made the theories compatible with the existing developments in the AFP; old theories are in the directory Attic
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Attic/Myhill.thy Mon Jul 25 13:33:38 2011 +0000
@@ -0,0 +1,331 @@
+theory Myhill
+ imports Myhill_2
+begin
+
+section {* Preliminaries \label{sec_prelim}*}
+
+subsection {* Finite automata and \mht \label{sec_fa_mh} *}
+
+text {*
+
+A {\em determinisitc finite automata (DFA)} $M$ is a 5-tuple
+$(Q, \Sigma, \delta, s, F)$, where:
+\begin{enumerate}
+ \item $Q$ is a finite set of {\em states}, also denoted $Q_M$.
+ \item $\Sigma$ is a finite set of {\em alphabets}, also denoted $\Sigma_M$.
+ \item $\delta$ is a {\em transition function} of type @{text "Q \<times> \<Sigma> \<Rightarrow> Q"} (a total function),
+ also denoted $\delta_M$.
+ \item @{text "s \<in> Q"} is a state called {\em initial state}, also denoted $s_M$.
+ \item @{text "F \<subseteq> Q"} is a set of states named {\em accepting states}, also denoted $F_M$.
+\end{enumerate}
+Therefore, we have $M = (Q_M, \Sigma_M, \delta_M, s_M, F_M)$. Every DFA $M$ can be interpreted as
+a function assigning states to strings, denoted $\dfa{M}$, the definition of which is as the following:
+\begin{equation}
+\begin{aligned}
+ \dfa{M}([]) &\equiv s_M \\
+ \dfa{M}(xa) &\equiv \delta_M(\dfa{M}(x), a)
+\end{aligned}
+\end{equation}
+A string @{text "x"} is said to be {\em accepted} (or {\em recognized}) by a DFA $M$ if
+$\dfa{M}(x) \in F_M$. The language recoginzed by DFA $M$, denoted
+$L(M)$, is defined as:
+\begin{equation}
+ L(M) \equiv \{x~|~\dfa{M}(x) \in F_M\}
+\end{equation}
+The standard way of specifying a laugage $\Lang$ as {\em regular} is by stipulating that:
+$\Lang = L(M)$ for some DFA $M$.
+
+For any DFA $M$, the DFA obtained by changing initial state to another $p \in Q_M$ is denoted $M_p$,
+which is defined as:
+\begin{equation}
+ M_p \ \equiv\ (Q_M, \Sigma_M, \delta_M, p, F_M)
+\end{equation}
+Two states $p, q \in Q_M$ are said to be {\em equivalent}, denoted $p \approx_M q$, iff.
+\begin{equation}\label{m_eq_def}
+ L(M_p) = L(M_q)
+\end{equation}
+It is obvious that $\approx_M$ is an equivalent relation over $Q_M$. and
+the partition induced by $\approx_M$ has $|Q_M|$ equivalent classes.
+By overloading $\approx_M$, an equivalent relation over strings can be defined:
+\begin{equation}
+ x \approx_M y ~ ~ \equiv ~ ~ \dfa{M}(x) \approx_M \dfa{M}(y)
+\end{equation}
+It can be proved that the the partition induced by $\approx_M$ also has $|Q_M|$ equivalent classes.
+It is also easy to show that: if $x \approx_M y$, then $x \approx_{L(M)} y$, and this means
+$\approx_M$ is a more refined equivalent relation than $\approx_{L(M)}$. Since partition induced by
+$\approx_M$ is finite, the one induced by $\approx_{L(M)}$ must also be finite, and this is
+one of the two directions of \mht:
+\begin{Lem}[\mht , Direction two]
+ If a language $\Lang$ is regular (i.e. $\Lang = L(M)$ for some DFA $M$), then
+ the partition induced by $\approx_\Lang$ is finite.
+\end{Lem}
+
+The other direction is:
+\begin{Lem}[\mht , Direction one]\label{auto_mh_d1}
+ If the partition induced by $\approx_\Lang$ is finite, then
+ $\Lang$ is regular (i.e. $\Lang = L(M)$ for some DFA $M$).
+\end{Lem}
+The $M$ we are seeking when prove lemma \ref{auto_mh_d2} can be constructed out of $\approx_\Lang$,
+denoted $M_\Lang$ and defined as the following:
+\begin{subequations}
+\begin{eqnarray}
+ Q_{M_\Lang} ~ & \equiv & ~ \{ \cls{x}{\approx_\Lang}~|~ x \in \Sigma^* \}\\
+ \Sigma_{M_\Lang} ~ & \equiv & ~ \Sigma_M \\
+ \delta_{M_\Lang} ~ & \equiv & ~ \left (\lambda (\cls{x}{\approx_\Lang}, a). \cls{xa}{\approx_\Lang} \right) \\
+ s_{M_\Lang} ~ & \equiv & ~ \cls{[]}{\approx_\Lang} \\
+ F_{M_\Lang} ~ & \equiv & ~ \{ \cls{x}{\approx_\Lang}~|~ x \in \Lang \}
+\end{eqnarray}
+\end{subequations}
+It can be proved that $Q_{M_\Lang}$ is indeed finite and $\Lang = L(M_\Lang)$, so lemma \ref{auto_mh_d1} holds.
+It can also be proved that $M_\Lang$ is the minimal DFA (therefore unique) which recoginzes $\Lang$.
+
+
+
+*}
+
+subsection {* The objective and the underlying intuition *}
+
+text {*
+ It is now obvious from section \ref{sec_fa_mh} that \mht\ can be established easily when
+ {\em reglar languages} are defined as ones recognized by finite automata.
+ Under the context where the use of finite automata is forbiden, the situation is quite different.
+ The theorem now has to be expressed as:
+ \begin{Thm}[\mht , Regular expression version]
+ A language $\Lang$ is regular (i.e. $\Lang = L(\re)$ for some {\em regular expression} $\re$)
+ iff. the partition induced by $\approx_\Lang$ is finite.
+ \end{Thm}
+ The proof of this version consists of two directions (if the use of automata are not allowed):
+ \begin{description}
+ \item [Direction one:]
+ generating a regular expression $\re$ out of the finite partition induced by $\approx_\Lang$,
+ such that $\Lang = L(\re)$.
+ \item [Direction two:]
+ showing the finiteness of the partition induced by $\approx_\Lang$, under the assmption
+ that $\Lang$ is recognized by some regular expression $\re$ (i.e. $\Lang = L(\re)$).
+ \end{description}
+ The development of these two directions consititutes the body of this paper.
+
+*}
+
+section {* Direction @{text "regular language \<Rightarrow>finite partition"} *}
+
+text {*
+ Although not used explicitly, the notion of finite autotmata and its relationship with
+ language partition, as outlined in section \ref{sec_fa_mh}, still servers as important intuitive
+ guides in the development of this paper.
+ For example, {\em Direction one} follows the {\em Brzozowski algebraic method}
+ used to convert finite autotmata to regular expressions, under the intuition that every
+ partition member $\cls{x}{\approx_\Lang}$ is a state in the DFA $M_\Lang$ constructed to prove
+ lemma \ref{auto_mh_d1} of section \ref{sec_fa_mh}.
+
+ The basic idea of Brzozowski method is to extract an equational system out of the
+ transition relationship of the automaton in question. In the equational system, every
+ automaton state is represented by an unknown, the solution of which is expected to be
+ a regular expresion characterizing the state in a certain sense. There are two choices of
+ how a automaton state can be characterized. The first is to characterize by the set of
+ strings leading from the state in question into accepting states.
+ The other choice is to characterize by the set of strings leading from initial state
+ into the state in question. For the second choice, the language recognized the automaton
+ can be characterized by the solution of initial state, while for the second choice,
+ the language recoginzed by the automaton can be characterized by
+ combining solutions of all accepting states by @{text "+"}. Because of the automaton
+ used as our intuitive guide, the $M_\Lang$, the states of which are
+ sets of strings leading from initial state, the second choice is used in this paper.
+
+ Supposing the automaton in Fig \ref{fig_auto_part_rel} is the $M_\Lang$ for some language $\Lang$,
+ and suppose $\Sigma = \{a, b, c, d, e\}$. Under the second choice, the equational system extracted is:
+ \begin{subequations}
+ \begin{eqnarray}
+ X_0 & = & X_1 \cdot c + X_2 \cdot d + \lambda \label{x_0_def_o} \\
+ X_1 & = & X_0 \cdot a + X_1 \cdot b + X_2 \cdot d \label{x_1_def_o} \\
+ X_2 & = & X_0 \cdot b + X_1 \cdot d + X_2 \cdot a \\
+ X_3 & = & \begin{aligned}
+ & X_0 \cdot (c + d + e) + X_1 \cdot (a + e) + X_2 \cdot (b + e) + \\
+ & X_3 \cdot (a + b + c + d + e)
+ \end{aligned}
+ \end{eqnarray}
+ \end{subequations}
+
+\begin{figure}[h!]
+\centering
+\begin{minipage}{0.5\textwidth}
+\scalebox{0.8}{
+\begin{tikzpicture}[ultra thick,>=latex]
+ \node[draw,circle,initial] (start) {$X_0$};
+ \node[draw,circle,accepting] at ($(start) + (3.5cm,2cm)$) (ac1) {$X_1$};
+ \node[draw,circle,accepting] at ($(start) + (3.5cm,-2cm)$) (ac2) {$X_2$};
+ \node[draw,circle] at ($(start) + (6.5cm,0cm)$) (ab) {$X_3$};
+
+ \path[->] (start) edge [bend left] node [midway, above] {$a$} (ac1);
+ \path[->] (start) edge [bend right] node [midway, below] {$b$} (ac2);
+ \path[->] (ac1) edge [loop above] node [midway, above] {$b$} (ac1);
+ \path[->] (ac2) edge [loop below] node [midway, below] {$a$} (ac2);
+ \path[->] (ac1) edge [bend right] node [midway, left] {$c$} (ac2);
+ \path[->] (ac2) edge [bend right] node [midway, right] {$c$} (ac1);
+ \path[->] (ac1) edge node [midway, sloped, above] {$d$} (start);
+ \path[->] (ac2) edge node [midway, sloped, above] {$d$} (start);
+
+ \path [draw, rounded corners,->,dashed] (start) -- ($(start) + (0cm, 3.7cm)$)
+ -- ($(ab) + (0cm, 3.7cm)$) node[midway,above,sloped]{$\Sigma - \{a, b\}$} -- (ab);
+ \path[->,dashed] (ac1) edge node [midway, above, sloped] {$\Sigma - \{b,c,d\}$} (ab);
+ \path[->,dashed] (ac2) edge node [midway, below, sloped] {$\Sigma - \{a,c,d\}$} (ab);
+ \path[->,dashed] (ab) edge [loop right] node [midway, right] {$\Sigma$} (ab);
+\end{tikzpicture}}
+\end{minipage}
+\caption{An example automaton}\label{fig_auto_part_rel}
+\end{figure}
+
+ Every $\cdot$-item on the right side of equations describes some state transtions, except
+ the $\lambda$ in \eqref{x_0_def_o}, which represents empty string @{text "[]"}.
+ The reason is that: every state is characterized by the
+ set of incoming strings leading from initial state. For non-initial state, every such
+ string can be splitted into a prefix leading into a preceding state and a single character suffix
+ transiting into from the preceding state. The exception happens at
+ initial state, where the empty string is a incoming string which can not be splitted. The $\lambda$
+ in \eqref{x_0_def_o} is introduce to repsent this indivisible string. There is one and only one
+ $\lambda$ in every equational system such obtained, becasue $[]$ can only be contaied in one
+ equivalent class (the intial state in $M_\Lang$) and equivalent classes are disjoint.
+
+ Suppose all unknowns ($X_0, X_1, X_2, X_3$) are solvable, the regular expression charactering
+ laugnage $\Lang$ is $X_1 + X_2$. This paper gives a procedure
+ by which arbitrarily picked unknown can be solved. The basic idea to solve $X_i$ is by
+ eliminating all variables other than $X_i$ from the equational system. If
+ $X_0$ is the one picked to be solved, variables $X_1, X_2, X_3$ have to be removed one by
+ one. The order to remove does not matter as long as the remaing equations are kept valid.
+ Suppose $X_1$ is the first one to remove, the action is to replace all occurences of $X_1$
+ in remaining equations by the right hand side of its characterizing equation, i.e.
+ the $ X_0 \cdot a + X_1 \cdot b + X_2 \cdot d$ in \eqref{x_1_def_o}. However, because
+ of the recursive occurence of $X_1$, this replacement does not really removed $X_1$. Arden's
+ lemma is invoked to transform recursive equations like \eqref{x_1_def_o} into non-recursive
+ ones. For example, the recursive equation \eqref{x_1_def_o} is transformed into the follwing
+ non-recursive one:
+ \begin{equation}
+ X_1 = (X_0 \cdot a + X_2 \cdot d) \cdot b^* = X_0 \cdot (a \cdot b^*) + X_2 \cdot (d \cdot b^*)
+ \end{equation}
+ which, by Arden's lemma, still characterizes $X_1$ correctly. By subsituting
+ $(X_0 \cdot a + X_2 \cdot d) \cdot b^*$ for all $X_1$ and removing \eqref{x_1_def_o},
+ we get:
+ \begin{subequations}
+ \begin{eqnarray}
+ X_0 & = & \begin{aligned}
+ & (X_0 \cdot (a \cdot b^*) + X_2 \cdot (d \cdot b^*)) \cdot c +
+ X_2 \cdot d + \lambda = \\
+ & X_0 \cdot (a \cdot b^* \cdot c) + X_2 \cdot (d \cdot b^* \cdot c) +
+ X_2 \cdot d + \lambda = \\
+ & X_0 \cdot (a \cdot b^* \cdot c) + X_2 \cdot (d \cdot b^* \cdot c + d) + \lambda
+ \end{aligned} \label{x_0_def_1} \\
+ X_2 & = & \begin{aligned}
+ & X_0 \cdot b + (X_0 \cdot (a \cdot b^*) + X_2 \cdot (d \cdot b^*)) \cdot d + X_2 \cdot a = \\
+ & X_0 \cdot b + X_0 \cdot (a \cdot b^* \cdot d) + X_2 \cdot (d \cdot b^* \cdot d) + X_2 \cdot a = \\
+ & X_0 \cdot (b + a \cdot b^* \cdot d) + X_2 \cdot (d \cdot b^* \cdot d + a)
+ \end{aligned} \\
+ X_3 & = & \begin{aligned}
+ & X_0 \cdot (c + d + e) + ((X_0 \cdot a + X_2 \cdot d) \cdot b^*) \cdot (a + e)\\
+ & + X_2 \cdot (b + e) + X_3 \cdot (a + b + c + d + e) \label{x_3_def_1}
+ \end{aligned}
+ \end{eqnarray}
+ \end{subequations}
+Suppose $X_3$ is the one to remove next, since $X_3$ dose not appear in either $X_0$ or $X_2$,
+the removal of equation \eqref{x_3_def_1} changes nothing in the rest equations. Therefore, we get:
+ \begin{subequations}
+ \begin{eqnarray}
+ X_0 & = & X_0 \cdot (a \cdot b^* \cdot c) + X_2 \cdot (d \cdot b^* \cdot c + d) + \lambda \label{x_0_def_2} \\
+ X_2 & = & X_0 \cdot (b + a \cdot b^* \cdot d) + X_2 \cdot (d \cdot b^* \cdot d + a) \label{x_2_def_2}
+ \end{eqnarray}
+ \end{subequations}
+Actually, since absorbing state like $X_3$ contributes nothing to the language $\Lang$, it could have been removed
+at the very beginning of this precedure without affecting the final result. Now, the last unknown to remove
+is $X_2$ and the Arden's transformaton of \eqref{x_2_def_2} is:
+\begin{equation} \label{x_2_ardened}
+ X_2 ~ = ~ (X_0 \cdot (b + a \cdot b^* \cdot d)) \cdot (d \cdot b^* \cdot d + a)^* =
+ X_0 \cdot ((b + a \cdot b^* \cdot d) \cdot (d \cdot b^* \cdot d + a)^*)
+\end{equation}
+By substituting the right hand side of \eqref{x_2_ardened} into \eqref{x_0_def_2}, we get:
+\begin{equation}
+\begin{aligned}
+ X_0 & = && X_0 \cdot (a \cdot b^* \cdot c) + \\
+ & && X_0 \cdot ((b + a \cdot b^* \cdot d) \cdot (d \cdot b^* \cdot d + a)^*) \cdot
+ (d \cdot b^* \cdot c + d) + \lambda \\
+ & = && X_0 \cdot ((a \cdot b^* \cdot c) + \\
+ & && \hspace{3em} ((b + a \cdot b^* \cdot d) \cdot (d \cdot b^* \cdot d + a)^*) \cdot
+ (d \cdot b^* \cdot c + d)) + \lambda
+\end{aligned}
+\end{equation}
+By applying Arden's transformation to this, we get the solution of $X_0$ as:
+\begin{equation}
+\begin{aligned}
+ X_0 = ((a \cdot b^* \cdot c) +
+ ((b + a \cdot b^* \cdot d) \cdot (d \cdot b^* \cdot d + a)^*) \cdot
+ (d \cdot b^* \cdot c + d))^*
+\end{aligned}
+\end{equation}
+Using the same method, solutions for $X_1$ and $X_2$ can be obtained as well and the
+regular expressoin for $\Lang$ is just $X_1 + X_2$. The formalization of this procedure
+consititues the first direction of the {\em regular expression} verion of
+\mht. Detailed explaination are given in {\bf paper.pdf} and more details and comments
+can be found in the formal scripts.
+*}
+
+section {* Direction @{text "finite partition \<Rightarrow> regular language"} *}
+
+text {*
+ It is well known in the theory of regular languages that
+ the existence of finite language partition amounts to the existence of
+ a minimal automaton, i.e. the $M_\Lang$ constructed in section \ref{sec_prelim}, which recoginzes the
+ same language $\Lang$. The standard way to prove the existence of finite language partition
+ is to construct a automaton out of the regular expression which recoginzes the same language, from
+ which the existence of finite language partition follows immediately. As discussed in the introducton of
+ {\bf paper.pdf} as well as in [5], the problem for this approach happens when automata
+ of sub regular expressions are combined to form the automaton of the mother regular expression,
+ no matter what kind of representation is used, the formalization is cubersome, as said
+ by Nipkow in [5]: `{\em a more abstract mathod is clearly desirable}'. In this section,
+ an {\em intrinsically abstract} method is given, which completely avoid the mentioning of automata,
+ let along any particular representations.
+ *}
+
+text {*
+ The main proof structure is a structural induction on regular expressions,
+ where base cases (cases for @{const "NULL"}, @{const "EMPTY"}, @{const "CHAR"}) are quite straightforward to
+ proof. Real difficulty lies in inductive cases. By inductive hypothesis, languages defined by
+ sub-expressions induce finite partitiions. Under such hypothsis, we need to prove that the language
+ defined by the composite regular expression gives rise to finite partion.
+ The basic idea is to attach a tag @{text "tag(x)"} to every string
+ @{text "x"}. The tagging fuction @{text "tag"} is carefully devised, which returns tags
+ made of equivalent classes of the partitions induced by subexpressoins, and therefore has a finite
+ range. Let @{text "Lang"} be the composite language, it is proved that:
+ \begin{quote}
+ If strings with the same tag are equivalent with respect to @{text "Lang"}, expressed as:
+ \[
+ @{text "tag(x) = tag(y) \<Longrightarrow> x \<approx>Lang y"}
+ \]
+ then the partition induced by @{text "Lang"} must be finite.
+ \end{quote}
+ There are two arguments for this. The first goes as the following:
+ \begin{enumerate}
+ \item First, the tagging function @{text "tag"} induces an equivalent relation @{text "(=tag=)"}
+ (defiintion of @{text "f_eq_rel"} and lemma @{text "equiv_f_eq_rel"}).
+ \item It is shown that: if the range of @{text "tag"} (denoted @{text "range(tag)"}) is finite,
+ the partition given rise by @{text "(=tag=)"} is finite (lemma @{text "finite_eq_f_rel"}).
+ Since tags are made from equivalent classes from component partitions, and the inductive
+ hypothesis ensures the finiteness of these partitions, it is not difficult to prove
+ the finiteness of @{text "range(tag)"}.
+ \item It is proved that if equivalent relation @{text "R1"} is more refined than @{text "R2"}
+ (expressed as @{text "R1 \<subseteq> R2"}),
+ and the partition induced by @{text "R1"} is finite, then the partition induced by @{text "R2"}
+ is finite as well (lemma @{text "refined_partition_finite"}).
+ \item The injectivity assumption @{text "tag(x) = tag(y) \<Longrightarrow> x \<approx>Lang y"} implies that
+ @{text "(=tag=)"} is more refined than @{text "(\<approx>Lang)"}.
+ \item Combining the points above, we have: the partition induced by language @{text "Lang"}
+ is finite (lemma @{text "tag_finite_imageD"}).
+ \end{enumerate}
+
+We could have followed another approach given in appendix II of Brzozowski's paper [?], where
+the set of derivatives of any regular expression can be proved to be finite.
+Since it is easy to show that strings with same derivative are equivalent with respect to the
+language, then the second direction follows. We believe that our
+apporoach is easy to formalize, with no need to do complicated derivation calculations
+and countings as in [???].
+*}
+
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Attic/MyhillNerode.thy Mon Jul 25 13:33:38 2011 +0000
@@ -0,0 +1,1816 @@
+theory MyhillNerode
+ imports "Main" "List_Prefix"
+begin
+
+text {* sequential composition of languages *}
+
+definition
+ lang_seq :: "string set \<Rightarrow> string set \<Rightarrow> string set" ("_ ; _" [100,100] 100)
+where
+ "L1 ; L2 = {s1 @ s2 | s1 s2. s1 \<in> L1 \<and> s2 \<in> L2}"
+
+lemma lang_seq_empty:
+ shows "{[]} ; L = L"
+ and "L ; {[]} = L"
+unfolding lang_seq_def by auto
+
+lemma lang_seq_null:
+ shows "{} ; L = {}"
+ and "L ; {} = {}"
+unfolding lang_seq_def by auto
+
+lemma lang_seq_append:
+ assumes a: "s1 \<in> L1"
+ and b: "s2 \<in> L2"
+ shows "s1@s2 \<in> L1 ; L2"
+unfolding lang_seq_def
+using a b by auto
+
+lemma lang_seq_union:
+ shows "(L1 \<union> L2); L3 = (L1; L3) \<union> (L2; L3)"
+ and "L1; (L2 \<union> L3) = (L1; L2) \<union> (L1; L3)"
+unfolding lang_seq_def by auto
+
+lemma lang_seq_assoc:
+ shows "(L1 ; L2) ; L3 = L1 ; (L2 ; L3)"
+unfolding lang_seq_def
+apply(auto)
+apply(metis)
+by (metis append_assoc)
+
+
+section {* Kleene star for languages defined as least fixed point *}
+
+inductive_set
+ Star :: "string set \<Rightarrow> string set" ("_\<star>" [101] 102)
+ for L :: "string set"
+where
+ start[intro]: "[] \<in> L\<star>"
+| step[intro]: "\<lbrakk>s1 \<in> L; s2 \<in> L\<star>\<rbrakk> \<Longrightarrow> s1@s2 \<in> L\<star>"
+
+lemma lang_star_empty:
+ shows "{}\<star> = {[]}"
+by (auto elim: Star.cases)
+
+lemma lang_star_cases:
+ shows "L\<star> = {[]} \<union> L ; L\<star>"
+proof
+ { fix x
+ have "x \<in> L\<star> \<Longrightarrow> x \<in> {[]} \<union> L ; L\<star>"
+ unfolding lang_seq_def
+ by (induct rule: Star.induct) (auto)
+ }
+ then show "L\<star> \<subseteq> {[]} \<union> L ; L\<star>" by auto
+next
+ show "{[]} \<union> L ; L\<star> \<subseteq> L\<star>"
+ unfolding lang_seq_def by auto
+qed
+
+lemma lang_star_cases':
+ shows "L\<star> = {[]} \<union> L\<star> ; L"
+proof
+ { fix x
+ have "x \<in> L\<star> \<Longrightarrow> x \<in> {[]} \<union> L\<star> ; L"
+ unfolding lang_seq_def
+ apply (induct rule: Star.induct)
+ apply simp
+ apply simp
+ apply (erule disjE)
+ apply (auto)[1]
+ apply (erule exE | erule conjE)+
+ apply (rule disjI2)
+ apply (rule_tac x = "s1 @ s1a" in exI)
+ by auto
+ }
+ then show "L\<star> \<subseteq> {[]} \<union> L\<star> ; L" by auto
+next
+ show "{[]} \<union> L\<star> ; L \<subseteq> L\<star>"
+ unfolding lang_seq_def
+ apply auto
+ apply (erule Star.induct)
+ apply auto
+ apply (drule step[of _ _ "[]"])
+ by (auto intro:start)
+qed
+
+lemma lang_star_simple:
+ shows "L \<subseteq> L\<star>"
+by (subst lang_star_cases)
+ (auto simp only: lang_seq_def)
+
+lemma lang_star_prop0_aux:
+ "s2 \<in> L\<star> \<Longrightarrow> \<forall> s1. s1 \<in> L \<longrightarrow> (\<exists> s3 s4. s3 \<in> L\<star> \<and> s4 \<in> L \<and> s1 @ s2 = s3 @ s4)"
+apply (erule Star.induct)
+apply (clarify, rule_tac x = "[]" in exI, rule_tac x = s1 in exI, simp add:start)
+apply (clarify, drule_tac x = s1 in spec)
+apply (drule mp, simp, clarify)
+apply (rule_tac x = "s1a @ s3" in exI, rule_tac x = s4 in exI)
+by auto
+
+lemma lang_star_prop0:
+ "\<lbrakk>s1 \<in> L; s2 \<in> L\<star>\<rbrakk> \<Longrightarrow> \<exists> s3 s4. s3 \<in> L\<star> \<and> s4 \<in> L \<and> s1 @ s2 = s3 @ s4"
+by (auto dest:lang_star_prop0_aux)
+
+lemma lang_star_prop1:
+ assumes asm: "L1; L2 \<subseteq> L2"
+ shows "L1\<star>; L2 \<subseteq> L2"
+proof -
+ { fix s1 s2
+ assume minor: "s2 \<in> L2"
+ assume major: "s1 \<in> L1\<star>"
+ then have "s1@s2 \<in> L2"
+ proof(induct rule: Star.induct)
+ case start
+ show "[]@s2 \<in> L2" using minor by simp
+ next
+ case (step s1 s1')
+ have "s1 \<in> L1" by fact
+ moreover
+ have "s1'@s2 \<in> L2" by fact
+ ultimately have "s1@(s1'@s2) \<in> L1; L2" by (auto simp add: lang_seq_def)
+ with asm have "s1@(s1'@s2) \<in> L2" by auto
+ then show "(s1@s1')@s2 \<in> L2" by simp
+ qed
+ }
+ then show "L1\<star>; L2 \<subseteq> L2" by (auto simp add: lang_seq_def)
+qed
+
+lemma lang_star_prop2_aux:
+ "s2 \<in> L2\<star> \<Longrightarrow> \<forall> s1. s1 \<in> L1 \<and> L1 ; L2 \<subseteq> L1 \<longrightarrow> s1 @ s2 \<in> L1"
+apply (erule Star.induct, simp)
+apply (clarify, drule_tac x = "s1a @ s1" in spec)
+by (auto simp:lang_seq_def)
+
+lemma lang_star_prop2:
+ "L1; L2 \<subseteq> L1 \<Longrightarrow> L1 ; L2\<star> \<subseteq> L1"
+by (auto dest!:lang_star_prop2_aux simp:lang_seq_def)
+
+lemma lang_star_seq_subseteq:
+ shows "L ; L\<star> \<subseteq> L\<star>"
+using lang_star_cases by blast
+
+lemma lang_star_double:
+ shows "L\<star>; L\<star> = L\<star>"
+proof
+ show "L\<star> ; L\<star> \<subseteq> L\<star>"
+ using lang_star_prop1 lang_star_seq_subseteq by blast
+next
+ have "L\<star> \<subseteq> L\<star> \<union> L\<star>; (L; L\<star>)" by auto
+ also have "\<dots> = L\<star>;{[]} \<union> L\<star>; (L; L\<star>)" by (simp add: lang_seq_empty)
+ also have "\<dots> = L\<star>; ({[]} \<union> L; L\<star>)" by (simp only: lang_seq_union)
+ also have "\<dots> = L\<star>; L\<star>" using lang_star_cases by simp
+ finally show "L\<star> \<subseteq> L\<star> ; L\<star>" by simp
+qed
+
+lemma lang_star_seq_subseteq':
+ shows "L\<star>; L \<subseteq> L\<star>"
+proof -
+ have "L \<subseteq> L\<star>" by (rule lang_star_simple)
+ then have "L\<star>; L \<subseteq> L\<star>; L\<star>" by (auto simp add: lang_seq_def)
+ then show "L\<star>; L \<subseteq> L\<star>" using lang_star_double by blast
+qed
+
+lemma
+ shows "L\<star> \<subseteq> L\<star>\<star>"
+by (rule lang_star_simple)
+
+
+section {* regular expressions *}
+
+datatype rexp =
+ NULL
+| EMPTY
+| CHAR char
+| SEQ rexp rexp
+| ALT rexp rexp
+| STAR rexp
+
+
+consts L:: "'a \<Rightarrow> string set"
+
+overloading L_rexp \<equiv> "L:: rexp \<Rightarrow> string set"
+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
+
+
+text{* ************ now is the codes writen by chunhan ************************************* *}
+
+section {* Arden's Lemma revised *}
+
+lemma arden_aux1:
+ assumes a: "X \<subseteq> X ; A \<union> B"
+ and b: "[] \<notin> A"
+ shows "x \<in> X \<Longrightarrow> x \<in> B ; A\<star>"
+apply (induct x taking:length rule:measure_induct)
+apply (subgoal_tac "x \<in> X ; A \<union> B")
+defer
+using a
+apply (auto)[1]
+apply simp
+apply (erule disjE)
+defer
+apply (auto simp add:lang_seq_def) [1]
+apply (subgoal_tac "\<exists> x1 x2. x = x1 @ x2 \<and> x1 \<in> X \<and> x2 \<in> A")
+defer
+apply (auto simp add:lang_seq_def) [1]
+apply (erule exE | erule conjE)+
+apply simp
+apply (drule_tac x = x1 in spec)
+apply (simp)
+using b
+apply -
+apply (auto)[1]
+apply (subgoal_tac "x1 @ x2 \<in> (B ; A\<star>) ; A")
+defer
+apply (auto simp add:lang_seq_def)[1]
+by (metis Un_absorb1 lang_seq_assoc lang_seq_union(2) lang_star_double lang_star_simple mem_def sup1CI)
+
+theorem ardens_revised:
+ assumes nemp: "[] \<notin> A"
+ shows "(X = X ; A \<union> B) \<longleftrightarrow> (X = B ; A\<star>)"
+apply(rule iffI)
+defer
+apply(simp)
+apply(subst lang_star_cases')
+apply(subst lang_seq_union)
+apply(simp add: lang_seq_empty)
+apply(simp add: lang_seq_assoc)
+apply(auto)[1]
+proof -
+ assume "X = X ; A \<union> B"
+ then have as1: "X ; A \<union> B \<subseteq> X" and as2: "X \<subseteq> X ; A \<union> B" by simp_all
+ from as1 have a: "X ; A \<subseteq> X" and b: "B \<subseteq> X" by simp_all
+ from b have "B; A\<star> \<subseteq> X ; A\<star>" by (auto simp add: lang_seq_def)
+ moreover
+ from a have "X ; A\<star> \<subseteq> X"
+
+by (rule lang_star_prop2)
+ ultimately have f1: "B ; A\<star> \<subseteq> X" by simp
+ from as2 nemp
+ have f2: "X \<subseteq> B; A\<star>" using arden_aux1 by auto
+ from f1 f2 show "X = B; A\<star>" by auto
+qed
+
+
+
+section {* equiv class' definition *}
+
+definition
+ equiv_str :: "string \<Rightarrow> string set \<Rightarrow> string \<Rightarrow> bool" ("_ \<equiv>_\<equiv> _" [100, 100, 100] 100)
+where
+ "x \<equiv>Lang\<equiv> y \<longleftrightarrow> (\<forall>z. x @ z \<in> Lang \<longleftrightarrow> y @ z \<in> Lang)"
+
+definition
+ equiv_class :: "string \<Rightarrow> (string set) \<Rightarrow> string set" ("\<lbrakk>_\<rbrakk>_" [100, 100] 100)
+where
+ "\<lbrakk>x\<rbrakk>Lang \<equiv> {y. x \<equiv>Lang\<equiv> y}"
+
+text {* Chunhan modifies Q to Quo *}
+
+definition
+ quot :: "string set \<Rightarrow> string set \<Rightarrow> (string set) set" ("_ Quo _" [100, 100] 100)
+where
+ "L1 Quo L2 \<equiv> { \<lbrakk>x\<rbrakk>L2 | x. x \<in> L1}"
+
+
+lemma lang_eqs_union_of_eqcls:
+ "Lang = \<Union> {X. X \<in> (UNIV Quo Lang) \<and> (\<forall> x \<in> X. x \<in> Lang)}"
+proof
+ show "Lang \<subseteq> \<Union>{X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang}"
+ proof
+ fix x
+ assume "x \<in> Lang"
+ thus "x \<in> \<Union>{X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang}"
+ proof (simp add:quot_def)
+ assume "(1)": "x \<in> Lang"
+ show "\<exists>xa. (\<exists>x. xa = \<lbrakk>x\<rbrakk>Lang) \<and> (\<forall>x\<in>xa. x \<in> Lang) \<and> x \<in> xa" (is "\<exists>xa.?P xa")
+ proof -
+ have "?P (\<lbrakk>x\<rbrakk>Lang)" using "(1)"
+ by (auto simp:equiv_class_def equiv_str_def dest: spec[where x = "[]"])
+ thus ?thesis by blast
+ qed
+ qed
+ qed
+next
+ show "\<Union>{X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang} \<subseteq> Lang"
+ by auto
+qed
+
+lemma empty_notin_CS: "{} \<notin> UNIV Quo Lang"
+apply (clarsimp simp:quot_def equiv_class_def)
+by (rule_tac x = x in exI, auto simp:equiv_str_def)
+
+lemma no_two_cls_inters:
+ "\<lbrakk>X \<in> UNIV Quo Lang; Y \<in> UNIV Quo Lang; X \<noteq> Y\<rbrakk> \<Longrightarrow> X \<inter> Y = {}"
+by (auto simp:quot_def equiv_class_def equiv_str_def)
+
+text {* equiv_class transition *}
+definition
+ CT :: "string set \<Rightarrow> char \<Rightarrow> string set \<Rightarrow> bool" ("_-_\<rightarrow>_" [99,99]99)
+where
+ "X-c\<rightarrow>Y \<equiv> ((X;{[c]}) \<subseteq> Y)"
+
+types t_equa_rhs = "(string set \<times> rexp) set"
+
+types t_equa = "(string set \<times> t_equa_rhs)"
+
+types t_equas = "t_equa set"
+
+text {*
+ "empty_rhs" generates "\<lambda>" for init-state, just like "\<lambda>" for final states
+ in Brzozowski method. But if the init-state is "{[]}" ("\<lambda>" itself) then
+ empty set is returned, see definition of "equation_rhs"
+*}
+
+definition
+ empty_rhs :: "string set \<Rightarrow> t_equa_rhs"
+where
+ "empty_rhs X \<equiv> if ([] \<in> X) then {({[]}, EMPTY)} else {}"
+
+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"
+
+definition
+ equation_rhs :: "(string set) set \<Rightarrow> string set \<Rightarrow> t_equa_rhs"
+where
+ "equation_rhs CS X \<equiv> if (X = {[]}) then {({[]}, EMPTY)}
+ else {(S, folds ALT NULL {CHAR c| c. S-c\<rightarrow>X} )| S. S \<in> CS} \<union> empty_rhs X"
+
+definition
+ equations :: "(string set) set \<Rightarrow> t_equas"
+where
+ "equations CS \<equiv> {(X, equation_rhs CS X) | X. X \<in> CS}"
+
+overloading L_rhs \<equiv> "L:: t_equa_rhs \<Rightarrow> string set"
+begin
+fun L_rhs:: "t_equa_rhs \<Rightarrow> string set"
+where
+ "L_rhs rhs = {x. \<exists> X r. (X, r) \<in> rhs \<and> x \<in> X;(L r)}"
+end
+
+definition
+ distinct_rhs :: "t_equa_rhs \<Rightarrow> bool"
+where
+ "distinct_rhs rhs \<equiv> \<forall> X reg\<^isub>1 reg\<^isub>2. (X, reg\<^isub>1) \<in> rhs \<and> (X, reg\<^isub>2) \<in> rhs \<longrightarrow> reg\<^isub>1 = reg\<^isub>2"
+
+definition
+ distinct_equas_rhs :: "t_equas \<Rightarrow> bool"
+where
+ "distinct_equas_rhs equas \<equiv> \<forall> X rhs. (X, rhs) \<in> equas \<longrightarrow> distinct_rhs rhs"
+
+definition
+ distinct_equas :: "t_equas \<Rightarrow> bool"
+where
+ "distinct_equas equas \<equiv> \<forall> X rhs rhs'. (X, rhs) \<in> equas \<and> (X, rhs') \<in> equas \<longrightarrow> rhs = rhs'"
+
+definition
+ seq_rhs_r :: "t_equa_rhs \<Rightarrow> rexp \<Rightarrow> t_equa_rhs"
+where
+ "seq_rhs_r rhs r \<equiv> (\<lambda>(X, reg). (X, SEQ reg r)) ` rhs"
+
+definition
+ del_x_paired :: "('a \<times> 'b) set \<Rightarrow> 'a \<Rightarrow> ('a \<times> 'b) set"
+where
+ "del_x_paired S x \<equiv> S - {X. X \<in> S \<and> fst X = x}"
+
+definition
+ arden_variate :: "string set \<Rightarrow> rexp \<Rightarrow> t_equa_rhs \<Rightarrow> t_equa_rhs"
+where
+ "arden_variate X r rhs \<equiv> seq_rhs_r (del_x_paired rhs X) (STAR r)"
+
+definition
+ no_EMPTY_rhs :: "t_equa_rhs \<Rightarrow> bool"
+where
+ "no_EMPTY_rhs rhs \<equiv> \<forall> X r. (X, r) \<in> rhs \<and> X \<noteq> {[]} \<longrightarrow> [] \<notin> L r"
+
+definition
+ no_EMPTY_equas :: "t_equas \<Rightarrow> bool"
+where
+ "no_EMPTY_equas equas \<equiv> \<forall> X rhs. (X, rhs) \<in> equas \<longrightarrow> no_EMPTY_rhs rhs"
+
+lemma fold_alt_null_eqs:
+ "finite rS \<Longrightarrow> x \<in> L (folds ALT NULL rS) = (\<exists> r \<in> rS. x \<in> L r)"
+apply (simp add:folds_def)
+apply (rule someI2_ex)
+apply (erule finite_imp_fold_graph)
+apply (erule fold_graph.induct)
+by auto (*??? how do this be in Isar ?? *)
+
+lemma seq_rhs_r_prop1:
+ "L (seq_rhs_r rhs r) = (L rhs);(L r)"
+apply (auto simp:seq_rhs_r_def image_def lang_seq_def)
+apply (rule_tac x = "s1 @ s1a" in exI, rule_tac x = "s2a" in exI, simp)
+apply (rule_tac x = a in exI, rule_tac x = b in exI, simp)
+apply (rule_tac x = s1 in exI, rule_tac x = s1a in exI, simp)
+apply (rule_tac x = X in exI, rule_tac x = "SEQ ra r" in exI, simp)
+apply (rule conjI)
+apply (rule_tac x = "(X, ra)" in bexI, simp+)
+apply (rule_tac x = s1a in exI, rule_tac x = "s2a @ s2" in exI, simp)
+apply (simp add:lang_seq_def)
+by (rule_tac x = s2a in exI, rule_tac x = s2 in exI, simp)
+
+lemma del_x_paired_prop1:
+ "\<lbrakk>distinct_rhs rhs; (X, r) \<in> rhs\<rbrakk> \<Longrightarrow> X ; L r \<union> L (del_x_paired rhs X) = L rhs"
+ apply (simp add:del_x_paired_def)
+ apply (simp add: distinct_rhs_def)
+ apply(auto simp add: lang_seq_def)
+ apply(metis)
+ done
+
+lemma arden_variate_prop:
+ assumes "(X, rx) \<in> rhs"
+ shows "(\<forall> Y. Y \<noteq> X \<longrightarrow> (\<exists> r. (Y, r) \<in> rhs) = (\<exists> r. (Y, r) \<in> (arden_variate X rx rhs)))"
+proof (rule allI, rule impI)
+ fix Y
+ assume "(1)": "Y \<noteq> X"
+ show "(\<exists>r. (Y, r) \<in> rhs) = (\<exists>r. (Y, r) \<in> arden_variate X rx rhs)"
+ proof
+ assume "(1_1)": "\<exists>r. (Y, r) \<in> rhs"
+ show "\<exists>r. (Y, r) \<in> arden_variate X rx rhs" (is "\<exists>r. ?P r")
+ proof -
+ from "(1_1)" obtain r where "(Y, r) \<in> rhs" ..
+ hence "?P (SEQ r (STAR rx))"
+ proof (simp add:arden_variate_def image_def)
+ have "(Y, r) \<in> rhs \<Longrightarrow> (Y, r) \<in> del_x_paired rhs X"
+ by (auto simp:del_x_paired_def "(1)")
+ thus "(Y, r) \<in> rhs \<Longrightarrow> (Y, SEQ r (STAR rx)) \<in> seq_rhs_r (del_x_paired rhs X) (STAR rx)"
+ by (auto simp:seq_rhs_r_def)
+ qed
+ thus ?thesis by blast
+ qed
+ next
+ assume "(2_1)": "\<exists>r. (Y, r) \<in> arden_variate X rx rhs"
+ thus "\<exists>r. (Y, r) \<in> rhs" (is "\<exists> r. ?P r")
+ by (auto simp:arden_variate_def del_x_paired_def seq_rhs_r_def image_def)
+ qed
+qed
+
+text {*
+ arden_variate_valid: proves variation from
+
+ "X = X;r + Y;ry + \<dots>" to "X = Y;(SEQ ry (STAR r)) + \<dots>"
+
+ holds the law of "language of left equiv language of right"
+*}
+lemma arden_variate_valid:
+ assumes X_not_empty: "X \<noteq> {[]}"
+ and l_eq_r: "X = L rhs"
+ and dist: "distinct_rhs rhs"
+ and no_empty: "no_EMPTY_rhs rhs"
+ and self_contained: "(X, r) \<in> rhs"
+ shows "X = L (arden_variate X r rhs)"
+proof -
+ have "[] \<notin> L r" using no_empty X_not_empty self_contained
+ by (auto simp:no_EMPTY_rhs_def)
+ hence ardens: "X = X;(L r) \<union> (L (del_x_paired rhs X)) \<longleftrightarrow> X = (L (del_x_paired rhs X)) ; (L r)\<star>"
+ by (rule ardens_revised)
+ have del_x: "X = X ; L r \<union> L (del_x_paired rhs X) \<longleftrightarrow> X = L rhs" using dist l_eq_r self_contained
+ by (auto dest!:del_x_paired_prop1)
+ show ?thesis
+ proof
+ show "X \<subseteq> L (arden_variate X r rhs)"
+ proof
+ fix x
+ assume "(1_1)": "x \<in> X" with l_eq_r ardens del_x
+ show "x \<in> L (arden_variate X r rhs)"
+ by (simp add:arden_variate_def seq_rhs_r_prop1 del:L_rhs.simps)
+ qed
+ next
+ show "L (arden_variate X r rhs) \<subseteq> X"
+ proof
+ fix x
+ assume "(2_1)": "x \<in> L (arden_variate X r rhs)" with ardens del_x l_eq_r
+ show "x \<in> X"
+ by (simp add:arden_variate_def seq_rhs_r_prop1 del:L_rhs.simps)
+ qed
+ qed
+qed
+
+text {*
+ merge_rhs {(x1, r1), (x2, r2}, (x4, r4), \<dots>} {(x1, r1'), (x3, r3'), \<dots>} =
+ {(x1, ALT r1 r1'}, (x2, r2), (x3, r3'), (x4, r4), \<dots>} *}
+definition
+ merge_rhs :: "t_equa_rhs \<Rightarrow> t_equa_rhs \<Rightarrow> t_equa_rhs"
+where
+ "merge_rhs rhs rhs' \<equiv> {(X, r). (\<exists> r1 r2. ((X,r1) \<in> rhs \<and> (X, r2) \<in> rhs') \<and> r = ALT r1 r2) \<or>
+ (\<exists> r1. (X, r1) \<in> rhs \<and> (\<not> (\<exists> r2. (X, r2) \<in> rhs')) \<and> r = r1) \<or>
+ (\<exists> r2. (X, r2) \<in> rhs' \<and> (\<not> (\<exists> r1. (X, r1) \<in> rhs)) \<and> r = r2) }"
+
+
+text {* rhs_subst rhs X=xrhs r: substitude all occurence of X in rhs((X,r) \<in> rhs) with xrhs *}
+definition
+ rhs_subst :: "t_equa_rhs \<Rightarrow> string set \<Rightarrow> t_equa_rhs \<Rightarrow> rexp \<Rightarrow> t_equa_rhs"
+where
+ "rhs_subst rhs X xrhs r \<equiv> merge_rhs (del_x_paired rhs X) (seq_rhs_r xrhs r)"
+
+definition
+ equas_subst_f :: "string set \<Rightarrow> t_equa_rhs \<Rightarrow> t_equa \<Rightarrow> t_equa"
+where
+ "equas_subst_f X xrhs equa \<equiv> let (Y, rhs) = equa in
+ if (\<exists> r. (X, r) \<in> rhs)
+ then (Y, rhs_subst rhs X xrhs (SOME r. (X, r) \<in> rhs))
+ else equa"
+
+definition
+ equas_subst :: "t_equas \<Rightarrow> string set \<Rightarrow> t_equa_rhs \<Rightarrow> t_equas"
+where
+ "equas_subst ES X xrhs \<equiv> del_x_paired (equas_subst_f X xrhs ` ES) X"
+
+lemma lang_seq_prop1:
+ "x \<in> X ; L r \<Longrightarrow> x \<in> X ; (L r \<union> L r')"
+by (auto simp:lang_seq_def)
+
+lemma lang_seq_prop1':
+ "x \<in> X; L r \<Longrightarrow> x \<in> X ; (L r' \<union> L r)"
+by (auto simp:lang_seq_def)
+
+lemma lang_seq_prop2:
+ "x \<in> X; (L r \<union> L r') \<Longrightarrow> x \<in> X;L r \<or> x \<in> X;L r'"
+by (auto simp:lang_seq_def)
+
+lemma merge_rhs_prop1:
+ shows "L (merge_rhs rhs rhs') = L rhs \<union> L rhs' "
+apply (auto simp add:merge_rhs_def dest!:lang_seq_prop2 intro:lang_seq_prop1)
+apply (rule_tac x = X in exI, rule_tac x = r1 in exI, simp)
+apply (case_tac "\<exists> r2. (X, r2) \<in> rhs'")
+apply (clarify, rule_tac x = X in exI, rule_tac x = "ALT r r2" in exI, simp add:lang_seq_prop1)
+apply (rule_tac x = X in exI, rule_tac x = r in exI, simp)
+apply (case_tac "\<exists> r1. (X, r1) \<in> rhs")
+apply (clarify, rule_tac x = X in exI, rule_tac x = "ALT r1 r" in exI, simp add:lang_seq_prop1')
+apply (rule_tac x = X in exI, rule_tac x = r in exI, simp)
+done
+
+lemma no_EMPTY_rhss_imp_merge_no_EMPTY:
+ "\<lbrakk>no_EMPTY_rhs rhs; no_EMPTY_rhs rhs'\<rbrakk> \<Longrightarrow> no_EMPTY_rhs (merge_rhs rhs rhs')"
+apply (simp add:no_EMPTY_rhs_def merge_rhs_def)
+apply (clarify, (erule conjE | erule exE | erule disjE)+)
+by auto
+
+lemma distinct_rhs_prop:
+ "\<lbrakk>distinct_rhs rhs; (X, r1) \<in> rhs; (X, r2) \<in> rhs\<rbrakk> \<Longrightarrow> r1 = r2"
+by (auto simp:distinct_rhs_def)
+
+lemma merge_rhs_prop2:
+ assumes dist_rhs: "distinct_rhs rhs"
+ and dist_rhs':"distinct_rhs rhs'"
+ shows "distinct_rhs (merge_rhs rhs rhs')"
+apply (auto simp:merge_rhs_def distinct_rhs_def)
+using dist_rhs
+apply (drule distinct_rhs_prop, simp+)
+using dist_rhs'
+apply (drule distinct_rhs_prop, simp+)
+using dist_rhs
+apply (drule distinct_rhs_prop, simp+)
+using dist_rhs'
+apply (drule distinct_rhs_prop, simp+)
+done
+
+lemma seq_rhs_r_holds_distinct:
+ "distinct_rhs rhs \<Longrightarrow> distinct_rhs (seq_rhs_r rhs r)"
+by (auto simp:distinct_rhs_def seq_rhs_r_def)
+
+lemma seq_rhs_r_prop0:
+ assumes l_eq_r: "X = L xrhs"
+ shows "L (seq_rhs_r xrhs r) = X ; L r "
+using l_eq_r
+by (auto simp only:seq_rhs_r_prop1)
+
+lemma rhs_subst_prop1:
+ assumes l_eq_r: "X = L xrhs"
+ and dist: "distinct_rhs rhs"
+ shows "(X, r) \<in> rhs \<Longrightarrow> L rhs = L (rhs_subst rhs X xrhs r)"
+apply (simp add:rhs_subst_def merge_rhs_prop1 del:L_rhs.simps)
+using l_eq_r
+apply (drule_tac r = r in seq_rhs_r_prop0, simp del:L_rhs.simps)
+using dist
+by (auto dest!:del_x_paired_prop1 simp del:L_rhs.simps)
+
+lemma del_x_paired_holds_distinct_rhs:
+ "distinct_rhs rhs \<Longrightarrow> distinct_rhs (del_x_paired rhs x)"
+by (auto simp:distinct_rhs_def del_x_paired_def)
+
+lemma rhs_subst_holds_distinct_rhs:
+ "\<lbrakk>distinct_rhs rhs; distinct_rhs xrhs\<rbrakk> \<Longrightarrow> distinct_rhs (rhs_subst rhs X xrhs r)"
+apply (drule_tac r = r and rhs = xrhs in seq_rhs_r_holds_distinct)
+apply (drule_tac x = X in del_x_paired_holds_distinct_rhs)
+by (auto dest:merge_rhs_prop2[where rhs = "del_x_paired rhs X"] simp:rhs_subst_def)
+
+section {* myhill-nerode theorem *}
+
+definition left_eq_cls :: "t_equas \<Rightarrow> (string set) set"
+where
+ "left_eq_cls ES \<equiv> {X. \<exists> rhs. (X, rhs) \<in> ES} "
+
+definition right_eq_cls :: "t_equas \<Rightarrow> (string set) set"
+where
+ "right_eq_cls ES \<equiv> {Y. \<exists> X rhs r. (X, rhs) \<in> ES \<and> (Y, r) \<in> rhs }"
+
+definition rhs_eq_cls :: "t_equa_rhs \<Rightarrow> (string set) set"
+where
+ "rhs_eq_cls rhs \<equiv> {Y. \<exists> r. (Y, r) \<in> rhs}"
+
+definition ardenable :: "t_equa \<Rightarrow> bool"
+where
+ "ardenable equa \<equiv> let (X, rhs) = equa in
+ distinct_rhs rhs \<and> no_EMPTY_rhs rhs \<and> X = L rhs"
+
+text {*
+ Inv: Invairance of the equation-system, during the decrease of the equation-system, Inv holds.
+*}
+definition Inv :: "string set \<Rightarrow> t_equas \<Rightarrow> bool"
+where
+ "Inv X ES \<equiv> finite ES \<and> (\<exists> rhs. (X, rhs) \<in> ES) \<and> distinct_equas ES \<and>
+ (\<forall> X xrhs. (X, xrhs) \<in> ES \<longrightarrow> ardenable (X, xrhs) \<and> X \<noteq> {} \<and> rhs_eq_cls xrhs \<subseteq> insert {[]} (left_eq_cls ES))"
+
+text {*
+ TCon: Termination Condition of the equation-system decreasion.
+*}
+definition TCon:: "'a set \<Rightarrow> bool"
+where
+ "TCon ES \<equiv> card ES = 1"
+
+
+text {*
+ The following is a iteration principle, and is the main framework for the proof:
+ 1: We can form the initial equation-system using "equations" defined above, and prove it has invariance Inv by lemma "init_ES_satisfy_Inv";
+ 2: We can decrease the number of the equation-system using ardens_lemma_revised and substitution ("equas_subst", defined above),
+ and prove it holds the property "step" in "wf_iter" by lemma "iteration_step"
+ and finally using property Inv and TCon to prove the myhill-nerode theorem
+
+*}
+lemma wf_iter [rule_format]:
+ fixes f
+ assumes step: "\<And> e. \<lbrakk>P e; \<not> Q e\<rbrakk> \<Longrightarrow> (\<exists> e'. P e' \<and> (f(e'), f(e)) \<in> less_than)"
+ shows pe: "P e \<longrightarrow> (\<exists> e'. P e' \<and> Q e')"
+proof(induct e rule: wf_induct
+ [OF wf_inv_image[OF wf_less_than, where f = "f"]], clarify)
+ fix x
+ assume h [rule_format]:
+ "\<forall>y. (y, x) \<in> inv_image less_than f \<longrightarrow> P y \<longrightarrow> (\<exists>e'. P e' \<and> Q e')"
+ and px: "P x"
+ show "\<exists>e'. P e' \<and> Q e'"
+ proof(cases "Q x")
+ assume "Q x" with px show ?thesis by blast
+ next
+ assume nq: "\<not> Q x"
+ from step [OF px nq]
+ obtain e' where pe': "P e'" and ltf: "(f e', f x) \<in> less_than" by auto
+ show ?thesis
+ proof(rule h)
+ from ltf show "(e', x) \<in> inv_image less_than f"
+ by (simp add:inv_image_def)
+ next
+ from pe' show "P e'" .
+ qed
+ qed
+qed
+
+
+text {* ******BEGIN: proving the initial equation-system satisfies Inv ****** *}
+
+lemma distinct_rhs_equations:
+ "(X, xrhs) \<in> equations (UNIV Quo Lang) \<Longrightarrow> distinct_rhs xrhs"
+by (auto simp: equations_def equation_rhs_def distinct_rhs_def empty_rhs_def dest:no_two_cls_inters)
+
+lemma every_nonempty_eqclass_has_strings:
+ "\<lbrakk>X \<in> (UNIV Quo Lang); X \<noteq> {[]}\<rbrakk> \<Longrightarrow> \<exists> clist. clist \<in> X \<and> clist \<noteq> []"
+by (auto simp:quot_def equiv_class_def equiv_str_def)
+
+lemma every_eqclass_is_derived_from_empty:
+ assumes not_empty: "X \<noteq> {[]}"
+ shows "X \<in> (UNIV Quo Lang) \<Longrightarrow> \<exists> clist. {[]};{clist} \<subseteq> X \<and> clist \<noteq> []"
+using not_empty
+apply (drule_tac every_nonempty_eqclass_has_strings, simp)
+by (auto intro:exI[where x = clist] simp:lang_seq_def)
+
+lemma equiv_str_in_CS:
+ "\<lbrakk>clist\<rbrakk>Lang \<in> (UNIV Quo Lang)"
+by (auto simp:quot_def)
+
+lemma has_str_imp_defined_by_str:
+ "\<lbrakk>str \<in> X; X \<in> UNIV Quo Lang\<rbrakk> \<Longrightarrow> X = \<lbrakk>str\<rbrakk>Lang"
+by (auto simp:quot_def equiv_class_def equiv_str_def)
+
+lemma every_eqclass_has_ascendent:
+ assumes has_str: "clist @ [c] \<in> X"
+ and in_CS: "X \<in> UNIV Quo Lang"
+ shows "\<exists> Y. Y \<in> UNIV Quo Lang \<and> Y-c\<rightarrow>X \<and> clist \<in> Y" (is "\<exists> Y. ?P Y")
+proof -
+ have "?P (\<lbrakk>clist\<rbrakk>Lang)"
+ proof -
+ have "\<lbrakk>clist\<rbrakk>Lang \<in> UNIV Quo Lang"
+ by (simp add:quot_def, rule_tac x = clist in exI, simp)
+ moreover have "\<lbrakk>clist\<rbrakk>Lang-c\<rightarrow>X"
+ proof -
+ have "X = \<lbrakk>(clist @ [c])\<rbrakk>Lang" using has_str in_CS
+ by (auto intro!:has_str_imp_defined_by_str)
+ moreover have "\<forall> sl. sl \<in> \<lbrakk>clist\<rbrakk>Lang \<longrightarrow> sl @ [c] \<in> \<lbrakk>(clist @ [c])\<rbrakk>Lang"
+ by (auto simp:equiv_class_def equiv_str_def)
+ ultimately show ?thesis unfolding CT_def lang_seq_def
+ by auto
+ qed
+ moreover have "clist \<in> \<lbrakk>clist\<rbrakk>Lang"
+ by (auto simp:equiv_str_def equiv_class_def)
+ ultimately show "?P (\<lbrakk>clist\<rbrakk>Lang)" by simp
+ qed
+ thus ?thesis by blast
+qed
+
+lemma finite_charset_rS:
+ "finite {CHAR c |c. Y-c\<rightarrow>X}"
+by (rule_tac A = UNIV and f = CHAR in finite_surj, auto)
+
+lemma l_eq_r_in_equations:
+ assumes X_in_equas: "(X, xrhs) \<in> equations (UNIV Quo Lang)"
+ shows "X = L xrhs"
+proof (cases "X = {[]}")
+ case True
+ thus ?thesis using X_in_equas
+ by (simp add:equations_def equation_rhs_def lang_seq_def)
+next
+ case False
+ show ?thesis
+ proof
+ show "X \<subseteq> L xrhs"
+ proof
+ fix x
+ assume "(1)": "x \<in> X"
+ show "x \<in> L xrhs"
+ proof (cases "x = []")
+ assume empty: "x = []"
+ hence "x \<in> L (empty_rhs X)" using "(1)"
+ by (auto simp:empty_rhs_def lang_seq_def)
+ thus ?thesis using X_in_equas False empty "(1)"
+ unfolding equations_def equation_rhs_def by auto
+ next
+ assume not_empty: "x \<noteq> []"
+ hence "\<exists> clist c. x = clist @ [c]" by (case_tac x rule:rev_cases, auto)
+ then obtain clist c where decom: "x = clist @ [c]" by blast
+ moreover have "\<And> Y. \<lbrakk>Y \<in> UNIV Quo Lang; Y-c\<rightarrow>X; clist \<in> Y\<rbrakk>
+ \<Longrightarrow> [c] \<in> L (folds ALT NULL {CHAR c |c. Y-c\<rightarrow>X})"
+ proof -
+ fix Y
+ assume Y_is_eq_cl: "Y \<in> UNIV Quo Lang"
+ and Y_CT_X: "Y-c\<rightarrow>X"
+ and clist_in_Y: "clist \<in> Y"
+ with finite_charset_rS
+ show "[c] \<in> L (folds ALT NULL {CHAR c |c. Y-c\<rightarrow>X})"
+ by (auto simp :fold_alt_null_eqs)
+ qed
+ hence "\<exists>Xa. Xa \<in> UNIV Quo Lang \<and> clist @ [c] \<in> Xa ; L (folds ALT NULL {CHAR c |c. Xa-c\<rightarrow>X})"
+ using X_in_equas False not_empty "(1)" decom
+ by (auto dest!:every_eqclass_has_ascendent simp:equations_def equation_rhs_def lang_seq_def)
+ then obtain Xa where
+ "Xa \<in> UNIV Quo Lang \<and> clist @ [c] \<in> Xa ; L (folds ALT NULL {CHAR c |c. Xa-c\<rightarrow>X})" by blast
+ hence "x \<in> L {(S, folds ALT NULL {CHAR c |c. S-c\<rightarrow>X}) |S. S \<in> UNIV Quo Lang}"
+ using X_in_equas "(1)" decom
+ by (auto simp add:equations_def equation_rhs_def intro!:exI[where x = Xa])
+ thus "x \<in> L xrhs" using X_in_equas False not_empty unfolding equations_def equation_rhs_def
+ by auto
+ qed
+ qed
+ next
+ show "L xrhs \<subseteq> X"
+ proof
+ fix x
+ assume "(2)": "x \<in> L xrhs"
+ have "(2_1)": "\<And> s1 s2 r Xa. \<lbrakk>s1 \<in> Xa; s2 \<in> L (folds ALT NULL {CHAR c |c. Xa-c\<rightarrow>X})\<rbrakk> \<Longrightarrow> s1 @ s2 \<in> X"
+ using finite_charset_rS
+ by (auto simp:CT_def lang_seq_def fold_alt_null_eqs)
+ have "(2_2)": "\<And> s1 s2 Xa r.\<lbrakk>s1 \<in> Xa; s2 \<in> L r; (Xa, r) \<in> empty_rhs X\<rbrakk> \<Longrightarrow> s1 @ s2 \<in> X"
+ by (simp add:empty_rhs_def split:if_splits)
+ show "x \<in> X" using X_in_equas False "(2)"
+ by (auto intro:"(2_1)" "(2_2)" simp:equations_def equation_rhs_def lang_seq_def)
+ qed
+ qed
+qed
+
+
+
+lemma no_EMPTY_equations:
+ "(X, xrhs) \<in> equations CS \<Longrightarrow> no_EMPTY_rhs xrhs"
+apply (clarsimp simp add:equations_def equation_rhs_def)
+apply (simp add:no_EMPTY_rhs_def empty_rhs_def, auto)
+apply (subgoal_tac "finite {CHAR c |c. Xa-c\<rightarrow>X}", drule_tac x = "[]" in fold_alt_null_eqs, clarsimp, rule finite_charset_rS)+
+done
+
+lemma init_ES_satisfy_ardenable:
+ "(X, xrhs) \<in> equations (UNIV Quo Lang) \<Longrightarrow> ardenable (X, xrhs)"
+ unfolding ardenable_def
+ by (auto intro:distinct_rhs_equations no_EMPTY_equations simp:l_eq_r_in_equations simp del:L_rhs.simps)
+
+lemma init_ES_satisfy_Inv:
+ assumes finite_CS: "finite (UNIV Quo Lang)"
+ and X_in_eq_cls: "X \<in> UNIV Quo Lang"
+ shows "Inv X (equations (UNIV Quo Lang))"
+proof -
+ have "finite (equations (UNIV Quo Lang))" using finite_CS
+ by (auto simp:equations_def)
+ moreover have "\<exists>rhs. (X, rhs) \<in> equations (UNIV Quo Lang)" using X_in_eq_cls
+ by (simp add:equations_def)
+ moreover have "distinct_equas (equations (UNIV Quo Lang))"
+ by (auto simp:distinct_equas_def equations_def)
+ moreover have "\<forall>X xrhs. (X, xrhs) \<in> equations (UNIV Quo Lang) \<longrightarrow>
+ rhs_eq_cls xrhs \<subseteq> insert {[]} (left_eq_cls (equations (UNIV Quo Lang)))"
+ apply (simp add:left_eq_cls_def equations_def rhs_eq_cls_def equation_rhs_def)
+ by (auto simp:empty_rhs_def split:if_splits)
+ moreover have "\<forall>X xrhs. (X, xrhs) \<in> equations (UNIV Quo Lang) \<longrightarrow> X \<noteq> {}"
+ by (clarsimp simp:equations_def empty_notin_CS intro:classical)
+ moreover have "\<forall>X xrhs. (X, xrhs) \<in> equations (UNIV Quo Lang) \<longrightarrow> ardenable (X, xrhs)"
+ by (auto intro!:init_ES_satisfy_ardenable)
+ ultimately show ?thesis by (simp add:Inv_def)
+qed
+
+
+text {* *********** END: proving the initial equation-system satisfies Inv ******* *}
+
+
+text {* ****** BEGIN: proving every equation-system's iteration step satisfies Inv ***** *}
+
+lemma not_T_aux: "\<lbrakk>\<not> TCon (insert a A); x = a\<rbrakk>
+ \<Longrightarrow> \<exists>y. x \<noteq> y \<and> y \<in> insert a A "
+apply (case_tac "insert a A = {a}")
+by (auto simp:TCon_def)
+
+lemma not_T_atleast_2[rule_format]:
+ "finite S \<Longrightarrow> \<forall> x. x \<in> S \<and> (\<not> TCon S)\<longrightarrow> (\<exists> y. x \<noteq> y \<and> y \<in> S)"
+apply (erule finite.induct, simp)
+apply (clarify, case_tac "x = a")
+by (erule not_T_aux, auto)
+
+lemma exist_another_equa:
+ "\<lbrakk>\<not> TCon ES; finite ES; distinct_equas ES; (X, rhl) \<in> ES\<rbrakk> \<Longrightarrow> \<exists> Y yrhl. (Y, yrhl) \<in> ES \<and> X \<noteq> Y"
+apply (drule not_T_atleast_2, simp)
+apply (clarsimp simp:distinct_equas_def)
+apply (drule_tac x= X in spec, drule_tac x = rhl in spec, drule_tac x = b in spec)
+by auto
+
+lemma Inv_mono_with_lambda:
+ assumes Inv_ES: "Inv X ES"
+ and X_noteq_Y: "X \<noteq> {[]}"
+ shows "Inv X (ES - {({[]}, yrhs)})"
+proof -
+ have "finite (ES - {({[]}, yrhs)})" using Inv_ES
+ by (simp add:Inv_def)
+ moreover have "\<exists>rhs. (X, rhs) \<in> ES - {({[]}, yrhs)}" using Inv_ES X_noteq_Y
+ by (simp add:Inv_def)
+ moreover have "distinct_equas (ES - {({[]}, yrhs)})" using Inv_ES X_noteq_Y
+ apply (clarsimp simp:Inv_def distinct_equas_def)
+ by (drule_tac x = Xa in spec, simp)
+ moreover have "\<forall>X xrhs.(X, xrhs) \<in> ES - {({[]}, yrhs)} \<longrightarrow>
+ ardenable (X, xrhs) \<and> X \<noteq> {}" using Inv_ES
+ by (clarify, simp add:Inv_def)
+ moreover
+ have "insert {[]} (left_eq_cls (ES - {({[]}, yrhs)})) = insert {[]} (left_eq_cls ES)"
+ by (auto simp:left_eq_cls_def)
+ hence "\<forall>X xrhs.(X, xrhs) \<in> ES - {({[]}, yrhs)} \<longrightarrow>
+ rhs_eq_cls xrhs \<subseteq> insert {[]} (left_eq_cls (ES - {({[]}, yrhs)}))"
+ using Inv_ES by (auto simp:Inv_def)
+ ultimately show ?thesis by (simp add:Inv_def)
+qed
+
+lemma non_empty_card_prop:
+ "finite ES \<Longrightarrow> \<forall>e. e \<in> ES \<longrightarrow> card ES - Suc 0 < card ES"
+apply (erule finite.induct, simp)
+apply (case_tac[!] "a \<in> A")
+by (auto simp:insert_absorb)
+
+lemma ardenable_prop:
+ assumes not_lambda: "Y \<noteq> {[]}"
+ and ardable: "ardenable (Y, yrhs)"
+ shows "\<exists> yrhs'. Y = L yrhs' \<and> distinct_rhs yrhs' \<and> rhs_eq_cls yrhs' = rhs_eq_cls yrhs - {Y}" (is "\<exists> yrhs'. ?P yrhs'")
+proof (cases "(\<exists> reg. (Y, reg) \<in> yrhs)")
+ case True
+ thus ?thesis
+ proof
+ fix reg
+ assume self_contained: "(Y, reg) \<in> yrhs"
+ show ?thesis
+ proof -
+ have "?P (arden_variate Y reg yrhs)"
+ proof -
+ have "Y = L (arden_variate Y reg yrhs)"
+ using self_contained not_lambda ardable
+ by (rule_tac arden_variate_valid, simp_all add:ardenable_def)
+ moreover have "distinct_rhs (arden_variate Y reg yrhs)"
+ using ardable
+ by (auto simp:distinct_rhs_def arden_variate_def seq_rhs_r_def del_x_paired_def ardenable_def)
+ moreover have "rhs_eq_cls (arden_variate Y reg yrhs) = rhs_eq_cls yrhs - {Y}"
+ proof -
+ have "\<And> rhs r. rhs_eq_cls (seq_rhs_r rhs r) = rhs_eq_cls rhs"
+ apply (auto simp:rhs_eq_cls_def seq_rhs_r_def image_def)
+ by (rule_tac x = "SEQ ra r" in exI, rule_tac x = "(x, ra)" in bexI, simp+)
+ moreover have "\<And> rhs X. rhs_eq_cls (del_x_paired rhs X) = rhs_eq_cls rhs - {X}"
+ by (auto simp:rhs_eq_cls_def del_x_paired_def)
+ ultimately show ?thesis by (simp add:arden_variate_def)
+ qed
+ ultimately show ?thesis by simp
+ qed
+ thus ?thesis by (rule_tac x= "arden_variate Y reg yrhs" in exI, simp)
+ qed
+ qed
+next
+ case False
+ hence "(2)": "rhs_eq_cls yrhs - {Y} = rhs_eq_cls yrhs"
+ by (auto simp:rhs_eq_cls_def)
+ show ?thesis
+ proof -
+ have "?P yrhs" using False ardable "(2)"
+ by (simp add:ardenable_def)
+ thus ?thesis by blast
+ qed
+qed
+
+lemma equas_subst_f_del_no_other:
+ assumes self_contained: "(Y, rhs) \<in> ES"
+ shows "\<exists> rhs'. (Y, rhs') \<in> (equas_subst_f X xrhs ` ES)" (is "\<exists> rhs'. ?P rhs'")
+proof -
+ have "\<exists> rhs'. equas_subst_f X xrhs (Y, rhs) = (Y, rhs')"
+ by (auto simp:equas_subst_f_def)
+ then obtain rhs' where "equas_subst_f X xrhs (Y, rhs) = (Y, rhs')" by blast
+ hence "?P rhs'" unfolding image_def using self_contained
+ by (auto intro:bexI[where x = "(Y, rhs)"])
+ thus ?thesis by blast
+qed
+
+lemma del_x_paired_del_only_x:
+ "\<lbrakk>X \<noteq> Y; (X, rhs) \<in> ES\<rbrakk> \<Longrightarrow> (X, rhs) \<in> del_x_paired ES Y"
+by (auto simp:del_x_paired_def)
+
+lemma equas_subst_del_no_other:
+ "\<lbrakk>(X, rhs) \<in> ES; X \<noteq> Y\<rbrakk> \<Longrightarrow> (\<exists>rhs. (X, rhs) \<in> equas_subst ES Y rhs')"
+unfolding equas_subst_def
+apply (drule_tac X = Y and xrhs = rhs' in equas_subst_f_del_no_other)
+by (erule exE, drule del_x_paired_del_only_x, auto)
+
+lemma equas_subst_holds_distinct:
+ "distinct_equas ES \<Longrightarrow> distinct_equas (equas_subst ES Y rhs')"
+apply (clarsimp simp add:equas_subst_def distinct_equas_def del_x_paired_def equas_subst_f_def)
+by (auto split:if_splits)
+
+lemma del_x_paired_dels:
+ "(X, rhs) \<in> ES \<Longrightarrow> {Y. Y \<in> ES \<and> fst Y = X} \<inter> ES \<noteq> {}"
+by (auto)
+
+lemma del_x_paired_subset:
+ "(X, rhs) \<in> ES \<Longrightarrow> ES - {Y. Y \<in> ES \<and> fst Y = X} \<subset> ES"
+apply (drule del_x_paired_dels)
+by auto
+
+lemma del_x_paired_card_less:
+ "\<lbrakk>finite ES; (X, rhs) \<in> ES\<rbrakk> \<Longrightarrow> card (del_x_paired ES X) < card ES"
+apply (simp add:del_x_paired_def)
+apply (drule del_x_paired_subset)
+by (auto intro:psubset_card_mono)
+
+lemma equas_subst_card_less:
+ "\<lbrakk>finite ES; (Y, yrhs) \<in> ES\<rbrakk> \<Longrightarrow> card (equas_subst ES Y rhs') < card ES"
+apply (simp add:equas_subst_def)
+apply (frule_tac h = "equas_subst_f Y rhs'" in finite_imageI)
+apply (drule_tac f = "equas_subst_f Y rhs'" in Finite_Set.card_image_le)
+apply (drule_tac X = Y and xrhs = rhs' in equas_subst_f_del_no_other,erule exE)
+by (drule del_x_paired_card_less, auto)
+
+lemma equas_subst_holds_distinct_rhs:
+ assumes dist': "distinct_rhs yrhs'"
+ and history: "\<forall>X xrhs. (X, xrhs) \<in> ES \<longrightarrow> ardenable (X, xrhs)"
+ and X_in : "(X, xrhs) \<in> equas_subst ES Y yrhs'"
+ shows "distinct_rhs xrhs"
+using X_in history
+apply (clarsimp simp:equas_subst_def del_x_paired_def)
+apply (drule_tac x = a in spec, drule_tac x = b in spec)
+apply (simp add:ardenable_def equas_subst_f_def)
+by (auto intro:rhs_subst_holds_distinct_rhs simp:dist' split:if_splits)
+
+lemma r_no_EMPTY_imp_seq_rhs_r_no_EMPTY:
+ "[] \<notin> L r \<Longrightarrow> no_EMPTY_rhs (seq_rhs_r rhs r)"
+by (auto simp:no_EMPTY_rhs_def seq_rhs_r_def lang_seq_def)
+
+lemma del_x_paired_holds_no_EMPTY:
+ "no_EMPTY_rhs yrhs \<Longrightarrow> no_EMPTY_rhs (del_x_paired yrhs Y)"
+by (auto simp:no_EMPTY_rhs_def del_x_paired_def)
+
+lemma rhs_subst_holds_no_EMPTY:
+ "\<lbrakk>no_EMPTY_rhs yrhs; (Y, r) \<in> yrhs; Y \<noteq> {[]}\<rbrakk> \<Longrightarrow> no_EMPTY_rhs (rhs_subst yrhs Y rhs' r)"
+apply (auto simp:rhs_subst_def intro!:no_EMPTY_rhss_imp_merge_no_EMPTY r_no_EMPTY_imp_seq_rhs_r_no_EMPTY del_x_paired_holds_no_EMPTY)
+by (auto simp:no_EMPTY_rhs_def)
+
+lemma equas_subst_holds_no_EMPTY:
+ assumes substor: "Y \<noteq> {[]}"
+ and history: "\<forall>X xrhs. (X, xrhs) \<in> ES \<longrightarrow> ardenable (X, xrhs)"
+ and X_in:"(X, xrhs) \<in> equas_subst ES Y yrhs'"
+ shows "no_EMPTY_rhs xrhs"
+proof-
+ from X_in have "\<exists> (Z, zrhs) \<in> ES. (X, xrhs) = equas_subst_f Y yrhs' (Z, zrhs)"
+ by (auto simp add:equas_subst_def del_x_paired_def)
+ then obtain Z zrhs where Z_in: "(Z, zrhs) \<in> ES"
+ and X_Z: "(X, xrhs) = equas_subst_f Y yrhs' (Z, zrhs)" by blast
+ hence dist_zrhs: "distinct_rhs zrhs" using history
+ by (auto simp:ardenable_def)
+ show ?thesis
+ proof (cases "\<exists> r. (Y, r) \<in> zrhs")
+ case True
+ then obtain r where Y_in_zrhs: "(Y, r) \<in> zrhs" ..
