theory LexerExt
  imports SpecExt 
begin


section {* The Lexer Functions by Sulzmann and Lu  *}

fun 
  mkeps :: "rexp \<Rightarrow> val"
where
  "mkeps(ONE) = Void"
| "mkeps(SEQ r1 r2) = Seq (mkeps r1) (mkeps r2)"
| "mkeps(ALT r1 r2) = (if nullable(r1) then Left (mkeps r1) else Right (mkeps r2))"
| "mkeps(STAR r) = Stars []"
| "mkeps(UPNTIMES r n) = Stars []"
| "mkeps(NTIMES r n) = Stars (replicate n (mkeps r))"
| "mkeps(FROMNTIMES r n) = Stars (replicate n (mkeps r))"
| "mkeps(NMTIMES r n m) = Stars (replicate n (mkeps r))"
  
fun injval :: "rexp \<Rightarrow> char \<Rightarrow> val \<Rightarrow> val"
where
  "injval (CHAR d) c Void = Char d"
| "injval (ALT r1 r2) c (Left v1) = Left(injval r1 c v1)"
| "injval (ALT r1 r2) c (Right v2) = Right(injval r2 c v2)"
| "injval (SEQ r1 r2) c (Seq v1 v2) = Seq (injval r1 c v1) v2"
| "injval (SEQ r1 r2) c (Left (Seq v1 v2)) = Seq (injval r1 c v1) v2"
| "injval (SEQ r1 r2) c (Right v2) = Seq (mkeps r1) (injval r2 c v2)"
| "injval (STAR r) c (Seq v (Stars vs)) = Stars ((injval r c v) # vs)" 
| "injval (NTIMES r n) c (Seq v (Stars vs)) = Stars ((injval r c v) # vs)" 
| "injval (FROMNTIMES r n) c (Seq v (Stars vs)) = Stars ((injval r c v) # vs)" 
| "injval (UPNTIMES r n) c (Seq v (Stars vs)) = Stars ((injval r c v) # vs)" 
| "injval (NMTIMES r n m) c (Seq v (Stars vs)) = Stars ((injval r c v) # vs)" 
  
fun 
  lexer :: "rexp \<Rightarrow> string \<Rightarrow> val option"
where
  "lexer r [] = (if nullable r then Some(mkeps r) else None)"
| "lexer r (c#s) = (case (lexer (der c r) s) of  
                    None \<Rightarrow> None
                  | Some(v) \<Rightarrow> Some(injval r c v))"



section {* Mkeps, Injval Properties *}

lemma mkeps_flat:
  assumes "nullable(r)" 
  shows "flat (mkeps r) = []"
using assms
  apply(induct rule: nullable.induct) 
         apply(auto)
  by presburger  
  
  
lemma mkeps_nullable:
  assumes "nullable(r)" 
  shows "\<Turnstile> mkeps r : r"
using assms
apply(induct rule: nullable.induct) 
         apply(auto intro: Prf.intros split: if_splits)
  using Prf.intros(8) apply force
     apply(subst append.simps(1)[symmetric])
     apply(rule Prf.intros)
       apply(simp)
      apply(simp)
       apply (simp add: mkeps_flat)
      apply(simp)
  using Prf.intros(9) apply force
    apply(subst append.simps(1)[symmetric])
     apply(rule Prf.intros)
       apply(simp)
      apply(simp)
       apply (simp add: mkeps_flat)
    apply(simp)
  using Prf.intros(11) apply force
    apply(subst append.simps(1)[symmetric])
     apply(rule Prf.intros)
       apply(simp)
      apply(simp)
    apply (simp add: mkeps_flat)
   apply(simp)
  apply(simp)
done
    

lemma Prf_injval_flat:
  assumes "\<Turnstile> v : der c r" 
  shows "flat (injval r c v) = c # (flat v)"
using assms
apply(induct arbitrary: v rule: der.induct)
apply(auto elim!: Prf_elims intro: mkeps_flat split: if_splits)
done

