--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Myhill_1.thy Thu Jan 27 12:35:06 2011 +0000
@@ -0,0 +1,1118 @@
+theory Myhill_1
+ imports Main List_Prefix Prefix_subtract
+begin
+
+(*
+text {*
+ \begin{figure}
+ \centering
+ \scalebox{0.95}{
+ \begin{tikzpicture}[->,>=latex,shorten >=1pt,auto,node distance=1.2cm, semithick]
+ \node[state,initial] (n1) {$1$};
+ \node[state,accepting] (n2) [right = 10em of n1] {$2$};
+
+ \path (n1) edge [bend left] node {$0$} (n2)
+ (n1) edge [loop above] node{$1$} (n1)
+ (n2) edge [loop above] node{$0$} (n2)
+ (n2) edge [bend left] node {$1$} (n1)
+ ;
+ \end{tikzpicture}}
+ \caption{An example automaton (or partition)}\label{fig:example_automata}
+ \end{figure}
+*}
+
+*)
+
+
+section {* Preliminary definitions *}
+
+text {* Sequential composition of two languages @{text "L1"} and @{text "L2"} *}
+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}"
+
+text {* Transitive closure of language @{text "L"}. *}
+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>"
+
+text {* Some properties of operator @{text ";;"}.*}
+
+lemma seq_union_distrib:
+ "(A \<union> B) ;; C = (A ;; C) \<union> (B ;; C)"
+by (auto simp:Seq_def)
+
+lemma seq_intro:
+ "\<lbrakk>x \<in> A; y \<in> B\<rbrakk> \<Longrightarrow> x @ y \<in> A ;; B "
+by (auto simp:Seq_def)
+
+lemma seq_assoc:
+ "(A ;; B) ;; C = A ;; (B ;; C)"
+apply(auto simp:Seq_def)
+apply blast
+by (metis append_assoc)
+
+lemma star_intro1[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_intro2: "y \<in> lang \<Longrightarrow> y \<in> lang\<star>"
+by (drule step[of y lang "[]"], auto simp:start)
+
+lemma star_intro3[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_intro2)
+
+lemma star_decom:
+ "\<lbrakk>x \<in> lang\<star>; x \<noteq> []\<rbrakk> \<Longrightarrow>(\<exists> a b. x = a @ b \<and> a \<noteq> [] \<and> a \<in> lang \<and> b \<in> lang\<star>)"
+by (induct x rule: Star.induct, simp, blast)
+
+lemma star_decom':
+ "\<lbrakk>x \<in> lang\<star>; x \<noteq> []\<rbrakk> \<Longrightarrow> \<exists>a b. x = a @ b \<and> a \<in> lang\<star> \<and> b \<in> lang"
+apply (induct x rule:Star.induct, simp)
+apply (case_tac "s2 = []")
+apply (rule_tac x = "[]" in exI, rule_tac x = s1 in exI, simp add:start)
+apply (simp, (erule exE| erule conjE)+)
+by (rule_tac x = "s1 @ a" in exI, rule_tac x = b in exI, simp add:step)
+
+text {* Ardens lemma expressed at the level of language, rather than the level of regular expression. *}
+
+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"
+ by (auto simp:Seq_def star_intro3 star_decom')
+ 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"
+ by (simp only:seq_assoc)
+ finally show "X = X ;; A \<union> B"
+ using eq by blast
+next
+ assume eq': "X = X ;; A \<union> B"
+ hence c1': "\<And> x. x \<in> B \<Longrightarrow> x \<in> X"
+ and c2': "\<And> x y. \<lbrakk>x \<in> X; y \<in> A\<rbrakk> \<Longrightarrow> x @ y \<in> X"
+ using Seq_def by auto
+ show "X = B ;; A\<star>"
+ proof
+ show "B ;; A\<star> \<subseteq> X"
+ proof-
+ { fix x y
+ have "\<lbrakk>y \<in> A\<star>; x \<in> X\<rbrakk> \<Longrightarrow> x @ y \<in> X "
+ apply (induct arbitrary:x rule:Star.induct, simp)
