(*<*)
theory Paper
imports "../Myhill" "LaTeXsugar" 
begin

declare [[show_question_marks = false]]

consts
 REL :: "(string \<times> string) \<Rightarrow> bool"
 UPLUS :: "'a set \<Rightarrow> 'a set \<Rightarrow> (nat \<times> 'a) set"

abbreviation
  "EClass x R \<equiv> R `` {x}"

abbreviation 
  "append_rexp2 r_itm r \<equiv> append_rexp r r_itm"


notation (latex output)
  str_eq_rel ("\<approx>\<^bsub>_\<^esub>") and
  str_eq ("_ \<approx>\<^bsub>_\<^esub> _") and
  Seq (infixr "\<cdot>" 100) and
  Star ("_\<^bsup>\<star>\<^esup>") and
  pow ("_\<^bsup>_\<^esup>" [100, 100] 100) and
  Suc ("_+1" [100] 100) and
  quotient ("_ \<^raw:\ensuremath{\!\sslash\!}> _" [90, 90] 90) and
  REL ("\<approx>") and
  UPLUS ("_ \<^raw:\ensuremath{\uplus}> _" [90, 90] 90) and
  L ("\<^raw:\ensuremath{\cal{L}}>'(_')" [0] 101) and
  Lam ("\<lambda>'(_')" [100] 100) and 
  Trn ("'(_, _')" [100, 100] 100) and 
  EClass ("\<lbrakk>_\<rbrakk>\<^bsub>_\<^esub>" [100, 100] 100) and
  transition ("_ \<^raw:\ensuremath{\stackrel{\text{>_\<^raw:}}{\Longmapsto}}> _" [100, 100, 100] 100) and
  Setalt ("\<^raw:\ensuremath{\bigplus}>_" [1000] 999) and
  append_rexp2 ("_ \<^raw:\ensuremath{\triangleleft}> _" [100, 100] 100) and
  append_rhs_rexp ("_ \<^raw:\ensuremath{\triangleleft}> _" [100, 100] 50)

(*>*)


section {* Introduction *}

text {*
  Regular languages are an important and well-understood subject in Computer
  Science, with many beautiful theorems and many useful algorithms. There is a
  wide range of textbooks on this subject, many of which are aimed at students
  and contain very detailed ``pencil-and-paper'' proofs
  (e.g.~\cite{Kozen97}). It seems natural to exercise theorem provers by
  formalising the theorems and by verifying formally the algorithms.

  There is however a problem: the typical approach to regular languages is to
  introduce finite automata and then define everything in terms of them.  For
  example, a regular language is normally defined as one whose strings are
  recognised by a finite deterministic automaton. This approach has many
  benefits. Among them is the fact that it is easy to convince oneself that
  regular languages are closed under complementation: one just has to exchange
  the accepting and non-accepting states in the corresponding automaton to
  obtain an automaton for the complement language.  The problem, however, lies with
  formalising such reasoning in a HOL-based theorem prover, in our case
  Isabelle/HOL. Automata are build up from states and transitions that 
  need to be represented as graphs, matrices or functions, none
  of which can be defined as inductive datatype. 

  In case of graphs and matrices, this means we have to build our own
  reasoning infrastructure for them, as neither Isabelle/HOL nor HOL4 nor
  HOLlight support them with libraries. Even worse, reasoning about graphs and
  matrices can be a real hassle in HOL-based theorem provers.  Consider for
  example the operation of sequencing two automata, say $A_1$ and $A_2$, by
  connecting the accepting states of $A_1$ to the initial state of $A_2$:  
  
  \begin{center}
  \begin{tabular}{ccc}
  \begin{tikzpicture}[scale=0.8]
  %\draw[step=2mm] (-1,-1) grid (1,1);
  
  \draw[rounded corners=1mm, very thick] (-1.0,-0.3) rectangle (-0.2,0.3);
  \draw[rounded corners=1mm, very thick] ( 0.2,-0.3) rectangle ( 1.0,0.3);

  \node (A) at (-1.0,0.0) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
  \node (B) at ( 0.2,0.0) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
  
  \node (C) at (-0.2, 0.13) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
  \node (D) at (-0.2,-0.13) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};

  \node (E) at (1.0, 0.2) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
  \node (F) at (1.0,-0.0) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
  \node (G) at (1.0,-0.2) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};

  \draw (-0.6,0.0) node {\footnotesize$A_1$};
  \draw ( 0.6,0.0) node {\footnotesize$A_2$};
  \end{tikzpicture}

  & 

  \raisebox{1.1mm}{\bf\Large$\;\;\;\Rightarrow\,\;\;$}

  &

  \begin{tikzpicture}[scale=0.8]
  %\draw[step=2mm] (-1,-1) grid (1,1);
  
  \draw[rounded corners=1mm, very thick] (-1.0,-0.3) rectangle (-0.2,0.3);
  \draw[rounded corners=1mm, very thick] ( 0.2,-0.3) rectangle ( 1.0,0.3);

  \node (A) at (-1.0,0.0) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
  \node (B) at ( 0.2,0.0) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
  
