theory Blexer

  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 sz where

  "sz ZERO = 0"

| "sz ONE = 0"

| "sz (CH _) = 0"

| "sz (SEQ r1 r2) = 1 + sz r1 + sz r2"

| "sz (ALT r1 r2) = 1 + sz r1 + sz r2"

| "sz (STAR r) = 1 + sz r"

| "sz (NTIMES r n) = 1 + (n + 1) + sz r"

fun 

  Stars_add :: "val \<Rightarrow> val \<Rightarrow> val"

where

  "Stars_add v (Stars vs) = Stars (v # vs)"

function (sequential)

  decode' :: "bit list \<Rightarrow> rexp \<Rightarrow> (val * bit list)"

where

  "decode' bs ZERO = (undefined, bs)"

| "decode' bs ONE = (Void, bs)"

| "decode' bs (CH d) = (Char d, bs)"

| "decode' [] (ALT r1 r2) = (Void, [])"

| "decode' (Z # bs) (ALT r1 r2) = (let (v, bs') = decode' bs r1 in (Left v, bs'))"

| "decode' (S # bs) (ALT r1 r2) = (let (v, bs') = decode' bs r2 in (Right v, bs'))"

| "decode' bs (SEQ r1 r2) = (let (v1, bs') = decode' bs r1 in

                             let (v2, bs'') = decode' bs' r2 in (Seq v1 v2, bs''))"

| "decode' [] (STAR r) = (Void, [])"

| "decode' (S # bs) (STAR r) = (Stars [], bs)"

| "decode' (Z # bs) (STAR r) = (let (v, bs') = decode' bs r in

                                    let (vs, bs'') = decode' bs' (STAR r) 

                                    in (Stars_add v vs, bs''))"

| "decode' bs (NTIMES r n) = decode' bs (STAR r)"

by pat_completeness auto

lemma decode'_smaller:

  assumes "decode'_dom (bs, r)"

  shows "length (snd (decode' bs r)) \<le> length bs"

using assms

apply(induct bs r)

apply(auto simp add: decode'.psimps split: prod.split)

using dual_order.trans apply blast

apply (meson dual_order.trans le_SucI)

  done

termination "decode'"  

apply(relation "inv_image (measure(%cs. sz cs) <*lex*> measure(%s. size s)) (%(ds,r). (r,ds))") 

apply(auto dest!: decode'_smaller)

   apply (metis less_Suc_eq_le snd_conv)

  done

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_NTIMES:

  assumes "\<forall>v\<in>set vs. \<Turnstile> v : r \<and> (\<forall>x. decode' (code v @ x) r = (v, x))" 

  shows "decode' (code (Stars vs) @ ds) (NTIMES r n) = (Stars vs, ds)"

  using assms

  apply(induct vs arbitrary: n r ds)

   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 rule: Prf.induct) 

  apply(auto)[6]

  using decode'_code_Stars apply blast

  apply(rule decode'_code_NTIMES)

  apply(simp)

  apply(auto)

  done

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)

datatype arexp = 

  AZERO

| AONE "bit list"

| ACHAR "bit list" char

| ASEQ "bit list" arexp arexp

| AALTs "bit list" "arexp list"

| ASTAR "bit list" arexp

| ANTIMES "bit list" arexp nat

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)"

| "asize (ANTIMES cs r n) = Suc (asize r) + n"

fun 

  erase :: "arexp \<Rightarrow> rexp"

where

  "erase AZERO = ZERO"

| "erase (AONE _) = ONE"

| "erase (ACHAR _ c) = CH c"

| "erase (AALTs _ []) = ZERO"

| "erase (AALTs _ [r]) = (erase r)"

| "erase (AALTs bs (r#rs)) = ALT (erase r) (erase (AALTs bs rs))"

| "erase (ASEQ _ r1 r2) = SEQ (erase r1) (erase r2)"

| "erase (ASTAR _ r) = STAR (erase r)"

| "erase (ANTIMES _ r n) = NTIMES (erase r) n"

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"

| "fuse bs (ANTIMES cs r n) = ANTIMES (bs @ cs) r n"

lemma fuse_append:

  shows "fuse (bs1 @ bs2) r = fuse bs1 (fuse bs2 r)"

  apply(induct r)

  apply(auto)

  done

fun intern :: "rexp \<Rightarrow> arexp" where

  "intern ZERO = AZERO"

| "intern ONE = AONE []"

| "intern (CH c) = ACHAR [] c"

| "intern (ALT r1 r2) = AALT [] (fuse [Z] (intern r1)) 