+ hence some: "(SOME r. (Y, r) \<in> zrhs) = r" using Z_in dist_zrhs
+ by (auto simp:distinct_rhs_def)
+ hence "no_EMPTY_rhs (rhs_subst zrhs Y yrhs' r)"
+ using substor Y_in_zrhs history Z_in
+ by (rule_tac rhs_subst_holds_no_EMPTY, auto simp:ardenable_def)
+ thus ?thesis using X_Z True some
+ by (simp add:equas_subst_def equas_subst_f_def)
+ next
+ case False
+ hence "(X, xrhs) = (Z, zrhs)" using Z_in X_Z
+ by (simp add:equas_subst_f_def)
+ thus ?thesis using history Z_in
+ by (auto simp:ardenable_def)
+ qed
+qed
+
+lemma equas_subst_f_holds_left_eq_right:
+ assumes substor: "Y = L rhs'"
+ and history: "\<forall>X xrhs. (X, xrhs) \<in> ES \<longrightarrow> distinct_rhs xrhs \<and> X = L xrhs"
+ and subst: "(X, xrhs) = equas_subst_f Y rhs' (Z, zrhs)"
+ and self_contained: "(Z, zrhs) \<in> ES"
+ shows "X = L xrhs"
+proof (cases "\<exists> r. (Y, r) \<in> zrhs")
+ case True
+ from True obtain r where "(1)":"(Y, r) \<in> zrhs" ..
+ show ?thesis
+ proof -
+ from history self_contained
+ have dist: "distinct_rhs zrhs" by auto
+ hence "(SOME r. (Y, r) \<in> zrhs) = r" using self_contained "(1)"
+ using distinct_rhs_def by (auto intro:some_equality)
+ moreover have "L zrhs = L (rhs_subst zrhs Y rhs' r)" using substor dist "(1)" self_contained
+ by (rule_tac rhs_subst_prop1, auto simp:distinct_equas_rhs_def)
+ ultimately show ?thesis using subst history self_contained
+ by (auto simp:equas_subst_f_def split:if_splits)
+ qed
+next
+ case False
+ thus ?thesis using history subst self_contained
+ by (auto simp:equas_subst_f_def)
+qed
+
+lemma equas_subst_holds_left_eq_right:
+ assumes history: "\<forall>X xrhs. (X, xrhs) \<in> ES \<longrightarrow> ardenable (X, xrhs)"
+ and substor: "Y = L rhs'"
+ and X_in : "(X, xrhs) \<in> equas_subst ES Y yrhs'"
+ shows "\<forall>X xrhs. (X, xrhs) \<in> equas_subst ES Y rhs' \<longrightarrow> X = L xrhs"
+apply (clarsimp simp add:equas_subst_def del_x_paired_def)
+using substor
+apply (drule_tac equas_subst_f_holds_left_eq_right)
+using history
+by (auto simp:ardenable_def)
+
+lemma equas_subst_holds_ardenable:
+ assumes substor: "Y = L yrhs'"
+ and history: "\<forall>X xrhs. (X, xrhs) \<in> ES \<longrightarrow> ardenable (X, xrhs)"
+ and X_in:"(X, xrhs) \<in> equas_subst ES Y yrhs'"
+ and dist': "distinct_rhs yrhs'"
+ and not_lambda: "Y \<noteq> {[]}"
+ shows "ardenable (X, xrhs)"
+proof -
+ have "distinct_rhs xrhs" using history X_in dist'
+ by (auto dest:equas_subst_holds_distinct_rhs)
+ moreover have "no_EMPTY_rhs xrhs" using history X_in not_lambda
+ by (auto intro:equas_subst_holds_no_EMPTY)
+ moreover have "X = L xrhs" using history substor X_in
+ by (auto dest: equas_subst_holds_left_eq_right simp del:L_rhs.simps)
+ ultimately show ?thesis using ardenable_def by simp
+qed
+
+lemma equas_subst_holds_cls_defined:
+ assumes X_in: "(X, xrhs) \<in> equas_subst ES Y yrhs'"
+ and Inv_ES: "Inv X' ES"
+ and subst: "(Y, yrhs) \<in> ES"
+ and cls_holds_but_Y: "rhs_eq_cls yrhs' = rhs_eq_cls yrhs - {Y}"
+ shows "rhs_eq_cls xrhs \<subseteq> insert {[]} (left_eq_cls (equas_subst ES Y yrhs'))"
+proof-
+ have tac: "\<lbrakk> A \<subseteq> B; C \<subseteq> D; E \<subseteq> A \<union> B\<rbrakk> \<Longrightarrow> E \<subseteq> B \<union> D" by auto
+ from X_in have "\<exists> (Z, zrhs) \<in> ES. (X, xrhs) = equas_subst_f Y yrhs' (Z, zrhs)"
+ by (auto simp add:equas_subst_def del_x_paired_def)
+ then obtain Z zrhs where Z_in: "(Z, zrhs) \<in> ES"
+ and X_Z: "(X, xrhs) = equas_subst_f Y yrhs' (Z, zrhs)" by blast
+ hence "rhs_eq_cls zrhs \<subseteq> insert {[]} (left_eq_cls ES)" using Inv_ES
+ by (auto simp:Inv_def)
+ moreover have "rhs_eq_cls yrhs' \<subseteq> insert {[]} (left_eq_cls ES) - {Y}"
+ using Inv_ES subst cls_holds_but_Y
+ by (auto simp:Inv_def)
+ moreover have "rhs_eq_cls xrhs \<subseteq> rhs_eq_cls zrhs \<union> rhs_eq_cls yrhs' - {Y}"
+ using X_Z cls_holds_but_Y
+ apply (clarsimp simp add:equas_subst_f_def rhs_subst_def split:if_splits)
+ by (auto simp:rhs_eq_cls_def merge_rhs_def del_x_paired_def seq_rhs_r_def)
+ moreover have "left_eq_cls (equas_subst ES Y yrhs') = left_eq_cls ES - {Y}" using subst
+ by (auto simp: left_eq_cls_def equas_subst_def del_x_paired_def equas_subst_f_def
+ dest: equas_subst_f_del_no_other
+ split: if_splits)
+ ultimately show ?thesis by blast
+qed
+
+lemma iteration_step:
+ assumes Inv_ES: "Inv X ES"
+ and not_T: "\<not> TCon ES"
+ shows "(\<exists> ES'. Inv X ES' \<and> (card ES', card ES) \<in> less_than)"
+proof -
+ from Inv_ES not_T have another: "\<exists>Y yrhs. (Y, yrhs) \<in> ES \<and> X \<noteq> Y" unfolding Inv_def
+ by (clarify, rule_tac exist_another_equa[where X = X], auto)
+ then obtain Y yrhs where subst: "(Y, yrhs) \<in> ES" and not_X: " X \<noteq> Y" by blast
+ show ?thesis (is "\<exists> ES'. ?P ES'")
+ proof (cases "Y = {[]}")
+ case True
+ --"in this situation, we pick a \"\<lambda>\" equation, thus directly remove it from the equation-system"
+ have "?P (ES - {(Y, yrhs)})"
+ proof
+ show "Inv X (ES - {(Y, yrhs)})" using True not_X
+ by (simp add:Inv_ES Inv_mono_with_lambda)
+ next
+ show "(card (ES - {(Y, yrhs)}), card ES) \<in> less_than" using Inv_ES subst
+ by (auto elim:non_empty_card_prop[rule_format] simp:Inv_def)
+ qed
+ thus ?thesis by blast
+ next
+ case False
+ --"in this situation, we pick a equation and using ardenable to get a
+ rhs without itself in it, then use equas_subst to form a new equation-system"
+ hence "\<exists> yrhs'. Y = L yrhs' \<and> distinct_rhs yrhs' \<and> rhs_eq_cls yrhs' = rhs_eq_cls yrhs - {Y}"
+ using subst Inv_ES
+ by (auto intro:ardenable_prop simp add:Inv_def simp del:L_rhs.simps)
+ then obtain yrhs' where Y'_l_eq_r: "Y = L yrhs'"
+ and dist_Y': "distinct_rhs yrhs'"
+ and cls_holds_but_Y: "rhs_eq_cls yrhs' = rhs_eq_cls yrhs - {Y}" by blast
+ hence "?P (equas_subst ES Y yrhs')"
+ proof -
+ have finite_del: "\<And> S x. finite S \<Longrightarrow> finite (del_x_paired S x)"
+ apply (rule_tac A = "del_x_paired S x" in finite_subset)
+ by (auto simp:del_x_paired_def)
+ have "finite (equas_subst ES Y yrhs')" using Inv_ES
+ by (auto intro!:finite_del simp:equas_subst_def Inv_def)
+ moreover have "\<exists>rhs. (X, rhs) \<in> equas_subst ES Y yrhs'" using Inv_ES not_X
+ by (auto intro:equas_subst_del_no_other simp:Inv_def)
+ moreover have "distinct_equas (equas_subst ES Y yrhs')" using Inv_ES
+ by (auto intro:equas_subst_holds_distinct simp:Inv_def)
+ moreover have "\<forall>X xrhs. (X, xrhs) \<in> equas_subst ES Y yrhs' \<longrightarrow> ardenable (X, xrhs)"
+ using Inv_ES dist_Y' False Y'_l_eq_r
+ apply (clarsimp simp:Inv_def)
+ by (rule equas_subst_holds_ardenable, simp_all)
+ moreover have "\<forall>X xrhs. (X, xrhs) \<in> equas_subst ES Y yrhs' \<longrightarrow> X \<noteq> {}" using Inv_ES
+ by (clarsimp simp:equas_subst_def Inv_def del_x_paired_def equas_subst_f_def split:if_splits, auto)
+ moreover have "\<forall>X xrhs. (X, xrhs) \<in> equas_subst ES Y yrhs' \<longrightarrow>
+ rhs_eq_cls xrhs \<subseteq> insert {[]} (left_eq_cls (equas_subst ES Y yrhs'))"
+ using Inv_ES subst cls_holds_but_Y
+ apply (rule_tac impI | rule_tac allI)+
+ by (erule equas_subst_holds_cls_defined, auto)
+ moreover have "(card (equas_subst ES Y yrhs'), card ES) \<in> less_than"using Inv_ES subst
+ by (simp add:equas_subst_card_less Inv_def)
+ ultimately show "?P (equas_subst ES Y yrhs')" by (auto simp:Inv_def)
+ qed
+ thus ?thesis by blast
+ qed
+qed
+
+text {* ***** END: proving every equation-system's iteration step satisfies Inv ************** *}
+
+lemma iteration_conc:
+ assumes history: "Inv X ES"
+ shows "\<exists> ES'. Inv X ES' \<and> TCon ES'" (is "\<exists> ES'. ?P ES'")
+proof (cases "TCon ES")
+ case True
+ hence "?P ES" using history by simp
+ thus ?thesis by blast
+next
+ case False
+ thus ?thesis using history iteration_step
+ by (rule_tac f = card in wf_iter, simp_all)
+qed
+
+lemma eqset_imp_iff': "A = B \<Longrightarrow> \<forall> x. x \<in> A \<longleftrightarrow> x \<in> B"
+apply (auto simp:mem_def)
+done
+
+lemma set_cases2:
+ "\<lbrakk>(A = {} \<Longrightarrow> R A); \<And> x. (A = {x}) \<Longrightarrow> R A; \<And> x y. \<lbrakk>x \<noteq> y; x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> R A\<rbrakk> \<Longrightarrow> R A"
+apply (case_tac "A = {}", simp)
+by (case_tac "\<exists> x. A = {x}", clarsimp, blast)
+
+lemma rhs_aux:"\<lbrakk>distinct_rhs rhs; {Y. \<exists>r. (Y, r) \<in> rhs} = {X}\<rbrakk> \<Longrightarrow> (\<exists>r. rhs = {(X, r)})"
+apply (rule_tac A = rhs in set_cases2, simp)
+apply (drule_tac x = X in eqset_imp_iff, clarsimp)
+apply (drule eqset_imp_iff',clarsimp)
+apply (frule_tac x = a in spec, drule_tac x = aa in spec)
+by (auto simp:distinct_rhs_def)
+
+lemma every_eqcl_has_reg:
+ assumes finite_CS: "finite (UNIV Quo Lang)"
+ and X_in_CS: "X \<in> (UNIV Quo Lang)"
+ shows "\<exists> (reg::rexp). L reg = X" (is "\<exists> r. ?E r")
+proof-
+ have "\<exists>ES'. Inv X ES' \<and> TCon ES'" using finite_CS X_in_CS
+ by (auto intro:init_ES_satisfy_Inv iteration_conc)
+ then obtain ES' where Inv_ES': "Inv X ES'"
+ and TCon_ES': "TCon ES'" by blast
+ from Inv_ES' TCon_ES'
+ have "\<exists> rhs. ES' = {(X, rhs)}"
+ apply (clarsimp simp:Inv_def TCon_def)
+ apply (rule_tac x = rhs in exI)
+ by (auto dest!:card_Suc_Diff1 simp:card_eq_0_iff)
+ then obtain rhs where ES'_single_equa: "ES' = {(X, rhs)}" ..
+ hence X_ardenable: "ardenable (X, rhs)" using Inv_ES'
+ by (simp add:Inv_def)
+
+ from X_ardenable have X_l_eq_r: "X = L rhs"
+ by (simp add:ardenable_def)
+ hence rhs_not_empty: "rhs \<noteq> {}" using Inv_ES' ES'_single_equa
+ by (auto simp:Inv_def ardenable_def)
+ have rhs_eq_cls: "rhs_eq_cls rhs \<subseteq> {X, {[]}}"
+ using Inv_ES' ES'_single_equa
+ by (auto simp:Inv_def ardenable_def left_eq_cls_def)
+ have X_not_empty: "X \<noteq> {}" using Inv_ES' ES'_single_equa
+ by (auto simp:Inv_def)
+ show ?thesis
+ proof (cases "X = {[]}")
+ case True
+ hence "?E EMPTY" by (simp)
+ thus ?thesis by blast
+ next
+ case False with X_ardenable
+ have "\<exists> rhs'. X = L rhs' \<and> rhs_eq_cls rhs' = rhs_eq_cls rhs - {X} \<and> distinct_rhs rhs'"
+ by (drule_tac ardenable_prop, auto)
+ then obtain rhs' where X_eq_rhs': "X = L rhs'"
+ and rhs'_eq_cls: "rhs_eq_cls rhs' = rhs_eq_cls rhs - {X}"
+ and rhs'_dist : "distinct_rhs rhs'" by blast
+ have "rhs_eq_cls rhs' \<subseteq> {{[]}}" using rhs_eq_cls False rhs'_eq_cls rhs_not_empty
+ by blast
+ hence "rhs_eq_cls rhs' = {{[]}}" using X_not_empty X_eq_rhs'
+ by (auto simp:rhs_eq_cls_def)
+ hence "\<exists> r. rhs' = {({[]}, r)}" using rhs'_dist
+ by (auto intro:rhs_aux simp:rhs_eq_cls_def)
+ then obtain r where "rhs' = {({[]}, r)}" ..
+ hence "?E r" using X_eq_rhs' by (auto simp add:lang_seq_def)
+ thus ?thesis by blast
+ qed
+qed
+
+text {* definition of a regular language *}
+
+abbreviation
+ reg :: "string set => bool"
+where
+ "reg L1 \<equiv> (\<exists>r::rexp. L r = L1)"
+
+theorem myhill_nerode:
+ assumes finite_CS: "finite (UNIV Quo Lang)"
+ shows "\<exists> (reg::rexp). Lang = L reg" (is "\<exists> r. ?P r")
+proof -
+ have has_r_each: "\<forall>C\<in>{X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang}. \<exists>(r::rexp). C = L r" using finite_CS
+ by (auto dest:every_eqcl_has_reg)
+ have "\<exists> (rS::rexp set). finite rS \<and>
+ (\<forall> C \<in> {X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang}. \<exists> r \<in> rS. C = L r) \<and>
+ (\<forall> r \<in> rS. \<exists> C \<in> {X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang}. C = L r)"
+ (is "\<exists> rS. ?Q rS")
+ proof-
+ have "\<And> C. C \<in> {X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang} \<Longrightarrow> C = L (SOME (ra::rexp). C = L ra)"
+ using has_r_each
+ apply (erule_tac x = C in ballE, erule_tac exE)
+ by (rule_tac a = r in someI2, simp+)
+ hence "?Q ((\<lambda> C. SOME r. C = L r) ` {X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang})" using has_r_each
+ using finite_CS by auto
+ thus ?thesis by blast
+ qed
+ then obtain rS where finite_rS : "finite rS"
+ and has_r_each': "\<forall> C \<in> {X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang}. \<exists> r \<in> (rS::rexp set). C = L r"
+ and has_cl_each: "\<forall> r \<in> (rS::rexp set). \<exists> C \<in> {X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang}. C = L r" by blast
+ have "?P (folds ALT NULL rS)"
+ proof
+ show "Lang \<subseteq> L (folds ALT NULL rS)" using finite_rS lang_eqs_union_of_eqcls[of Lang] has_r_each'
+ apply (clarsimp simp:fold_alt_null_eqs) by blast
+ next
+ show "L (folds ALT NULL rS) \<subseteq> Lang" using finite_rS lang_eqs_union_of_eqcls[of Lang] has_cl_each
+ by (clarsimp simp:fold_alt_null_eqs)
+ qed
+ thus ?thesis by blast
+qed
+
+
+text {* tests by Christian *}
+
+(* Alternative definition for Quo *)
+definition
+ QUOT :: "string set \<Rightarrow> (string set) set"
+where
+ "QUOT Lang \<equiv> (\<Union>x. {\<lbrakk>x\<rbrakk>Lang})"
+
+lemma test:
+ "UNIV Quo Lang = QUOT Lang"
+by (auto simp add: quot_def QUOT_def)
+
+lemma test1:
+ assumes finite_CS: "finite (QUOT Lang)"
+ shows "reg Lang"
+using finite_CS
+unfolding test[symmetric]
+by (auto dest: myhill_nerode)
+
+lemma cons_one: "x @ y \<in> {z} \<Longrightarrow> x @ y = z"
+by simp
+
+lemma quot_lambda: "QUOT {[]} = {{[]}, UNIV - {[]}}"
+proof
+ show "QUOT {[]} \<subseteq> {{[]}, UNIV - {[]}}"
+ proof
+ fix x
+ assume in_quot: "x \<in> QUOT {[]}"
+ show "x \<in> {{[]}, UNIV - {[]}}"
+ proof -
+ from in_quot
+ have "x = {[]} \<or> x = UNIV - {[]}"
+ unfolding QUOT_def equiv_class_def
+ proof
+ fix xa
+ assume in_univ: "xa \<in> UNIV"
+ and in_eqiv: "x \<in> {{y. xa \<equiv>{[]}\<equiv> y}}"
+ show "x = {[]} \<or> x = UNIV - {[]}"
+ proof(cases "xa = []")
+ case True
+ hence "{y. xa \<equiv>{[]}\<equiv> y} = {[]}" using in_eqiv
+ by (auto simp add:equiv_str_def)
+ thus ?thesis using in_eqiv by (rule_tac disjI1, simp)
+ next
+ case False
+ hence "{y. xa \<equiv>{[]}\<equiv> y} = UNIV - {[]}" using in_eqiv
+ by (auto simp:equiv_str_def)
+ thus ?thesis using in_eqiv by (rule_tac disjI2, simp)
+ qed
+ qed
+ thus ?thesis by simp
+ qed
+ qed
+next
+ show "{{[]}, UNIV - {[]}} \<subseteq> QUOT {[]}"
+ proof
+ fix x
+ assume in_res: "x \<in> {{[]}, (UNIV::string set) - {[]}}"
+ show "x \<in> (QUOT {[]})"
+ proof -
+ have "x = {[]} \<Longrightarrow> x \<in> QUOT {[]}"
+ apply (simp add:QUOT_def equiv_class_def equiv_str_def)
+ by (rule_tac x = "[]" in exI, auto)
+ moreover have "x = UNIV - {[]} \<Longrightarrow> x \<in> QUOT {[]}"
+ apply (simp add:QUOT_def equiv_class_def equiv_str_def)
+ by (rule_tac x = "''a''" in exI, auto)
+ ultimately show ?thesis using in_res by blast
+ qed
+ qed
+qed
+
+lemma quot_single_aux: "\<lbrakk>x \<noteq> []; x \<noteq> [c]\<rbrakk> \<Longrightarrow> x @ z \<noteq> [c]"
+by (induct x, auto)
+
+lemma quot_single: "\<And> (c::char). QUOT {[c]} = {{[]}, {[c]}, UNIV - {[], [c]}}"
+proof -
+ fix c::"char"
+ have exist_another: "\<exists> a. a \<noteq> c"
+ apply (case_tac "c = CHR ''a''")
+ apply (rule_tac x = "CHR ''b''" in exI, simp)
+ by (rule_tac x = "CHR ''a''" in exI, simp)
+ show "QUOT {[c]} = {{[]}, {[c]}, UNIV - {[], [c]}}"
+ proof
+ show "QUOT {[c]} \<subseteq> {{[]},{[c]}, UNIV - {[], [c]}}"
+ proof
+ fix x
+ assume in_quot: "x \<in> QUOT {[c]}"
+ show "x \<in> {{[]}, {[c]}, UNIV - {[],[c]}}"
+ proof -
+ from in_quot
+ have "x = {[]} \<or> x = {[c]} \<or> x = UNIV - {[],[c]}"
+ unfolding QUOT_def equiv_class_def
+ proof
+ fix xa
+ assume in_eqiv: "x \<in> {{y. xa \<equiv>{[c]}\<equiv> y}}"
+ show "x = {[]} \<or> x = {[c]} \<or> x = UNIV - {[], [c]}"
+ proof-
+ have "xa = [] \<Longrightarrow> x = {[]}" using in_eqiv
+ by (auto simp add:equiv_str_def)
+ moreover have "xa = [c] \<Longrightarrow> x = {[c]}"
+ proof -
+ have "xa = [c] \<Longrightarrow> {y. xa \<equiv>{[c]}\<equiv> y} = {[c]}" using in_eqiv
+ apply (simp add:equiv_str_def)
+ apply (rule set_ext, rule iffI, simp)
+ apply (drule_tac x = "[]" in spec, auto)
+ done
+ thus "xa = [c] \<Longrightarrow> x = {[c]}" using in_eqiv by simp
+ qed
+ moreover have "\<lbrakk>xa \<noteq> []; xa \<noteq> [c]\<rbrakk> \<Longrightarrow> x = UNIV - {[],[c]}"
+ proof -
+ have "\<lbrakk>xa \<noteq> []; xa \<noteq> [c]\<rbrakk> \<Longrightarrow> {y. xa \<equiv>{[c]}\<equiv> y} = UNIV - {[],[c]}"
+ apply (clarsimp simp add:equiv_str_def)
+ apply (rule set_ext, rule iffI, simp)
+ apply (rule conjI)
+ apply (drule_tac x = "[c]" in spec, simp)
+ apply (drule_tac x = "[]" in spec, simp)
+ by (auto dest:quot_single_aux)
+ thus "\<lbrakk>xa \<noteq> []; xa \<noteq> [c]\<rbrakk> \<Longrightarrow> x = UNIV - {[],[c]}" using in_eqiv by simp
+ qed
+ ultimately show ?thesis by blast
+ qed
+ qed
+ thus ?thesis by simp
+ qed
+ qed
+ next
+ show "{{[]}, {[c]}, UNIV - {[],[c]}} \<subseteq> QUOT {[c]}"
+ proof
+ fix x
+ assume in_res: "x \<in> {{[]},{[c]}, (UNIV::string set) - {[],[c]}}"
+ show "x \<in> (QUOT {[c]})"
+ proof -
+ have "x = {[]} \<Longrightarrow> x \<in> QUOT {[c]}"
+ apply (simp add:QUOT_def equiv_class_def equiv_str_def)
+ by (rule_tac x = "[]" in exI, auto)
+ moreover have "x = {[c]} \<Longrightarrow> x \<in> QUOT {[c]}"
+ apply (simp add:QUOT_def equiv_class_def equiv_str_def)
+ apply (rule_tac x = "[c]" in exI, simp)
+ apply (rule set_ext, rule iffI, simp+)
+ by (drule_tac x = "[]" in spec, simp)
+ moreover have "x = UNIV - {[],[c]} \<Longrightarrow> x \<in> QUOT {[c]}"
+ using exist_another
+ apply (clarsimp simp add:QUOT_def equiv_class_def equiv_str_def)
+ apply (rule_tac x = "[a]" in exI, simp)
+ apply (rule set_ext, rule iffI, simp)
+ apply (clarsimp simp:quot_single_aux, simp)
+ apply (rule conjI)
+ apply (drule_tac x = "[c]" in spec, simp)
+ by (drule_tac x = "[]" in spec, simp)
+ ultimately show ?thesis using in_res by blast
+ qed
+ qed
+ qed
+qed
+
+lemma eq_class_imp_eq_str:
+ "\<lbrakk>x\<rbrakk>lang = \<lbrakk>y\<rbrakk>lang \<Longrightarrow> x \<equiv>lang\<equiv> y"
+by (auto simp:equiv_class_def equiv_str_def)
+
+lemma finite_tag_image:
+ "finite (range tag) \<Longrightarrow> finite (((op `) tag) ` S)"
+apply (rule_tac B = "Pow (tag ` UNIV)" in finite_subset)
+by (auto simp add:image_def Pow_def)
+
+lemma str_inj_imps:
+ assumes str_inj: "\<And> m n. tag m = tag (n::string) \<Longrightarrow> m \<equiv>lang\<equiv> n"
+ shows "inj_on ((op `) tag) (QUOT lang)"
+proof (clarsimp simp add:inj_on_def QUOT_def)
+ fix x y
+ assume eq_tag: "tag ` \<lbrakk>x\<rbrakk>lang = tag ` \<lbrakk>y\<rbrakk>lang"
+ show "\<lbrakk>x\<rbrakk>lang = \<lbrakk>y\<rbrakk>lang"
+ proof -
+ have aux1:"\<And>a b. a \<in> (\<lbrakk>b\<rbrakk>lang) \<Longrightarrow> (a \<equiv>lang\<equiv> b)"
+ by (simp add:equiv_class_def equiv_str_def)
+ have aux2: "\<And> A B f. \<lbrakk>f ` A = f ` B; A \<noteq> {}\<rbrakk> \<Longrightarrow> \<exists> a b. a \<in> A \<and> b \<in> B \<and> f a = f b"
+ by auto
+ have aux3: "\<And> a l. \<lbrakk>a\<rbrakk>l \<noteq> {}"
+ by (auto simp:equiv_class_def equiv_str_def)
+ show ?thesis using eq_tag
+ apply (drule_tac aux2, simp add:aux3, clarsimp)
+ apply (drule_tac str_inj, (drule_tac aux1)+)
+ by (simp add:equiv_str_def equiv_class_def)
+ qed
+qed
+
+definition tag_str_ALT :: "string set \<Rightarrow> string set \<Rightarrow> string \<Rightarrow> (string set \<times> string set)"
+where
+ "tag_str_ALT L\<^isub>1 L\<^isub>2 x \<equiv> (\<lbrakk>x\<rbrakk>L\<^isub>1, \<lbrakk>x\<rbrakk>L\<^isub>2)"
+
+lemma tag_str_alt_range_finite:
+ assumes finite1: "finite (QUOT L\<^isub>1)"
+ and finite2: "finite (QUOT L\<^isub>2)"
+ shows "finite (range (tag_str_ALT L\<^isub>1 L\<^isub>2))"
+proof -
+ have "{y. \<exists>x. y = (\<lbrakk>x\<rbrakk>L\<^isub>1, \<lbrakk>x\<rbrakk>L\<^isub>2)} \<subseteq> (QUOT L\<^isub>1) \<times> (QUOT L\<^isub>2)"
+ by (auto simp:QUOT_def)
+ thus ?thesis using finite1 finite2
+ by (auto simp: image_def tag_str_ALT_def UNION_def
+ intro: finite_subset[where B = "(QUOT L\<^isub>1) \<times> (QUOT L\<^isub>2)"])
+qed
+
+lemma tag_str_alt_inj:
+ "tag_str_ALT L\<^isub>1 L\<^isub>2 x = tag_str_ALT L\<^isub>1 L\<^isub>2 y \<Longrightarrow> x \<equiv>(L\<^isub>1 \<union> L\<^isub>2)\<equiv> y"
+apply (simp add:tag_str_ALT_def equiv_class_def equiv_str_def)
+by blast
+
+lemma quot_alt:
+ assumes finite1: "finite (QUOT L\<^isub>1)"
+ and finite2: "finite (QUOT L\<^isub>2)"
+ shows "finite (QUOT (L\<^isub>1 \<union> L\<^isub>2))"
+proof (rule_tac f = "(op `) (tag_str_ALT L\<^isub>1 L\<^isub>2)" in finite_imageD)
+ show "finite (op ` (tag_str_ALT L\<^isub>1 L\<^isub>2) ` QUOT (L\<^isub>1 \<union> L\<^isub>2))"
+ using finite_tag_image tag_str_alt_range_finite finite1 finite2
+ by auto
+next
+ show "inj_on (op ` (tag_str_ALT L\<^isub>1 L\<^isub>2)) (QUOT (L\<^isub>1 \<union> L\<^isub>2))"
+ apply (rule_tac str_inj_imps)
+ by (erule_tac tag_str_alt_inj)
+qed
+
+(* list_diff:: list substract, once different return tailer *)
+fun list_diff :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" (infix "-" 51)
+where
+ "list_diff [] xs = []" |
+ "list_diff (x#xs) [] = x#xs" |
+ "list_diff (x#xs) (y#ys) = (if x = y then list_diff xs ys else (x#xs))"
+
+lemma [simp]: "(x @ y) - x = y"
+apply (induct x)
+by (case_tac y, simp+)
+
+lemma [simp]: "x - x = []"
+by (induct x, auto)
+
+lemma [simp]: "x = xa @ y \<Longrightarrow> x - xa = y "
+by (induct x, auto)
+
+lemma [simp]: "x - [] = x"
+by (induct x, auto)
+
+lemma [simp]: "xa \<le> x \<Longrightarrow> (x @ y) - xa = (x - xa) @ y"
+by (auto elim:prefixE)
+
+definition tag_str_SEQ:: "string set \<Rightarrow> string set \<Rightarrow> string \<Rightarrow> (string set \<times> string set set)"
+where
+ "tag_str_SEQ L\<^isub>1 L\<^isub>2 x \<equiv> if (\<exists> xa \<le> x. xa \<in> L\<^isub>1)
+ then (\<lbrakk>x\<rbrakk>L\<^isub>1, {\<lbrakk>(x - xa)\<rbrakk>L\<^isub>2 | xa. xa \<le> x \<and> xa \<in> L\<^isub>1})
+ else (\<lbrakk>x\<rbrakk>L\<^isub>1, {})"
+
+lemma tag_seq_eq_E:
+ "tag_str_SEQ L\<^isub>1 L\<^isub>2 x = tag_str_SEQ L\<^isub>1 L\<^isub>2 y \<Longrightarrow>
+ ((\<exists> xa \<le> x. xa \<in> L\<^isub>1) \<and> \<lbrakk>x\<rbrakk>L\<^isub>1 = \<lbrakk>y\<rbrakk>L\<^isub>1 \<and>
+ {\<lbrakk>(x - xa)\<rbrakk>L\<^isub>2 | xa. xa \<le> x \<and> xa \<in> L\<^isub>1} = {\<lbrakk>(y - ya)\<rbrakk>L\<^isub>2 | ya. ya \<le> y \<and> ya \<in> L\<^isub>1} ) \<or>
+ ((\<forall> xa \<le> x. xa \<notin> L\<^isub>1) \<and> \<lbrakk>x\<rbrakk>L\<^isub>1 = \<lbrakk>y\<rbrakk>L\<^isub>1)"
+by (simp add:tag_str_SEQ_def split:if_splits, blast)
+
+lemma tag_str_seq_range_finite:
+ assumes finite1: "finite (QUOT L\<^isub>1)"
+ and finite2: "finite (QUOT L\<^isub>2)"
+ shows "finite (range (tag_str_SEQ L\<^isub>1 L\<^isub>2))"
+proof -
+ have "(range (tag_str_SEQ L\<^isub>1 L\<^isub>2)) \<subseteq> (QUOT L\<^isub>1) \<times> (Pow (QUOT L\<^isub>2))"
+ by (auto simp:image_def tag_str_SEQ_def QUOT_def)
+ thus ?thesis using finite1 finite2
+ by (rule_tac B = "(QUOT L\<^isub>1) \<times> (Pow (QUOT L\<^isub>2))" in finite_subset, auto)
+qed
+
+lemma app_in_seq_decom[rule_format]:
+ "\<forall> x. x @ z \<in> L\<^isub>1 ; L\<^isub>2 \<longrightarrow> (\<exists> xa \<le> x. xa \<in> L\<^isub>1 \<and> (x - xa) @ z \<in> L\<^isub>2) \<or>
+ (\<exists> za \<le> z. (x @ za) \<in> L\<^isub>1 \<and> (z - za) \<in> L\<^isub>2)"
+apply (induct z)
+apply (simp, rule allI, rule impI, rule disjI1)
+apply (clarsimp simp add:lang_seq_def)
+apply (rule_tac x = s1 in exI, simp)
+apply (rule allI | rule impI)+
+apply (drule_tac x = "x @ [a]" in spec, simp)
+apply (erule exE | erule conjE | erule disjE)+
+apply (rule disjI2, rule_tac x = "[a]" in exI, simp)
+apply (rule disjI1, rule_tac x = xa in exI, simp)
+apply (erule exE | erule conjE)+
+apply (rule disjI2, rule_tac x = "a # za" in exI, simp)
+done
+
+lemma tag_str_seq_inj:
+ assumes tag_eq: "tag_str_SEQ L\<^isub>1 L\<^isub>2 x = tag_str_SEQ L\<^isub>1 L\<^isub>2 y"
+ shows "(x::string) \<equiv>(L\<^isub>1 ; L\<^isub>2)\<equiv> y"
+proof -
+ have aux: "\<And> x y z. \<lbrakk>tag_str_SEQ L\<^isub>1 L\<^isub>2 x = tag_str_SEQ L\<^isub>1 L\<^isub>2 y; x @ z \<in> L\<^isub>1 ; L\<^isub>2\<rbrakk>
+ \<Longrightarrow> y @ z \<in> L\<^isub>1 ; L\<^isub>2"
+ proof (drule app_in_seq_decom, erule disjE)
+ fix x y z
+ assume tag_eq: "tag_str_SEQ L\<^isub>1 L\<^isub>2 x = tag_str_SEQ L\<^isub>1 L\<^isub>2 y"
+ and x_gets_l2: "\<exists>xa\<le>x. xa \<in> L\<^isub>1 \<and> (x - xa) @ z \<in> L\<^isub>2"
+ from x_gets_l2 have "\<exists> xa \<le> x. xa \<in> L\<^isub>1" by blast
+ hence xy_l2:"{\<lbrakk>(x - xa)\<rbrakk>L\<^isub>2 | xa. xa \<le> x \<and> xa \<in> L\<^isub>1} = {\<lbrakk>(y - ya)\<rbrakk>L\<^isub>2 | ya. ya \<le> y \<and> ya \<in> L\<^isub>1}"
+ using tag_eq tag_seq_eq_E by blast
+ from x_gets_l2 obtain xa where xa_le_x: "xa \<le> x"
+ and xa_in_l1: "xa \<in> L\<^isub>1"
+ and rest_in_l2: "(x - xa) @ z \<in> L\<^isub>2" by blast
+ hence "\<exists> ya. \<lbrakk>(x - xa)\<rbrakk>L\<^isub>2 = \<lbrakk>(y - ya)\<rbrakk>L\<^isub>2 \<and> ya \<le> y \<and> ya \<in> L\<^isub>1" using xy_l2 by auto
+ then obtain ya where ya_le_x: "ya \<le> y"
+ and ya_in_l1: "ya \<in> L\<^isub>1"
+ and rest_eq: "\<lbrakk>(x - xa)\<rbrakk>L\<^isub>2 = \<lbrakk>(y - ya)\<rbrakk>L\<^isub>2" by blast
+ from rest_eq rest_in_l2 have "(y - ya) @ z \<in> L\<^isub>2"
+ by (auto simp:equiv_class_def equiv_str_def)
+ hence "ya @ ((y - ya) @ z) \<in> L\<^isub>1 ; L\<^isub>2" using ya_in_l1
+ by (auto simp:lang_seq_def)
+ thus "y @ z \<in> L\<^isub>1 ; L\<^isub>2" using ya_le_x
+ by (erule_tac prefixE, simp)
+ next
+ fix x y z
+ assume tag_eq: "tag_str_SEQ L\<^isub>1 L\<^isub>2 x = tag_str_SEQ L\<^isub>1 L\<^isub>2 y"
+ and x_gets_l1: "\<exists>za\<le>z. x @ za \<in> L\<^isub>1 \<and> z - za \<in> L\<^isub>2"
+ from tag_eq tag_seq_eq_E have x_y_eq: "\<lbrakk>x\<rbrakk>L\<^isub>1 = \<lbrakk>y\<rbrakk>L\<^isub>1" by blast
+ from x_gets_l1 obtain za where za_le_z: "za \<le> z"
+ and x_za_in_l1: "(x @ za) \<in> L\<^isub>1"
+ and rest_in_l2: "z - za \<in> L\<^isub>2" by blast
+ from x_y_eq x_za_in_l1 have y_za_in_l1: "y @ za \<in> L\<^isub>1"
+ by (auto simp:equiv_class_def equiv_str_def)
+ hence "(y @ za) @ (z - za) \<in> L\<^isub>1 ; L\<^isub>2" using rest_in_l2
+ apply (simp add:lang_seq_def)
+ by (rule_tac x = "y @ za" in exI, rule_tac x = "z - za" in exI, simp)
+ thus "y @ z \<in> L\<^isub>1 ; L\<^isub>2" using za_le_z
+ by (erule_tac prefixE, simp)
+ qed
+ show ?thesis using tag_eq
+ apply (simp add:equiv_str_def)
+ by (auto intro:aux)
+qed
+
+lemma quot_seq:
+ assumes finite1: "finite (QUOT L\<^isub>1)"
+ and finite2: "finite (QUOT L\<^isub>2)"
+ shows "finite (QUOT (L\<^isub>1;L\<^isub>2))"
+proof (rule_tac f = "(op `) (tag_str_SEQ L\<^isub>1 L\<^isub>2)" in finite_imageD)
+ show "finite (op ` (tag_str_SEQ L\<^isub>1 L\<^isub>2) ` QUOT (L\<^isub>1 ; L\<^isub>2))"
+ using finite_tag_image tag_str_seq_range_finite finite1 finite2
+ by auto
+next
+ show "inj_on (op ` (tag_str_SEQ L\<^isub>1 L\<^isub>2)) (QUOT (L\<^isub>1 ; L\<^isub>2))"
+ apply (rule_tac str_inj_imps)
+ by (erule_tac tag_str_seq_inj)
+qed
+
+(****************** the STAR case ************************)
+
+lemma app_eq_elim[rule_format]:
+ "\<And> a. \<forall> b x y. a @ b = x @ y \<longrightarrow> (\<exists> aa ab. a = aa @ ab \<and> x = aa \<and> y = ab @ b) \<or>
+ (\<exists> ba bb. b = ba @ bb \<and> x = a @ ba \<and> y = bb \<and> ba \<noteq> [])"
+ apply (induct_tac a rule:List.induct, simp)
+ apply (rule allI | rule impI)+
+ by (case_tac x, auto)
+
+definition tag_str_STAR:: "string set \<Rightarrow> string \<Rightarrow> string set set"
+where
+ "tag_str_STAR L\<^isub>1 x \<equiv> if (x = [])
+ then {}
+ else {\<lbrakk>x\<^isub>2\<rbrakk>L\<^isub>1 | x\<^isub>1 x\<^isub>2. x = x\<^isub>1 @ x\<^isub>2 \<and> x\<^isub>1 \<in> L\<^isub>1\<star> \<and> x\<^isub>2 \<noteq> []}"
+
+lemma tag_str_star_range_finite:
+ assumes finite1: "finite (QUOT L\<^isub>1)"
+ shows "finite (range (tag_str_STAR L\<^isub>1))"
+proof -
+ have "range (tag_str_STAR L\<^isub>1) \<subseteq> Pow (QUOT L\<^isub>1)"
+ by (auto simp:image_def tag_str_STAR_def QUOT_def)
+ thus ?thesis using finite1
+ by (rule_tac B = "Pow (QUOT L\<^isub>1)" in finite_subset, auto)
+qed
+
+lemma star_prop[rule_format]: "x \<in> lang\<star> \<Longrightarrow> \<forall> y. y \<in> lang\<star> \<longrightarrow> x @ y \<in> lang\<star>"
+by (erule Star.induct, auto)
+
+lemma star_prop2: "y \<in> lang \<Longrightarrow> y \<in> lang\<star>"
+by (drule step[of y lang "[]"], auto simp:start)
+
+lemma star_prop3[rule_format]: "x \<in> lang\<star> \<Longrightarrow> \<forall>y . y \<in> lang \<longrightarrow> x @ y \<in> lang\<star>"
+by (erule Star.induct, auto intro:star_prop2)
+
+lemma postfix_prop: "y >>= (x @ y) \<Longrightarrow> x = []"
+by (erule postfixE, induct x arbitrary:y, auto)
+
+lemma inj_aux:
+ "\<lbrakk>(m @ z) \<in> L\<^isub>1\<star>; m \<equiv>L\<^isub>1\<equiv> yb; xa @ m = x; xa \<in> L\<^isub>1\<star>; m \<noteq> [];
+ \<forall> xa xb. x = xa @ xb \<and> xa \<in> L\<^isub>1\<star> \<and> xb \<noteq> [] \<and> xb @ z \<in> L\<^isub>1\<star> \<longrightarrow> xb >>= m\<rbrakk>
+ \<Longrightarrow> (yb @ z) \<in> L\<^isub>1\<star>"
+proof-
+ have "\<And>s. s \<in> L\<^isub>1\<star> \<Longrightarrow> \<forall> m z yb. (s = m @ z \<and> m \<equiv>L\<^isub>1\<equiv> yb \<and> x = xa @ m \<and> xa \<in> L\<^isub>1\<star> \<and> m \<noteq> [] \<and>
+ (\<forall> xa xb. x = xa @ xb \<and> xa \<in> L\<^isub>1\<star> \<and> xb \<noteq> [] \<and> xb @ z \<in> L\<^isub>1\<star> \<longrightarrow> xb >>= m) \<longrightarrow> (yb @ z) \<in> L\<^isub>1\<star>)"
+ apply (erule Star.induct, simp)
+ apply (rule allI | rule impI | erule conjE)+
+ apply (drule app_eq_elim)
+ apply (erule disjE | erule exE | erule conjE)+
+ apply simp
+ apply (simp (no_asm) only:append_assoc[THEN sym])
+ apply (rule step)
+ apply (simp add:equiv_str_def)
+ apply simp
+
+ apply (erule exE | erule conjE)+
+ apply (rotate_tac 3)
+ apply (frule_tac x = "xa @ s1" in spec)
+ apply (rotate_tac 12)
+ apply (drule_tac x = ba in spec)
+ apply (erule impE)
+ apply ( simp add:star_prop3)
+ apply (simp)
+ apply (drule postfix_prop)
+ apply simp
+ done
+ thus "\<lbrakk>(m @ z) \<in> L\<^isub>1\<star>; m \<equiv>L\<^isub>1\<equiv> yb; xa @ m = x; xa \<in> L\<^isub>1\<star>; m \<noteq> [];
+ \<forall> xa xb. x = xa @ xb \<and> xa \<in> L\<^isub>1\<star> \<and> xb \<noteq> [] \<and> xb @ z \<in> L\<^isub>1\<star> \<longrightarrow> xb >>= m\<rbrakk>
+ \<Longrightarrow> (yb @ z) \<in> L\<^isub>1\<star>" by blast
+qed
+
+
+lemma min_postfix_exists[rule_format]:
+ "finite A \<Longrightarrow> A \<noteq> {} \<and> (\<forall> a \<in> A. \<forall> b \<in> A. ((b >>= a) \<or> (a >>= b)))
+ \<longrightarrow> (\<exists> min. (min \<in> A \<and> (\<forall> a \<in> A. a >>= min)))"
+apply (erule finite.induct)
+apply simp
+apply simp
+apply (case_tac "A = {}")
+apply simp
+apply clarsimp
+apply (case_tac "a >>= min")
+apply (rule_tac x = min in exI, simp)
+apply (rule_tac x = a in exI, simp)
+apply clarify
+apply (rotate_tac 5)
+apply (erule_tac x = aa in ballE) defer apply simp
+apply (erule_tac ys = min in postfix_trans)
+apply (erule_tac x = min in ballE)
+by simp+
+
+lemma tag_str_star_inj:
+ "tag_str_STAR L\<^isub>1 x = tag_str_STAR L\<^isub>1 (y::string) \<Longrightarrow> x \<equiv>L\<^isub>1\<star>\<equiv> y"
+proof -
+ have aux: "\<And> x y z. \<lbrakk>tag_str_STAR L\<^isub>1 x = tag_str_STAR L\<^isub>1 y; x @ z \<in> L\<^isub>1\<star>\<rbrakk> \<Longrightarrow> y @ z \<in> L\<^isub>1\<star>"
+ proof-
+ fix x y z
+ assume tag_eq: "tag_str_STAR L\<^isub>1 x = tag_str_STAR L\<^isub>1 y"
+ and x_z: "x @ z \<in> L\<^isub>1\<star>"
+ show "y @ z \<in> L\<^isub>1\<star>"
+ proof (cases "x = []")
+ case True
+ with tag_eq have "y = []" by (simp add:tag_str_STAR_def split:if_splits, blast)
+ thus ?thesis using x_z True by simp
+ next
+ case False
+ hence not_empty: "{xb. \<exists> xa. x = xa @ xb \<and> xa \<in> L\<^isub>1\<star> \<and> xb \<noteq> [] \<and> xb @ z \<in> L\<^isub>1\<star>} \<noteq> {}" using x_z
+ by (simp, rule_tac x = x in exI, rule_tac x = "[]" in exI, simp add:start)
+ have finite_set: "finite {xb. \<exists> xa. x = xa @ xb \<and> xa \<in> L\<^isub>1\<star> \<and> xb \<noteq> [] \<and> xb @ z \<in> L\<^isub>1\<star>}"
+ apply (rule_tac B = "{xb. \<exists> xa. x = xa @ xb}" in finite_subset)
+ apply auto
+ apply (induct x, simp)
+ apply (subgoal_tac "{xb. \<exists>xa. a # x = xa @ xb} = insert (a # x) {xb. \<exists>xa. x = xa @ xb}")
+ apply auto
+ by (case_tac xaa, simp+)
+ have comparable: "\<forall> a \<in> {xb. \<exists> xa. x = xa @ xb \<and> xa \<in> L\<^isub>1\<star> \<and> xb \<noteq> [] \<and> xb @ z \<in> L\<^isub>1\<star>}.
+ \<forall> b \<in> {xb. \<exists> xa. x = xa @ xb \<and> xa \<in> L\<^isub>1\<star> \<and> xb \<noteq> [] \<and> xb @ z \<in> L\<^isub>1\<star>}.
+ ((b >>= a) \<or> (a >>= b))"
+ by (auto simp:postfix_def, drule app_eq_elim, blast)
+ hence "\<exists> min. min \<in> {xb. \<exists> xa. x = xa @ xb \<and> xa \<in> L\<^isub>1\<star> \<and> xb \<noteq> [] \<and> xb @ z \<in> L\<^isub>1\<star>}
+ \<and> (\<forall> xa xb. x = xa @ xb \<and> xa \<in> L\<^isub>1\<star> \<and> xb \<noteq> [] \<and> xb @ z \<in> L\<^isub>1\<star> \<longrightarrow> xb >>= min)"
+ using finite_set not_empty comparable
+ apply (drule_tac min_postfix_exists, simp)
+ by (erule exE, rule_tac x = min in exI, auto)
+ then obtain min xa where x_decom: "x = xa @ min \<and> xa \<in> L\<^isub>1\<star>"
+ and min_not_empty: "min \<noteq> []"
+ and min_z_in_star: "min @ z \<in> L\<^isub>1\<star>"
+ and is_min: "\<forall> xa xb. x = xa @ xb \<and> xa \<in> L\<^isub>1\<star> \<and> xb \<noteq> [] \<and> xb @ z \<in> L\<^isub>1\<star> \<longrightarrow> xb >>= min" by blast
+ from x_decom min_not_empty have "\<lbrakk>min\<rbrakk>L\<^isub>1 \<in> tag_str_STAR L\<^isub>1 x" by (auto simp:tag_str_STAR_def)
+ hence "\<exists> yb. \<lbrakk>yb\<rbrakk>L\<^isub>1 \<in> tag_str_STAR L\<^isub>1 y \<and> \<lbrakk>min\<rbrakk>L\<^isub>1 = \<lbrakk>yb\<rbrakk>L\<^isub>1" using tag_eq by auto
+ hence "\<exists> ya yb. y = ya @ yb \<and> ya \<in> L\<^isub>1\<star> \<and> min \<equiv>L\<^isub>1\<equiv> yb \<and> yb \<noteq> [] "
+ by (simp add:tag_str_STAR_def equiv_class_def equiv_str_def split:if_splits, blast)
+ then obtain ya yb where y_decom: "y = ya @ yb"
+ and ya_in_star: "ya \<in> L\<^isub>1\<star>"
+ and yb_not_empty: "yb \<noteq> []"
+ and min_yb_eq: "min \<equiv>L\<^isub>1\<equiv> yb" by blast
+ from min_z_in_star min_yb_eq min_not_empty is_min x_decom
+ have "yb @ z \<in> L\<^isub>1\<star>"
+ by (rule_tac x = x and xa = xa in inj_aux, simp+)
+ thus ?thesis using ya_in_star y_decom
+ by (auto dest:star_prop)
+ qed
+ qed
+ show "tag_str_STAR L\<^isub>1 x = tag_str_STAR L\<^isub>1 (y::string) \<Longrightarrow> x \<equiv>L\<^isub>1\<star>\<equiv> y"
+ by (auto intro:aux simp:equiv_str_def)
+qed
+
+lemma quot_star:
+ assumes finite1: "finite (QUOT L\<^isub>1)"
+ shows "finite (QUOT (L\<^isub>1\<star>))"
+proof (rule_tac f = "(op `) (tag_str_STAR L\<^isub>1)" in finite_imageD)
+ show "finite (op ` (tag_str_STAR L\<^isub>1) ` QUOT (L\<^isub>1\<star>))"
+ using finite_tag_image tag_str_star_range_finite finite1
+ by auto
+next
+ show "inj_on (op ` (tag_str_STAR L\<^isub>1)) (QUOT (L\<^isub>1\<star>))"
+ apply (rule_tac str_inj_imps)
+ by (erule_tac tag_str_star_inj)
+qed
+
+lemma other_direction:
+ "Lang = L (r::rexp) \<Longrightarrow> finite (QUOT Lang)"
+apply (induct arbitrary:Lang rule:rexp.induct)
+apply (simp add:QUOT_def equiv_class_def equiv_str_def)
+by (simp_all add:quot_lambda quot_single quot_seq quot_alt quot_star)
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Attic/Prelude.thy Mon Jul 25 13:33:38 2011 +0000
@@ -0,0 +1,19 @@
+theory Prelude
+imports Main
+begin
+
+(*
+To make the theory work under Isabelle 2009 and 2011
+
+Isabelle 2009: set_ext
+Isabelle 2011: set_eqI
+
+*)
+
+
+lemma set_eq_intro:
+ "(\<And>x. (x \<in> A) = (x \<in> B)) \<Longrightarrow> A = B"
+by blast
+
+
+end
\ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Attic/old/Closure.thy Mon Jul 25 13:33:38 2011 +0000
@@ -0,0 +1,158 @@
+(* Author: Christian Urban, Xingyuan Zhang, Chunhan Wu *)
+theory Closure
+imports Derivs
+begin
+
+section {* Closure properties of regular languages *}
+
+abbreviation
+ regular :: "lang \<Rightarrow> bool"
+where
+ "regular A \<equiv> \<exists>r. A = L_rexp r"
+
+subsection {* Closure under set operations *}
+
+lemma closure_union[intro]:
+ assumes "regular A" "regular B"
+ shows "regular (A \<union> B)"
+proof -
+ from assms obtain r1 r2::rexp where "L_rexp r1 = A" "L_rexp r2 = B" by auto
+ then have "A \<union> B = L_rexp (ALT r1 r2)" by simp
+ then show "regular (A \<union> B)" by blast
+qed
+
+lemma closure_seq[intro]:
+ assumes "regular A" "regular B"
+ shows "regular (A \<cdot> B)"
+proof -
+ from assms obtain r1 r2::rexp where "L_rexp r1 = A" "L_rexp r2 = B" by auto
+ then have "A \<cdot> B = L_rexp (SEQ r1 r2)" by simp
+ then show "regular (A \<cdot> B)" by blast
+qed
+
+lemma closure_star[intro]:
+ assumes "regular A"
+ shows "regular (A\<star>)"
+proof -
+ from assms obtain r::rexp where "L_rexp r = A" by auto
+ then have "A\<star> = L_rexp (STAR r)" by simp
+ then show "regular (A\<star>)" by blast
+qed
+
+text {* Closure under complementation is proved via the
+ Myhill-Nerode theorem *}
+
+lemma closure_complement[intro]:
+ assumes "regular A"
+ shows "regular (- A)"
+proof -
+ from assms have "finite (UNIV // \<approx>A)" by (simp add: Myhill_Nerode)
+ then have "finite (UNIV // \<approx>(-A))" by (simp add: str_eq_rel_def)
+ then show "regular (- A)" by (simp add: Myhill_Nerode)
+qed
+
+lemma closure_difference[intro]:
+ assumes "regular A" "regular B"
+ shows "regular (A - B)"
+proof -
+ have "A - B = - (- A \<union> B)" by blast
+ moreover
+ have "regular (- (- A \<union> B))"
+ using assms by blast
+ ultimately show "regular (A - B)" by simp
+qed
+
+lemma closure_intersection[intro]:
+ assumes "regular A" "regular B"
+ shows "regular (A \<inter> B)"
+proof -
+ have "A \<inter> B = - (- A \<union> - B)" by blast
+ moreover
+ have "regular (- (- A \<union> - B))"
+ using assms by blast
+ ultimately show "regular (A \<inter> B)" by simp
+qed
+
+subsection {* Closure under string reversal *}
+
+fun
+ Rev :: "rexp \<Rightarrow> rexp"
+where
+ "Rev NULL = NULL"
+| "Rev EMPTY = EMPTY"
+| "Rev (CHAR c) = CHAR c"
+| "Rev (ALT r1 r2) = ALT (Rev r1) (Rev r2)"
+| "Rev (SEQ r1 r2) = SEQ (Rev r2) (Rev r1)"
+| "Rev (STAR r) = STAR (Rev r)"
+
+lemma rev_seq[simp]:
+ shows "rev ` (B \<cdot> A) = (rev ` A) \<cdot> (rev ` B)"
+unfolding Seq_def image_def
+by (auto) (metis rev_append)+
+
+lemma rev_star1:
+ assumes a: "s \<in> (rev ` A)\<star>"
+ shows "s \<in> rev ` (A\<star>)"
+using a
+proof(induct rule: star_induct)
+ case (step s1 s2)
+ have inj: "inj (rev::string \<Rightarrow> string)" unfolding inj_on_def by auto
+ have "s1 \<in> rev ` A" "s2 \<in> rev ` (A\<star>)" by fact+
+ then obtain x1 x2 where "x1 \<in> A" "x2 \<in> A\<star>" and eqs: "s1 = rev x1" "s2 = rev x2" by auto
+ then have "x1 \<in> A\<star>" "x2 \<in> A\<star>" by (auto intro: star_intro2)
+ then have "x2 @ x1 \<in> A\<star>" by (auto intro: star_intro1)
+ then have "rev (x2 @ x1) \<in> rev ` A\<star>" using inj by (simp only: inj_image_mem_iff)
+ then show "s1 @ s2 \<in> rev ` A\<star>" using eqs by simp
+qed (auto)
+
+lemma rev_star2:
+ assumes a: "s \<in> A\<star>"
+ shows "rev s \<in> (rev ` A)\<star>"
+using a
+proof(induct rule: star_induct)
+ case (step s1 s2)
+ have inj: "inj (rev::string \<Rightarrow> string)" unfolding inj_on_def by auto
+ have "s1 \<in> A"by fact
+ then have "rev s1 \<in> rev ` A" using inj by (simp only: inj_image_mem_iff)
+ then have "rev s1 \<in> (rev ` A)\<star>" by (auto intro: star_intro2)
+ moreover
+ have "rev s2 \<in> (rev ` A)\<star>" by fact
+ ultimately show "rev (s1 @ s2) \<in> (rev ` A)\<star>" by (auto intro: star_intro1)
+qed (auto)
+
+lemma rev_star[simp]:
+ shows " rev ` (A\<star>) = (rev ` A)\<star>"
+using rev_star1 rev_star2 by auto
+
+lemma rev_lang:
+ shows "rev ` (L_rexp r) = L_rexp (Rev r)"
+by (induct r) (simp_all add: image_Un)
+
+lemma closure_reversal[intro]:
+ assumes "regular A"
+ shows "regular (rev ` A)"
+proof -
+ from assms obtain r::rexp where "A = L_rexp r" by auto
+ then have "L_rexp (Rev r) = rev ` A" by (simp add: rev_lang)
+ then show "regular (rev` A)" by blast
+qed
+
+subsection {* Closure under left-quotients *}
+
+lemma closure_left_quotient:
+ assumes "regular A"
+ shows "regular (Ders_set B A)"
+proof -
+ from assms obtain r::rexp where eq: "L_rexp r = A" by auto
+ have fin: "finite (pders_set B r)" by (rule finite_pders_set)
+
+ have "Ders_set B (L_rexp r) = (\<Union> L_rexp ` (pders_set B r))"
+ by (simp add: Ders_set_pders_set)
+ also have "\<dots> = L_rexp (\<Uplus>(pders_set B r))" using fin by simp
+ finally have "Ders_set B A = L_rexp (\<Uplus>(pders_set B r))" using eq
+ by simp
+ then show "regular (Ders_set B A)" by auto
+qed
+
+
+end
\ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Attic/old/Derivs.thy Mon Jul 25 13:33:38 2011 +0000
@@ -0,0 +1,492 @@
+theory Derivs
+imports Myhill_2
+begin
+
+section {* Left-Quotients and Derivatives *}
+
+subsection {* Left-Quotients *}
+
+definition
+ Delta :: "lang \<Rightarrow> lang"
+where
+ "Delta A = (if [] \<in> A then {[]} else {})"
+
+definition
+ Der :: "char \<Rightarrow> lang \<Rightarrow> lang"
+where
+ "Der c A \<equiv> {s. [c] @ s \<in> A}"
+
+definition
+ Ders :: "string \<Rightarrow> lang \<Rightarrow> lang"
+where
+ "Ders s A \<equiv> {s'. s @ s' \<in> A}"
+
+definition
+ Ders_set :: "lang \<Rightarrow> lang \<Rightarrow> lang"
+where
+ "Ders_set A B \<equiv> {s' | s s'. s @ s' \<in> B \<and> s \<in> A}"
+
+lemma Ders_set_Ders:
+ shows "Ders_set A B = (\<Union>s \<in> A. Ders s B)"
+unfolding Ders_set_def Ders_def
+by auto
+
+lemma Der_null [simp]:
+ shows "Der c {} = {}"
+unfolding Der_def
+by auto
+
+lemma Der_empty [simp]:
+ shows "Der c {[]} = {}"
+unfolding Der_def
+by auto
+
+lemma Der_char [simp]:
+ shows "Der c {[d]} = (if c = d then {[]} else {})"
+unfolding Der_def
+by auto
+
+lemma Der_union [simp]:
+ shows "Der c (A \<union> B) = Der c A \<union> Der c B"
+unfolding Der_def
+by auto
+
+lemma Der_seq [simp]:
+ shows "Der c (A \<cdot> B) = (Der c A) \<cdot> B \<union> (Delta A \<cdot> Der c B)"
+unfolding Der_def Delta_def
+unfolding Seq_def
+by (auto simp add: Cons_eq_append_conv)
+
+lemma Der_star [simp]:
+ shows "Der c (A\<star>) = (Der c A) \<cdot> A\<star>"
+proof -
+ have incl: "Delta A \<cdot> Der c (A\<star>) \<subseteq> (Der c A) \<cdot> A\<star>"
+ unfolding Der_def Delta_def Seq_def
+ apply(auto)
+ apply(drule star_decom)
+ apply(auto simp add: Cons_eq_append_conv)
+ done
+
+ have "Der c (A\<star>) = Der c ({[]} \<union> A \<cdot> A\<star>)"
+ by (simp only: star_cases[symmetric])
+ also have "... = Der c (A \<cdot> A\<star>)"
+ by (simp only: Der_union Der_empty) (simp)
+ also have "... = (Der c A) \<cdot> A\<star> \<union> (Delta A \<cdot> Der c (A\<star>))"
+ by simp
+ also have "... = (Der c A) \<cdot> A\<star>"
+ using incl by auto
+ finally show "Der c (A\<star>) = (Der c A) \<cdot> A\<star>" .