  \node (C) at (-0.2, 0.13) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
  \node (D) at (-0.2,-0.13) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};

  \node (E) at (1.0, 0.2) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
  \node (F) at (1.0,-0.0) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
  \node (G) at (1.0,-0.2) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
  
  \draw (C) to [very thick, bend left=45] (B);
  \draw (D) to [very thick, bend right=45] (B);

  \draw (-0.6,0.0) node {\footnotesize$A_1$};
  \draw ( 0.6,0.0) node {\footnotesize$A_2$};
  \end{tikzpicture}

  \end{tabular}
  \end{center}

  \noindent
  On ``paper'' we can define the corresponding graph in terms of the disjoint 
  union of the state nodes. Unfortunately in HOL, the standard definition for disjoint 
  union, namely 
  %
  \begin{equation}\label{disjointunion}
  @{term "UPLUS A\<^isub>1 A\<^isub>2 \<equiv> {(1, x) | x. x \<in> A\<^isub>1} \<union> {(2, y) | y. y \<in> A\<^isub>2}"}
  \end{equation}

  \noindent
  changes the type---the disjoint union is not a set, but a set of pairs. 
  Using this definition for disjoint unions means we do not have a single type for automata
  and hence will not be able to state certain properties about \emph{all}
  automata, since there is no type quantification available in HOL. An
  alternative, which provides us with a single type for automata, is to give every 
  state node an identity, for example a natural
  number, and then be careful to rename these identities apart whenever
  connecting two automata. This results in clunky proofs
  establishing that properties are invariant under renaming. Similarly,
  connecting two automata represented as matrices results in very adhoc
  constructions, which are not pleasant to reason about.

  Functions are much better supported in Isabelle/HOL, but they still lead to similar
  problems as with graphs.  Composing, for example, two non-deterministic automata in parallel
  requires also the formalisation of disjoint unions. Nipkow \cite{Nipkow98} 
  dismisses for this the option of using identities, because it leads according to 
  him to ``messy proofs''. He
  opts for a variant of \eqref{disjointunion} using bit lists, but writes 

  \begin{quote}
  \it%
  \begin{tabular}{@ {}l@ {}p{0.88\textwidth}@ {}}
  `` & All lemmas appear obvious given a picture of the composition of automata\ldots
       Yet their proofs require a painful amount of detail.''
  \end{tabular}
  \end{quote}

  \noindent
  and
  
  \begin{quote}
  \it%
  \begin{tabular}{@ {}l@ {}p{0.88\textwidth}@ {}}
  `` & If the reader finds the above treatment in terms of bit lists revoltingly
       concrete, I cannot disagree. A more abstract approach is clearly desirable.''
  \end{tabular}
  \end{quote}


  \noindent
  Moreover, it is not so clear how to conveniently impose a finiteness condition 
  upon functions in order to represent \emph{finite} automata. The best is
  probably to resort to more advanced reasoning frameworks, such as \emph{locales}
  or \emph{type classes},
  which are not avaiable in \emph{all} HOL-based theorem provers.

  Because of these problems to do with representing automata, there seems
  to be no substantial formalisation of automata theory and regular languages 
  carried out in HOL-based theorem provers. Nipkow establishes in 
  \cite{Nipkow98} the link between regular expressions and automata in
  the context of lexing. The only larger formalisations of automata theory 
  are carried out in Nuprl \cite{Constable00} and in Coq (for example 
  \cite{Filliatre97}).
  
  In this paper, we will not attempt to formalise automata theory in
  Isabelle/HOL, but take a completely different approach to regular
  languages. Instead of defining a regular language as one where there exists
  an automaton that recognises all strings of the language, we define a
  regular language as:

  \begin{definition}
  A language @{text A} is \emph{regular}, provided there is a regular expression that matches all
  strings of @{text "A"}.
  \end{definition}
  
  \noindent
  The reason is that regular expressions, unlike graphs, matrices and functons, can
  be easily defined as inductive datatype. Consequently a corresponding reasoning 
  infrastructure comes for free. This has recently been exploited in HOL4 with a formalisation
  of regular expression matching based on derivatives \cite{OwensSlind08} and
  with an equivalence checker for regular expressions in Isabelle/HOL \cite{KraussNipkow11}.  
  The purpose of this paper is to
  show that a central result about regular languages---the Myhill-Nerode theorem---can 
  be recreated by only using regular expressions. This theorem gives necessary
  and sufficient conditions for when a language is regular. As a corollary of this
  theorem we can easily establish the usual closure properties, including 
  complementation, for regular languages.\smallskip
  