                                (fuse [S]  (intern r2))"

| "intern (SEQ r1 r2) = ASEQ [] (intern r1) (intern r2)"

| "intern (STAR r) = ASTAR [] (intern r)"

| "intern (NTIMES r n) = ANTIMES [] (intern r) n"

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)"

| "retrieve (ANTIMES bs r 0) (Stars []) = bs @ [S]"

| "retrieve (ANTIMES bs r (Suc n)) (Stars (v#vs)) = 

     bs @ [Z] @ retrieve r v @ retrieve (ANTIMES [] r n) (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"

| "bnullable (ANTIMES bs r n) = (if n  = 0 then True else bnullable r)"

abbreviation

  bnullables :: "arexp list \<Rightarrow> bool"

where

  "bnullables rs \<equiv> (\<exists>r \<in> set rs. bnullable r)"

function (sequential)

  bmkeps :: "arexp \<Rightarrow> bit list" 

where

  "bmkeps(AONE bs) = bs"

| "bmkeps(ASEQ bs r1 r2) = bs @ (bmkeps r1) @ (bmkeps r2)"

| "bmkeps(AALTs bs (r#rs)) = 

    (if bnullable(r) then (bs @ bmkeps r) else (bmkeps (AALTs bs rs)))"

| "bmkeps(ASTAR bs r) = bs @ [S]"

| "bmkeps(ANTIMES bs r n) = 

    (if n = 0 then bs @ [S] else bs @ [Z] @ (bmkeps r) @ bmkeps(ANTIMES [] r (n - 1)))"

apply(pat_completeness)

apply(auto)

done

termination "bmkeps"  

apply(relation "measure asize") 

        apply(auto)

  using asize.elims by force

fun

   bmkepss :: "arexp list \<Rightarrow> bit list"

where

  "bmkepss (r # rs) = (if bnullable(r) then (bmkeps r) else (bmkepss rs))"

lemma bmkepss1:

  assumes "\<not> bnullables rs1"

  shows "bmkepss (rs1 @ rs2) = bmkepss rs2"

  using assms

  by(induct rs1) (auto)

lemma bmkepss2:

  assumes "bnullables rs1"

  shows "bmkepss (rs1 @ rs2) = bmkepss rs1"

  using assms

  by (induct rs1) (auto)

fun

 bder :: "char \<Rightarrow> arexp \<Rightarrow> arexp"

where

  "bder c (AZERO) = AZERO"

| "bder c (AONE bs) = AZERO"

| "bder c (ACHAR bs d) = (if c = d then AONE bs else AZERO)"

| "bder c (AALTs bs rs) = AALTs bs (map (bder c) rs)"

| "bder c (ASEQ bs r1 r2) = 

     (if bnullable r1

      then AALT bs (ASEQ [] (bder c r1) r2) (fuse (bmkeps r1) (bder c r2))

      else ASEQ bs (bder c r1) r2)"

| "bder c (ASTAR bs r) = ASEQ (bs @ [Z]) (bder c r) (ASTAR [] r)"

| "bder c (ANTIMES bs r n) = (if n = 0 then AZERO else ASEQ (bs @ [Z]) (bder c r) (ANTIMES [] r (n - 1)))"

fun 

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

where

  "bders r [] = r"

| "bders r (c#s) = bders (bder c r) s"

lemma bders_append:

  "bders c (s1 @ s2) = bders (bders c s1) s2"

  apply(induct s1 arbitrary: c s2)

  apply(simp_all)

  done

lemma bnullable_correctness:

  shows "nullable (erase r) = bnullable r"

  apply(induct r rule: erase.induct)

  apply(simp_all)

  done

lemma erase_fuse:

  shows "erase (fuse bs r) = erase r"

  apply(induct r rule: erase.induct)

  apply(simp_all)

  done

lemma erase_intern [simp]:

  shows "erase (intern r) = r"

  apply(induct r)

  apply(simp_all add: erase_fuse)

  done

lemma erase_bder [simp]:

  shows "erase (bder a r) = der a (erase r)"

  apply(induct r rule: erase.induct)

