(*  Author: Tobias Nipkow *)

section "Regular expressions"

theory Regular_Exp

imports Regular_Set

begin

datatype (atoms: 'a) rexp =

  is_Zero: Zero |

  is_One: One |

  Atom 'a |

  Plus "('a rexp)" "('a rexp)" |

  Times "('a rexp)" "('a rexp)" |

  Star "('a rexp)"

primrec lang :: "'a rexp => 'a lang" where

"lang Zero = {}" |

"lang One = {[]}" |

"lang (Atom a) = {[a]}" |

"lang (Plus r s) = (lang r) Un (lang s)" |

"lang (Times r s) = conc (lang r) (lang s)" |

"lang (Star r) = star(lang r)"

primrec nullable :: "'a rexp \<Rightarrow> bool" where

"nullable Zero = False" |

"nullable One = True" |

"nullable (Atom c) = False" |

"nullable (Plus r1 r2) = (nullable r1 \<or> nullable r2)" |

"nullable (Times r1 r2) = (nullable r1 \<and> nullable r2)" |

"nullable (Star r) = True"

lemma nullable_iff: "nullable r \<longleftrightarrow> [] \<in> lang r"

by (induct r) (auto simp add: conc_def split: if_splits)

text{* Composition on rhs usually complicates matters: *}

lemma map_map_rexp:

  "map_rexp f (map_rexp g r) = map_rexp (\<lambda>r. f (g r)) r"

  unfolding rexp.map_comp o_def ..

lemma map_rexp_ident[simp]: "map_rexp (\<lambda>x. x) = (\<lambda>r. r)"

  unfolding id_def[symmetric] fun_eq_iff rexp.map_id id_apply by (intro allI refl)

lemma atoms_lang: "w : lang r \<Longrightarrow> set w \<subseteq> atoms r"

proof(induction r arbitrary: w)

  case Times thus ?case by fastforce

next

  case Star thus ?case by (fastforce simp add: star_conv_concat)

qed auto

lemma lang_eq_ext: "(lang r = lang s) =

  (\<forall>w \<in> lists(atoms r \<union> atoms s). w \<in> lang r \<longleftrightarrow> w \<in> lang s)"

  by (auto simp: atoms_lang[unfolded subset_iff])

lemma lang_eq_ext_Nil_fold_Deriv:

  fixes r s

  defines "\<BB> \<equiv> {(fold Deriv w (lang r), fold Deriv w (lang s))| w. w\<in>lists (atoms r \<union> atoms s)}"

  shows "lang r = lang s \<longleftrightarrow> (\<forall>(K, L) \<in> \<BB>. [] \<in> K \<longleftrightarrow> [] \<in> L)"

  unfolding lang_eq_ext \<BB>_def by (subst (1 2) in_fold_Deriv[of "[]", simplified, symmetric]) auto

subsection {* Term ordering *}

instantiation rexp :: (order) "{order}"

begin

fun le_rexp :: "('a::order) rexp \<Rightarrow> ('a::order) rexp \<Rightarrow> bool"

where

  "le_rexp Zero _ = True"

| "le_rexp _ Zero = False"

| "le_rexp One _ = True"

| "le_rexp _ One = False"

| "le_rexp (Atom a) (Atom b) = (a <= b)"

| "le_rexp (Atom _) _ = True"

| "le_rexp _ (Atom _) = False"

| "le_rexp (Star r) (Star s) = le_rexp r s"

| "le_rexp (Star _) _ = True"

| "le_rexp _ (Star _) = False"

| "le_rexp (Plus r r') (Plus s s') =

    (if r = s then le_rexp r' s' else le_rexp r s)"

| "le_rexp (Plus _ _) _ = True"

| "le_rexp _ (Plus _ _) = False"

| "le_rexp (Times r r') (Times s s') =

    (if r = s then le_rexp r' s' else le_rexp r s)"