+qed
+
+
+lemma Ders_singleton:
+ shows "Ders [c] A = Der c A"
+unfolding Der_def Ders_def
+by simp
+
+lemma Ders_append:
+ shows "Ders (s1 @ s2) A = Ders s2 (Ders s1 A)"
+unfolding Ders_def by simp
+
+lemma MN_Rel_Ders:
+ shows "x \<approx>A y \<longleftrightarrow> Ders x A = Ders y A"
+unfolding Ders_def str_eq_def str_eq_rel_def
+by auto
+
+subsection {* Brozowsky's derivatives of regular expressions *}
+
+fun
+ nullable :: "rexp \<Rightarrow> bool"
+where
+ "nullable (NULL) = False"
+| "nullable (EMPTY) = True"
+| "nullable (CHAR c) = False"
+| "nullable (ALT r1 r2) = (nullable r1 \<or> nullable r2)"
+| "nullable (SEQ r1 r2) = (nullable r1 \<and> nullable r2)"
+| "nullable (STAR r) = True"
+
+fun
+ der :: "char \<Rightarrow> rexp \<Rightarrow> rexp"
+where
+ "der c (NULL) = NULL"
+| "der c (EMPTY) = NULL"
+| "der c (CHAR c') = (if c = c' then EMPTY else NULL)"
+| "der c (ALT r1 r2) = ALT (der c r1) (der c r2)"
+| "der c (SEQ r1 r2) = ALT (SEQ (der c r1) r2) (if nullable r1 then der c r2 else NULL)"
+| "der c (STAR r) = SEQ (der c r) (STAR r)"
+
+function
+ ders :: "string \<Rightarrow> rexp \<Rightarrow> rexp"
+where
+ "ders [] r = r"
+| "ders (s @ [c]) r = der c (ders s r)"
+by (auto) (metis rev_cases)
+
+termination
+ by (relation "measure (length o fst)") (auto)
+
+lemma Delta_nullable:
+ shows "Delta (L_rexp r) = (if nullable r then {[]} else {})"
+unfolding Delta_def
+by (induct r) (auto simp add: Seq_def split: if_splits)
+
+lemma Der_der:
+ fixes r::rexp
+ shows "Der c (L_rexp r) = L_rexp (der c r)"
+by (induct r) (simp_all add: Delta_nullable)
+
+lemma Ders_ders:
+ fixes r::rexp
+ shows "Ders s (L_rexp r) = L_rexp (ders s r)"
+apply(induct s rule: rev_induct)
+apply(simp add: Ders_def)
+apply(simp only: ders.simps)
+apply(simp only: Ders_append)
+apply(simp only: Ders_singleton)
+apply(simp only: Der_der)
+done
+
+
+subsection {* Antimirov's Partial Derivatives *}
+
+abbreviation
+ "SEQS R r \<equiv> {SEQ r' r | r'. r' \<in> R}"
+
+fun
+ pder :: "char \<Rightarrow> rexp \<Rightarrow> rexp set"
+where
+ "pder c NULL = {NULL}"
+| "pder c EMPTY = {NULL}"
+| "pder c (CHAR c') = (if c = c' then {EMPTY} else {NULL})"
+| "pder c (ALT r1 r2) = (pder c r1) \<union> (pder c r2)"
+| "pder c (SEQ r1 r2) = SEQS (pder c r1) r2 \<union> (if nullable r1 then pder c r2 else {})"
+| "pder c (STAR r) = SEQS (pder c r) (STAR r)"
+
+abbreviation
+ "pder_set c R \<equiv> \<Union>r \<in> R. pder c r"
+
+function
+ pders :: "string \<Rightarrow> rexp \<Rightarrow> rexp set"
+where
+ "pders [] r = {r}"
+| "pders (s @ [c]) r = pder_set c (pders s r)"
+by (auto) (metis rev_cases)
+
+termination
+ by (relation "measure (length o fst)") (auto)
+
+abbreviation
+ "pders_set A r \<equiv> \<Union>s \<in> A. pders s r"
+
+lemma pders_append:
+ "pders (s1 @ s2) r = \<Union> (pders s2) ` (pders s1 r)"
+apply(induct s2 arbitrary: s1 r rule: rev_induct)
+apply(simp)
+apply(subst append_assoc[symmetric])
+apply(simp only: pders.simps)
+apply(auto)
+done
+
+lemma pders_singleton:
+ "pders [c] r = pder c r"
+apply(subst append_Nil[symmetric])
+apply(simp only: pders.simps)
+apply(simp)
+done
+
+lemma pder_set_lang:
+ shows "(\<Union> (L_rexp ` pder_set c R)) = (\<Union>r \<in> R. (\<Union>L_rexp ` (pder c r)))"
+unfolding image_def
+by auto
+
+lemma
+ shows seq_UNION_left: "B \<cdot> (\<Union>n\<in>C. A n) = (\<Union>n\<in>C. B \<cdot> A n)"
+ and seq_UNION_right: "(\<Union>n\<in>C. A n) \<cdot> B = (\<Union>n\<in>C. A n \<cdot> B)"
+unfolding Seq_def by auto
+
+lemma Der_pder:
+ fixes r::rexp
+ shows "Der c (L_rexp r) = \<Union> L_rexp ` (pder c r)"
+by (induct r) (auto simp add: Delta_nullable seq_UNION_right)
+
+lemma Ders_pders:
+ fixes r::rexp
+ shows "Ders s (L_rexp r) = \<Union> L_rexp ` (pders s r)"
+proof (induct s rule: rev_induct)
+ case (snoc c s)
+ have ih: "Ders s (L_rexp r) = \<Union> L_rexp ` (pders s r)" by fact
+ have "Ders (s @ [c]) (L_rexp r) = Ders [c] (Ders s (L_rexp r))"
+ by (simp add: Ders_append)
+ also have "\<dots> = Der c (\<Union> L_rexp ` (pders s r))" using ih
+ by (simp add: Ders_singleton)
+ also have "\<dots> = (\<Union>r\<in>pders s r. Der c (L_rexp r))"
+ unfolding Der_def image_def by auto
+ also have "\<dots> = (\<Union>r\<in>pders s r. (\<Union> L_rexp ` (pder c r)))"
+ by (simp add: Der_pder)
+ also have "\<dots> = (\<Union>L_rexp ` (pder_set c (pders s r)))"
+ by (simp add: pder_set_lang)
+ also have "\<dots> = (\<Union>L_rexp ` (pders (s @ [c]) r))"
+ by simp
+ finally show "Ders (s @ [c]) (L_rexp r) = \<Union> L_rexp ` pders (s @ [c]) r" .
+qed (simp add: Ders_def)
+
+lemma Ders_set_pders_set:
+ fixes r::rexp
+ shows "Ders_set A (L_rexp r) = (\<Union> L_rexp ` (pders_set A r))"
+by (simp add: Ders_set_Ders Ders_pders)
+
+lemma pders_NULL [simp]:
+ shows "pders s NULL = {NULL}"
+by (induct s rule: rev_induct) (simp_all)
+
+lemma pders_EMPTY [simp]:
+ shows "pders s EMPTY = (if s = [] then {EMPTY} else {NULL})"
+by (induct s rule: rev_induct) (auto)
+
+lemma pders_CHAR [simp]:
+ shows "pders s (CHAR c) = (if s = [] then {CHAR c} else (if s = [c] then {EMPTY} else {NULL}))"
+by (induct s rule: rev_induct) (auto)
+
+lemma pders_ALT [simp]:
+ shows "pders s (ALT r1 r2) = (if s = [] then {ALT r1 r2} else (pders s r1) \<union> (pders s r2))"
+by (induct s rule: rev_induct) (auto)
+
+definition
+ "Suf s \<equiv> {v. v \<noteq> [] \<and> (\<exists>u. u @ v = s)}"
+
+lemma Suf:
+ shows "Suf (s @ [c]) = (Suf s) \<cdot> {[c]} \<union> {[c]}"
+unfolding Suf_def Seq_def
+by (auto simp add: append_eq_append_conv2 append_eq_Cons_conv)
+
+lemma Suf_Union:
+ shows "(\<Union>v \<in> Suf s \<cdot> {[c]}. P v) = (\<Union>v \<in> Suf s. P (v @ [c]))"
+by (auto simp add: Seq_def)
+
+lemma inclusion1:
+ shows "pder_set c (SEQS R r2) \<subseteq> SEQS (pder_set c R) r2 \<union> (pder c r2)"
+apply(auto simp add: if_splits)
+apply(blast)
+done
+
+lemma pders_SEQ:
+ shows "pders s (SEQ r1 r2) \<subseteq> SEQS (pders s r1) r2 \<union> (\<Union>v \<in> Suf s. pders v r2)"
+proof (induct s rule: rev_induct)
+ case (snoc c s)
+ have ih: "pders s (SEQ r1 r2) \<subseteq> SEQS (pders s r1) r2 \<union> (\<Union>v \<in> Suf s. pders v r2)"
+ by fact
+ have "pders (s @ [c]) (SEQ r1 r2) = pder_set c (pders s (SEQ r1 r2))" by simp
+ also have "\<dots> \<subseteq> pder_set c (SEQS (pders s r1) r2 \<union> (\<Union>v \<in> Suf s. pders v r2))"
+ using ih by (auto) (blast)
+ also have "\<dots> = pder_set c (SEQS (pders s r1) r2) \<union> pder_set c (\<Union>v \<in> Suf s. pders v r2)"
+ by (simp)
+ also have "\<dots> = pder_set c (SEQS (pders s r1) r2) \<union> (\<Union>v \<in> Suf s. pder_set c (pders v r2))"
+ by (simp)
+ also have "\<dots> \<subseteq> pder_set c (SEQS (pders s r1) r2) \<union> (pder c r2) \<union> (\<Union>v \<in> Suf s. pders (v @ [c]) r2)"
+ by (auto)
+ also have "\<dots> \<subseteq> SEQS (pder_set c (pders s r1)) r2 \<union> (pder c r2) \<union> (\<Union>v \<in> Suf s. pders (v @ [c]) r2)"
+ using inclusion1 by blast
+ also have "\<dots> = SEQS (pders (s @ [c]) r1) r2 \<union> (\<Union>v \<in> Suf (s @ [c]). pders v r2)"
+ apply(subst (2) pders.simps)
+ apply(simp only: Suf)
+ apply(simp add: Suf_Union pders_singleton)
+ apply(auto)
+ done
+ finally show ?case .
+qed (simp)
+
+lemma pders_STAR:
+ assumes a: "s \<noteq> []"
+ shows "pders s (STAR r) \<subseteq> (\<Union>v \<in> Suf s. SEQS (pders v r) (STAR r))"
+using a
+proof (induct s rule: rev_induct)
+ case (snoc c s)
+ have ih: "s \<noteq> [] \<Longrightarrow> pders s (STAR r) \<subseteq> (\<Union>v\<in>Suf s. SEQS (pders v r) (STAR r))" by fact
+ { assume asm: "s \<noteq> []"
+ have "pders (s @ [c]) (STAR r) = pder_set c (pders s (STAR r))" by simp
+ also have "\<dots> \<subseteq> (pder_set c (\<Union>v\<in>Suf s. SEQS (pders v r) (STAR r)))"
+ using ih[OF asm] by blast
+ also have "\<dots> = (\<Union>v\<in>Suf s. pder_set c (SEQS (pders v r) (STAR r)))"
+ by simp
+ also have "\<dots> \<subseteq> (\<Union>v\<in>Suf s. (SEQS (pder_set c (pders v r)) (STAR r) \<union> pder c (STAR r)))"
+ using inclusion1 by (auto split: if_splits)
+ also have "\<dots> = (\<Union>v\<in>Suf s. (SEQS (pder_set c (pders v r)) (STAR r))) \<union> pder c (STAR r)"
+ using asm by (auto simp add: Suf_def)
+ also have "\<dots> = (\<Union>v\<in>Suf s. (SEQS (pders (v @ [c]) r) (STAR r))) \<union> (SEQS (pder c r) (STAR r))"
+ by simp
+ also have "\<dots> = (\<Union>v\<in>Suf (s @ [c]). (SEQS (pders v r) (STAR r)))"
+ apply(simp only: Suf)
+ apply(simp add: Suf_Union pders_singleton)
+ apply(auto)
+ done
+ finally have ?case .
+ }
+ moreover
+ { assume asm: "s = []"
+ then have ?case
+ apply(simp add: pders_singleton Suf_def)
+ apply(auto)
+ apply(rule_tac x="[c]" in exI)
+ apply(simp add: pders_singleton)
+ done
+ }
+ ultimately show ?case by blast
+qed (simp)
+
+abbreviation
+ "UNIV1 \<equiv> UNIV - {[]}"
+
+lemma pders_set_NULL:
+ shows "pders_set UNIV1 NULL = {NULL}"
+by auto
+
+lemma pders_set_EMPTY:
+ shows "pders_set UNIV1 EMPTY = {NULL}"
+by (auto split: if_splits)
+
+lemma pders_set_CHAR:
+ shows "pders_set UNIV1 (CHAR c) \<subseteq> {EMPTY, NULL}"
+by (auto split: if_splits)
+
+lemma pders_set_ALT:
+ shows "pders_set UNIV1 (ALT r1 r2) = pders_set UNIV1 r1 \<union> pders_set UNIV1 r2"
+by auto
+
+lemma pders_set_SEQ_aux:
+ assumes a: "s \<in> UNIV1"
+ shows "pders_set (Suf s) r2 \<subseteq> pders_set UNIV1 r2"
+using a by (auto simp add: Suf_def)
+
+lemma pders_set_SEQ:
+ shows "pders_set UNIV1 (SEQ r1 r2) \<subseteq> SEQS (pders_set UNIV1 r1) r2 \<union> pders_set UNIV1 r2"
+apply(rule UN_least)
+apply(rule subset_trans)
+apply(rule pders_SEQ)
+apply(simp)
+apply(rule conjI)
+apply(auto)[1]
+apply(rule subset_trans)
+apply(rule pders_set_SEQ_aux)
+apply(auto)
+done
+
+lemma pders_set_STAR:
+ shows "pders_set UNIV1 (STAR r) \<subseteq> SEQS (pders_set UNIV1 r) (STAR r)"
+apply(rule UN_least)
+apply(rule subset_trans)
+apply(rule pders_STAR)
+apply(simp)
+apply(simp add: Suf_def)
+apply(auto)
+done
+
+lemma finite_SEQS:
+ assumes a: "finite A"
+ shows "finite (SEQS A r)"
+using a by (auto)
+
+lemma finite_pders_set_UNIV1:
+ shows "finite (pders_set UNIV1 r)"
+apply(induct r)
+apply(simp)
+apply(simp only: pders_set_EMPTY)
+apply(simp)
+apply(rule finite_subset)
+apply(rule pders_set_CHAR)
+apply(simp)
+apply(rule finite_subset)
+apply(rule pders_set_SEQ)
+apply(simp only: finite_SEQS finite_Un)
+apply(simp)
+apply(simp only: pders_set_ALT)
+apply(simp)
+apply(rule finite_subset)
+apply(rule pders_set_STAR)
+apply(simp only: finite_SEQS)
+done
+
+lemma pders_set_UNIV_UNIV1:
+ shows "pders_set UNIV r = pders [] r \<union> pders_set UNIV1 r"
+apply(auto)
+apply(rule_tac x="[]" in exI)
+apply(simp)
+done
+
+lemma finite_pders_set_UNIV:
+ shows "finite (pders_set UNIV r)"
+unfolding pders_set_UNIV_UNIV1
+by (simp add: finite_pders_set_UNIV1)
+
+lemma finite_pders_set:
+ shows "finite (pders_set A r)"
+apply(rule rev_finite_subset)
+apply(rule_tac r="r" in finite_pders_set_UNIV)
+apply(auto)
+done
+
+lemma finite_pders:
+ shows "finite (pders s r)"
+using finite_pders_set[where A="{s}" and r="r"]
+by simp
+
+lemma finite_pders2:
+ shows "finite {pders s r | s. s \<in> A}"
+proof -
+ have "{pders s r | s. s \<in> A} \<subseteq> Pow (pders_set A r)" by auto
+ moreover
+ have "finite (Pow (pders_set A r))"
+ using finite_pders_set by simp
+ ultimately
+ show "finite {pders s r | s. s \<in> A}"
+ by(rule finite_subset)
+qed
+
+
+lemma Myhill_Nerode3:
+ fixes r::"rexp"
+ shows "finite (UNIV // \<approx>(L_rexp r))"
+proof -
+ have "finite (UNIV // =(\<lambda>x. pders x r)=)"
+ proof -
+ have "range (\<lambda>x. pders x r) = {pders s r | s. s \<in> UNIV}" by auto
+ moreover
+ have "finite {pders s r | s. s \<in> UNIV}" by (rule finite_pders2)
+ ultimately
+ have "finite (range (\<lambda>x. pders x r))"
+ by simp
+ then show "finite (UNIV // =(\<lambda>x. pders x r)=)"
+ by (rule finite_eq_tag_rel)
+ qed
+ moreover
+ have "=(\<lambda>x. pders x r)= \<subseteq> \<approx>(L_rexp r)"
+ unfolding tag_eq_rel_def
+ unfolding str_eq_def2
+ unfolding MN_Rel_Ders
+ unfolding Ders_pders
+ by auto
+ moreover
+ have "equiv UNIV =(\<lambda>x. pders x r)="
+ unfolding equiv_def refl_on_def sym_def trans_def
+ unfolding tag_eq_rel_def
+ by auto
+ moreover
+ have "equiv UNIV (\<approx>(L_rexp r))"
+ unfolding equiv_def refl_on_def sym_def trans_def
+ unfolding str_eq_rel_def
+ by auto
+ ultimately show "finite (UNIV // \<approx>(L_rexp r))"
+ by (rule refined_partition_finite)
+qed
+
+
+section {* Relating derivatives and partial derivatives *}
+
+lemma
+ shows "(\<Union> L_rexp ` (pder c r)) = L_rexp (der c r)"
+unfolding Der_der[symmetric] Der_pder by simp
+
+lemma
+ shows "(\<Union> L_rexp ` (pders s r)) = L_rexp (ders s r)"
+unfolding Ders_ders[symmetric] Ders_pders by simp
+
+end
\ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Attic/old/Folds.thy Mon Jul 25 13:33:38 2011 +0000
@@ -0,0 +1,22 @@
+theory Folds
+imports Main
+begin
+
+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"
+
+
+end
\ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Attic/old/My.thy Mon Jul 25 13:33:38 2011 +0000
@@ -0,0 +1,389 @@
+theory My
+imports Main Infinite_Set
+begin
+
+
+definition
+ Seq :: "string set \<Rightarrow> string set \<Rightarrow> string set" ("_ ;; _" [100,100] 100)
+where
+ "L1 ;; L2 = {s1 @ s2 | s1 s2. s1 \<in> L1 \<and> s2 \<in> L2}"
+
+inductive_set
+ Star :: "string set \<Rightarrow> string set" ("_\<star>" [101] 102)
+ for L :: "string set"
+where
+ start[intro]: "[] \<in> L\<star>"
+| step[intro]: "\<lbrakk>s1 \<in> L; s2 \<in> L\<star>\<rbrakk> \<Longrightarrow> s1@s2 \<in> L\<star>"
+
+lemma lang_star_cases:
+ shows "L\<star> = {[]} \<union> L ;; L\<star>"
+unfolding Seq_def
+by (auto) (metis Star.simps)
+
+lemma lang_star_cases2:
+ shows "L ;; L\<star> = L\<star> ;; L"
+sorry
+
+
+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" sorry
+ then have "B ;; A\<star> = B ;; ({[]} \<union> A\<star> ;; A)" unfolding Seq_def by simp
+ also have "\<dots> = B \<union> B ;; (A\<star> ;; A)" unfolding Seq_def by auto
+ also have "\<dots> = B \<union> (B ;; A\<star>) ;; A" unfolding Seq_def
+ by (auto) (metis append_assoc)+
+ finally show "X = X ;; A \<union> B" using eq by auto
+next
+ assume "X = X ;; A \<union> B"
+ then have "B \<subseteq> X" "X ;; A \<subseteq> X" by auto
+ show "X = B ;; A\<star>" sorry
+qed
+
+datatype rexp =
+ NULL
+| EMPTY
+| CHAR char
+| SEQ rexp rexp
+| ALT rexp rexp
+| STAR rexp
+
+fun
+ Sem :: "rexp \<Rightarrow> string set" ("\<lparr>_\<rparr>" [0] 1000)
+where
+ "\<lparr>NULL\<rparr> = {}"
+ | "\<lparr>EMPTY\<rparr> = {[]}"
+ | "\<lparr>CHAR c\<rparr> = {[c]}"
+ | "\<lparr>SEQ r1 r2\<rparr> = \<lparr>r1\<rparr> ;; \<lparr>r2\<rparr>"
+ | "\<lparr>ALT r1 r2\<rparr> = \<lparr>r1\<rparr> \<union> \<lparr>r2\<rparr>"
+ | "\<lparr>STAR r\<rparr> = \<lparr>r\<rparr>\<star>"
+
+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"
+
+lemma folds_simp_null [simp]:
+ "finite rs \<Longrightarrow> x \<in> \<lparr>folds ALT NULL rs\<rparr> \<longleftrightarrow> (\<exists>r \<in> rs. x \<in> \<lparr>r\<rparr>)"
+apply (simp add: folds_def)
+apply (rule someI2_ex)
+apply (erule finite_imp_fold_graph)
+apply (erule fold_graph.induct)
+apply (auto)
+done
+
+lemma folds_simp_empty [simp]:
+ "finite rs \<Longrightarrow> x \<in> \<lparr>folds ALT EMPTY rs\<rparr> \<longleftrightarrow> (\<exists>r \<in> rs. x \<in> \<lparr>r\<rparr>) \<or> x = []"
+apply (simp add: folds_def)
+apply (rule someI2_ex)
+apply (erule finite_imp_fold_graph)
+apply (erule fold_graph.induct)
+apply (auto)
+done
+
+lemma [simp]:
+ shows "(x, y) \<in> {(x, y). P x y} \<longleftrightarrow> P x y"
+by simp
+
+definition
+ str_eq ("_ \<approx>_ _")
+where
+ "x \<approx>Lang y \<equiv> (\<forall>z. x @ z \<in> Lang \<longleftrightarrow> y @ z \<in> Lang)"
+
+definition
+ str_eq_rel ("\<approx>_")
+where
+ "\<approx>Lang \<equiv> {(x, y). x \<approx>Lang y}"
+
+definition
+ final :: "string set \<Rightarrow> string set \<Rightarrow> bool"
+where
+ "final X Lang \<equiv> (X \<in> UNIV // \<approx>Lang) \<and> (\<forall>s \<in> X. s \<in> Lang)"
+
+lemma lang_is_union_of_finals:
+ "Lang = \<Union> {X. final X Lang}"
+proof -
+ have "Lang \<subseteq> \<Union> {X. final X Lang}"
+ unfolding final_def
+ unfolding quotient_def Image_def
+ unfolding str_eq_rel_def
+ apply(simp)
+ apply(auto)
+ apply(rule_tac x="(\<approx>Lang) `` {x}" in exI)
+ unfolding Image_def
+ unfolding str_eq_rel_def
+ apply(auto)
+ unfolding str_eq_def
+ apply(auto)
+ apply(drule_tac x="[]" in spec)
+ apply(simp)
+ done
+ moreover
+ have "\<Union>{X. final X Lang} \<subseteq> Lang"
+ unfolding final_def by auto
+ ultimately
+ show "Lang = \<Union> {X. final X Lang}"
+ by blast
+qed
+
+lemma all_rexp:
+ "\<lbrakk>finite (UNIV // \<approx>Lang); X \<in> (UNIV // \<approx>Lang)\<rbrakk> \<Longrightarrow> \<exists>r. X = \<lparr>r\<rparr>"
+sorry
+
+lemma final_rexp:
+ "\<lbrakk>finite (UNIV // (\<approx>Lang)); final X Lang\<rbrakk> \<Longrightarrow> \<exists>r. X = \<lparr>r\<rparr>"
+unfolding final_def
+using all_rexp by blast
+
+lemma finite_f_one_to_one:
+ assumes "finite B"
+ and "\<forall>x \<in> B. \<exists>y. f y = x"
+ shows "\<exists>rs. finite rs \<and> (B = {f y | y . y \<in> rs})"
+using assms
+by (induct) (auto)
+
+lemma finite_final:
+ assumes "finite (UNIV // (\<approx>Lang))"
+ shows "finite {X. final X Lang}"
+using assms
+proof -
+ have "{X. final X Lang} \<subseteq> (UNIV // (\<approx>Lang))"
+ unfolding final_def by auto
+ with assms show "finite {X. final X Lang}"
+ using finite_subset by auto
+qed
+
+lemma finite_regular_aux:
+ fixes Lang :: "string set"
+ assumes "finite (UNIV // (\<approx>Lang))"
+ shows "\<exists>rs. Lang = \<lparr>folds ALT NULL rs\<rparr>"
+apply(subst lang_is_union_of_finals)
+using assms
+apply -
+apply(drule finite_final)
+apply(drule_tac f="Sem" in finite_f_one_to_one)
+apply(clarify)
+apply(drule final_rexp[OF assms])
+apply(auto)[1]
+apply(clarify)
+apply(rule_tac x="rs" in exI)
+apply(simp)
+apply(rule set_eqI)
+apply(auto)
+done
+
+lemma finite_regular:
+ fixes Lang :: "string set"
+ assumes "finite (UNIV // (\<approx>Lang))"
+ shows "\<exists>r. Lang = \<lparr>r\<rparr>"
+using assms finite_regular_aux
+by auto
+
+
+
+section {* other direction *}
+
+
+lemma inj_image_lang:
+ fixes f::"string \<Rightarrow> 'a"
+ assumes str_inj: "\<And>x y. f x = f y \<Longrightarrow> x \<approx>Lang y"
+ shows "inj_on (image f) (UNIV // (\<approx>Lang))"
+proof -
+ { fix x y::string
+ assume eq_tag: "f ` {z. x \<approx>Lang z} = f ` {z. y \<approx>Lang z}"
+ moreover
+ have "{z. x \<approx>Lang z} \<noteq> {}" unfolding str_eq_def by auto
+ ultimately obtain a b where "x \<approx>Lang a" "y \<approx>Lang b" "f a = f b" by blast
+ then have "x \<approx>Lang a" "y \<approx>Lang b" "a \<approx>Lang b" using str_inj by auto
+ then have "x \<approx>Lang y" unfolding str_eq_def by simp
+ then have "{z. x \<approx>Lang z} = {z. y \<approx>Lang z}" unfolding str_eq_def by simp
+ }
+ then have "\<forall>x\<in>UNIV // \<approx>Lang. \<forall>y\<in>UNIV // \<approx>Lang. f ` x = f ` y \<longrightarrow> x = y"
+ unfolding quotient_def Image_def str_eq_rel_def by simp
+ then show "inj_on (image f) (UNIV // (\<approx>Lang))"
+ unfolding inj_on_def by simp
+qed
+
+
+lemma finite_range_image:
+ assumes fin: "finite (range f)"
+ shows "finite ((image f) ` X)"
+proof -
+ from fin have "finite (Pow (f ` UNIV))" by auto
+ moreover
+ have "(image f) ` X \<subseteq> Pow (f ` UNIV)" by auto
+ ultimately show "finite ((image f) ` X)" using finite_subset by auto
+qed
+
+definition
+ tag1 :: "string set \<Rightarrow> string set \<Rightarrow> string \<Rightarrow> (string set \<times> string set)"
+where
+ "tag1 L\<^isub>1 L\<^isub>2 \<equiv> \<lambda>x. ((\<approx>L\<^isub>1) `` {x}, (\<approx>L\<^isub>2) `` {x})"
+
+lemma tag1_range_finite:
+ assumes finite1: "finite (UNIV // \<approx>L\<^isub>1)"
+ and finite2: "finite (UNIV // \<approx>L\<^isub>2)"
+ shows "finite (range (tag1 L\<^isub>1 L\<^isub>2))"
+proof -
+ have "finite (UNIV // \<approx>L\<^isub>1 \<times> UNIV // \<approx>L\<^isub>2)" using finite1 finite2 by auto
+ moreover
+ have "range (tag1 L\<^isub>1 L\<^isub>2) \<subseteq> (UNIV // \<approx>L\<^isub>1) \<times> (UNIV // \<approx>L\<^isub>2)"
+ unfolding tag1_def quotient_def by auto
+ ultimately show "finite (range (tag1 L\<^isub>1 L\<^isub>2))"
+ using finite_subset by blast
+qed
+
+lemma tag1_inj:
+ "tag1 L\<^isub>1 L\<^isub>2 x = tag1 L\<^isub>1 L\<^isub>2 y \<Longrightarrow> x \<approx>(L\<^isub>1 \<union> L\<^isub>2) y"
+unfolding tag1_def Image_def str_eq_rel_def str_eq_def
+by auto
+
+lemma quot_alt_cu:
+ fixes L\<^isub>1 L\<^isub>2::"string set"
+ assumes fin1: "finite (UNIV // \<approx>L\<^isub>1)"
+ and fin2: "finite (UNIV // \<approx>L\<^isub>2)"
+ shows "finite (UNIV // \<approx>(L\<^isub>1 \<union> L\<^isub>2))"
+proof -
+ have "finite (range (tag1 L\<^isub>1 L\<^isub>2))"
+ using fin1 fin2 tag1_range_finite by simp
+ then have "finite (image (tag1 L\<^isub>1 L\<^isub>2) ` (UNIV // \<approx>(L\<^isub>1 \<union> L\<^isub>2)))"
+ using finite_range_image by blast
+ moreover
+ have "\<And>x y. tag1 L\<^isub>1 L\<^isub>2 x = tag1 L\<^isub>1 L\<^isub>2 y \<Longrightarrow> x \<approx>(L\<^isub>1 \<union> L\<^isub>2) y"
+ using tag1_inj by simp
+ then have "inj_on (image (tag1 L\<^isub>1 L\<^isub>2)) (UNIV // \<approx>(L\<^isub>1 \<union> L\<^isub>2))"
+ using inj_image_lang by blast
+ ultimately
+ show "finite (UNIV // \<approx>(L\<^isub>1 \<union> L\<^isub>2))" by (rule finite_imageD)
+qed
+
+
+section {* finite \<Rightarrow> regular *}
+
+definition
+ transitions :: "string set \<Rightarrow> string set \<Rightarrow> rexp set" ("_\<Turnstile>\<Rightarrow>_")
+where
+ "Y \<Turnstile>\<Rightarrow> X \<equiv> {CHAR c | c. Y ;; {[c]} \<subseteq> X}"
+
+definition
+ transitions_rexp ("_ \<turnstile>\<rightarrow> _")
+where
+ "Y \<turnstile>\<rightarrow> X \<equiv> if [] \<in> X then folds ALT EMPTY (Y \<Turnstile>\<Rightarrow>X) else folds ALT NULL (Y \<Turnstile>\<Rightarrow>X)"
+
+definition
+ "rhs CS X \<equiv> if X = {[]} then {({[]}, EMPTY)} else {(Y, Y \<turnstile>\<rightarrow>X) | Y. Y \<in> CS}"
+
+definition
+ "rhs_sem CS X \<equiv> \<Union> {(Y;; \<lparr>r\<rparr>) | Y r . (Y, r) \<in> rhs CS X}"
+
+definition
+ "eqs CS \<equiv> (\<Union>X \<in> CS. {(X, rhs CS X)})"
+
+definition
+ "eqs_sem CS \<equiv> (\<Union>X \<in> CS. {(X, rhs_sem CS X)})"
+
+lemma [simp]:
+ shows "finite (Y \<Turnstile>\<Rightarrow> X)"
+unfolding transitions_def
+by auto
+
+
+lemma defined_by_str:
+ assumes "s \<in> X"
+ and "X \<in> UNIV // (\<approx>Lang)"
+ shows "X = (\<approx>Lang) `` {s}"
+using assms
+unfolding quotient_def Image_def
+unfolding str_eq_rel_def str_eq_def
+by auto
+
+lemma every_eqclass_has_transition:
+ assumes has_str: "s @ [c] \<in> X"
+ and in_CS: "X \<in> UNIV // (\<approx>Lang)"
+ obtains Y where "Y \<in> UNIV // (\<approx>Lang)" and "Y ;; {[c]} \<subseteq> X" and "s \<in> Y"
+proof -
+ def Y \<equiv> "(\<approx>Lang) `` {s}"
+ have "Y \<in> UNIV // (\<approx>Lang)"
+ unfolding Y_def quotient_def by auto
+ moreover
+ have "X = (\<approx>Lang) `` {s @ [c]}"
+ using has_str in_CS defined_by_str by blast
+ then have "Y ;; {[c]} \<subseteq> X"
+ unfolding Y_def Image_def Seq_def
+ unfolding str_eq_rel_def
+ by (auto) (simp add: str_eq_def)
+ moreover
+ have "s \<in> Y" unfolding Y_def
+ unfolding Image_def str_eq_rel_def str_eq_def by simp
+ (*moreover
+ have "True" by simp FIXME *)
+ ultimately show thesis by (blast intro: that)
+qed
+
+lemma test:
+ assumes "[] \<in> X"
+ shows "[] \<in> \<lparr>Y \<turnstile>\<rightarrow> X\<rparr>"
+using assms
+by (simp add: transitions_rexp_def)
+
+lemma rhs_sem:
+ assumes "X \<in> (UNIV // (\<approx>Lang))"
+ shows "X \<subseteq> rhs_sem (UNIV // (\<approx>Lang)) X"
+apply(case_tac "X = {[]}")
+apply(simp)
+apply(simp add: rhs_sem_def rhs_def Seq_def)
+apply(rule subsetI)
+apply(case_tac "x = []")
+apply(simp add: rhs_sem_def rhs_def)
+apply(rule_tac x = "X" in exI)
+apply(simp)
+apply(rule_tac x = "X" in exI)
+apply(simp add: assms)
+apply(simp add: transitions_rexp_def)
+oops
+
+
+(*
+fun
+ power :: "string \<Rightarrow> nat \<Rightarrow> string" (infixr "\<Up>" 100)
+where
+ "s \<Up> 0 = s"
+| "s \<Up> (Suc n) = s @ (s \<Up> n)"
+
+definition
+ "Lone = {(''0'' \<Up> n) @ (''1'' \<Up> n) | n. True }"
+
+lemma
+ "infinite (UNIV // (\<approx>Lone))"
+unfolding infinite_iff_countable_subset
+apply(rule_tac x="\<lambda>n. {(''0'' \<Up> n) @ (''1'' \<Up> i) | i. i \<in> {..n} }" in exI)
+apply(auto)
+prefer 2
+unfolding Lone_def
+unfolding quotient_def
+unfolding Image_def
+apply(simp)
+unfolding str_eq_rel_def
+unfolding str_eq_def
+apply(auto)
+apply(rule_tac x="''0'' \<Up> n" in exI)
+apply(auto)
+unfolding infinite_nat_iff_unbounded
+unfolding Lone_def
+*)
+
+
+
+text {* Derivatives *}
+
+definition
+ DERS :: "string \<Rightarrow> string set \<Rightarrow> string set"
+where
+ "DERS s L \<equiv> {s'. s @ s' \<in> L}"
+
+lemma
+ shows "x \<approx>L y \<longleftrightarrow> DERS x L = DERS y L"
+unfolding DERS_def str_eq_def
+by auto
\ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Attic/old/Myhill_1.thy Mon Jul 25 13:33:38 2011 +0000
@@ -0,0 +1,776 @@
+theory Myhill_1
+imports Regular
+ "~~/src/HOL/Library/While_Combinator"
+begin
+
+section {* Direction @{text "finite partition \<Rightarrow> regular language"} *}
+
+lemma Pair_Collect[simp]:
+ shows "(x, y) \<in> {(x, y). P x y} \<longleftrightarrow> P x y"
+by simp
+
+text {* Myhill-Nerode relation *}
+
+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)}"
+
+definition
+ finals :: "lang \<Rightarrow> lang set"
+where
+ "finals A \<equiv> {\<approx>A `` {s} | s . s \<in> A}"
+
+lemma lang_is_union_of_finals:
+ shows "A = \<Union> finals A"
+unfolding finals_def
+unfolding Image_def
+unfolding str_eq_rel_def
+by (auto) (metis append_Nil2)
+
+lemma finals_in_partitions:
+ shows "finals A \<subseteq> (UNIV // \<approx>A)"
+unfolding finals_def quotient_def
+by auto
+
+section {* Equational systems *}
+
+text {* The two kinds of terms in the rhs of equations. *}
+
+datatype trm =
+ Lam "rexp" (* Lambda-marker *)
+ | Trn "lang" "rexp" (* Transition *)
+
+fun
+ L_trm::"trm \<Rightarrow> lang"
+where
+ "L_trm (Lam r) = L_rexp r"
+| "L_trm (Trn X r) = X \<cdot> L_rexp r"
+
+fun
+ L_rhs::"trm set \<Rightarrow> lang"
+where
+ "L_rhs rhs = \<Union> (L_trm ` rhs)"
+
+lemma L_rhs_set:
+ shows "L_rhs {Trn X r | r. P r} = \<Union>{L_trm (Trn X r) | r. P r}"
+by (auto)
+
+lemma L_rhs_union_distrib:
+ fixes A B::"trm set"
+ shows "L_rhs A \<union> L_rhs B = L_rhs (A \<union> B)"
+by simp
+
+
+text {* Transitions between equivalence classes *}
+
+definition
+ transition :: "lang \<Rightarrow> char \<Rightarrow> lang \<Rightarrow> bool" ("_ \<Turnstile>_\<Rightarrow>_" [100,100,100] 100)
+where
+ "Y \<Turnstile>c\<Rightarrow> X \<equiv> Y \<cdot> {[c]} \<subseteq> X"
+
+text {* Initial equational system *}
+
+definition
+ "Init_rhs CS X \<equiv>
+ if ([] \<in> X) then
+ {Lam EMPTY} \<union> {Trn Y (CHAR c) | Y c. Y \<in> CS \<and> Y \<Turnstile>c\<Rightarrow> X}
+ else
+ {Trn Y (CHAR c)| Y c. Y \<in> CS \<and> Y \<Turnstile>c\<Rightarrow> X}"
+
+definition
+ "Init CS \<equiv> {(X, Init_rhs CS X) | X. X \<in> CS}"
+
+
+section {* Arden Operation on equations *}
+
+fun
+ Append_rexp :: "rexp \<Rightarrow> trm \<Rightarrow> trm"
+where
+ "Append_rexp r (Lam rexp) = Lam (SEQ rexp r)"
+| "Append_rexp r (Trn X rexp) = Trn X (SEQ rexp r)"
+
+
+definition
+ "Append_rexp_rhs rhs rexp \<equiv> (Append_rexp rexp) ` rhs"
+
+definition
+ "Arden X rhs \<equiv>
+ Append_rexp_rhs (rhs - {Trn X r | r. Trn X r \<in> rhs}) (STAR (\<Uplus> {r. Trn X r \<in> rhs}))"
+
+
+section {* Substitution Operation on equations *}
+
+definition
+ "Subst rhs X xrhs \<equiv>
+ (rhs - {Trn X r | r. Trn X r \<in> rhs}) \<union> (Append_rexp_rhs xrhs (\<Uplus> {r. Trn X r \<in> rhs}))"
+
+definition
+ Subst_all :: "(lang \<times> trm set) set \<Rightarrow> lang \<Rightarrow> trm set \<Rightarrow> (lang \<times> trm set) set"
+where
+ "Subst_all ES X xrhs \<equiv> {(Y, Subst yrhs X xrhs) | Y yrhs. (Y, yrhs) \<in> ES}"
+
+definition
+ "Remove ES X xrhs \<equiv>
+ Subst_all (ES - {(X, xrhs)}) X (Arden X xrhs)"
+
+
+section {* While-combinator *}
+
+definition
+ "Iter X ES \<equiv> (let (Y, yrhs) = SOME (Y, yrhs). (Y, yrhs) \<in> ES \<and> X \<noteq> Y
+ in Remove ES Y yrhs)"
+
+lemma IterI2:
+ assumes "(Y, yrhs) \<in> ES"
+ and "X \<noteq> Y"
+ and "\<And>Y yrhs. \<lbrakk>(Y, yrhs) \<in> ES; X \<noteq> Y\<rbrakk> \<Longrightarrow> Q (Remove ES Y yrhs)"
+ shows "Q (Iter X ES)"
+unfolding Iter_def using assms
+by (rule_tac a="(Y, yrhs)" in someI2) (auto)
+
+abbreviation
+ "Cond ES \<equiv> card ES \<noteq> 1"
+
+definition
+ "Solve X ES \<equiv> while Cond (Iter X) ES"
+
+
+section {* Invariants *}
+
+definition
+ "distinctness ES \<equiv>
+ \<forall> X rhs rhs'. (X, rhs) \<in> ES \<and> (X, rhs') \<in> ES \<longrightarrow> rhs = rhs'"
+
+definition
+ "soundness ES \<equiv> \<forall>(X, rhs) \<in> ES. X = L_rhs rhs"
+
+definition
+ "ardenable rhs \<equiv> (\<forall> Y r. Trn Y r \<in> rhs \<longrightarrow> [] \<notin> L_rexp r)"
+
+definition
+ "ardenable_all ES \<equiv> \<forall>(X, rhs) \<in> ES. ardenable rhs"
+
+definition
+ "finite_rhs ES \<equiv> \<forall>(X, rhs) \<in> ES. finite rhs"
+
+lemma finite_rhs_def2:
+ "finite_rhs ES = (\<forall> X rhs. (X, rhs) \<in> ES \<longrightarrow> finite rhs)"
+unfolding finite_rhs_def by auto
+
+definition
+ "rhss rhs \<equiv> {X | X r. Trn X r \<in> rhs}"
+
+definition
+ "lhss ES \<equiv> {Y | Y yrhs. (Y, yrhs) \<in> ES}"
+
+definition
+ "validity ES \<equiv> \<forall>(X, rhs) \<in> ES. rhss rhs \<subseteq> lhss ES"
+
+lemma rhss_union_distrib:
+ shows "rhss (A \<union> B) = rhss A \<union> rhss B"
+by (auto simp add: rhss_def)
+
+lemma lhss_union_distrib:
+ shows "lhss (A \<union> B) = lhss A \<union> lhss B"
+by (auto simp add: lhss_def)
+
+
+definition
+ "invariant ES \<equiv> finite ES
+ \<and> finite_rhs ES
+ \<and> soundness ES
+ \<and> distinctness ES
+ \<and> ardenable_all ES
+ \<and> validity ES"
+
+
+lemma invariantI:
+ assumes "soundness ES" "finite ES" "distinctness ES" "ardenable_all ES"
+ "finite_rhs ES" "validity ES"
+ shows "invariant ES"
+using assms by (simp add: invariant_def)
+
+
+subsection {* The proof of this direction *}
+
+lemma finite_Trn:
+ assumes fin: "finite rhs"
+ shows "finite {r. Trn Y r \<in> rhs}"
+proof -
+ have "finite {Trn Y r | Y r. Trn Y r \<in> rhs}"
+ by (rule rev_finite_subset[OF fin]) (auto)
+ then have "finite ((\<lambda>(Y, r). Trn Y r) ` {(Y, r) | Y r. Trn Y r \<in> rhs})"
+ by (simp add: image_Collect)
+ then have "finite {(Y, r) | Y r. Trn Y r \<in> rhs}"
+ by (erule_tac finite_imageD) (simp add: inj_on_def)
+ then show "finite {r. Trn Y r \<in> rhs}"
+ by (erule_tac f="snd" in finite_surj) (auto simp add: image_def)
+qed
+
+lemma finite_Lam:
+ assumes fin: "finite rhs"
+ shows "finite {r. Lam r \<in> rhs}"
+proof -
+ have "finite {Lam r | r. Lam r \<in> rhs}"
+ by (rule rev_finite_subset[OF fin]) (auto)
+ then show "finite {r. Lam r \<in> rhs}"
+ apply(simp add: image_Collect[symmetric])
+ apply(erule finite_imageD)
+ apply(auto simp add: inj_on_def)
+ done
+qed
+
+lemma trm_soundness:
+ assumes finite:"finite rhs"
+ shows "L_rhs ({Trn X r| r. Trn X r \<in> rhs}) = X \<cdot> (L_rexp (\<Uplus>{r. Trn X r \<in> rhs}))"
+proof -
+ have "finite {r. Trn X r \<in> rhs}"
+ by (rule finite_Trn[OF finite])
+ then show "L_rhs ({Trn X r| r. Trn X r \<in> rhs}) = X \<cdot> (L_rexp (\<Uplus>{r. Trn X r \<in> rhs}))"
+ by (simp only: L_rhs_set L_trm.simps) (auto simp add: Seq_def)
+qed
+
+lemma lang_of_append_rexp:
+ "L_trm (Append_rexp r trm) = L_trm trm \<cdot> L_rexp r"
+by (induct rule: Append_rexp.induct)
+ (auto simp add: seq_assoc)
+
+lemma lang_of_append_rexp_rhs:
+ "L_rhs (Append_rexp_rhs rhs r) = L_rhs rhs \<cdot> L_rexp r"
+unfolding Append_rexp_rhs_def
+by (auto simp add: Seq_def lang_of_append_rexp)
+
+
+subsubsection {* Intial Equational System *}
+
+lemma defined_by_str:
+ assumes "s \<in> X" "X \<in> UNIV // \<approx>A"
+ shows "X = \<approx>A `` {s}"
+using assms
+unfolding quotient_def Image_def str_eq_rel_def
+by auto
+
+lemma every_eqclass_has_transition:
+ assumes has_str: "s @ [c] \<in> X"
+ and in_CS: "X \<in> UNIV // \<approx>A"
+ obtains Y where "Y \<in> UNIV // \<approx>A" and "Y \<cdot> {[c]} \<subseteq> X" and "s \<in> Y"
+proof -
+ def Y \<equiv> "\<approx>A `` {s}"
+ have "Y \<in> UNIV // \<approx>A"
+ unfolding Y_def quotient_def by auto
+ moreover
+ have "X = \<approx>A `` {s @ [c]}"
+ using has_str in_CS defined_by_str by blast
+ then have "Y \<cdot> {[c]} \<subseteq> X"
+ unfolding Y_def Image_def Seq_def
+ unfolding str_eq_rel_def
+ by clarsimp
+ moreover
+ have "s \<in> Y" unfolding Y_def
+ unfolding Image_def str_eq_rel_def by simp
+ ultimately show thesis using that by blast
+qed
+
+lemma l_eq_r_in_eqs:
+ assumes X_in_eqs: "(X, rhs) \<in> Init (UNIV // \<approx>A)"
+ shows "X = L_rhs rhs"
+proof
+ show "X \<subseteq> L_rhs rhs"
+ proof
+ fix x
+ assume in_X: "x \<in> X"
+ { assume empty: "x = []"
+ then have "x \<in> L_rhs rhs" using X_in_eqs in_X
+ unfolding Init_def Init_rhs_def
+ by auto
+ }
+ moreover
+ { assume not_empty: "x \<noteq> []"
+ then obtain s c where decom: "x = s @ [c]"
+ using rev_cases by blast
+ have "X \<in> UNIV // \<approx>A" using X_in_eqs unfolding Init_def by auto
+ then obtain Y where "Y \<in> UNIV // \<approx>A" "Y \<cdot> {[c]} \<subseteq> X" "s \<in> Y"
+ using decom in_X every_eqclass_has_transition by blast
+ then have "x \<in> L_rhs {Trn Y (CHAR c)| Y c. Y \<in> UNIV // \<approx>A \<and> Y \<Turnstile>c\<Rightarrow> X}"
+ unfolding transition_def
+ using decom by (force simp add: Seq_def)
+ then have "x \<in> L_rhs rhs" using X_in_eqs in_X
+ unfolding Init_def Init_rhs_def by simp
+ }
+ ultimately show "x \<in> L_rhs rhs" by blast
+ qed
+next
+ show "L_rhs rhs \<subseteq> X" using X_in_eqs
+ unfolding Init_def Init_rhs_def transition_def
+ by auto
+qed
+
+lemma test:
+ assumes X_in_eqs: "(X, rhs) \<in> Init (UNIV // \<approx>A)"
+ shows "X = \<Union> (L_trm ` rhs)"
+using assms l_eq_r_in_eqs by (simp)
+
+lemma finite_Init_rhs:
+ assumes finite: "finite CS"
+ shows "finite (Init_rhs CS X)"
+proof-
+ def S \<equiv> "{(Y, c)| Y c. Y \<in> CS \<and> Y \<cdot> {[c]} \<subseteq> X}"
+ def h \<equiv> "\<lambda> (Y, c). Trn Y (CHAR c)"
+ have "finite (CS \<times> (UNIV::char set))" using finite by auto
+ then have "finite S" using S_def
+ by (rule_tac B = "CS \<times> UNIV" in finite_subset) (auto)
+ moreover have "{Trn Y (CHAR c) |Y c. Y \<in> CS \<and> Y \<cdot> {[c]} \<subseteq> X} = h ` S"
+ unfolding S_def h_def image_def by auto
+ ultimately
+ have "finite {Trn Y (CHAR c) |Y c. Y \<in> CS \<and> Y \<cdot> {[c]} \<subseteq> X}" by auto
+ then show "finite (Init_rhs CS X)" unfolding Init_rhs_def transition_def by simp
+qed
+
+lemma Init_ES_satisfies_invariant:
+ assumes finite_CS: "finite (UNIV // \<approx>A)"
+ shows "invariant (Init (UNIV // \<approx>A))"
+proof (rule invariantI)
+ show "soundness (Init (UNIV // \<approx>A))"
+ unfolding soundness_def
+ using l_eq_r_in_eqs by auto
+ show "finite (Init (UNIV // \<approx>A))" using finite_CS
+ unfolding Init_def by simp
+ show "distinctness (Init (UNIV // \<approx>A))"
+ unfolding distinctness_def Init_def by simp
+ show "ardenable_all (Init (UNIV // \<approx>A))"
+ unfolding ardenable_all_def Init_def Init_rhs_def ardenable_def
+ by auto
+ show "finite_rhs (Init (UNIV // \<approx>A))"
+ using finite_Init_rhs[OF finite_CS]
+ unfolding finite_rhs_def Init_def by auto
+ show "validity (Init (UNIV // \<approx>A))"
+ unfolding validity_def Init_def Init_rhs_def rhss_def lhss_def
+ by auto
+qed
+
+subsubsection {* Interation step *}
+
+lemma Arden_keeps_eq:
+ assumes l_eq_r: "X = L_rhs rhs"
+ and not_empty: "ardenable rhs"
+ and finite: "finite rhs"
+ shows "X = L_rhs (Arden X rhs)"
+proof -
+ def A \<equiv> "L_rexp (\<Uplus>{r. Trn X r \<in> rhs})"
+ def b \<equiv> "{Trn X r | r. Trn X r \<in> rhs}"
+ def B \<equiv> "L_rhs (rhs - b)"
+ have not_empty2: "[] \<notin> A"
+ using finite_Trn[OF finite] not_empty
+ unfolding A_def ardenable_def by simp
+ have "X = L_rhs rhs" using l_eq_r by simp
+ also have "\<dots> = L_rhs (b \<union> (rhs - b))" unfolding b_def by auto
+ also have "\<dots> = L_rhs b \<union> B" unfolding B_def by (simp only: L_rhs_union_distrib)
+ also have "\<dots> = X \<cdot> A \<union> B"
+ unfolding b_def
+ unfolding trm_soundness[OF finite]
+ unfolding A_def
+ by blast
+ finally have "X = X \<cdot> A \<union> B" .