lemma Prf_injval:
  assumes "\<Turnstile> v : der c r" 
  shows "\<Turnstile> (injval r c v) : r"
using assms
apply(induct r arbitrary: c v rule: rexp.induct)
apply(auto intro!: Prf.intros mkeps_nullable elim!: Prf_elims split: if_splits)[6]
    apply(simp add: Prf_injval_flat)
   apply(simp)
  apply(case_tac x2)
    apply(simp)
  apply(erule Prf_elims)
   apply(simp)
   apply(erule Prf_elims)
   apply(simp)
  apply(erule Prf_elims)
   apply(simp)
  using Prf.intros(7) Prf_injval_flat apply auto[1]
    apply(simp)
    apply(case_tac x2)
     apply(simp)
    apply(erule Prf_elims)
    apply(simp)
    apply(erule Prf_elims)
   apply(simp)
  apply(erule Prf_elims)
   apply(simp)
    apply(subst append.simps(2)[symmetric])
    apply(rule Prf.intros)
      apply(simp add: Prf_injval_flat)
     apply(simp)
    apply(simp)
    prefer 2
   apply(simp)
   apply(case_tac "x3a < x2")
    apply(simp)
    apply(erule Prf_elims)
   apply(simp)
    apply(case_tac x2)
    apply(simp)
    apply(case_tac x3a)
     apply(simp)
    apply(erule Prf_elims)
    apply(simp)
    apply(erule Prf_elims)
    apply(simp)
    apply(erule Prf_elims)
    apply(simp)
  using Prf.intros(12) Prf_injval_flat apply auto[1]
   apply(simp)
    apply(erule Prf_elims)
   apply(simp)
    apply(erule Prf_elims)
    apply(simp)
    apply(subst append.simps(2)[symmetric])
    apply(rule Prf.intros)
     apply(simp add: Prf_injval_flat)
     apply(simp)
     apply(simp)
    apply(simp)
   apply(simp)
  using Prf.intros(12) Prf_injval_flat apply auto[1]
    apply(case_tac x2)
   apply(simp)
    apply(erule Prf_elims)
   apply(simp)
    apply(erule Prf_elims)
    apply(simp_all)
    apply (simp add: Prf.intros(10) Prf_injval_flat)
  using Prf.intros(10) Prf_injval_flat apply auto[1]
  apply(erule Prf_elims)
  apply(simp)
    apply(erule Prf_elims)
    apply(simp_all)
    apply(subst append.simps(2)[symmetric])
    apply(rule Prf.intros)
     apply(simp add: Prf_injval_flat)
     apply(simp)
   apply(simp)
done



text {*
  Mkeps and injval produce, or preserve, Posix values.
*}

lemma Posix_mkeps:
  assumes "nullable r"
  shows "[] \<in> r \<rightarrow> mkeps r"
using assms
apply(induct r rule: nullable.induct)
apply(auto intro: Posix.intros simp add: nullable_correctness Sequ_def)
apply(subst append.simps(1)[symmetric])
apply(rule Posix.intros)
      apply(auto)
  done
    