  \noindent
  {\bf Contributions:} 
  There is an extensive literature on regular languages.
  To our knowledge, our proof of the Myhill-Nerode theorem is the
  first that is based on regular expressions, only. We prove the part of this theorem 
  stating that a regular expression has only finitely many partitions using certain 
  tagging-functions. Again to our best knowledge, these tagging functions have
  not been used before to establish the Myhill-Nerode theorem.
*}

section {* Preliminaries *}

text {*
  Strings in Isabelle/HOL are lists of characters with the \emph{empty string}
  being represented by the empty list, written @{term "[]"}.  \emph{Languages}
  are sets of strings. The language containing all strings is written in
  Isabelle/HOL as @{term "UNIV::string set"}. The concatenation of two languages 
  is written @{term "A ;; B"} and a language raised to the power @{text n} is written 
  @{term "A \<up> n"}. They are defined as usual

  \begin{center}
  @{thm Seq_def[THEN eq_reflection, where A1="A" and B1="B"]}
  \hspace{7mm}
  @{thm pow.simps(1)[THEN eq_reflection, where A1="A"]}
  \hspace{7mm}
  @{thm pow.simps(2)[THEN eq_reflection, where A1="A" and n1="n"]}
  \end{center}

  \noindent
  where @{text "@"} is the usual list-append operation. The Kleene-star of a language @{text A}
  is defined as the union over all powers, namely @{thm Star_def}. In the paper
  we will make use of the following properties of these constructions.
  
  \begin{proposition}\label{langprops}\mbox{}\\
  \begin{tabular}{@ {}ll}
  (i)   & @{thm star_cases}     \\ 
  (ii)  & @{thm[mode=IfThen] pow_length}\\
  (iii) & @{thm seq_Union_left} \\ 
  \end{tabular}
  \end{proposition}

  \noindent
  In @{text "(ii)"} we use the notation @{term "length s"} for the length of a
  string.  This property states that if @{term "[] \<notin> A"} then the lengths of
  the strings in @{term "A \<up> (Suc n)"} must be longer than @{text n}.  We omit
  the proofs for these properties, but invite the reader to consult our
  formalisation.\footnote{Available at ???}

  The notation in Isabelle/HOL for the quotient of a language @{text A} according to an 
  equivalence relation @{term REL} is @{term "A // REL"}. We will write 
  @{text "\<lbrakk>x\<rbrakk>\<^isub>\<approx>"} for the equivalence class defined 
  as @{text "{y | y \<approx> x}"}.


  Central to our proof will be the solution of equational systems
  involving equivalence classes of languages. For this we will use Arden's lemma \cite{Brzozowski64}
  which solves equations of the form @{term "X = A ;; X \<union> B"} provided
  @{term "[] \<notin> A"}. However we will need the following ``reverse'' 
  version of Arden's lemma.

  \begin{lemma}[Reverse Arden's Lemma]\label{arden}\mbox{}\\
  If @{thm (prem 1) arden} then
  @{thm (lhs) arden} has the unique solution
  @{thm (rhs) arden}.
  \end{lemma}

  \begin{proof}
  For the right-to-left direction we assume @{thm (rhs) arden} and show
  that @{thm (lhs) arden} holds. From Prop.~\ref{langprops}@{text "(i)"} 
  we have @{term "A\<star> = {[]} \<union> A ;; A\<star>"},
  which is equal to @{term "A\<star> = {[]} \<union> A\<star> ;; A"}. Adding @{text B} to both 
  sides gives @{term "B ;; A\<star> = B ;; ({[]} \<union> A\<star> ;; A)"}, whose right-hand side
  is equal to @{term "(B ;; A\<star>) ;; A \<union> B"}. This completes this direction. 

  For the other direction we assume @{thm (lhs) arden}. By a simple induction
  on @{text n}, we can establish the property

  \begin{center}
  @{text "(*)"}\hspace{5mm} @{thm (concl) arden_helper}
  \end{center}
  
  \noindent
  Using this property we can show that @{term "B ;; (A \<up> n) \<subseteq> X"} holds for
  all @{text n}. From this we can infer @{term "B ;; A\<star> \<subseteq> X"} using the definition
  of @{text "\<star>"}.
  For the inclusion in the other direction we assume a string @{text s}
  with length @{text k} is element in @{text X}. Since @{thm (prem 1) arden}
  we know by Prop.~\ref{langprops}@{text "(ii)"} that 
  @{term "s \<notin> X ;; (A \<up> Suc k)"} since its length is only @{text k}
  (the strings in @{term "X ;; (A \<up> Suc k)"} are all longer). 
  From @{text "(*)"} it follows then that
  @{term s} must be element in @{term "(\<Union>m\<in>{0..k}. B ;; (A \<up> m))"}. This in turn
  implies that @{term s} is in @{term "(\<Union>n. B ;; (A \<up> n))"}. Using Prop.~\ref{langprops}@{text "(iii)"} 
  this is equal to @{term "B ;; A\<star>"}, as we needed to show.\qed
  \end{proof}

  \noindent
  Regular expressions are defined as the inductive datatype

  \begin{center}
  @{text r} @{text "::="}
  @{term NULL}\hspace{1.5mm}@{text"|"}\hspace{1.5mm} 
  @{term EMPTY}\hspace{1.5mm}@{text"|"}\hspace{1.5mm} 
  @{term "CHAR c"}\hspace{1.5mm}@{text"|"}\hspace{1.5mm} 
  @{term "SEQ r r"}\hspace{1.5mm}@{text"|"}\hspace{1.5mm} 
  @{term "ALT r r"}\hspace{1.5mm}@{text"|"}\hspace{1.5mm} 
  @{term "STAR r"}
  \end{center}

