imports "Lexer" 

datatype rrexp = 

  RZERO

| RONE 

| RCHAR char

| RSEQ rrexp rrexp

| RALTS "rrexp list"

| RSTAR rrexp

| RNTIMES rrexp nat

abbreviation

  "RALT r1 r2 \<equiv> RALTS [r1, r2]"

fun

 rnullable :: "rrexp \<Rightarrow> bool"

where

  "rnullable (RZERO) = False"

| "rnullable (RONE) = True"

| "rnullable (RCHAR c) = False"

| "rnullable (RALTS rs) = (\<exists>r \<in> set rs. rnullable r)"

| "rnullable (RSEQ r1 r2) = (rnullable r1 \<and> rnullable r2)"

| "rnullable (RSTAR r) = True"

| "rnullable (RNTIMES r n) = (if n = 0 then True else rnullable r)"

fun

 rder :: "char \<Rightarrow> rrexp \<Rightarrow> rrexp"

where

  "rder c (RZERO) = RZERO"

| "rder c (RONE) = RZERO"

| "rder c (RCHAR d) = (if c = d then RONE else RZERO)"

| "rder c (RALTS rs) = RALTS (map (rder c) rs)"

| "rder c (RSEQ r1 r2) = 

     (if rnullable r1

      then RALT   (RSEQ (rder c r1) r2) (rder c r2)

      else RSEQ   (rder c r1) r2)"

| "rder c (RSTAR r) = RSEQ  (rder c r) (RSTAR r)"   

| "rder c (RNTIMES r n) = (if n = 0 then RZERO else RSEQ (rder c r) (RNTIMES r (n - 1)))"

fun 

  rders :: "rrexp \<Rightarrow> string \<Rightarrow> rrexp"

where

  "rders r [] = r"

| "rders r (c#s) = rders (rder c r) s"

fun rdistinct :: "'a list \<Rightarrow>'a set \<Rightarrow> 'a list"

  where

  "rdistinct [] acc = []"

| "rdistinct (x#xs)  acc = 

     (if x \<in> acc then rdistinct xs  acc 

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

lemma rdistinct1:

  assumes "a \<in> acc"

  shows "a \<notin> set (rdistinct rs acc)"

  using assms

  apply(induct rs arbitrary: acc a)

   apply(auto)

  done

lemma rdistinct_does_the_job:

  shows "distinct (rdistinct rs s)"

  apply(induct rs s rule: rdistinct.induct)

  apply(auto simp add: rdistinct1)

  done

lemma rdistinct_concat:

  assumes "set rs \<subseteq> rset"

  shows "rdistinct (rs @ rsa) rset = rdistinct rsa rset"

  using assms

  apply(induct rs)

  apply simp+

  done

lemma distinct_not_exist:

  assumes "a \<notin> set rs"

  shows "rdistinct rs rset = rdistinct rs (insert a rset)"

  using assms

  apply(induct rs arbitrary: rset)

  apply(auto)

  done

lemma rdistinct_on_distinct:

  shows "distinct rs \<Longrightarrow> rdistinct rs {} = rs"

  apply(induct rs)

  apply simp

  using distinct_not_exist by fastforce

lemma distinct_rdistinct_append:

  assumes "distinct rs1" "\<forall>r \<in> set rs1. r \<notin> acc"

  shows "rdistinct (rs1 @ rsa) acc = rs1 @ (rdistinct rsa (acc \<union> set rs1))"

  using assms

  apply(induct rs1 arbitrary: rsa acc)

   apply(auto)[1]

  apply(auto)[1]

  apply(drule_tac x="rsa" in meta_spec)

  apply(drule_tac x="{a} \<union> acc" in meta_spec)

  apply(simp)

  apply(drule meta_mp)

   apply(auto)[1]

  apply(simp)

  done

lemma rdistinct_set_equality1:

  shows "set (rdistinct rs acc) = set rs - acc"

  apply(induct rs acc rule: rdistinct.induct)

  apply(auto)

  done

lemma rdistinct_set_equality:

  shows "set (rdistinct rs {}) = set rs"

  by (simp add: rdistinct_set_equality1)

fun rflts :: "rrexp list \<Rightarrow> rrexp list"

  where 

  "rflts [] = []"

| "rflts (RZERO # rs) = rflts rs"

| "rflts ((RALTS rs1) # rs) = rs1 @ rflts rs"