+ then have "X = B \<cdot> A\<star>"
+ by (simp add: arden[OF not_empty2])
+ also have "\<dots> = L_rhs (Arden X rhs)"
+ unfolding Arden_def A_def B_def b_def
+ by (simp only: lang_of_append_rexp_rhs L_rexp.simps)
+ finally show "X = L_rhs (Arden X rhs)" by simp
+qed
+
+lemma Append_keeps_finite:
+ "finite rhs \<Longrightarrow> finite (Append_rexp_rhs rhs r)"
+by (auto simp:Append_rexp_rhs_def)
+
+lemma Arden_keeps_finite:
+ "finite rhs \<Longrightarrow> finite (Arden X rhs)"
+by (auto simp:Arden_def Append_keeps_finite)
+
+lemma Append_keeps_nonempty:
+ "ardenable rhs \<Longrightarrow> ardenable (Append_rexp_rhs rhs r)"
+apply (auto simp:ardenable_def Append_rexp_rhs_def)
+by (case_tac x, auto simp:Seq_def)
+
+lemma nonempty_set_sub:
+ "ardenable rhs \<Longrightarrow> ardenable (rhs - A)"
+by (auto simp:ardenable_def)
+
+lemma nonempty_set_union:
+ "\<lbrakk>ardenable rhs; ardenable rhs'\<rbrakk> \<Longrightarrow> ardenable (rhs \<union> rhs')"
+by (auto simp:ardenable_def)
+
+lemma Arden_keeps_nonempty:
+ "ardenable rhs \<Longrightarrow> ardenable (Arden X rhs)"
+by (simp only:Arden_def Append_keeps_nonempty nonempty_set_sub)
+
+
+lemma Subst_keeps_nonempty:
+ "\<lbrakk>ardenable rhs; ardenable xrhs\<rbrakk> \<Longrightarrow> ardenable (Subst rhs X xrhs)"
+by (simp only: Subst_def Append_keeps_nonempty nonempty_set_union nonempty_set_sub)
+
+lemma Subst_keeps_eq:
+ assumes substor: "X = L_rhs xrhs"
+ and finite: "finite rhs"
+ shows "L_rhs (Subst rhs X xrhs) = L_rhs rhs" (is "?Left = ?Right")
+proof-
+ def A \<equiv> "L_rhs (rhs - {Trn X r | r. Trn X r \<in> rhs})"
+ have "?Left = A \<union> L_rhs (Append_rexp_rhs xrhs (\<Uplus>{r. Trn X r \<in> rhs}))"
+ unfolding Subst_def
+ unfolding L_rhs_union_distrib[symmetric]
+ by (simp add: A_def)
+ moreover have "?Right = A \<union> L_rhs {Trn X r | r. Trn X r \<in> rhs}"
+ proof-
+ have "rhs = (rhs - {Trn X r | r. Trn X r \<in> rhs}) \<union> ({Trn X r | r. Trn X r \<in> rhs})" by auto
+ thus ?thesis
+ unfolding A_def
+ unfolding L_rhs_union_distrib
+ by simp
+ qed
+ moreover have "L_rhs (Append_rexp_rhs xrhs (\<Uplus>{r. Trn X r \<in> rhs})) = L_rhs {Trn X r | r. Trn X r \<in> rhs}"
+ using finite substor by (simp only: lang_of_append_rexp_rhs trm_soundness)
+ ultimately show ?thesis by simp
+qed
+
+lemma Subst_keeps_finite_rhs:
+ "\<lbrakk>finite rhs; finite yrhs\<rbrakk> \<Longrightarrow> finite (Subst rhs Y yrhs)"
+by (auto simp: Subst_def Append_keeps_finite)
+
+lemma Subst_all_keeps_finite:
+ assumes finite: "finite ES"
+ shows "finite (Subst_all ES Y yrhs)"
+proof -
+ def eqns \<equiv> "{(X::lang, rhs) |X rhs. (X, rhs) \<in> ES}"
+ def h \<equiv> "\<lambda>(X::lang, rhs). (X, Subst rhs Y yrhs)"
+ have "finite (h ` eqns)" using finite h_def eqns_def by auto
+ moreover
+ have "Subst_all ES Y yrhs = h ` eqns" unfolding h_def eqns_def Subst_all_def by auto
+ ultimately
+ show "finite (Subst_all ES Y yrhs)" by simp
+qed
+
+lemma Subst_all_keeps_finite_rhs:
+ "\<lbrakk>finite_rhs ES; finite yrhs\<rbrakk> \<Longrightarrow> finite_rhs (Subst_all ES Y yrhs)"
+by (auto intro:Subst_keeps_finite_rhs simp add:Subst_all_def finite_rhs_def)
+
+lemma append_rhs_keeps_cls:
+ "rhss (Append_rexp_rhs rhs r) = rhss rhs"
+apply (auto simp:rhss_def Append_rexp_rhs_def)
+apply (case_tac xa, auto simp:image_def)
+by (rule_tac x = "SEQ ra r" in exI, rule_tac x = "Trn x ra" in bexI, simp+)
+
+lemma Arden_removes_cl:
+ "rhss (Arden Y yrhs) = rhss yrhs - {Y}"
+apply (simp add:Arden_def append_rhs_keeps_cls)
+by (auto simp:rhss_def)
+
+lemma lhss_keeps_cls:
+ "lhss (Subst_all ES Y yrhs) = lhss ES"
+by (auto simp:lhss_def Subst_all_def)
+
+lemma Subst_updates_cls:
+ "X \<notin> rhss xrhs \<Longrightarrow>
+ rhss (Subst rhs X xrhs) = rhss rhs \<union> rhss xrhs - {X}"
+apply (simp only:Subst_def append_rhs_keeps_cls rhss_union_distrib)
+by (auto simp:rhss_def)
+
+lemma Subst_all_keeps_validity:
+ assumes sc: "validity (ES \<union> {(Y, yrhs)})" (is "validity ?A")
+ shows "validity (Subst_all ES Y (Arden Y yrhs))" (is "validity ?B")
+proof -
+ { fix X xrhs'
+ assume "(X, xrhs') \<in> ?B"
+ then obtain xrhs
+ where xrhs_xrhs': "xrhs' = Subst xrhs Y (Arden Y yrhs)"
+ and X_in: "(X, xrhs) \<in> ES" by (simp add:Subst_all_def, blast)
+ have "rhss xrhs' \<subseteq> lhss ?B"
+ proof-
+ have "lhss ?B = lhss ES" by (auto simp add:lhss_def Subst_all_def)
+ moreover have "rhss xrhs' \<subseteq> lhss ES"
+ proof-
+ have "rhss xrhs' \<subseteq> rhss xrhs \<union> rhss (Arden Y yrhs) - {Y}"
+ proof-
+ have "Y \<notin> rhss (Arden Y yrhs)"
+ using Arden_removes_cl by simp
+ thus ?thesis using xrhs_xrhs' by (auto simp:Subst_updates_cls)
+ qed
+ moreover have "rhss xrhs \<subseteq> lhss ES \<union> {Y}" using X_in sc
+ apply (simp only:validity_def lhss_union_distrib)
+ by (drule_tac x = "(X, xrhs)" in bspec, auto simp:lhss_def)
+ moreover have "rhss (Arden Y yrhs) \<subseteq> lhss ES \<union> {Y}"
+ using sc
+ by (auto simp add:Arden_removes_cl validity_def lhss_def)
+ ultimately show ?thesis by auto
+ qed
+ ultimately show ?thesis by simp
+ qed
+ } thus ?thesis by (auto simp only:Subst_all_def validity_def)
+qed
+
+lemma Subst_all_satisfies_invariant:
+ assumes invariant_ES: "invariant (ES \<union> {(Y, yrhs)})"
+ shows "invariant (Subst_all ES Y (Arden Y yrhs))"
+proof (rule invariantI)
+ have Y_eq_yrhs: "Y = L_rhs yrhs"
+ using invariant_ES by (simp only:invariant_def soundness_def, blast)
+ have finite_yrhs: "finite yrhs"
+ using invariant_ES by (auto simp:invariant_def finite_rhs_def)
+ have nonempty_yrhs: "ardenable yrhs"
+ using invariant_ES by (auto simp:invariant_def ardenable_all_def)
+ show "soundness (Subst_all ES Y (Arden Y yrhs))"
+ proof -
+ have "Y = L_rhs (Arden Y yrhs)"
+ using Y_eq_yrhs invariant_ES finite_yrhs
+ using finite_Trn[OF finite_yrhs]
+ apply(rule_tac Arden_keeps_eq)
+ apply(simp_all)
+ unfolding invariant_def ardenable_all_def ardenable_def
+ apply(auto)
+ done
+ thus ?thesis using invariant_ES
+ unfolding invariant_def finite_rhs_def2 soundness_def Subst_all_def
+ by (auto simp add: Subst_keeps_eq simp del: L_rhs.simps)
+ qed
+ show "finite (Subst_all ES Y (Arden Y yrhs))"
+ using invariant_ES by (simp add:invariant_def Subst_all_keeps_finite)
+ show "distinctness (Subst_all ES Y (Arden Y yrhs))"
+ using invariant_ES
+ unfolding distinctness_def Subst_all_def invariant_def by auto
+ show "ardenable_all (Subst_all ES Y (Arden Y yrhs))"
+ proof -
+ { fix X rhs
+ assume "(X, rhs) \<in> ES"
+ hence "ardenable rhs" using invariant_ES
+ by (auto simp add:invariant_def ardenable_all_def)
+ with nonempty_yrhs
+ have "ardenable (Subst rhs Y (Arden Y yrhs))"
+ by (simp add:nonempty_yrhs
+ Subst_keeps_nonempty Arden_keeps_nonempty)
+ } thus ?thesis by (auto simp add:ardenable_all_def Subst_all_def)
+ qed
+ show "finite_rhs (Subst_all ES Y (Arden Y yrhs))"
+ proof-
+ have "finite_rhs ES" using invariant_ES
+ by (simp add:invariant_def finite_rhs_def)
+ moreover have "finite (Arden Y yrhs)"
+ proof -
+ have "finite yrhs" using invariant_ES
+ by (auto simp:invariant_def finite_rhs_def)
+ thus ?thesis using Arden_keeps_finite by simp
+ qed
+ ultimately show ?thesis
+ by (simp add:Subst_all_keeps_finite_rhs)
+ qed
+ show "validity (Subst_all ES Y (Arden Y yrhs))"
+ using invariant_ES Subst_all_keeps_validity by (simp add:invariant_def)
+qed
+
+lemma Remove_in_card_measure:
+ assumes finite: "finite ES"
+ and in_ES: "(X, rhs) \<in> ES"
+ shows "(Remove ES X rhs, ES) \<in> measure card"
+proof -
+ def f \<equiv> "\<lambda> x. ((fst x)::lang, Subst (snd x) X (Arden X rhs))"
+ def ES' \<equiv> "ES - {(X, rhs)}"
+ have "Subst_all ES' X (Arden X rhs) = f ` ES'"
+ apply (auto simp: Subst_all_def f_def image_def)
+ by (rule_tac x = "(Y, yrhs)" in bexI, simp+)
+ then have "card (Subst_all ES' X (Arden X rhs)) \<le> card ES'"
+ unfolding ES'_def using finite by (auto intro: card_image_le)
+ also have "\<dots> < card ES" unfolding ES'_def
+ using in_ES finite by (rule_tac card_Diff1_less)
+ finally show "(Remove ES X rhs, ES) \<in> measure card"
+ unfolding Remove_def ES'_def by simp
+qed
+
+
+lemma Subst_all_cls_remains:
+ "(X, xrhs) \<in> ES \<Longrightarrow> \<exists> xrhs'. (X, xrhs') \<in> (Subst_all ES Y yrhs)"
+by (auto simp: Subst_all_def)
+
+lemma card_noteq_1_has_more:
+ assumes card:"Cond ES"
+ and e_in: "(X, xrhs) \<in> ES"
+ and finite: "finite ES"
+ shows "\<exists>(Y, yrhs) \<in> ES. (X, xrhs) \<noteq> (Y, yrhs)"
+proof-
+ have "card ES > 1" using card e_in finite
+ by (cases "card ES") (auto)
+ then have "card (ES - {(X, xrhs)}) > 0"
+ using finite e_in by auto
+ then have "(ES - {(X, xrhs)}) \<noteq> {}" using finite by (rule_tac notI, simp)
+ then show "\<exists>(Y, yrhs) \<in> ES. (X, xrhs) \<noteq> (Y, yrhs)"
+ by auto
+qed
+
+lemma iteration_step_measure:
+ assumes Inv_ES: "invariant ES"
+ and X_in_ES: "(X, xrhs) \<in> ES"
+ and Cnd: "Cond ES "
+ shows "(Iter X ES, ES) \<in> measure card"
+proof -
+ have fin: "finite ES" using Inv_ES unfolding invariant_def by simp
+ then obtain Y yrhs
+ where Y_in_ES: "(Y, yrhs) \<in> ES" and not_eq: "(X, xrhs) \<noteq> (Y, yrhs)"
+ using Cnd X_in_ES by (drule_tac card_noteq_1_has_more) (auto)
+ then have "(Y, yrhs) \<in> ES " "X \<noteq> Y"
+ using X_in_ES Inv_ES unfolding invariant_def distinctness_def
+ by auto
+ then show "(Iter X ES, ES) \<in> measure card"
+ apply(rule IterI2)
+ apply(rule Remove_in_card_measure)
+ apply(simp_all add: fin)
+ done
+qed
+
+lemma iteration_step_invariant:
+ assumes Inv_ES: "invariant ES"
+ and X_in_ES: "(X, xrhs) \<in> ES"
+ and Cnd: "Cond ES"
+ shows "invariant (Iter X ES)"
+proof -
+ have finite_ES: "finite ES" using Inv_ES by (simp add: invariant_def)
+ then obtain Y yrhs
+ where Y_in_ES: "(Y, yrhs) \<in> ES" and not_eq: "(X, xrhs) \<noteq> (Y, yrhs)"
+ using Cnd X_in_ES by (drule_tac card_noteq_1_has_more) (auto)
+ then have "(Y, yrhs) \<in> ES" "X \<noteq> Y"
+ using X_in_ES Inv_ES unfolding invariant_def distinctness_def
+ by auto
+ then show "invariant (Iter X ES)"
+ proof(rule IterI2)
+ fix Y yrhs
+ assume h: "(Y, yrhs) \<in> ES" "X \<noteq> Y"
+ then have "ES - {(Y, yrhs)} \<union> {(Y, yrhs)} = ES" by auto
+ then show "invariant (Remove ES Y yrhs)" unfolding Remove_def
+ using Inv_ES
+ by (rule_tac Subst_all_satisfies_invariant) (simp)
+ qed
+qed
+
+lemma iteration_step_ex:
+ assumes Inv_ES: "invariant ES"
+ and X_in_ES: "(X, xrhs) \<in> ES"
+ and Cnd: "Cond ES"
+ shows "\<exists>xrhs'. (X, xrhs') \<in> (Iter X ES)"
+proof -
+ have finite_ES: "finite ES" using Inv_ES by (simp add: invariant_def)
+ then obtain Y yrhs
+ where "(Y, yrhs) \<in> ES" "(X, xrhs) \<noteq> (Y, yrhs)"
+ using Cnd X_in_ES by (drule_tac card_noteq_1_has_more) (auto)
+ then have "(Y, yrhs) \<in> ES " "X \<noteq> Y"
+ using X_in_ES Inv_ES unfolding invariant_def distinctness_def
+ by auto
+ then show "\<exists>xrhs'. (X, xrhs') \<in> (Iter X ES)"
+ apply(rule IterI2)
+ unfolding Remove_def
+ apply(rule Subst_all_cls_remains)
+ using X_in_ES
+ apply(auto)
+ done
+qed
+
+
+subsubsection {* Conclusion of the proof *}
+
+lemma Solve:
+ assumes fin: "finite (UNIV // \<approx>A)"
+ and X_in: "X \<in> (UNIV // \<approx>A)"
+ shows "\<exists>rhs. Solve X (Init (UNIV // \<approx>A)) = {(X, rhs)} \<and> invariant {(X, rhs)}"
+proof -
+ def Inv \<equiv> "\<lambda>ES. invariant ES \<and> (\<exists>rhs. (X, rhs) \<in> ES)"
+ have "Inv (Init (UNIV // \<approx>A))" unfolding Inv_def
+ using fin X_in by (simp add: Init_ES_satisfies_invariant, simp add: Init_def)
+ moreover
+ { fix ES
+ assume inv: "Inv ES" and crd: "Cond ES"
+ then have "Inv (Iter X ES)"
+ unfolding Inv_def
+ by (auto simp add: iteration_step_invariant iteration_step_ex) }
+ moreover
+ { fix ES
+ assume inv: "Inv ES" and not_crd: "\<not>Cond ES"
+ from inv obtain rhs where "(X, rhs) \<in> ES" unfolding Inv_def by auto
+ moreover
+ from not_crd have "card ES = 1" by simp
+ ultimately
+ have "ES = {(X, rhs)}" by (auto simp add: card_Suc_eq)
+ then have "\<exists>rhs'. ES = {(X, rhs')} \<and> invariant {(X, rhs')}" using inv
+ unfolding Inv_def by auto }
+ moreover
+ have "wf (measure card)" by simp
+ moreover
+ { fix ES
+ assume inv: "Inv ES" and crd: "Cond ES"
+ then have "(Iter X ES, ES) \<in> measure card"
+ unfolding Inv_def
+ apply(clarify)
+ apply(rule_tac iteration_step_measure)
+ apply(auto)
+ done }
+ ultimately
+ show "\<exists>rhs. Solve X (Init (UNIV // \<approx>A)) = {(X, rhs)} \<and> invariant {(X, rhs)}"
+ unfolding Solve_def by (rule while_rule)
+qed
+
+lemma every_eqcl_has_reg:
+ assumes finite_CS: "finite (UNIV // \<approx>A)"
+ and X_in_CS: "X \<in> (UNIV // \<approx>A)"
+ shows "\<exists>r. X = L_rexp r"
+proof -
+ from finite_CS X_in_CS
+ obtain xrhs where Inv_ES: "invariant {(X, xrhs)}"
+ using Solve by metis
+
+ def A \<equiv> "Arden X xrhs"
+ have "rhss xrhs \<subseteq> {X}" using Inv_ES
+ unfolding validity_def invariant_def rhss_def lhss_def
+ by auto
+ then have "rhss A = {}" unfolding A_def
+ by (simp add: Arden_removes_cl)
+ then have eq: "{Lam r | r. Lam r \<in> A} = A" unfolding rhss_def
+ by (auto, case_tac x, auto)
+
+ have "finite A" using Inv_ES unfolding A_def invariant_def finite_rhs_def
+ using Arden_keeps_finite by auto
+ then have fin: "finite {r. Lam r \<in> A}" by (rule finite_Lam)
+
+ have "X = L_rhs xrhs" using Inv_ES unfolding invariant_def soundness_def
+ by simp
+ then have "X = L_rhs A" using Inv_ES
+ unfolding A_def invariant_def ardenable_all_def finite_rhs_def
+ by (rule_tac Arden_keeps_eq) (simp_all add: finite_Trn)
+ then have "X = L_rhs {Lam r | r. Lam r \<in> A}" using eq by simp
+ then have "X = L_rexp (\<Uplus>{r. Lam r \<in> A})" using fin by auto
+ then show "\<exists>r. X = L_rexp r" by blast
+qed
+
+lemma bchoice_finite_set:
+ assumes a: "\<forall>x \<in> S. \<exists>y. x = f y"
+ and b: "finite S"
+ shows "\<exists>ys. (\<Union> S) = \<Union>(f ` ys) \<and> finite ys"
+using bchoice[OF a] b
+apply(erule_tac exE)
+apply(rule_tac x="fa ` S" in exI)
+apply(auto)
+done
+
+theorem Myhill_Nerode1:
+ assumes finite_CS: "finite (UNIV // \<approx>A)"
+ shows "\<exists>r. A = L_rexp r"
+proof -
+ have fin: "finite (finals A)"
+ using finals_in_partitions finite_CS by (rule finite_subset)
+ have "\<forall>X \<in> (UNIV // \<approx>A). \<exists>r. X = L_rexp r"
+ using finite_CS every_eqcl_has_reg by blast
+ then have a: "\<forall>X \<in> finals A. \<exists>r. X = L_rexp r"
+ using finals_in_partitions by auto
+ then obtain rs::"rexp set" where "\<Union> (finals A) = \<Union>(L_rexp ` rs)" "finite rs"
+ using fin by (auto dest: bchoice_finite_set)
+ then have "A = L_rexp (\<Uplus>rs)"
+ unfolding lang_is_union_of_finals[symmetric] by simp
+ then show "\<exists>r. A = L_rexp r" by blast
+qed
+
+
+end
\ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Attic/old/Myhill_2.thy Mon Jul 25 13:33:38 2011 +0000
@@ -0,0 +1,473 @@
+theory Myhill_2
+ imports Myhill_1 Prefix_subtract
+ "~~/src/HOL/Library/List_Prefix"
+begin
+
+section {* Direction @{text "regular language \<Rightarrow> finite partition"} *}
+
+definition
+ str_eq :: "string \<Rightarrow> lang \<Rightarrow> string \<Rightarrow> bool" ("_ \<approx>_ _")
+where
+ "x \<approx>A y \<equiv> (x, y) \<in> (\<approx>A)"
+
+lemma str_eq_def2:
+ shows "\<approx>A = {(x, y) | x y. x \<approx>A y}"
+unfolding str_eq_def
+by simp
+
+definition
+ tag_eq_rel :: "(string \<Rightarrow> 'b) \<Rightarrow> (string \<times> string) set" ("=_=")
+where
+ "=tag= \<equiv> {(x, y). tag x = tag y}"
+
+lemma finite_eq_tag_rel:
+ assumes rng_fnt: "finite (range tag)"
+ shows "finite (UNIV // =tag=)"
+proof -
+ let "?f" = "\<lambda>X. tag ` X" and ?A = "(UNIV // =tag=)"
+ have "finite (?f ` ?A)"
+ proof -
+ have "range ?f \<subseteq> (Pow (range tag))" unfolding Pow_def by auto
+ moreover
+ have "finite (Pow (range tag))" using rng_fnt by simp
+ ultimately
+ have "finite (range ?f)" unfolding image_def by (blast intro: finite_subset)
+ moreover
+ have "?f ` ?A \<subseteq> range ?f" by auto
+ ultimately show "finite (?f ` ?A)" by (rule rev_finite_subset)
+ qed
+ moreover
+ have "inj_on ?f ?A"
+ proof -
+ { fix X Y
+ assume X_in: "X \<in> ?A"
+ and Y_in: "Y \<in> ?A"
+ and tag_eq: "?f X = ?f Y"
+ then obtain x y
+ where "x \<in> X" "y \<in> Y" "tag x = tag y"
+ unfolding quotient_def Image_def image_def tag_eq_rel_def
+ by (simp) (blast)
+ with X_in Y_in
+ have "X = Y"
+ unfolding quotient_def tag_eq_rel_def by auto
+ }
+ then show "inj_on ?f ?A" unfolding inj_on_def by auto
+ qed
+ ultimately show "finite (UNIV // =tag=)" by (rule finite_imageD)
+qed
+
+lemma refined_partition_finite:
+ assumes fnt: "finite (UNIV // R1)"
+ and refined: "R1 \<subseteq> R2"
+ and eq1: "equiv UNIV R1" and eq2: "equiv UNIV R2"
+ shows "finite (UNIV // R2)"
+proof -
+ let ?f = "\<lambda>X. {R1 `` {x} | x. x \<in> X}"
+ and ?A = "UNIV // R2" and ?B = "UNIV // R1"
+ have "?f ` ?A \<subseteq> Pow ?B"
+ unfolding image_def Pow_def quotient_def by auto
+ moreover
+ have "finite (Pow ?B)" using fnt by simp
+ ultimately
+ have "finite (?f ` ?A)" by (rule finite_subset)
+ moreover
+ have "inj_on ?f ?A"
+ proof -
+ { fix X Y
+ assume X_in: "X \<in> ?A" and Y_in: "Y \<in> ?A" and eq_f: "?f X = ?f Y"
+ from quotientE [OF X_in]
+ obtain x where "X = R2 `` {x}" by blast
+ with equiv_class_self[OF eq2] have x_in: "x \<in> X" by simp
+ then have "R1 ``{x} \<in> ?f X" by auto
+ with eq_f have "R1 `` {x} \<in> ?f Y" by simp
+ then obtain y
+ where y_in: "y \<in> Y" and eq_r1_xy: "R1 `` {x} = R1 `` {y}" by auto
+ with eq_equiv_class[OF _ eq1]
+ have "(x, y) \<in> R1" by blast
+ with refined have "(x, y) \<in> R2" by auto
+ with quotient_eqI [OF eq2 X_in Y_in x_in y_in]
+ have "X = Y" .
+ }
+ then show "inj_on ?f ?A" unfolding inj_on_def by blast
+ qed
+ ultimately show "finite (UNIV // R2)" by (rule finite_imageD)
+qed
+
+lemma tag_finite_imageD:
+ assumes rng_fnt: "finite (range tag)"
+ and same_tag_eqvt: "\<And>m n. tag m = tag n \<Longrightarrow> m \<approx>A n"
+ shows "finite (UNIV // \<approx>A)"
+proof (rule_tac refined_partition_finite [of "=tag="])
+ show "finite (UNIV // =tag=)" by (rule finite_eq_tag_rel[OF rng_fnt])
+next
+ from same_tag_eqvt
+ show "=tag= \<subseteq> \<approx>A" unfolding tag_eq_rel_def str_eq_def
+ by auto
+next
+ show "equiv UNIV =tag="
+ unfolding equiv_def tag_eq_rel_def refl_on_def sym_def trans_def
+ by auto
+next
+ show "equiv UNIV (\<approx>A)"
+ unfolding equiv_def str_eq_rel_def sym_def refl_on_def trans_def
+ by blast
+qed
+
+
+subsection {* The proof *}
+
+subsubsection {* The base case for @{const "NULL"} *}
+
+lemma quot_null_eq:
+ shows "UNIV // \<approx>{} = {UNIV}"
+unfolding quotient_def Image_def str_eq_rel_def by auto
+
+lemma quot_null_finiteI [intro]:
+ shows "finite (UNIV // \<approx>{})"
+unfolding quot_null_eq by simp
+
+
+subsubsection {* The base case for @{const "EMPTY"} *}
+
+lemma quot_empty_subset:
+ shows "UNIV // \<approx>{[]} \<subseteq> {{[]}, UNIV - {[]}}"
+proof
+ fix x
+ assume "x \<in> UNIV // \<approx>{[]}"
+ then obtain y where h: "x = {z. (y, z) \<in> \<approx>{[]}}"
+ unfolding quotient_def Image_def by blast
+ show "x \<in> {{[]}, UNIV - {[]}}"
+ proof (cases "y = []")
+ case True with h
+ have "x = {[]}" by (auto simp: str_eq_rel_def)
+ thus ?thesis by simp
+ next
+ case False with h
+ have "x = UNIV - {[]}" by (auto simp: str_eq_rel_def)
+ thus ?thesis by simp
+ qed
+qed
+
+lemma quot_empty_finiteI [intro]:
+ shows "finite (UNIV // \<approx>{[]})"
+by (rule finite_subset[OF quot_empty_subset]) (simp)
+
+
+subsubsection {* The base case for @{const "CHAR"} *}
+
+lemma quot_char_subset:
+ "UNIV // (\<approx>{[c]}) \<subseteq> {{[]},{[c]}, UNIV - {[], [c]}}"
+proof
+ fix x
+ assume "x \<in> UNIV // \<approx>{[c]}"
+ then obtain y where h: "x = {z. (y, z) \<in> \<approx>{[c]}}"
+ unfolding quotient_def Image_def by blast
+ show "x \<in> {{[]},{[c]}, UNIV - {[], [c]}}"
+ proof -
+ { assume "y = []" hence "x = {[]}" using h
+ by (auto simp:str_eq_rel_def) }
+ moreover
+ { assume "y = [c]" hence "x = {[c]}" using h
+ by (auto dest!:spec[where x = "[]"] simp:str_eq_rel_def) }
+ moreover
+ { assume "y \<noteq> []" and "y \<noteq> [c]"
+ hence "\<forall> z. (y @ z) \<noteq> [c]" by (case_tac y, auto)
+ moreover have "\<And> p. (p \<noteq> [] \<and> p \<noteq> [c]) = (\<forall> q. p @ q \<noteq> [c])"
+ by (case_tac p, auto)
+ ultimately have "x = UNIV - {[],[c]}" using h
+ by (auto simp add:str_eq_rel_def)
+ }
+ ultimately show ?thesis by blast
+ qed
+qed
+
+lemma quot_char_finiteI [intro]:
+ shows "finite (UNIV // \<approx>{[c]})"
+by (rule finite_subset[OF quot_char_subset]) (simp)
+
+
+subsubsection {* The inductive case for @{const ALT} *}
+
+definition
+ tag_str_ALT :: "lang \<Rightarrow> lang \<Rightarrow> string \<Rightarrow> (lang \<times> lang)"
+where
+ "tag_str_ALT A B \<equiv> (\<lambda>x. (\<approx>A `` {x}, \<approx>B `` {x}))"
+
+lemma quot_union_finiteI [intro]:
+ assumes finite1: "finite (UNIV // \<approx>A)"
+ and finite2: "finite (UNIV // \<approx>B)"
+ shows "finite (UNIV // \<approx>(A \<union> B))"
+proof (rule_tac tag = "tag_str_ALT A B" in tag_finite_imageD)
+ have "finite ((UNIV // \<approx>A) \<times> (UNIV // \<approx>B))"
+ using finite1 finite2 by auto
+ then show "finite (range (tag_str_ALT A B))"
+ unfolding tag_str_ALT_def quotient_def
+ by (rule rev_finite_subset) (auto)
+next
+ show "\<And>x y. tag_str_ALT A B x = tag_str_ALT A B y \<Longrightarrow> x \<approx>(A \<union> B) y"
+ unfolding tag_str_ALT_def
+ unfolding str_eq_def
+ unfolding str_eq_rel_def
+ by auto
+qed
+
+
+subsubsection {* The inductive case for @{text "SEQ"}*}
+
+definition
+ tag_str_SEQ :: "lang \<Rightarrow> lang \<Rightarrow> string \<Rightarrow> (lang \<times> lang set)"
+where
+ "tag_str_SEQ L1 L2 \<equiv>
+ (\<lambda>x. (\<approx>L1 `` {x}, {(\<approx>L2 `` {x - xa}) | xa. xa \<le> x \<and> xa \<in> L1}))"
+
+lemma Seq_in_cases:
+ assumes "x @ z \<in> A \<cdot> B"
+ shows "(\<exists> x' \<le> x. x' \<in> A \<and> (x - x') @ z \<in> B) \<or>
+ (\<exists> z' \<le> z. (x @ z') \<in> A \<and> (z - z') \<in> B)"
+using assms
+unfolding Seq_def prefix_def
+by (auto simp add: append_eq_append_conv2)
+
+lemma tag_str_SEQ_injI:
+ assumes eq_tag: "tag_str_SEQ A B x = tag_str_SEQ A B y"
+ shows "x \<approx>(A \<cdot> B) y"
+proof -
+ { fix x y z
+ assume xz_in_seq: "x @ z \<in> A \<cdot> B"
+ and tag_xy: "tag_str_SEQ A B x = tag_str_SEQ A B y"
+ have"y @ z \<in> A \<cdot> B"
+ proof -
+ { (* first case with x' in A and (x - x') @ z in B *)
+ fix x'
+ assume h1: "x' \<le> x" and h2: "x' \<in> A" and h3: "(x - x') @ z \<in> B"
+ obtain y'
+ where "y' \<le> y"
+ and "y' \<in> A"
+ and "(y - y') @ z \<in> B"
+ proof -
+ have "{\<approx>B `` {x - x'} |x'. x' \<le> x \<and> x' \<in> A} =
+ {\<approx>B `` {y - y'} |y'. y' \<le> y \<and> y' \<in> A}" (is "?Left = ?Right")
+ using tag_xy unfolding tag_str_SEQ_def by simp
+ moreover
+ have "\<approx>B `` {x - x'} \<in> ?Left" using h1 h2 by auto
+ ultimately
+ have "\<approx>B `` {x - x'} \<in> ?Right" by simp
+ then obtain y'
+ where eq_xy': "\<approx>B `` {x - x'} = \<approx>B `` {y - y'}"
+ and pref_y': "y' \<le> y" and y'_in: "y' \<in> A"
+ by simp blast
+
+ have "(x - x') \<approx>B (y - y')" using eq_xy'
+ unfolding Image_def str_eq_rel_def str_eq_def by auto
+ with h3 have "(y - y') @ z \<in> B"
+ unfolding str_eq_rel_def str_eq_def by simp
+ with pref_y' y'_in
+ show ?thesis using that by blast
+ qed
+ then have "y @ z \<in> A \<cdot> B" by (erule_tac prefixE) (auto simp: Seq_def)
+ }
+ moreover
+ { (* second case with x @ z' in A and z - z' in B *)
+ fix z'
+ assume h1: "z' \<le> z" and h2: "(x @ z') \<in> A" and h3: "z - z' \<in> B"
+ have "\<approx>A `` {x} = \<approx>A `` {y}"
+ using tag_xy unfolding tag_str_SEQ_def by simp
+ with h2 have "y @ z' \<in> A"
+ unfolding Image_def str_eq_rel_def str_eq_def by auto
+ with h1 h3 have "y @ z \<in> A \<cdot> B"
+ unfolding prefix_def Seq_def
+ by (auto) (metis append_assoc)
+ }
+ ultimately show "y @ z \<in> A \<cdot> B"
+ using Seq_in_cases [OF xz_in_seq] by blast
+ qed
+ }
+ from this [OF _ eq_tag] and this [OF _ eq_tag [THEN sym]]
+ show "x \<approx>(A \<cdot> B) y" unfolding str_eq_def str_eq_rel_def by blast
+qed
+
+lemma quot_seq_finiteI [intro]:
+ fixes L1 L2::"lang"
+ assumes fin1: "finite (UNIV // \<approx>L1)"
+ and fin2: "finite (UNIV // \<approx>L2)"
+ shows "finite (UNIV // \<approx>(L1 \<cdot> L2))"
+proof (rule_tac tag = "tag_str_SEQ L1 L2" in tag_finite_imageD)
+ show "\<And>x y. tag_str_SEQ L1 L2 x = tag_str_SEQ L1 L2 y \<Longrightarrow> x \<approx>(L1 \<cdot> L2) y"
+ by (rule tag_str_SEQ_injI)
+next
+ have *: "finite ((UNIV // \<approx>L1) \<times> (Pow (UNIV // \<approx>L2)))"
+ using fin1 fin2 by auto
+ show "finite (range (tag_str_SEQ L1 L2))"
+ unfolding tag_str_SEQ_def
+ apply(rule finite_subset[OF _ *])
+ unfolding quotient_def
+ by auto
+qed
+
+
+subsubsection {* The inductive case for @{const "STAR"} *}
+
+definition
+ tag_str_STAR :: "lang \<Rightarrow> string \<Rightarrow> lang set"
+where
+ "tag_str_STAR L1 \<equiv> (\<lambda>x. {\<approx>L1 `` {x - xa} | xa. xa < x \<and> xa \<in> L1\<star>})"
+
+text {* A technical lemma. *}
+lemma finite_set_has_max: "\<lbrakk>finite A; A \<noteq> {}\<rbrakk> \<Longrightarrow>
+ (\<exists> max \<in> A. \<forall> a \<in> A. f a <= (f max :: nat))"
+proof (induct rule:finite.induct)
+ case emptyI thus ?case by simp
+next
+ case (insertI A a)
+ show ?case
+ proof (cases "A = {}")
+ case True thus ?thesis by (rule_tac x = a in bexI, auto)
+ next
+ case False
+ with insertI.hyps and False
+ obtain max
+ where h1: "max \<in> A"
+ and h2: "\<forall>a\<in>A. f a \<le> f max" by blast
+ show ?thesis
+ proof (cases "f a \<le> f max")
+ assume "f a \<le> f max"
+ with h1 h2 show ?thesis by (rule_tac x = max in bexI, auto)
+ next
+ assume "\<not> (f a \<le> f max)"
+ thus ?thesis using h2 by (rule_tac x = a in bexI, auto)
+ qed
+ qed
+qed
+
+
+text {* The following is a technical lemma, which helps to show the range finiteness of tag function. *}
+
+lemma finite_strict_prefix_set: "finite {xa. xa < (x::string)}"
+apply (induct x rule:rev_induct, simp)
+apply (subgoal_tac "{xa. xa < xs @ [x]} = {xa. xa < xs} \<union> {xs}")
+by (auto simp:strict_prefix_def)
+
+
+lemma tag_str_STAR_injI:
+ assumes eq_tag: "tag_str_STAR L\<^isub>1 v = tag_str_STAR L\<^isub>1 w"
+ shows "v \<approx>(L\<^isub>1\<star>) w"
+proof-
+ { fix x y z
+ assume xz_in_star: "x @ z \<in> L\<^isub>1\<star>"
+ and tag_xy: "tag_str_STAR L\<^isub>1 x = tag_str_STAR L\<^isub>1 y"
+ have "y @ z \<in> L\<^isub>1\<star>"
+ proof(cases "x = []")
+ case True
+ with tag_xy have "y = []"
+ by (auto simp add: tag_str_STAR_def strict_prefix_def)
+ thus ?thesis using xz_in_star True by simp
+ next
+ case False
+ let ?S = "{xa. xa < x \<and> xa \<in> L\<^isub>1\<star> \<and> (x - xa) @ z \<in> L\<^isub>1\<star>}"
+ have "finite ?S"
+ by (rule_tac B = "{xa. xa < x}" in finite_subset,
+ auto simp:finite_strict_prefix_set)
+ moreover have "?S \<noteq> {}" using False xz_in_star
+ by (simp, rule_tac x = "[]" in exI, auto simp:strict_prefix_def)
+ ultimately have "\<exists> xa_max \<in> ?S. \<forall> xa \<in> ?S. length xa \<le> length xa_max"
+ using finite_set_has_max by blast
+ then obtain xa_max
+ where h1: "xa_max < x"
+ and h2: "xa_max \<in> L\<^isub>1\<star>"
+ and h3: "(x - xa_max) @ z \<in> L\<^isub>1\<star>"
+ and h4:"\<forall> xa < x. xa \<in> L\<^isub>1\<star> \<and> (x - xa) @ z \<in> L\<^isub>1\<star>
+ \<longrightarrow> length xa \<le> length xa_max"
+ by blast
+ obtain ya
+ where h5: "ya < y" and h6: "ya \<in> L\<^isub>1\<star>"
+ and eq_xya: "(x - xa_max) \<approx>L\<^isub>1 (y - ya)"
+ proof-
+ from tag_xy have "{\<approx>L\<^isub>1 `` {x - xa} |xa. xa < x \<and> xa \<in> L\<^isub>1\<star>} =
+ {\<approx>L\<^isub>1 `` {y - xa} |xa. xa < y \<and> xa \<in> L\<^isub>1\<star>}" (is "?left = ?right")
+ by (auto simp:tag_str_STAR_def)
+ moreover have "\<approx>L\<^isub>1 `` {x - xa_max} \<in> ?left" using h1 h2 by auto
+ ultimately have "\<approx>L\<^isub>1 `` {x - xa_max} \<in> ?right" by simp
+ thus ?thesis using that
+ apply (simp add:Image_def str_eq_rel_def str_eq_def) by blast
+ qed
+ have "(y - ya) @ z \<in> L\<^isub>1\<star>"
+ proof-
+ obtain za zb where eq_zab: "z = za @ zb"
+ and l_za: "(y - ya)@za \<in> L\<^isub>1" and ls_zb: "zb \<in> L\<^isub>1\<star>"
+ proof -
+ from h1 have "(x - xa_max) @ z \<noteq> []"
+ by (auto simp:strict_prefix_def elim:prefixE)
+ from star_decom [OF h3 this]
+ obtain a b where a_in: "a \<in> L\<^isub>1"
+ and a_neq: "a \<noteq> []" and b_in: "b \<in> L\<^isub>1\<star>"
+ and ab_max: "(x - xa_max) @ z = a @ b" by blast
+ let ?za = "a - (x - xa_max)" and ?zb = "b"
+ have pfx: "(x - xa_max) \<le> a" (is "?P1")
+ and eq_z: "z = ?za @ ?zb" (is "?P2")
+ proof -
+ have "((x - xa_max) \<le> a \<and> (a - (x - xa_max)) @ b = z) \<or>
+ (a < (x - xa_max) \<and> ((x - xa_max) - a) @ z = b)"
+ using append_eq_dest[OF ab_max] by (auto simp:strict_prefix_def)
+ moreover {
+ assume np: "a < (x - xa_max)"
+ and b_eqs: "((x - xa_max) - a) @ z = b"
+ have "False"
+ proof -
+ let ?xa_max' = "xa_max @ a"
+ have "?xa_max' < x"
+ using np h1 by (clarsimp simp:strict_prefix_def diff_prefix)
+ moreover have "?xa_max' \<in> L\<^isub>1\<star>"
+ using a_in h2 by (simp add:star_intro3)
+ moreover have "(x - ?xa_max') @ z \<in> L\<^isub>1\<star>"
+ using b_eqs b_in np h1 by (simp add:diff_diff_append)
+ moreover have "\<not> (length ?xa_max' \<le> length xa_max)"
+ using a_neq by simp
+ ultimately show ?thesis using h4 by blast
+ qed }
+ ultimately show ?P1 and ?P2 by auto
+ qed
+ hence "(x - xa_max)@?za \<in> L\<^isub>1" using a_in by (auto elim:prefixE)
+ with eq_xya have "(y - ya) @ ?za \<in> L\<^isub>1"
+ by (auto simp:str_eq_def str_eq_rel_def)
+ with eq_z and b_in
+ show ?thesis using that by blast
+ qed
+ have "((y - ya) @ za) @ zb \<in> L\<^isub>1\<star>" using l_za ls_zb by blast
+ with eq_zab show ?thesis by simp
+ qed
+ with h5 h6 show ?thesis
+ by (drule_tac star_intro1) (auto simp:strict_prefix_def elim: prefixE)
+ qed
+ }
+ from this [OF _ eq_tag] and this [OF _ eq_tag [THEN sym]]
+ show ?thesis unfolding str_eq_def str_eq_rel_def by blast
+qed
+
+lemma quot_star_finiteI [intro]:
+ assumes finite1: "finite (UNIV // \<approx>A)"
+ shows "finite (UNIV // \<approx>(A\<star>))"
+proof (rule_tac tag = "tag_str_STAR A" in tag_finite_imageD)
+ show "\<And>x y. tag_str_STAR A x = tag_str_STAR A y \<Longrightarrow> x \<approx>(A\<star>) y"
+ by (rule tag_str_STAR_injI)
+next
+ have *: "finite (Pow (UNIV // \<approx>A))"
+ using finite1 by auto
+ show "finite (range (tag_str_STAR A))"
+ unfolding tag_str_STAR_def
+ apply(rule finite_subset[OF _ *])
+ unfolding quotient_def
+ by auto
+qed
+
+subsubsection{* The conclusion *}
+
+lemma Myhill_Nerode2:
+ shows "finite (UNIV // \<approx>(L_rexp r))"
+by (induct r) (auto)
+
+
+theorem Myhill_Nerode:
+ shows "(\<exists>r. A = L_rexp r) \<longleftrightarrow> finite (UNIV // \<approx>A)"
+using Myhill_Nerode1 Myhill_Nerode2 by auto
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Attic/old/Prefix_subtract.thy Mon Jul 25 13:33:38 2011 +0000
@@ -0,0 +1,60 @@
+theory Prefix_subtract
+ imports Main "~~/src/HOL/Library/List_Prefix"
+begin
+
+
+section {* A small theory of prefix subtraction *}
+
+text {*
+ The notion of @{text "prefix_subtract"} makes
+ the second direction of the Myhill-Nerode theorem
+ more readable.
+*}
+
+instantiation list :: (type) minus
+begin
+
+fun minus_list :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
+where
+ "minus_list [] xs = []"
+| "minus_list (x#xs) [] = x#xs"
+| "minus_list (x#xs) (y#ys) = (if x = y then minus_list xs ys else (x#xs))"
+
+instance by default
+
+end
+
+lemma [simp]: "x - [] = x"
+by (induct x) (auto)
+
+lemma [simp]: "(x @ y) - x = y"
+by (induct x) (auto)
+
+lemma [simp]: "x - x = []"
+by (induct x) (auto)
+
+lemma [simp]: "x = z @ y \<Longrightarrow> x - z = y "
+by (induct x) (auto)
+
+lemma diff_prefix:
+ "\<lbrakk>c \<le> a - b; b \<le> a\<rbrakk> \<Longrightarrow> b @ c \<le> a"
+by (auto elim: prefixE)
+
+lemma diff_diff_append:
+ "\<lbrakk>c < a - b; b < a\<rbrakk> \<Longrightarrow> (a - b) - c = a - (b @ c)"
+apply (clarsimp simp:strict_prefix_def)
+by (drule diff_prefix, auto elim:prefixE)
+
+lemma append_eq_cases:
+ assumes a: "x @ y = m @ n"
+ shows "x \<le> m \<or> m \<le> x"
+unfolding prefix_def using a
+by (auto simp add: append_eq_append_conv2)
+
+lemma append_eq_dest:
+ assumes a: "x @ y = m @ n"
+ shows "(x \<le> m \<and> (m - x) @ n = y) \<or> (m \<le> x \<and> (x - m) @ y = n)"
+using append_eq_cases[OF a] a
+by (auto elim: prefixE)
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Attic/old/Regular.thy Mon Jul 25 13:33:38 2011 +0000
@@ -0,0 +1,302 @@
+(* Author: Christian Urban, Xingyuan Zhang, Chunhan Wu *)
+theory Regular
+imports Main Folds
+begin
+
+section {* Preliminary definitions *}
+
+type_synonym lang = "string set"
+
+
+text {* Sequential composition of two languages *}
+
+definition
+ Seq :: "lang \<Rightarrow> lang \<Rightarrow> lang" (infixr "\<cdot>" 100)
+where
+ "A \<cdot> 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 "\<cdot>"}. *}
+
+lemma seq_add_left:
+ assumes a: "A = B"
+ shows "C \<cdot> A = C \<cdot> B"
+using a by simp
+
+lemma seq_union_distrib_right:
+ shows "(A \<union> B) \<cdot> C = (A \<cdot> C) \<union> (B \<cdot> C)"
+unfolding Seq_def by auto
+
+lemma seq_union_distrib_left:
+ shows "C \<cdot> (A \<union> B) = (C \<cdot> A) \<union> (C \<cdot> B)"
+unfolding Seq_def by auto
+
+lemma seq_intro:
+ assumes a: "x \<in> A" "y \<in> B"
+ shows "x @ y \<in> A \<cdot> B "
+using a by (auto simp: Seq_def)
+
+lemma seq_assoc:
+ shows "(A \<cdot> B) \<cdot> C = A \<cdot> (B \<cdot> C)"
+unfolding Seq_def
+apply(auto)
+apply(blast)
+by (metis append_assoc)
+
+lemma seq_empty [simp]:
+ shows "A \<cdot> {[]} = A"
+ and "{[]} \<cdot> A = A"
+by (simp_all add: Seq_def)
+
+lemma seq_null [simp]:
+ shows "A \<cdot> {} = {}"
+ and "{} \<cdot> 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 \<cdot> (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 \<cdot> A\<star>"
+proof
+ { fix x
+ have "x \<in> A\<star> \<Longrightarrow> x \<in> {[]} \<union> A \<cdot> A\<star>"
+ unfolding Seq_def
+ by (induct rule: star_induct) (auto)
+ }
+ then show "A\<star> \<subseteq> {[]} \<union> A \<cdot> A\<star>" by auto
+next
+ show "{[]} \<union> A \<cdot> 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
+by (induct rule: star_induct) (blast)+
+
+lemma seq_Union_left:
+ shows "B \<cdot> (\<Union>n. A \<up> n) = (\<Union>n. B \<cdot> (A \<up> n))"
+unfolding Seq_def by auto
+
+lemma seq_Union_right:
+ shows "(\<Union>n. A \<up> n) \<cdot> B = (\<Union>n. (A \<up> n) \<cdot> B)"
+unfolding Seq_def by auto
+
+lemma seq_pow_comm:
+ shows "A \<cdot> (A \<up> n) = (A \<up> n) \<cdot> A"
+by (induct n) (simp_all add: seq_assoc[symmetric])
+
+lemma seq_star_comm:
+ shows "A \<cdot> A\<star> = A\<star> \<cdot> A"
+unfolding Star_def seq_Union_left
+unfolding seq_pow_comm 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 \<cdot> (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 modified version of Arden's lemma *}
+
+text {* A helper lemma for Arden *}
+
+lemma arden_helper:
+ assumes eq: "X = X \<cdot> A \<union> B"
+ shows "X = X \<cdot> (A \<up> Suc n) \<union> (\<Union>m\<in>{0..n}. B \<cdot> (A \<up> m))"
+proof (induct n)
+ case 0
+ show "X = X \<cdot> (A \<up> Suc 0) \<union> (\<Union>(m::nat)\<in>{0..0}. B \<cdot> (A \<up> m))"
+ using eq by simp
+next
+ case (Suc n)
+ have ih: "X = X \<cdot> (A \<up> Suc n) \<union> (\<Union>m\<in>{0..n}. B \<cdot> (A \<up> m))" by fact
+ also have "\<dots> = (X \<cdot> A \<union> B) \<cdot> (A \<up> Suc n) \<union> (\<Union>m\<in>{0..n}. B \<cdot> (A \<up> m))" using eq by simp
+ also have "\<dots> = X \<cdot> (A \<up> Suc (Suc n)) \<union> (B \<cdot> (A \<up> Suc n)) \<union> (\<Union>m\<in>{0..n}. B \<cdot> (A \<up> m))"
+ by (simp add: seq_union_distrib_right seq_assoc)
+ also have "\<dots> = X \<cdot> (A \<up> Suc (Suc n)) \<union> (\<Union>m\<in>{0..Suc n}. B \<cdot> (A \<up> m))"
+ by (auto simp add: le_Suc_eq)
+ finally show "X = X \<cdot> (A \<up> Suc (Suc n)) \<union> (\<Union>m\<in>{0..Suc n}. B \<cdot> (A \<up> m))" .
+qed
+
+theorem arden:
+ assumes nemp: "[] \<notin> A"
+ shows "X = X \<cdot> A \<union> B \<longleftrightarrow> X = B \<cdot> A\<star>"
+proof
+ assume eq: "X = B \<cdot> A\<star>"
+ have "A\<star> = {[]} \<union> A\<star> \<cdot> A"
+ unfolding seq_star_comm[symmetric]
+ by (rule star_cases)
+ then have "B \<cdot> A\<star> = B \<cdot> ({[]} \<union> A\<star> \<cdot> A)"
+ by (rule seq_add_left)
+ also have "\<dots> = B \<union> B \<cdot> (A\<star> \<cdot> A)"
+ unfolding seq_union_distrib_left by simp
+ also have "\<dots> = B \<union> (B \<cdot> A\<star>) \<cdot> A"
+ by (simp only: seq_assoc)
+ finally show "X = X \<cdot> A \<union> B"
+ using eq by blast
+next
+ assume eq: "X = X \<cdot> A \<union> B"
+ { fix n::nat
+ have "B \<cdot> (A \<up> n) \<subseteq> X" using arden_helper[OF eq, of "n"] by auto }
+ then have "B \<cdot> 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 \<cdot> (A \<up> Suc k)"
+ using seq_pow_length[OF nemp] by blast
+ assume "s \<in> X"
+ then have "s \<in> X \<cdot> (A \<up> Suc k) \<union> (\<Union>m\<in>{0..k}. B \<cdot> (A \<up> m))"
+ using arden_helper[OF eq, of "k"] by auto
+ then have "s \<in> (\<Union>m\<in>{0..k}. B \<cdot> (A \<up> m))" using not_in by auto
+ moreover
+ have "(\<Union>m\<in>{0..k}. B \<cdot> (A \<up> m)) \<subseteq> (\<Union>n. B \<cdot> (A \<up> n))" by auto
+ ultimately
+ have "s \<in> B \<cdot> A\<star>"
+ unfolding seq_Union_left Star_def by auto }
+ then have "X \<subseteq> B \<cdot> A\<star>" by auto
+ ultimately
+ show "X = B \<cdot> A\<star>" by simp
+qed
+
+
+section {* Regular Expressions *}
+
+datatype rexp =
+ NULL
+| EMPTY
+| CHAR char
+| SEQ rexp rexp
+| ALT rexp rexp
+| STAR rexp
+
+fun
+ L_rexp :: "rexp \<Rightarrow> lang"
+where
+ "L_rexp (NULL) = {}"
+ | "L_rexp (EMPTY) = {[]}"
+ | "L_rexp (CHAR c) = {[c]}"
+ | "L_rexp (SEQ r1 r2) = (L_rexp r1) \<cdot> (L_rexp r2)"
+ | "L_rexp (ALT r1 r2) = (L_rexp r1) \<union> (L_rexp r2)"
+ | "L_rexp (STAR r) = (L_rexp r)\<star>"
+
+text {* ALT-combination for a set of regular expressions *}
+
+abbreviation
+ Setalt ("\<Uplus>_" [1000] 999)
+where
+ "\<Uplus>A \<equiv> folds ALT NULL A"
+
+text {*
+ For finite sets, @{term Setalt} is preserved under @{term L_exp}.
+*}
+
+lemma folds_alt_simp [simp]:
+ fixes rs::"rexp set"
+ assumes a: "finite rs"
+ shows "L_rexp (\<Uplus>rs) = \<Union> (L_rexp ` rs)"
+unfolding folds_def
+apply(rule set_eqI)
+apply(rule someI2_ex)
+apply(rule_tac finite_imp_fold_graph[OF a])
+apply(erule fold_graph.induct)
+apply(auto)
+done
+
+end
\ No newline at end of file
--- a/Closure.thy Fri Jun 03 13:59:21 2011 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,158 +0,0 @@
-(* Author: Christian Urban, Xingyuan Zhang, Chunhan Wu *)
-theory Closure
-imports Derivs
-begin
-
-section {* Closure properties of regular languages *}
-
-abbreviation
- regular :: "lang \<Rightarrow> bool"
-where
- "regular A \<equiv> \<exists>r. A = L_rexp r"
-
-subsection {* Closure under set operations *}
-
-lemma closure_union[intro]:
- assumes "regular A" "regular B"
- shows "regular (A \<union> B)"
-proof -
- from assms obtain r1 r2::rexp where "L_rexp r1 = A" "L_rexp r2 = B" by auto
- then have "A \<union> B = L_rexp (ALT r1 r2)" by simp
- then show "regular (A \<union> B)" by blast
-qed
-
-lemma closure_seq[intro]:
- assumes "regular A" "regular B"
- shows "regular (A \<cdot> B)"
-proof -
- from assms obtain r1 r2::rexp where "L_rexp r1 = A" "L_rexp r2 = B" by auto
- then have "A \<cdot> B = L_rexp (SEQ r1 r2)" by simp
- then show "regular (A \<cdot> B)" by blast
-qed
-
-lemma closure_star[intro]:
- assumes "regular A"
- shows "regular (A\<star>)"
-proof -
- from assms obtain r::rexp where "L_rexp r = A" by auto
- then have "A\<star> = L_rexp (STAR r)" by simp
- then show "regular (A\<star>)" by blast
-qed
-
-text {* Closure under complementation is proved via the
- Myhill-Nerode theorem *}
-
-lemma closure_complement[intro]:
- assumes "regular A"
- shows "regular (- A)"
-proof -
- from assms have "finite (UNIV // \<approx>A)" by (simp add: Myhill_Nerode)
- then have "finite (UNIV // \<approx>(-A))" by (simp add: str_eq_rel_def)
- then show "regular (- A)" by (simp add: Myhill_Nerode)
-qed
-
-lemma closure_difference[intro]:
- assumes "regular A" "regular B"
- shows "regular (A - B)"
-proof -
- have "A - B = - (- A \<union> B)" by blast
- moreover
- have "regular (- (- A \<union> B))"
- using assms by blast
- ultimately show "regular (A - B)" by simp
-qed
-
-lemma closure_intersection[intro]:
- assumes "regular A" "regular B"
- shows "regular (A \<inter> B)"
-proof -
- have "A \<inter> B = - (- A \<union> - B)" by blast
- moreover
- have "regular (- (- A \<union> - B))"
- using assms by blast
- ultimately show "regular (A \<inter> B)" by simp
-qed
-
-subsection {* Closure under string reversal *}
-
-fun
- Rev :: "rexp \<Rightarrow> rexp"
-where
- "Rev NULL = NULL"
-| "Rev EMPTY = EMPTY"
-| "Rev (CHAR c) = CHAR c"
-| "Rev (ALT r1 r2) = ALT (Rev r1) (Rev r2)"
-| "Rev (SEQ r1 r2) = SEQ (Rev r2) (Rev r1)"
-| "Rev (STAR r) = STAR (Rev r)"
-
-lemma rev_seq[simp]:
- shows "rev ` (B \<cdot> A) = (rev ` A) \<cdot> (rev ` B)"
-unfolding Seq_def image_def
-by (auto) (metis rev_append)+
-
-lemma rev_star1:
- assumes a: "s \<in> (rev ` A)\<star>"
- shows "s \<in> rev ` (A\<star>)"
-using a
-proof(induct rule: star_induct)
- case (step s1 s2)
- have inj: "inj (rev::string \<Rightarrow> string)" unfolding inj_on_def by auto
- have "s1 \<in> rev ` A" "s2 \<in> rev ` (A\<star>)" by fact+
- then obtain x1 x2 where "x1 \<in> A" "x2 \<in> A\<star>" and eqs: "s1 = rev x1" "s2 = rev x2" by auto
- then have "x1 \<in> A\<star>" "x2 \<in> A\<star>" by (auto intro: star_intro2)
- then have "x2 @ x1 \<in> A\<star>" by (auto intro: star_intro1)
- then have "rev (x2 @ x1) \<in> rev ` A\<star>" using inj by (simp only: inj_image_mem_iff)
- then show "s1 @ s2 \<in> rev ` A\<star>" using eqs by simp
-qed (auto)
-
-lemma rev_star2:
- assumes a: "s \<in> A\<star>"
- shows "rev s \<in> (rev ` A)\<star>"
-using a
-proof(induct rule: star_induct)
- case (step s1 s2)
- have inj: "inj (rev::string \<Rightarrow> string)" unfolding inj_on_def by auto
- have "s1 \<in> A"by fact
- then have "rev s1 \<in> rev ` A" using inj by (simp only: inj_image_mem_iff)
- then have "rev s1 \<in> (rev ` A)\<star>" by (auto intro: star_intro2)
- moreover
- have "rev s2 \<in> (rev ` A)\<star>" by fact
- ultimately show "rev (s1 @ s2) \<in> (rev ` A)\<star>" by (auto intro: star_intro1)
-qed (auto)
-
-lemma rev_star[simp]:
- shows " rev ` (A\<star>) = (rev ` A)\<star>"
-using rev_star1 rev_star2 by auto
-
-lemma rev_lang:
- shows "rev ` (L_rexp r) = L_rexp (Rev r)"
-by (induct r) (simp_all add: image_Un)
-
-lemma closure_reversal[intro]:
- assumes "regular A"
- shows "regular (rev ` A)"
-proof -
- from assms obtain r::rexp where "A = L_rexp r" by auto
- then have "L_rexp (Rev r) = rev ` A" by (simp add: rev_lang)
- then show "regular (rev` A)" by blast
-qed
-
-subsection {* Closure under left-quotients *}
-
-lemma closure_left_quotient:
- assumes "regular A"
- shows "regular (Ders_set B A)"
-proof -
- from assms obtain r::rexp where eq: "L_rexp r = A" by auto
- have fin: "finite (pders_set B r)" by (rule finite_pders_set)
-
- have "Ders_set B (L_rexp r) = (\<Union> L_rexp ` (pders_set B r))"
- by (simp add: Ders_set_pders_set)
- also have "\<dots> = L_rexp (\<Uplus>(pders_set B r))" using fin by simp
- finally have "Ders_set B A = L_rexp (\<Uplus>(pders_set B r))" using eq
- by simp
- then show "regular (Ders_set B A)" by auto
-qed
-
-
-end
\ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Closures.thy Mon Jul 25 13:33:38 2011 +0000
@@ -0,0 +1,161 @@
+(* Author: Christian Urban, Xingyuan Zhang, Chunhan Wu *)
+theory Closures
+imports Derivatives
+begin
+
+section {* Closure properties of regular languages *}
+
+abbreviation
+ regular :: "'a lang \<Rightarrow> bool"
+where
+ "regular A \<equiv> \<exists>r. A = lang r"
+
+subsection {* Closure under set operations *}
+
+lemma closure_union [intro]:
+ assumes "regular A" "regular B"
+ shows "regular (A \<union> B)"
+proof -
+ from assms obtain r1 r2::"'a rexp" where "lang r1 = A" "lang r2 = B" by auto
+ then have "A \<union> B = lang (Plus r1 r2)" by simp
+ then show "regular (A \<union> B)" by blast
+qed
+
+lemma closure_seq [intro]:
+ assumes "regular A" "regular B"
+ shows "regular (A \<cdot> B)"
+proof -
+ from assms obtain r1 r2::"'a rexp" where "lang r1 = A" "lang r2 = B" by auto
+ then have "A \<cdot> B = lang (Times r1 r2)" by simp
+ then show "regular (A \<cdot> B)" by blast
+qed
+
+lemma closure_star [intro]:
+ assumes "regular A"
+ shows "regular (A\<star>)"
+proof -
+ from assms obtain r::"'a rexp" where "lang r = A" by auto
+ then have "A\<star> = lang (Star r)" by simp
+ then show "regular (A\<star>)" by blast
+qed
+
+text {* Closure under complementation is proved via the
+ Myhill-Nerode theorem *}
+
+lemma closure_complement [intro]:
+ fixes A::"('a::finite) lang"
+ assumes "regular A"
+ shows "regular (- A)"
+proof -
+ from assms have "finite (UNIV // \<approx>A)" by (simp add: Myhill_Nerode)
+ then have "finite (UNIV // \<approx>(-A))" by (simp add: str_eq_rel_def)
+ then show "regular (- A)" by (simp add: Myhill_Nerode)
+qed
+
+lemma closure_difference [intro]:
+ fixes A::"('a::finite) lang"
+ assumes "regular A" "regular B"
+ shows "regular (A - B)"
+proof -
+ have "A - B = - (- A \<union> B)" by blast
+ moreover
+ have "regular (- (- A \<union> B))"
+ using assms by blast
+ ultimately show "regular (A - B)" by simp
+qed
+
+lemma closure_intersection [intro]:
+ fixes A::"('a::finite) lang"
+ assumes "regular A" "regular B"
+ shows "regular (A \<inter> B)"
+proof -
+ have "A \<inter> B = - (- A \<union> - B)" by blast
+ moreover
+ have "regular (- (- A \<union> - B))"
+ using assms by blast
+ ultimately show "regular (A \<inter> B)" by simp
+qed
+
+subsection {* Closure under string reversal *}
+
+fun
+ Rev :: "'a rexp \<Rightarrow> 'a rexp"
+where
+ "Rev Zero = Zero"
+| "Rev One = One"
+| "Rev (Atom c) = Atom c"
+| "Rev (Plus r1 r2) = Plus (Rev r1) (Rev r2)"
+| "Rev (Times r1 r2) = Times (Rev r2) (Rev r1)"
+| "Rev (Star r) = Star (Rev r)"
+
+lemma rev_seq[simp]:
+ shows "rev ` (B \<cdot> A) = (rev ` A) \<cdot> (rev ` B)"
+unfolding conc_def image_def
+by (auto) (metis rev_append)+
+
+lemma rev_star1:
+ assumes a: "s \<in> (rev ` A)\<star>"
+ shows "s \<in> rev ` (A\<star>)"
+using a
+proof(induct rule: star_induct)
+ case (append s1 s2)
+ have inj: "inj (rev::'a list \<Rightarrow> 'a list)" unfolding inj_on_def by auto
+ have "s1 \<in> rev ` A" "s2 \<in> rev ` (A\<star>)" by fact+
+ then obtain x1 x2 where "x1 \<in> A" "x2 \<in> A\<star>" and eqs: "s1 = rev x1" "s2 = rev x2" by auto
+ then have "x1 \<in> A\<star>" "x2 \<in> A\<star>" by (auto)
+ then have "x2 @ x1 \<in> A\<star>" by (auto)
+ then have "rev (x2 @ x1) \<in> rev ` A\<star>" using inj by (simp only: inj_image_mem_iff)
+ then show "s1 @ s2 \<in> rev ` A\<star>" using eqs by simp
+qed (auto)
+
+lemma rev_star2:
+ assumes a: "s \<in> A\<star>"
+ shows "rev s \<in> (rev ` A)\<star>"
+using a
+proof(induct rule: star_induct)
+ case (append s1 s2)
+ have inj: "inj (rev::'a list \<Rightarrow> 'a list)" unfolding inj_on_def by auto
+ have "s1 \<in> A"by fact
+ then have "rev s1 \<in> rev ` A" using inj by (simp only: inj_image_mem_iff)
+ then have "rev s1 \<in> (rev ` A)\<star>" by (auto)
+ moreover
+ have "rev s2 \<in> (rev ` A)\<star>" by fact
+ ultimately show "rev (s1 @ s2) \<in> (rev ` A)\<star>" by (auto)
+qed (auto)
+
+lemma rev_star [simp]:
+ shows " rev ` (A\<star>) = (rev ` A)\<star>"
+using rev_star1 rev_star2 by auto
+
+lemma rev_lang:
+ shows "rev ` (lang r) = lang (Rev r)"
+by (induct r) (simp_all add: image_Un)
+
+lemma closure_reversal [intro]:
+ assumes "regular A"
+ shows "regular (rev ` A)"
+proof -
+ from assms obtain r::"'a rexp" where "A = lang r" by auto
+ then have "lang (Rev r) = rev ` A" by (simp add: rev_lang)
+ then show "regular (rev` A)" by blast
+qed
+
+subsection {* Closure under left-quotients *}
+
+lemma closure_left_quotient:
+ assumes "regular A"
+ shows "regular (Ders_set B A)"
+proof -
+ from assms obtain r::"'a rexp" where eq: "lang r = A" by auto
+ have fin: "finite (pders_set B r)" by (rule finite_pders_set)
+
+ have "Ders_set B (lang r) = (\<Union> lang ` (pders_set B r))"
+ by (simp add: Ders_set_pders_set)
+ also have "\<dots> = lang (\<Uplus>(pders_set B r))" using fin by simp
+ finally have "Ders_set B A = lang (\<Uplus>(pders_set B r))" using eq
+ by simp
+ then show "regular (Ders_set B A)" by auto
+qed
+
+
+end
\ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Derivatives.thy Mon Jul 25 13:33:38 2011 +0000
@@ -0,0 +1,490 @@
+theory Derivatives
+imports Myhill_2
+begin
+
+section {* Left-Quotients and Derivatives *}
+
+subsection {* Left-Quotients *}
+
+definition
+ Delta :: "'a lang \<Rightarrow> 'a lang"
+where
+ "Delta A = (if [] \<in> A then {[]} else {})"
+
+definition
+ Der :: "'a \<Rightarrow> 'a lang \<Rightarrow> 'a lang"
+where
+ "Der c A \<equiv> {s. [c] @ s \<in> A}"
+
+definition
+ Ders :: "'a list \<Rightarrow> 'a lang \<Rightarrow> 'a lang"
+where
+ "Ders s A \<equiv> {s'. s @ s' \<in> A}"
+
+definition
+ Ders_set :: "'a lang \<Rightarrow> 'a lang \<Rightarrow> 'a lang"
+where
+ "Ders_set A B \<equiv> {s' | s s'. s @ s' \<in> B \<and> s \<in> A}"
+
+lemma Ders_set_Ders:
+ shows "Ders_set A B = (\<Union>s \<in> A. Ders s B)"
+unfolding Ders_set_def Ders_def
+by auto
+
+lemma Der_zero [simp]:
+ shows "Der c {} = {}"
+unfolding Der_def
+by auto
+
+lemma Der_one [simp]:
+ shows "Der c {[]} = {}"
+unfolding Der_def
+by auto
+
+lemma Der_atom [simp]:
+ shows "Der c {[d]} = (if c = d then {[]} else {})"
+unfolding Der_def
+by auto
+
+lemma Der_union [simp]:
+ shows "Der c (A \<union> B) = Der c A \<union> Der c B"
+unfolding Der_def
+by auto
+
+lemma Der_conc [simp]:
+ shows "Der c (A \<cdot> B) = (Der c A) \<cdot> B \<union> (Delta A \<cdot> Der c B)"
+unfolding Der_def Delta_def conc_def
+by (auto simp add: Cons_eq_append_conv)
+
+lemma Der_star [simp]:
+ shows "Der c (A\<star>) = (Der c A) \<cdot> A\<star>"
+proof -
+ have incl: "Delta A \<cdot> Der c (A\<star>) \<subseteq> (Der c A) \<cdot> A\<star>"
+ unfolding Der_def Delta_def
+ apply(auto)
+ apply(drule star_decom)
+ apply(auto simp add: Cons_eq_append_conv)
+ done
+
+ have "Der c (A\<star>) = Der c ({[]} \<union> A \<cdot> A\<star>)"
+ by (simp only: star_cases[symmetric])
+ also have "... = Der c (A \<cdot> A\<star>)"
+ by (simp only: Der_union Der_one) (simp)
+ also have "... = (Der c A) \<cdot> A\<star> \<union> (Delta A \<cdot> Der c (A\<star>))"
+ by simp
+ also have "... = (Der c A) \<cdot> A\<star>"
+ using incl by auto
+ finally show "Der c (A\<star>) = (Der c A) \<cdot> A\<star>" .
