 (map (bder c) (flts [ a1, a2])) =
         bsimp_AALTs x51 (flts (map (bder c) [ a1, a2]))"
  using assms
  apply -
  apply(subst manipulate_flts)
  apply(case_tac "a1")
       apply(simp)
      apply(simp)
     apply(case_tac "x32 = c")
      prefer 2
  apply(simp)
     prefer 2
     apply(case_tac "bnullable x42")
      apply(simp)
       apply(simp)

  apply(case_tac "a2")
            apply(simp)
         apply(simp)
        apply(case_tac "x32 = c")
         prefer 2
  apply(simp)
        apply(simp)
       apply(case_tac "bnullable x42a")
        apply(simp)
       apply(subst go_inside_flts)
  apply(simp)
        apply(simp)
       apply(simp)
      apply(simp)
      apply (simp add: WWW4)
      apply(simp)
      apply (simp add: WWW4)
  apply (simp add: go_inside_flts)
  apply (metis (no_types, lifting) go_inside_flts k0 list.simps(8) list.simps(9))
  by (smt bder.simps(6) flts.simps(1) flts.simps(6) flts.simps(7) go_inside_flts k0 list.inject list.simps(9))

lemma big0:
  shows "bsimp (AALT x51 (AALTs bs1 as1) (AALTs bs2 as2)) =
         bsimp (AALTs x51 ((map (fuse bs1) as1) @ (map (fuse bs2) as2)))"
  by (smt WWW3 bsimp.simps(2) k0 k00 list.simps(8) list.simps(9) map_append)

lemma bignA:
  shows "bsimp (AALTs x51 (AALTs bs1 as1 # as2)) =
         bsimp (AALTs x51 ((map (fuse bs1) as1) @ as2))"
  apply(simp)
  apply(subst k0)
  apply(subst WWW3)
  apply(simp add: flts_append)
  done

lemma XXX4ab:
  shows "good (bders_simp (bsimp r) s)  \<or> bders_simp (bsimp r) s = AZERO"
  apply(induct s arbitrary: r rule:  rev_induct)
   apply(simp)
  apply (simp add: good1)
  apply(simp add: bders_simp_append)
  apply (simp add: good1)
  done

lemma XXX4:
  assumes "good a"
  shows "bders_simp a s = bsimp (bders a s)"
  using  assms
  apply(induct s arbitrary: a rule: rev_induct)
   apply(simp)
   apply (simp add: test2)
  apply(simp add: bders_append bders_simp_append)
  oops

lemma MAINMAIN:
  "blexer r s = blexer_simp r s"
  apply(induct s arbitrary: r)
  apply(simp add: blexer_def blexer_simp_def)
  apply(simp add: blexer_def blexer_simp_def del: bders.simps bders_simp.simps)
  apply(auto simp del: bders.simps bders_simp.simps)
    prefer 2
  apply (metis b4 bders.simps(2) bders_simp.simps(2))
   prefer 2
  apply (metis b4 bders.simps(2))
  apply(subst bmkeps_simp)
   apply(simp)
  apply(case_tac s)
   apply(simp only: bders.simps)
   apply(subst bders_simp.simps)
  apply(simp)
  oops

lemma
  fixes n :: nat
  shows "(\<Sum>i \<in> {0..n}. i) = n * (n + 1) div 2"
  apply(induct n)
  apply(simp)
  apply(simp)
  done

end


theory GDPRGlobal imports SDL        (* Christoph Benzmüller & Xavier Parent, 2018 *)

begin (*** GDPR Example ***)
 consts process_data_lawfully::\<sigma> erase_data::\<sigma> kill_boss::\<sigma>

 axiomatization where
  (* It is an obligation to process data lawfully. *)
    A1: "\<lfloor>\<^bold>O\<^bold>\<langle>process_data_lawfully\<^bold>\<rangle>\<rfloor>"
  (* Given a situation where data is processed unlawfully. *) and
    A3: "\<lfloor>\<^bold>\<not>process_data_lawfully\<rfloor>"

 (*** Some Experiments ***)
  lemma True  nitpick [satisfy] nunchaku [satisfy] oops  (* Consistency-check: Is there a model? *)
  lemma False sledgehammer      oops  (* Inconsistency-check: Can Falsum be derived? *)

  lemma "\<lfloor>\<^bold>O\<^bold>\<langle>erase_data\<^bold>\<rangle>\<rfloor>"   sledgehammer nitpick oops  (* Should the data be erased? *)
  lemma "\<lfloor>\<^bold>O\<^bold>\<langle>\<^bold>\<not>erase_data\<^bold>\<rangle>\<rfloor>"  sledgehammer nitpick oops  (* Should the data be kept? *)
  lemma "\<lfloor>\<^bold>O\<^bold>\<langle>kill_boss\<^bold>\<rangle>\<rfloor>"    sledgehammer nitpick oops  (* Should the boss be killed? *)
end