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

lemma rflts_def_idiot:

  shows "\<lbrakk> a \<noteq> RZERO; \<nexists>rs1. a = RALTS rs1\<rbrakk> \<Longrightarrow> rflts (a # rs) = a # rflts rs"

  apply(case_tac a)

  apply simp_all

  done

lemma rflts_def_idiot2:

  shows "\<lbrakk>a \<noteq> RZERO; \<nexists>rs1. a = RALTS rs1; a \<in> set rs\<rbrakk> \<Longrightarrow> a \<in> set (rflts rs)"

  apply(induct rs rule: rflts.induct)

  apply(auto)

  done

lemma flts_append:

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

  apply(induct rs1)

   apply simp

  apply(case_tac a)

       apply simp+

  done

fun rsimp_ALTs :: " rrexp list \<Rightarrow> rrexp"

  where

  "rsimp_ALTs  [] = RZERO"

| "rsimp_ALTs [r] =  r"

| "rsimp_ALTs rs = RALTS rs"

lemma rsimpalts_conscons:

  shows "rsimp_ALTs (r1 # rsa @ r2 # rsb) = RALTS (r1 # rsa @ r2 # rsb)"

  by (metis Nil_is_append_conv list.exhaust rsimp_ALTs.simps(3))

lemma rsimp_alts_equal:

  shows "rsimp_ALTs (rsa @ a # rsb @ a # rsc) = RALTS (rsa @ a # rsb @ a # rsc) "

  by (metis append_Cons append_Nil neq_Nil_conv rsimpalts_conscons)

fun rsimp_SEQ :: " rrexp \<Rightarrow> rrexp \<Rightarrow> rrexp"

  where

  "rsimp_SEQ  RZERO _ = RZERO"

| "rsimp_SEQ  _ RZERO = RZERO"

| "rsimp_SEQ RONE r2 = r2"

| "rsimp_SEQ r1 r2 = RSEQ r1 r2"

fun rsimp :: "rrexp \<Rightarrow> rrexp" 

  where

  "rsimp (RSEQ r1 r2) = rsimp_SEQ  (rsimp r1) (rsimp r2)"

| "rsimp (RALTS rs) = rsimp_ALTs  (rdistinct (rflts (map rsimp rs))  {}) "

| "rsimp r = r"

fun 

  rders_simp :: "rrexp \<Rightarrow> string \<Rightarrow> rrexp"

where

  "rders_simp r [] = r"

| "rders_simp r (c#s) = rders_simp (rsimp (rder c r)) s"

fun rsize :: "rrexp \<Rightarrow> nat" where

  "rsize RZERO = 1"

| "rsize (RONE) = 1" 

| "rsize (RCHAR  c) = 1"

| "rsize (RALTS  rs) = Suc (sum_list (map rsize rs))"

| "rsize (RSEQ  r1 r2) = Suc (rsize r1 + rsize r2)"

| "rsize (RSTAR  r) = Suc (rsize r)"

| "rsize (RNTIMES  r n) = Suc (rsize r) + n"

abbreviation rsizes where

  "rsizes rs \<equiv> sum_list (map rsize rs)"

         apply simp_all

  apply(case_tac r2)

         apply simp_all

         apply simp_all

apply(case_tac r2)

         apply simp_all

fun nonalt :: "rrexp \<Rightarrow> bool"

  where

  "nonalt (RALTS  rs) = False"

| "nonalt r = True"

fun good :: "rrexp \<Rightarrow> bool" where

  "good RZERO = False"

| "good (RONE) = True" 

| "good (RCHAR c) = True"

| "good (RALTS []) = False"

| "good (RALTS [r]) = False"

| "good (RALTS (r1 # r2 # rs)) = ((distinct ( (r1 # r2 # rs))) \<and>(\<forall>r' \<in> set (r1 # r2 # rs). good r' \<and> nonalt r'))"

| "good (RSEQ RZERO _) = False"

| "good (RSEQ RONE _) = False"

| "good (RSEQ  _ RZERO) = False"

| "good (RSEQ r1 r2) = (good r1 \<and> good r2)"

| "good (RSTAR r) = True"

| "good (RNTIMES r n) = True"

   apply(simp)

  by simp

  apply(induct rs arbitrary: rule: rflts.induct)

         apply(simp_all)

apply(case_tac r2)

         apply(simp_all)

fun nonnested :: "rrexp \<Rightarrow> bool"

  where

  "nonnested (RALTS []) = True"

| "nonnested (RALTS ((RALTS rs1) # rs2)) = False"

| "nonnested (RALTS (r # rs2)) = nonnested (RALTS rs2)"

| "nonnested r = True"

lemma  k00:

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

  apply(induct rs1 arbitrary: rs2)

   apply(auto)

  by (metis append.assoc k0)

lemma k0b:

  assumes "nonalt r" "r \<noteq> RZERO"

  shows "rflts [r] = [r]"

  using assms

  apply(case_tac  r)

  apply(simp_all)

  done