+qed
+
+
+lemma Ders_singleton:
+ shows "Ders [c] A = Der c A"
+unfolding Der_def Ders_def
+by simp
+
+lemma Ders_append:
+ shows "Ders (s1 @ s2) A = Ders s2 (Ders s1 A)"
+unfolding Ders_def by simp
+
+
+text {* Relating the Myhill-Nerode relation with left-quotients. *}
+
+lemma MN_Rel_Ders:
+ shows "x \<approx>A y \<longleftrightarrow> Ders x A = Ders y A"
+unfolding Ders_def str_eq_def str_eq_rel_def
+by auto
+
+
+subsection {* Brozowsky's derivatives of regular expressions *}
+
+fun
+ 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"
+
+fun
+ der :: "'a \<Rightarrow> 'a rexp \<Rightarrow> 'a rexp"
+where
+ "der c (Zero) = Zero"
+| "der c (One) = Zero"
+| "der c (Atom c') = (if c = c' then One else Zero)"
+| "der c (Plus r1 r2) = Plus (der c r1) (der c r2)"
+| "der c (Times r1 r2) = Plus (Times (der c r1) r2) (if nullable r1 then der c r2 else Zero)"
+| "der c (Star r) = Times (der c r) (Star r)"
+
+function
+ ders :: "'a list \<Rightarrow> 'a rexp \<Rightarrow> 'a rexp"
+where
+ "ders [] r = r"
+| "ders (s @ [c]) r = der c (ders s r)"
+by (auto) (metis rev_cases)
+
+termination
+ by (relation "measure (length o fst)") (auto)
+
+lemma Delta_nullable:
+ shows "Delta (lang r) = (if nullable r then {[]} else {})"
+unfolding Delta_def
+by (induct r) (auto simp add: conc_def split: if_splits)
+
+lemma Der_der:
+ shows "Der c (lang r) = lang (der c r)"
+by (induct r) (simp_all add: Delta_nullable)
+
+lemma Ders_ders:
+ shows "Ders s (lang r) = lang (ders s r)"
+apply(induct s rule: rev_induct)
+apply(simp add: Ders_def)
+apply(simp only: ders.simps)
+apply(simp only: Ders_append)
+apply(simp only: Ders_singleton)
+apply(simp only: Der_der)
+done
+
+
+subsection {* Antimirov's Partial Derivatives *}
+
+abbreviation
+ "Times_set rs r \<equiv> {Times r' r | r'. r' \<in> rs}"
+
+fun
+ pder :: "'a \<Rightarrow> 'a rexp \<Rightarrow> ('a rexp) set"
+where
+ "pder c Zero = {Zero}"
+| "pder c One = {Zero}"
+| "pder c (Atom c') = (if c = c' then {One} else {Zero})"
+| "pder c (Plus r1 r2) = (pder c r1) \<union> (pder c r2)"
+| "pder c (Times r1 r2) = Times_set (pder c r1) r2 \<union> (if nullable r1 then pder c r2 else {})"
+| "pder c (Star r) = Times_set (pder c r) (Star r)"
+
+abbreviation
+ "pder_set c rs \<equiv> \<Union>r \<in> rs. pder c r"
+
+function
+ pders :: "'a list \<Rightarrow> 'a rexp \<Rightarrow> ('a rexp) set"
+where
+ "pders [] r = {r}"
+| "pders (s @ [c]) r = pder_set c (pders s r)"
+by (auto) (metis rev_cases)
+
+termination
+ by (relation "measure (length o fst)") (auto)
+
+abbreviation
+ "pders_set A r \<equiv> \<Union>s \<in> A. pders s r"
+
+lemma pders_append:
+ "pders (s1 @ s2) r = \<Union> (pders s2) ` (pders s1 r)"
+apply(induct s2 arbitrary: s1 r rule: rev_induct)
+apply(simp)
+apply(subst append_assoc[symmetric])
+apply(simp only: pders.simps)
+apply(auto)
+done
+
+lemma pders_singleton:
+ "pders [c] r = pder c r"
+apply(subst append_Nil[symmetric])
+apply(simp only: pders.simps)
+apply(simp)
+done
+
+lemma pders_set_lang:
+ shows "(\<Union> (lang ` pder_set c rs)) = (\<Union>r \<in> rs. (\<Union>lang ` (pder c r)))"
+unfolding image_def
+by auto
+
+lemma pders_Zero [simp]:
+ shows "pders s Zero = {Zero}"
+by (induct s rule: rev_induct) (simp_all)
+
+lemma pders_One [simp]:
+ shows "pders s One = (if s = [] then {One} else {Zero})"
+by (induct s rule: rev_induct) (auto)
+
+lemma pders_Atom [simp]:
+ shows "pders s (Atom c) = (if s = [] then {Atom c} else (if s = [c] then {One} else {Zero}))"
+by (induct s rule: rev_induct) (auto)
+
+lemma pders_Plus [simp]:
+ shows "pders s (Plus r1 r2) = (if s = [] then {Plus r1 r2} else (pders s r1) \<union> (pders s r2))"
+by (induct s rule: rev_induct) (auto)
+
+text {* Non-empty suffixes of a string *}
+
+definition
+ "Suf s \<equiv> {v. v \<noteq> [] \<and> (\<exists>u. u @ v = s)}"
+
+lemma Suf:
+ shows "Suf (s @ [c]) = (Suf s) \<cdot> {[c]} \<union> {[c]}"
+unfolding Suf_def conc_def
+by (auto simp add: append_eq_append_conv2 append_eq_Cons_conv)
+
+lemma Suf_Union:
+ shows "(\<Union>v \<in> Suf s \<cdot> {[c]}. P v) = (\<Union>v \<in> Suf s. P (v @ [c]))"
+by (auto simp add: conc_def)
+
+lemma pders_Times:
+ shows "pders s (Times r1 r2) \<subseteq> Times_set (pders s r1) r2 \<union> (\<Union>v \<in> Suf s. pders v r2)"
+proof (induct s rule: rev_induct)
+ case (snoc c s)
+ have ih: "pders s (Times r1 r2) \<subseteq> Times_set (pders s r1) r2 \<union> (\<Union>v \<in> Suf s. pders v r2)"
+ by fact
+ have "pders (s @ [c]) (Times r1 r2) = pder_set c (pders s (Times r1 r2))" by simp
+ also have "\<dots> \<subseteq> pder_set c (Times_set (pders s r1) r2 \<union> (\<Union>v \<in> Suf s. pders v r2))"
+ using ih by (auto) (blast)
+ also have "\<dots> = pder_set c (Times_set (pders s r1) r2) \<union> pder_set c (\<Union>v \<in> Suf s. pders v r2)"
+ by (simp)
+ also have "\<dots> = pder_set c (Times_set (pders s r1) r2) \<union> (\<Union>v \<in> Suf s. pder_set c (pders v r2))"
+ by (simp)
+ also have "\<dots> \<subseteq> pder_set c (Times_set (pders s r1) r2) \<union> (pder c r2) \<union> (\<Union>v \<in> Suf s. pders (v @ [c]) r2)"
+ by (auto)
+ also have "\<dots> \<subseteq> Times_set (pder_set c (pders s r1)) r2 \<union> (pder c r2) \<union> (\<Union>v \<in> Suf s. pders (v @ [c]) r2)"
+ by (auto simp add: if_splits) (blast)
+ also have "\<dots> = Times_set (pders (s @ [c]) r1) r2 \<union> (\<Union>v \<in> Suf (s @ [c]). pders v r2)"
+ apply(subst (2) pders.simps)
+ apply(simp only: Suf)
+ apply(simp add: Suf_Union pders_singleton)
+ apply(auto)
+ done
+ finally show ?case .
+qed (simp)
+
+lemma pders_Star:
+ assumes a: "s \<noteq> []"
+ shows "pders s (Star r) \<subseteq> (\<Union>v \<in> Suf s. Times_set (pders v r) (Star r))"
+using a
+proof (induct s rule: rev_induct)
+ case (snoc c s)
+ have ih: "s \<noteq> [] \<Longrightarrow> pders s (Star r) \<subseteq> (\<Union>v\<in>Suf s. Times_set (pders v r) (Star r))" by fact
+ { assume asm: "s \<noteq> []"
+ have "pders (s @ [c]) (Star r) = pder_set c (pders s (Star r))" by simp
+ also have "\<dots> \<subseteq> (pder_set c (\<Union>v\<in>Suf s. Times_set (pders v r) (Star r)))"
+ using ih[OF asm] by blast
+ also have "\<dots> = (\<Union>v\<in>Suf s. pder_set c (Times_set (pders v r) (Star r)))"
+ by simp
+ also have "\<dots> \<subseteq> (\<Union>v\<in>Suf s. (Times_set (pder_set c (pders v r)) (Star r) \<union> pder c (Star r)))"
+ by (auto split: if_splits)
+ also have "\<dots> = (\<Union>v\<in>Suf s. (Times_set (pder_set c (pders v r)) (Star r))) \<union> pder c (Star r)"
+ using asm by (auto simp add: Suf_def)
+ also have "\<dots> = (\<Union>v\<in>Suf s. (Times_set (pders (v @ [c]) r) (Star r))) \<union> (Times_set (pder c r) (Star r))"
+ by simp
+ also have "\<dots> = (\<Union>v\<in>Suf (s @ [c]). (Times_set (pders v r) (Star r)))"
+ apply(simp only: Suf)
+ apply(simp add: Suf_Union pders_singleton)
+ apply(auto)
+ done
+ finally have ?case .
+ }
+ moreover
+ { assume asm: "s = []"
+ then have ?case
+ apply(simp add: pders_singleton Suf_def)
+ apply(auto)
+ apply(rule_tac x="[c]" in exI)
+ apply(simp add: pders_singleton)
+ done
+ }
+ ultimately show ?case by blast
+qed (simp)
+
+abbreviation
+ "UNIV1 \<equiv> UNIV - {[]}"
+
+lemma pders_set_Zero:
+ shows "pders_set UNIV1 Zero = {Zero}"
+by auto
+
+lemma pders_set_One:
+ shows "pders_set UNIV1 One = {Zero}"
+by (auto split: if_splits)
+
+lemma pders_set_Atom:
+ shows "pders_set UNIV1 (Atom c) \<subseteq> {One, Zero}"
+by (auto split: if_splits)
+
+lemma pders_set_Plus:
+ shows "pders_set UNIV1 (Plus r1 r2) = pders_set UNIV1 r1 \<union> pders_set UNIV1 r2"
+by auto
+
+lemma pders_set_Times_aux:
+ assumes a: "s \<in> UNIV1"
+ shows "pders_set (Suf s) r2 \<subseteq> pders_set UNIV1 r2"
+using a by (auto simp add: Suf_def)
+
+lemma pders_set_Times:
+ shows "pders_set UNIV1 (Times r1 r2) \<subseteq> Times_set (pders_set UNIV1 r1) r2 \<union> pders_set UNIV1 r2"
+apply(rule UN_least)
+apply(rule subset_trans)
+apply(rule pders_Times)
+apply(simp)
+apply(rule conjI)
+apply(auto)[1]
+apply(rule subset_trans)
+apply(rule pders_set_Times_aux)
+apply(auto)
+done
+
+lemma pders_set_Star:
+ shows "pders_set UNIV1 (Star r) \<subseteq> Times_set (pders_set UNIV1 r) (Star r)"
+apply(rule UN_least)
+apply(rule subset_trans)
+apply(rule pders_Star)
+apply(simp)
+apply(simp add: Suf_def)
+apply(auto)
+done
+
+lemma finite_Times_set:
+ assumes a: "finite A"
+ shows "finite (Times_set A r)"
+using a by (auto)
+
+lemma finite_pders_set_UNIV1:
+ shows "finite (pders_set UNIV1 r)"
+apply(induct r)
+apply(simp)
+apply(simp only: pders_set_One)
+apply(simp)
+apply(rule finite_subset)
+apply(rule pders_set_Atom)
+apply(simp)
+apply(simp only: pders_set_Plus)
+apply(simp)
+apply(rule finite_subset)
+apply(rule pders_set_Times)
+apply(simp only: finite_Times_set finite_Un)
+apply(simp)
+apply(rule finite_subset)
+apply(rule pders_set_Star)
+apply(simp only: finite_Times_set)
+done
+
+lemma pders_set_UNIV_UNIV1:
+ shows "pders_set UNIV r = pders [] r \<union> pders_set UNIV1 r"
+apply(auto)
+apply(rule_tac x="[]" in exI)
+apply(simp)
+done
+
+lemma finite_pders_set_UNIV:
+ shows "finite (pders_set UNIV r)"
+unfolding pders_set_UNIV_UNIV1
+by (simp add: finite_pders_set_UNIV1)
+
+lemma finite_pders_set:
+ shows "finite (pders_set A r)"
+apply(rule rev_finite_subset)
+apply(rule_tac r="r" in finite_pders_set_UNIV)
+apply(auto)
+done
+
+lemma finite_pders:
+ shows "finite (pders s r)"
+using finite_pders_set[where A="{s}" and r="r"]
+by simp
+
+lemma finite_pders2:
+ shows "finite {pders s r | s. s \<in> A}"
+proof -
+ have "{pders s r | s. s \<in> A} \<subseteq> Pow (pders_set A r)" by auto
+ moreover
+ have "finite (Pow (pders_set A r))"
+ using finite_pders_set by simp
+ ultimately
+ show "finite {pders s r | s. s \<in> A}"
+ by(rule finite_subset)
+qed
+
+
+subsection {* Relating left-quotients and partial derivatives *}
+
+lemma Der_pder:
+ shows "Der c (lang r) = \<Union> lang ` (pder c r)"
+by (induct r) (auto simp add: Delta_nullable conc_UNION_distrib)
+
+lemma Ders_pders:
+ shows "Ders s (lang r) = \<Union> lang ` (pders s r)"
+proof (induct s rule: rev_induct)
+ case (snoc c s)
+ have ih: "Ders s (lang r) = \<Union> lang ` (pders s r)" by fact
+ have "Ders (s @ [c]) (lang r) = Ders [c] (Ders s (lang r))"
+ by (simp add: Ders_append)
+ also have "\<dots> = Der c (\<Union> lang ` (pders s r))" using ih
+ by (simp add: Ders_singleton)
+ also have "\<dots> = (\<Union>r\<in>pders s r. Der c (lang r))"
+ unfolding Der_def image_def by auto
+ also have "\<dots> = (\<Union>r\<in>pders s r. (\<Union> lang ` (pder c r)))"
+ by (simp add: Der_pder)
+ also have "\<dots> = (\<Union>lang ` (pder_set c (pders s r)))"
+ by (simp add: pders_set_lang)
+ also have "\<dots> = (\<Union>lang ` (pders (s @ [c]) r))"
+ by simp
+ finally show "Ders (s @ [c]) (lang r) = \<Union> lang ` pders (s @ [c]) r" .
+qed (simp add: Ders_def)
+
+lemma Ders_set_pders_set:
+ shows "Ders_set A (lang r) = (\<Union> lang ` (pders_set A r))"
+by (simp add: Ders_set_Ders Ders_pders)
+
+
+subsection {* Relating derivatives and partial derivatives *}
+
+lemma
+ shows "(\<Union> lang ` (pder c r)) = lang (der c r)"
+unfolding Der_der[symmetric] Der_pder by simp
+
+lemma
+ shows "(\<Union> lang ` (pders s r)) = lang (ders s r)"
+unfolding Ders_ders[symmetric] Ders_pders by simp
+
+
+
+subsection {*
+ The second direction of the Myhill-Nerode theorem using
+ partial derivatives.
+*}
+
+lemma Myhill_Nerode3:
+ fixes r::"'a rexp"
+ shows "finite (UNIV // \<approx>(lang r))"
+proof -
+ have "finite (UNIV // =(\<lambda>x. pders x r)=)"
+ proof -
+ have "range (\<lambda>x. pders x r) = {pders s r | s. s \<in> UNIV}" by auto
+ moreover
+ have "finite {pders s r | s. s \<in> UNIV}" by (rule finite_pders2)
+ ultimately
+ have "finite (range (\<lambda>x. pders x r))"
+ by simp
+ then show "finite (UNIV // =(\<lambda>x. pders x r)=)"
+ by (rule finite_eq_tag_rel)
+ qed
+ moreover
+ have "=(\<lambda>x. pders x r)= \<subseteq> \<approx>(lang r)"
+ unfolding tag_eq_rel_def
+ unfolding str_eq_def2
+ unfolding MN_Rel_Ders
+ unfolding Ders_pders
+ by auto
+ moreover
+ have "equiv UNIV =(\<lambda>x. pders x r)="
+ unfolding equiv_def refl_on_def sym_def trans_def
+ unfolding tag_eq_rel_def
+ by auto
+ moreover
+ have "equiv UNIV (\<approx>(lang r))"
+ unfolding equiv_def refl_on_def sym_def trans_def
+ unfolding str_eq_rel_def
+ by auto
+ ultimately show "finite (UNIV // \<approx>(lang r))"
+ by (rule refined_partition_finite)
+qed
+
+end
\ No newline at end of file
--- a/Derivs.thy Fri Jun 03 13:59:21 2011 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,492 +0,0 @@
-theory Derivs
-imports Myhill_2
-begin
-
-section {* Left-Quotients and Derivatives *}
-
-subsection {* Left-Quotients *}
-
-definition
- Delta :: "lang \<Rightarrow> lang"
-where
- "Delta A = (if [] \<in> A then {[]} else {})"
-
-definition
- Der :: "char \<Rightarrow> lang \<Rightarrow> lang"
-where
- "Der c A \<equiv> {s. [c] @ s \<in> A}"
-
-definition
- Ders :: "string \<Rightarrow> lang \<Rightarrow> lang"
-where
- "Ders s A \<equiv> {s'. s @ s' \<in> A}"
-
-definition
- Ders_set :: "lang \<Rightarrow> lang \<Rightarrow> lang"
-where
- "Ders_set A B \<equiv> {s' | s s'. s @ s' \<in> B \<and> s \<in> A}"
-
-lemma Ders_set_Ders:
- shows "Ders_set A B = (\<Union>s \<in> A. Ders s B)"
-unfolding Ders_set_def Ders_def
-by auto
-
-lemma Der_null [simp]:
- shows "Der c {} = {}"
-unfolding Der_def
-by auto
-
-lemma Der_empty [simp]:
- shows "Der c {[]} = {}"
-unfolding Der_def
-by auto
-
-lemma Der_char [simp]:
- shows "Der c {[d]} = (if c = d then {[]} else {})"
-unfolding Der_def
-by auto
-
-lemma Der_union [simp]:
- shows "Der c (A \<union> B) = Der c A \<union> Der c B"
-unfolding Der_def
-by auto
-
-lemma Der_seq [simp]:
- shows "Der c (A \<cdot> B) = (Der c A) \<cdot> B \<union> (Delta A \<cdot> Der c B)"
-unfolding Der_def Delta_def
-unfolding Seq_def
-by (auto simp add: Cons_eq_append_conv)
-
-lemma Der_star [simp]:
- shows "Der c (A\<star>) = (Der c A) \<cdot> A\<star>"
-proof -
- have incl: "Delta A \<cdot> Der c (A\<star>) \<subseteq> (Der c A) \<cdot> A\<star>"
- unfolding Der_def Delta_def Seq_def
- apply(auto)
- apply(drule star_decom)
- apply(auto simp add: Cons_eq_append_conv)
- done
-
- have "Der c (A\<star>) = Der c ({[]} \<union> A \<cdot> A\<star>)"
- by (simp only: star_cases[symmetric])
- also have "... = Der c (A \<cdot> A\<star>)"
- by (simp only: Der_union Der_empty) (simp)
- also have "... = (Der c A) \<cdot> A\<star> \<union> (Delta A \<cdot> Der c (A\<star>))"
- by simp
- also have "... = (Der c A) \<cdot> A\<star>"
- using incl by auto
- finally show "Der c (A\<star>) = (Der c A) \<cdot> A\<star>" .
-qed
-
-
-lemma Ders_singleton:
- shows "Ders [c] A = Der c A"
-unfolding Der_def Ders_def
-by simp
-
-lemma Ders_append:
- shows "Ders (s1 @ s2) A = Ders s2 (Ders s1 A)"
-unfolding Ders_def by simp
-
-lemma MN_Rel_Ders:
- shows "x \<approx>A y \<longleftrightarrow> Ders x A = Ders y A"
-unfolding Ders_def str_eq_def str_eq_rel_def
-by auto
-
-subsection {* Brozowsky's derivatives of regular expressions *}
-
-fun
- nullable :: "rexp \<Rightarrow> bool"
-where
- "nullable (NULL) = False"
-| "nullable (EMPTY) = True"
-| "nullable (CHAR c) = False"
-| "nullable (ALT r1 r2) = (nullable r1 \<or> nullable r2)"
-| "nullable (SEQ r1 r2) = (nullable r1 \<and> nullable r2)"
-| "nullable (STAR r) = True"
-
-fun
- der :: "char \<Rightarrow> rexp \<Rightarrow> rexp"
-where
- "der c (NULL) = NULL"
-| "der c (EMPTY) = NULL"
-| "der c (CHAR c') = (if c = c' then EMPTY else NULL)"
-| "der c (ALT r1 r2) = ALT (der c r1) (der c r2)"
-| "der c (SEQ r1 r2) = ALT (SEQ (der c r1) r2) (if nullable r1 then der c r2 else NULL)"
-| "der c (STAR r) = SEQ (der c r) (STAR r)"
-
-function
- ders :: "string \<Rightarrow> rexp \<Rightarrow> rexp"
-where
- "ders [] r = r"
-| "ders (s @ [c]) r = der c (ders s r)"
-by (auto) (metis rev_cases)
-
-termination
- by (relation "measure (length o fst)") (auto)
-
-lemma Delta_nullable:
- shows "Delta (L_rexp r) = (if nullable r then {[]} else {})"
-unfolding Delta_def
-by (induct r) (auto simp add: Seq_def split: if_splits)
-
-lemma Der_der:
- fixes r::rexp
- shows "Der c (L_rexp r) = L_rexp (der c r)"
-by (induct r) (simp_all add: Delta_nullable)
-
-lemma Ders_ders:
- fixes r::rexp
- shows "Ders s (L_rexp r) = L_rexp (ders s r)"
-apply(induct s rule: rev_induct)
-apply(simp add: Ders_def)
-apply(simp only: ders.simps)
-apply(simp only: Ders_append)
-apply(simp only: Ders_singleton)
-apply(simp only: Der_der)
-done
-
-
-subsection {* Antimirov's Partial Derivatives *}
-
-abbreviation
- "SEQS R r \<equiv> {SEQ r' r | r'. r' \<in> R}"
-
-fun
- pder :: "char \<Rightarrow> rexp \<Rightarrow> rexp set"
-where
- "pder c NULL = {NULL}"
-| "pder c EMPTY = {NULL}"
-| "pder c (CHAR c') = (if c = c' then {EMPTY} else {NULL})"
-| "pder c (ALT r1 r2) = (pder c r1) \<union> (pder c r2)"
-| "pder c (SEQ r1 r2) = SEQS (pder c r1) r2 \<union> (if nullable r1 then pder c r2 else {})"
-| "pder c (STAR r) = SEQS (pder c r) (STAR r)"
-
-abbreviation
- "pder_set c R \<equiv> \<Union>r \<in> R. pder c r"
-
-function
- pders :: "string \<Rightarrow> rexp \<Rightarrow> rexp set"
-where
- "pders [] r = {r}"
-| "pders (s @ [c]) r = pder_set c (pders s r)"
-by (auto) (metis rev_cases)
-
-termination
- by (relation "measure (length o fst)") (auto)
-
-abbreviation
- "pders_set A r \<equiv> \<Union>s \<in> A. pders s r"
-
-lemma pders_append:
- "pders (s1 @ s2) r = \<Union> (pders s2) ` (pders s1 r)"
-apply(induct s2 arbitrary: s1 r rule: rev_induct)
-apply(simp)
-apply(subst append_assoc[symmetric])
-apply(simp only: pders.simps)
-apply(auto)
-done
-
-lemma pders_singleton:
- "pders [c] r = pder c r"
-apply(subst append_Nil[symmetric])
-apply(simp only: pders.simps)
-apply(simp)
-done
-
-lemma pder_set_lang:
- shows "(\<Union> (L_rexp ` pder_set c R)) = (\<Union>r \<in> R. (\<Union>L_rexp ` (pder c r)))"
-unfolding image_def
-by auto
-
-lemma
- shows seq_UNION_left: "B \<cdot> (\<Union>n\<in>C. A n) = (\<Union>n\<in>C. B \<cdot> A n)"
- and seq_UNION_right: "(\<Union>n\<in>C. A n) \<cdot> B = (\<Union>n\<in>C. A n \<cdot> B)"
-unfolding Seq_def by auto
-
-lemma Der_pder:
- fixes r::rexp
- shows "Der c (L_rexp r) = \<Union> L_rexp ` (pder c r)"
-by (induct r) (auto simp add: Delta_nullable seq_UNION_right)
-
-lemma Ders_pders:
- fixes r::rexp
- shows "Ders s (L_rexp r) = \<Union> L_rexp ` (pders s r)"
-proof (induct s rule: rev_induct)
- case (snoc c s)
- have ih: "Ders s (L_rexp r) = \<Union> L_rexp ` (pders s r)" by fact
- have "Ders (s @ [c]) (L_rexp r) = Ders [c] (Ders s (L_rexp r))"
- by (simp add: Ders_append)
- also have "\<dots> = Der c (\<Union> L_rexp ` (pders s r))" using ih
- by (simp add: Ders_singleton)
- also have "\<dots> = (\<Union>r\<in>pders s r. Der c (L_rexp r))"
- unfolding Der_def image_def by auto
- also have "\<dots> = (\<Union>r\<in>pders s r. (\<Union> L_rexp ` (pder c r)))"
- by (simp add: Der_pder)
- also have "\<dots> = (\<Union>L_rexp ` (pder_set c (pders s r)))"
- by (simp add: pder_set_lang)
- also have "\<dots> = (\<Union>L_rexp ` (pders (s @ [c]) r))"
- by simp
- finally show "Ders (s @ [c]) (L_rexp r) = \<Union> L_rexp ` pders (s @ [c]) r" .
-qed (simp add: Ders_def)
-
-lemma Ders_set_pders_set:
- fixes r::rexp
- shows "Ders_set A (L_rexp r) = (\<Union> L_rexp ` (pders_set A r))"
-by (simp add: Ders_set_Ders Ders_pders)
-
-lemma pders_NULL [simp]:
- shows "pders s NULL = {NULL}"
-by (induct s rule: rev_induct) (simp_all)
-
-lemma pders_EMPTY [simp]:
- shows "pders s EMPTY = (if s = [] then {EMPTY} else {NULL})"
-by (induct s rule: rev_induct) (auto)
-
-lemma pders_CHAR [simp]:
- shows "pders s (CHAR c) = (if s = [] then {CHAR c} else (if s = [c] then {EMPTY} else {NULL}))"
-by (induct s rule: rev_induct) (auto)
-
-lemma pders_ALT [simp]:
- shows "pders s (ALT r1 r2) = (if s = [] then {ALT r1 r2} else (pders s r1) \<union> (pders s r2))"
-by (induct s rule: rev_induct) (auto)
-
-definition
- "Suf s \<equiv> {v. v \<noteq> [] \<and> (\<exists>u. u @ v = s)}"
-
-lemma Suf:
- shows "Suf (s @ [c]) = (Suf s) \<cdot> {[c]} \<union> {[c]}"
-unfolding Suf_def Seq_def
-by (auto simp add: append_eq_append_conv2 append_eq_Cons_conv)
-
-lemma Suf_Union:
- shows "(\<Union>v \<in> Suf s \<cdot> {[c]}. P v) = (\<Union>v \<in> Suf s. P (v @ [c]))"
-by (auto simp add: Seq_def)
-
-lemma inclusion1:
- shows "pder_set c (SEQS R r2) \<subseteq> SEQS (pder_set c R) r2 \<union> (pder c r2)"
-apply(auto simp add: if_splits)
-apply(blast)
-done
-
-lemma pders_SEQ:
- shows "pders s (SEQ r1 r2) \<subseteq> SEQS (pders s r1) r2 \<union> (\<Union>v \<in> Suf s. pders v r2)"
-proof (induct s rule: rev_induct)
- case (snoc c s)
- have ih: "pders s (SEQ r1 r2) \<subseteq> SEQS (pders s r1) r2 \<union> (\<Union>v \<in> Suf s. pders v r2)"
- by fact
- have "pders (s @ [c]) (SEQ r1 r2) = pder_set c (pders s (SEQ r1 r2))" by simp
- also have "\<dots> \<subseteq> pder_set c (SEQS (pders s r1) r2 \<union> (\<Union>v \<in> Suf s. pders v r2))"
- using ih by (auto) (blast)
- also have "\<dots> = pder_set c (SEQS (pders s r1) r2) \<union> pder_set c (\<Union>v \<in> Suf s. pders v r2)"
- by (simp)
- also have "\<dots> = pder_set c (SEQS (pders s r1) r2) \<union> (\<Union>v \<in> Suf s. pder_set c (pders v r2))"
- by (simp)
- also have "\<dots> \<subseteq> pder_set c (SEQS (pders s r1) r2) \<union> (pder c r2) \<union> (\<Union>v \<in> Suf s. pders (v @ [c]) r2)"
- by (auto)
- also have "\<dots> \<subseteq> SEQS (pder_set c (pders s r1)) r2 \<union> (pder c r2) \<union> (\<Union>v \<in> Suf s. pders (v @ [c]) r2)"
- using inclusion1 by blast
- also have "\<dots> = SEQS (pders (s @ [c]) r1) r2 \<union> (\<Union>v \<in> Suf (s @ [c]). pders v r2)"
- apply(subst (2) pders.simps)
- apply(simp only: Suf)
- apply(simp add: Suf_Union pders_singleton)
- apply(auto)
- done
- finally show ?case .
-qed (simp)
-
-lemma pders_STAR:
- assumes a: "s \<noteq> []"
- shows "pders s (STAR r) \<subseteq> (\<Union>v \<in> Suf s. SEQS (pders v r) (STAR r))"
-using a
-proof (induct s rule: rev_induct)
- case (snoc c s)
- have ih: "s \<noteq> [] \<Longrightarrow> pders s (STAR r) \<subseteq> (\<Union>v\<in>Suf s. SEQS (pders v r) (STAR r))" by fact
- { assume asm: "s \<noteq> []"
- have "pders (s @ [c]) (STAR r) = pder_set c (pders s (STAR r))" by simp
- also have "\<dots> \<subseteq> (pder_set c (\<Union>v\<in>Suf s. SEQS (pders v r) (STAR r)))"
- using ih[OF asm] by blast
- also have "\<dots> = (\<Union>v\<in>Suf s. pder_set c (SEQS (pders v r) (STAR r)))"
- by simp
- also have "\<dots> \<subseteq> (\<Union>v\<in>Suf s. (SEQS (pder_set c (pders v r)) (STAR r) \<union> pder c (STAR r)))"
- using inclusion1 by (auto split: if_splits)
- also have "\<dots> = (\<Union>v\<in>Suf s. (SEQS (pder_set c (pders v r)) (STAR r))) \<union> pder c (STAR r)"
- using asm by (auto simp add: Suf_def)
- also have "\<dots> = (\<Union>v\<in>Suf s. (SEQS (pders (v @ [c]) r) (STAR r))) \<union> (SEQS (pder c r) (STAR r))"
- by simp
- also have "\<dots> = (\<Union>v\<in>Suf (s @ [c]). (SEQS (pders v r) (STAR r)))"
- apply(simp only: Suf)
- apply(simp add: Suf_Union pders_singleton)
- apply(auto)
- done
- finally have ?case .
- }
- moreover
- { assume asm: "s = []"
- then have ?case
- apply(simp add: pders_singleton Suf_def)
- apply(auto)
- apply(rule_tac x="[c]" in exI)
- apply(simp add: pders_singleton)
- done
- }
- ultimately show ?case by blast
-qed (simp)
-
-abbreviation
- "UNIV1 \<equiv> UNIV - {[]}"
-
-lemma pders_set_NULL:
- shows "pders_set UNIV1 NULL = {NULL}"
-by auto
-
-lemma pders_set_EMPTY:
- shows "pders_set UNIV1 EMPTY = {NULL}"
-by (auto split: if_splits)
-
-lemma pders_set_CHAR:
- shows "pders_set UNIV1 (CHAR c) \<subseteq> {EMPTY, NULL}"
-by (auto split: if_splits)
-
-lemma pders_set_ALT:
- shows "pders_set UNIV1 (ALT r1 r2) = pders_set UNIV1 r1 \<union> pders_set UNIV1 r2"
-by auto
-
-lemma pders_set_SEQ_aux:
- assumes a: "s \<in> UNIV1"
- shows "pders_set (Suf s) r2 \<subseteq> pders_set UNIV1 r2"
-using a by (auto simp add: Suf_def)
-
-lemma pders_set_SEQ:
- shows "pders_set UNIV1 (SEQ r1 r2) \<subseteq> SEQS (pders_set UNIV1 r1) r2 \<union> pders_set UNIV1 r2"
-apply(rule UN_least)
-apply(rule subset_trans)
-apply(rule pders_SEQ)
-apply(simp)
-apply(rule conjI)
-apply(auto)[1]
-apply(rule subset_trans)
-apply(rule pders_set_SEQ_aux)
-apply(auto)
-done
-
-lemma pders_set_STAR:
- shows "pders_set UNIV1 (STAR r) \<subseteq> SEQS (pders_set UNIV1 r) (STAR r)"
-apply(rule UN_least)
-apply(rule subset_trans)
-apply(rule pders_STAR)
-apply(simp)
-apply(simp add: Suf_def)
-apply(auto)
-done
-
-lemma finite_SEQS:
- assumes a: "finite A"
- shows "finite (SEQS A r)"
-using a by (auto)
-
-lemma finite_pders_set_UNIV1:
- shows "finite (pders_set UNIV1 r)"
-apply(induct r)
-apply(simp)
-apply(simp only: pders_set_EMPTY)
-apply(simp)
-apply(rule finite_subset)
-apply(rule pders_set_CHAR)
-apply(simp)
-apply(rule finite_subset)
-apply(rule pders_set_SEQ)
-apply(simp only: finite_SEQS finite_Un)
-apply(simp)
-apply(simp only: pders_set_ALT)
-apply(simp)
-apply(rule finite_subset)
-apply(rule pders_set_STAR)
-apply(simp only: finite_SEQS)
-done
-
-lemma pders_set_UNIV_UNIV1:
- shows "pders_set UNIV r = pders [] r \<union> pders_set UNIV1 r"
-apply(auto)
-apply(rule_tac x="[]" in exI)
-apply(simp)
-done
-
-lemma finite_pders_set_UNIV:
- shows "finite (pders_set UNIV r)"
-unfolding pders_set_UNIV_UNIV1
-by (simp add: finite_pders_set_UNIV1)
-
-lemma finite_pders_set:
- shows "finite (pders_set A r)"
-apply(rule rev_finite_subset)
-apply(rule_tac r="r" in finite_pders_set_UNIV)
-apply(auto)
-done
-
-lemma finite_pders:
- shows "finite (pders s r)"
-using finite_pders_set[where A="{s}" and r="r"]
-by simp
-
-lemma finite_pders2:
- shows "finite {pders s r | s. s \<in> A}"
-proof -
- have "{pders s r | s. s \<in> A} \<subseteq> Pow (pders_set A r)" by auto
- moreover
- have "finite (Pow (pders_set A r))"
- using finite_pders_set by simp
- ultimately
- show "finite {pders s r | s. s \<in> A}"
- by(rule finite_subset)
-qed
-
-
-lemma Myhill_Nerode3:
- fixes r::"rexp"
- shows "finite (UNIV // \<approx>(L_rexp r))"
-proof -
- have "finite (UNIV // =(\<lambda>x. pders x r)=)"
- proof -
- have "range (\<lambda>x. pders x r) = {pders s r | s. s \<in> UNIV}" by auto
- moreover
- have "finite {pders s r | s. s \<in> UNIV}" by (rule finite_pders2)
- ultimately
- have "finite (range (\<lambda>x. pders x r))"
- by simp
- then show "finite (UNIV // =(\<lambda>x. pders x r)=)"
- by (rule finite_eq_tag_rel)
- qed
- moreover
- have "=(\<lambda>x. pders x r)= \<subseteq> \<approx>(L_rexp r)"
- unfolding tag_eq_rel_def
- unfolding str_eq_def2
- unfolding MN_Rel_Ders
- unfolding Ders_pders
- by auto
- moreover
- have "equiv UNIV =(\<lambda>x. pders x r)="
- unfolding equiv_def refl_on_def sym_def trans_def
- unfolding tag_eq_rel_def
- by auto
- moreover
- have "equiv UNIV (\<approx>(L_rexp r))"
- unfolding equiv_def refl_on_def sym_def trans_def
- unfolding str_eq_rel_def
- by auto
- ultimately show "finite (UNIV // \<approx>(L_rexp r))"
- by (rule refined_partition_finite)
-qed
-
-
-section {* Relating derivatives and partial derivatives *}
-
-lemma
- shows "(\<Union> L_rexp ` (pder c r)) = L_rexp (der c r)"
-unfolding Der_der[symmetric] Der_pder by simp
-
-lemma
- shows "(\<Union> L_rexp ` (pders s r)) = L_rexp (ders s r)"
-unfolding Ders_ders[symmetric] Ders_pders by simp
-
-end
\ No newline at end of file
--- a/Journal/Paper.thy Fri Jun 03 13:59:21 2011 +0000
+++ b/Journal/Paper.thy Mon Jul 25 13:33:38 2011 +0000
@@ -93,6 +93,7 @@
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.
+ Some of the popular theorem provers are based on Higher-Order Logic (HOL).
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
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/More_Regular_Set.thy Mon Jul 25 13:33:38 2011 +0000
@@ -0,0 +1,170 @@
+(* Author: Christian Urban, Xingyuan Zhang, Chunhan Wu *)
+theory More_Regular_Set
+imports "Regular_Exp" "Folds"
+begin
+
+text {* Some properties of operator @{text "@@"}. *}
+
+notation
+ conc (infixr "\<cdot>" 100) and
+ star ("_\<star>" [101] 102)
+
+lemma conc_add_left:
+ assumes a: "A = B"
+ shows "C \<cdot> A = C \<cdot> B"
+using a by simp
+
+lemma star_cases:
+ shows "A\<star> = {[]} \<union> A \<cdot> A\<star>"
+proof
+ { fix x
+ have "x \<in> A\<star> \<Longrightarrow> x \<in> {[]} \<union> A \<cdot> A\<star>"
+ unfolding conc_def
+ by (induct rule: star_induct) (auto)
+ }
+ then show "A\<star> \<subseteq> {[]} \<union> A \<cdot> A\<star>" by auto
+next
+ show "{[]} \<union> A \<cdot> A\<star> \<subseteq> A\<star>"
+ unfolding conc_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
+by (induct rule: star_induct) (blast)+
+
+lemma seq_pow_comm:
+ shows "A \<cdot> (A ^^ n) = (A ^^ n) \<cdot> A"
+by (induct n) (simp_all add: conc_assoc[symmetric])
+
+lemma seq_star_comm:
+ shows "A \<cdot> A\<star> = A\<star> \<cdot> A"
+unfolding star_def seq_pow_comm conc_UNION_distrib
+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 ^^ Suc n"
+ shows "n < length s"
+using b
+proof (induct n arbitrary: s)
+ case 0
+ have "s \<in> A ^^ 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 ^^ Suc n \<Longrightarrow> n < length s" by fact
+ have "s \<in> A ^^ Suc (Suc n)" by fact
+ then obtain s1 s2 where eq: "s = s1 @ s2" and *: "s1 \<in> A" and **: "s2 \<in> A ^^ Suc n"
+ by (auto simp add: conc_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 \<cdot> (A ^^ Suc n)"
+ shows "n < length s"
+proof -
+ from b obtain s1 s2 where eq: "s = s1 @ s2" and *: "s2 \<in> A ^^ 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 modified version of Arden's lemma *}
+
+text {* A helper lemma for Arden *}
+
+lemma arden_helper:
+ assumes eq: "X = X \<cdot> A \<union> B"
+ shows "X = X \<cdot> (A ^^ Suc n) \<union> (\<Union>m\<in>{0..n}. B \<cdot> (A ^^ m))"
+proof (induct n)
+ case 0
+ show "X = X \<cdot> (A ^^ Suc 0) \<union> (\<Union>(m::nat)\<in>{0..0}. B \<cdot> (A ^^ m))"
+ using eq by simp
+next
+ case (Suc n)
+ have ih: "X = X \<cdot> (A ^^ Suc n) \<union> (\<Union>m\<in>{0..n}. B \<cdot> (A ^^ m))" by fact
+ also have "\<dots> = (X \<cdot> A \<union> B) \<cdot> (A ^^ Suc n) \<union> (\<Union>m\<in>{0..n}. B \<cdot> (A ^^ m))" using eq by simp
+ also have "\<dots> = X \<cdot> (A ^^ Suc (Suc n)) \<union> (B \<cdot> (A ^^ Suc n)) \<union> (\<Union>m\<in>{0..n}. B \<cdot> (A ^^ m))"
+ by (simp add: conc_Un_distrib conc_assoc)
+ also have "\<dots> = X \<cdot> (A ^^ Suc (Suc n)) \<union> (\<Union>m\<in>{0..Suc n}. B \<cdot> (A ^^ m))"
+ by (auto simp add: le_Suc_eq)
+ finally show "X = X \<cdot> (A ^^ Suc (Suc n)) \<union> (\<Union>m\<in>{0..Suc n}. B \<cdot> (A ^^ m))" .
+qed
+
+theorem arden:
+ assumes nemp: "[] \<notin> A"
+ shows "X = X \<cdot> A \<union> B \<longleftrightarrow> X = B \<cdot> A\<star>"
+proof
+ assume eq: "X = B \<cdot> A\<star>"
+ have "A\<star> = {[]} \<union> A\<star> \<cdot> A"
+ unfolding seq_star_comm[symmetric]
+ by (rule star_cases)
+ then have "B \<cdot> A\<star> = B \<cdot> ({[]} \<union> A\<star> \<cdot> A)"
+ by (rule conc_add_left)
+ also have "\<dots> = B \<union> B \<cdot> (A\<star> \<cdot> A)"
+ unfolding conc_Un_distrib by simp
+ also have "\<dots> = B \<union> (B \<cdot> A\<star>) \<cdot> A"
+ by (simp only: conc_assoc)
+ finally show "X = X \<cdot> A \<union> B"
+ using eq by blast
+next
+ assume eq: "X = X \<cdot> A \<union> B"
+ { fix n::nat
+ have "B \<cdot> (A ^^ n) \<subseteq> X" using arden_helper[OF eq, of "n"] by auto }
+ then have "B \<cdot> A\<star> \<subseteq> X"
+ unfolding conc_def star_def UNION_def by auto
+ moreover
+ { fix s::"'a list"
+ obtain k where "k = length s" by auto
+ then have not_in: "s \<notin> X \<cdot> (A ^^ Suc k)"
+ using seq_pow_length[OF nemp] by blast
+ assume "s \<in> X"
+ then have "s \<in> X \<cdot> (A ^^ Suc k) \<union> (\<Union>m\<in>{0..k}. B \<cdot> (A ^^ m))"
+ using arden_helper[OF eq, of "k"] by auto
+ then have "s \<in> (\<Union>m\<in>{0..k}. B \<cdot> (A ^^ m))" using not_in by auto
+ moreover
+ have "(\<Union>m\<in>{0..k}. B \<cdot> (A ^^ m)) \<subseteq> (\<Union>n. B \<cdot> (A ^^ n))" by auto
+ ultimately
+ have "s \<in> B \<cdot> A\<star>"
+ unfolding conc_Un_distrib star_def by auto }
+ then have "X \<subseteq> B \<cdot> A\<star>" by auto
+ ultimately
+ show "X = B \<cdot> A\<star>" by simp
+qed
+
+
+text {* Plus-combination for a set of regular expressions *}
+
+abbreviation
+ Setalt ("\<Uplus>_" [1000] 999)
+where
+ "\<Uplus>A \<equiv> folds Plus Zero A"
+
+text {*
+ For finite sets, @{term Setalt} is preserved under @{term lang}.
+*}
+
+lemma folds_alt_simp [simp]:
+ fixes rs::"('a rexp) set"
+ assumes a: "finite rs"
+ shows "lang (\<Uplus>rs) = \<Union> (lang ` rs)"
+unfolding folds_def
+apply(rule set_eqI)
+apply(rule someI2_ex)
+apply(rule_tac finite_imp_fold_graph[OF a])
+apply(erule fold_graph.induct)
+apply(auto)
+done
+
+end
\ No newline at end of file
--- a/My.thy Fri Jun 03 13:59:21 2011 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,389 +0,0 @@
-theory My
-imports Main Infinite_Set
-begin
-
-
-definition
- Seq :: "string set \<Rightarrow> string set \<Rightarrow> string set" ("_ ;; _" [100,100] 100)
-where
- "L1 ;; L2 = {s1 @ s2 | s1 s2. s1 \<in> L1 \<and> s2 \<in> L2}"
-
-inductive_set
- Star :: "string set \<Rightarrow> string set" ("_\<star>" [101] 102)
- for L :: "string set"
-where
- start[intro]: "[] \<in> L\<star>"
-| step[intro]: "\<lbrakk>s1 \<in> L; s2 \<in> L\<star>\<rbrakk> \<Longrightarrow> s1@s2 \<in> L\<star>"
-
-lemma lang_star_cases:
- shows "L\<star> = {[]} \<union> L ;; L\<star>"
-unfolding Seq_def
-by (auto) (metis Star.simps)
-
-lemma lang_star_cases2:
- shows "L ;; L\<star> = L\<star> ;; L"
-sorry
-
-
-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" sorry
- then have "B ;; A\<star> = B ;; ({[]} \<union> A\<star> ;; A)" unfolding Seq_def by simp
- also have "\<dots> = B \<union> B ;; (A\<star> ;; A)" unfolding Seq_def by auto
- also have "\<dots> = B \<union> (B ;; A\<star>) ;; A" unfolding Seq_def
- by (auto) (metis append_assoc)+
- finally show "X = X ;; A \<union> B" using eq by auto
-next
- assume "X = X ;; A \<union> B"
- then have "B \<subseteq> X" "X ;; A \<subseteq> X" by auto
- show "X = B ;; A\<star>" sorry
-qed
-
-datatype rexp =
- NULL
-| EMPTY
-| CHAR char
-| SEQ rexp rexp
-| ALT rexp rexp
-| STAR rexp
-
-fun
- Sem :: "rexp \<Rightarrow> string set" ("\<lparr>_\<rparr>" [0] 1000)
-where
- "\<lparr>NULL\<rparr> = {}"
- | "\<lparr>EMPTY\<rparr> = {[]}"
- | "\<lparr>CHAR c\<rparr> = {[c]}"
- | "\<lparr>SEQ r1 r2\<rparr> = \<lparr>r1\<rparr> ;; \<lparr>r2\<rparr>"
- | "\<lparr>ALT r1 r2\<rparr> = \<lparr>r1\<rparr> \<union> \<lparr>r2\<rparr>"
- | "\<lparr>STAR r\<rparr> = \<lparr>r\<rparr>\<star>"
-
-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"
-
-lemma folds_simp_null [simp]:
- "finite rs \<Longrightarrow> x \<in> \<lparr>folds ALT NULL rs\<rparr> \<longleftrightarrow> (\<exists>r \<in> rs. x \<in> \<lparr>r\<rparr>)"
-apply (simp add: folds_def)
-apply (rule someI2_ex)
-apply (erule finite_imp_fold_graph)
-apply (erule fold_graph.induct)
-apply (auto)
-done
-
-lemma folds_simp_empty [simp]:
- "finite rs \<Longrightarrow> x \<in> \<lparr>folds ALT EMPTY rs\<rparr> \<longleftrightarrow> (\<exists>r \<in> rs. x \<in> \<lparr>r\<rparr>) \<or> x = []"
-apply (simp add: folds_def)
-apply (rule someI2_ex)
-apply (erule finite_imp_fold_graph)
-apply (erule fold_graph.induct)
-apply (auto)
-done
-
-lemma [simp]:
- shows "(x, y) \<in> {(x, y). P x y} \<longleftrightarrow> P x y"
-by simp
-
-definition
- str_eq ("_ \<approx>_ _")
-where
- "x \<approx>Lang y \<equiv> (\<forall>z. x @ z \<in> Lang \<longleftrightarrow> y @ z \<in> Lang)"
-
-definition
- str_eq_rel ("\<approx>_")
-where
- "\<approx>Lang \<equiv> {(x, y). x \<approx>Lang y}"
-
-definition
- final :: "string set \<Rightarrow> string set \<Rightarrow> bool"
-where
- "final X Lang \<equiv> (X \<in> UNIV // \<approx>Lang) \<and> (\<forall>s \<in> X. s \<in> Lang)"
-
-lemma lang_is_union_of_finals:
- "Lang = \<Union> {X. final X Lang}"
-proof -
- have "Lang \<subseteq> \<Union> {X. final X Lang}"
- unfolding final_def
- unfolding quotient_def Image_def
- unfolding str_eq_rel_def
- apply(simp)
- apply(auto)
- apply(rule_tac x="(\<approx>Lang) `` {x}" in exI)
- unfolding Image_def
- unfolding str_eq_rel_def
- apply(auto)
- unfolding str_eq_def
- apply(auto)
- apply(drule_tac x="[]" in spec)
- apply(simp)
- done
- moreover
- have "\<Union>{X. final X Lang} \<subseteq> Lang"
- unfolding final_def by auto
- ultimately
- show "Lang = \<Union> {X. final X Lang}"
- by blast
-qed
-
-lemma all_rexp:
- "\<lbrakk>finite (UNIV // \<approx>Lang); X \<in> (UNIV // \<approx>Lang)\<rbrakk> \<Longrightarrow> \<exists>r. X = \<lparr>r\<rparr>"
-sorry
-
-lemma final_rexp:
- "\<lbrakk>finite (UNIV // (\<approx>Lang)); final X Lang\<rbrakk> \<Longrightarrow> \<exists>r. X = \<lparr>r\<rparr>"
-unfolding final_def
-using all_rexp by blast
-
-lemma finite_f_one_to_one:
- assumes "finite B"
- and "\<forall>x \<in> B. \<exists>y. f y = x"
- shows "\<exists>rs. finite rs \<and> (B = {f y | y . y \<in> rs})"
-using assms
-by (induct) (auto)
-
-lemma finite_final:
- assumes "finite (UNIV // (\<approx>Lang))"
- shows "finite {X. final X Lang}"
-using assms
-proof -
- have "{X. final X Lang} \<subseteq> (UNIV // (\<approx>Lang))"
- unfolding final_def by auto
- with assms show "finite {X. final X Lang}"
- using finite_subset by auto
-qed
-
-lemma finite_regular_aux:
- fixes Lang :: "string set"
- assumes "finite (UNIV // (\<approx>Lang))"
- shows "\<exists>rs. Lang = \<lparr>folds ALT NULL rs\<rparr>"
-apply(subst lang_is_union_of_finals)
-using assms
-apply -
-apply(drule finite_final)
-apply(drule_tac f="Sem" in finite_f_one_to_one)
-apply(clarify)
-apply(drule final_rexp[OF assms])
-apply(auto)[1]
-apply(clarify)
-apply(rule_tac x="rs" in exI)
-apply(simp)
-apply(rule set_eqI)
-apply(auto)
-done
-
-lemma finite_regular:
- fixes Lang :: "string set"
- assumes "finite (UNIV // (\<approx>Lang))"
- shows "\<exists>r. Lang = \<lparr>r\<rparr>"
-using assms finite_regular_aux
-by auto
-
-
-
-section {* other direction *}
-
-
-lemma inj_image_lang:
- fixes f::"string \<Rightarrow> 'a"
- assumes str_inj: "\<And>x y. f x = f y \<Longrightarrow> x \<approx>Lang y"
- shows "inj_on (image f) (UNIV // (\<approx>Lang))"
-proof -
- { fix x y::string
- assume eq_tag: "f ` {z. x \<approx>Lang z} = f ` {z. y \<approx>Lang z}"
- moreover
- have "{z. x \<approx>Lang z} \<noteq> {}" unfolding str_eq_def by auto
- ultimately obtain a b where "x \<approx>Lang a" "y \<approx>Lang b" "f a = f b" by blast
- then have "x \<approx>Lang a" "y \<approx>Lang b" "a \<approx>Lang b" using str_inj by auto
- then have "x \<approx>Lang y" unfolding str_eq_def by simp
- then have "{z. x \<approx>Lang z} = {z. y \<approx>Lang z}" unfolding str_eq_def by simp
- }
- then have "\<forall>x\<in>UNIV // \<approx>Lang. \<forall>y\<in>UNIV // \<approx>Lang. f ` x = f ` y \<longrightarrow> x = y"
- unfolding quotient_def Image_def str_eq_rel_def by simp
- then show "inj_on (image f) (UNIV // (\<approx>Lang))"
- unfolding inj_on_def by simp
-qed
-
-
-lemma finite_range_image:
- assumes fin: "finite (range f)"
- shows "finite ((image f) ` X)"
-proof -
- from fin have "finite (Pow (f ` UNIV))" by auto
- moreover
- have "(image f) ` X \<subseteq> Pow (f ` UNIV)" by auto
- ultimately show "finite ((image f) ` X)" using finite_subset by auto
-qed
-
-definition
- tag1 :: "string set \<Rightarrow> string set \<Rightarrow> string \<Rightarrow> (string set \<times> string set)"
-where
- "tag1 L\<^isub>1 L\<^isub>2 \<equiv> \<lambda>x. ((\<approx>L\<^isub>1) `` {x}, (\<approx>L\<^isub>2) `` {x})"
-
-lemma tag1_range_finite:
- assumes finite1: "finite (UNIV // \<approx>L\<^isub>1)"
- and finite2: "finite (UNIV // \<approx>L\<^isub>2)"
- shows "finite (range (tag1 L\<^isub>1 L\<^isub>2))"
-proof -
- have "finite (UNIV // \<approx>L\<^isub>1 \<times> UNIV // \<approx>L\<^isub>2)" using finite1 finite2 by auto
- moreover
- have "range (tag1 L\<^isub>1 L\<^isub>2) \<subseteq> (UNIV // \<approx>L\<^isub>1) \<times> (UNIV // \<approx>L\<^isub>2)"
- unfolding tag1_def quotient_def by auto
- ultimately show "finite (range (tag1 L\<^isub>1 L\<^isub>2))"
- using finite_subset by blast
-qed
-
-lemma tag1_inj:
- "tag1 L\<^isub>1 L\<^isub>2 x = tag1 L\<^isub>1 L\<^isub>2 y \<Longrightarrow> x \<approx>(L\<^isub>1 \<union> L\<^isub>2) y"
-unfolding tag1_def Image_def str_eq_rel_def str_eq_def
-by auto
-
-lemma quot_alt_cu:
- fixes L\<^isub>1 L\<^isub>2::"string set"
- assumes fin1: "finite (UNIV // \<approx>L\<^isub>1)"
- and fin2: "finite (UNIV // \<approx>L\<^isub>2)"
- shows "finite (UNIV // \<approx>(L\<^isub>1 \<union> L\<^isub>2))"
-proof -
- have "finite (range (tag1 L\<^isub>1 L\<^isub>2))"
- using fin1 fin2 tag1_range_finite by simp
- then have "finite (image (tag1 L\<^isub>1 L\<^isub>2) ` (UNIV // \<approx>(L\<^isub>1 \<union> L\<^isub>2)))"
- using finite_range_image by blast
- moreover
- have "\<And>x y. tag1 L\<^isub>1 L\<^isub>2 x = tag1 L\<^isub>1 L\<^isub>2 y \<Longrightarrow> x \<approx>(L\<^isub>1 \<union> L\<^isub>2) y"
- using tag1_inj by simp
- then have "inj_on (image (tag1 L\<^isub>1 L\<^isub>2)) (UNIV // \<approx>(L\<^isub>1 \<union> L\<^isub>2))"
- using inj_image_lang by blast
- ultimately
- show "finite (UNIV // \<approx>(L\<^isub>1 \<union> L\<^isub>2))" by (rule finite_imageD)
-qed
-
-
-section {* finite \<Rightarrow> regular *}
-
-definition
- transitions :: "string set \<Rightarrow> string set \<Rightarrow> rexp set" ("_\<Turnstile>\<Rightarrow>_")
-where
- "Y \<Turnstile>\<Rightarrow> X \<equiv> {CHAR c | c. Y ;; {[c]} \<subseteq> X}"
-
-definition
- transitions_rexp ("_ \<turnstile>\<rightarrow> _")
-where
- "Y \<turnstile>\<rightarrow> X \<equiv> if [] \<in> X then folds ALT EMPTY (Y \<Turnstile>\<Rightarrow>X) else folds ALT NULL (Y \<Turnstile>\<Rightarrow>X)"
-
-definition
- "rhs CS X \<equiv> if X = {[]} then {({[]}, EMPTY)} else {(Y, Y \<turnstile>\<rightarrow>X) | Y. Y \<in> CS}"
-
-definition
- "rhs_sem CS X \<equiv> \<Union> {(Y;; \<lparr>r\<rparr>) | Y r . (Y, r) \<in> rhs CS X}"
-
-definition
- "eqs CS \<equiv> (\<Union>X \<in> CS. {(X, rhs CS X)})"
-
-definition
- "eqs_sem CS \<equiv> (\<Union>X \<in> CS. {(X, rhs_sem CS X)})"
-
-lemma [simp]:
- shows "finite (Y \<Turnstile>\<Rightarrow> X)"
-unfolding transitions_def
-by auto
-
-
-lemma defined_by_str:
- assumes "s \<in> X"
- and "X \<in> UNIV // (\<approx>Lang)"
- shows "X = (\<approx>Lang) `` {s}"
-using assms
-unfolding quotient_def Image_def
-unfolding str_eq_rel_def str_eq_def
-by auto
-
-lemma every_eqclass_has_transition:
- assumes has_str: "s @ [c] \<in> X"
- and in_CS: "X \<in> UNIV // (\<approx>Lang)"
- obtains Y where "Y \<in> UNIV // (\<approx>Lang)" and "Y ;; {[c]} \<subseteq> X" and "s \<in> Y"
-proof -
- def Y \<equiv> "(\<approx>Lang) `` {s}"
- have "Y \<in> UNIV // (\<approx>Lang)"
- unfolding Y_def quotient_def by auto
- moreover
- have "X = (\<approx>Lang) `` {s @ [c]}"
- using has_str in_CS defined_by_str by blast
- then have "Y ;; {[c]} \<subseteq> X"
- unfolding Y_def Image_def Seq_def
- unfolding str_eq_rel_def
- by (auto) (simp add: str_eq_def)
- moreover
- have "s \<in> Y" unfolding Y_def
- unfolding Image_def str_eq_rel_def str_eq_def by simp
- (*moreover
- have "True" by simp FIXME *)
- ultimately show thesis by (blast intro: that)
-qed
-
-lemma test:
- assumes "[] \<in> X"
- shows "[] \<in> \<lparr>Y \<turnstile>\<rightarrow> X\<rparr>"
-using assms
-by (simp add: transitions_rexp_def)
-
-lemma rhs_sem:
- assumes "X \<in> (UNIV // (\<approx>Lang))"
- shows "X \<subseteq> rhs_sem (UNIV // (\<approx>Lang)) X"
-apply(case_tac "X = {[]}")
-apply(simp)
-apply(simp add: rhs_sem_def rhs_def Seq_def)
-apply(rule subsetI)
-apply(case_tac "x = []")
-apply(simp add: rhs_sem_def rhs_def)
-apply(rule_tac x = "X" in exI)
-apply(simp)
-apply(rule_tac x = "X" in exI)
-apply(simp add: assms)
-apply(simp add: transitions_rexp_def)
-oops
-
-
-(*
-fun
- power :: "string \<Rightarrow> nat \<Rightarrow> string" (infixr "\<Up>" 100)
-where
- "s \<Up> 0 = s"
-| "s \<Up> (Suc n) = s @ (s \<Up> n)"
-
-definition
- "Lone = {(''0'' \<Up> n) @ (''1'' \<Up> n) | n. True }"
-
-lemma
- "infinite (UNIV // (\<approx>Lone))"
-unfolding infinite_iff_countable_subset
-apply(rule_tac x="\<lambda>n. {(''0'' \<Up> n) @ (''1'' \<Up> i) | i. i \<in> {..n} }" in exI)
-apply(auto)
-prefer 2
-unfolding Lone_def
-unfolding quotient_def
-unfolding Image_def
-apply(simp)
-unfolding str_eq_rel_def
-unfolding str_eq_def
-apply(auto)
-apply(rule_tac x="''0'' \<Up> n" in exI)
-apply(auto)
-unfolding infinite_nat_iff_unbounded
-unfolding Lone_def
-*)
-
-
-
-text {* Derivatives *}
-
-definition
- DERS :: "string \<Rightarrow> string set \<Rightarrow> string set"
-where
- "DERS s L \<equiv> {s'. s @ s' \<in> L}"
-
-lemma
- shows "x \<approx>L y \<longleftrightarrow> DERS x L = DERS y L"
-unfolding DERS_def str_eq_def
-by auto
\ No newline at end of file
--- a/Myhill.thy Fri Jun 03 13:59:21 2011 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,331 +0,0 @@
-theory Myhill
- imports Myhill_2
-begin
-
-section {* Preliminaries \label{sec_prelim}*}
-
-subsection {* Finite automata and \mht \label{sec_fa_mh} *}
-
-text {*
-
-A {\em determinisitc finite automata (DFA)} $M$ is a 5-tuple
-$(Q, \Sigma, \delta, s, F)$, where:
-\begin{enumerate}
- \item $Q$ is a finite set of {\em states}, also denoted $Q_M$.
- \item $\Sigma$ is a finite set of {\em alphabets}, also denoted $\Sigma_M$.
- \item $\delta$ is a {\em transition function} of type @{text "Q \<times> \<Sigma> \<Rightarrow> Q"} (a total function),
- also denoted $\delta_M$.
- \item @{text "s \<in> Q"} is a state called {\em initial state}, also denoted $s_M$.
- \item @{text "F \<subseteq> Q"} is a set of states named {\em accepting states}, also denoted $F_M$.
-\end{enumerate}
-Therefore, we have $M = (Q_M, \Sigma_M, \delta_M, s_M, F_M)$. Every DFA $M$ can be interpreted as
-a function assigning states to strings, denoted $\dfa{M}$, the definition of which is as the following:
-\begin{equation}
-\begin{aligned}
- \dfa{M}([]) &\equiv s_M \\
- \dfa{M}(xa) &\equiv \delta_M(\dfa{M}(x), a)
-\end{aligned}
-\end{equation}
-A string @{text "x"} is said to be {\em accepted} (or {\em recognized}) by a DFA $M$ if
-$\dfa{M}(x) \in F_M$. The language recoginzed by DFA $M$, denoted
-$L(M)$, is defined as:
-\begin{equation}
- L(M) \equiv \{x~|~\dfa{M}(x) \in F_M\}
-\end{equation}
-The standard way of specifying a laugage $\Lang$ as {\em regular} is by stipulating that:
-$\Lang = L(M)$ for some DFA $M$.
-
-For any DFA $M$, the DFA obtained by changing initial state to another $p \in Q_M$ is denoted $M_p$,
-which is defined as:
-\begin{equation}
- M_p \ \equiv\ (Q_M, \Sigma_M, \delta_M, p, F_M)
-\end{equation}
-Two states $p, q \in Q_M$ are said to be {\em equivalent}, denoted $p \approx_M q$, iff.
-\begin{equation}\label{m_eq_def}
- L(M_p) = L(M_q)
-\end{equation}
-It is obvious that $\approx_M$ is an equivalent relation over $Q_M$. and
-the partition induced by $\approx_M$ has $|Q_M|$ equivalent classes.
-By overloading $\approx_M$, an equivalent relation over strings can be defined:
-\begin{equation}
- x \approx_M y ~ ~ \equiv ~ ~ \dfa{M}(x) \approx_M \dfa{M}(y)
-\end{equation}
-It can be proved that the the partition induced by $\approx_M$ also has $|Q_M|$ equivalent classes.
-It is also easy to show that: if $x \approx_M y$, then $x \approx_{L(M)} y$, and this means
-$\approx_M$ is a more refined equivalent relation than $\approx_{L(M)}$. Since partition induced by
-$\approx_M$ is finite, the one induced by $\approx_{L(M)}$ must also be finite, and this is
-one of the two directions of \mht:
-\begin{Lem}[\mht , Direction two]
- If a language $\Lang$ is regular (i.e. $\Lang = L(M)$ for some DFA $M$), then
- the partition induced by $\approx_\Lang$ is finite.
-\end{Lem}
-
-The other direction is:
-\begin{Lem}[\mht , Direction one]\label{auto_mh_d1}
- If the partition induced by $\approx_\Lang$ is finite, then
- $\Lang$ is regular (i.e. $\Lang = L(M)$ for some DFA $M$).
-\end{Lem}
-The $M$ we are seeking when prove lemma \ref{auto_mh_d2} can be constructed out of $\approx_\Lang$,
-denoted $M_\Lang$ and defined as the following:
-\begin{subequations}
-\begin{eqnarray}
- Q_{M_\Lang} ~ & \equiv & ~ \{ \cls{x}{\approx_\Lang}~|~ x \in \Sigma^* \}\\
- \Sigma_{M_\Lang} ~ & \equiv & ~ \Sigma_M \\
- \delta_{M_\Lang} ~ & \equiv & ~ \left (\lambda (\cls{x}{\approx_\Lang}, a). \cls{xa}{\approx_\Lang} \right) \\
- s_{M_\Lang} ~ & \equiv & ~ \cls{[]}{\approx_\Lang} \\
- F_{M_\Lang} ~ & \equiv & ~ \{ \cls{x}{\approx_\Lang}~|~ x \in \Lang \}
-\end{eqnarray}
-\end{subequations}
-It can be proved that $Q_{M_\Lang}$ is indeed finite and $\Lang = L(M_\Lang)$, so lemma \ref{auto_mh_d1} holds.
-It can also be proved that $M_\Lang$ is the minimal DFA (therefore unique) which recoginzes $\Lang$.
-
-
-
-*}
-
-subsection {* The objective and the underlying intuition *}
-
-text {*
- It is now obvious from section \ref{sec_fa_mh} that \mht\ can be established easily when
- {\em reglar languages} are defined as ones recognized by finite automata.
- Under the context where the use of finite automata is forbiden, the situation is quite different.
- The theorem now has to be expressed as:
- \begin{Thm}[\mht , Regular expression version]
- A language $\Lang$ is regular (i.e. $\Lang = L(\re)$ for some {\em regular expression} $\re$)
- iff. the partition induced by $\approx_\Lang$ is finite.
- \end{Thm}
- The proof of this version consists of two directions (if the use of automata are not allowed):
- \begin{description}
- \item [Direction one:]
- generating a regular expression $\re$ out of the finite partition induced by $\approx_\Lang$,
- such that $\Lang = L(\re)$.
- \item [Direction two:]
- showing the finiteness of the partition induced by $\approx_\Lang$, under the assmption
- that $\Lang$ is recognized by some regular expression $\re$ (i.e. $\Lang = L(\re)$).
- \end{description}
- The development of these two directions consititutes the body of this paper.
-
-*}
-
-section {* Direction @{text "regular language \<Rightarrow>finite partition"} *}
-
-text {*
- Although not used explicitly, the notion of finite autotmata and its relationship with
- language partition, as outlined in section \ref{sec_fa_mh}, still servers as important intuitive
- guides in the development of this paper.