  \noindent
  and the language matched by a regular expression is defined as

  \begin{center}
  \begin{tabular}{c@ {\hspace{10mm}}c}
  \begin{tabular}{rcl}
  @{thm (lhs) L_rexp.simps(1)} & @{text "\<equiv>"} & @{thm (rhs) L_rexp.simps(1)}\\
  @{thm (lhs) L_rexp.simps(2)} & @{text "\<equiv>"} & @{thm (rhs) L_rexp.simps(2)}\\
  @{thm (lhs) L_rexp.simps(3)[where c="c"]} & @{text "\<equiv>"} & @{thm (rhs) L_rexp.simps(3)[where c="c"]}\\
  \end{tabular}
  &
  \begin{tabular}{rcl}
  @{thm (lhs) L_rexp.simps(4)[where ?r1.0="r\<^isub>1" and ?r2.0="r\<^isub>2"]} & @{text "\<equiv>"} &
       @{thm (rhs) L_rexp.simps(4)[where ?r1.0="r\<^isub>1" and ?r2.0="r\<^isub>2"]}\\
  @{thm (lhs) L_rexp.simps(5)[where ?r1.0="r\<^isub>1" and ?r2.0="r\<^isub>2"]} & @{text "\<equiv>"} &
       @{thm (rhs) L_rexp.simps(5)[where ?r1.0="r\<^isub>1" and ?r2.0="r\<^isub>2"]}\\
  @{thm (lhs) L_rexp.simps(6)[where r="r"]} & @{text "\<equiv>"} &
      @{thm (rhs) L_rexp.simps(6)[where r="r"]}\\
  \end{tabular}
  \end{tabular}
  \end{center}

  Given a finite set of regular expressions @{text rs}, we will make use of the operation of generating 
  a regular expression that matches all languages of @{text rs}. We only need to know the existence
  of such a regular expression and therefore we use Isabelle/HOL's @{const "fold_graph"} and Hilbert's
  @{text "\<epsilon>"} to define @{term "\<Uplus>rs"}. This operation, roughly speaking, folds @{const ALT} over the 
  set @{text rs} with @{const NULL} for the empty set. We can prove that for a finite set @{text rs}

  \begin{center}
  @{thm (lhs) folds_alt_simp} @{text "= \<Union> (\<calL> ` rs)"}
  \end{center}

  \noindent
  holds, whereby @{text "\<calL> ` rs"} stands for the 
  image of the set @{text rs} under function @{text "\<calL>"}.
*}

section {* The Myhill-Nerode Theorem, First Part *}

text {*
  The key definition in the Myhill-Nerode theorem is the
  \emph{Myhill-Nerode relation}, which states that w.r.t.~a language two 
  strings are related, provided there is no distinguishing extension in this
  language. This can be defined as:

  \begin{definition}[Myhill-Nerode Relation]\mbox{}\\
  @{thm str_eq_def[simplified str_eq_rel_def Pair_Collect]}
  \end{definition}

  \noindent
  It is easy to see that @{term "\<approx>A"} is an equivalence relation, which
  partitions the set of all strings, @{text "UNIV"}, into a set of disjoint
  equivalence classes. To illustrate this quotient construction, let us give an 
  example: consider the regular language containing just
  the string @{text "[c]"}. The relation @{term "\<approx>({[c]})"} partitions @{text UNIV}
  into three equivalence classes @{text "X\<^isub>1"}, @{text "X\<^isub>2"} and  @{text "X\<^isub>3"}
  as follows
  
  \begin{center}
  @{text "X\<^isub>1 = {[]}"}\hspace{5mm}
  @{text "X\<^isub>2 = {[c]}"}\hspace{5mm}
  @{text "X\<^isub>3 = UNIV - {[], [c]}"}
  \end{center}

  One direction of the Myhill-Nerode theorem establishes 
  that if there are finitely many equivalence classes, like in the example above, then 
  the language is regular. In our setting we therefore have to show:
  
  \begin{theorem}\label{myhillnerodeone}
  @{thm[mode=IfThen] Myhill_Nerode1}
  \end{theorem}

  \noindent
  To prove this theorem, we first define the set @{term "finals A"} as those equivalence
  classes from @{term "UNIV // \<approx>A"} that contain strings of @{text A}, namely
  %
  \begin{equation} 
  @{thm finals_def}
  \end{equation}

  \noindent
  In our running example, @{text "X\<^isub>2"} is the only equivalence class in @{term "finals {[c]}"}.
  It is straightforward to show that in general @{thm lang_is_union_of_finals} and 
  @{thm finals_in_partitions} hold. 
  Therefore if we know that there exists a regular expression for every
  equivalence class in \mbox{@{term "finals A"}} (which by assumption must be
  a finite set), then we can use @{text "\<bigplus>"} to obtain a regular expression 
  that matches every string in @{text A}.