lemma Posix_injval:
  assumes "s \<in> (der c r) \<rightarrow> v" 
  shows "(c # s) \<in> r \<rightarrow> (injval r c v)"
using assms
proof(induct r arbitrary: s v rule: rexp.induct)
  case ZERO
  have "s \<in> der c ZERO \<rightarrow> v" by fact
  then have "s \<in> ZERO \<rightarrow> v" by simp
  then have "False" by cases
  then show "(c # s) \<in> ZERO \<rightarrow> (injval ZERO c v)" by simp
next
  case ONE
  have "s \<in> der c ONE \<rightarrow> v" by fact
  then have "s \<in> ZERO \<rightarrow> v" by simp
  then have "False" by cases
  then show "(c # s) \<in> ONE \<rightarrow> (injval ONE c v)" by simp
next 
  case (CHAR d)
  consider (eq) "c = d" | (ineq) "c \<noteq> d" by blast
  then show "(c # s) \<in> (CHAR d) \<rightarrow> (injval (CHAR d) c v)"
  proof (cases)
    case eq
    have "s \<in> der c (CHAR d) \<rightarrow> v" by fact
    then have "s \<in> ONE \<rightarrow> v" using eq by simp
    then have eqs: "s = [] \<and> v = Void" by cases simp
    show "(c # s) \<in> CHAR d \<rightarrow> injval (CHAR d) c v" using eq eqs 
    by (auto intro: Posix.intros)
  next
    case ineq
    have "s \<in> der c (CHAR d) \<rightarrow> v" by fact
    then have "s \<in> ZERO \<rightarrow> v" using ineq by simp
    then have "False" by cases
    then show "(c # s) \<in> CHAR d \<rightarrow> injval (CHAR d) c v" by simp
  qed
next
  case (ALT r1 r2)
  have IH1: "\<And>s v. s \<in> der c r1 \<rightarrow> v \<Longrightarrow> (c # s) \<in> r1 \<rightarrow> injval r1 c v" by fact
  have IH2: "\<And>s v. s \<in> der c r2 \<rightarrow> v \<Longrightarrow> (c # s) \<in> r2 \<rightarrow> injval r2 c v" by fact
  have "s \<in> der c (ALT r1 r2) \<rightarrow> v" by fact
  then have "s \<in> ALT (der c r1) (der c r2) \<rightarrow> v" by simp
  then consider (left) v' where "v = Left v'" "s \<in> der c r1 \<rightarrow> v'" 
              | (right) v' where "v = Right v'" "s \<notin> L (der c r1)" "s \<in> der c r2 \<rightarrow> v'" 
              by cases auto
  then show "(c # s) \<in> ALT r1 r2 \<rightarrow> injval (ALT r1 r2) c v"
  proof (cases)
    case left
    have "s \<in> der c r1 \<rightarrow> v'" by fact
    then have "(c # s) \<in> r1 \<rightarrow> injval r1 c v'" using IH1 by simp
    then have "(c # s) \<in> ALT r1 r2 \<rightarrow> injval (ALT r1 r2) c (Left v')" by (auto intro: Posix.intros)
    then show "(c # s) \<in> ALT r1 r2 \<rightarrow> injval (ALT r1 r2) c v" using left by simp
  next 
    case right
    have "s \<notin> L (der c r1)" by fact
    then have "c # s \<notin> L r1" by (simp add: der_correctness Der_def)
    moreover 
    have "s \<in> der c r2 \<rightarrow> v'" by fact
    then have "(c # s) \<in> r2 \<rightarrow> injval r2 c v'" using IH2 by simp
    ultimately have "(c # s) \<in> ALT r1 r2 \<rightarrow> injval (ALT r1 r2) c (Right v')" 
      by (auto intro: Posix.intros)
    then show "(c # s) \<in> ALT r1 r2 \<rightarrow> injval (ALT r1 r2) c v" using right by simp
  qed
next
  case (SEQ r1 r2)
  have IH1: "\<And>s v. s \<in> der c r1 \<rightarrow> v \<Longrightarrow> (c # s) \<in> r1 \<rightarrow> injval r1 c v" by fact
  have IH2: "\<And>s v. s \<in> der c r2 \<rightarrow> v \<Longrightarrow> (c # s) \<in> r2 \<rightarrow> injval r2 c v" by fact
  have "s \<in> der c (SEQ r1 r2) \<rightarrow> v" by fact
  then consider 
        (left_nullable) v1 v2 s1 s2 where 
        "v = Left (Seq v1 v2)"  "s = s1 @ s2" 
        "s1 \<in> der c r1 \<rightarrow> v1" "s2 \<in> r2 \<rightarrow> v2" "nullable r1" 
        "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r1) \<and> s\<^sub>4 \<in> L r2)"
      | (right_nullable) v1 s1 s2 where 
        "v = Right v1" "s = s1 @ s2"  
        "s \<in> der c r2 \<rightarrow> v1" "nullable r1" "s1 @ s2 \<notin> L (SEQ (der c r1) r2)"
      | (not_nullable) v1 v2 s1 s2 where
