theory Myhill_1

begin

(*

text {*

     \begin{figure}

    \centering

    \scalebox{0.95}{

    \begin{tikzpicture}[->,>=latex,shorten >=1pt,auto,node distance=1.2cm, semithick]

        \node[state,initial] (n1)                   {$1$};

        \node[state,accepting] (n2) [right = 10em of n1]   {$2$};

        \path (n1) edge [bend left] node {$0$} (n2)

            (n1) edge [loop above] node{$1$} (n1)

            (n2) edge [loop above] node{$0$} (n2)

            (n2) edge [bend left]  node {$1$} (n1)

            ;

    \end{tikzpicture}}

    \caption{An example automaton (or partition)}\label{fig:example_automata}

    \end{figure}

*}

*)

section {* Preliminary definitions *}

types lang = "string set"

text {*  Sequential composition of two languages *}

definition 

  Seq :: "lang \<Rightarrow> lang \<Rightarrow> lang" (infixr ";;" 100)

where 

  "A ;; B = {s\<^isub>1 @ s\<^isub>2 | s\<^isub>1 s\<^isub>2. s\<^isub>1 \<in> A \<and> s\<^isub>2 \<in> B}"

text {* Some properties of operator @{text ";;"}. *}

lemma seq_add_left:

  assumes a: "A = B"

  shows "C ;; A = C ;; B"

using a by simp

lemma seq_union_distrib_right:

  shows "(A \<union> B) ;; C = (A ;; C) \<union> (B ;; C)"

unfolding Seq_def by auto

lemma seq_union_distrib_left:

  shows "C ;; (A \<union> B) = (C ;; A) \<union> (C ;; B)"

unfolding Seq_def by  auto

lemma seq_intro:

  assumes a: "x \<in> A" "y \<in> B"

  shows "x @ y \<in> A ;; B "

using a by (auto simp: Seq_def)

lemma seq_assoc:

  shows "(A ;; B) ;; C = A ;; (B ;; C)"

unfolding Seq_def

apply(auto)

apply(blast)

by (metis append_assoc)

lemma seq_empty [simp]:

  shows "A ;; {[]} = A"

  and   "{[]} ;; A = A"

by (simp_all add: Seq_def)

text {* Power and Star of a language *}

fun 

  pow :: "lang \<Rightarrow> nat \<Rightarrow> lang" (infixl "\<up>" 100)

where

  "A \<up> 0 = {[]}"

| "A \<up> (Suc n) =  A ;; (A \<up> n)" 

definition

  Star :: "lang \<Rightarrow> lang" ("_\<star>" [101] 102)

where

  "A\<star> \<equiv> (\<Union>n. A \<up> n)"

lemma star_start[intro]:

  shows "[] \<in> A\<star>"

proof -

  have "[] \<in> A \<up> 0" by auto

  then show "[] \<in> A\<star>" unfolding Star_def by blast

qed

lemma star_step [intro]:

  assumes a: "s1 \<in> A" 

  and     b: "s2 \<in> A\<star>"

  shows "s1 @ s2 \<in> A\<star>"

proof -

  from b obtain n where "s2 \<in> A \<up> n" unfolding Star_def by auto

  then have "s1 @ s2 \<in> A \<up> (Suc n)" using a by (auto simp add: Seq_def)

  then show "s1 @ s2 \<in> A\<star>" unfolding Star_def by blast

qed

lemma star_induct[consumes 1, case_names start step]:

  assumes a: "x \<in> A\<star>" 

  and     b: "P []"

  and     c: "\<And>s1 s2. \<lbrakk>s1 \<in> A; s2 \<in> A\<star>; P s2\<rbrakk> \<Longrightarrow> P (s1 @ s2)"

  shows "P x"

proof -

  from a obtain n where "x \<in> A \<up> n" unfolding Star_def by auto

  then show "P x"

    by (induct n arbitrary: x)

       (auto intro!: b c simp add: Seq_def Star_def)

qed

lemma star_intro1:

  assumes a: "x \<in> A\<star>"

  and     b: "y \<in> A\<star>"

  shows "x @ y \<in> A\<star>"

using a b

by (induct rule: star_induct) (auto)

lemma star_intro2: 

  assumes a: "y \<in> A"

  shows "y \<in> A\<star>"

proof -

  from a have "y @ [] \<in> A\<star>" by blast

  then show "y \<in> A\<star>" by simp

qed

lemma star_intro3:

  assumes a: "x \<in> A\<star>"

  and     b: "y \<in> A"

  shows "x @ y \<in> A\<star>"

using a b by (blast intro: star_intro1 star_intro2)

lemma star_cases:

  shows "A\<star> =  {[]} \<union> A ;; A\<star>"

proof

  { fix x

    have "x \<in> A\<star> \<Longrightarrow> x \<in> {[]} \<union> A ;; A\<star>"

      unfolding Seq_def

    by (induct rule: star_induct) (auto)