  Our proof of Thm.~\ref{myhillnerodeone} relies on a method that can calculate a
  regular expression for \emph{every} equivalence class, not just the ones 
  in @{term "finals A"}. We
  first define the notion of \emph{one-character-transition} between 
  two equivalence classes
  %
  \begin{equation} 
  @{thm transition_def}
  \end{equation}

  \noindent
  which means that if we concatenate the character @{text c} to the end of all 
  strings in the equivalence class @{text Y}, we obtain a subset of 
  @{text X}. Note that we do not define an automaton here, we merely relate two sets
  (with respect to a character). In our concrete example we have 
  @{term "X\<^isub>1 \<Turnstile>c\<Rightarrow> X\<^isub>2"}, @{term "X\<^isub>1 \<Turnstile>d\<Rightarrow> X\<^isub>3"} with @{text d} being any 
  other character than @{text c}, and @{term "X\<^isub>3 \<Turnstile>d\<Rightarrow> X\<^isub>3"} for any @{text d}.
  
  Next we build an \emph{initial} equational system that
  contains an equation for each equivalence class. Suppose we have 
  the equivalence classes @{text "X\<^isub>1,\<dots>,X\<^isub>n"}, there must be one and only one that
  contains the empty string @{text "[]"} (since equivalence classes are disjoint).
  Let us assume @{text "[] \<in> X\<^isub>1"}. We build the following equational system
  
  \begin{center}
  \begin{tabular}{rcl}
  @{text "X\<^isub>1"} & @{text "="} & @{text "(Y\<^isub>1\<^isub>1, CHAR c\<^isub>1\<^isub>1) + \<dots> + (Y\<^isub>1\<^isub>p, CHAR c\<^isub>1\<^isub>p) + \<lambda>(EMPTY)"} \\
  @{text "X\<^isub>2"} & @{text "="} & @{text "(Y\<^isub>2\<^isub>1, CHAR c\<^isub>2\<^isub>1) + \<dots> + (Y\<^isub>2\<^isub>o, CHAR c\<^isub>2\<^isub>o)"} \\
  & $\vdots$ \\
  @{text "X\<^isub>n"} & @{text "="} & @{text "(Y\<^isub>n\<^isub>1, CHAR c\<^isub>n\<^isub>1) + \<dots> + (Y\<^isub>n\<^isub>q, CHAR c\<^isub>n\<^isub>q)"}\\
  \end{tabular}
  \end{center}

  \noindent
  where the terms @{text "(Y\<^isub>i\<^isub>j, CHAR c\<^isub>i\<^isub>j)"}
  stand for all transitions @{term "Y\<^isub>i\<^isub>j \<Turnstile>c\<^isub>i\<^isub>j\<Rightarrow>
  X\<^isub>i"}.   There can only be
  finitely many such terms in a right-hand side since there are only finitely many
  equivalence classes and only finitely many characters.  The term @{text
  "\<lambda>(EMPTY)"} in the first equation acts as a marker for the equivalence class
  containing @{text "[]"}.\footnote{Note that we mark, roughly speaking, the
  single ``initial'' state in the equational system, which is different from
  the method by Brzozowski \cite{Brzozowski64}, where he marks the
  ``terminal'' states. We are forced to set up the equational system in our
  way, because the Myhill-Nerode relation determines the ``direction'' of the
  transitions. The successor ``state'' of an equivalence class @{text Y} can
  be reached by adding characters to the end of @{text Y}. This is also the
  reason why we have to use our reverse version of Arden's lemma.}
  Overloading the function @{text \<calL>} for the two kinds of terms in the
  equational system, we have
  
  \begin{center}
  @{text "\<calL>(Y, r) \<equiv>"} %
  @{thm (rhs) L_rhs_item.simps(2)[where X="Y" and r="r", THEN eq_reflection]}\hspace{10mm}
  @{thm L_rhs_item.simps(1)[where r="r", THEN eq_reflection]}
  \end{center}

  \noindent
  and we can prove for @{text "X\<^isub>2\<^isub>.\<^isub>.\<^isub>n"} that the following equations
  %
  \begin{equation}\label{inv1}
  @{text "X\<^isub>i = \<calL>(Y\<^isub>i\<^isub>1, CHAR c\<^isub>i\<^isub>1) \<union> \<dots> \<union> \<calL>(Y\<^isub>i\<^isub>q, CHAR c\<^isub>i\<^isub>q)"}.
  \end{equation}

  \noindent
  hold. Similarly for @{text "X\<^isub>1"} we can show the following equation
  %
  \begin{equation}\label{inv2}
  @{text "X\<^isub>1 = \<calL>(Y\<^isub>i\<^isub>1, CHAR c\<^isub>i\<^isub>1) \<union> \<dots> \<union> \<calL>(Y\<^isub>i\<^isub>p, CHAR c\<^isub>i\<^isub>p) \<union> \<calL>(\<lambda>(EMPTY))"}.
  \end{equation}

  \noindent
  The reason for adding the @{text \<lambda>}-marker to our initial equational system is 
  to obtain this equation: it only holds with the marker, since none of 
  the other terms contain the empty string. 