  apply(simp_all add: erase_fuse bnullable_correctness)

  done

lemma erase_bders [simp]:

  shows "erase (bders r s) = ders s (erase r)"

  apply(induct s arbitrary: r )

  apply(simp_all)

  done

lemma bnullable_fuse:

  shows "bnullable (fuse bs r) = bnullable r"

  apply(induct r arbitrary: bs)

  apply(auto)

  done

lemma retrieve_encode_STARS:

  assumes "\<forall>v\<in>set vs. \<Turnstile> v : r \<and> code v = retrieve (intern r) v"

  shows "code (Stars vs) = retrieve (ASTAR [] (intern r)) (Stars vs)"

  using assms

  apply(induct vs)

  apply(simp_all)

  done

lemma retrieve_encode_NTIMES:

  assumes "\<forall>v\<in>set vs. \<Turnstile> v : r \<and> code v = retrieve (intern r) v" "length vs = n"

  shows "code (Stars vs) = retrieve (ANTIMES [] (intern r) n) (Stars vs)"

  using assms

  apply(induct vs arbitrary: n)

   apply(simp_all)

  by force

lemma retrieve_fuse2:

  assumes "\<Turnstile> v : (erase r)"

  shows "retrieve (fuse bs r) v = bs @ retrieve r v"

  using assms

  apply(induct r arbitrary: v bs)

  apply(auto elim: Prf_elims)[4]

  apply(case_tac x2a)

  apply(simp)

  using Prf_elims(1) apply blast

  apply(case_tac x2a)

  apply(simp)

  apply(simp)

  apply(case_tac list)

  apply(simp)

  apply(simp)

  apply (smt (verit, best) Prf_elims(3) append_assoc retrieve.simps(4) retrieve.simps(5))

  apply(simp)

  using retrieve_encode_STARS

  apply(auto elim!: Prf_elims)[1]

  apply(case_tac vs)

  apply(simp)

   apply(simp)

  (* NTIMES *)

  apply(auto elim!: Prf_elims)[1]

  apply(case_tac vs1)

   apply(simp_all)

  apply(case_tac vs2)

   apply(simp_all)

  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)

  apply(subst retrieve_encode_NTIMES)

    apply(auto)

  done 

lemma retrieve_AALTs_bnullable1:

  assumes "bnullable r"

  shows "retrieve (AALTs bs (r # rs)) (mkeps (erase (AALTs bs (r # rs))))

         = bs @ retrieve r (mkeps (erase r))"

  using assms

  apply(case_tac rs)

  apply(auto simp add: bnullable_correctness)

  done

lemma retrieve_AALTs_bnullable2:

  assumes "\<not>bnullable r" "bnullables rs"

  shows "retrieve (AALTs bs (r # rs)) (mkeps (erase (AALTs bs (r # rs))))

         = retrieve (AALTs bs rs) (mkeps (erase (AALTs bs rs)))"

  using assms

  apply(induct rs arbitrary: r bs)

  apply(auto)

  using bnullable_correctness apply blast

  apply(case_tac rs)

  apply(auto)

  using bnullable_correctness apply blast

  apply(case_tac rs)

  apply(auto)

  done

lemma bmkeps_retrieve_AALTs: 

  assumes "\<forall>r \<in> set rs. bnullable r \<longrightarrow> bmkeps r = retrieve r (mkeps (erase r))" 

          "bnullables rs"

  shows "bs @ bmkepss rs = retrieve (AALTs bs rs) (mkeps (erase (AALTs bs rs)))"

 using assms

  apply(induct rs arbitrary: bs)

  apply(auto)

  using retrieve_AALTs_bnullable1 apply presburger

  apply (metis retrieve_AALTs_bnullable2)

  apply (simp add: retrieve_AALTs_bnullable1)

  by (metis retrieve_AALTs_bnullable2)

lemma bmkeps_retrieve_ANTIMES: 

  assumes "if n = 0 then True else bmkeps r = retrieve r (mkeps (erase r))" 

  and     "bnullable (ANTIMES bs r n)"

  shows "bmkeps (ANTIMES bs r n) = retrieve (ANTIMES bs r n) (Stars (replicate n (mkeps (erase r))))"

 using assms

  apply(induct n arbitrary: r bs)

  apply(auto)[1]

  apply(simp)