+ by (auto simp only:append_assoc[THEN sym] dest:c2')
+ } thus ?thesis using c1' by (auto simp:Seq_def)
+ qed
+ next
+ show "X \<subseteq> B ;; A\<star>"
+ proof-
+ { fix x
+ have "x \<in> X \<Longrightarrow> x \<in> B ;; A\<star>"
+ proof (induct x taking:length rule:measure_induct)
+ fix z
+ assume hyps:
+ "\<forall>y. length y < length z \<longrightarrow> y \<in> X \<longrightarrow> y \<in> B ;; A\<star>"
+ and z_in: "z \<in> X"
+ show "z \<in> B ;; A\<star>"
+ proof (cases "z \<in> B")
+ case True thus ?thesis by (auto simp:Seq_def start)
+ next
+ case False hence "z \<in> X ;; A" using eq' z_in by auto
+ then obtain za zb where za_in: "za \<in> X"
+ and zab: "z = za @ zb \<and> zb \<in> A" and zbne: "zb \<noteq> []"
+ using nemp unfolding Seq_def by blast
+ from zbne zab have "length za < length z" by auto
+ with za_in hyps have "za \<in> B ;; A\<star>" by blast
+ hence "za @ zb \<in> B ;; A\<star>" using zab
+ by (clarsimp simp:Seq_def, blast dest:star_intro3)
+ thus ?thesis using zab by simp
+ qed
+ qed
+ } thus ?thesis by blast
+ qed
+ qed
+qed
+
+
+text {* The syntax of regular expressions is defined by the datatype @{text "rexp"}. *}
+datatype rexp =
+ NULL
+| EMPTY
+| CHAR char
+| SEQ rexp rexp
+| ALT rexp rexp
+| STAR rexp
+
+
+text {*
+ The following @{text "L"} is an overloaded operator, where @{text "L(x)"} evaluates to
+ the language represented by the syntactic object @{text "x"}.
+*}
+consts L:: "'a \<Rightarrow> string set"
+
+
+text {*
+ The @{text "L(rexp)"} for regular expression @{text "rexp"} is defined by the
+ following overloading function @{text "L_rexp"}.
+*}
+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 {*
+ To obtain equational system out of finite set of equivalent classes, a fold operation
+ on finite set @{text "folds"} is defined. The use of @{text "SOME"} makes @{text "fold"}
+ more robust than the @{text "fold"} in Isabelle library. The expression @{text "folds f"}
+ makes sense when @{text "f"} is not @{text "associative"} and @{text "commutitive"},
+ while @{text "fold f"} does not.
+*}
+
+definition
+ folds :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b"
+where
+ "folds f z S \<equiv> SOME x. fold_graph f z S x"
+
+text {*
+ The following lemma assures that the arbitrary choice made by the @{text "SOME"} in @{text "folds"}
+ does not affect the @{text "L"}-value of the resultant regular expression.
+ *}
+lemma folds_alt_simp [simp]:
+ "finite rs \<Longrightarrow> L (folds ALT NULL rs) = \<Union> (L ` rs)"
+apply (rule set_ext, simp add:folds_def)
+apply (rule someI2_ex, erule finite_imp_fold_graph)
+by (erule fold_graph.induct, auto)
+
+(* Just a technical lemma. *)
+lemma [simp]:
+ shows "(x, y) \<in> {(x, y). P x y} \<longleftrightarrow> P x y"
+by simp
+
+text {*
+ @{text "\<approx>L"} is an equivalent class defined by language @{text "Lang"}.
+*}
+definition
+ str_eq_rel ("\<approx>_")
+where
+ "\<approx>Lang \<equiv> {(x, y). (\<forall>z. x @ z \<in> Lang \<longleftrightarrow> y @ z \<in> Lang)}"
+
+text {*
+ Among equivlant clases of @{text "\<approx>Lang"}, the set @{text "finals(Lang)"} singles out
+ those which contains strings from @{text "Lang"}.