  }

  then show "A\<star> \<subseteq> {[]} \<union> A ;; A\<star>" by auto

next

  show "{[]} \<union> A ;; A\<star> \<subseteq> A\<star>"

    unfolding Seq_def by auto

qed

lemma star_decom: 

  assumes a: "x \<in> A\<star>" "x \<noteq> []"

  shows "\<exists>a b. x = a @ b \<and> a \<noteq> [] \<and> a \<in> A \<and> b \<in> A\<star>"

using a

apply(induct rule: star_induct)

apply(simp)

apply(blast)

done

lemma

  shows seq_Union_left:  "B ;; (\<Union>n. A \<up> n) = (\<Union>n. B ;; (A \<up> n))"

  and   seq_Union_right: "(\<Union>n. A \<up> n) ;; B = (\<Union>n. (A \<up> n) ;; B)"

unfolding Seq_def by auto

lemma seq_pow_comm:

  shows "A ;; (A \<up> n) = (A \<up> n) ;; A"

by (induct n) (simp_all add: seq_assoc[symmetric])

lemma seq_star_comm:

  shows "A ;; A\<star> = A\<star> ;; A"

unfolding Star_def

unfolding seq_Union_left

unfolding seq_pow_comm

unfolding seq_Union_right 

by simp

text {* Two lemmas about the length of strings in @{text "A \<up> n"} *}

lemma pow_length:

  assumes a: "[] \<notin> A"

  and     b: "s \<in> A \<up> Suc n"

  shows "n < length s"

using b

proof (induct n arbitrary: s)

  case 0

  have "s \<in> A \<up> Suc 0" by fact

  with a have "s \<noteq> []" by auto

  then show "0 < length s" by auto

next

  case (Suc n)

  have ih: "\<And>s. s \<in> A \<up> Suc n \<Longrightarrow> n < length s" by fact

  have "s \<in> A \<up> Suc (Suc n)" by fact

  then obtain s1 s2 where eq: "s = s1 @ s2" and *: "s1 \<in> A" and **: "s2 \<in> A \<up> Suc n"

    by (auto simp add: Seq_def)

  from ih ** have "n < length s2" by simp

  moreover have "0 < length s1" using * a by auto

  ultimately show "Suc n < length s" unfolding eq 

    by (simp only: length_append)

qed

lemma seq_pow_length:

  assumes a: "[] \<notin> A"

  and     b: "s \<in> B ;; (A \<up> Suc n)"

  shows "n < length s"

proof -

  from b obtain s1 s2 where eq: "s = s1 @ s2" and *: "s2 \<in> A \<up> Suc n"

    unfolding Seq_def by auto

  from * have " n < length s2" by (rule pow_length[OF a])

  then show "n < length s" using eq by simp

qed

section {* A slightly modified version of Arden's lemma *}

text {*  A helper lemma for Arden *}

lemma ardens_helper:

  assumes eq: "X = X ;; A \<union> B"

  shows "X = X ;; (A \<up> Suc n) \<union> (\<Union>m\<in>{0..n}. B ;; (A \<up> m))"

proof (induct n)

  case 0 

  show "X = X ;; (A \<up> Suc 0) \<union> (\<Union>(m::nat)\<in>{0..0}. B ;; (A \<up> m))"

    using eq by simp

next

  case (Suc n)

  have ih: "X = X ;; (A \<up> Suc n) \<union> (\<Union>m\<in>{0..n}. B ;; (A \<up> m))" by fact

  also have "\<dots> = (X ;; A \<union> B) ;; (A \<up> Suc n) \<union> (\<Union>m\<in>{0..n}. B ;; (A \<up> m))" using eq by simp

  also have "\<dots> = X ;; (A \<up> Suc (Suc n)) \<union> (B ;; (A \<up> Suc n)) \<union> (\<Union>m\<in>{0..n}. B ;; (A \<up> m))"

    by (simp add: seq_union_distrib_right seq_assoc)

  also have "\<dots> = X ;; (A \<up> Suc (Suc n)) \<union> (\<Union>m\<in>{0..Suc n}. B ;; (A \<up> m))"

    by (auto simp add: le_Suc_eq)

  finally show "X = X ;; (A \<up> Suc (Suc n)) \<union> (\<Union>m\<in>{0..Suc n}. B ;; (A \<up> m))" .