  done

lemma bmkeps_retrieve:

  assumes "bnullable r"

  shows "bmkeps r = retrieve r (mkeps (erase r))"

  using assms

  apply(induct r rule: bmkeps.induct)

        apply(auto)

  apply (simp add: retrieve_AALTs_bnullable1)

  using retrieve_AALTs_bnullable1 apply force

  apply(metis retrieve_AALTs_bnullable2)

  by (metis Cons_eq_appendI One_nat_def Suc_diff_1 eq_Nil_appendI replicate_Suc retrieve.simps(10))  

lemma bder_retrieve:

  assumes "\<Turnstile> v : der c (erase r)"

  shows "retrieve (bder c r) v = retrieve r (injval (erase r) c v)"

  using assms  

  apply(induct r arbitrary: v rule: erase.induct)

  using Prf_elims(1) apply auto[1]

  using Prf_elims(1) apply auto[1]

  apply(auto)[1]

  apply (metis Prf_elims(4) injval.simps(1) retrieve.simps(1) retrieve.simps(2))

  using Prf_elims(1) apply blast

  (* AALTs case *)

  apply(simp)

  apply(erule Prf_elims)

  apply(simp)

  apply(simp)

  apply(rename_tac "r\<^sub>1" "r\<^sub>2" rs v)

  apply(erule Prf_elims)

  apply(simp)

  apply(simp)

  apply(case_tac rs)

  apply(simp)

  apply(simp)

  using Prf_elims(3) apply fastforce

  (* ASEQ case *) 

  apply(simp)

  apply(case_tac "nullable (erase r1)")

  apply(simp)

  apply(erule Prf_elims)

  using Prf_elims(2) bnullable_correctness apply force

  apply (simp add: bmkeps_retrieve bnullable_correctness retrieve_fuse2)

  apply (simp add: bmkeps_retrieve bnullable_correctness retrieve_fuse2)

  using Prf_elims(2) apply force

  (* ASTAR case *)  

  apply(rename_tac bs r v)

  apply(simp)  

  apply(erule Prf_elims)

  apply(clarify)

  apply(erule Prf_elims)

  apply(clarify)

   apply (simp add: retrieve_fuse2)

  (* ANTIMES case *)

  apply(auto)  

  apply(erule Prf_elims)

  apply(erule Prf_elims)

  apply(clarify)

  apply(erule Prf_elims)

  apply(clarify)

  by (metis (full_types) Suc_pred append_assoc injval.simps(8) retrieve.simps(10) retrieve.simps(6))

lemma MAIN_decode:

  assumes "\<Turnstile> v : ders s r"

  shows "Some (flex r id s v) = decode (retrieve (bders (intern r) s) v) r"

  using assms

proof (induct s arbitrary: v rule: rev_induct)

  case Nil

  have "\<Turnstile> v : ders [] r" by fact

  then have "\<Turnstile> v : r" by simp

  then have "Some v = decode (retrieve (intern r) v) r"

    using decode_code retrieve_code by auto

  then show "Some (flex r id [] v) = decode (retrieve (bders (intern r) []) v) r"

    by simp

next

  case (snoc c s v)

  have IH: "\<And>v. \<Turnstile> v : ders s r \<Longrightarrow> 

     Some (flex r id s v) = decode (retrieve (bders (intern r) s) v) r" by fact

  have asm: "\<Turnstile> v : ders (s @ [c]) r" by fact

  then have asm2: "\<Turnstile> injval (ders s r) c v : ders s r" 

    by (simp add: Prf_injval ders_append)

  have "Some (flex r id (s @ [c]) v) = Some (flex r id s (injval (ders s r) c v))"

    by (simp add: flex_append)

  also have "... = decode (retrieve (bders (intern r) s) (injval (ders s r) c v)) r"

    using asm2 IH by simp

  also have "... = decode (retrieve (bder c (bders (intern r) s)) v) r"

    using asm by (simp_all add: bder_retrieve ders_append)

  finally show "Some (flex r id (s @ [c]) v) = 

                 decode (retrieve (bders (intern r) (s @ [c])) v) r" by (simp add: bders_append)

qed

definition blexer where

 "blexer r s \<equiv> if bnullable (bders (intern r) s) then 

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

lemma blexer_correctness:

  shows "blexer r s = lexer r s"

proof -