+*}
+
+definition
+ "finals Lang \<equiv> {\<approx>Lang `` {x} | x . x \<in> Lang}"
+
+text {*
+ The following lemma show the relationshipt between @{text "finals(Lang)"} and @{text "Lang"}.
+*}
+lemma lang_is_union_of_finals:
+ "Lang = \<Union> finals(Lang)"
+proof
+ show "Lang \<subseteq> \<Union> (finals Lang)"
+ proof
+ fix x
+ assume "x \<in> Lang"
+ thus "x \<in> \<Union> (finals Lang)"
+ apply (simp add:finals_def, rule_tac x = "(\<approx>Lang) `` {x}" in exI)
+ by (auto simp:Image_def str_eq_rel_def)
+ qed
+next
+ show "\<Union> (finals Lang) \<subseteq> Lang"
+ apply (clarsimp simp:finals_def str_eq_rel_def)
+ by (drule_tac x = "[]" in spec, auto)
+qed
+
+section {* Direction @{text "finite partition \<Rightarrow> regular language"}*}
+
+text {*
+ The relationship between equivalent classes can be described by an
+ equational system.
+ For example, in equational system \eqref{example_eqns}, $X_0, X_1$ are equivalent
+ classes. The first equation says every string in $X_0$ is obtained either by
+ appending one $b$ to a string in $X_0$ or by appending one $a$ to a string in
+ $X_1$ or just be an empty string (represented by the regular expression $\lambda$). Similary,
+ the second equation tells how the strings inside $X_1$ are composed.
+ \begin{equation}\label{example_eqns}
+ \begin{aligned}
+ X_0 & = X_0 b + X_1 a + \lambda \\
+ X_1 & = X_0 a + X_1 b
+ \end{aligned}
+ \end{equation}
+ The summands on the right hand side is represented by the following data type
+ @{text "rhs_item"}, mnemonic for 'right hand side item'.
+ Generally, there are two kinds of right hand side items, one kind corresponds to
+ pure regular expressions, like the $\lambda$ in \eqref{example_eqns}, the other kind corresponds to
+ transitions from one one equivalent class to another, like the $X_0 b, X_1 a$ etc.
+ *}
+
+datatype rhs_item =
+ Lam "rexp" (* Lambda *)
+ | Trn "(string set)" "rexp" (* Transition *)
+
+text {*
+ In this formalization, pure regular expressions like $\lambda$ is
+ repsented by @{text "Lam(EMPTY)"}, while transitions like $X_0 a$ is represented by $Trn~X_0~(CHAR~a)$.
+ *}
+
+text {*
+ The functions @{text "the_r"} and @{text "the_Trn"} are used to extract
+ subcomponents from right hand side items.
+ *}
+
+fun the_r :: "rhs_item \<Rightarrow> rexp"
+where "the_r (Lam r) = r"
+
+fun the_Trn:: "rhs_item \<Rightarrow> (string set \<times> rexp)"
+where "the_Trn (Trn Y r) = (Y, r)"
+
+text {*
+ Every right hand side item @{text "itm"} defines a string set given
+ @{text "L(itm)"}, defined as:
+*}
+overloading L_rhs_e \<equiv> "L:: rhs_item \<Rightarrow> string set"
+begin
+ fun L_rhs_e:: "rhs_item \<Rightarrow> string set"
+ where
+ "L_rhs_e (Lam r) = L r" |
+ "L_rhs_e (Trn X r) = X ;; L r"
+end
+
+text {*
+ The right hand side of every equation is represented by a set of
+ items. The string set defined by such a set @{text "itms"} is given
+ by @{text "L(itms)"}, defined as:
+*}
+
+overloading L_rhs \<equiv> "L:: rhs_item set \<Rightarrow> string set"
+begin
+ fun L_rhs:: "rhs_item set \<Rightarrow> string set"
+ where "L_rhs rhs = \<Union> (L ` rhs)"
+end
+
+text {*