  Our representation for the equations in Isabelle/HOL are pairs,
  where the first component is an equivalence class and the second component
  is a set of terms. Given a set of equivalence
  classes @{text CS}, our initial equational system @{term "Init CS"} is thus 
  formally defined as
  %
  \begin{equation}\label{initcs}
  \mbox{\begin{tabular}{rcl}     
  @{thm (lhs) Init_rhs_def} & @{text "\<equiv>"} & 
  @{text "if"}~@{term "[] \<in> X"}\\
  & & @{text "then"}~@{term "{Trn Y (CHAR c) | Y c. Y \<in> CS \<and> Y \<Turnstile>c\<Rightarrow> X} \<union> {Lam EMPTY}"}\\
  & & @{text "else"}~@{term "{Trn Y (CHAR c)| Y c. Y \<in> CS \<and> Y \<Turnstile>c\<Rightarrow> X}"}\\
  @{thm (lhs) Init_def}     & @{text "\<equiv>"} & @{thm (rhs) Init_def}
  \end{tabular}}
  \end{equation}

  

  \noindent
  Because we use sets of terms 
  for representing the right-hand sides of equations, we can 
  prove \eqref{inv1} and \eqref{inv2} more concisely as
  %
  \begin{lemma}\label{inv}
  If @{thm (prem 1) test} then @{text "X = \<Union> \<calL> ` rhs"}.
  \end{lemma}

  \noindent
  Our proof of Thm.~\ref{myhillnerodeone} will proceed by transforming the
  initial equational system into one in \emph{solved form} maintaining the invariant
  in Lemma \ref{inv}. From the solved form we will be able to read
  off the regular expressions. 

  In order to transform an equational system into solved form, we have two 
  operations: one that takes an equation of the form @{text "X = rhs"} and removes
  the recursive occurences of @{text X} in the @{text rhs} using our variant of Arden's 
  Lemma. The other operation takes an equation @{text "X = rhs"}
  and substitutes @{text X} throughout the rest of the equational system
  adjusting the remaining regular expressions approriately. To define this adjustment 
  we define the \emph{append-operation} 

  \begin{center}
  @{thm append_rexp.simps(2)[where X="Y" and r="r\<^isub>1" and rexp="r\<^isub>2", THEN eq_reflection]}\hspace{10mm}
  @{thm append_rexp.simps(1)[where r="r\<^isub>1" and rexp="r\<^isub>2", THEN eq_reflection]}
  \end{center}

  \noindent
  which we also lift to entire right-hand sides of equations, written as
  @{thm (lhs) append_rhs_rexp_def[where rexp="r"]}. With this we can define
  the \emph{arden-operation} for an equation of the form @{text "X = rhs"} as:
  
  \begin{center}
  \begin{tabular}{rc@ {\hspace{2mm}}r@ {\hspace{1mm}}l}
  @{thm (lhs) Arden_def} & @{text "\<equiv>"}~~\mbox{} & \multicolumn{2}{@ {\hspace{-2mm}}l}{@{text "let"}}\\ 
   & & @{text "rhs' ="} & @{term "rhs - {Trn X r | r. Trn X r \<in> rhs}"} \\
   & & @{text "r' ="}   & @{term "STAR (\<Uplus> {r. Trn X r \<in> rhs})"}\\
   & &  \multicolumn{2}{@ {\hspace{-2mm}}l}{@{text "in"}~~@{term "append_rhs_rexp rhs' r'"}}\\ 
  \end{tabular}
  \end{center}

  \noindent
  In this definition, we first delete all terms of the form @{text "(X, r)"} from @{text rhs};
  then we calculate the combinded regular expressions for all @{text r} coming 
  from the deleted @{text "(X, r)"}, and take the @{const STAR} of it;
  finally we append this regular expression to @{text rhs'}. It can be easily seen 
  that this operation mimics Arden's lemma on the level of equations.  
  The \emph{substituion-operation} takes an equation
  of the form @{text "X = xrhs"} and substitutes it into the right-hand side @{text rhs}.

  \begin{center}
  \begin{tabular}{rc@ {\hspace{2mm}}r@ {\hspace{1mm}}l}
  @{thm (lhs) Subst_def} & @{text "\<equiv>"}~~\mbox{} & \multicolumn{2}{@ {\hspace{-2mm}}l}{@{text "let"}}\\ 
   & & @{text "rhs' ="} & @{term "rhs - {Trn X r | r. Trn X r \<in> rhs}"} \\
   & & @{text "r' ="}   & @{term "\<Uplus> {r. Trn X r \<in> rhs}"}\\
   & &  \multicolumn{2}{@ {\hspace{-2mm}}l}{@{text "in"}~~@{term "rhs' \<union> append_rhs_rexp xrhs r'"}}\\ 
  \end{tabular}
  \end{center}

  \noindent
  We again delete first all occurence of @{text "(X, r)"} in @{text rhs}; we then calculate
  the regular expression corresponding to the deleted terms; finally we append this
  regular expression to @{text "xrhs"} and union it up with @{text rhs'}. When we use
  the substitution operation we will arrange it so that @{text "xrhs"} does not contain
  any occurence of @{text X}.

