theory Rsimp imports "Lexer"

begin

datatype rrexp = 

  RZERO

| RONE 

| RCHAR char

| RSEQ rrexp rrexp

| RALTS "rrexp list"

| RSTAR rrexp

abbreviation

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

fun

  RL :: "rrexp \<Rightarrow> string set"

where

  "RL (RZERO) = {}"

| "RL (RONE) = {[]}"

| "RL (RCHAR c) = {[c]}"

| "RL (RSEQ r1 r2) = (RL r1) ;; (RL r2)"

| "RL (RALTS rs) = (\<Union> (set (map RL rs)))"

| "RL (RSTAR r) = (RL r)\<star>"

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"

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)"

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)))"

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"

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

  where

  "nonalt (RALTS  rs) = False"

| "nonalt r = True"

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

  where

  "rsimp_ALTs  [] = RZERO"

| "rsimp_ALTs [r] =  r"

| "rsimp_ALTs rs = RALTS rs"

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)"

abbreviation rsizes where

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

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"

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"

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

  where

  "nonazero RZERO = False"

| "nonazero r = True"

lemma basic_rsimp_SEQ_property1:

  shows "rsimp_SEQ RONE r = r"

  apply(induct r)

       apply simp+

  done

lemma basic_rsimp_SEQ_property3:

  shows "rsimp_SEQ r RZERO = RZERO"  

  using rsimp_SEQ.elims by blast

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)

end