- For example, {\em Direction one} follows the {\em Brzozowski algebraic method}
- used to convert finite autotmata to regular expressions, under the intuition that every
- partition member $\cls{x}{\approx_\Lang}$ is a state in the DFA $M_\Lang$ constructed to prove
- lemma \ref{auto_mh_d1} of section \ref{sec_fa_mh}.
-
- The basic idea of Brzozowski method is to extract an equational system out of the
- transition relationship of the automaton in question. In the equational system, every
- automaton state is represented by an unknown, the solution of which is expected to be
- a regular expresion characterizing the state in a certain sense. There are two choices of
- how a automaton state can be characterized. The first is to characterize by the set of
- strings leading from the state in question into accepting states.
- The other choice is to characterize by the set of strings leading from initial state
- into the state in question. For the second choice, the language recognized the automaton
- can be characterized by the solution of initial state, while for the second choice,
- the language recoginzed by the automaton can be characterized by
- combining solutions of all accepting states by @{text "+"}. Because of the automaton
- used as our intuitive guide, the $M_\Lang$, the states of which are
- sets of strings leading from initial state, the second choice is used in this paper.
-
- Supposing the automaton in Fig \ref{fig_auto_part_rel} is the $M_\Lang$ for some language $\Lang$,
- and suppose $\Sigma = \{a, b, c, d, e\}$. Under the second choice, the equational system extracted is:
- \begin{subequations}
- \begin{eqnarray}
- X_0 & = & X_1 \cdot c + X_2 \cdot d + \lambda \label{x_0_def_o} \\
- X_1 & = & X_0 \cdot a + X_1 \cdot b + X_2 \cdot d \label{x_1_def_o} \\
- X_2 & = & X_0 \cdot b + X_1 \cdot d + X_2 \cdot a \\
- X_3 & = & \begin{aligned}
- & X_0 \cdot (c + d + e) + X_1 \cdot (a + e) + X_2 \cdot (b + e) + \\
- & X_3 \cdot (a + b + c + d + e)
- \end{aligned}
- \end{eqnarray}
- \end{subequations}
-
-\begin{figure}[h!]
-\centering
-\begin{minipage}{0.5\textwidth}
-\scalebox{0.8}{
-\begin{tikzpicture}[ultra thick,>=latex]
- \node[draw,circle,initial] (start) {$X_0$};
- \node[draw,circle,accepting] at ($(start) + (3.5cm,2cm)$) (ac1) {$X_1$};
- \node[draw,circle,accepting] at ($(start) + (3.5cm,-2cm)$) (ac2) {$X_2$};
- \node[draw,circle] at ($(start) + (6.5cm,0cm)$) (ab) {$X_3$};
-
- \path[->] (start) edge [bend left] node [midway, above] {$a$} (ac1);
- \path[->] (start) edge [bend right] node [midway, below] {$b$} (ac2);
- \path[->] (ac1) edge [loop above] node [midway, above] {$b$} (ac1);
- \path[->] (ac2) edge [loop below] node [midway, below] {$a$} (ac2);
- \path[->] (ac1) edge [bend right] node [midway, left] {$c$} (ac2);
- \path[->] (ac2) edge [bend right] node [midway, right] {$c$} (ac1);
- \path[->] (ac1) edge node [midway, sloped, above] {$d$} (start);
- \path[->] (ac2) edge node [midway, sloped, above] {$d$} (start);
-
- \path [draw, rounded corners,->,dashed] (start) -- ($(start) + (0cm, 3.7cm)$)
- -- ($(ab) + (0cm, 3.7cm)$) node[midway,above,sloped]{$\Sigma - \{a, b\}$} -- (ab);
- \path[->,dashed] (ac1) edge node [midway, above, sloped] {$\Sigma - \{b,c,d\}$} (ab);
- \path[->,dashed] (ac2) edge node [midway, below, sloped] {$\Sigma - \{a,c,d\}$} (ab);
- \path[->,dashed] (ab) edge [loop right] node [midway, right] {$\Sigma$} (ab);
-\end{tikzpicture}}
-\end{minipage}
-\caption{An example automaton}\label{fig_auto_part_rel}
-\end{figure}
-
- Every $\cdot$-item on the right side of equations describes some state transtions, except
- the $\lambda$ in \eqref{x_0_def_o}, which represents empty string @{text "[]"}.
- The reason is that: every state is characterized by the
- set of incoming strings leading from initial state. For non-initial state, every such
- string can be splitted into a prefix leading into a preceding state and a single character suffix
- transiting into from the preceding state. The exception happens at
- initial state, where the empty string is a incoming string which can not be splitted. The $\lambda$
- in \eqref{x_0_def_o} is introduce to repsent this indivisible string. There is one and only one
- $\lambda$ in every equational system such obtained, becasue $[]$ can only be contaied in one
- equivalent class (the intial state in $M_\Lang$) and equivalent classes are disjoint.
-
- Suppose all unknowns ($X_0, X_1, X_2, X_3$) are solvable, the regular expression charactering
- laugnage $\Lang$ is $X_1 + X_2$. This paper gives a procedure
- by which arbitrarily picked unknown can be solved. The basic idea to solve $X_i$ is by
- eliminating all variables other than $X_i$ from the equational system. If
- $X_0$ is the one picked to be solved, variables $X_1, X_2, X_3$ have to be removed one by
- one. The order to remove does not matter as long as the remaing equations are kept valid.
- Suppose $X_1$ is the first one to remove, the action is to replace all occurences of $X_1$
- in remaining equations by the right hand side of its characterizing equation, i.e.
- the $ X_0 \cdot a + X_1 \cdot b + X_2 \cdot d$ in \eqref{x_1_def_o}. However, because
- of the recursive occurence of $X_1$, this replacement does not really removed $X_1$. Arden's
- lemma is invoked to transform recursive equations like \eqref{x_1_def_o} into non-recursive
- ones. For example, the recursive equation \eqref{x_1_def_o} is transformed into the follwing
- non-recursive one:
- \begin{equation}
- X_1 = (X_0 \cdot a + X_2 \cdot d) \cdot b^* = X_0 \cdot (a \cdot b^*) + X_2 \cdot (d \cdot b^*)
- \end{equation}
- which, by Arden's lemma, still characterizes $X_1$ correctly. By subsituting
- $(X_0 \cdot a + X_2 \cdot d) \cdot b^*$ for all $X_1$ and removing \eqref{x_1_def_o},
- we get:
- \begin{subequations}
- \begin{eqnarray}
- X_0 & = & \begin{aligned}
- & (X_0 \cdot (a \cdot b^*) + X_2 \cdot (d \cdot b^*)) \cdot c +
- X_2 \cdot d + \lambda = \\
- & X_0 \cdot (a \cdot b^* \cdot c) + X_2 \cdot (d \cdot b^* \cdot c) +
- X_2 \cdot d + \lambda = \\
- & X_0 \cdot (a \cdot b^* \cdot c) + X_2 \cdot (d \cdot b^* \cdot c + d) + \lambda
- \end{aligned} \label{x_0_def_1} \\
- X_2 & = & \begin{aligned}
- & X_0 \cdot b + (X_0 \cdot (a \cdot b^*) + X_2 \cdot (d \cdot b^*)) \cdot d + X_2 \cdot a = \\
- & X_0 \cdot b + X_0 \cdot (a \cdot b^* \cdot d) + X_2 \cdot (d \cdot b^* \cdot d) + X_2 \cdot a = \\
- & X_0 \cdot (b + a \cdot b^* \cdot d) + X_2 \cdot (d \cdot b^* \cdot d + a)
- \end{aligned} \\
- X_3 & = & \begin{aligned}
- & X_0 \cdot (c + d + e) + ((X_0 \cdot a + X_2 \cdot d) \cdot b^*) \cdot (a + e)\\
- & + X_2 \cdot (b + e) + X_3 \cdot (a + b + c + d + e) \label{x_3_def_1}
- \end{aligned}
- \end{eqnarray}
- \end{subequations}
-Suppose $X_3$ is the one to remove next, since $X_3$ dose not appear in either $X_0$ or $X_2$,
-the removal of equation \eqref{x_3_def_1} changes nothing in the rest equations. Therefore, we get:
- \begin{subequations}
- \begin{eqnarray}
- X_0 & = & X_0 \cdot (a \cdot b^* \cdot c) + X_2 \cdot (d \cdot b^* \cdot c + d) + \lambda \label{x_0_def_2} \\
- X_2 & = & X_0 \cdot (b + a \cdot b^* \cdot d) + X_2 \cdot (d \cdot b^* \cdot d + a) \label{x_2_def_2}
- \end{eqnarray}
- \end{subequations}
-Actually, since absorbing state like $X_3$ contributes nothing to the language $\Lang$, it could have been removed
-at the very beginning of this precedure without affecting the final result. Now, the last unknown to remove
-is $X_2$ and the Arden's transformaton of \eqref{x_2_def_2} is:
-\begin{equation} \label{x_2_ardened}
- X_2 ~ = ~ (X_0 \cdot (b + a \cdot b^* \cdot d)) \cdot (d \cdot b^* \cdot d + a)^* =
- X_0 \cdot ((b + a \cdot b^* \cdot d) \cdot (d \cdot b^* \cdot d + a)^*)
-\end{equation}
-By substituting the right hand side of \eqref{x_2_ardened} into \eqref{x_0_def_2}, we get:
-\begin{equation}
-\begin{aligned}
- X_0 & = && X_0 \cdot (a \cdot b^* \cdot c) + \\
- & && X_0 \cdot ((b + a \cdot b^* \cdot d) \cdot (d \cdot b^* \cdot d + a)^*) \cdot
- (d \cdot b^* \cdot c + d) + \lambda \\
- & = && X_0 \cdot ((a \cdot b^* \cdot c) + \\
- & && \hspace{3em} ((b + a \cdot b^* \cdot d) \cdot (d \cdot b^* \cdot d + a)^*) \cdot
- (d \cdot b^* \cdot c + d)) + \lambda
-\end{aligned}
-\end{equation}
-By applying Arden's transformation to this, we get the solution of $X_0$ as:
-\begin{equation}
-\begin{aligned}
- X_0 = ((a \cdot b^* \cdot c) +
- ((b + a \cdot b^* \cdot d) \cdot (d \cdot b^* \cdot d + a)^*) \cdot
- (d \cdot b^* \cdot c + d))^*
-\end{aligned}
-\end{equation}
-Using the same method, solutions for $X_1$ and $X_2$ can be obtained as well and the
-regular expressoin for $\Lang$ is just $X_1 + X_2$. The formalization of this procedure
-consititues the first direction of the {\em regular expression} verion of
-\mht. Detailed explaination are given in {\bf paper.pdf} and more details and comments
-can be found in the formal scripts.
-*}
-
-section {* Direction @{text "finite partition \<Rightarrow> regular language"} *}
-
-text {*
- It is well known in the theory of regular languages that
- the existence of finite language partition amounts to the existence of
- a minimal automaton, i.e. the $M_\Lang$ constructed in section \ref{sec_prelim}, which recoginzes the
- same language $\Lang$. The standard way to prove the existence of finite language partition
- is to construct a automaton out of the regular expression which recoginzes the same language, from
- which the existence of finite language partition follows immediately. As discussed in the introducton of
- {\bf paper.pdf} as well as in [5], the problem for this approach happens when automata
- of sub regular expressions are combined to form the automaton of the mother regular expression,
- no matter what kind of representation is used, the formalization is cubersome, as said
- by Nipkow in [5]: `{\em a more abstract mathod is clearly desirable}'. In this section,
- an {\em intrinsically abstract} method is given, which completely avoid the mentioning of automata,
- let along any particular representations.
- *}
-
-text {*
- The main proof structure is a structural induction on regular expressions,
- where base cases (cases for @{const "NULL"}, @{const "EMPTY"}, @{const "CHAR"}) are quite straightforward to
- proof. Real difficulty lies in inductive cases. By inductive hypothesis, languages defined by
- sub-expressions induce finite partitiions. Under such hypothsis, we need to prove that the language
- defined by the composite regular expression gives rise to finite partion.
- The basic idea is to attach a tag @{text "tag(x)"} to every string
- @{text "x"}. The tagging fuction @{text "tag"} is carefully devised, which returns tags
- made of equivalent classes of the partitions induced by subexpressoins, and therefore has a finite
- range. Let @{text "Lang"} be the composite language, it is proved that:
- \begin{quote}
- If strings with the same tag are equivalent with respect to @{text "Lang"}, expressed as:
- \[
- @{text "tag(x) = tag(y) \<Longrightarrow> x \<approx>Lang y"}
- \]
- then the partition induced by @{text "Lang"} must be finite.
- \end{quote}
- There are two arguments for this. The first goes as the following:
- \begin{enumerate}
- \item First, the tagging function @{text "tag"} induces an equivalent relation @{text "(=tag=)"}
- (defiintion of @{text "f_eq_rel"} and lemma @{text "equiv_f_eq_rel"}).
- \item It is shown that: if the range of @{text "tag"} (denoted @{text "range(tag)"}) is finite,
- the partition given rise by @{text "(=tag=)"} is finite (lemma @{text "finite_eq_f_rel"}).
- Since tags are made from equivalent classes from component partitions, and the inductive
- hypothesis ensures the finiteness of these partitions, it is not difficult to prove
- the finiteness of @{text "range(tag)"}.
- \item It is proved that if equivalent relation @{text "R1"} is more refined than @{text "R2"}
- (expressed as @{text "R1 \<subseteq> R2"}),
- and the partition induced by @{text "R1"} is finite, then the partition induced by @{text "R2"}
- is finite as well (lemma @{text "refined_partition_finite"}).
- \item The injectivity assumption @{text "tag(x) = tag(y) \<Longrightarrow> x \<approx>Lang y"} implies that
- @{text "(=tag=)"} is more refined than @{text "(\<approx>Lang)"}.
- \item Combining the points above, we have: the partition induced by language @{text "Lang"}
- is finite (lemma @{text "tag_finite_imageD"}).
- \end{enumerate}
-
-We could have followed another approach given in appendix II of Brzozowski's paper [?], where
-the set of derivatives of any regular expression can be proved to be finite.
-Since it is easy to show that strings with same derivative are equivalent with respect to the
-language, then the second direction follows. We believe that our
-apporoach is easy to formalize, with no need to do complicated derivation calculations
-and countings as in [???].
-*}
-
-
-end
--- a/MyhillNerode.thy Fri Jun 03 13:59:21 2011 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,1816 +0,0 @@
-theory MyhillNerode
- imports "Main" "List_Prefix"
-begin
-
-text {* sequential composition of languages *}
-
-definition
- lang_seq :: "string set \<Rightarrow> string set \<Rightarrow> string set" ("_ ; _" [100,100] 100)
-where
- "L1 ; L2 = {s1 @ s2 | s1 s2. s1 \<in> L1 \<and> s2 \<in> L2}"
-
-lemma lang_seq_empty:
- shows "{[]} ; L = L"
- and "L ; {[]} = L"
-unfolding lang_seq_def by auto
-
-lemma lang_seq_null:
- shows "{} ; L = {}"
- and "L ; {} = {}"
-unfolding lang_seq_def by auto
-
-lemma lang_seq_append:
- assumes a: "s1 \<in> L1"
- and b: "s2 \<in> L2"
- shows "s1@s2 \<in> L1 ; L2"
-unfolding lang_seq_def
-using a b by auto
-
-lemma lang_seq_union:
- shows "(L1 \<union> L2); L3 = (L1; L3) \<union> (L2; L3)"
- and "L1; (L2 \<union> L3) = (L1; L2) \<union> (L1; L3)"
-unfolding lang_seq_def by auto
-
-lemma lang_seq_assoc:
- shows "(L1 ; L2) ; L3 = L1 ; (L2 ; L3)"
-unfolding lang_seq_def
-apply(auto)
-apply(metis)
-by (metis append_assoc)
-
-
-section {* Kleene star for languages defined as least fixed point *}
-
-inductive_set
- Star :: "string set \<Rightarrow> string set" ("_\<star>" [101] 102)
- for L :: "string set"
-where
- start[intro]: "[] \<in> L\<star>"
-| step[intro]: "\<lbrakk>s1 \<in> L; s2 \<in> L\<star>\<rbrakk> \<Longrightarrow> s1@s2 \<in> L\<star>"
-
-lemma lang_star_empty:
- shows "{}\<star> = {[]}"
-by (auto elim: Star.cases)
-
-lemma lang_star_cases:
- shows "L\<star> = {[]} \<union> L ; L\<star>"
-proof
- { fix x
- have "x \<in> L\<star> \<Longrightarrow> x \<in> {[]} \<union> L ; L\<star>"
- unfolding lang_seq_def
- by (induct rule: Star.induct) (auto)
- }
- then show "L\<star> \<subseteq> {[]} \<union> L ; L\<star>" by auto
-next
- show "{[]} \<union> L ; L\<star> \<subseteq> L\<star>"
- unfolding lang_seq_def by auto
-qed
-
-lemma lang_star_cases':
- shows "L\<star> = {[]} \<union> L\<star> ; L"
-proof
- { fix x
- have "x \<in> L\<star> \<Longrightarrow> x \<in> {[]} \<union> L\<star> ; L"
- unfolding lang_seq_def
- apply (induct rule: Star.induct)
- apply simp
- apply simp
- apply (erule disjE)
- apply (auto)[1]
- apply (erule exE | erule conjE)+
- apply (rule disjI2)
- apply (rule_tac x = "s1 @ s1a" in exI)
- by auto
- }
- then show "L\<star> \<subseteq> {[]} \<union> L\<star> ; L" by auto
-next
- show "{[]} \<union> L\<star> ; L \<subseteq> L\<star>"
- unfolding lang_seq_def
- apply auto
- apply (erule Star.induct)
- apply auto
- apply (drule step[of _ _ "[]"])
- by (auto intro:start)
-qed
-
-lemma lang_star_simple:
- shows "L \<subseteq> L\<star>"
-by (subst lang_star_cases)
- (auto simp only: lang_seq_def)
-
-lemma lang_star_prop0_aux:
- "s2 \<in> L\<star> \<Longrightarrow> \<forall> s1. s1 \<in> L \<longrightarrow> (\<exists> s3 s4. s3 \<in> L\<star> \<and> s4 \<in> L \<and> s1 @ s2 = s3 @ s4)"
-apply (erule Star.induct)
-apply (clarify, rule_tac x = "[]" in exI, rule_tac x = s1 in exI, simp add:start)
-apply (clarify, drule_tac x = s1 in spec)
-apply (drule mp, simp, clarify)
-apply (rule_tac x = "s1a @ s3" in exI, rule_tac x = s4 in exI)
-by auto
-
-lemma lang_star_prop0:
- "\<lbrakk>s1 \<in> L; s2 \<in> L\<star>\<rbrakk> \<Longrightarrow> \<exists> s3 s4. s3 \<in> L\<star> \<and> s4 \<in> L \<and> s1 @ s2 = s3 @ s4"
-by (auto dest:lang_star_prop0_aux)
-
-lemma lang_star_prop1:
- assumes asm: "L1; L2 \<subseteq> L2"
- shows "L1\<star>; L2 \<subseteq> L2"
-proof -
- { fix s1 s2
- assume minor: "s2 \<in> L2"
- assume major: "s1 \<in> L1\<star>"
- then have "s1@s2 \<in> L2"
- proof(induct rule: Star.induct)
- case start
- show "[]@s2 \<in> L2" using minor by simp
- next
- case (step s1 s1')
- have "s1 \<in> L1" by fact
- moreover
- have "s1'@s2 \<in> L2" by fact
- ultimately have "s1@(s1'@s2) \<in> L1; L2" by (auto simp add: lang_seq_def)
- with asm have "s1@(s1'@s2) \<in> L2" by auto
- then show "(s1@s1')@s2 \<in> L2" by simp
- qed
- }
- then show "L1\<star>; L2 \<subseteq> L2" by (auto simp add: lang_seq_def)
-qed
-
-lemma lang_star_prop2_aux:
- "s2 \<in> L2\<star> \<Longrightarrow> \<forall> s1. s1 \<in> L1 \<and> L1 ; L2 \<subseteq> L1 \<longrightarrow> s1 @ s2 \<in> L1"
-apply (erule Star.induct, simp)
-apply (clarify, drule_tac x = "s1a @ s1" in spec)
-by (auto simp:lang_seq_def)
-
-lemma lang_star_prop2:
- "L1; L2 \<subseteq> L1 \<Longrightarrow> L1 ; L2\<star> \<subseteq> L1"
-by (auto dest!:lang_star_prop2_aux simp:lang_seq_def)
-
-lemma lang_star_seq_subseteq:
- shows "L ; L\<star> \<subseteq> L\<star>"
-using lang_star_cases by blast
-
-lemma lang_star_double:
- shows "L\<star>; L\<star> = L\<star>"
-proof
- show "L\<star> ; L\<star> \<subseteq> L\<star>"
- using lang_star_prop1 lang_star_seq_subseteq by blast
-next
- have "L\<star> \<subseteq> L\<star> \<union> L\<star>; (L; L\<star>)" by auto
- also have "\<dots> = L\<star>;{[]} \<union> L\<star>; (L; L\<star>)" by (simp add: lang_seq_empty)
- also have "\<dots> = L\<star>; ({[]} \<union> L; L\<star>)" by (simp only: lang_seq_union)
- also have "\<dots> = L\<star>; L\<star>" using lang_star_cases by simp
- finally show "L\<star> \<subseteq> L\<star> ; L\<star>" by simp
-qed
-
-lemma lang_star_seq_subseteq':
- shows "L\<star>; L \<subseteq> L\<star>"
-proof -
- have "L \<subseteq> L\<star>" by (rule lang_star_simple)
- then have "L\<star>; L \<subseteq> L\<star>; L\<star>" by (auto simp add: lang_seq_def)
- then show "L\<star>; L \<subseteq> L\<star>" using lang_star_double by blast
-qed
-
-lemma
- shows "L\<star> \<subseteq> L\<star>\<star>"
-by (rule lang_star_simple)
-
-
-section {* regular expressions *}
-
-datatype rexp =
- NULL
-| EMPTY
-| CHAR char
-| SEQ rexp rexp
-| ALT rexp rexp
-| STAR rexp
-
-
-consts L:: "'a \<Rightarrow> string set"
-
-overloading L_rexp \<equiv> "L:: rexp \<Rightarrow> string set"
-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
-
-
-text{* ************ now is the codes writen by chunhan ************************************* *}
-
-section {* Arden's Lemma revised *}
-
-lemma arden_aux1:
- assumes a: "X \<subseteq> X ; A \<union> B"
- and b: "[] \<notin> A"
- shows "x \<in> X \<Longrightarrow> x \<in> B ; A\<star>"
-apply (induct x taking:length rule:measure_induct)
-apply (subgoal_tac "x \<in> X ; A \<union> B")
-defer
-using a
-apply (auto)[1]
-apply simp
-apply (erule disjE)
-defer
-apply (auto simp add:lang_seq_def) [1]
-apply (subgoal_tac "\<exists> x1 x2. x = x1 @ x2 \<and> x1 \<in> X \<and> x2 \<in> A")
-defer
-apply (auto simp add:lang_seq_def) [1]
-apply (erule exE | erule conjE)+
-apply simp
-apply (drule_tac x = x1 in spec)
-apply (simp)
-using b
-apply -
-apply (auto)[1]
-apply (subgoal_tac "x1 @ x2 \<in> (B ; A\<star>) ; A")
-defer
-apply (auto simp add:lang_seq_def)[1]
-by (metis Un_absorb1 lang_seq_assoc lang_seq_union(2) lang_star_double lang_star_simple mem_def sup1CI)
-
-theorem ardens_revised:
- assumes nemp: "[] \<notin> A"
- shows "(X = X ; A \<union> B) \<longleftrightarrow> (X = B ; A\<star>)"
-apply(rule iffI)
-defer
-apply(simp)
-apply(subst lang_star_cases')
-apply(subst lang_seq_union)
-apply(simp add: lang_seq_empty)
-apply(simp add: lang_seq_assoc)
-apply(auto)[1]
-proof -
- assume "X = X ; A \<union> B"
- then have as1: "X ; A \<union> B \<subseteq> X" and as2: "X \<subseteq> X ; A \<union> B" by simp_all
- from as1 have a: "X ; A \<subseteq> X" and b: "B \<subseteq> X" by simp_all
- from b have "B; A\<star> \<subseteq> X ; A\<star>" by (auto simp add: lang_seq_def)
- moreover
- from a have "X ; A\<star> \<subseteq> X"
-
-by (rule lang_star_prop2)
- ultimately have f1: "B ; A\<star> \<subseteq> X" by simp
- from as2 nemp
- have f2: "X \<subseteq> B; A\<star>" using arden_aux1 by auto
- from f1 f2 show "X = B; A\<star>" by auto
-qed
-
-
-
-section {* equiv class' definition *}
-
-definition
- equiv_str :: "string \<Rightarrow> string set \<Rightarrow> string \<Rightarrow> bool" ("_ \<equiv>_\<equiv> _" [100, 100, 100] 100)
-where
- "x \<equiv>Lang\<equiv> y \<longleftrightarrow> (\<forall>z. x @ z \<in> Lang \<longleftrightarrow> y @ z \<in> Lang)"
-
-definition
- equiv_class :: "string \<Rightarrow> (string set) \<Rightarrow> string set" ("\<lbrakk>_\<rbrakk>_" [100, 100] 100)
-where
- "\<lbrakk>x\<rbrakk>Lang \<equiv> {y. x \<equiv>Lang\<equiv> y}"
-
-text {* Chunhan modifies Q to Quo *}
-
-definition
- quot :: "string set \<Rightarrow> string set \<Rightarrow> (string set) set" ("_ Quo _" [100, 100] 100)
-where
- "L1 Quo L2 \<equiv> { \<lbrakk>x\<rbrakk>L2 | x. x \<in> L1}"
-
-
-lemma lang_eqs_union_of_eqcls:
- "Lang = \<Union> {X. X \<in> (UNIV Quo Lang) \<and> (\<forall> x \<in> X. x \<in> Lang)}"
-proof
- show "Lang \<subseteq> \<Union>{X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang}"
- proof
- fix x
- assume "x \<in> Lang"
- thus "x \<in> \<Union>{X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang}"
- proof (simp add:quot_def)
- assume "(1)": "x \<in> Lang"
- show "\<exists>xa. (\<exists>x. xa = \<lbrakk>x\<rbrakk>Lang) \<and> (\<forall>x\<in>xa. x \<in> Lang) \<and> x \<in> xa" (is "\<exists>xa.?P xa")
- proof -
- have "?P (\<lbrakk>x\<rbrakk>Lang)" using "(1)"
- by (auto simp:equiv_class_def equiv_str_def dest: spec[where x = "[]"])
- thus ?thesis by blast
- qed
- qed
- qed
-next
- show "\<Union>{X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang} \<subseteq> Lang"
- by auto
-qed
-
-lemma empty_notin_CS: "{} \<notin> UNIV Quo Lang"
-apply (clarsimp simp:quot_def equiv_class_def)
-by (rule_tac x = x in exI, auto simp:equiv_str_def)
-
-lemma no_two_cls_inters:
- "\<lbrakk>X \<in> UNIV Quo Lang; Y \<in> UNIV Quo Lang; X \<noteq> Y\<rbrakk> \<Longrightarrow> X \<inter> Y = {}"
-by (auto simp:quot_def equiv_class_def equiv_str_def)
-
-text {* equiv_class transition *}
-definition
- CT :: "string set \<Rightarrow> char \<Rightarrow> string set \<Rightarrow> bool" ("_-_\<rightarrow>_" [99,99]99)
-where
- "X-c\<rightarrow>Y \<equiv> ((X;{[c]}) \<subseteq> Y)"
-
-types t_equa_rhs = "(string set \<times> rexp) set"
-
-types t_equa = "(string set \<times> t_equa_rhs)"
-
-types t_equas = "t_equa set"
-
-text {*
- "empty_rhs" generates "\<lambda>" for init-state, just like "\<lambda>" for final states
- in Brzozowski method. But if the init-state is "{[]}" ("\<lambda>" itself) then
- empty set is returned, see definition of "equation_rhs"
-*}
-
-definition
- empty_rhs :: "string set \<Rightarrow> t_equa_rhs"
-where
- "empty_rhs X \<equiv> if ([] \<in> X) then {({[]}, EMPTY)} else {}"
-
-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"
-
-definition
- equation_rhs :: "(string set) set \<Rightarrow> string set \<Rightarrow> t_equa_rhs"
-where
- "equation_rhs CS X \<equiv> if (X = {[]}) then {({[]}, EMPTY)}
- else {(S, folds ALT NULL {CHAR c| c. S-c\<rightarrow>X} )| S. S \<in> CS} \<union> empty_rhs X"
-
-definition
- equations :: "(string set) set \<Rightarrow> t_equas"
-where
- "equations CS \<equiv> {(X, equation_rhs CS X) | X. X \<in> CS}"
-
-overloading L_rhs \<equiv> "L:: t_equa_rhs \<Rightarrow> string set"
-begin
-fun L_rhs:: "t_equa_rhs \<Rightarrow> string set"
-where
- "L_rhs rhs = {x. \<exists> X r. (X, r) \<in> rhs \<and> x \<in> X;(L r)}"
-end
-
-definition
- distinct_rhs :: "t_equa_rhs \<Rightarrow> bool"
-where
- "distinct_rhs rhs \<equiv> \<forall> X reg\<^isub>1 reg\<^isub>2. (X, reg\<^isub>1) \<in> rhs \<and> (X, reg\<^isub>2) \<in> rhs \<longrightarrow> reg\<^isub>1 = reg\<^isub>2"
-
-definition
- distinct_equas_rhs :: "t_equas \<Rightarrow> bool"
-where
- "distinct_equas_rhs equas \<equiv> \<forall> X rhs. (X, rhs) \<in> equas \<longrightarrow> distinct_rhs rhs"
-
-definition
- distinct_equas :: "t_equas \<Rightarrow> bool"
-where
- "distinct_equas equas \<equiv> \<forall> X rhs rhs'. (X, rhs) \<in> equas \<and> (X, rhs') \<in> equas \<longrightarrow> rhs = rhs'"
-
-definition
- seq_rhs_r :: "t_equa_rhs \<Rightarrow> rexp \<Rightarrow> t_equa_rhs"
-where
- "seq_rhs_r rhs r \<equiv> (\<lambda>(X, reg). (X, SEQ reg r)) ` rhs"
-
-definition
- del_x_paired :: "('a \<times> 'b) set \<Rightarrow> 'a \<Rightarrow> ('a \<times> 'b) set"
-where
- "del_x_paired S x \<equiv> S - {X. X \<in> S \<and> fst X = x}"
-
-definition
- arden_variate :: "string set \<Rightarrow> rexp \<Rightarrow> t_equa_rhs \<Rightarrow> t_equa_rhs"
-where
- "arden_variate X r rhs \<equiv> seq_rhs_r (del_x_paired rhs X) (STAR r)"
-
-definition
- no_EMPTY_rhs :: "t_equa_rhs \<Rightarrow> bool"
-where
- "no_EMPTY_rhs rhs \<equiv> \<forall> X r. (X, r) \<in> rhs \<and> X \<noteq> {[]} \<longrightarrow> [] \<notin> L r"
-
-definition
- no_EMPTY_equas :: "t_equas \<Rightarrow> bool"
-where
- "no_EMPTY_equas equas \<equiv> \<forall> X rhs. (X, rhs) \<in> equas \<longrightarrow> no_EMPTY_rhs rhs"
-
-lemma fold_alt_null_eqs:
- "finite rS \<Longrightarrow> x \<in> L (folds ALT NULL rS) = (\<exists> r \<in> rS. x \<in> L r)"
-apply (simp add:folds_def)
-apply (rule someI2_ex)
-apply (erule finite_imp_fold_graph)
-apply (erule fold_graph.induct)
-by auto (*??? how do this be in Isar ?? *)
-
-lemma seq_rhs_r_prop1:
- "L (seq_rhs_r rhs r) = (L rhs);(L r)"
-apply (auto simp:seq_rhs_r_def image_def lang_seq_def)
-apply (rule_tac x = "s1 @ s1a" in exI, rule_tac x = "s2a" in exI, simp)
-apply (rule_tac x = a in exI, rule_tac x = b in exI, simp)
-apply (rule_tac x = s1 in exI, rule_tac x = s1a in exI, simp)
-apply (rule_tac x = X in exI, rule_tac x = "SEQ ra r" in exI, simp)
-apply (rule conjI)
-apply (rule_tac x = "(X, ra)" in bexI, simp+)
-apply (rule_tac x = s1a in exI, rule_tac x = "s2a @ s2" in exI, simp)
-apply (simp add:lang_seq_def)
-by (rule_tac x = s2a in exI, rule_tac x = s2 in exI, simp)
-
-lemma del_x_paired_prop1:
- "\<lbrakk>distinct_rhs rhs; (X, r) \<in> rhs\<rbrakk> \<Longrightarrow> X ; L r \<union> L (del_x_paired rhs X) = L rhs"
- apply (simp add:del_x_paired_def)
- apply (simp add: distinct_rhs_def)
- apply(auto simp add: lang_seq_def)
- apply(metis)
- done
-
-lemma arden_variate_prop:
- assumes "(X, rx) \<in> rhs"
- shows "(\<forall> Y. Y \<noteq> X \<longrightarrow> (\<exists> r. (Y, r) \<in> rhs) = (\<exists> r. (Y, r) \<in> (arden_variate X rx rhs)))"
-proof (rule allI, rule impI)
- fix Y
- assume "(1)": "Y \<noteq> X"
- show "(\<exists>r. (Y, r) \<in> rhs) = (\<exists>r. (Y, r) \<in> arden_variate X rx rhs)"
- proof
- assume "(1_1)": "\<exists>r. (Y, r) \<in> rhs"
- show "\<exists>r. (Y, r) \<in> arden_variate X rx rhs" (is "\<exists>r. ?P r")
- proof -
- from "(1_1)" obtain r where "(Y, r) \<in> rhs" ..
- hence "?P (SEQ r (STAR rx))"
- proof (simp add:arden_variate_def image_def)
- have "(Y, r) \<in> rhs \<Longrightarrow> (Y, r) \<in> del_x_paired rhs X"
- by (auto simp:del_x_paired_def "(1)")
- thus "(Y, r) \<in> rhs \<Longrightarrow> (Y, SEQ r (STAR rx)) \<in> seq_rhs_r (del_x_paired rhs X) (STAR rx)"
- by (auto simp:seq_rhs_r_def)
- qed
- thus ?thesis by blast
- qed
- next
- assume "(2_1)": "\<exists>r. (Y, r) \<in> arden_variate X rx rhs"
- thus "\<exists>r. (Y, r) \<in> rhs" (is "\<exists> r. ?P r")
- by (auto simp:arden_variate_def del_x_paired_def seq_rhs_r_def image_def)
- qed
-qed
-
-text {*
- arden_variate_valid: proves variation from
-
- "X = X;r + Y;ry + \<dots>" to "X = Y;(SEQ ry (STAR r)) + \<dots>"
-
- holds the law of "language of left equiv language of right"
-*}
-lemma arden_variate_valid:
- assumes X_not_empty: "X \<noteq> {[]}"
- and l_eq_r: "X = L rhs"
- and dist: "distinct_rhs rhs"
- and no_empty: "no_EMPTY_rhs rhs"
- and self_contained: "(X, r) \<in> rhs"
- shows "X = L (arden_variate X r rhs)"
-proof -
- have "[] \<notin> L r" using no_empty X_not_empty self_contained
- by (auto simp:no_EMPTY_rhs_def)
- hence ardens: "X = X;(L r) \<union> (L (del_x_paired rhs X)) \<longleftrightarrow> X = (L (del_x_paired rhs X)) ; (L r)\<star>"
- by (rule ardens_revised)
- have del_x: "X = X ; L r \<union> L (del_x_paired rhs X) \<longleftrightarrow> X = L rhs" using dist l_eq_r self_contained
- by (auto dest!:del_x_paired_prop1)
- show ?thesis
- proof
- show "X \<subseteq> L (arden_variate X r rhs)"
- proof
- fix x
- assume "(1_1)": "x \<in> X" with l_eq_r ardens del_x
- show "x \<in> L (arden_variate X r rhs)"
- by (simp add:arden_variate_def seq_rhs_r_prop1 del:L_rhs.simps)
- qed
- next
- show "L (arden_variate X r rhs) \<subseteq> X"
- proof
- fix x
- assume "(2_1)": "x \<in> L (arden_variate X r rhs)" with ardens del_x l_eq_r
- show "x \<in> X"
- by (simp add:arden_variate_def seq_rhs_r_prop1 del:L_rhs.simps)
- qed
- qed
-qed
-
-text {*
- merge_rhs {(x1, r1), (x2, r2}, (x4, r4), \<dots>} {(x1, r1'), (x3, r3'), \<dots>} =
- {(x1, ALT r1 r1'}, (x2, r2), (x3, r3'), (x4, r4), \<dots>} *}
-definition
- merge_rhs :: "t_equa_rhs \<Rightarrow> t_equa_rhs \<Rightarrow> t_equa_rhs"
-where
- "merge_rhs rhs rhs' \<equiv> {(X, r). (\<exists> r1 r2. ((X,r1) \<in> rhs \<and> (X, r2) \<in> rhs') \<and> r = ALT r1 r2) \<or>
- (\<exists> r1. (X, r1) \<in> rhs \<and> (\<not> (\<exists> r2. (X, r2) \<in> rhs')) \<and> r = r1) \<or>
- (\<exists> r2. (X, r2) \<in> rhs' \<and> (\<not> (\<exists> r1. (X, r1) \<in> rhs)) \<and> r = r2) }"
-
-
-text {* rhs_subst rhs X=xrhs r: substitude all occurence of X in rhs((X,r) \<in> rhs) with xrhs *}
-definition
- rhs_subst :: "t_equa_rhs \<Rightarrow> string set \<Rightarrow> t_equa_rhs \<Rightarrow> rexp \<Rightarrow> t_equa_rhs"
-where
- "rhs_subst rhs X xrhs r \<equiv> merge_rhs (del_x_paired rhs X) (seq_rhs_r xrhs r)"
-
-definition
- equas_subst_f :: "string set \<Rightarrow> t_equa_rhs \<Rightarrow> t_equa \<Rightarrow> t_equa"
-where
- "equas_subst_f X xrhs equa \<equiv> let (Y, rhs) = equa in
- if (\<exists> r. (X, r) \<in> rhs)
- then (Y, rhs_subst rhs X xrhs (SOME r. (X, r) \<in> rhs))
- else equa"
-
-definition
- equas_subst :: "t_equas \<Rightarrow> string set \<Rightarrow> t_equa_rhs \<Rightarrow> t_equas"
-where
- "equas_subst ES X xrhs \<equiv> del_x_paired (equas_subst_f X xrhs ` ES) X"
-
-lemma lang_seq_prop1:
- "x \<in> X ; L r \<Longrightarrow> x \<in> X ; (L r \<union> L r')"
-by (auto simp:lang_seq_def)
-
-lemma lang_seq_prop1':
- "x \<in> X; L r \<Longrightarrow> x \<in> X ; (L r' \<union> L r)"
-by (auto simp:lang_seq_def)
-
-lemma lang_seq_prop2:
- "x \<in> X; (L r \<union> L r') \<Longrightarrow> x \<in> X;L r \<or> x \<in> X;L r'"
-by (auto simp:lang_seq_def)
-
-lemma merge_rhs_prop1:
- shows "L (merge_rhs rhs rhs') = L rhs \<union> L rhs' "
-apply (auto simp add:merge_rhs_def dest!:lang_seq_prop2 intro:lang_seq_prop1)
-apply (rule_tac x = X in exI, rule_tac x = r1 in exI, simp)
-apply (case_tac "\<exists> r2. (X, r2) \<in> rhs'")
-apply (clarify, rule_tac x = X in exI, rule_tac x = "ALT r r2" in exI, simp add:lang_seq_prop1)
-apply (rule_tac x = X in exI, rule_tac x = r in exI, simp)
-apply (case_tac "\<exists> r1. (X, r1) \<in> rhs")
-apply (clarify, rule_tac x = X in exI, rule_tac x = "ALT r1 r" in exI, simp add:lang_seq_prop1')
-apply (rule_tac x = X in exI, rule_tac x = r in exI, simp)
-done
-
-lemma no_EMPTY_rhss_imp_merge_no_EMPTY:
- "\<lbrakk>no_EMPTY_rhs rhs; no_EMPTY_rhs rhs'\<rbrakk> \<Longrightarrow> no_EMPTY_rhs (merge_rhs rhs rhs')"
-apply (simp add:no_EMPTY_rhs_def merge_rhs_def)
-apply (clarify, (erule conjE | erule exE | erule disjE)+)
-by auto
-
-lemma distinct_rhs_prop:
- "\<lbrakk>distinct_rhs rhs; (X, r1) \<in> rhs; (X, r2) \<in> rhs\<rbrakk> \<Longrightarrow> r1 = r2"
-by (auto simp:distinct_rhs_def)
-
-lemma merge_rhs_prop2:
- assumes dist_rhs: "distinct_rhs rhs"
- and dist_rhs':"distinct_rhs rhs'"
- shows "distinct_rhs (merge_rhs rhs rhs')"
-apply (auto simp:merge_rhs_def distinct_rhs_def)
-using dist_rhs
-apply (drule distinct_rhs_prop, simp+)
-using dist_rhs'
-apply (drule distinct_rhs_prop, simp+)
-using dist_rhs
-apply (drule distinct_rhs_prop, simp+)
-using dist_rhs'
-apply (drule distinct_rhs_prop, simp+)
-done
-
-lemma seq_rhs_r_holds_distinct:
- "distinct_rhs rhs \<Longrightarrow> distinct_rhs (seq_rhs_r rhs r)"
-by (auto simp:distinct_rhs_def seq_rhs_r_def)
-
-lemma seq_rhs_r_prop0:
- assumes l_eq_r: "X = L xrhs"
- shows "L (seq_rhs_r xrhs r) = X ; L r "
-using l_eq_r
-by (auto simp only:seq_rhs_r_prop1)
-
-lemma rhs_subst_prop1:
- assumes l_eq_r: "X = L xrhs"
- and dist: "distinct_rhs rhs"
- shows "(X, r) \<in> rhs \<Longrightarrow> L rhs = L (rhs_subst rhs X xrhs r)"
-apply (simp add:rhs_subst_def merge_rhs_prop1 del:L_rhs.simps)
-using l_eq_r
-apply (drule_tac r = r in seq_rhs_r_prop0, simp del:L_rhs.simps)
-using dist
-by (auto dest!:del_x_paired_prop1 simp del:L_rhs.simps)
-
-lemma del_x_paired_holds_distinct_rhs:
- "distinct_rhs rhs \<Longrightarrow> distinct_rhs (del_x_paired rhs x)"
-by (auto simp:distinct_rhs_def del_x_paired_def)
-
-lemma rhs_subst_holds_distinct_rhs:
- "\<lbrakk>distinct_rhs rhs; distinct_rhs xrhs\<rbrakk> \<Longrightarrow> distinct_rhs (rhs_subst rhs X xrhs r)"
-apply (drule_tac r = r and rhs = xrhs in seq_rhs_r_holds_distinct)
-apply (drule_tac x = X in del_x_paired_holds_distinct_rhs)
-by (auto dest:merge_rhs_prop2[where rhs = "del_x_paired rhs X"] simp:rhs_subst_def)
-
-section {* myhill-nerode theorem *}
-
-definition left_eq_cls :: "t_equas \<Rightarrow> (string set) set"
-where
- "left_eq_cls ES \<equiv> {X. \<exists> rhs. (X, rhs) \<in> ES} "
-
-definition right_eq_cls :: "t_equas \<Rightarrow> (string set) set"
-where
- "right_eq_cls ES \<equiv> {Y. \<exists> X rhs r. (X, rhs) \<in> ES \<and> (Y, r) \<in> rhs }"
-
-definition rhs_eq_cls :: "t_equa_rhs \<Rightarrow> (string set) set"
-where
- "rhs_eq_cls rhs \<equiv> {Y. \<exists> r. (Y, r) \<in> rhs}"
-
-definition ardenable :: "t_equa \<Rightarrow> bool"
-where
- "ardenable equa \<equiv> let (X, rhs) = equa in
- distinct_rhs rhs \<and> no_EMPTY_rhs rhs \<and> X = L rhs"
-
-text {*
- Inv: Invairance of the equation-system, during the decrease of the equation-system, Inv holds.
-*}
-definition Inv :: "string set \<Rightarrow> t_equas \<Rightarrow> bool"
-where
- "Inv X ES \<equiv> finite ES \<and> (\<exists> rhs. (X, rhs) \<in> ES) \<and> distinct_equas ES \<and>
- (\<forall> X xrhs. (X, xrhs) \<in> ES \<longrightarrow> ardenable (X, xrhs) \<and> X \<noteq> {} \<and> rhs_eq_cls xrhs \<subseteq> insert {[]} (left_eq_cls ES))"
-
-text {*
- TCon: Termination Condition of the equation-system decreasion.
-*}
-definition TCon:: "'a set \<Rightarrow> bool"
-where
- "TCon ES \<equiv> card ES = 1"
-
-
-text {*
- The following is a iteration principle, and is the main framework for the proof:
- 1: We can form the initial equation-system using "equations" defined above, and prove it has invariance Inv by lemma "init_ES_satisfy_Inv";
- 2: We can decrease the number of the equation-system using ardens_lemma_revised and substitution ("equas_subst", defined above),
- and prove it holds the property "step" in "wf_iter" by lemma "iteration_step"
- and finally using property Inv and TCon to prove the myhill-nerode theorem
-
-*}
-lemma wf_iter [rule_format]:
- fixes f
- assumes step: "\<And> e. \<lbrakk>P e; \<not> Q e\<rbrakk> \<Longrightarrow> (\<exists> e'. P e' \<and> (f(e'), f(e)) \<in> less_than)"
- shows pe: "P e \<longrightarrow> (\<exists> e'. P e' \<and> Q e')"
-proof(induct e rule: wf_induct
- [OF wf_inv_image[OF wf_less_than, where f = "f"]], clarify)
- fix x
- assume h [rule_format]:
- "\<forall>y. (y, x) \<in> inv_image less_than f \<longrightarrow> P y \<longrightarrow> (\<exists>e'. P e' \<and> Q e')"
- and px: "P x"
- show "\<exists>e'. P e' \<and> Q e'"
- proof(cases "Q x")
- assume "Q x" with px show ?thesis by blast
- next
- assume nq: "\<not> Q x"
- from step [OF px nq]
- obtain e' where pe': "P e'" and ltf: "(f e', f x) \<in> less_than" by auto
- show ?thesis
- proof(rule h)
- from ltf show "(e', x) \<in> inv_image less_than f"
- by (simp add:inv_image_def)
- next
- from pe' show "P e'" .
- qed
- qed
-qed
-
-
-text {* ******BEGIN: proving the initial equation-system satisfies Inv ****** *}
-
-lemma distinct_rhs_equations:
- "(X, xrhs) \<in> equations (UNIV Quo Lang) \<Longrightarrow> distinct_rhs xrhs"
-by (auto simp: equations_def equation_rhs_def distinct_rhs_def empty_rhs_def dest:no_two_cls_inters)
-
-lemma every_nonempty_eqclass_has_strings:
- "\<lbrakk>X \<in> (UNIV Quo Lang); X \<noteq> {[]}\<rbrakk> \<Longrightarrow> \<exists> clist. clist \<in> X \<and> clist \<noteq> []"
-by (auto simp:quot_def equiv_class_def equiv_str_def)
-
-lemma every_eqclass_is_derived_from_empty:
- assumes not_empty: "X \<noteq> {[]}"
- shows "X \<in> (UNIV Quo Lang) \<Longrightarrow> \<exists> clist. {[]};{clist} \<subseteq> X \<and> clist \<noteq> []"
-using not_empty
-apply (drule_tac every_nonempty_eqclass_has_strings, simp)
-by (auto intro:exI[where x = clist] simp:lang_seq_def)
-
-lemma equiv_str_in_CS:
- "\<lbrakk>clist\<rbrakk>Lang \<in> (UNIV Quo Lang)"
-by (auto simp:quot_def)
-
-lemma has_str_imp_defined_by_str:
- "\<lbrakk>str \<in> X; X \<in> UNIV Quo Lang\<rbrakk> \<Longrightarrow> X = \<lbrakk>str\<rbrakk>Lang"
-by (auto simp:quot_def equiv_class_def equiv_str_def)
-
-lemma every_eqclass_has_ascendent:
- assumes has_str: "clist @ [c] \<in> X"
- and in_CS: "X \<in> UNIV Quo Lang"
- shows "\<exists> Y. Y \<in> UNIV Quo Lang \<and> Y-c\<rightarrow>X \<and> clist \<in> Y" (is "\<exists> Y. ?P Y")
-proof -
- have "?P (\<lbrakk>clist\<rbrakk>Lang)"
- proof -
- have "\<lbrakk>clist\<rbrakk>Lang \<in> UNIV Quo Lang"
- by (simp add:quot_def, rule_tac x = clist in exI, simp)
- moreover have "\<lbrakk>clist\<rbrakk>Lang-c\<rightarrow>X"
- proof -
- have "X = \<lbrakk>(clist @ [c])\<rbrakk>Lang" using has_str in_CS
- by (auto intro!:has_str_imp_defined_by_str)
- moreover have "\<forall> sl. sl \<in> \<lbrakk>clist\<rbrakk>Lang \<longrightarrow> sl @ [c] \<in> \<lbrakk>(clist @ [c])\<rbrakk>Lang"
- by (auto simp:equiv_class_def equiv_str_def)
- ultimately show ?thesis unfolding CT_def lang_seq_def
- by auto
- qed
- moreover have "clist \<in> \<lbrakk>clist\<rbrakk>Lang"
- by (auto simp:equiv_str_def equiv_class_def)
- ultimately show "?P (\<lbrakk>clist\<rbrakk>Lang)" by simp
- qed
- thus ?thesis by blast
-qed
-
-lemma finite_charset_rS:
- "finite {CHAR c |c. Y-c\<rightarrow>X}"
-by (rule_tac A = UNIV and f = CHAR in finite_surj, auto)
-
-lemma l_eq_r_in_equations:
- assumes X_in_equas: "(X, xrhs) \<in> equations (UNIV Quo Lang)"
- shows "X = L xrhs"
-proof (cases "X = {[]}")
- case True
- thus ?thesis using X_in_equas
- by (simp add:equations_def equation_rhs_def lang_seq_def)
-next
- case False
- show ?thesis
- proof
- show "X \<subseteq> L xrhs"
- proof
- fix x
- assume "(1)": "x \<in> X"
- show "x \<in> L xrhs"
- proof (cases "x = []")
- assume empty: "x = []"
- hence "x \<in> L (empty_rhs X)" using "(1)"
- by (auto simp:empty_rhs_def lang_seq_def)
- thus ?thesis using X_in_equas False empty "(1)"
- unfolding equations_def equation_rhs_def by auto
- next
- assume not_empty: "x \<noteq> []"
- hence "\<exists> clist c. x = clist @ [c]" by (case_tac x rule:rev_cases, auto)
- then obtain clist c where decom: "x = clist @ [c]" by blast
- moreover have "\<And> Y. \<lbrakk>Y \<in> UNIV Quo Lang; Y-c\<rightarrow>X; clist \<in> Y\<rbrakk>
- \<Longrightarrow> [c] \<in> L (folds ALT NULL {CHAR c |c. Y-c\<rightarrow>X})"
- proof -
- fix Y
- assume Y_is_eq_cl: "Y \<in> UNIV Quo Lang"
- and Y_CT_X: "Y-c\<rightarrow>X"
- and clist_in_Y: "clist \<in> Y"
- with finite_charset_rS
- show "[c] \<in> L (folds ALT NULL {CHAR c |c. Y-c\<rightarrow>X})"
- by (auto simp :fold_alt_null_eqs)
- qed
- hence "\<exists>Xa. Xa \<in> UNIV Quo Lang \<and> clist @ [c] \<in> Xa ; L (folds ALT NULL {CHAR c |c. Xa-c\<rightarrow>X})"
- using X_in_equas False not_empty "(1)" decom
- by (auto dest!:every_eqclass_has_ascendent simp:equations_def equation_rhs_def lang_seq_def)
- then obtain Xa where
- "Xa \<in> UNIV Quo Lang \<and> clist @ [c] \<in> Xa ; L (folds ALT NULL {CHAR c |c. Xa-c\<rightarrow>X})" by blast
- hence "x \<in> L {(S, folds ALT NULL {CHAR c |c. S-c\<rightarrow>X}) |S. S \<in> UNIV Quo Lang}"
- using X_in_equas "(1)" decom
- by (auto simp add:equations_def equation_rhs_def intro!:exI[where x = Xa])
- thus "x \<in> L xrhs" using X_in_equas False not_empty unfolding equations_def equation_rhs_def
- by auto
- qed
- qed
- next
- show "L xrhs \<subseteq> X"
- proof
- fix x
- assume "(2)": "x \<in> L xrhs"
- have "(2_1)": "\<And> s1 s2 r Xa. \<lbrakk>s1 \<in> Xa; s2 \<in> L (folds ALT NULL {CHAR c |c. Xa-c\<rightarrow>X})\<rbrakk> \<Longrightarrow> s1 @ s2 \<in> X"
- using finite_charset_rS
- by (auto simp:CT_def lang_seq_def fold_alt_null_eqs)
- have "(2_2)": "\<And> s1 s2 Xa r.\<lbrakk>s1 \<in> Xa; s2 \<in> L r; (Xa, r) \<in> empty_rhs X\<rbrakk> \<Longrightarrow> s1 @ s2 \<in> X"
- by (simp add:empty_rhs_def split:if_splits)
- show "x \<in> X" using X_in_equas False "(2)"
- by (auto intro:"(2_1)" "(2_2)" simp:equations_def equation_rhs_def lang_seq_def)
- qed
- qed
-qed
-
-
-
-lemma no_EMPTY_equations:
- "(X, xrhs) \<in> equations CS \<Longrightarrow> no_EMPTY_rhs xrhs"
-apply (clarsimp simp add:equations_def equation_rhs_def)
-apply (simp add:no_EMPTY_rhs_def empty_rhs_def, auto)
-apply (subgoal_tac "finite {CHAR c |c. Xa-c\<rightarrow>X}", drule_tac x = "[]" in fold_alt_null_eqs, clarsimp, rule finite_charset_rS)+
-done
-
-lemma init_ES_satisfy_ardenable:
- "(X, xrhs) \<in> equations (UNIV Quo Lang) \<Longrightarrow> ardenable (X, xrhs)"
- unfolding ardenable_def
- by (auto intro:distinct_rhs_equations no_EMPTY_equations simp:l_eq_r_in_equations simp del:L_rhs.simps)
-
-lemma init_ES_satisfy_Inv:
- assumes finite_CS: "finite (UNIV Quo Lang)"
- and X_in_eq_cls: "X \<in> UNIV Quo Lang"
- shows "Inv X (equations (UNIV Quo Lang))"
-proof -
- have "finite (equations (UNIV Quo Lang))" using finite_CS
- by (auto simp:equations_def)
- moreover have "\<exists>rhs. (X, rhs) \<in> equations (UNIV Quo Lang)" using X_in_eq_cls
- by (simp add:equations_def)
- moreover have "distinct_equas (equations (UNIV Quo Lang))"
- by (auto simp:distinct_equas_def equations_def)
- moreover have "\<forall>X xrhs. (X, xrhs) \<in> equations (UNIV Quo Lang) \<longrightarrow>
- rhs_eq_cls xrhs \<subseteq> insert {[]} (left_eq_cls (equations (UNIV Quo Lang)))"
- apply (simp add:left_eq_cls_def equations_def rhs_eq_cls_def equation_rhs_def)
- by (auto simp:empty_rhs_def split:if_splits)
- moreover have "\<forall>X xrhs. (X, xrhs) \<in> equations (UNIV Quo Lang) \<longrightarrow> X \<noteq> {}"
- by (clarsimp simp:equations_def empty_notin_CS intro:classical)
- moreover have "\<forall>X xrhs. (X, xrhs) \<in> equations (UNIV Quo Lang) \<longrightarrow> ardenable (X, xrhs)"
- by (auto intro!:init_ES_satisfy_ardenable)
- ultimately show ?thesis by (simp add:Inv_def)
-qed
-
-
-text {* *********** END: proving the initial equation-system satisfies Inv ******* *}
-
-
-text {* ****** BEGIN: proving every equation-system's iteration step satisfies Inv ***** *}
-
-lemma not_T_aux: "\<lbrakk>\<not> TCon (insert a A); x = a\<rbrakk>
- \<Longrightarrow> \<exists>y. x \<noteq> y \<and> y \<in> insert a A "
-apply (case_tac "insert a A = {a}")
-by (auto simp:TCon_def)
-
-lemma not_T_atleast_2[rule_format]:
- "finite S \<Longrightarrow> \<forall> x. x \<in> S \<and> (\<not> TCon S)\<longrightarrow> (\<exists> y. x \<noteq> y \<and> y \<in> S)"
-apply (erule finite.induct, simp)
-apply (clarify, case_tac "x = a")
-by (erule not_T_aux, auto)
-
-lemma exist_another_equa:
- "\<lbrakk>\<not> TCon ES; finite ES; distinct_equas ES; (X, rhl) \<in> ES\<rbrakk> \<Longrightarrow> \<exists> Y yrhl. (Y, yrhl) \<in> ES \<and> X \<noteq> Y"
-apply (drule not_T_atleast_2, simp)
-apply (clarsimp simp:distinct_equas_def)
-apply (drule_tac x= X in spec, drule_tac x = rhl in spec, drule_tac x = b in spec)
-by auto
-
-lemma Inv_mono_with_lambda:
- assumes Inv_ES: "Inv X ES"
- and X_noteq_Y: "X \<noteq> {[]}"
- shows "Inv X (ES - {({[]}, yrhs)})"
-proof -
- have "finite (ES - {({[]}, yrhs)})" using Inv_ES
- by (simp add:Inv_def)
- moreover have "\<exists>rhs. (X, rhs) \<in> ES - {({[]}, yrhs)}" using Inv_ES X_noteq_Y
- by (simp add:Inv_def)
- moreover have "distinct_equas (ES - {({[]}, yrhs)})" using Inv_ES X_noteq_Y
- apply (clarsimp simp:Inv_def distinct_equas_def)
- by (drule_tac x = Xa in spec, simp)
- moreover have "\<forall>X xrhs.(X, xrhs) \<in> ES - {({[]}, yrhs)} \<longrightarrow>
- ardenable (X, xrhs) \<and> X \<noteq> {}" using Inv_ES
- by (clarify, simp add:Inv_def)
- moreover
- have "insert {[]} (left_eq_cls (ES - {({[]}, yrhs)})) = insert {[]} (left_eq_cls ES)"
- by (auto simp:left_eq_cls_def)
- hence "\<forall>X xrhs.(X, xrhs) \<in> ES - {({[]}, yrhs)} \<longrightarrow>
- rhs_eq_cls xrhs \<subseteq> insert {[]} (left_eq_cls (ES - {({[]}, yrhs)}))"
- using Inv_ES by (auto simp:Inv_def)
- ultimately show ?thesis by (simp add:Inv_def)
-qed
-
-lemma non_empty_card_prop:
- "finite ES \<Longrightarrow> \<forall>e. e \<in> ES \<longrightarrow> card ES - Suc 0 < card ES"
-apply (erule finite.induct, simp)
-apply (case_tac[!] "a \<in> A")
-by (auto simp:insert_absorb)
-
-lemma ardenable_prop:
- assumes not_lambda: "Y \<noteq> {[]}"
- and ardable: "ardenable (Y, yrhs)"
- shows "\<exists> yrhs'. Y = L yrhs' \<and> distinct_rhs yrhs' \<and> rhs_eq_cls yrhs' = rhs_eq_cls yrhs - {Y}" (is "\<exists> yrhs'. ?P yrhs'")
-proof (cases "(\<exists> reg. (Y, reg) \<in> yrhs)")
- case True
- thus ?thesis
- proof
- fix reg
- assume self_contained: "(Y, reg) \<in> yrhs"
- show ?thesis
- proof -
- have "?P (arden_variate Y reg yrhs)"
- proof -
- have "Y = L (arden_variate Y reg yrhs)"
- using self_contained not_lambda ardable
- by (rule_tac arden_variate_valid, simp_all add:ardenable_def)
- moreover have "distinct_rhs (arden_variate Y reg yrhs)"
- using ardable
- by (auto simp:distinct_rhs_def arden_variate_def seq_rhs_r_def del_x_paired_def ardenable_def)
- moreover have "rhs_eq_cls (arden_variate Y reg yrhs) = rhs_eq_cls yrhs - {Y}"
- proof -
- have "\<And> rhs r. rhs_eq_cls (seq_rhs_r rhs r) = rhs_eq_cls rhs"
- apply (auto simp:rhs_eq_cls_def seq_rhs_r_def image_def)
- by (rule_tac x = "SEQ ra r" in exI, rule_tac x = "(x, ra)" in bexI, simp+)
- moreover have "\<And> rhs X. rhs_eq_cls (del_x_paired rhs X) = rhs_eq_cls rhs - {X}"
- by (auto simp:rhs_eq_cls_def del_x_paired_def)
- ultimately show ?thesis by (simp add:arden_variate_def)
- qed
- ultimately show ?thesis by simp
- qed
- thus ?thesis by (rule_tac x= "arden_variate Y reg yrhs" in exI, simp)
- qed
- qed
-next
- case False
- hence "(2)": "rhs_eq_cls yrhs - {Y} = rhs_eq_cls yrhs"
- by (auto simp:rhs_eq_cls_def)
- show ?thesis
- proof -
- have "?P yrhs" using False ardable "(2)"
- by (simp add:ardenable_def)
- thus ?thesis by blast
- qed
-qed
-
-lemma equas_subst_f_del_no_other:
- assumes self_contained: "(Y, rhs) \<in> ES"
- shows "\<exists> rhs'. (Y, rhs') \<in> (equas_subst_f X xrhs ` ES)" (is "\<exists> rhs'. ?P rhs'")
-proof -
- have "\<exists> rhs'. equas_subst_f X xrhs (Y, rhs) = (Y, rhs')"
- by (auto simp:equas_subst_f_def)
- then obtain rhs' where "equas_subst_f X xrhs (Y, rhs) = (Y, rhs')" by blast
- hence "?P rhs'" unfolding image_def using self_contained
- by (auto intro:bexI[where x = "(Y, rhs)"])
- thus ?thesis by blast
-qed
-
-lemma del_x_paired_del_only_x:
- "\<lbrakk>X \<noteq> Y; (X, rhs) \<in> ES\<rbrakk> \<Longrightarrow> (X, rhs) \<in> del_x_paired ES Y"
-by (auto simp:del_x_paired_def)
-
-lemma equas_subst_del_no_other:
- "\<lbrakk>(X, rhs) \<in> ES; X \<noteq> Y\<rbrakk> \<Longrightarrow> (\<exists>rhs. (X, rhs) \<in> equas_subst ES Y rhs')"
-unfolding equas_subst_def
-apply (drule_tac X = Y and xrhs = rhs' in equas_subst_f_del_no_other)
-by (erule exE, drule del_x_paired_del_only_x, auto)
-
-lemma equas_subst_holds_distinct:
- "distinct_equas ES \<Longrightarrow> distinct_equas (equas_subst ES Y rhs')"
-apply (clarsimp simp add:equas_subst_def distinct_equas_def del_x_paired_def equas_subst_f_def)
-by (auto split:if_splits)
-
-lemma del_x_paired_dels:
- "(X, rhs) \<in> ES \<Longrightarrow> {Y. Y \<in> ES \<and> fst Y = X} \<inter> ES \<noteq> {}"
-by (auto)
-
-lemma del_x_paired_subset:
- "(X, rhs) \<in> ES \<Longrightarrow> ES - {Y. Y \<in> ES \<and> fst Y = X} \<subset> ES"
-apply (drule del_x_paired_dels)
-by auto
-
-lemma del_x_paired_card_less:
- "\<lbrakk>finite ES; (X, rhs) \<in> ES\<rbrakk> \<Longrightarrow> card (del_x_paired ES X) < card ES"
-apply (simp add:del_x_paired_def)
-apply (drule del_x_paired_subset)
-by (auto intro:psubset_card_mono)
-
-lemma equas_subst_card_less:
- "\<lbrakk>finite ES; (Y, yrhs) \<in> ES\<rbrakk> \<Longrightarrow> card (equas_subst ES Y rhs') < card ES"
-apply (simp add:equas_subst_def)
-apply (frule_tac h = "equas_subst_f Y rhs'" in finite_imageI)
-apply (drule_tac f = "equas_subst_f Y rhs'" in Finite_Set.card_image_le)
-apply (drule_tac X = Y and xrhs = rhs' in equas_subst_f_del_no_other,erule exE)
-by (drule del_x_paired_card_less, auto)
-
-lemma equas_subst_holds_distinct_rhs:
- assumes dist': "distinct_rhs yrhs'"
- and history: "\<forall>X xrhs. (X, xrhs) \<in> ES \<longrightarrow> ardenable (X, xrhs)"
- and X_in : "(X, xrhs) \<in> equas_subst ES Y yrhs'"
- shows "distinct_rhs xrhs"
-using X_in history
-apply (clarsimp simp:equas_subst_def del_x_paired_def)
-apply (drule_tac x = a in spec, drule_tac x = b in spec)
-apply (simp add:ardenable_def equas_subst_f_def)
-by (auto intro:rhs_subst_holds_distinct_rhs simp:dist' split:if_splits)
-
-lemma r_no_EMPTY_imp_seq_rhs_r_no_EMPTY:
- "[] \<notin> L r \<Longrightarrow> no_EMPTY_rhs (seq_rhs_r rhs r)"
-by (auto simp:no_EMPTY_rhs_def seq_rhs_r_def lang_seq_def)
-
-lemma del_x_paired_holds_no_EMPTY:
- "no_EMPTY_rhs yrhs \<Longrightarrow> no_EMPTY_rhs (del_x_paired yrhs Y)"
-by (auto simp:no_EMPTY_rhs_def del_x_paired_def)
-
-lemma rhs_subst_holds_no_EMPTY:
- "\<lbrakk>no_EMPTY_rhs yrhs; (Y, r) \<in> yrhs; Y \<noteq> {[]}\<rbrakk> \<Longrightarrow> no_EMPTY_rhs (rhs_subst yrhs Y rhs' r)"
-apply (auto simp:rhs_subst_def intro!:no_EMPTY_rhss_imp_merge_no_EMPTY r_no_EMPTY_imp_seq_rhs_r_no_EMPTY del_x_paired_holds_no_EMPTY)
-by (auto simp:no_EMPTY_rhs_def)
-
-lemma equas_subst_holds_no_EMPTY:
- assumes substor: "Y \<noteq> {[]}"
- and history: "\<forall>X xrhs. (X, xrhs) \<in> ES \<longrightarrow> ardenable (X, xrhs)"
- and X_in:"(X, xrhs) \<in> equas_subst ES Y yrhs'"
- shows "no_EMPTY_rhs xrhs"
-proof-
- from X_in have "\<exists> (Z, zrhs) \<in> ES. (X, xrhs) = equas_subst_f Y yrhs' (Z, zrhs)"
- by (auto simp add:equas_subst_def del_x_paired_def)
- then obtain Z zrhs where Z_in: "(Z, zrhs) \<in> ES"
- and X_Z: "(X, xrhs) = equas_subst_f Y yrhs' (Z, zrhs)" by blast
- hence dist_zrhs: "distinct_rhs zrhs" using history
- by (auto simp:ardenable_def)
- show ?thesis
- proof (cases "\<exists> r. (Y, r) \<in> zrhs")
- case True
- then obtain r where Y_in_zrhs: "(Y, r) \<in> zrhs" ..