  With these two operation in place, we can define the operation that removes one equation
  from an equational systems @{text ES}. The operation @{const Subst_all}
  substitutes an equation @{text "X = xrhs"} throughout an equational system @{text ES}; 
  @{const Remove} then completely removes such an equation from @{text ES} by substituting 
  it to the rest of the equational system, but first eliminating all recursive occurences
  of @{text X} by applying @{const Arden} to @{text "xrhs"}.

  \begin{center}
  \begin{tabular}{rcl}
  @{thm (lhs) Subst_all_def} & @{text "\<equiv>"} & @{thm (rhs) Subst_all_def}\\
  @{thm (lhs) Remove_def}    & @{text "\<equiv>"} & @{thm (rhs) Remove_def}
  \end{tabular}
  \end{center}

  \noindent
  Finially, we can define how an equational system should be solved. For this 
  we will need to iterate the process of eliminating equations until only one equation
  will be left in the system. However, we not just want to have any equation
  as being the last one, but the one involving the equivalence class for 
  which we want to calculate the regular 
  expression. Let us suppose the equivalence class is @{text X}. 
  Since @{text X} is the one to be solved, in every iteration step we have to pick an
  equation to be eliminated that which is different from @{text X}. In this way 
  @{text X} is kept to the final step. The choice is implemented by Hilbert's choice 
  operator, written @{text SOME} in the definition below.
  
  \begin{center}
  \begin{tabular}{rc@ {\hspace{4mm}}r@ {\hspace{1mm}}l}
  @{thm (lhs) Iter_def} & @{text "\<equiv>"}~~\mbox{} & \multicolumn{2}{@ {\hspace{-4mm}}l}{@{text "let"}}\\ 
   & & @{text "(Y, yrhs) ="} & @{term "SOME (Y, yrhs). (Y, yrhs) \<in> ES \<and> X \<noteq> Y"} \\
   & &  \multicolumn{2}{@ {\hspace{-4mm}}l}{@{text "in"}~~@{term "Remove ES Y yrhs"}}\\ 
  \end{tabular}
  \end{center}

  \noindent
  The last definition we need applies @{term Iter} over and over again until a condition 
  @{text COND} is \emph{not} satisfied anymore. The condition states that there
  are more than one equation left in the equational system @{text ES}. For this
  we use Isabelle/HOL's @{text while}-operator as follows:
  
  \begin{center}
  @{thm Solve_def}
  \end{center}

  \noindent
  We are not concerned here with the definition of this operator
  (see \cite{BerghoferNipkow00}), but note that we eliminate
  in each @{const Iter}-step a single equation, and therefore 
  have a well-founded termination order by taking the cardinality 
  of the equational system @{text ES}. This enables us to prove
  properties about our definition of @{const Solve} when we ``call'' it with
  the equivalence class @{text X} and the initial equational system 
  @{term "Init (UNIV // \<approx>A)"} from
  \eqref{initcs}:


  \begin{center}
  \begin{tabular}{l}
  @{term "invariant (Init (UNIV // \<approx>A))"} \\
  @{term "\<forall>ES. invariant ES \<and> Cond ES \<longrightarrow> invariant (Iter X ES)"}\\
  @{term "\<forall>ES. invariant ES \<and> Cond ES \<longrightarrow> card (Iter X ES) < card ES"}\\
  @{term "\<forall>ES. invariant ES \<and> \<not> Cond ES \<longrightarrow> P ES"}\\
  \hline
  \multicolumn{1}{c}{@{term "P (Solve X (Init (UNIV // \<approx>A)))"}}
  \end{tabular}
  \end{center}

  \noindent
  This principle states that given an invariant (which we will specify below) 
  we can prove a property
  @{text "P"} involving @{const Solve}. For this we have to discharge the following
  proof obligations: first the
  initial equational system satisfies the invariant; second that the iteration
  step @{text "Iter"} preserves the the invariant as long as the condition @{term Cond} holds;
  third that @{text "Iter"} decreases the termination order, and fourth that
  once the condition does not hold anymore then the property @{text P} must hold.

  The property @{term P} in our proof will state that @{term "Solve X (Init (UNIV // \<approx>A))"}
  returns with a single equation @{text "X = xrhs"}, for some @{text "xrhs"} and
  that this equational system still satisfies the invariant. In order to get
  the proof through, the invariant is composed of the following six properties:

  \begin{center}
  \begin{tabular}{@ {}rcl@ {\hspace{-13mm}}l @ {}}
  @{text "invariant ES"} & @{text "\<equiv>"} &
      @{term "finite ES"} & @{text "(finiteness)"}\\
  & @{text "\<and>"} & @{thm (rhs) finite_rhs_def} & @{text "(finiteness rhs)"}\\
  & @{text "\<and>"} & @{text "\<forall>(X, rhs)\<in>ES. X = \<Union>\<calL> ` rhs"} & @{text "(soundness)"}\\
  & @{text "\<and>"} & @{thm (rhs) distinct_equas_def}\\
  &             &  & @{text "(distinctness)"}\\
  & @{text "\<and>"} & @{thm (rhs) ardenable_def} & @{text "(ardenable)"}\\   
  & @{text "\<and>"} & @{thm (rhs) valid_eqs_def} & @{text "(validity)"}\\
  \end{tabular}
  \end{center}
 
  \noindent
  The first two ensure that the equational system is always finite (number of equations
  and number of terms in each equation); \ldots

