theory BasicIdentities imports 

"Lexer"  "PDerivs"

begin

datatype rrexp = 

  RZERO

| RONE 

| RCHAR char

| RSEQ rrexp rrexp

| RALTS "rrexp list"

| RSTAR rrexp

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"

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)))"

lemma rdistinct_concat:

  shows "set rs \<subseteq> rset \<Longrightarrow> rdistinct (rs @ rsa) rset = rdistinct rsa rset"

  apply(induct rs)

   apply simp+

  done

lemma rdistinct_concat2:

  shows "\<forall>r \<in> set rs. r \<in> rset \<Longrightarrow> rdistinct (rs @ rsa) rset = rdistinct rsa rset"

  by (simp add: rdistinct_concat subsetI)

lemma distinct_not_exist:

  shows "a \<notin> set rs \<Longrightarrow> rdistinct rs rset = rdistinct rs (insert a rset)"

  apply(induct rs arbitrary: rset)

   apply simp

  apply(case_tac "aa \<in> rset")

   apply simp

  apply(subgoal_tac "a \<noteq> aa")

   prefer 2

  apply simp

  apply simp

  done

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

  where

  "rsimp_ALTs  [] = RZERO"

| "rsimp_ALTs [r] =  r"

| "rsimp_ALTs rs = RALTS rs"

lemma rsimpalts_gte2elems:

  shows "size rlist \<ge> 2 \<Longrightarrow> rsimp_ALTs rlist = RALTS rlist"

  apply(induct rlist)

   apply simp

  apply(induct rlist)

   apply simp

  apply (metis Suc_le_length_iff rsimp_ALTs.simps(3))

  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))

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)"

lemma rder_rsimp_ALTs_commute:

  shows "  (rder x (rsimp_ALTs rs)) = rsimp_ALTs (map (rder x) rs)"

  apply(induct rs)

   apply simp

  apply(case_tac rs)

   apply simp

  apply auto

  done

lemma rsimp_aalts_smaller:

  shows "rsize (rsimp_ALTs  rs) \<le> rsize (RALTS rs)"

  apply(induct rs)

   apply simp

  apply simp

  apply(case_tac "rs = []")

   apply simp

  apply(subgoal_tac "\<exists>rsp ap. rs = ap # rsp")

   apply(erule exE)+

   apply simp

  apply simp

  by(meson neq_Nil_conv)

lemma rSEQ_mono:

  shows "rsize (rsimp_SEQ r1 r2) \<le>rsize ( RSEQ r1 r2)"

  apply auto

  apply(induct r1)

       apply auto

      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 r2)

  apply simp_all

  done

lemma ralts_cap_mono:

  shows "rsize (RALTS rs) \<le> Suc ( sum_list (map rsize rs)) "

  by simp

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_mono:

  shows "sum_list (map rsize (rflts rs))\<le> sum_list (map rsize rs)"

  apply(induct rs)

  apply simp

  apply(case_tac "a = RZERO")

   apply simp

  apply(case_tac "\<exists>rs1. a = RALTS rs1")

  apply(erule exE)

   apply simp

  apply(subgoal_tac "rflts (a # rs) = a # (rflts rs)")

  prefer 2

  using rflts_def_idiot apply blast

  apply simp

  done

lemma rdistinct_smaller: shows "sum_list (map rsize (rdistinct rs ss)) \<le>

sum_list (map rsize rs )"

  apply (induct rs arbitrary: ss)

   apply simp

  by (simp add: trans_le_add2)

lemma rdistinct_phi_smaller: "sum_list (map rsize (rdistinct rs {})) \<le> sum_list (map rsize rs)"

  by (simp add: rdistinct_smaller)

lemma rsimp_alts_mono :

  shows "\<And>x. (\<And>xa. xa \<in> set x \<Longrightarrow> rsize (rsimp xa) \<le> rsize xa)  \<Longrightarrow>

rsize (rsimp_ALTs (rdistinct (rflts (map rsimp x)) {})) \<le> Suc (sum_list (map rsize x))"

  apply(subgoal_tac "rsize (rsimp_ALTs (rdistinct (rflts (map rsimp x)) {} )) 