        "v = Seq v1 v2" "s = s1 @ s2" 
        "s1 \<in> der c r1 \<rightarrow> v1" "s2 \<in> r2 \<rightarrow> v2" "\<not>nullable r1" 
        "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r1) \<and> s\<^sub>4 \<in> L r2)"
        by (force split: if_splits elim!: Posix_elims simp add: Sequ_def der_correctness Der_def)   
  then show "(c # s) \<in> SEQ r1 r2 \<rightarrow> injval (SEQ r1 r2) c v" 
    proof (cases)
      case left_nullable
      have "s1 \<in> der c r1 \<rightarrow> v1" by fact
      then have "(c # s1) \<in> r1 \<rightarrow> injval r1 c v1" using IH1 by simp
      moreover
      have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r1) \<and> s\<^sub>4 \<in> L r2)" by fact
      then have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (c # s1) @ s\<^sub>3 \<in> L r1 \<and> s\<^sub>4 \<in> L r2)" by (simp add: der_correctness Der_def)
      ultimately have "((c # s1) @ s2) \<in> SEQ r1 r2 \<rightarrow> Seq (injval r1 c v1) v2" using left_nullable by (rule_tac Posix.intros)
      then show "(c # s) \<in> SEQ r1 r2 \<rightarrow> injval (SEQ r1 r2) c v" using left_nullable by simp
    next
      case right_nullable
      have "nullable r1" by fact
      then have "[] \<in> r1 \<rightarrow> (mkeps r1)" by (rule Posix_mkeps)
      moreover
      have "s \<in> der c r2 \<rightarrow> v1" by fact
      then have "(c # s) \<in> r2 \<rightarrow> (injval r2 c v1)" using IH2 by simp
      moreover
      have "s1 @ s2 \<notin> L (SEQ (der c r1) r2)" by fact
      then have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = c # s \<and> [] @ s\<^sub>3 \<in> L r1 \<and> s\<^sub>4 \<in> L r2)" using right_nullable
        by(auto simp add: der_correctness Der_def append_eq_Cons_conv Sequ_def)
      ultimately have "([] @ (c # s)) \<in> SEQ r1 r2 \<rightarrow> Seq (mkeps r1) (injval r2 c v1)"
      by(rule Posix.intros)
      then show "(c # s) \<in> SEQ r1 r2 \<rightarrow> injval (SEQ r1 r2) c v" using right_nullable by simp
    next
      case not_nullable
      have "s1 \<in> der c r1 \<rightarrow> v1" by fact
      then have "(c # s1) \<in> r1 \<rightarrow> injval r1 c v1" using IH1 by simp
      moreover
      have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r1) \<and> s\<^sub>4 \<in> L r2)" by fact
      then have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (c # s1) @ s\<^sub>3 \<in> L r1 \<and> s\<^sub>4 \<in> L r2)" by (simp add: der_correctness Der_def)
      ultimately have "((c # s1) @ s2) \<in> SEQ r1 r2 \<rightarrow> Seq (injval r1 c v1) v2" using not_nullable 
        by (rule_tac Posix.intros) (simp_all) 
      then show "(c # s) \<in> SEQ r1 r2 \<rightarrow> injval (SEQ r1 r2) c v" using not_nullable by simp
    qed
next
case (UPNTIMES r n s v)
  have IH: "\<And>s v. s \<in> der c r \<rightarrow> v \<Longrightarrow> (c # s) \<in> r \<rightarrow> injval r c v" by fact
  have "s \<in> der c (UPNTIMES r n) \<rightarrow> v" by fact
  then consider
      (cons) v1 vs s1 s2 where 
        "v = Seq v1 (Stars vs)" "s = s1 @ s2" 
        "s1 \<in> der c r \<rightarrow> v1" "s2 \<in> (UPNTIMES r (n - 1)) \<rightarrow> (Stars vs)" "0 < n"
        "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r) \<and> s\<^sub>4 \<in> L (UPNTIMES r (n - 1)))" 
    (* here *)
    apply(auto elim: Posix_elims simp add: der_correctness Der_def intro: Posix.intros split: if_splits)
    apply(erule Posix_elims)
    apply(simp)
    apply(subgoal_tac "\<exists>vss. v2 = Stars vss")
    apply(clarify)
    apply(drule_tac x="v1" in meta_spec)
    apply(drule_tac x="vss" in meta_spec)
    apply(drule_tac x="s1" in meta_spec)
    apply(drule_tac x="s2" in meta_spec)
     apply(simp add: der_correctness Der_def)
    apply(erule Posix_elims)
     apply(auto)
      done
    then show "(c # s) \<in> (UPNTIMES r n) \<rightarrow> injval (UPNTIMES r n) c v" 