  It is straightforward to prove that the inital equational system satisfies the
  invariant.

  \begin{lemma}
  @{thm[mode=IfThen] Init_ES_satisfies_invariant}
  \end{lemma}

  \begin{proof}
  Finiteness is given by the assumption and the way how we set up the 
  initial equational system. Soundness is proved in Lem.~\ref{inv}. Distinctness
  follows from the fact that the equivalence classes are disjoint. The ardenable
  property also follows from the setup of the equational system as does 
  validity.\qed
  \end{proof}

  \begin{lemma}
  @{thm[mode=IfThen] iteration_step_invariant[where xrhs="rhs"]}
  \end{lemma}

  \begin{proof} 
  ???
  \end{proof}

  \begin{lemma}
  @{thm[mode=IfThen] iteration_step_measure[simplified (no_asm), where xrhs="rhs"]}
  \end{lemma}

  \begin{proof}
  By assumption we know that @{text "ES"} is finite and has more than one element.
  Therefore there must be an element @{term "(Y, yrhs) \<in> ES"} with 
  @{term "(Y, yrhs) \<noteq> (X, rhs)"}. Using the distictness property we can infer
  that @{term "Y \<noteq> X"}. We further know that @{text "Remove ES Y yrhs"}
  removes the equation @{text "Y = yrhs"} from the system, and therefore 
  the cardinality of @{const Iter} strictly decreases.\qed
  \end{proof}

  \begin{lemma}
  If @{thm (prem 1) Solve} and @{thm (prem 2) Solve} then there exists
  a @{text rhs} such that  @{term "Solve X (Init (UNIV // \<approx>A)) = {(X, rhs)}"}
  and @{term "invariant {(X, rhs)}"}.
  \end{lemma}

  \begin{proof} 
  ???
  \end{proof}

  \noindent
  With this lemma in place we can show that for every equivalence class in @{term "UNIV // \<approx>A"}
  there exists a regular expression.

  \begin{lemma}\label{every_eqcl_has_reg}
  @{thm[mode=IfThen] every_eqcl_has_reg}
  \end{lemma}

  \begin{proof}
  By the preceeding Lemma, we know that there exists a @{text "rhs"} such
  that @{term "Solve X (Init (UNIV // \<approx>A))"} returns the equation @{text "X = rhs"},
  and that the invariant holds for this equation. That means we 
  know @{text "X = \<Union>\<calL> ` rhs"}. We further know that
  this is equal to \mbox{@{text "\<Union>\<calL> ` (Arden X rhs)"}} using the properties in the 
  invariant and Lem.~???. Using the validity property for the equation @{text "X = rhs"},
  we can infer that @{term "rhss rhs \<subseteq> {X}"} and because the arden operation
  removes that @{text X} from @{text rhs}, that @{term "rhss (Arden X rhs) = {}"}.
  That means @{term "Arden X rhs"} can only consist of terms of the form @{term "Lam r"}.
  So we can collect those (finitely many) regular expressions and have @{term "X = L (\<Uplus>rs)"}.
  With this we can conclude the proof.\qed
  \end{proof}

  \noindent
  Lem.~\ref{every_eqcl_has_reg} allows us to finally give a proof for the first direction
  of the Myhill-Nerode theorem.

  \begin{proof}[of Thm.~\ref{myhillnerodeone}]
  By Lem.~\ref{every_eqcl_has_reg} we know that there exists a regular language for
  every equivalence class in @{term "UNIV // \<approx>A"}. Since @{text "finals A"} is
  a subset of  @{term "UNIV // \<approx>A"}, we also know that for every equvalence class
  in @{term "finals A"} there exists a regular language. Moreover by assumption 
  we know that @{term "finals A"} must be finite, and therefore there must be a finite
  set of regular expressions @{text "rs"} such that

  \begin{center}
  @{term "\<Union>(finals A) = L (\<Uplus>rs)"}
  \end{center}

  \noindent
  Since the left-hand side is equal to @{text A}, we can use @{term "\<Uplus>rs"} 
  as the regular expression that is needed in the theorem.\qed
  \end{proof}
*}




section {* Myhill-Nerode, Second Part *}

text {*

  \begin{theorem}
  Given @{text "r"} is a regular expressions, then @{thm rexp_imp_finite}.
  \end{theorem}  

  \begin{proof}
  By induction on the structure of @{text r}. The cases for @{const NULL}, @{const EMPTY}
  and @{const CHAR} are straightforward, because we can easily establish

  \begin{center}
  \begin{tabular}{l}
  @{thm quot_null_eq}\\
  @{thm quot_empty_subset}\\
  @{thm quot_char_subset}
  \end{tabular}
  \end{center}

  \end{proof}
*}


section {* Conclusion and Related Work *}

text {*
  In this paper we took the view that a regular language as one where there exists 
  a regular expression that matches all its strings. For us it was important to find 
  out how far we can push this point of view. Having formalised the Myhill-Nerode
  theorem means pushed very far. Having the Myhill-Nerode theorem means we can 
  formalise much of the textbook results in this subject. 


*}


(*<*)
end
(*>*)