(* The class instance stuff is by Dmitriy Traytel *)

definition less_eq_rexp where "r \<le> s \<equiv> le_rexp r s"

definition less_rexp where "r < s \<equiv> le_rexp r s \<and> r \<noteq> s"

lemma le_rexp_Zero: "le_rexp r Zero \<Longrightarrow> r = Zero"

by (induction r) auto

lemma le_rexp_refl: "le_rexp r r"

by (induction r) auto

lemma le_rexp_antisym: "\<lbrakk>le_rexp r s; le_rexp s r\<rbrakk> \<Longrightarrow> r = s"

by (induction r s rule: le_rexp.induct) (auto dest: le_rexp_Zero)

lemma le_rexp_trans: "\<lbrakk>le_rexp r s; le_rexp s t\<rbrakk> \<Longrightarrow> le_rexp r t"

proof (induction r s arbitrary: t rule: le_rexp.induct)

  fix v t assume "le_rexp (Atom v) t" thus "le_rexp One t" by (cases t) auto

next

  fix s1 s2 t assume "le_rexp (Plus s1 s2) t" thus "le_rexp One t" by (cases t) auto

next

  fix s1 s2 t assume "le_rexp (Times s1 s2) t" thus "le_rexp One t" by (cases t) auto

next

  fix s t assume "le_rexp (Star s) t" thus "le_rexp One t" by (cases t) auto

next

  fix v u t assume "le_rexp (Atom v) (Atom u)" "le_rexp (Atom u) t"

  thus "le_rexp (Atom v) t" by (cases t) auto

next

  fix v s1 s2 t assume "le_rexp (Plus s1 s2) t" thus "le_rexp (Atom v) t" by (cases t) auto

next

  fix v s1 s2 t assume "le_rexp (Times s1 s2) t" thus "le_rexp (Atom v) t" by (cases t) auto

next

  fix v s t assume "le_rexp (Star s) t" thus "le_rexp (Atom v) t" by (cases t) auto

next

  fix r s t

  assume IH: "\<And>t. le_rexp r s \<Longrightarrow> le_rexp s t \<Longrightarrow> le_rexp r t"

    and "le_rexp (Star r) (Star s)" "le_rexp (Star s) t"

  thus "le_rexp (Star r) t" by (cases t) auto

next

  fix r s1 s2 t assume "le_rexp (Plus s1 s2) t" thus "le_rexp (Star r) t" by (cases t) auto

next

  fix r s1 s2 t assume "le_rexp (Times s1 s2) t" thus "le_rexp (Star r) t" by (cases t) auto

next

  fix r1 r2 s1 s2 t

  assume "\<And>t. r1 = s1 \<Longrightarrow> le_rexp r2 s2 \<Longrightarrow> le_rexp s2 t \<Longrightarrow> le_rexp r2 t"

         "\<And>t. r1 \<noteq> s1 \<Longrightarrow> le_rexp r1 s1 \<Longrightarrow> le_rexp s1 t \<Longrightarrow> le_rexp r1 t"

         "le_rexp (Plus r1 r2) (Plus s1 s2)" "le_rexp (Plus s1 s2) t"

  thus "le_rexp (Plus r1 r2) t" by (cases t) (auto split: split_if_asm intro: le_rexp_antisym)

next

  fix r1 r2 s1 s2 t assume "le_rexp (Times s1 s2) t" thus "le_rexp (Plus r1 r2) t" by (cases t) auto

next

  fix r1 r2 s1 s2 t

  assume "\<And>t. r1 = s1 \<Longrightarrow> le_rexp r2 s2 \<Longrightarrow> le_rexp s2 t \<Longrightarrow> le_rexp r2 t"

         "\<And>t. r1 \<noteq> s1 \<Longrightarrow> le_rexp r1 s1 \<Longrightarrow> le_rexp s1 t \<Longrightarrow> le_rexp r1 t"

         "le_rexp (Times r1 r2) (Times s1 s2)" "le_rexp (Times s1 s2) t"

  thus "le_rexp (Times r1 r2) t" by (cases t) (auto split: split_if_asm intro: le_rexp_antisym)

qed auto

instance proof

qed (auto simp add: less_eq_rexp_def less_rexp_def

       intro: le_rexp_refl le_rexp_antisym le_rexp_trans)

end

instantiation rexp :: (linorder) "{linorder}"

begin

lemma le_rexp_total: "le_rexp (r :: 'a :: linorder rexp) s \<or> le_rexp s r"

by (induction r s rule: le_rexp.induct) auto

instance proof

qed (unfold less_eq_rexp_def less_rexp_def, rule le_rexp_total)

end

end