    proof (cases)
      case cons
          have "s1 \<in> der c r \<rightarrow> v1" by fact
          then have "(c # s1) \<in> r \<rightarrow> injval r c v1" using IH by simp
        moreover
          have "s2 \<in> (UPNTIMES r (n - 1)) \<rightarrow> Stars vs" by fact
        moreover 
          have "(c # s1) \<in> r \<rightarrow> injval r c v1" by fact 
          then have "flat (injval r c v1) = (c # s1)" by (rule Posix1)
          then have "flat (injval r c v1) \<noteq> []" by simp
        moreover 
          have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r) \<and> s\<^sub>4 \<in> L (UPNTIMES r (n - 1)))" by fact
          then have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (c # s1) @ s\<^sub>3 \<in> L r \<and> s\<^sub>4 \<in> L (UPNTIMES r (n - 1)))" 
            by (simp add: der_correctness Der_def)
        ultimately 
        have "((c # s1) @ s2) \<in> UPNTIMES r n \<rightarrow> Stars (injval r c v1 # vs)" 
           thm Posix.intros
           apply (rule_tac Posix.intros)
               apply(simp_all)
              apply(case_tac n)
            apply(simp)
           using Posix_elims(1) UPNTIMES.prems apply auto[1]
             apply(simp)
             done
        then show "(c # s) \<in> UPNTIMES r n \<rightarrow> injval (UPNTIMES r n) c v" using cons by(simp)
      qed
    next
      case (STAR r)
  have IH: "\<And>s v. s \<in> der c r \<rightarrow> v \<Longrightarrow> (c # s) \<in> r \<rightarrow> injval r c v" by fact
  have "s \<in> der c (STAR r) \<rightarrow> v" by fact
  then consider
      (cons) v1 vs s1 s2 where 
        "v = Seq v1 (Stars vs)" "s = s1 @ s2" 
        "s1 \<in> der c r \<rightarrow> v1" "s2 \<in> (STAR r) \<rightarrow> (Stars vs)"
        "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r) \<and> s\<^sub>4 \<in> L (STAR r))" 
        apply(auto elim!: Posix_elims(1-5) simp add: der_correctness Der_def intro: Posix.intros)
        apply(rotate_tac 3)
        apply(erule_tac Posix_elims(6))
        apply (simp add: Posix.intros(6))
        using Posix.intros(7) by blast
    then show "(c # s) \<in> STAR r \<rightarrow> injval (STAR r) c v" 
    proof (cases)
      case cons
          have "s1 \<in> der c r \<rightarrow> v1" by fact
          then have "(c # s1) \<in> r \<rightarrow> injval r c v1" using IH by simp
        moreover
          have "s2 \<in> STAR r \<rightarrow> Stars vs" by fact
        moreover 
          have "(c # s1) \<in> r \<rightarrow> injval r c v1" by fact 
          then have "flat (injval r c v1) = (c # s1)" by (rule Posix1)
          then have "flat (injval r c v1) \<noteq> []" by simp
        moreover 
          have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r) \<and> s\<^sub>4 \<in> L (STAR r))" by fact
          then have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (c # s1) @ s\<^sub>3 \<in> L r \<and> s\<^sub>4 \<in> L (STAR r))" 
            by (simp add: der_correctness Der_def)
        ultimately 
        have "((c # s1) @ s2) \<in> STAR r \<rightarrow> Stars (injval r c v1 # vs)" by (rule Posix.intros)
        then show "(c # s) \<in> STAR r \<rightarrow> injval (STAR r) c v" using cons by(simp)
    qed
  next
    case (NTIMES r n s v)
     have IH: "\<And>s v. s \<in> der c r \<rightarrow> v \<Longrightarrow> (c # s) \<in> r \<rightarrow> injval r c v" by fact
  have "s \<in> der c (NTIMES r n) \<rightarrow> v" by fact
  then consider
      (cons) v1 vs s1 s2 where 
        "v = Seq v1 (Stars vs)" "s = s1 @ s2" 
        "s1 \<in> der c r \<rightarrow> v1" "s2 \<in> (NTIMES r (n - 1)) \<rightarrow> (Stars vs)" "0 < n"
        "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r) \<and> s\<^sub>4 \<in> L (NTIMES r (n - 1)))" 
    apply(auto elim: Posix_elims simp add: der_correctness Der_def intro: Posix.intros split: if_splits)