+ Given a set of equivalent classses @{text "CS"} and one equivalent class @{text "X"} among
+ @{text "CS"}, the term @{text "init_rhs CS X"} is used to extract the right hand side of
+ the equation describing the formation of @{text "X"}. The definition of @{text "init_rhs"}
+ is:
+ *}
+
+definition
+ "init_rhs CS X \<equiv>
+ if ([] \<in> X) then
+ {Lam(EMPTY)} \<union> {Trn Y (CHAR c) | Y c. Y \<in> CS \<and> Y ;; {[c]} \<subseteq> X}
+ else
+ {Trn Y (CHAR c)| Y c. Y \<in> CS \<and> Y ;; {[c]} \<subseteq> X}"
+
+text {*
+ In the definition of @{text "init_rhs"}, the term
+ @{text "{Trn Y (CHAR c)| Y c. Y \<in> CS \<and> Y ;; {[c]} \<subseteq> X}"} appearing on both branches
+ describes the formation of strings in @{text "X"} out of transitions, while
+ the term @{text "{Lam(EMPTY)}"} describes the empty string which is intrinsically contained in
+ @{text "X"} rather than by transition. This @{text "{Lam(EMPTY)}"} corresponds to
+ the $\lambda$ in \eqref{example_eqns}.
+
+ With the help of @{text "init_rhs"}, the equitional system descrbing the formation of every
+ equivalent class inside @{text "CS"} is given by the following @{text "eqs(CS)"}.
+ *}
+
+definition "eqs CS \<equiv> {(X, init_rhs CS X) | X. X \<in> CS}"
+(************ arden's lemma variation ********************)
+
+text {*
+ The following @{text "items_of rhs X"} returns all @{text "X"}-items in @{text "rhs"}.
+ *}
+definition
+ "items_of rhs X \<equiv> {Trn X r | r. (Trn X r) \<in> rhs}"
+
+text {*
+ The following @{text "rexp_of rhs X"} combines all regular expressions in @{text "X"}-items
+ using @{text "ALT"} to form a single regular expression.
+ It will be used later to implement @{text "arden_variate"} and @{text "rhs_subst"}.
+ *}
+
+definition
+ "rexp_of rhs X \<equiv> folds ALT NULL ((snd o the_Trn) ` items_of rhs X)"
+
+text {*
+ The following @{text "lam_of rhs"} returns all pure regular expression items in @{text "rhs"}.
+ *}
+
+definition
+ "lam_of rhs \<equiv> {Lam r | r. Lam r \<in> rhs}"
+
+text {*
+ The following @{text "rexp_of_lam rhs"} combines pure regular expression items in @{text "rhs"}
+ using @{text "ALT"} to form a single regular expression.
+ When all variables inside @{text "rhs"} are eliminated, @{text "rexp_of_lam rhs"}
+ is used to compute compute the regular expression corresponds to @{text "rhs"}.
+ *}
+
+definition
+ "rexp_of_lam rhs \<equiv> folds ALT NULL (the_r ` lam_of rhs)"
+
+text {*
+ The following @{text "attach_rexp rexp' itm"} attach
+ the regular expression @{text "rexp'"} to
+ the right of right hand side item @{text "itm"}.
+ *}
+
+fun attach_rexp :: "rexp \<Rightarrow> rhs_item \<Rightarrow> rhs_item"
+where
+ "attach_rexp rexp' (Lam rexp) = Lam (SEQ rexp rexp')"
+| "attach_rexp rexp' (Trn X rexp) = Trn X (SEQ rexp rexp')"
+
+text {*
+ The following @{text "append_rhs_rexp rhs rexp"} attaches
+ @{text "rexp"} to every item in @{text "rhs"}.
+ *}
+
+definition
+ "append_rhs_rexp rhs rexp \<equiv> (attach_rexp rexp) ` rhs"
+
+text {*
+ With the help of the two functions immediately above, Ardens'
+ transformation on right hand side @{text "rhs"} is implemented
+ by the following function @{text "arden_variate X rhs"}.