qed

theorem ardens_revised:

  assumes nemp: "[] \<notin> A"

  shows "X = X ;; A \<union> B \<longleftrightarrow> X = B ;; A\<star>"

proof

  assume eq: "X = B ;; A\<star>"

  have "A\<star> = {[]} \<union> A\<star> ;; A" 

    unfolding seq_star_comm[symmetric]

    by (rule star_cases)

  then have "B ;; A\<star> = B ;; ({[]} \<union> A\<star> ;; A)"

    by (rule seq_add_left)

  also have "\<dots> = B \<union> B ;; (A\<star> ;; A)"

    unfolding seq_union_distrib_left by simp

  also have "\<dots> = B \<union> (B ;; A\<star>) ;; A" 

    by (simp only: seq_assoc)

  finally show "X = X ;; A \<union> B" 

    using eq by blast 

next

  assume eq: "X = X ;; A \<union> B"

  { fix n::nat

    have "B ;; (A \<up> n) \<subseteq> X" using ardens_helper[OF eq, of "n"] by auto }

  then have "B ;; A\<star> \<subseteq> X" 

    unfolding Seq_def Star_def UNION_def

    by auto

  moreover

  { fix s::string

    obtain k where "k = length s" by auto

    then have not_in: "s \<notin> X ;; (A \<up> Suc k)" 

      using seq_pow_length[OF nemp] by blast

    assume "s \<in> X"

    then have "s \<in> X ;; (A \<up> Suc k) \<union> (\<Union>m\<in>{0..k}. B ;; (A \<up> m))"

      using ardens_helper[OF eq, of "k"] by auto

    then have "s \<in> (\<Union>m\<in>{0..k}. B ;; (A \<up> m))" using not_in by auto

    moreover

    have "(\<Union>m\<in>{0..k}. B ;; (A \<up> m)) \<subseteq> (\<Union>n. B ;; (A \<up> n))" by auto

    ultimately 

    have "s \<in> B ;; A\<star>" 

      unfolding seq_Union_left Star_def

      by auto }

  then have "X \<subseteq> B ;; A\<star>" by auto

  ultimately 

  show "X = B ;; A\<star>" by simp

qed

section {* Regular Expressions *}

datatype rexp =

  NULL

| EMPTY

| CHAR char

| SEQ rexp rexp

| ALT rexp rexp

| STAR rexp

text {* 

  The following @{text "L"} is an overloaded operator, where @{text "L(x)"} evaluates to 

  the language represented by the syntactic object @{text "x"}.

*}

consts L:: "'a \<Rightarrow> lang"

text {* The @{text "L (rexp)"} for regular expressions. *}

overloading L_rexp \<equiv> "L::  rexp \<Rightarrow> lang"

begin

fun

  L_rexp :: "rexp \<Rightarrow> string set"

where

    "L_rexp (NULL) = {}"

  | "L_rexp (EMPTY) = {[]}"

  | "L_rexp (CHAR c) = {[c]}"

  | "L_rexp (SEQ r1 r2) = (L_rexp r1) ;; (L_rexp r2)"

  | "L_rexp (ALT r1 r2) = (L_rexp r1) \<union> (L_rexp r2)"

  | "L_rexp (STAR r) = (L_rexp r)\<star>"

end

section {* Folds for Sets *}

text {*

  To obtain equational system out of finite set of equivalence classes, a fold operation

  on finite sets @{text "folds"} is defined. The use of @{text "SOME"} makes @{text "folds"}

  more robust than the @{text "fold"} in the Isabelle library. The expression @{text "folds f"}

  makes sense when @{text "f"} is not @{text "associative"} and @{text "commutitive"},

  while @{text "fold f"} does not.  

*}

definition 

  folds :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b"

where

  "folds f z S \<equiv> SOME x. fold_graph f z S x"

text {* 

  The following lemma ensures that the arbitrary choice made by the 

  @{text "SOME"} in @{text "folds"} does not affect the @{text "L"}-value 

  of the resultant regular expression. 

  *}

lemma folds_alt_simp [simp]:

  assumes a: "finite rs"

  shows "L (folds ALT NULL rs) = \<Union> (L ` rs)"

apply(simp add: folds_def)

apply(rule someI2_ex)

apply(rule_tac finite_imp_fold_graph[OF a])

apply(erule fold_graph.induct)

apply(auto)

done

text {* Just a technical lemma for collections and pairs *}

  shows "(x, y) \<in> {(x, y). P x y} \<longleftrightarrow> P x y"

by simp

text {*

  @{text "\<approx>A"} is an equivalence class defined by language @{text "A"}.

*}

definition

  str_eq_rel :: "lang \<Rightarrow> (string \<times> string) set" ("\<approx>_" [100] 100)

where

  "\<approx>A \<equiv> {(x, y).  (\<forall>z. x @ z \<in> A \<longleftrightarrow> y @ z \<in> A)}"

text {* 

  Among the equivalence clases of @{text "\<approx>A"}, the set @{text "finals A"} singles out 

  those which contains the strings from @{text "A"}.