                    \<le> rsize (RALTS (rdistinct (rflts (map rsimp x)) {} ))")

  prefer 2

  using rsimp_aalts_smaller apply auto[1]

  apply(subgoal_tac "rsize (RALTS (rdistinct (rflts (map rsimp x)) {})) \<le>Suc( sum_list (map rsize (rdistinct (rflts (map rsimp x)) {})))")

  prefer 2

  using ralts_cap_mono apply blast

  apply(subgoal_tac "sum_list (map rsize (rdistinct (rflts (map rsimp x)) {})) \<le>

                     sum_list (map rsize ( (rflts (map rsimp x))))")

  prefer 2

  using rdistinct_smaller apply presburger

  apply(subgoal_tac "sum_list (map rsize (rflts (map rsimp x))) \<le> 

                     sum_list (map rsize (map rsimp x))")

  prefer 2

  using rflts_mono apply blast

  apply(subgoal_tac "sum_list (map rsize (map rsimp x)) \<le> sum_list (map rsize x)")

  prefer 2

  apply (simp add: sum_list_mono)

  by linarith

lemma rsimp_mono:

  shows "rsize (rsimp r) \<le> rsize r"

  apply(induct r)

  apply simp_all

  apply(subgoal_tac "rsize (rsimp_SEQ (rsimp r1) (rsimp r2)) \<le> rsize (RSEQ (rsimp r1) (rsimp r2))")

    apply force

  using rSEQ_mono

   apply presburger

  using rsimp_alts_mono by auto

lemma idiot:

  shows "rsimp_SEQ RONE r = r"

  apply(case_tac r)

       apply simp_all

  done

lemma no_alt_short_list_after_simp:

  shows "RALTS rs = rsimp r \<Longrightarrow> rsimp_ALTs rs = RALTS rs"

  sorry

lemma no_further_dB_after_simp:

  shows "RALTS rs = rsimp r \<Longrightarrow> rdistinct rs {} = rs"

  sorry

lemma idiot2:

  shows " \<lbrakk>r1 \<noteq> RZERO; r1 \<noteq> RONE;r2 \<noteq> RZERO\<rbrakk>

    \<Longrightarrow> rsimp_SEQ r1 r2 = RSEQ r1 r2"

  apply(case_tac r1)

       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 r2)

       apply simp_all

  done

lemma rders__onechar:

  shows " (rders_simp r [c]) =  (rsimp (rders r [c]))"

  by simp

lemma rders_append:

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

  apply(induct s1 arbitrary: c s2)

  apply(simp_all)

  done

lemma rders_simp_append:

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

  apply(induct s1 arbitrary: c s2)

   apply(simp_all)

  done

lemma inside_simp_removal:

  shows " rsimp (rder x (rsimp r)) = rsimp (rder x r)"

  sorry

lemma set_related_list:

  shows "distinct rs  \<Longrightarrow> length rs = card (set rs)"

  by (simp add: distinct_card)

(*this section deals with the property of distinctBy: creates a list without duplicates*)

lemma rdistinct_never_added_twice:

  shows "rdistinct (a # rs) {a} = rdistinct rs {a}"

  by force

lemma rdistinct_does_the_job:

  shows "distinct (rdistinct rs s)"

  apply(induct rs arbitrary: s)

   apply simp

  apply simp

  sorry

lemma rders_simp_same_simpders:

  shows "s \<noteq> [] \<Longrightarrow> rders_simp r s = rsimp (rders r s)"

  apply(induct s rule: rev_induct)

   apply simp

  apply(case_tac "xs = []")

   apply simp

  apply(simp add: rders_append rders_simp_append)

  using inside_simp_removal by blast

lemma rders_simp_one_char:

  shows "rders_simp r [c] = rsimp (rder c r)"

  apply auto

  done

lemma rsimp_idem:

  shows "rsimp (rsimp r) = rsimp r"

  sorry

corollary rsimp_inner_idem1:

  shows "rsimp r = RSEQ r1 r2 \<Longrightarrow> rsimp r1 = r1 \<and> rsimp r2 = r2"

  sorry

corollary rsimp_inner_idem2:

  shows "rsimp r = RALTS rs \<Longrightarrow> \<forall>r' \<in> (set rs). rsimp r' = r'"

  sorry

corollary rsimp_inner_idem3:

  shows "rsimp r = RALTS rs \<Longrightarrow> map rsimp rs = rs"

  by (meson map_idI rsimp_inner_idem2)

corollary rsimp_inner_idem4:

  shows "rsimp r = RALTS rs \<Longrightarrow> rflts rs = rs"

  sorry

lemma head_one_more_simp:

  shows "map rsimp (r # rs) = map rsimp (( rsimp r) # rs)"

  by (simp add: rsimp_idem)

lemma head_one_more_dersimp:

  shows "map rsimp ((rder x (rders_simp r s) # rs)) = map rsimp ((rders_simp r (s@[x]) ) # rs)"

  using head_one_more_simp rders_simp_append rders_simp_one_char by presburger

lemma ders_simp_nullability:

  shows "rnullable (rders r s) = rnullable (rders_simp r s)"

  sorry

lemma  first_elem_seqder:

  shows "\<not>rnullable r1p \<Longrightarrow> map rsimp (rder x (RSEQ r1p r2)

                   # rs) = map rsimp ((RSEQ (rder x r1p) r2) # rs) "

  by auto

lemma first_elem_seqder1:

  shows  "\<not>rnullable (rders_simp r xs) \<Longrightarrow> map rsimp ( (rder x (RSEQ (rders_simp r xs) r2)) # rs) = 

                                          map rsimp ( (RSEQ (rsimp (rder x (rders_simp r xs))) r2) # rs)"

  by (simp add: rsimp_idem)

lemma first_elem_seqdersimps:

  shows "\<not>rnullable (rders_simp r xs) \<Longrightarrow> map rsimp ( (rder x (RSEQ (rders_simp r xs) r2)) # rs) = 

                                          map rsimp ( (RSEQ (rders_simp r (xs @ [x])) r2) # rs)"

  using first_elem_seqder1 rders_simp_append by auto

lemma seq_ders_closed_form1:

  shows "\<exists>Ss. rders_simp (RSEQ r1 r2) [c] = rsimp (RALTS ((RSEQ (rders_simp r1 [c]) r2) # 

(map ( rders_simp r2 ) Ss)))"

  apply(case_tac "rnullable r1")

   apply(subgoal_tac " rders_simp (RSEQ r1 r2) [c] = 

rsimp (RALTS ((RSEQ (rders_simp r1 [c]) r2) # (map (rders_simp r2) [[c]])))")

    prefer 2

    apply (simp add: rsimp_idem)

   apply(rule_tac x = "[[c]]" in exI)

   apply simp

  apply(subgoal_tac  " rders_simp (RSEQ r1 r2) [c] = 

rsimp (RALTS ((RSEQ (rders_simp r1 [c]) r2) # (map (rders_simp r2) [])))")

   apply blast

  apply(simp add: rsimp_idem)

  sorry

lemma simp_flatten:

  shows "rsimp (RALTS ((RALTS rsa) # rsb)) = rsimp (RALTS (rsa @ rsb))"

  sorry

fun vsuf :: "char list \<Rightarrow> rrexp \<Rightarrow> char list list" where

"vsuf [] _ = []"

|"vsuf (c#cs) r1 = (if (rnullable r1) then  (vsuf cs (rder c r1)) @ [c # cs]

                                      else  (vsuf cs (rder c r1))

                   ) "

fun star_update :: "char \<Rightarrow> rrexp \<Rightarrow> char list list \<Rightarrow> char list list" where

"star_update c r [] = []"

|"star_update c r (s # Ss) = (if (rnullable (rders_simp r s)) 

                                then (s@[c]) # [c] # (star_update c r Ss) 

                               else   (s@[c]) # (star_update c r Ss) )"

fun star_updates :: "char list \<Rightarrow> rrexp \<Rightarrow> char list list \<Rightarrow> char list list"

  where

"star_updates [] r Ss = Ss"

| "star_updates (c # cs) r Ss = star_updates cs r (star_update c r Ss)"

(*some basic facts about rsimp*)

end