+ After this transformation, the recursive occurent of @{text "X"}
+ in @{text "rhs"} will be eliminated, while the
+ string set defined by @{text "rhs"} is kept unchanged.
+ *}
+definition
+ "arden_variate X rhs \<equiv>
+ append_rhs_rexp (rhs - items_of rhs X) (STAR (rexp_of rhs X))"
+
+
+(*********** substitution of ES *************)
+
+text {*
+ Suppose the equation defining @{text "X"} is $X = xrhs$,
+ the purpose of @{text "rhs_subst"} is to substitute all occurences of @{text "X"} in
+ @{text "rhs"} by @{text "xrhs"}.
+ A litte thought may reveal that the final result
+ should be: first append $(a_1 | a_2 | \ldots | a_n)$ to every item of @{text "xrhs"} and then
+ union the result with all non-@{text "X"}-items of @{text "rhs"}.
+ *}
+definition
+ "rhs_subst rhs X xrhs \<equiv>
+ (rhs - (items_of rhs X)) \<union> (append_rhs_rexp xrhs (rexp_of rhs X))"
+
+text {*
+ Suppose the equation defining @{text "X"} is $X = xrhs$, the follwing
+ @{text "eqs_subst ES X xrhs"} substitute @{text "xrhs"} into every equation
+ of the equational system @{text "ES"}.
+ *}
+
+definition
+ "eqs_subst ES X xrhs \<equiv> {(Y, rhs_subst yrhs X xrhs) | Y yrhs. (Y, yrhs) \<in> ES}"
+
+text {*
+ The computation of regular expressions for equivalent classes is accomplished
+ using a iteration principle given by the following lemma.
+ *}
+
+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 {*
+ The @{text "P"} in lemma @{text "wf_iter"} is an invaiant kept throughout the iteration procedure.
+ The particular invariant used to solve our problem is defined by function @{text "Inv(ES)"},
+ an invariant over equal system @{text "ES"}.
+ Every definition starting next till @{text "Inv"} stipulates a property to be satisfied by @{text "ES"}.
+*}
+
+text {*
+ Every variable is defined at most onece in @{text "ES"}.
+ *}
+definition
+ "distinct_equas ES \<equiv>
+ \<forall> X rhs rhs'. (X, rhs) \<in> ES \<and> (X, rhs') \<in> ES \<longrightarrow> rhs = rhs'"
+text {*
+ Every equation in @{text "ES"} (represented by @{text "(X, rhs)"}) is valid, i.e. @{text "(X = L rhs)"}.
+ *}
+definition
+ "valid_eqns ES \<equiv> \<forall> X rhs. (X, rhs) \<in> ES \<longrightarrow> (X = L rhs)"
+
+text {*
+ The following @{text "rhs_nonempty rhs"} requires regular expressions occuring in transitional
+ items of @{text "rhs"} does not contain empty string. This is necessary for
+ the application of Arden's transformation to @{text "rhs"}.
+ *}
+definition
+ "rhs_nonempty rhs \<equiv> (\<forall> Y r. Trn Y r \<in> rhs \<longrightarrow> [] \<notin> L r)"
+
+text {*
+ The following @{text "ardenable ES"} requires that Arden's transformation is applicable
+ to every equation of equational system @{text "ES"}.
+ *}
+definition
+ "ardenable ES \<equiv> \<forall> X rhs. (X, rhs) \<in> ES \<longrightarrow> rhs_nonempty rhs"
+
+(* The following non_empty seems useless. *)
+definition
+ "non_empty ES \<equiv> \<forall> X rhs. (X, rhs) \<in> ES \<longrightarrow> X \<noteq> {}"
+
+text {*
+ The following @{text "finite_rhs ES"} requires every equation in @{text "rhs"} be finite.
+ *}
+definition
+ "finite_rhs ES \<equiv> \<forall> X rhs. (X, rhs) \<in> ES \<longrightarrow> finite rhs"
+
+text {*
+ The following @{text "classes_of rhs"} returns all variables (or equivalent classes)
+ occuring in @{text "rhs"}.