    apply(erule Posix_elims)
    apply(simp)
    apply(subgoal_tac "\<exists>vss. v2 = Stars vss")
    apply(clarify)
    apply(drule_tac x="v1" in meta_spec)
    apply(drule_tac x="vss" in meta_spec)
    apply(drule_tac x="s1" in meta_spec)
    apply(drule_tac x="s2" in meta_spec)
     apply(simp add: der_correctness Der_def)
    apply(erule Posix_elims)
     apply(auto)
      done
    then show "(c # s) \<in> (NTIMES r n) \<rightarrow> injval (NTIMES r n) c v" 
    proof (cases)
      case cons
          have "s1 \<in> der c r \<rightarrow> v1" by fact
          then have "(c # s1) \<in> r \<rightarrow> injval r c v1" using IH by simp
        moreover
          have "s2 \<in> (NTIMES r (n - 1)) \<rightarrow> Stars vs" by fact
        moreover 
          have "(c # s1) \<in> r \<rightarrow> injval r c v1" by fact 
          then have "flat (injval r c v1) = (c # s1)" by (rule Posix1)
          then have "flat (injval r c v1) \<noteq> []" by simp
        moreover 
          have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r) \<and> s\<^sub>4 \<in> L (NTIMES r (n - 1)))" by fact
          then have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (c # s1) @ s\<^sub>3 \<in> L r \<and> s\<^sub>4 \<in> L (NTIMES r (n - 1)))" 
            by (simp add: der_correctness Der_def)
        ultimately 
        have "((c # s1) @ s2) \<in> NTIMES r n \<rightarrow> Stars (injval r c v1 # vs)" 
           apply (rule_tac Posix.intros)
               apply(simp_all)
              apply(case_tac n)
            apply(simp)
           using Posix_elims(1) NTIMES.prems apply auto[1]
             apply(simp)
             done
        then show "(c # s) \<in> NTIMES r n \<rightarrow> injval (NTIMES r n) c v" using cons by(simp)
      qed  
  next
    case (FROMNTIMES r n s v)
  have IH: "\<And>s v. s \<in> der c r \<rightarrow> v \<Longrightarrow> (c # s) \<in> r \<rightarrow> injval r c v" by fact
  have "s \<in> der c (FROMNTIMES r n) \<rightarrow> v" by fact
  then consider
      (cons) v1 vs s1 s2 where 
        "v = Seq v1 (Stars vs)" "s = s1 @ s2" 
        "s1 \<in> der c r \<rightarrow> v1" "s2 \<in> (FROMNTIMES r (n - 1)) \<rightarrow> (Stars vs)" "0 < n"
        "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r) \<and> s\<^sub>4 \<in> L (FROMNTIMES r (n - 1)))"
     | (null) v1 vs s1 s2 where 
        "v = Seq v1 (Stars vs)" "s = s1 @ s2"  "s2 \<in> (STAR r) \<rightarrow> (Stars vs)" 
        "s1 \<in> der c r \<rightarrow> v1" "n = 0"
         "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r) \<and> s\<^sub>4 \<in> L (STAR r))"  
    apply(auto elim: Posix_elims simp add: der_correctness Der_def intro: Posix.intros split: if_splits)
    prefer 2
    apply(erule Posix_elims)
    apply(simp)
    apply(subgoal_tac "\<exists>vss. v2 = Stars vss")
    apply(clarify)
    apply(drule_tac x="v1" in meta_spec)
    apply(drule_tac x="vss" in meta_spec)
    apply(drule_tac x="s1" in meta_spec)
    apply(drule_tac x="s2" in meta_spec)
     apply(simp add: der_correctness Der_def)
      apply(rotate_tac 5)
    apply(erule Posix_elims)
      apply(auto)[2]
    apply(erule Posix_elims)
      apply(simp)
     apply blast
    apply(erule Posix_elims)
    apply(auto)
      apply(auto elim: Posix_elims simp add: der_correctness Der_def intro: Posix.intros split: if_splits)      
    apply(subgoal_tac "\<exists>vss. v2 = Stars vss")
     apply(clarify)
    apply simp
      apply(rotate_tac 6)
    apply(erule Posix_elims)
      apply(auto)[2]
    done
    then show "(c # s) \<in> (FROMNTIMES r n) \<rightarrow> injval (FROMNTIMES r n) c v" 
    proof (cases)
      case cons
          have "s1 \<in> der c r \<rightarrow> v1" by fact
          then have "(c # s1) \<in> r \<rightarrow> injval r c v1" using IH by simp
        moreover