- hence some: "(SOME r. (Y, r) \<in> zrhs) = r" using Z_in dist_zrhs
- by (auto simp:distinct_rhs_def)
- hence "no_EMPTY_rhs (rhs_subst zrhs Y yrhs' r)"
- using substor Y_in_zrhs history Z_in
- by (rule_tac rhs_subst_holds_no_EMPTY, auto simp:ardenable_def)
- thus ?thesis using X_Z True some
- by (simp add:equas_subst_def equas_subst_f_def)
- next
- case False
- hence "(X, xrhs) = (Z, zrhs)" using Z_in X_Z
- by (simp add:equas_subst_f_def)
- thus ?thesis using history Z_in
- by (auto simp:ardenable_def)
- qed
-qed
-
-lemma equas_subst_f_holds_left_eq_right:
- assumes substor: "Y = L rhs'"
- and history: "\<forall>X xrhs. (X, xrhs) \<in> ES \<longrightarrow> distinct_rhs xrhs \<and> X = L xrhs"
- and subst: "(X, xrhs) = equas_subst_f Y rhs' (Z, zrhs)"
- and self_contained: "(Z, zrhs) \<in> ES"
- shows "X = L xrhs"
-proof (cases "\<exists> r. (Y, r) \<in> zrhs")
- case True
- from True obtain r where "(1)":"(Y, r) \<in> zrhs" ..
- show ?thesis
- proof -
- from history self_contained
- have dist: "distinct_rhs zrhs" by auto
- hence "(SOME r. (Y, r) \<in> zrhs) = r" using self_contained "(1)"
- using distinct_rhs_def by (auto intro:some_equality)
- moreover have "L zrhs = L (rhs_subst zrhs Y rhs' r)" using substor dist "(1)" self_contained
- by (rule_tac rhs_subst_prop1, auto simp:distinct_equas_rhs_def)
- ultimately show ?thesis using subst history self_contained
- by (auto simp:equas_subst_f_def split:if_splits)
- qed
-next
- case False
- thus ?thesis using history subst self_contained
- by (auto simp:equas_subst_f_def)
-qed
-
-lemma equas_subst_holds_left_eq_right:
- assumes history: "\<forall>X xrhs. (X, xrhs) \<in> ES \<longrightarrow> ardenable (X, xrhs)"
- and substor: "Y = L rhs'"
- and X_in : "(X, xrhs) \<in> equas_subst ES Y yrhs'"
- shows "\<forall>X xrhs. (X, xrhs) \<in> equas_subst ES Y rhs' \<longrightarrow> X = L xrhs"
-apply (clarsimp simp add:equas_subst_def del_x_paired_def)
-using substor
-apply (drule_tac equas_subst_f_holds_left_eq_right)
-using history
-by (auto simp:ardenable_def)
-
-lemma equas_subst_holds_ardenable:
- assumes substor: "Y = L yrhs'"
- and history: "\<forall>X xrhs. (X, xrhs) \<in> ES \<longrightarrow> ardenable (X, xrhs)"
- and X_in:"(X, xrhs) \<in> equas_subst ES Y yrhs'"
- and dist': "distinct_rhs yrhs'"
- and not_lambda: "Y \<noteq> {[]}"
- shows "ardenable (X, xrhs)"
-proof -
- have "distinct_rhs xrhs" using history X_in dist'
- by (auto dest:equas_subst_holds_distinct_rhs)
- moreover have "no_EMPTY_rhs xrhs" using history X_in not_lambda
- by (auto intro:equas_subst_holds_no_EMPTY)
- moreover have "X = L xrhs" using history substor X_in
- by (auto dest: equas_subst_holds_left_eq_right simp del:L_rhs.simps)
- ultimately show ?thesis using ardenable_def by simp
-qed
-
-lemma equas_subst_holds_cls_defined:
- assumes X_in: "(X, xrhs) \<in> equas_subst ES Y yrhs'"
- and Inv_ES: "Inv X' ES"
- and subst: "(Y, yrhs) \<in> ES"
- and cls_holds_but_Y: "rhs_eq_cls yrhs' = rhs_eq_cls yrhs - {Y}"
- shows "rhs_eq_cls xrhs \<subseteq> insert {[]} (left_eq_cls (equas_subst ES Y yrhs'))"
-proof-
- have tac: "\<lbrakk> A \<subseteq> B; C \<subseteq> D; E \<subseteq> A \<union> B\<rbrakk> \<Longrightarrow> E \<subseteq> B \<union> D" by auto
- from X_in have "\<exists> (Z, zrhs) \<in> ES. (X, xrhs) = equas_subst_f Y yrhs' (Z, zrhs)"
- by (auto simp add:equas_subst_def del_x_paired_def)
- then obtain Z zrhs where Z_in: "(Z, zrhs) \<in> ES"
- and X_Z: "(X, xrhs) = equas_subst_f Y yrhs' (Z, zrhs)" by blast
- hence "rhs_eq_cls zrhs \<subseteq> insert {[]} (left_eq_cls ES)" using Inv_ES
- by (auto simp:Inv_def)
- moreover have "rhs_eq_cls yrhs' \<subseteq> insert {[]} (left_eq_cls ES) - {Y}"
- using Inv_ES subst cls_holds_but_Y
- by (auto simp:Inv_def)
- moreover have "rhs_eq_cls xrhs \<subseteq> rhs_eq_cls zrhs \<union> rhs_eq_cls yrhs' - {Y}"
- using X_Z cls_holds_but_Y
- apply (clarsimp simp add:equas_subst_f_def rhs_subst_def split:if_splits)
- by (auto simp:rhs_eq_cls_def merge_rhs_def del_x_paired_def seq_rhs_r_def)
- moreover have "left_eq_cls (equas_subst ES Y yrhs') = left_eq_cls ES - {Y}" using subst
- by (auto simp: left_eq_cls_def equas_subst_def del_x_paired_def equas_subst_f_def
- dest: equas_subst_f_del_no_other
- split: if_splits)
- ultimately show ?thesis by blast
-qed
-
-lemma iteration_step:
- assumes Inv_ES: "Inv X ES"
- and not_T: "\<not> TCon ES"
- shows "(\<exists> ES'. Inv X ES' \<and> (card ES', card ES) \<in> less_than)"
-proof -
- from Inv_ES not_T have another: "\<exists>Y yrhs. (Y, yrhs) \<in> ES \<and> X \<noteq> Y" unfolding Inv_def
- by (clarify, rule_tac exist_another_equa[where X = X], auto)
- then obtain Y yrhs where subst: "(Y, yrhs) \<in> ES" and not_X: " X \<noteq> Y" by blast
- show ?thesis (is "\<exists> ES'. ?P ES'")
- proof (cases "Y = {[]}")
- case True
- --"in this situation, we pick a \"\<lambda>\" equation, thus directly remove it from the equation-system"
- have "?P (ES - {(Y, yrhs)})"
- proof
- show "Inv X (ES - {(Y, yrhs)})" using True not_X
- by (simp add:Inv_ES Inv_mono_with_lambda)
- next
- show "(card (ES - {(Y, yrhs)}), card ES) \<in> less_than" using Inv_ES subst
- by (auto elim:non_empty_card_prop[rule_format] simp:Inv_def)
- qed
- thus ?thesis by blast
- next
- case False
- --"in this situation, we pick a equation and using ardenable to get a
- rhs without itself in it, then use equas_subst to form a new equation-system"
- hence "\<exists> yrhs'. Y = L yrhs' \<and> distinct_rhs yrhs' \<and> rhs_eq_cls yrhs' = rhs_eq_cls yrhs - {Y}"
- using subst Inv_ES
- by (auto intro:ardenable_prop simp add:Inv_def simp del:L_rhs.simps)
- then obtain yrhs' where Y'_l_eq_r: "Y = L yrhs'"
- and dist_Y': "distinct_rhs yrhs'"
- and cls_holds_but_Y: "rhs_eq_cls yrhs' = rhs_eq_cls yrhs - {Y}" by blast
- hence "?P (equas_subst ES Y yrhs')"
- proof -
- have finite_del: "\<And> S x. finite S \<Longrightarrow> finite (del_x_paired S x)"
- apply (rule_tac A = "del_x_paired S x" in finite_subset)
- by (auto simp:del_x_paired_def)
- have "finite (equas_subst ES Y yrhs')" using Inv_ES
- by (auto intro!:finite_del simp:equas_subst_def Inv_def)
- moreover have "\<exists>rhs. (X, rhs) \<in> equas_subst ES Y yrhs'" using Inv_ES not_X
- by (auto intro:equas_subst_del_no_other simp:Inv_def)
- moreover have "distinct_equas (equas_subst ES Y yrhs')" using Inv_ES
- by (auto intro:equas_subst_holds_distinct simp:Inv_def)
- moreover have "\<forall>X xrhs. (X, xrhs) \<in> equas_subst ES Y yrhs' \<longrightarrow> ardenable (X, xrhs)"
- using Inv_ES dist_Y' False Y'_l_eq_r
- apply (clarsimp simp:Inv_def)
- by (rule equas_subst_holds_ardenable, simp_all)
- moreover have "\<forall>X xrhs. (X, xrhs) \<in> equas_subst ES Y yrhs' \<longrightarrow> X \<noteq> {}" using Inv_ES
- by (clarsimp simp:equas_subst_def Inv_def del_x_paired_def equas_subst_f_def split:if_splits, auto)
- moreover have "\<forall>X xrhs. (X, xrhs) \<in> equas_subst ES Y yrhs' \<longrightarrow>
- rhs_eq_cls xrhs \<subseteq> insert {[]} (left_eq_cls (equas_subst ES Y yrhs'))"
- using Inv_ES subst cls_holds_but_Y
- apply (rule_tac impI | rule_tac allI)+
- by (erule equas_subst_holds_cls_defined, auto)
- moreover have "(card (equas_subst ES Y yrhs'), card ES) \<in> less_than"using Inv_ES subst
- by (simp add:equas_subst_card_less Inv_def)
- ultimately show "?P (equas_subst ES Y yrhs')" by (auto simp:Inv_def)
- qed
- thus ?thesis by blast
- qed
-qed
-
-text {* ***** END: proving every equation-system's iteration step satisfies Inv ************** *}
-
-lemma iteration_conc:
- assumes history: "Inv X ES"
- shows "\<exists> ES'. Inv X ES' \<and> TCon ES'" (is "\<exists> ES'. ?P ES'")
-proof (cases "TCon ES")
- case True
- hence "?P ES" using history by simp
- thus ?thesis by blast
-next
- case False
- thus ?thesis using history iteration_step
- by (rule_tac f = card in wf_iter, simp_all)
-qed
-
-lemma eqset_imp_iff': "A = B \<Longrightarrow> \<forall> x. x \<in> A \<longleftrightarrow> x \<in> B"
-apply (auto simp:mem_def)
-done
-
-lemma set_cases2:
- "\<lbrakk>(A = {} \<Longrightarrow> R A); \<And> x. (A = {x}) \<Longrightarrow> R A; \<And> x y. \<lbrakk>x \<noteq> y; x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> R A\<rbrakk> \<Longrightarrow> R A"
-apply (case_tac "A = {}", simp)
-by (case_tac "\<exists> x. A = {x}", clarsimp, blast)
-
-lemma rhs_aux:"\<lbrakk>distinct_rhs rhs; {Y. \<exists>r. (Y, r) \<in> rhs} = {X}\<rbrakk> \<Longrightarrow> (\<exists>r. rhs = {(X, r)})"
-apply (rule_tac A = rhs in set_cases2, simp)
-apply (drule_tac x = X in eqset_imp_iff, clarsimp)
-apply (drule eqset_imp_iff',clarsimp)
-apply (frule_tac x = a in spec, drule_tac x = aa in spec)
-by (auto simp:distinct_rhs_def)
-
-lemma every_eqcl_has_reg:
- assumes finite_CS: "finite (UNIV Quo Lang)"
- and X_in_CS: "X \<in> (UNIV Quo Lang)"
- shows "\<exists> (reg::rexp). L reg = X" (is "\<exists> r. ?E r")
-proof-
- have "\<exists>ES'. Inv X ES' \<and> TCon ES'" using finite_CS X_in_CS
- by (auto intro:init_ES_satisfy_Inv iteration_conc)
- then obtain ES' where Inv_ES': "Inv X ES'"
- and TCon_ES': "TCon ES'" by blast
- from Inv_ES' TCon_ES'
- have "\<exists> rhs. ES' = {(X, rhs)}"
- apply (clarsimp simp:Inv_def TCon_def)
- apply (rule_tac x = rhs in exI)
- by (auto dest!:card_Suc_Diff1 simp:card_eq_0_iff)
- then obtain rhs where ES'_single_equa: "ES' = {(X, rhs)}" ..
- hence X_ardenable: "ardenable (X, rhs)" using Inv_ES'
- by (simp add:Inv_def)
-
- from X_ardenable have X_l_eq_r: "X = L rhs"
- by (simp add:ardenable_def)
- hence rhs_not_empty: "rhs \<noteq> {}" using Inv_ES' ES'_single_equa
- by (auto simp:Inv_def ardenable_def)
- have rhs_eq_cls: "rhs_eq_cls rhs \<subseteq> {X, {[]}}"
- using Inv_ES' ES'_single_equa
- by (auto simp:Inv_def ardenable_def left_eq_cls_def)
- have X_not_empty: "X \<noteq> {}" using Inv_ES' ES'_single_equa
- by (auto simp:Inv_def)
- show ?thesis
- proof (cases "X = {[]}")
- case True
- hence "?E EMPTY" by (simp)
- thus ?thesis by blast
- next
- case False with X_ardenable
- have "\<exists> rhs'. X = L rhs' \<and> rhs_eq_cls rhs' = rhs_eq_cls rhs - {X} \<and> distinct_rhs rhs'"
- by (drule_tac ardenable_prop, auto)
- then obtain rhs' where X_eq_rhs': "X = L rhs'"
- and rhs'_eq_cls: "rhs_eq_cls rhs' = rhs_eq_cls rhs - {X}"
- and rhs'_dist : "distinct_rhs rhs'" by blast
- have "rhs_eq_cls rhs' \<subseteq> {{[]}}" using rhs_eq_cls False rhs'_eq_cls rhs_not_empty
- by blast
- hence "rhs_eq_cls rhs' = {{[]}}" using X_not_empty X_eq_rhs'
- by (auto simp:rhs_eq_cls_def)
- hence "\<exists> r. rhs' = {({[]}, r)}" using rhs'_dist
- by (auto intro:rhs_aux simp:rhs_eq_cls_def)
- then obtain r where "rhs' = {({[]}, r)}" ..
- hence "?E r" using X_eq_rhs' by (auto simp add:lang_seq_def)
- thus ?thesis by blast
- qed
-qed
-
-text {* definition of a regular language *}
-
-abbreviation
- reg :: "string set => bool"
-where
- "reg L1 \<equiv> (\<exists>r::rexp. L r = L1)"
-
-theorem myhill_nerode:
- assumes finite_CS: "finite (UNIV Quo Lang)"
- shows "\<exists> (reg::rexp). Lang = L reg" (is "\<exists> r. ?P r")
-proof -
- have has_r_each: "\<forall>C\<in>{X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang}. \<exists>(r::rexp). C = L r" using finite_CS
- by (auto dest:every_eqcl_has_reg)
- have "\<exists> (rS::rexp set). finite rS \<and>
- (\<forall> C \<in> {X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang}. \<exists> r \<in> rS. C = L r) \<and>
- (\<forall> r \<in> rS. \<exists> C \<in> {X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang}. C = L r)"
- (is "\<exists> rS. ?Q rS")
- proof-
- have "\<And> C. C \<in> {X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang} \<Longrightarrow> C = L (SOME (ra::rexp). C = L ra)"
- using has_r_each
- apply (erule_tac x = C in ballE, erule_tac exE)
- by (rule_tac a = r in someI2, simp+)
- hence "?Q ((\<lambda> C. SOME r. C = L r) ` {X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang})" using has_r_each
- using finite_CS by auto
- thus ?thesis by blast
- qed
- then obtain rS where finite_rS : "finite rS"
- and has_r_each': "\<forall> C \<in> {X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang}. \<exists> r \<in> (rS::rexp set). C = L r"
- and has_cl_each: "\<forall> r \<in> (rS::rexp set). \<exists> C \<in> {X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang}. C = L r" by blast
- have "?P (folds ALT NULL rS)"
- proof
- show "Lang \<subseteq> L (folds ALT NULL rS)" using finite_rS lang_eqs_union_of_eqcls[of Lang] has_r_each'
- apply (clarsimp simp:fold_alt_null_eqs) by blast
- next
- show "L (folds ALT NULL rS) \<subseteq> Lang" using finite_rS lang_eqs_union_of_eqcls[of Lang] has_cl_each
- by (clarsimp simp:fold_alt_null_eqs)
- qed
- thus ?thesis by blast
-qed
-
-
-text {* tests by Christian *}
-
-(* Alternative definition for Quo *)
-definition
- QUOT :: "string set \<Rightarrow> (string set) set"
-where
- "QUOT Lang \<equiv> (\<Union>x. {\<lbrakk>x\<rbrakk>Lang})"
-
-lemma test:
- "UNIV Quo Lang = QUOT Lang"
-by (auto simp add: quot_def QUOT_def)
-
-lemma test1:
- assumes finite_CS: "finite (QUOT Lang)"
- shows "reg Lang"
-using finite_CS
-unfolding test[symmetric]
-by (auto dest: myhill_nerode)
-
-lemma cons_one: "x @ y \<in> {z} \<Longrightarrow> x @ y = z"
-by simp
-
-lemma quot_lambda: "QUOT {[]} = {{[]}, UNIV - {[]}}"
-proof
- show "QUOT {[]} \<subseteq> {{[]}, UNIV - {[]}}"
- proof
- fix x
- assume in_quot: "x \<in> QUOT {[]}"
- show "x \<in> {{[]}, UNIV - {[]}}"
- proof -
- from in_quot
- have "x = {[]} \<or> x = UNIV - {[]}"
- unfolding QUOT_def equiv_class_def
- proof
- fix xa
- assume in_univ: "xa \<in> UNIV"
- and in_eqiv: "x \<in> {{y. xa \<equiv>{[]}\<equiv> y}}"
- show "x = {[]} \<or> x = UNIV - {[]}"
- proof(cases "xa = []")
- case True
- hence "{y. xa \<equiv>{[]}\<equiv> y} = {[]}" using in_eqiv
- by (auto simp add:equiv_str_def)
- thus ?thesis using in_eqiv by (rule_tac disjI1, simp)
- next
- case False
- hence "{y. xa \<equiv>{[]}\<equiv> y} = UNIV - {[]}" using in_eqiv
- by (auto simp:equiv_str_def)
- thus ?thesis using in_eqiv by (rule_tac disjI2, simp)
- qed
- qed
- thus ?thesis by simp
- qed
- qed
-next
- show "{{[]}, UNIV - {[]}} \<subseteq> QUOT {[]}"
- proof
- fix x
- assume in_res: "x \<in> {{[]}, (UNIV::string set) - {[]}}"
- show "x \<in> (QUOT {[]})"
- proof -
- have "x = {[]} \<Longrightarrow> x \<in> QUOT {[]}"
- apply (simp add:QUOT_def equiv_class_def equiv_str_def)
- by (rule_tac x = "[]" in exI, auto)
- moreover have "x = UNIV - {[]} \<Longrightarrow> x \<in> QUOT {[]}"
- apply (simp add:QUOT_def equiv_class_def equiv_str_def)
- by (rule_tac x = "''a''" in exI, auto)
- ultimately show ?thesis using in_res by blast
- qed
- qed
-qed
-
-lemma quot_single_aux: "\<lbrakk>x \<noteq> []; x \<noteq> [c]\<rbrakk> \<Longrightarrow> x @ z \<noteq> [c]"
-by (induct x, auto)
-
-lemma quot_single: "\<And> (c::char). QUOT {[c]} = {{[]}, {[c]}, UNIV - {[], [c]}}"
-proof -
- fix c::"char"
- have exist_another: "\<exists> a. a \<noteq> c"
- apply (case_tac "c = CHR ''a''")
- apply (rule_tac x = "CHR ''b''" in exI, simp)
- by (rule_tac x = "CHR ''a''" in exI, simp)
- show "QUOT {[c]} = {{[]}, {[c]}, UNIV - {[], [c]}}"
- proof
- show "QUOT {[c]} \<subseteq> {{[]},{[c]}, UNIV - {[], [c]}}"
- proof
- fix x
- assume in_quot: "x \<in> QUOT {[c]}"
- show "x \<in> {{[]}, {[c]}, UNIV - {[],[c]}}"
- proof -
- from in_quot
- have "x = {[]} \<or> x = {[c]} \<or> x = UNIV - {[],[c]}"
- unfolding QUOT_def equiv_class_def
- proof
- fix xa
- assume in_eqiv: "x \<in> {{y. xa \<equiv>{[c]}\<equiv> y}}"
- show "x = {[]} \<or> x = {[c]} \<or> x = UNIV - {[], [c]}"
- proof-
- have "xa = [] \<Longrightarrow> x = {[]}" using in_eqiv
- by (auto simp add:equiv_str_def)
- moreover have "xa = [c] \<Longrightarrow> x = {[c]}"
- proof -
- have "xa = [c] \<Longrightarrow> {y. xa \<equiv>{[c]}\<equiv> y} = {[c]}" using in_eqiv
- apply (simp add:equiv_str_def)
- apply (rule set_ext, rule iffI, simp)
- apply (drule_tac x = "[]" in spec, auto)
- done
- thus "xa = [c] \<Longrightarrow> x = {[c]}" using in_eqiv by simp
- qed
- moreover have "\<lbrakk>xa \<noteq> []; xa \<noteq> [c]\<rbrakk> \<Longrightarrow> x = UNIV - {[],[c]}"
- proof -
- have "\<lbrakk>xa \<noteq> []; xa \<noteq> [c]\<rbrakk> \<Longrightarrow> {y. xa \<equiv>{[c]}\<equiv> y} = UNIV - {[],[c]}"
- apply (clarsimp simp add:equiv_str_def)
- apply (rule set_ext, rule iffI, simp)
- apply (rule conjI)
- apply (drule_tac x = "[c]" in spec, simp)
- apply (drule_tac x = "[]" in spec, simp)
- by (auto dest:quot_single_aux)
- thus "\<lbrakk>xa \<noteq> []; xa \<noteq> [c]\<rbrakk> \<Longrightarrow> x = UNIV - {[],[c]}" using in_eqiv by simp
- qed
- ultimately show ?thesis by blast
- qed
- qed
- thus ?thesis by simp
- qed
- qed
- next
- show "{{[]}, {[c]}, UNIV - {[],[c]}} \<subseteq> QUOT {[c]}"
- proof
- fix x
- assume in_res: "x \<in> {{[]},{[c]}, (UNIV::string set) - {[],[c]}}"
- show "x \<in> (QUOT {[c]})"
- proof -
- have "x = {[]} \<Longrightarrow> x \<in> QUOT {[c]}"
- apply (simp add:QUOT_def equiv_class_def equiv_str_def)
- by (rule_tac x = "[]" in exI, auto)
- moreover have "x = {[c]} \<Longrightarrow> x \<in> QUOT {[c]}"
- apply (simp add:QUOT_def equiv_class_def equiv_str_def)
- apply (rule_tac x = "[c]" in exI, simp)
- apply (rule set_ext, rule iffI, simp+)
- by (drule_tac x = "[]" in spec, simp)
- moreover have "x = UNIV - {[],[c]} \<Longrightarrow> x \<in> QUOT {[c]}"
- using exist_another
- apply (clarsimp simp add:QUOT_def equiv_class_def equiv_str_def)
- apply (rule_tac x = "[a]" in exI, simp)
- apply (rule set_ext, rule iffI, simp)
- apply (clarsimp simp:quot_single_aux, simp)
- apply (rule conjI)
- apply (drule_tac x = "[c]" in spec, simp)
- by (drule_tac x = "[]" in spec, simp)
- ultimately show ?thesis using in_res by blast
- qed
- qed
- qed
-qed
-
-lemma eq_class_imp_eq_str:
- "\<lbrakk>x\<rbrakk>lang = \<lbrakk>y\<rbrakk>lang \<Longrightarrow> x \<equiv>lang\<equiv> y"
-by (auto simp:equiv_class_def equiv_str_def)
-
-lemma finite_tag_image:
- "finite (range tag) \<Longrightarrow> finite (((op `) tag) ` S)"
-apply (rule_tac B = "Pow (tag ` UNIV)" in finite_subset)
-by (auto simp add:image_def Pow_def)
-
-lemma str_inj_imps:
- assumes str_inj: "\<And> m n. tag m = tag (n::string) \<Longrightarrow> m \<equiv>lang\<equiv> n"
- shows "inj_on ((op `) tag) (QUOT lang)"
-proof (clarsimp simp add:inj_on_def QUOT_def)
- fix x y
- assume eq_tag: "tag ` \<lbrakk>x\<rbrakk>lang = tag ` \<lbrakk>y\<rbrakk>lang"
- show "\<lbrakk>x\<rbrakk>lang = \<lbrakk>y\<rbrakk>lang"
- proof -
- have aux1:"\<And>a b. a \<in> (\<lbrakk>b\<rbrakk>lang) \<Longrightarrow> (a \<equiv>lang\<equiv> b)"
- by (simp add:equiv_class_def equiv_str_def)
- have aux2: "\<And> A B f. \<lbrakk>f ` A = f ` B; A \<noteq> {}\<rbrakk> \<Longrightarrow> \<exists> a b. a \<in> A \<and> b \<in> B \<and> f a = f b"
- by auto
- have aux3: "\<And> a l. \<lbrakk>a\<rbrakk>l \<noteq> {}"
- by (auto simp:equiv_class_def equiv_str_def)
- show ?thesis using eq_tag
- apply (drule_tac aux2, simp add:aux3, clarsimp)
- apply (drule_tac str_inj, (drule_tac aux1)+)
- by (simp add:equiv_str_def equiv_class_def)
- qed
-qed
-
-definition tag_str_ALT :: "string set \<Rightarrow> string set \<Rightarrow> string \<Rightarrow> (string set \<times> string set)"
-where
- "tag_str_ALT L\<^isub>1 L\<^isub>2 x \<equiv> (\<lbrakk>x\<rbrakk>L\<^isub>1, \<lbrakk>x\<rbrakk>L\<^isub>2)"
-
-lemma tag_str_alt_range_finite:
- assumes finite1: "finite (QUOT L\<^isub>1)"
- and finite2: "finite (QUOT L\<^isub>2)"
- shows "finite (range (tag_str_ALT L\<^isub>1 L\<^isub>2))"
-proof -
- have "{y. \<exists>x. y = (\<lbrakk>x\<rbrakk>L\<^isub>1, \<lbrakk>x\<rbrakk>L\<^isub>2)} \<subseteq> (QUOT L\<^isub>1) \<times> (QUOT L\<^isub>2)"
- by (auto simp:QUOT_def)
- thus ?thesis using finite1 finite2
- by (auto simp: image_def tag_str_ALT_def UNION_def
- intro: finite_subset[where B = "(QUOT L\<^isub>1) \<times> (QUOT L\<^isub>2)"])
-qed
-
-lemma tag_str_alt_inj:
- "tag_str_ALT L\<^isub>1 L\<^isub>2 x = tag_str_ALT L\<^isub>1 L\<^isub>2 y \<Longrightarrow> x \<equiv>(L\<^isub>1 \<union> L\<^isub>2)\<equiv> y"
-apply (simp add:tag_str_ALT_def equiv_class_def equiv_str_def)
-by blast
-
-lemma quot_alt:
- assumes finite1: "finite (QUOT L\<^isub>1)"
- and finite2: "finite (QUOT L\<^isub>2)"
- shows "finite (QUOT (L\<^isub>1 \<union> L\<^isub>2))"
-proof (rule_tac f = "(op `) (tag_str_ALT L\<^isub>1 L\<^isub>2)" in finite_imageD)
- show "finite (op ` (tag_str_ALT L\<^isub>1 L\<^isub>2) ` QUOT (L\<^isub>1 \<union> L\<^isub>2))"
- using finite_tag_image tag_str_alt_range_finite finite1 finite2
- by auto
-next
- show "inj_on (op ` (tag_str_ALT L\<^isub>1 L\<^isub>2)) (QUOT (L\<^isub>1 \<union> L\<^isub>2))"
- apply (rule_tac str_inj_imps)
- by (erule_tac tag_str_alt_inj)
-qed
-
-(* list_diff:: list substract, once different return tailer *)
-fun list_diff :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" (infix "-" 51)
-where
- "list_diff [] xs = []" |
- "list_diff (x#xs) [] = x#xs" |
- "list_diff (x#xs) (y#ys) = (if x = y then list_diff xs ys else (x#xs))"
-
-lemma [simp]: "(x @ y) - x = y"
-apply (induct x)
-by (case_tac y, simp+)
-
-lemma [simp]: "x - x = []"
-by (induct x, auto)
-
-lemma [simp]: "x = xa @ y \<Longrightarrow> x - xa = y "
-by (induct x, auto)
-
-lemma [simp]: "x - [] = x"
-by (induct x, auto)
-
-lemma [simp]: "xa \<le> x \<Longrightarrow> (x @ y) - xa = (x - xa) @ y"
-by (auto elim:prefixE)
-
-definition tag_str_SEQ:: "string set \<Rightarrow> string set \<Rightarrow> string \<Rightarrow> (string set \<times> string set set)"
-where
- "tag_str_SEQ L\<^isub>1 L\<^isub>2 x \<equiv> if (\<exists> xa \<le> x. xa \<in> L\<^isub>1)
- then (\<lbrakk>x\<rbrakk>L\<^isub>1, {\<lbrakk>(x - xa)\<rbrakk>L\<^isub>2 | xa. xa \<le> x \<and> xa \<in> L\<^isub>1})
- else (\<lbrakk>x\<rbrakk>L\<^isub>1, {})"
-
-lemma tag_seq_eq_E:
- "tag_str_SEQ L\<^isub>1 L\<^isub>2 x = tag_str_SEQ L\<^isub>1 L\<^isub>2 y \<Longrightarrow>
- ((\<exists> xa \<le> x. xa \<in> L\<^isub>1) \<and> \<lbrakk>x\<rbrakk>L\<^isub>1 = \<lbrakk>y\<rbrakk>L\<^isub>1 \<and>
- {\<lbrakk>(x - xa)\<rbrakk>L\<^isub>2 | xa. xa \<le> x \<and> xa \<in> L\<^isub>1} = {\<lbrakk>(y - ya)\<rbrakk>L\<^isub>2 | ya. ya \<le> y \<and> ya \<in> L\<^isub>1} ) \<or>
- ((\<forall> xa \<le> x. xa \<notin> L\<^isub>1) \<and> \<lbrakk>x\<rbrakk>L\<^isub>1 = \<lbrakk>y\<rbrakk>L\<^isub>1)"
-by (simp add:tag_str_SEQ_def split:if_splits, blast)
-
-lemma tag_str_seq_range_finite:
- assumes finite1: "finite (QUOT L\<^isub>1)"
- and finite2: "finite (QUOT L\<^isub>2)"
- shows "finite (range (tag_str_SEQ L\<^isub>1 L\<^isub>2))"
-proof -
- have "(range (tag_str_SEQ L\<^isub>1 L\<^isub>2)) \<subseteq> (QUOT L\<^isub>1) \<times> (Pow (QUOT L\<^isub>2))"
- by (auto simp:image_def tag_str_SEQ_def QUOT_def)
- thus ?thesis using finite1 finite2
- by (rule_tac B = "(QUOT L\<^isub>1) \<times> (Pow (QUOT L\<^isub>2))" in finite_subset, auto)
-qed
-
-lemma app_in_seq_decom[rule_format]:
- "\<forall> x. x @ z \<in> L\<^isub>1 ; L\<^isub>2 \<longrightarrow> (\<exists> xa \<le> x. xa \<in> L\<^isub>1 \<and> (x - xa) @ z \<in> L\<^isub>2) \<or>
- (\<exists> za \<le> z. (x @ za) \<in> L\<^isub>1 \<and> (z - za) \<in> L\<^isub>2)"
-apply (induct z)
-apply (simp, rule allI, rule impI, rule disjI1)
-apply (clarsimp simp add:lang_seq_def)
-apply (rule_tac x = s1 in exI, simp)
-apply (rule allI | rule impI)+
-apply (drule_tac x = "x @ [a]" in spec, simp)
-apply (erule exE | erule conjE | erule disjE)+
-apply (rule disjI2, rule_tac x = "[a]" in exI, simp)
-apply (rule disjI1, rule_tac x = xa in exI, simp)
-apply (erule exE | erule conjE)+
-apply (rule disjI2, rule_tac x = "a # za" in exI, simp)
-done
-
-lemma tag_str_seq_inj:
- assumes tag_eq: "tag_str_SEQ L\<^isub>1 L\<^isub>2 x = tag_str_SEQ L\<^isub>1 L\<^isub>2 y"
- shows "(x::string) \<equiv>(L\<^isub>1 ; L\<^isub>2)\<equiv> y"
-proof -
- have aux: "\<And> x y z. \<lbrakk>tag_str_SEQ L\<^isub>1 L\<^isub>2 x = tag_str_SEQ L\<^isub>1 L\<^isub>2 y; x @ z \<in> L\<^isub>1 ; L\<^isub>2\<rbrakk>
- \<Longrightarrow> y @ z \<in> L\<^isub>1 ; L\<^isub>2"
- proof (drule app_in_seq_decom, erule disjE)
- fix x y z
- assume tag_eq: "tag_str_SEQ L\<^isub>1 L\<^isub>2 x = tag_str_SEQ L\<^isub>1 L\<^isub>2 y"
- and x_gets_l2: "\<exists>xa\<le>x. xa \<in> L\<^isub>1 \<and> (x - xa) @ z \<in> L\<^isub>2"
- from x_gets_l2 have "\<exists> xa \<le> x. xa \<in> L\<^isub>1" by blast
- hence xy_l2:"{\<lbrakk>(x - xa)\<rbrakk>L\<^isub>2 | xa. xa \<le> x \<and> xa \<in> L\<^isub>1} = {\<lbrakk>(y - ya)\<rbrakk>L\<^isub>2 | ya. ya \<le> y \<and> ya \<in> L\<^isub>1}"
- using tag_eq tag_seq_eq_E by blast
- from x_gets_l2 obtain xa where xa_le_x: "xa \<le> x"
- and xa_in_l1: "xa \<in> L\<^isub>1"
- and rest_in_l2: "(x - xa) @ z \<in> L\<^isub>2" by blast
- hence "\<exists> ya. \<lbrakk>(x - xa)\<rbrakk>L\<^isub>2 = \<lbrakk>(y - ya)\<rbrakk>L\<^isub>2 \<and> ya \<le> y \<and> ya \<in> L\<^isub>1" using xy_l2 by auto
- then obtain ya where ya_le_x: "ya \<le> y"
- and ya_in_l1: "ya \<in> L\<^isub>1"
- and rest_eq: "\<lbrakk>(x - xa)\<rbrakk>L\<^isub>2 = \<lbrakk>(y - ya)\<rbrakk>L\<^isub>2" by blast
- from rest_eq rest_in_l2 have "(y - ya) @ z \<in> L\<^isub>2"
- by (auto simp:equiv_class_def equiv_str_def)
- hence "ya @ ((y - ya) @ z) \<in> L\<^isub>1 ; L\<^isub>2" using ya_in_l1
- by (auto simp:lang_seq_def)
- thus "y @ z \<in> L\<^isub>1 ; L\<^isub>2" using ya_le_x
- by (erule_tac prefixE, simp)
- next
- fix x y z
- assume tag_eq: "tag_str_SEQ L\<^isub>1 L\<^isub>2 x = tag_str_SEQ L\<^isub>1 L\<^isub>2 y"
- and x_gets_l1: "\<exists>za\<le>z. x @ za \<in> L\<^isub>1 \<and> z - za \<in> L\<^isub>2"
- from tag_eq tag_seq_eq_E have x_y_eq: "\<lbrakk>x\<rbrakk>L\<^isub>1 = \<lbrakk>y\<rbrakk>L\<^isub>1" by blast
- from x_gets_l1 obtain za where za_le_z: "za \<le> z"
- and x_za_in_l1: "(x @ za) \<in> L\<^isub>1"
- and rest_in_l2: "z - za \<in> L\<^isub>2" by blast
- from x_y_eq x_za_in_l1 have y_za_in_l1: "y @ za \<in> L\<^isub>1"
- by (auto simp:equiv_class_def equiv_str_def)
- hence "(y @ za) @ (z - za) \<in> L\<^isub>1 ; L\<^isub>2" using rest_in_l2
- apply (simp add:lang_seq_def)
- by (rule_tac x = "y @ za" in exI, rule_tac x = "z - za" in exI, simp)
- thus "y @ z \<in> L\<^isub>1 ; L\<^isub>2" using za_le_z
- by (erule_tac prefixE, simp)
- qed
- show ?thesis using tag_eq
- apply (simp add:equiv_str_def)
- by (auto intro:aux)
-qed
-
-lemma quot_seq:
- assumes finite1: "finite (QUOT L\<^isub>1)"
- and finite2: "finite (QUOT L\<^isub>2)"
- shows "finite (QUOT (L\<^isub>1;L\<^isub>2))"
-proof (rule_tac f = "(op `) (tag_str_SEQ L\<^isub>1 L\<^isub>2)" in finite_imageD)
- show "finite (op ` (tag_str_SEQ L\<^isub>1 L\<^isub>2) ` QUOT (L\<^isub>1 ; L\<^isub>2))"
- using finite_tag_image tag_str_seq_range_finite finite1 finite2
- by auto
-next
- show "inj_on (op ` (tag_str_SEQ L\<^isub>1 L\<^isub>2)) (QUOT (L\<^isub>1 ; L\<^isub>2))"
- apply (rule_tac str_inj_imps)
- by (erule_tac tag_str_seq_inj)
-qed
-
-(****************** the STAR case ************************)
-
-lemma app_eq_elim[rule_format]:
- "\<And> a. \<forall> b x y. a @ b = x @ y \<longrightarrow> (\<exists> aa ab. a = aa @ ab \<and> x = aa \<and> y = ab @ b) \<or>
- (\<exists> ba bb. b = ba @ bb \<and> x = a @ ba \<and> y = bb \<and> ba \<noteq> [])"
- apply (induct_tac a rule:List.induct, simp)
- apply (rule allI | rule impI)+
- by (case_tac x, auto)
-
-definition tag_str_STAR:: "string set \<Rightarrow> string \<Rightarrow> string set set"
-where
- "tag_str_STAR L\<^isub>1 x \<equiv> if (x = [])
- then {}
- else {\<lbrakk>x\<^isub>2\<rbrakk>L\<^isub>1 | x\<^isub>1 x\<^isub>2. x = x\<^isub>1 @ x\<^isub>2 \<and> x\<^isub>1 \<in> L\<^isub>1\<star> \<and> x\<^isub>2 \<noteq> []}"
-
-lemma tag_str_star_range_finite:
- assumes finite1: "finite (QUOT L\<^isub>1)"
- shows "finite (range (tag_str_STAR L\<^isub>1))"
-proof -
- have "range (tag_str_STAR L\<^isub>1) \<subseteq> Pow (QUOT L\<^isub>1)"
- by (auto simp:image_def tag_str_STAR_def QUOT_def)
- thus ?thesis using finite1
- by (rule_tac B = "Pow (QUOT L\<^isub>1)" in finite_subset, auto)
-qed
-
-lemma star_prop[rule_format]: "x \<in> lang\<star> \<Longrightarrow> \<forall> y. y \<in> lang\<star> \<longrightarrow> x @ y \<in> lang\<star>"
-by (erule Star.induct, auto)
-
-lemma star_prop2: "y \<in> lang \<Longrightarrow> y \<in> lang\<star>"
-by (drule step[of y lang "[]"], auto simp:start)
-
-lemma star_prop3[rule_format]: "x \<in> lang\<star> \<Longrightarrow> \<forall>y . y \<in> lang \<longrightarrow> x @ y \<in> lang\<star>"
-by (erule Star.induct, auto intro:star_prop2)
-
-lemma postfix_prop: "y >>= (x @ y) \<Longrightarrow> x = []"
-by (erule postfixE, induct x arbitrary:y, auto)
-
-lemma inj_aux:
- "\<lbrakk>(m @ z) \<in> L\<^isub>1\<star>; m \<equiv>L\<^isub>1\<equiv> yb; xa @ m = x; xa \<in> L\<^isub>1\<star>; m \<noteq> [];
- \<forall> xa xb. x = xa @ xb \<and> xa \<in> L\<^isub>1\<star> \<and> xb \<noteq> [] \<and> xb @ z \<in> L\<^isub>1\<star> \<longrightarrow> xb >>= m\<rbrakk>
- \<Longrightarrow> (yb @ z) \<in> L\<^isub>1\<star>"
-proof-
- have "\<And>s. s \<in> L\<^isub>1\<star> \<Longrightarrow> \<forall> m z yb. (s = m @ z \<and> m \<equiv>L\<^isub>1\<equiv> yb \<and> x = xa @ m \<and> xa \<in> L\<^isub>1\<star> \<and> m \<noteq> [] \<and>
- (\<forall> xa xb. x = xa @ xb \<and> xa \<in> L\<^isub>1\<star> \<and> xb \<noteq> [] \<and> xb @ z \<in> L\<^isub>1\<star> \<longrightarrow> xb >>= m) \<longrightarrow> (yb @ z) \<in> L\<^isub>1\<star>)"
- apply (erule Star.induct, simp)
- apply (rule allI | rule impI | erule conjE)+
- apply (drule app_eq_elim)
- apply (erule disjE | erule exE | erule conjE)+
- apply simp
- apply (simp (no_asm) only:append_assoc[THEN sym])
- apply (rule step)
- apply (simp add:equiv_str_def)
- apply simp
-
- apply (erule exE | erule conjE)+
- apply (rotate_tac 3)
- apply (frule_tac x = "xa @ s1" in spec)
- apply (rotate_tac 12)
- apply (drule_tac x = ba in spec)
- apply (erule impE)
- apply ( simp add:star_prop3)
- apply (simp)
- apply (drule postfix_prop)
- apply simp
- done
- thus "\<lbrakk>(m @ z) \<in> L\<^isub>1\<star>; m \<equiv>L\<^isub>1\<equiv> yb; xa @ m = x; xa \<in> L\<^isub>1\<star>; m \<noteq> [];
- \<forall> xa xb. x = xa @ xb \<and> xa \<in> L\<^isub>1\<star> \<and> xb \<noteq> [] \<and> xb @ z \<in> L\<^isub>1\<star> \<longrightarrow> xb >>= m\<rbrakk>
- \<Longrightarrow> (yb @ z) \<in> L\<^isub>1\<star>" by blast
-qed
-
-
-lemma min_postfix_exists[rule_format]:
- "finite A \<Longrightarrow> A \<noteq> {} \<and> (\<forall> a \<in> A. \<forall> b \<in> A. ((b >>= a) \<or> (a >>= b)))
- \<longrightarrow> (\<exists> min. (min \<in> A \<and> (\<forall> a \<in> A. a >>= min)))"
-apply (erule finite.induct)
-apply simp
-apply simp
-apply (case_tac "A = {}")
-apply simp
-apply clarsimp
-apply (case_tac "a >>= min")
-apply (rule_tac x = min in exI, simp)
-apply (rule_tac x = a in exI, simp)
-apply clarify
-apply (rotate_tac 5)
-apply (erule_tac x = aa in ballE) defer apply simp
-apply (erule_tac ys = min in postfix_trans)
-apply (erule_tac x = min in ballE)
-by simp+
-
-lemma tag_str_star_inj:
- "tag_str_STAR L\<^isub>1 x = tag_str_STAR L\<^isub>1 (y::string) \<Longrightarrow> x \<equiv>L\<^isub>1\<star>\<equiv> y"
-proof -
- have aux: "\<And> x y z. \<lbrakk>tag_str_STAR L\<^isub>1 x = tag_str_STAR L\<^isub>1 y; x @ z \<in> L\<^isub>1\<star>\<rbrakk> \<Longrightarrow> y @ z \<in> L\<^isub>1\<star>"
- proof-
- fix x y z
- assume tag_eq: "tag_str_STAR L\<^isub>1 x = tag_str_STAR L\<^isub>1 y"
- and x_z: "x @ z \<in> L\<^isub>1\<star>"
- show "y @ z \<in> L\<^isub>1\<star>"
- proof (cases "x = []")
- case True
- with tag_eq have "y = []" by (simp add:tag_str_STAR_def split:if_splits, blast)
- thus ?thesis using x_z True by simp
- next
- case False
- hence not_empty: "{xb. \<exists> xa. x = xa @ xb \<and> xa \<in> L\<^isub>1\<star> \<and> xb \<noteq> [] \<and> xb @ z \<in> L\<^isub>1\<star>} \<noteq> {}" using x_z
- by (simp, rule_tac x = x in exI, rule_tac x = "[]" in exI, simp add:start)
- have finite_set: "finite {xb. \<exists> xa. x = xa @ xb \<and> xa \<in> L\<^isub>1\<star> \<and> xb \<noteq> [] \<and> xb @ z \<in> L\<^isub>1\<star>}"
- apply (rule_tac B = "{xb. \<exists> xa. x = xa @ xb}" in finite_subset)
- apply auto
- apply (induct x, simp)
- apply (subgoal_tac "{xb. \<exists>xa. a # x = xa @ xb} = insert (a # x) {xb. \<exists>xa. x = xa @ xb}")
- apply auto
- by (case_tac xaa, simp+)
- have comparable: "\<forall> a \<in> {xb. \<exists> xa. x = xa @ xb \<and> xa \<in> L\<^isub>1\<star> \<and> xb \<noteq> [] \<and> xb @ z \<in> L\<^isub>1\<star>}.
- \<forall> b \<in> {xb. \<exists> xa. x = xa @ xb \<and> xa \<in> L\<^isub>1\<star> \<and> xb \<noteq> [] \<and> xb @ z \<in> L\<^isub>1\<star>}.
- ((b >>= a) \<or> (a >>= b))"
- by (auto simp:postfix_def, drule app_eq_elim, blast)
- hence "\<exists> min. min \<in> {xb. \<exists> xa. x = xa @ xb \<and> xa \<in> L\<^isub>1\<star> \<and> xb \<noteq> [] \<and> xb @ z \<in> L\<^isub>1\<star>}
- \<and> (\<forall> xa xb. x = xa @ xb \<and> xa \<in> L\<^isub>1\<star> \<and> xb \<noteq> [] \<and> xb @ z \<in> L\<^isub>1\<star> \<longrightarrow> xb >>= min)"
- using finite_set not_empty comparable
- apply (drule_tac min_postfix_exists, simp)
- by (erule exE, rule_tac x = min in exI, auto)
- then obtain min xa where x_decom: "x = xa @ min \<and> xa \<in> L\<^isub>1\<star>"
- and min_not_empty: "min \<noteq> []"
- and min_z_in_star: "min @ z \<in> L\<^isub>1\<star>"
- and is_min: "\<forall> xa xb. x = xa @ xb \<and> xa \<in> L\<^isub>1\<star> \<and> xb \<noteq> [] \<and> xb @ z \<in> L\<^isub>1\<star> \<longrightarrow> xb >>= min" by blast
- from x_decom min_not_empty have "\<lbrakk>min\<rbrakk>L\<^isub>1 \<in> tag_str_STAR L\<^isub>1 x" by (auto simp:tag_str_STAR_def)
- hence "\<exists> yb. \<lbrakk>yb\<rbrakk>L\<^isub>1 \<in> tag_str_STAR L\<^isub>1 y \<and> \<lbrakk>min\<rbrakk>L\<^isub>1 = \<lbrakk>yb\<rbrakk>L\<^isub>1" using tag_eq by auto
- hence "\<exists> ya yb. y = ya @ yb \<and> ya \<in> L\<^isub>1\<star> \<and> min \<equiv>L\<^isub>1\<equiv> yb \<and> yb \<noteq> [] "
- by (simp add:tag_str_STAR_def equiv_class_def equiv_str_def split:if_splits, blast)
- then obtain ya yb where y_decom: "y = ya @ yb"
- and ya_in_star: "ya \<in> L\<^isub>1\<star>"
- and yb_not_empty: "yb \<noteq> []"
- and min_yb_eq: "min \<equiv>L\<^isub>1\<equiv> yb" by blast
- from min_z_in_star min_yb_eq min_not_empty is_min x_decom
- have "yb @ z \<in> L\<^isub>1\<star>"
- by (rule_tac x = x and xa = xa in inj_aux, simp+)
- thus ?thesis using ya_in_star y_decom
- by (auto dest:star_prop)
- qed
- qed
- show "tag_str_STAR L\<^isub>1 x = tag_str_STAR L\<^isub>1 (y::string) \<Longrightarrow> x \<equiv>L\<^isub>1\<star>\<equiv> y"
- by (auto intro:aux simp:equiv_str_def)
-qed
-
-lemma quot_star:
- assumes finite1: "finite (QUOT L\<^isub>1)"
- shows "finite (QUOT (L\<^isub>1\<star>))"
-proof (rule_tac f = "(op `) (tag_str_STAR L\<^isub>1)" in finite_imageD)
- show "finite (op ` (tag_str_STAR L\<^isub>1) ` QUOT (L\<^isub>1\<star>))"
- using finite_tag_image tag_str_star_range_finite finite1
- by auto
-next
- show "inj_on (op ` (tag_str_STAR L\<^isub>1)) (QUOT (L\<^isub>1\<star>))"
- apply (rule_tac str_inj_imps)
- by (erule_tac tag_str_star_inj)
-qed
-
-lemma other_direction:
- "Lang = L (r::rexp) \<Longrightarrow> finite (QUOT Lang)"
-apply (induct arbitrary:Lang rule:rexp.induct)
-apply (simp add:QUOT_def equiv_class_def equiv_str_def)
-by (simp_all add:quot_lambda quot_single quot_seq quot_alt quot_star)
-
-end
--- a/Myhill_1.thy Fri Jun 03 13:59:21 2011 +0000
+++ b/Myhill_1.thy Mon Jul 25 13:33:38 2011 +0000
@@ -1,5 +1,5 @@
theory Myhill_1
-imports Regular
+imports More_Regular_Set
"~~/src/HOL/Library/While_Combinator"
begin
@@ -12,12 +12,12 @@
text {* Myhill-Nerode relation *}
definition
- str_eq_rel :: "lang \<Rightarrow> (string \<times> string) set" ("\<approx>_" [100] 100)
+ str_eq_rel :: "'a lang \<Rightarrow> ('a list \<times> 'a list) set" ("\<approx>_" [100] 100)
where
"\<approx>A \<equiv> {(x, y). (\<forall>z. x @ z \<in> A \<longleftrightarrow> y @ z \<in> A)}"
definition
- finals :: "lang \<Rightarrow> lang set"
+ finals :: "'a lang \<Rightarrow> 'a lang set"
where
"finals A \<equiv> {\<approx>A `` {s} | s . s \<in> A}"
@@ -37,35 +37,35 @@
text {* The two kinds of terms in the rhs of equations. *}
-datatype trm =
- Lam "rexp" (* Lambda-marker *)
- | Trn "lang" "rexp" (* Transition *)
+datatype 'a trm =
+ Lam "'a rexp" (* Lambda-marker *)
+ | Trn "'a lang" "'a rexp" (* Transition *)
fun
- L_trm::"trm \<Rightarrow> lang"
+ lang_trm::"'a trm \<Rightarrow> 'a lang"
where
- "L_trm (Lam r) = L_rexp r"
-| "L_trm (Trn X r) = X \<cdot> L_rexp r"
+ "lang_trm (Lam r) = lang r"
+| "lang_trm (Trn X r) = X \<cdot> lang r"
fun
- L_rhs::"trm set \<Rightarrow> lang"
+ lang_rhs::"('a trm) set \<Rightarrow> 'a lang"
where
- "L_rhs rhs = \<Union> (L_trm ` rhs)"
+ "lang_rhs rhs = \<Union> (lang_trm ` rhs)"
-lemma L_rhs_set:
- shows "L_rhs {Trn X r | r. P r} = \<Union>{L_trm (Trn X r) | r. P r}"
+lemma lang_rhs_set:
+ shows "lang_rhs {Trn X r | r. P r} = \<Union>{lang_trm (Trn X r) | r. P r}"
by (auto)
-lemma L_rhs_union_distrib:
- fixes A B::"trm set"
- shows "L_rhs A \<union> L_rhs B = L_rhs (A \<union> B)"
+lemma lang_rhs_union_distrib:
+ fixes A B::"('a trm) set"
+ shows "lang_rhs A \<union> lang_rhs B = lang_rhs (A \<union> B)"
by simp
text {* Transitions between equivalence classes *}
definition
- transition :: "lang \<Rightarrow> char \<Rightarrow> lang \<Rightarrow> bool" ("_ \<Turnstile>_\<Rightarrow>_" [100,100,100] 100)
+ transition :: "'a lang \<Rightarrow> 'a \<Rightarrow> 'a lang \<Rightarrow> bool" ("_ \<Turnstile>_\<Rightarrow>_" [100,100,100] 100)
where
"Y \<Turnstile>c\<Rightarrow> X \<equiv> Y \<cdot> {[c]} \<subseteq> X"
@@ -74,9 +74,9 @@
definition
"Init_rhs CS X \<equiv>
if ([] \<in> X) then
- {Lam EMPTY} \<union> {Trn Y (CHAR c) | Y c. Y \<in> CS \<and> Y \<Turnstile>c\<Rightarrow> X}
+ {Lam One} \<union> {Trn Y (Atom c) | Y c. Y \<in> CS \<and> Y \<Turnstile>c\<Rightarrow> X}
else
- {Trn Y (CHAR c)| Y c. Y \<in> CS \<and> Y \<Turnstile>c\<Rightarrow> X}"
+ {Trn Y (Atom c)| Y c. Y \<in> CS \<and> Y \<Turnstile>c\<Rightarrow> X}"
definition
"Init CS \<equiv> {(X, Init_rhs CS X) | X. X \<in> CS}"
@@ -85,10 +85,10 @@
section {* Arden Operation on equations *}
fun
- Append_rexp :: "rexp \<Rightarrow> trm \<Rightarrow> trm"
+ Append_rexp :: "'a rexp \<Rightarrow> 'a trm \<Rightarrow> 'a trm"
where
- "Append_rexp r (Lam rexp) = Lam (SEQ rexp r)"
-| "Append_rexp r (Trn X rexp) = Trn X (SEQ rexp r)"
+ "Append_rexp r (Lam rexp) = Lam (Times rexp r)"
+| "Append_rexp r (Trn X rexp) = Trn X (Times rexp r)"
definition
@@ -96,7 +96,7 @@
definition
"Arden X rhs \<equiv>
- Append_rexp_rhs (rhs - {Trn X r | r. Trn X r \<in> rhs}) (STAR (\<Uplus> {r. Trn X r \<in> rhs}))"
+ Append_rexp_rhs (rhs - {Trn X r | r. Trn X r \<in> rhs}) (Star (\<Uplus> {r. Trn X r \<in> rhs}))"
section {* Substitution Operation on equations *}
@@ -106,7 +106,7 @@
(rhs - {Trn X r | r. Trn X r \<in> rhs}) \<union> (Append_rexp_rhs xrhs (\<Uplus> {r. Trn X r \<in> rhs}))"
definition
- Subst_all :: "(lang \<times> trm set) set \<Rightarrow> lang \<Rightarrow> trm set \<Rightarrow> (lang \<times> trm set) set"
+ Subst_all :: "('a lang \<times> ('a trm) set) set \<Rightarrow> 'a lang \<Rightarrow> ('a trm) set \<Rightarrow> ('a lang \<times> ('a trm) set) set"
where
"Subst_all ES X xrhs \<equiv> {(Y, Subst yrhs X xrhs) | Y yrhs. (Y, yrhs) \<in> ES}"
@@ -143,10 +143,10 @@
\<forall> X rhs rhs'. (X, rhs) \<in> ES \<and> (X, rhs') \<in> ES \<longrightarrow> rhs = rhs'"
definition
- "soundness ES \<equiv> \<forall>(X, rhs) \<in> ES. X = L_rhs rhs"
+ "soundness ES \<equiv> \<forall>(X, rhs) \<in> ES. X = lang_rhs rhs"
definition
- "ardenable rhs \<equiv> (\<forall> Y r. Trn Y r \<in> rhs \<longrightarrow> [] \<notin> L_rexp r)"
+ "ardenable rhs \<equiv> (\<forall> Y r. Trn Y r \<in> rhs \<longrightarrow> [] \<notin> lang r)"
definition
"ardenable_all ES \<equiv> \<forall>(X, rhs) \<in> ES. ardenable rhs"
@@ -223,23 +223,23 @@
lemma trm_soundness:
assumes finite:"finite rhs"
- shows "L_rhs ({Trn X r| r. Trn X r \<in> rhs}) = X \<cdot> (L_rexp (\<Uplus>{r. Trn X r \<in> rhs}))"
+ shows "lang_rhs ({Trn X r| r. Trn X r \<in> rhs}) = X \<cdot> (lang (\<Uplus>{r. Trn X r \<in> rhs}))"
proof -
have "finite {r. Trn X r \<in> rhs}"
by (rule finite_Trn[OF finite])
- then show "L_rhs ({Trn X r| r. Trn X r \<in> rhs}) = X \<cdot> (L_rexp (\<Uplus>{r. Trn X r \<in> rhs}))"
- by (simp only: L_rhs_set L_trm.simps) (auto simp add: Seq_def)
+ then show "lang_rhs ({Trn X r| r. Trn X r \<in> rhs}) = X \<cdot> (lang (\<Uplus>{r. Trn X r \<in> rhs}))"
+ by (simp only: lang_rhs_set lang_trm.simps) (auto simp add: conc_def)
qed
lemma lang_of_append_rexp:
- "L_trm (Append_rexp r trm) = L_trm trm \<cdot> L_rexp r"
+ "lang_trm (Append_rexp r trm) = lang_trm trm \<cdot> lang r"
by (induct rule: Append_rexp.induct)
- (auto simp add: seq_assoc)
+ (auto simp add: conc_assoc)
lemma lang_of_append_rexp_rhs:
- "L_rhs (Append_rexp_rhs rhs r) = L_rhs rhs \<cdot> L_rexp r"
+ "lang_rhs (Append_rexp_rhs rhs r) = lang_rhs rhs \<cdot> lang r"
unfolding Append_rexp_rhs_def
-by (auto simp add: Seq_def lang_of_append_rexp)
+by (auto simp add: conc_def lang_of_append_rexp)
subsubsection {* Intial Equational System *}
@@ -263,7 +263,7 @@
have "X = \<approx>A `` {s @ [c]}"
using has_str in_CS defined_by_str by blast
then have "Y \<cdot> {[c]} \<subseteq> X"
- unfolding Y_def Image_def Seq_def
+ unfolding Y_def Image_def conc_def
unfolding str_eq_rel_def
by clarsimp
moreover
@@ -274,14 +274,14 @@
lemma l_eq_r_in_eqs:
assumes X_in_eqs: "(X, rhs) \<in> Init (UNIV // \<approx>A)"
- shows "X = L_rhs rhs"
+ shows "X = lang_rhs rhs"
proof
- show "X \<subseteq> L_rhs rhs"
+ show "X \<subseteq> lang_rhs rhs"
proof
fix x
assume in_X: "x \<in> X"
{ assume empty: "x = []"
- then have "x \<in> L_rhs rhs" using X_in_eqs in_X
+ then have "x \<in> lang_rhs rhs" using X_in_eqs in_X
unfolding Init_def Init_rhs_def
by auto
}
@@ -291,43 +291,42 @@
using rev_cases by blast
have "X \<in> UNIV // \<approx>A" using X_in_eqs unfolding Init_def by auto
then obtain Y where "Y \<in> UNIV // \<approx>A" "Y \<cdot> {[c]} \<subseteq> X" "s \<in> Y"
- using decom in_X every_eqclass_has_transition by blast
- then have "x \<in> L_rhs {Trn Y (CHAR c)| Y c. Y \<in> UNIV // \<approx>A \<and> Y \<Turnstile>c\<Rightarrow> X}"
+ using decom in_X every_eqclass_has_transition by metis
+ then have "x \<in> lang_rhs {Trn Y (Atom c)| Y c. Y \<in> UNIV // \<approx>A \<and> Y \<Turnstile>c\<Rightarrow> X}"
unfolding transition_def
- using decom by (force simp add: Seq_def)
- then have "x \<in> L_rhs rhs" using X_in_eqs in_X
+ using decom by (force simp add: conc_def)
+ then have "x \<in> lang_rhs rhs" using X_in_eqs in_X
unfolding Init_def Init_rhs_def by simp
}
- ultimately show "x \<in> L_rhs rhs" by blast
+ ultimately show "x \<in> lang_rhs rhs" by blast
qed
next
- show "L_rhs rhs \<subseteq> X" using X_in_eqs
+ show "lang_rhs rhs \<subseteq> X" using X_in_eqs
unfolding Init_def Init_rhs_def transition_def
by auto
qed
-lemma test:
- assumes X_in_eqs: "(X, rhs) \<in> Init (UNIV // \<approx>A)"
- shows "X = \<Union> (L_trm ` rhs)"
-using assms l_eq_r_in_eqs by (simp)
lemma finite_Init_rhs:
+ fixes CS::"(('a::finite) lang) set"
assumes finite: "finite CS"
shows "finite (Init_rhs CS X)"
proof-
- def S \<equiv> "{(Y, c)| Y c. Y \<in> CS \<and> Y \<cdot> {[c]} \<subseteq> X}"
- def h \<equiv> "\<lambda> (Y, c). Trn Y (CHAR c)"
- have "finite (CS \<times> (UNIV::char set))" using finite by auto
+ def S \<equiv> "{(Y, c)| Y c::'a. Y \<in> CS \<and> Y \<cdot> {[c]} \<subseteq> X}"
+ def h \<equiv> "\<lambda> (Y, c::'a). Trn Y (Atom c)"
+ have "finite (CS \<times> (UNIV::('a::finite) set))" using finite by auto
then have "finite S" using S_def
by (rule_tac B = "CS \<times> UNIV" in finite_subset) (auto)
- moreover have "{Trn Y (CHAR c) |Y c. Y \<in> CS \<and> Y \<cdot> {[c]} \<subseteq> X} = h ` S"
+ moreover have "{Trn Y (Atom c) |Y c::'a. Y \<in> CS \<and> Y \<cdot> {[c]} \<subseteq> X} = h ` S"
unfolding S_def h_def image_def by auto
ultimately
- have "finite {Trn Y (CHAR c) |Y c. Y \<in> CS \<and> Y \<cdot> {[c]} \<subseteq> X}" by auto
+ have "finite {Trn Y (Atom c) |Y c. Y \<in> CS \<and> Y \<cdot> {[c]} \<subseteq> X}" by auto
then show "finite (Init_rhs CS X)" unfolding Init_rhs_def transition_def by simp
qed
+
lemma Init_ES_satisfies_invariant:
+ fixes A::"(('a::finite) lang)"
assumes finite_CS: "finite (UNIV // \<approx>A)"
shows "invariant (Init (UNIV // \<approx>A))"
proof (rule invariantI)
@@ -352,20 +351,20 @@
subsubsection {* Interation step *}
lemma Arden_keeps_eq:
- assumes l_eq_r: "X = L_rhs rhs"
+ assumes l_eq_r: "X = lang_rhs rhs"
and not_empty: "ardenable rhs"
and finite: "finite rhs"
- shows "X = L_rhs (Arden X rhs)"
+ shows "X = lang_rhs (Arden X rhs)"
proof -
- def A \<equiv> "L_rexp (\<Uplus>{r. Trn X r \<in> rhs})"
+ def A \<equiv> "lang (\<Uplus>{r. Trn X r \<in> rhs})"
def b \<equiv> "{Trn X r | r. Trn X r \<in> rhs}"
- def B \<equiv> "L_rhs (rhs - b)"
+ def B \<equiv> "lang_rhs (rhs - b)"
have not_empty2: "[] \<notin> A"
using finite_Trn[OF finite] not_empty
unfolding A_def ardenable_def by simp
- have "X = L_rhs rhs" using l_eq_r by simp
- also have "\<dots> = L_rhs (b \<union> (rhs - b))" unfolding b_def by auto
- also have "\<dots> = L_rhs b \<union> B" unfolding B_def by (simp only: L_rhs_union_distrib)
+ have "X = lang_rhs rhs" using l_eq_r by simp
+ also have "\<dots> = lang_rhs (b \<union> (rhs - b))" unfolding b_def by auto
+ also have "\<dots> = lang_rhs b \<union> B" unfolding B_def by (simp only: lang_rhs_union_distrib)
also have "\<dots> = X \<cdot> A \<union> B"
unfolding b_def
unfolding trm_soundness[OF finite]
@@ -374,24 +373,24 @@
finally have "X = X \<cdot> A \<union> B" .