*}

definition 

  finals :: "lang \<Rightarrow> lang set"

where

  "finals A \<equiv> {\<approx>A `` {x} | x . x \<in> A}"

text {* 

  The following lemma establishes the relationshipt between 

  @{text "finals A"} and @{text "A"}.

*}

lemma lang_is_union_of_finals: 

  shows "A = \<Union> finals A"

unfolding finals_def

unfolding Image_def

unfolding str_eq_rel_def

apply(auto)

apply(drule_tac x = "[]" in spec)

apply(auto)

done

section {* Direction @{text "finite partition \<Rightarrow> regular language"}*}

text {* 

  The relationship between equivalent classes can be described by an

  equational system.  For example, in equational system \eqref{example_eqns},

  $X_0, X_1$ are equivalent classes. The first equation says every string in

  $X_0$ is obtained either by appending one $b$ to a string in $X_0$ or by

  appending one $a$ to a string in $X_1$ or just be an empty string

  (represented by the regular expression $\lambda$). Similary, the second

  equation tells how the strings inside $X_1$ are composed.

  \begin{equation}\label{example_eqns}

    \begin{aligned}

      X_0 & = X_0 b + X_1 a + \lambda \\

      X_1 & = X_0 a + X_1 b

    \end{aligned}

  \end{equation}

  \noindent

  The summands on the right hand side is represented by the following data

  type @{text "rhs_item"}, mnemonic for 'right hand side item'.  Generally,

  there are two kinds of right hand side items, one kind corresponds to pure

  regular expressions, like the $\lambda$ in \eqref{example_eqns}, the other

  kind corresponds to transitions from one one equivalent class to another,

  like the $X_0 b, X_1 a$ etc.

*}

datatype rhs_item = 

   Lam "rexp"            (* Lambda *)

 | Trn "lang" "rexp"     (* Transition *)

text {*

  In this formalization, pure regular expressions like $\lambda$ is 

  repsented by @{text "Lam(EMPTY)"}, while transitions like $X_0 a$ is 

  represented by @{term "Trn X\<^isub>0 (CHAR a)"}.

*}

text {*

  The functions @{text "the_r"} and @{text "the_Trn"} are used to extract

  subcomponents from right hand side items.

  *}

fun 

  the_r :: "rhs_item \<Rightarrow> rexp"

where 

  "the_r (Lam r) = r"

fun 

  the_Trn:: "rhs_item \<Rightarrow> (lang \<times> rexp)"

where 

  "the_Trn (Trn Y r) = (Y, r)"

text {*

  Every right-hand side item @{text "itm"} defines a language given 

  by @{text "L(itm)"}, defined as:

*}

overloading L_rhs_e \<equiv> "L:: rhs_item \<Rightarrow> lang"

begin

  fun L_rhs_e:: "rhs_item \<Rightarrow> lang"

  where

    "L_rhs_e (Lam r) = L r" 

  | "L_rhs_e (Trn X r) = X ;; L r"

end

text {*

  The right hand side of every equation is represented by a set of

  items. The string set defined by such a set @{text "itms"} is given

  by @{text "L(itms)"}, defined as:

*}

overloading L_rhs \<equiv> "L:: rhs_item set \<Rightarrow> lang"

begin

   fun L_rhs:: "rhs_item set \<Rightarrow> lang"

   where 

     "L_rhs rhs = \<Union> (L ` rhs)"

end

text {* 

  Given a set of equivalence classes @{text "CS"} and one equivalence class @{text "X"} among

  @{text "CS"}, the term @{text "init_rhs CS X"} is used to extract the right hand side of

  the equation describing the formation of @{text "X"}. The definition of @{text "init_rhs"}

  is:

*}

definition 

where

definition

  "init_rhs CS X \<equiv>  

      if ([] \<in> X) then 

      else 

text {*

  In the definition of @{text "init_rhs"}, the term 

  @{text "{Trn Y (CHAR c)| Y c. Y \<in> CS \<and> Y ;; {[c]} \<subseteq> X}"} appearing on both branches

  describes the formation of strings in @{text "X"} out of transitions, while 

  the term @{text "{Lam(EMPTY)}"} describes the empty string which is intrinsically contained in

  @{text "X"} rather than by transition. This @{text "{Lam(EMPTY)}"} corresponds to 

  the $\lambda$ in \eqref{example_eqns}.

  With the help of @{text "init_rhs"}, the equitional system descrbing the formation of every

  equivalent class inside @{text "CS"} is given by the following @{text "eqs(CS)"}.