          have "s2 \<in> (FROMNTIMES r (n - 1)) \<rightarrow> Stars vs" by fact
        moreover 
          have "(c # s1) \<in> r \<rightarrow> injval r c v1" by fact 
          then have "flat (injval r c v1) = (c # s1)" by (rule Posix1)
          then have "flat (injval r c v1) \<noteq> []" by simp
        moreover 
          have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r) \<and> s\<^sub>4 \<in> L (FROMNTIMES r (n - 1)))" by fact
          then have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (c # s1) @ s\<^sub>3 \<in> L r \<and> s\<^sub>4 \<in> L (FROMNTIMES r (n - 1)))" 
            by (simp add: der_correctness Der_def)
        ultimately 
        have "((c # s1) @ s2) \<in> FROMNTIMES r n \<rightarrow> Stars (injval r c v1 # vs)" 
           apply (rule_tac Posix.intros)
               apply(simp_all)
              apply(case_tac n)
            apply(simp)
          using Posix_elims(1) FROMNTIMES.prems apply auto[1]
          using cons(5) apply blast
             apply(simp)
             done
        then show "(c # s) \<in> FROMNTIMES r n \<rightarrow> injval (FROMNTIMES r n) c v" using cons by(simp)
      next 
       case null
          have "s1 \<in> der c r \<rightarrow> v1" by fact
          then have "(c # s1) \<in> r \<rightarrow> injval r c v1" using IH by simp
          moreover 
            have "s2 \<in> STAR r \<rightarrow> Stars vs" by fact
          moreover 
          have "(c # s1) \<in> r \<rightarrow> injval r c v1" by fact 
          then have "flat (injval r c v1) = (c # s1)" by (rule Posix1)
          then have "flat (injval r c v1) \<noteq> []" by simp
          moreover
             moreover 
          have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r) \<and> s\<^sub>4 \<in> L (STAR r))" by fact
          then have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (c # s1) @ s\<^sub>3 \<in> L r \<and> s\<^sub>4 \<in> L (STAR r))" 
            by (simp add: der_correctness Der_def)
        ultimately 
        have "((c # s1) @ s2) \<in> FROMNTIMES r 0 \<rightarrow> Stars (injval r c v1 # vs)" 
           apply (rule_tac Posix.intros) back
             apply(simp_all)
           done
        then show "(c # s) \<in> FROMNTIMES r n \<rightarrow> injval (FROMNTIMES r n) c v" using null 
          apply(simp)
          done  
      qed  
  next
    case (NMTIMES r n m s v)
      have IH: "\<And>s v. s \<in> der c r \<rightarrow> v \<Longrightarrow> (c # s) \<in> r \<rightarrow> injval r c v" by fact
  have "s \<in> der c (NMTIMES r n m) \<rightarrow> v" by fact
  then consider
      (cons) v1 vs s1 s2 where 
        "v = Seq v1 (Stars vs)" "s = s1 @ s2" 
        "s1 \<in> der c r \<rightarrow> v1" "s2 \<in> (NMTIMES r (n - 1) (m - 1)) \<rightarrow> (Stars vs)" "0 < n" "n \<le> m"
        "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r) \<and> s\<^sub>4 \<in> L (NMTIMES r (n - 1) (m - 1)))"
     | (null) v1 vs s1 s2 where 
        "v = Seq v1 (Stars vs)" "s = s1 @ s2"  "s2 \<in> (UPNTIMES r (m - 1)) \<rightarrow> (Stars vs)" 
        "s1 \<in> der c r \<rightarrow> v1" "n = 0" "0 < m"
         "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r) \<and> s\<^sub>4 \<in> L (UPNTIMES r (m - 1)))"  
    apply(auto elim: Posix_elims simp add: der_correctness Der_def intro: Posix.intros split: if_splits)
    prefer 2
    apply(erule Posix_elims)
    apply(simp)
    apply(subgoal_tac "\<exists>vss. v2 = Stars vss")
    apply(clarify)
    apply(drule_tac x="v1" in meta_spec)
    apply(drule_tac x="vss" in meta_spec)
    apply(drule_tac x="s1" in meta_spec)
    apply(drule_tac x="s2" in meta_spec)
     apply(simp add: der_correctness Der_def)
      apply(rotate_tac 5)
    apply(erule Posix_elims)
      apply(auto)[2]
    apply(erule Posix_elims)
      apply(simp)
     apply blast
      