then have "X = B \<cdot> A\<star>"
by (simp add: arden[OF not_empty2])
- also have "\<dots> = L_rhs (Arden X rhs)"
+ also have "\<dots> = lang_rhs (Arden X rhs)"
unfolding Arden_def A_def B_def b_def
- by (simp only: lang_of_append_rexp_rhs L_rexp.simps)
- finally show "X = L_rhs (Arden X rhs)" by simp
+ by (simp only: lang_of_append_rexp_rhs lang.simps)
+ finally show "X = lang_rhs (Arden X rhs)" by simp
qed
lemma Append_keeps_finite:
"finite rhs \<Longrightarrow> finite (Append_rexp_rhs rhs r)"
-by (auto simp:Append_rexp_rhs_def)
+by (auto simp: Append_rexp_rhs_def)
lemma Arden_keeps_finite:
"finite rhs \<Longrightarrow> finite (Arden X rhs)"
-by (auto simp:Arden_def Append_keeps_finite)
+by (auto simp: Arden_def Append_keeps_finite)
lemma Append_keeps_nonempty:
"ardenable rhs \<Longrightarrow> ardenable (Append_rexp_rhs rhs r)"
-apply (auto simp:ardenable_def Append_rexp_rhs_def)
-by (case_tac x, auto simp:Seq_def)
+apply (auto simp: ardenable_def Append_rexp_rhs_def)
+by (case_tac x, auto simp: conc_def)
lemma nonempty_set_sub:
"ardenable rhs \<Longrightarrow> ardenable (rhs - A)"
@@ -411,24 +410,25 @@
by (simp only: Subst_def Append_keeps_nonempty nonempty_set_union nonempty_set_sub)
lemma Subst_keeps_eq:
- assumes substor: "X = L_rhs xrhs"
+ assumes substor: "X = lang_rhs xrhs"
and finite: "finite rhs"
- shows "L_rhs (Subst rhs X xrhs) = L_rhs rhs" (is "?Left = ?Right")
+ shows "lang_rhs (Subst rhs X xrhs) = lang_rhs rhs" (is "?Left = ?Right")
proof-
- def A \<equiv> "L_rhs (rhs - {Trn X r | r. Trn X r \<in> rhs})"
- have "?Left = A \<union> L_rhs (Append_rexp_rhs xrhs (\<Uplus>{r. Trn X r \<in> rhs}))"
+ def A \<equiv> "lang_rhs (rhs - {Trn X r | r. Trn X r \<in> rhs})"
+ have "?Left = A \<union> lang_rhs (Append_rexp_rhs xrhs (\<Uplus>{r. Trn X r \<in> rhs}))"
unfolding Subst_def
- unfolding L_rhs_union_distrib[symmetric]
+ unfolding lang_rhs_union_distrib[symmetric]
by (simp add: A_def)
- moreover have "?Right = A \<union> L_rhs {Trn X r | r. Trn X r \<in> rhs}"
+ moreover have "?Right = A \<union> lang_rhs {Trn X r | r. Trn X r \<in> rhs}"
proof-
have "rhs = (rhs - {Trn X r | r. Trn X r \<in> rhs}) \<union> ({Trn X r | r. Trn X r \<in> rhs})" by auto
thus ?thesis
unfolding A_def
- unfolding L_rhs_union_distrib
+ unfolding lang_rhs_union_distrib
by simp
qed
- moreover have "L_rhs (Append_rexp_rhs xrhs (\<Uplus>{r. Trn X r \<in> rhs})) = L_rhs {Trn X r | r. Trn X r \<in> rhs}"
+ moreover
+ have "lang_rhs (Append_rexp_rhs xrhs (\<Uplus>{r. Trn X r \<in> rhs})) = lang_rhs {Trn X r | r. Trn X r \<in> rhs}"
using finite substor by (simp only: lang_of_append_rexp_rhs trm_soundness)
ultimately show ?thesis by simp
qed
@@ -441,8 +441,8 @@
assumes finite: "finite ES"
shows "finite (Subst_all ES Y yrhs)"
proof -
- def eqns \<equiv> "{(X::lang, rhs) |X rhs. (X, rhs) \<in> ES}"
- def h \<equiv> "\<lambda>(X::lang, rhs). (X, Subst rhs Y yrhs)"
+ def eqns \<equiv> "{(X::'a lang, rhs) |X rhs. (X, rhs) \<in> ES}"
+ def h \<equiv> "\<lambda>(X::'a lang, rhs). (X, Subst rhs Y yrhs)"
have "finite (h ` eqns)" using finite h_def eqns_def by auto
moreover
have "Subst_all ES Y yrhs = h ` eqns" unfolding h_def eqns_def Subst_all_def by auto
@@ -456,24 +456,24 @@
lemma append_rhs_keeps_cls:
"rhss (Append_rexp_rhs rhs r) = rhss rhs"
-apply (auto simp:rhss_def Append_rexp_rhs_def)
-apply (case_tac xa, auto simp:image_def)
-by (rule_tac x = "SEQ ra r" in exI, rule_tac x = "Trn x ra" in bexI, simp+)
+apply (auto simp: rhss_def Append_rexp_rhs_def)
+apply (case_tac xa, auto simp: image_def)
+by (rule_tac x = "Times ra r" in exI, rule_tac x = "Trn x ra" in bexI, simp+)
lemma Arden_removes_cl:
"rhss (Arden Y yrhs) = rhss yrhs - {Y}"
apply (simp add:Arden_def append_rhs_keeps_cls)
-by (auto simp:rhss_def)
+by (auto simp: rhss_def)
lemma lhss_keeps_cls:
"lhss (Subst_all ES Y yrhs) = lhss ES"
-by (auto simp:lhss_def Subst_all_def)
+by (auto simp: lhss_def Subst_all_def)
lemma Subst_updates_cls:
"X \<notin> rhss xrhs \<Longrightarrow>
rhss (Subst rhs X xrhs) = rhss rhs \<union> rhss xrhs - {X}"
apply (simp only:Subst_def append_rhs_keeps_cls rhss_union_distrib)
-by (auto simp:rhss_def)
+by (auto simp: rhss_def)
lemma Subst_all_keeps_validity:
assumes sc: "validity (ES \<union> {(Y, yrhs)})" (is "validity ?A")
@@ -490,17 +490,17 @@
moreover have "rhss xrhs' \<subseteq> lhss ES"
proof-
have "rhss xrhs' \<subseteq> rhss xrhs \<union> rhss (Arden Y yrhs) - {Y}"
- proof-
+ proof -
have "Y \<notin> rhss (Arden Y yrhs)"
- using Arden_removes_cl by simp
- thus ?thesis using xrhs_xrhs' by (auto simp:Subst_updates_cls)
+ using Arden_removes_cl by auto
+ thus ?thesis using xrhs_xrhs' by (auto simp: Subst_updates_cls)
qed
moreover have "rhss xrhs \<subseteq> lhss ES \<union> {Y}" using X_in sc
apply (simp only:validity_def lhss_union_distrib)
by (drule_tac x = "(X, xrhs)" in bspec, auto simp:lhss_def)
moreover have "rhss (Arden Y yrhs) \<subseteq> lhss ES \<union> {Y}"
using sc
- by (auto simp add:Arden_removes_cl validity_def lhss_def)
+ by (auto simp add: Arden_removes_cl validity_def lhss_def)
ultimately show ?thesis by auto
qed
ultimately show ?thesis by simp
@@ -512,7 +512,7 @@
assumes invariant_ES: "invariant (ES \<union> {(Y, yrhs)})"
shows "invariant (Subst_all ES Y (Arden Y yrhs))"
proof (rule invariantI)
- have Y_eq_yrhs: "Y = L_rhs yrhs"
+ have Y_eq_yrhs: "Y = lang_rhs yrhs"
using invariant_ES by (simp only:invariant_def soundness_def, blast)
have finite_yrhs: "finite yrhs"
using invariant_ES by (auto simp:invariant_def finite_rhs_def)
@@ -520,7 +520,7 @@
using invariant_ES by (auto simp:invariant_def ardenable_all_def)
show "soundness (Subst_all ES Y (Arden Y yrhs))"
proof -
- have "Y = L_rhs (Arden Y yrhs)"
+ have "Y = lang_rhs (Arden Y yrhs)"
using Y_eq_yrhs invariant_ES finite_yrhs
using finite_Trn[OF finite_yrhs]
apply(rule_tac Arden_keeps_eq)
@@ -530,7 +530,7 @@
done
thus ?thesis using invariant_ES
unfolding invariant_def finite_rhs_def2 soundness_def Subst_all_def
- by (auto simp add: Subst_keeps_eq simp del: L_rhs.simps)
+ by (auto simp add: Subst_keeps_eq simp del: lang_rhs.simps)
qed
show "finite (Subst_all ES Y (Arden Y yrhs))"
using invariant_ES by (simp add:invariant_def Subst_all_keeps_finite)
@@ -557,13 +557,13 @@
proof -
have "finite yrhs" using invariant_ES
by (auto simp:invariant_def finite_rhs_def)
- thus ?thesis using Arden_keeps_finite by simp
+ thus ?thesis using Arden_keeps_finite by auto
qed
ultimately show ?thesis
by (simp add:Subst_all_keeps_finite_rhs)
qed
show "validity (Subst_all ES Y (Arden Y yrhs))"
- using invariant_ES Subst_all_keeps_validity by (simp add:invariant_def)
+ using invariant_ES Subst_all_keeps_validity by (auto simp add: invariant_def)
qed
lemma Remove_in_card_measure:
@@ -571,7 +571,7 @@
and in_ES: "(X, rhs) \<in> ES"
shows "(Remove ES X rhs, ES) \<in> measure card"
proof -
- def f \<equiv> "\<lambda> x. ((fst x)::lang, Subst (snd x) X (Arden X rhs))"
+ def f \<equiv> "\<lambda> x. ((fst x)::'a lang, Subst (snd x) X (Arden X rhs))"
def ES' \<equiv> "ES - {(X, rhs)}"
have "Subst_all ES' X (Arden X rhs) = f ` ES'"
apply (auto simp: Subst_all_def f_def image_def)
@@ -674,6 +674,7 @@
subsubsection {* Conclusion of the proof *}
lemma Solve:
+ fixes A::"('a::finite) lang"
assumes fin: "finite (UNIV // \<approx>A)"
and X_in: "X \<in> (UNIV // \<approx>A)"
shows "\<exists>rhs. Solve X (Init (UNIV // \<approx>A)) = {(X, rhs)} \<and> invariant {(X, rhs)}"
@@ -714,9 +715,10 @@
qed
lemma every_eqcl_has_reg:
+ fixes A::"('a::finite) lang"
assumes finite_CS: "finite (UNIV // \<approx>A)"
and X_in_CS: "X \<in> (UNIV // \<approx>A)"
- shows "\<exists>r. X = L_rexp r"
+ shows "\<exists>r. X = lang r"
proof -
from finite_CS X_in_CS
obtain xrhs where Inv_ES: "invariant {(X, xrhs)}"
@@ -735,14 +737,14 @@
using Arden_keeps_finite by auto
then have fin: "finite {r. Lam r \<in> A}" by (rule finite_Lam)
- have "X = L_rhs xrhs" using Inv_ES unfolding invariant_def soundness_def
+ have "X = lang_rhs xrhs" using Inv_ES unfolding invariant_def soundness_def
by simp
- then have "X = L_rhs A" using Inv_ES
+ then have "X = lang_rhs A" using Inv_ES
unfolding A_def invariant_def ardenable_all_def finite_rhs_def
by (rule_tac Arden_keeps_eq) (simp_all add: finite_Trn)
- then have "X = L_rhs {Lam r | r. Lam r \<in> A}" using eq by simp
- then have "X = L_rexp (\<Uplus>{r. Lam r \<in> A})" using fin by auto
- then show "\<exists>r. X = L_rexp r" by blast
+ then have "X = lang_rhs {Lam r | r. Lam r \<in> A}" using eq by simp
+ then have "X = lang (\<Uplus>{r. Lam r \<in> A})" using fin by auto
+ then show "\<exists>r. X = lang r" by blast
qed
lemma bchoice_finite_set:
@@ -756,20 +758,21 @@
done
theorem Myhill_Nerode1:
+ fixes A::"('a::finite) lang"
assumes finite_CS: "finite (UNIV // \<approx>A)"
- shows "\<exists>r. A = L_rexp r"
+ shows "\<exists>r. A = lang r"
proof -
have fin: "finite (finals A)"
using finals_in_partitions finite_CS by (rule finite_subset)
- have "\<forall>X \<in> (UNIV // \<approx>A). \<exists>r. X = L_rexp r"
+ have "\<forall>X \<in> (UNIV // \<approx>A). \<exists>r. X = lang r"
using finite_CS every_eqcl_has_reg by blast
- then have a: "\<forall>X \<in> finals A. \<exists>r. X = L_rexp r"
+ then have a: "\<forall>X \<in> finals A. \<exists>r. X = lang r"
using finals_in_partitions by auto
- then obtain rs::"rexp set" where "\<Union> (finals A) = \<Union>(L_rexp ` rs)" "finite rs"
+ then obtain rs::"('a rexp) set" where "\<Union> (finals A) = \<Union>(lang ` rs)" "finite rs"
using fin by (auto dest: bchoice_finite_set)
- then have "A = L_rexp (\<Uplus>rs)"
+ then have "A = lang (\<Uplus>rs)"
unfolding lang_is_union_of_finals[symmetric] by simp
- then show "\<exists>r. A = L_rexp r" by blast
+ then show "\<exists>r. A = lang r" by blast
qed
--- a/Myhill_2.thy Fri Jun 03 13:59:21 2011 +0000
+++ b/Myhill_2.thy Mon Jul 25 13:33:38 2011 +0000
@@ -6,7 +6,7 @@
section {* Direction @{text "regular language \<Rightarrow> finite partition"} *}
definition
- str_eq :: "string \<Rightarrow> lang \<Rightarrow> string \<Rightarrow> bool" ("_ \<approx>_ _")
+ str_eq :: "'a list \<Rightarrow> 'a lang \<Rightarrow> 'a list \<Rightarrow> bool" ("_ \<approx>_ _")
where
"x \<approx>A y \<equiv> (x, y) \<in> (\<approx>A)"
@@ -16,7 +16,7 @@
by simp
definition
- tag_eq_rel :: "(string \<Rightarrow> 'b) \<Rightarrow> (string \<times> string) set" ("=_=")
+ tag_eq_rel :: "('a list \<Rightarrow> 'b) \<Rightarrow> ('a list \<times> 'a list) set" ("=_=")
where
"=tag= \<equiv> {(x, y). tag x = tag y}"
@@ -116,20 +116,20 @@
subsection {* The proof *}
-subsubsection {* The base case for @{const "NULL"} *}
+subsubsection {* The base case for @{const "Zero"} *}
-lemma quot_null_eq:
+lemma quot_zero_eq:
shows "UNIV // \<approx>{} = {UNIV}"
unfolding quotient_def Image_def str_eq_rel_def by auto
-lemma quot_null_finiteI [intro]:
+lemma quot_zero_finiteI [intro]:
shows "finite (UNIV // \<approx>{})"
-unfolding quot_null_eq by simp
+unfolding quot_zero_eq by simp
-subsubsection {* The base case for @{const "EMPTY"} *}
+subsubsection {* The base case for @{const "One"} *}
-lemma quot_empty_subset:
+lemma quot_one_subset:
shows "UNIV // \<approx>{[]} \<subseteq> {{[]}, UNIV - {[]}}"
proof
fix x
@@ -148,14 +148,14 @@
qed
qed
-lemma quot_empty_finiteI [intro]:
+lemma quot_one_finiteI [intro]:
shows "finite (UNIV // \<approx>{[]})"
-by (rule finite_subset[OF quot_empty_subset]) (simp)
+by (rule finite_subset[OF quot_one_subset]) (simp)
-subsubsection {* The base case for @{const "CHAR"} *}
+subsubsection {* The base case for @{const "Atom"} *}
-lemma quot_char_subset:
+lemma quot_atom_subset:
"UNIV // (\<approx>{[c]}) \<subseteq> {{[]},{[c]}, UNIV - {[], [c]}}"
proof
fix x
@@ -181,43 +181,43 @@
qed
qed
-lemma quot_char_finiteI [intro]:
+lemma quot_atom_finiteI [intro]:
shows "finite (UNIV // \<approx>{[c]})"
-by (rule finite_subset[OF quot_char_subset]) (simp)
+by (rule finite_subset[OF quot_atom_subset]) (simp)
-subsubsection {* The inductive case for @{const ALT} *}
+subsubsection {* The inductive case for @{const Plus} *}
definition
- tag_str_ALT :: "lang \<Rightarrow> lang \<Rightarrow> string \<Rightarrow> (lang \<times> lang)"
+ tag_str_Plus :: "'a lang \<Rightarrow> 'a lang \<Rightarrow> 'a list \<Rightarrow> ('a lang \<times> 'a lang)"
where
- "tag_str_ALT A B \<equiv> (\<lambda>x. (\<approx>A `` {x}, \<approx>B `` {x}))"
+ "tag_str_Plus A B \<equiv> (\<lambda>x. (\<approx>A `` {x}, \<approx>B `` {x}))"
lemma quot_union_finiteI [intro]:
assumes finite1: "finite (UNIV // \<approx>A)"
and finite2: "finite (UNIV // \<approx>B)"
shows "finite (UNIV // \<approx>(A \<union> B))"
-proof (rule_tac tag = "tag_str_ALT A B" in tag_finite_imageD)
+proof (rule_tac tag = "tag_str_Plus A B" in tag_finite_imageD)
have "finite ((UNIV // \<approx>A) \<times> (UNIV // \<approx>B))"
using finite1 finite2 by auto
- then show "finite (range (tag_str_ALT A B))"
- unfolding tag_str_ALT_def quotient_def
+ then show "finite (range (tag_str_Plus A B))"
+ unfolding tag_str_Plus_def quotient_def
by (rule rev_finite_subset) (auto)
next
- show "\<And>x y. tag_str_ALT A B x = tag_str_ALT A B y \<Longrightarrow> x \<approx>(A \<union> B) y"
- unfolding tag_str_ALT_def
+ show "\<And>x y. tag_str_Plus A B x = tag_str_Plus A B y \<Longrightarrow> x \<approx>(A \<union> B) y"
+ unfolding tag_str_Plus_def
unfolding str_eq_def
unfolding str_eq_rel_def
by auto
qed
-subsubsection {* The inductive case for @{text "SEQ"}*}
+subsubsection {* The inductive case for @{text "Times"}*}
definition
- tag_str_SEQ :: "lang \<Rightarrow> lang \<Rightarrow> string \<Rightarrow> (lang \<times> lang set)"
+ tag_str_Times :: "'a lang \<Rightarrow> 'a lang \<Rightarrow> 'a list \<Rightarrow> ('a lang \<times> 'a lang set)"
where
- "tag_str_SEQ L1 L2 \<equiv>
+ "tag_str_Times L1 L2 \<equiv>
(\<lambda>x. (\<approx>L1 `` {x}, {(\<approx>L2 `` {x - xa}) | xa. xa \<le> x \<and> xa \<in> L1}))"
lemma Seq_in_cases:
@@ -225,16 +225,16 @@
shows "(\<exists> x' \<le> x. x' \<in> A \<and> (x - x') @ z \<in> B) \<or>
(\<exists> z' \<le> z. (x @ z') \<in> A \<and> (z - z') \<in> B)"
using assms
-unfolding Seq_def prefix_def
+unfolding conc_def prefix_def
by (auto simp add: append_eq_append_conv2)
-lemma tag_str_SEQ_injI:
- assumes eq_tag: "tag_str_SEQ A B x = tag_str_SEQ A B y"
+lemma tag_str_Times_injI:
+ assumes eq_tag: "tag_str_Times A B x = tag_str_Times A B y"
shows "x \<approx>(A \<cdot> B) y"
proof -
{ fix x y z
assume xz_in_seq: "x @ z \<in> A \<cdot> B"
- and tag_xy: "tag_str_SEQ A B x = tag_str_SEQ A B y"
+ and tag_xy: "tag_str_Times A B x = tag_str_Times A B y"
have"y @ z \<in> A \<cdot> B"
proof -
{ (* first case with x' in A and (x - x') @ z in B *)
@@ -247,7 +247,7 @@
proof -
have "{\<approx>B `` {x - x'} |x'. x' \<le> x \<and> x' \<in> A} =
{\<approx>B `` {y - y'} |y'. y' \<le> y \<and> y' \<in> A}" (is "?Left = ?Right")
- using tag_xy unfolding tag_str_SEQ_def by simp
+ using tag_xy unfolding tag_str_Times_def by simp
moreover
have "\<approx>B `` {x - x'} \<in> ?Left" using h1 h2 by auto
ultimately
@@ -271,11 +271,11 @@
fix z'
assume h1: "z' \<le> z" and h2: "(x @ z') \<in> A" and h3: "z - z' \<in> B"
have "\<approx>A `` {x} = \<approx>A `` {y}"
- using tag_xy unfolding tag_str_SEQ_def by simp
+ using tag_xy unfolding tag_str_Times_def by simp
with h2 have "y @ z' \<in> A"
unfolding Image_def str_eq_rel_def str_eq_def by auto
with h1 h3 have "y @ z \<in> A \<cdot> B"
- unfolding prefix_def Seq_def
+ unfolding prefix_def conc_def
by (auto) (metis append_assoc)
}
ultimately show "y @ z \<in> A \<cdot> B"
@@ -287,30 +287,30 @@
qed
lemma quot_seq_finiteI [intro]:
- fixes L1 L2::"lang"
+ fixes L1 L2::"'a lang"
assumes fin1: "finite (UNIV // \<approx>L1)"
and fin2: "finite (UNIV // \<approx>L2)"
shows "finite (UNIV // \<approx>(L1 \<cdot> L2))"
-proof (rule_tac tag = "tag_str_SEQ L1 L2" in tag_finite_imageD)
- show "\<And>x y. tag_str_SEQ L1 L2 x = tag_str_SEQ L1 L2 y \<Longrightarrow> x \<approx>(L1 \<cdot> L2) y"
- by (rule tag_str_SEQ_injI)
+proof (rule_tac tag = "tag_str_Times L1 L2" in tag_finite_imageD)
+ show "\<And>x y. tag_str_Times L1 L2 x = tag_str_Times L1 L2 y \<Longrightarrow> x \<approx>(L1 \<cdot> L2) y"
+ by (rule tag_str_Times_injI)
next
have *: "finite ((UNIV // \<approx>L1) \<times> (Pow (UNIV // \<approx>L2)))"
using fin1 fin2 by auto
- show "finite (range (tag_str_SEQ L1 L2))"
- unfolding tag_str_SEQ_def
+ show "finite (range (tag_str_Times L1 L2))"
+ unfolding tag_str_Times_def
apply(rule finite_subset[OF _ *])
unfolding quotient_def
by auto
qed
-subsubsection {* The inductive case for @{const "STAR"} *}
+subsubsection {* The inductive case for @{const "Star"} *}
definition
- tag_str_STAR :: "lang \<Rightarrow> string \<Rightarrow> lang set"
+ tag_str_Star :: "'a lang \<Rightarrow> 'a list \<Rightarrow> ('a lang) set"
where
- "tag_str_STAR L1 \<equiv> (\<lambda>x. {\<approx>L1 `` {x - xa} | xa. xa < x \<and> xa \<in> L1\<star>})"
+ "tag_str_Star L1 \<equiv> (\<lambda>x. {\<approx>L1 `` {x - xa} | xa. xa < x \<and> xa \<in> L1\<star>})"
text {* A technical lemma. *}
lemma finite_set_has_max: "\<lbrakk>finite A; A \<noteq> {}\<rbrakk> \<Longrightarrow>
@@ -342,31 +342,33 @@
text {* The following is a technical lemma, which helps to show the range finiteness of tag function. *}
-lemma finite_strict_prefix_set: "finite {xa. xa < (x::string)}"
+lemma finite_strict_prefix_set:
+ shows "finite {xa. xa < (x::'a list)}"
apply (induct x rule:rev_induct, simp)
apply (subgoal_tac "{xa. xa < xs @ [x]} = {xa. xa < xs} \<union> {xs}")
by (auto simp:strict_prefix_def)
-lemma tag_str_STAR_injI:
- assumes eq_tag: "tag_str_STAR L\<^isub>1 v = tag_str_STAR L\<^isub>1 w"
+lemma tag_str_Star_injI:
+ fixes L\<^isub>1::"('a::finite) lang"
+ assumes eq_tag: "tag_str_Star L\<^isub>1 v = tag_str_Star L\<^isub>1 w"
shows "v \<approx>(L\<^isub>1\<star>) w"
proof-
{ fix x y z
assume xz_in_star: "x @ z \<in> L\<^isub>1\<star>"
- and tag_xy: "tag_str_STAR L\<^isub>1 x = tag_str_STAR L\<^isub>1 y"
+ and tag_xy: "tag_str_Star L\<^isub>1 x = tag_str_Star L\<^isub>1 y"
have "y @ z \<in> L\<^isub>1\<star>"
proof(cases "x = []")
case True
with tag_xy have "y = []"
- by (auto simp add: tag_str_STAR_def strict_prefix_def)
+ by (auto simp add: tag_str_Star_def strict_prefix_def)
thus ?thesis using xz_in_star True by simp
next
case False
- let ?S = "{xa. xa < x \<and> xa \<in> L\<^isub>1\<star> \<and> (x - xa) @ z \<in> L\<^isub>1\<star>}"
+ let ?S = "{xa::('a::finite) list. xa < x \<and> xa \<in> L\<^isub>1\<star> \<and> (x - xa) @ z \<in> L\<^isub>1\<star>}"
have "finite ?S"
- by (rule_tac B = "{xa. xa < x}" in finite_subset,
- auto simp:finite_strict_prefix_set)
+ by (rule_tac B = "{xa. xa < x}" in finite_subset)
+ (auto simp: finite_strict_prefix_set)
moreover have "?S \<noteq> {}" using False xz_in_star
by (simp, rule_tac x = "[]" in exI, auto simp:strict_prefix_def)
ultimately have "\<exists> xa_max \<in> ?S. \<forall> xa \<in> ?S. length xa \<le> length xa_max"
@@ -384,7 +386,7 @@
proof-
from tag_xy have "{\<approx>L\<^isub>1 `` {x - xa} |xa. xa < x \<and> xa \<in> L\<^isub>1\<star>} =
{\<approx>L\<^isub>1 `` {y - xa} |xa. xa < y \<and> xa \<in> L\<^isub>1\<star>}" (is "?left = ?right")
- by (auto simp:tag_str_STAR_def)
+ by (auto simp:tag_str_Star_def)
moreover have "\<approx>L\<^isub>1 `` {x - xa_max} \<in> ?left" using h1 h2 by auto
ultimately have "\<approx>L\<^isub>1 `` {x - xa_max} \<in> ?right" by simp
thus ?thesis using that
@@ -417,7 +419,7 @@
have "?xa_max' < x"
using np h1 by (clarsimp simp:strict_prefix_def diff_prefix)
moreover have "?xa_max' \<in> L\<^isub>1\<star>"
- using a_in h2 by (simp add:star_intro3)
+ using a_in h2 by (auto)
moreover have "(x - ?xa_max') @ z \<in> L\<^isub>1\<star>"
using b_eqs b_in np h1 by (simp add:diff_diff_append)
moreover have "\<not> (length ?xa_max' \<le> length xa_max)"
@@ -432,11 +434,12 @@
with eq_z and b_in
show ?thesis using that by blast
qed
- have "((y - ya) @ za) @ zb \<in> L\<^isub>1\<star>" using l_za ls_zb by blast
+ have "((y - ya) @ za) @ zb \<in> L\<^isub>1\<star>" using l_za ls_zb
+ by (rule_tac append_in_starI) (auto)
with eq_zab show ?thesis by simp
qed
- with h5 h6 show ?thesis
- by (drule_tac star_intro1) (auto simp:strict_prefix_def elim: prefixE)
+ with h5 h6 show ?thesis
+ by (auto simp:strict_prefix_def elim: prefixE)
qed
}
from this [OF _ eq_tag] and this [OF _ eq_tag [THEN sym]]
@@ -444,16 +447,17 @@
qed
lemma quot_star_finiteI [intro]:
+ fixes A::"('a::finite) lang"
assumes finite1: "finite (UNIV // \<approx>A)"
shows "finite (UNIV // \<approx>(A\<star>))"
-proof (rule_tac tag = "tag_str_STAR A" in tag_finite_imageD)
- show "\<And>x y. tag_str_STAR A x = tag_str_STAR A y \<Longrightarrow> x \<approx>(A\<star>) y"
- by (rule tag_str_STAR_injI)
+proof (rule_tac tag = "tag_str_Star A" in tag_finite_imageD)
+ show "\<And>x y. tag_str_Star A x = tag_str_Star A y \<Longrightarrow> x \<approx>(A\<star>) y"
+ by (rule tag_str_Star_injI)
next
have *: "finite (Pow (UNIV // \<approx>A))"
using finite1 by auto
- show "finite (range (tag_str_STAR A))"
- unfolding tag_str_STAR_def
+ show "finite (range (tag_str_Star A))"
+ unfolding tag_str_Star_def
apply(rule finite_subset[OF _ *])
unfolding quotient_def
by auto
@@ -462,12 +466,13 @@
subsubsection{* The conclusion *}
lemma Myhill_Nerode2:
- shows "finite (UNIV // \<approx>(L_rexp r))"
+ fixes r::"('a::finite) rexp"
+ shows "finite (UNIV // \<approx>(lang r))"
by (induct r) (auto)
-
theorem Myhill_Nerode:
- shows "(\<exists>r. A = L_rexp r) \<longleftrightarrow> finite (UNIV // \<approx>A)"
+ fixes A::"('a::finite) lang"
+ shows "(\<exists>r. A = lang r) \<longleftrightarrow> finite (UNIV // \<approx>A)"
using Myhill_Nerode1 Myhill_Nerode2 by auto
end
--- a/Paper/Paper.thy Fri Jun 03 13:59:21 2011 +0000
+++ b/Paper/Paper.thy Mon Jul 25 13:33:38 2011 +0000
@@ -15,17 +15,33 @@
abbreviation
"Append_rexp2 r_itm r == Append_rexp r r_itm"
+abbreviation
+ "pow" (infixl "\<up>" 100)
+where
+ "A \<up> n \<equiv> A ^^ n"
+
+
+abbreviation "NULL \<equiv> Zero"
+abbreviation "EMPTY \<equiv> One"
+abbreviation "CHAR \<equiv> Atom"
+abbreviation "ALT \<equiv> Plus"
+abbreviation "SEQ \<equiv> Times"
+abbreviation "STAR \<equiv> Star"
+
+
+ML {* @{term "op ^^"} *}
+
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
+ conc (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_rexp ("\<^raw:\ensuremath{\cal{L}}>'(_')" [0] 101) and
+ lang ("\<^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
@@ -35,18 +51,27 @@
Append_rexp_rhs ("_ \<^raw:\ensuremath{\triangleleft}> _" [100, 100] 50) and
uminus ("\<^raw:\ensuremath{\overline{>_\<^raw:}}>" [100] 100) and
- tag_str_ALT ("tag\<^isub>A\<^isub>L\<^isub>T _ _" [100, 100] 100) and
- tag_str_ALT ("tag\<^isub>A\<^isub>L\<^isub>T _ _ _" [100, 100, 100] 100) and
- tag_str_SEQ ("tag\<^isub>S\<^isub>E\<^isub>Q _ _" [100, 100] 100) and
- tag_str_SEQ ("tag\<^isub>S\<^isub>E\<^isub>Q _ _ _" [100, 100, 100] 100) and
- tag_str_STAR ("tag\<^isub>S\<^isub>T\<^isub>A\<^isub>R _" [100] 100) and
- tag_str_STAR ("tag\<^isub>S\<^isub>T\<^isub>A\<^isub>R _ _" [100, 100] 100)
+ tag_str_Plus ("tag\<^isub>A\<^isub>L\<^isub>T _ _" [100, 100] 100) and
+ tag_str_Plus ("tag\<^isub>A\<^isub>L\<^isub>T _ _ _" [100, 100, 100] 100) and
+ tag_str_Times ("tag\<^isub>S\<^isub>E\<^isub>Q _ _" [100, 100] 100) and
+ tag_str_Times ("tag\<^isub>S\<^isub>E\<^isub>Q _ _ _" [100, 100, 100] 100) and
+ tag_str_Star ("tag\<^isub>S\<^isub>T\<^isub>A\<^isub>R _" [100] 100) and
+ tag_str_Star ("tag\<^isub>S\<^isub>T\<^isub>A\<^isub>R _ _" [100, 100] 100)
lemma meta_eq_app:
shows "f \<equiv> \<lambda>x. g x \<Longrightarrow> f x \<equiv> g x"
by auto
+lemma conc_def':
+ "A \<cdot> B = {s\<^isub>1 @ s\<^isub>2 | s\<^isub>1 s\<^isub>2. s\<^isub>1 \<in> A \<and> s\<^isub>2 \<in> B}"
+unfolding conc_def by simp
+
(* THEOREMS *)
+
+lemma conc_Union_left:
+ shows "B \<cdot> (\<Union>n. A \<up> n) = (\<Union>n. B \<cdot> (A \<up> n))"
+unfolding conc_def by auto
+
notation (Rule output)
"==>" ("\<^raw:\mbox{}\inferrule{\mbox{>_\<^raw:}}>\<^raw:{\mbox{>_\<^raw:}}>")
@@ -277,23 +302,23 @@
@{term "A \<up> n"}. They are defined as usual
\begin{center}
- @{thm Seq_def[THEN eq_reflection, where A1="A" and B1="B"]}
+ @{thm conc_def'[THEN eq_reflection, where A1="A" and B1="B"]}
\hspace{7mm}
- @{thm pow.simps(1)[THEN eq_reflection, where A1="A"]}
+ @{thm lang_pow.simps(1)[THEN eq_reflection, where A1="A"]}
\hspace{7mm}
- @{thm pow.simps(2)[THEN eq_reflection, where A1="A" and n1="n"]}
+ @{thm lang_pow.simps(2)[THEN eq_reflection, where A1="A" and n1="n"]}
\end{center}
\noindent
where @{text "@"} is the 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
+ 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} \\
+ (iii) & @{thm conc_Union_left} \\
\end{tabular}
\end{proposition}
@@ -372,18 +397,18 @@
\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"]}\\
+ @{thm (lhs) lang.simps(1)} & @{text "\<equiv>"} & @{thm (rhs) lang.simps(1)}\\
+ @{thm (lhs) lang.simps(2)} & @{text "\<equiv>"} & @{thm (rhs) lang.simps(2)}\\
+ @{thm (lhs) lang.simps(3)[where a="c"]} & @{text "\<equiv>"} & @{thm (rhs) lang.simps(3)[where a="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"]}\\
+ @{thm (lhs) lang.simps(4)[where ?r="r\<^isub>1" and ?s="r\<^isub>2"]} & @{text "\<equiv>"} &
+ @{thm (rhs) lang.simps(4)[where ?r="r\<^isub>1" and ?s="r\<^isub>2"]}\\
+ @{thm (lhs) lang.simps(5)[where ?r="r\<^isub>1" and ?s="r\<^isub>2"]} & @{text "\<equiv>"} &
+ @{thm (rhs) lang.simps(5)[where ?r="r\<^isub>1" and ?s="r\<^isub>2"]}\\
+ @{thm (lhs) lang.simps(6)[where r="r"]} & @{text "\<equiv>"} &
+ @{thm (rhs) lang.simps(6)[where r="r"]}\\
\end{tabular}
\end{tabular}
\end{center}
@@ -528,8 +553,8 @@
\begin{center}
@{text "\<calL>(Y, r) \<equiv>"} %
- @{thm (rhs) L_trm.simps(2)[where X="Y" and r="r", THEN eq_reflection]}\hspace{10mm}
- @{thm L_trm.simps(1)[where r="r", THEN eq_reflection]}
+ @{thm (rhs) lang_trm.simps(2)[where X="Y" and r="r", THEN eq_reflection]}\hspace{10mm}
+ @{thm lang_trm.simps(1)[where r="r", THEN eq_reflection]}
\end{center}
\noindent
@@ -568,9 +593,14 @@
@{thm (lhs) Init_def} & @{text "\<equiv>"} & @{thm (rhs) Init_def}
\end{tabular}}
\end{equation}
+*}(*<*)
-
+lemma test:
+ assumes X_in_eqs: "(X, rhs) \<in> Init (UNIV // \<approx>A)"
+ shows "X = \<Union> (lang_trm ` rhs)"
+using assms l_eq_r_in_eqs by (simp)
+(*>*)text {*
\noindent
Because we use sets of terms
for representing the right-hand sides of equations, we can
@@ -926,9 +956,9 @@
\begin{center}
\begin{tabular}{l}
- @{thm quot_null_eq}\\
- @{thm quot_empty_subset}\\
- @{thm quot_char_subset}
+ @{thm quot_zero_eq}\\
+ @{thm quot_one_subset}\\
+ @{thm quot_atom_subset}
\end{tabular}
\end{center}
@@ -1021,7 +1051,7 @@
We take as tagging-function
%
\begin{center}
- @{thm tag_str_ALT_def[where A="A" and B="B", THEN meta_eq_app]}
+ @{thm tag_str_Plus_def[where A="A" and B="B", THEN meta_eq_app]}
\end{center}
\noindent
@@ -1154,7 +1184,7 @@
following tagging-function
%
\begin{center}
- @{thm tag_str_SEQ_def[where ?L1.0="A" and ?L2.0="B", THEN meta_eq_app]}
+ @{thm tag_str_Times_def[where ?L1.0="A" and ?L2.0="B", THEN meta_eq_app]}
\end{center}
\noindent
@@ -1268,7 +1298,7 @@
the following tagging-function:
%
\begin{center}
- @{thm tag_str_STAR_def[where ?L1.0="A", THEN meta_eq_app]}\smallskip
+ @{thm tag_str_Star_def[where ?L1.0="A", THEN meta_eq_app]}\smallskip
\end{center}
\begin{proof}[@{const STAR}-Case]
--- a/Paper/ROOT.ML Fri Jun 03 13:59:21 2011 +0000
+++ b/Paper/ROOT.ML Mon Jul 25 13:33:38 2011 +0000
@@ -1,3 +1,3 @@
-no_document use_thy "../Myhill";
+no_document use_thy "../Myhill_2";
use_thy "Paper"
\ No newline at end of file
--- a/Prelude.thy Fri Jun 03 13:59:21 2011 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,19 +0,0 @@
-theory Prelude
-imports Main
-begin
-
-(*
-To make the theory work under Isabelle 2009 and 2011
-
-Isabelle 2009: set_ext
-Isabelle 2011: set_eqI
-
-*)
-
-
-lemma set_eq_intro:
- "(\<And>x. (x \<in> A) = (x \<in> B)) \<Longrightarrow> A = B"
-by blast
-
-
-end
\ No newline at end of file
--- a/Regular.thy Fri Jun 03 13:59:21 2011 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,302 +0,0 @@
-(* Author: Christian Urban, Xingyuan Zhang, Chunhan Wu *)
-theory Regular
-imports Main Folds
-begin
-
-section {* Preliminary definitions *}
-
-type_synonym lang = "string set"
-
-
-text {* Sequential composition of two languages *}
-
-definition
- Seq :: "lang \<Rightarrow> lang \<Rightarrow> lang" (infixr "\<cdot>" 100)
-where
- "A \<cdot> 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 "\<cdot>"}. *}
-
-lemma seq_add_left:
- assumes a: "A = B"
- shows "C \<cdot> A = C \<cdot> B"
-using a by simp
-
-lemma seq_union_distrib_right:
- shows "(A \<union> B) \<cdot> C = (A \<cdot> C) \<union> (B \<cdot> C)"
-unfolding Seq_def by auto
-
-lemma seq_union_distrib_left:
- shows "C \<cdot> (A \<union> B) = (C \<cdot> A) \<union> (C \<cdot> B)"
-unfolding Seq_def by auto
-
-lemma seq_intro:
- assumes a: "x \<in> A" "y \<in> B"
- shows "x @ y \<in> A \<cdot> B "
-using a by (auto simp: Seq_def)
-
-lemma seq_assoc:
- shows "(A \<cdot> B) \<cdot> C = A \<cdot> (B \<cdot> C)"
-unfolding Seq_def
-apply(auto)
-apply(blast)
-by (metis append_assoc)
-
-lemma seq_empty [simp]:
- shows "A \<cdot> {[]} = A"
- and "{[]} \<cdot> A = A"
-by (simp_all add: Seq_def)
-
-lemma seq_null [simp]:
- shows "A \<cdot> {} = {}"
- and "{} \<cdot> 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 \<cdot> (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 \<cdot> A\<star>"
-proof
- { fix x
- have "x \<in> A\<star> \<Longrightarrow> x \<in> {[]} \<union> A \<cdot> A\<star>"
- unfolding Seq_def
- by (induct rule: star_induct) (auto)
- }
- then show "A\<star> \<subseteq> {[]} \<union> A \<cdot> A\<star>" by auto
-next
- show "{[]} \<union> A \<cdot> 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
-by (induct rule: star_induct) (blast)+
-
-lemma seq_Union_left:
- shows "B \<cdot> (\<Union>n. A \<up> n) = (\<Union>n. B \<cdot> (A \<up> n))"
-unfolding Seq_def by auto
-
-lemma seq_Union_right:
- shows "(\<Union>n. A \<up> n) \<cdot> B = (\<Union>n. (A \<up> n) \<cdot> B)"
-unfolding Seq_def by auto
-
-lemma seq_pow_comm:
- shows "A \<cdot> (A \<up> n) = (A \<up> n) \<cdot> A"
-by (induct n) (simp_all add: seq_assoc[symmetric])
-
-lemma seq_star_comm:
- shows "A \<cdot> A\<star> = A\<star> \<cdot> A"
-unfolding Star_def seq_Union_left
-unfolding seq_pow_comm 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 \<cdot> (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 modified version of Arden's lemma *}
-
-text {* A helper lemma for Arden *}
-
-lemma arden_helper:
- assumes eq: "X = X \<cdot> A \<union> B"
- shows "X = X \<cdot> (A \<up> Suc n) \<union> (\<Union>m\<in>{0..n}. B \<cdot> (A \<up> m))"
-proof (induct n)
- case 0
- show "X = X \<cdot> (A \<up> Suc 0) \<union> (\<Union>(m::nat)\<in>{0..0}. B \<cdot> (A \<up> m))"
- using eq by simp
-next
- case (Suc n)
- have ih: "X = X \<cdot> (A \<up> Suc n) \<union> (\<Union>m\<in>{0..n}. B \<cdot> (A \<up> m))" by fact
- also have "\<dots> = (X \<cdot> A \<union> B) \<cdot> (A \<up> Suc n) \<union> (\<Union>m\<in>{0..n}. B \<cdot> (A \<up> m))" using eq by simp
- also have "\<dots> = X \<cdot> (A \<up> Suc (Suc n)) \<union> (B \<cdot> (A \<up> Suc n)) \<union> (\<Union>m\<in>{0..n}. B \<cdot> (A \<up> m))"
- by (simp add: seq_union_distrib_right seq_assoc)
- also have "\<dots> = X \<cdot> (A \<up> Suc (Suc n)) \<union> (\<Union>m\<in>{0..Suc n}. B \<cdot> (A \<up> m))"
- by (auto simp add: le_Suc_eq)
- finally show "X = X \<cdot> (A \<up> Suc (Suc n)) \<union> (\<Union>m\<in>{0..Suc n}. B \<cdot> (A \<up> m))" .
-qed
-
-theorem arden:
- assumes nemp: "[] \<notin> A"
- shows "X = X \<cdot> A \<union> B \<longleftrightarrow> X = B \<cdot> A\<star>"
-proof
- assume eq: "X = B \<cdot> A\<star>"
- have "A\<star> = {[]} \<union> A\<star> \<cdot> A"
- unfolding seq_star_comm[symmetric]
- by (rule star_cases)
- then have "B \<cdot> A\<star> = B \<cdot> ({[]} \<union> A\<star> \<cdot> A)"
- by (rule seq_add_left)
- also have "\<dots> = B \<union> B \<cdot> (A\<star> \<cdot> A)"
- unfolding seq_union_distrib_left by simp
- also have "\<dots> = B \<union> (B \<cdot> A\<star>) \<cdot> A"
- by (simp only: seq_assoc)
- finally show "X = X \<cdot> A \<union> B"
- using eq by blast
-next
- assume eq: "X = X \<cdot> A \<union> B"
- { fix n::nat
- have "B \<cdot> (A \<up> n) \<subseteq> X" using arden_helper[OF eq, of "n"] by auto }
- then have "B \<cdot> 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 \<cdot> (A \<up> Suc k)"
- using seq_pow_length[OF nemp] by blast
- assume "s \<in> X"
- then have "s \<in> X \<cdot> (A \<up> Suc k) \<union> (\<Union>m\<in>{0..k}. B \<cdot> (A \<up> m))"
- using arden_helper[OF eq, of "k"] by auto
- then have "s \<in> (\<Union>m\<in>{0..k}. B \<cdot> (A \<up> m))" using not_in by auto
- moreover
- have "(\<Union>m\<in>{0..k}. B \<cdot> (A \<up> m)) \<subseteq> (\<Union>n. B \<cdot> (A \<up> n))" by auto
- ultimately
- have "s \<in> B \<cdot> A\<star>"
- unfolding seq_Union_left Star_def by auto }
- then have "X \<subseteq> B \<cdot> A\<star>" by auto
- ultimately
- show "X = B \<cdot> A\<star>" by simp
-qed
-
-
-section {* Regular Expressions *}
-
-datatype rexp =
- NULL
-| EMPTY
-| CHAR char
-| SEQ rexp rexp
-| ALT rexp rexp
-| STAR rexp
-
-fun
- L_rexp :: "rexp \<Rightarrow> lang"
-where
- "L_rexp (NULL) = {}"
- | "L_rexp (EMPTY) = {[]}"
- | "L_rexp (CHAR c) = {[c]}"
- | "L_rexp (SEQ r1 r2) = (L_rexp r1) \<cdot> (L_rexp r2)"
- | "L_rexp (ALT r1 r2) = (L_rexp r1) \<union> (L_rexp r2)"
- | "L_rexp (STAR r) = (L_rexp r)\<star>"
-
-text {* ALT-combination for a set of regular expressions *}
-
-abbreviation
- Setalt ("\<Uplus>_" [1000] 999)
-where
- "\<Uplus>A \<equiv> folds ALT NULL A"
-
-text {*
- For finite sets, @{term Setalt} is preserved under @{term L_exp}.
-*}
-
-lemma folds_alt_simp [simp]:
- fixes rs::"rexp set"
- assumes a: "finite rs"
- shows "L_rexp (\<Uplus>rs) = \<Union> (L_rexp ` rs)"
-unfolding folds_def
-apply(rule set_eqI)
-apply(rule someI2_ex)
-apply(rule_tac finite_imp_fold_graph[OF a])
-apply(erule fold_graph.induct)
-apply(auto)
-done
-
-end
\ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Regular_Exp.thy Mon Jul 25 13:33:38 2011 +0000
@@ -0,0 +1,36 @@
+(* Author: Tobias Nipkow
+ Copyright 1998 TUM
+*)
+
+header "Regular expressions"
+
+theory Regular_Exp
+imports Regular_Set
+begin
+
+datatype 'a rexp =
+ Zero |
+ One |
+ Atom 'a |
+ Plus "('a rexp)" "('a rexp)" |
+ Times "('a rexp)" "('a rexp)" |
+ Star "('a rexp)"
+
+primrec lang :: "'a rexp => 'a list set" 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 atoms :: "'a rexp \<Rightarrow> 'a set"
+where
+"atoms Zero = {}" |
+"atoms One = {}" |
+"atoms (Atom a) = {a}" |
+"atoms (Plus r s) = atoms r \<union> atoms s" |
+"atoms (Times r s) = atoms r \<union> atoms s" |
+"atoms (Star r) = atoms r"
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Regular_Set.thy Mon Jul 25 13:33:38 2011 +0000
@@ -0,0 +1,348 @@
+(* Author: Tobias Nipkow, Alex Krauss *)
+
+header "Regular sets"
+
+theory Regular_Set
+imports Main
+begin
+
+type_synonym 'a lang = "'a list set"
+
+definition conc :: "'a lang \<Rightarrow> 'a lang \<Rightarrow> 'a lang" (infixr "@@" 75) where
+"A @@ B = {xs@ys | xs ys. xs:A & ys:B}"
+
+overloading lang_pow == "compow :: nat \<Rightarrow> 'a lang \<Rightarrow> 'a lang"
+begin
+ primrec lang_pow :: "nat \<Rightarrow> 'a lang \<Rightarrow> 'a lang" where
+ "lang_pow 0 A = {[]}" |
+ "lang_pow (Suc n) A = A @@ (lang_pow n A)"
+end
+
+definition star :: "'a lang \<Rightarrow> 'a lang" where
+"star A = (\<Union>n. A ^^ n)"
+
+
+
+definition deriv :: "'a \<Rightarrow> 'a lang \<Rightarrow> 'a lang"
+where "deriv x L = { xs. x#xs \<in> L }"
+
+
+coinductive bisimilar :: "'a list set \<Rightarrow> 'a list set \<Rightarrow> bool" where
+"([] \<in> K \<longleftrightarrow> [] \<in> L)
+ \<Longrightarrow> (\<And>x. bisimilar (deriv x K) (deriv x L))
+ \<Longrightarrow> bisimilar K L"
+
+
+subsection{* @{term "op @@"} *}
+
+lemma concI[simp,intro]: "u : A \<Longrightarrow> v : B \<Longrightarrow> u@v : A @@ B"
+by (auto simp add: conc_def)
+
+lemma concE[elim]:
+assumes "w \<in> A @@ B"
+obtains u v where "u \<in> A" "v \<in> B" "w = u@v"
+using assms by (auto simp: conc_def)
+
+lemma conc_mono: "A \<subseteq> C \<Longrightarrow> B \<subseteq> D \<Longrightarrow> A @@ B \<subseteq> C @@ D"
+by (auto simp: conc_def)
+
+lemma conc_empty[simp]: shows "{} @@ A = {}" and "A @@ {} = {}"
+by auto
+
+lemma conc_epsilon[simp]: shows "{[]} @@ A = A" and "A @@ {[]} = A"
+by (simp_all add:conc_def)
+
+lemma conc_assoc: "(A @@ B) @@ C = A @@ (B @@ C)"
+by (auto elim!: concE) (simp only: append_assoc[symmetric] concI)
+
+lemma conc_Un_distrib:
+shows "A @@ (B \<union> C) = A @@ B \<union> A @@ C"
+and "(A \<union> B) @@ C = A @@ C \<union> B @@ C"
+by auto
+
+lemma conc_UNION_distrib:
+shows "A @@ UNION I M = UNION I (%i. A @@ M i)"
+and "UNION I M @@ A = UNION I (%i. M i @@ A)"
+by auto
+
+
+subsection{* @{term "A ^^ n"} *}
+
+lemma lang_pow_add: "A ^^ (n + m) = A ^^ n @@ A ^^ m"
+by (induct n) (auto simp: conc_assoc)
+
+lemma lang_pow_empty: "{} ^^ n = (if n = 0 then {[]} else {})"
+by (induct n) auto
+
+lemma lang_pow_empty_Suc[simp]: "({}::'a lang) ^^ Suc n = {}"
+by (simp add: lang_pow_empty)
+
+
+lemma length_lang_pow_ub:
+ "ALL w : A. length w \<le> k \<Longrightarrow> w : A^^n \<Longrightarrow> length w \<le> k*n"
+by(induct n arbitrary: w) (fastsimp simp: conc_def)+
+
+lemma length_lang_pow_lb:
+ "ALL w : A. length w \<ge> k \<Longrightarrow> w : A^^n \<Longrightarrow> length w \<ge> k*n"
+by(induct n arbitrary: w) (fastsimp simp: conc_def)+
+
+
+subsection{* @{const star} *}
+
+lemma star_if_lang_pow[simp]: "w : A ^^ n \<Longrightarrow> w : star A"
+by (auto simp: star_def)
+
+lemma Nil_in_star[iff]: "[] : star A"
+proof (rule star_if_lang_pow)
+ show "[] : A ^^ 0" by simp
+qed
+
+lemma star_if_lang[simp]: assumes "w : A" shows "w : star A"
+proof (rule star_if_lang_pow)
+ show "w : A ^^ 1" using `w : A` by simp
+qed
+
+lemma append_in_starI[simp]:
+assumes "u : star A" and "v : star A" shows "u@v : star A"
+proof -
+ from `u : star A` obtain m where "u : A ^^ m" by (auto simp: star_def)
+ moreover
+ from `v : star A` obtain n where "v : A ^^ n" by (auto simp: star_def)
+ ultimately have "u@v : A ^^ (m+n)" by (simp add: lang_pow_add)
+ thus ?thesis by simp
+qed
+
+lemma conc_star_star: "star A @@ star A = star A"
+by (auto simp: conc_def)
+
+lemma star_induct[consumes 1, case_names Nil append, induct set: star]:
+assumes "w : star A"
+ and "P []"
+ and step: "!!u v. u : A \<Longrightarrow> v : star A \<Longrightarrow> P v \<Longrightarrow> P (u@v)"
+shows "P w"
+proof -
+ { fix n have "w : A ^^ n \<Longrightarrow> P w"
+ by (induct n arbitrary: w) (auto intro: `P []` step star_if_lang_pow) }
+ with `w : star A` show "P w" by (auto simp: star_def)
+qed
+
+lemma star_empty[simp]: "star {} = {[]}"
+by (auto elim: star_induct)
+
+lemma star_epsilon[simp]: "star {[]} = {[]}"
+by (auto elim: star_induct)
+
+lemma star_idemp[simp]: "star (star A) = star A"
+by (auto elim: star_induct)
+
+lemma star_unfold_left: "star A = A @@ star A \<union> {[]}" (is "?L = ?R")
+proof
+ show "?L \<subseteq> ?R" by (rule, erule star_induct) auto
+qed auto
+
+lemma concat_in_star: "set ws \<subseteq> A \<Longrightarrow> concat ws : star A"
+by (induct ws) simp_all
+
+lemma in_star_iff_concat:
+ "w : star A = (EX ws. set ws \<subseteq> A & w = concat ws)"
+ (is "_ = (EX ws. ?R w ws)")
+proof
+ assume "w : star A" thus "EX ws. ?R w ws"
+ proof induct
+ case Nil have "?R [] []" by simp
+ thus ?case ..
+ next
+ case (append u v)
+ moreover
+ then obtain ws where "set ws \<subseteq> A \<and> v = concat ws" by blast
+ ultimately have "?R (u@v) (u#ws)" by auto
+ thus ?case ..
+ qed
+next
+ assume "EX us. ?R w us" thus "w : star A"
+ by (auto simp: concat_in_star)
+qed
+
+lemma star_conv_concat: "star A = {concat ws|ws. set ws \<subseteq> A}"
+by (fastsimp simp: in_star_iff_concat)
+
+lemma star_insert_eps[simp]: "star (insert [] A) = star(A)"
+proof-
+ { fix us
+ have "set us \<subseteq> insert [] A \<Longrightarrow> EX vs. concat us = concat vs \<and> set vs \<subseteq> A"
+ (is "?P \<Longrightarrow> EX vs. ?Q vs")
+ proof
+ let ?vs = "filter (%u. u \<noteq> []) us"
+ show "?P \<Longrightarrow> ?Q ?vs" by (induct us) auto
+ qed
+ } thus ?thesis by (auto simp: star_conv_concat)
+qed
+
+lemma Arden:
+assumes "[] \<notin> A" and "X = A @@ X \<union> B"
+shows "X = star A @@ B"
+proof -
+ { fix n have "X = A^^(n+1)@@X \<union> (\<Union>i\<le>n. A^^i@@B)"
+ proof(induct n)
+ case 0 show ?case using `X = A @@ X \<union> B` by simp
+ next
+ case (Suc n)
+ have "X = A@@X Un B" by(rule assms(2))
+ also have "\<dots> = A@@(A^^(n+1)@@X \<union> (\<Union>i\<le>n. A^^i@@B)) Un B"
+ by(subst Suc)(rule refl)
+ also have "\<dots> = A^^(n+2)@@X \<union> (\<Union>i\<le>n. A^^(i+1)@@B) Un B"
+ by(simp add:conc_UNION_distrib conc_assoc conc_Un_distrib)
+ also have "\<dots> = A^^(n+2)@@X \<union> (UN i : {1..n+1}. A^^i@@B) \<union> B"
+ by(subst UN_le_add_shift)(rule refl)
+ also have "\<dots> = A^^(n+2)@@X \<union> (UN i : {1..n+1}. A^^i@@B) \<union> A^^0@@B"
+ by(simp)
+ also have "\<dots> = A^^(n+2)@@X \<union> (\<Union>i\<le>n+1. A^^i@@B)"
+ by(auto simp: UN_le_eq_Un0)
+ finally show ?case by simp
+ qed
+ } note 1 = this
+ { fix w assume "w : X"
+ let ?n = "size w"
+ from `[] \<notin> A` have "ALL u : A. length u \<ge> 1"
+ by (metis Suc_eq_plus1 add_leD2 le_0_eq length_0_conv not_less_eq_eq)
+ hence "ALL u : A^^(?n+1). length u \<ge> ?n+1"
+ by (metis length_lang_pow_lb nat_mult_1)
+ hence "ALL u : A^^(?n+1)@@X. length u \<ge> ?n+1"
+ by(auto simp only: conc_def length_append)
+ hence "w \<notin> A^^(?n+1)@@X" by auto
+ hence "w : star A @@ B" using `w : X` 1[of ?n]
+ by(auto simp add: star_def conc_UNION_distrib)
+ } moreover
+ { fix w assume "w : star A @@ B"
+ hence "EX n. w : A^^n @@ B" by(auto simp: conc_def star_def)
+ hence "w : X" using 1 by blast
+ } ultimately show ?thesis by blast
+qed
+
+subsection{* @{const deriv} *}
+
+lemma deriv_empty[simp]: "deriv a {} = {}"
+and deriv_epsilon[simp]: "deriv a {[]} = {}"
+and deriv_char[simp]: "deriv a {[b]} = (if a = b then {[]} else {})"
+and deriv_union[simp]: "deriv a (A \<union> B) = deriv a A \<union> deriv a B"
+by (auto simp: deriv_def)
+
+lemma deriv_conc_subset:
+"deriv a A @@ B \<subseteq> deriv a (A @@ B)" (is "?L \<subseteq> ?R")
+proof
+ fix w assume "w \<in> ?L"
+ then obtain u v where "w = u @ v" "a # u \<in> A" "v \<in> B"
+ by (auto simp: deriv_def)
+ then have "a # w \<in> A @@ B"
+ by (auto intro: concI[of "a # u", simplified])
+ thus "w \<in> ?R" by (auto simp: deriv_def)
+qed
+
+lemma deriv_conc1:
+assumes "[] \<notin> A"
+shows "deriv a (A @@ B) = deriv a A @@ B" (is "?L = ?R")
+proof
+ show "?L \<subseteq> ?R"
+ proof
+ fix w assume "w \<in> ?L"
+ then have "a # w \<in> A @@ B" by (simp add: deriv_def)
+ then obtain x y
+ where aw: "a # w = x @ y" "x \<in> A" "y \<in> B" by auto
+ with `[] \<notin> A` obtain x' where "x = a # x'"
+ by (cases x) auto
+ with aw have "w = x' @ y" "x' \<in> deriv a A"
+ by (auto simp: deriv_def)
+ with `y \<in> B` show "w \<in> ?R" by simp
+ qed
+ show "?R \<subseteq> ?L" by (fact deriv_conc_subset)
+qed
+
+lemma deriv_conc2:
+assumes "[] \<in> A"
+shows "deriv a (A @@ B) = deriv a A @@ B \<union> deriv a B"
+(is "?L = ?R")
+proof
+ show "?L \<subseteq> ?R"
+ proof
+ fix w assume "w \<in> ?L"
+ then have "a # w \<in> A @@ B" by (simp add: deriv_def)
+ then obtain x y
+ where aw: "a # w = x @ y" "x \<in> A" "y \<in> B" by auto
+ show "w \<in> ?R"
+ proof (cases x)
+ case Nil
+ with aw have "w \<in> deriv a B" by (auto simp: deriv_def)
+ thus ?thesis ..
+ next
+ case (Cons b x')
+ with aw have "w = x' @ y" "x' \<in> deriv a A"
+ by (auto simp: deriv_def)
+ with `y \<in> B` show "w \<in> ?R" by simp
+ qed
+ qed
+
+ from concI[OF `[] \<in> A`, simplified]
+ have "B \<subseteq> A @@ B" ..
+ then have "deriv a B \<subseteq> deriv a (A @@ B)" by (auto simp: deriv_def)
+ with deriv_conc_subset[of a A B]
+ show "?R \<subseteq> ?L" by auto
+qed
+
+lemma wlog_no_eps:
+assumes PB: "\<And>B. [] \<notin> B \<Longrightarrow> P B"
+assumes preserved: "\<And>A. P A \<Longrightarrow> P (insert [] A)"
+shows "P A"
+proof -
+ let ?B = "A - {[]}"
+ have "P ?B" by (rule PB) auto
+ thus "P A"
+ proof cases
+ assume "[] \<in> A"
+ then have "(insert [] ?B) = A" by auto
+ with preserved[OF `P ?B`]
+ show ?thesis by simp
+ qed auto
+qed
+
+lemma deriv_insert_eps[simp]:
+"deriv a (insert [] A) = deriv a A"
+by (auto simp: deriv_def)
+
+lemma deriv_star[simp]: "deriv a (star A) = deriv a A @@ star A"
+proof (induct A rule: wlog_no_eps)
+ fix B :: "'a list set" assume "[] \<notin> B"
+ thus "deriv a (star B) = deriv a B @@ star B"
+ by (subst star_unfold_left) (simp add: deriv_conc1)
+qed auto
+
+
+subsection{* @{const bisimilar} *}
+
+lemma equal_if_bisimilar:
+assumes "bisimilar K L" shows "K = L"
+proof (rule set_eqI)
+ fix w
+ from `bisimilar K L` show "w \<in> K \<longleftrightarrow> w \<in> L"
+ proof (induct w arbitrary: K L)
+ case Nil thus ?case by (auto elim: bisimilar.cases)
+ next
+ case (Cons a w K L)
+ from `bisimilar K L` have "bisimilar (deriv a K) (deriv a L)"
+ by (auto elim: bisimilar.cases)
+ then have "w \<in> deriv a K \<longleftrightarrow> w \<in> deriv a L" by (rule Cons(1))
+ thus ?case by (auto simp: deriv_def)
+ qed
+qed
+
+lemma language_coinduct:
+fixes R (infixl "\<sim>" 50)
+assumes "K \<sim> L"
+assumes "\<And>K L. K \<sim> L \<Longrightarrow> ([] \<in> K \<longleftrightarrow> [] \<in> L)"
+assumes "\<And>K L x. K \<sim> L \<Longrightarrow> deriv x K \<sim> deriv x L"
+shows "K = L"
+apply (rule equal_if_bisimilar)
+apply (rule bisimilar.coinduct[of R, OF `K \<sim> L`])
+apply (auto simp: assms)
+done
+
+end