    apply(erule Posix_elims)
    apply(auto)
      apply(auto elim: Posix_elims simp add: der_correctness Der_def intro: Posix.intros split: if_splits)      
    apply(subgoal_tac "\<exists>vss. v2 = Stars vss")
     apply(clarify)
    apply simp
      apply(rotate_tac 6)
    apply(erule Posix_elims)
      apply(auto)[2]
    done
    then show "(c # s) \<in> (NMTIMES r n m) \<rightarrow> injval (NMTIMES r n m) c v" 
    proof (cases)
      case cons
          have "s1 \<in> der c r \<rightarrow> v1" by fact
          then have "(c # s1) \<in> r \<rightarrow> injval r c v1" using IH by simp
        moreover
          have "s2 \<in> (NMTIMES r (n - 1) (m - 1)) \<rightarrow> Stars vs" by fact
        moreover 
          have "(c # s1) \<in> r \<rightarrow> injval r c v1" by fact 
          then have "flat (injval r c v1) = (c # s1)" by (rule Posix1)
          then have "flat (injval r c v1) \<noteq> []" by simp
        moreover 
          have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r) \<and> s\<^sub>4 \<in> L (NMTIMES r (n - 1) (m - 1)))" by fact
          then have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (c # s1) @ s\<^sub>3 \<in> L r \<and> s\<^sub>4 \<in> L (NMTIMES r (n - 1) (m - 1)))" 
            by (simp add: der_correctness Der_def)
        ultimately 
        have "((c # s1) @ s2) \<in> NMTIMES r n m \<rightarrow> Stars (injval r c v1 # vs)" 
           apply (rule_tac Posix.intros)
               apply(simp_all)
              apply(case_tac n)
            apply(simp)
          using Posix_elims(1) NMTIMES.prems apply auto[1]
          using cons(5) apply blast
           apply(simp)
            apply(rule cons)
             done
        then show "(c # s) \<in> NMTIMES r n m \<rightarrow> injval (NMTIMES r n m) c v" using cons by(simp)
      next 
       case null
          have "s1 \<in> der c r \<rightarrow> v1" by fact
          then have "(c # s1) \<in> r \<rightarrow> injval r c v1" using IH by simp
          moreover 
            have "s2 \<in> UPNTIMES r (m - 1) \<rightarrow> Stars vs" by fact
          moreover 
          have "(c # s1) \<in> r \<rightarrow> injval r c v1" by fact 
          then have "flat (injval r c v1) = (c # s1)" by (rule Posix1)
          then have "flat (injval r c v1) \<noteq> []" by simp
          moreover
             moreover 
          have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r) \<and> s\<^sub>4 \<in> L (UPNTIMES r (m - 1)))" by fact
          then have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (c # s1) @ s\<^sub>3 \<in> L r \<and> s\<^sub>4 \<in> L (UPNTIMES r (m - 1)))" 
            by (simp add: der_correctness Der_def)
        ultimately 
        have "((c # s1) @ s2) \<in> NMTIMES r 0 m \<rightarrow> Stars (injval r c v1 # vs)" 
           apply (rule_tac Posix.intros) back
              apply(simp_all)
              apply(rule null)
           done
        then show "(c # s) \<in> NMTIMES r n m \<rightarrow> injval (NMTIMES r n m) c v" using null 
          apply(simp)
          done  
      qed    
qed

section {* Lexer Correctness *}

lemma lexer_correct_None:
  shows "s \<notin> L r \<longleftrightarrow> lexer r s = None"
apply(induct s arbitrary: r)
apply(simp add: nullable_correctness)
apply(drule_tac x="der a r" in meta_spec)
apply(auto simp add: der_correctness Der_def)
done

lemma lexer_correct_Some:
  shows "s \<in> L r \<longleftrightarrow> (\<exists>v. lexer r s = Some(v) \<and> s \<in> r \<rightarrow> v)"
apply(induct s arbitrary: r)
apply(auto simp add: Posix_mkeps nullable_correctness)[1]
apply(drule_tac x="der a r" in meta_spec)
apply(simp add: der_correctness Der_def)
apply(rule iffI)
apply(auto intro: Posix_injval simp add: Posix1(1))
done 

lemma lexer_correctness:
  shows "(lexer r s = Some v) \<longleftrightarrow> s \<in> r \<rightarrow> v"
  and   "(lexer r s = None) \<longleftrightarrow> \<not>(\<exists>v. s \<in> r \<rightarrow> v)"
using Posix1(1) Posix_determ lexer_correct_None lexer_correct_Some apply fastforce
using Posix1(1) lexer_correct_None lexer_correct_Some by blast

end