+ *}
+definition
+ "classes_of rhs \<equiv> {X. \<exists> r. Trn X r \<in> rhs}"
+
+text {*
+ The following @{text "lefts_of ES"} returns all variables
+ defined by equational system @{text "ES"}.
+ *}
+definition
+ "lefts_of ES \<equiv> {Y | Y yrhs. (Y, yrhs) \<in> ES}"
+
+text {*
+ The following @{text "self_contained ES"} requires that every
+ variable occuring on the right hand side of equations is already defined by some
+ equation in @{text "ES"}.
+ *}
+definition
+ "self_contained ES \<equiv> \<forall> (X, xrhs) \<in> ES. classes_of xrhs \<subseteq> lefts_of ES"
+
+
+text {*
+ The invariant @{text "Inv(ES)"} is a conjunction of all the previously defined constaints.
+ *}
+definition
+ "Inv ES \<equiv> valid_eqns ES \<and> finite ES \<and> distinct_equas ES \<and> ardenable ES \<and>
+ non_empty ES \<and> finite_rhs ES \<and> self_contained ES"
+
+subsection {* The proof of this direction *}
+
+subsubsection {* Basic properties *}
+
+text {*
+ The following are some basic properties of the above definitions.
+*}
+
+lemma L_rhs_union_distrib:
+ " L (A::rhs_item set) \<union> L B = L (A \<union> B)"
+by simp
+
+lemma finite_snd_Trn:
+ assumes finite:"finite rhs"
+ shows "finite {r\<^isub>2. Trn Y r\<^isub>2 \<in> rhs}" (is "finite ?B")
+proof-
+ def rhs' \<equiv> "{e \<in> rhs. \<exists> r. e = Trn Y r}"
+ have "?B = (snd o the_Trn) ` rhs'" using rhs'_def by (auto simp:image_def)
+ moreover have "finite rhs'" using finite rhs'_def by auto
+ ultimately show ?thesis by simp
+qed
+
+lemma rexp_of_empty:
+ assumes finite:"finite rhs"
+ and nonempty:"rhs_nonempty rhs"
+ shows "[] \<notin> L (rexp_of rhs X)"
+using finite nonempty rhs_nonempty_def
+by (drule_tac finite_snd_Trn[where Y = X], auto simp:rexp_of_def items_of_def)
+
+lemma [intro!]:
+ "P (Trn X r) \<Longrightarrow> (\<exists>a. (\<exists>r. a = Trn X r \<and> P a))" by auto
+
+lemma finite_items_of:
+ "finite rhs \<Longrightarrow> finite (items_of rhs X)"
+by (auto simp:items_of_def intro:finite_subset)
+
+lemma lang_of_rexp_of:
+ assumes finite:"finite rhs"
+ shows "L (items_of rhs X) = X ;; (L (rexp_of rhs X))"
+proof -
+ have "finite ((snd \<circ> the_Trn) ` items_of rhs X)" using finite_items_of[OF finite] by auto
+ thus ?thesis
+ apply (auto simp:rexp_of_def Seq_def items_of_def)
+ apply (rule_tac x = s1 in exI, rule_tac x = s2 in exI, auto)
+ by (rule_tac x= "Trn X r" in exI, auto simp:Seq_def)
+qed
+
+lemma rexp_of_lam_eq_lam_set:
+ assumes finite: "finite rhs"
+ shows "L (rexp_of_lam rhs) = L (lam_of rhs)"
+proof -
+ have "finite (the_r ` {Lam r |r. Lam r \<in> rhs})" using finite
+ by (rule_tac finite_imageI, auto intro:finite_subset)
+ thus ?thesis by (auto simp:rexp_of_lam_def lam_of_def)
+qed
+
+lemma [simp]:
+ " L (attach_rexp r xb) = L xb ;; L r"
+apply (cases xb, auto simp:Seq_def)
+by (rule_tac x = "s1 @ s1a" in exI, rule_tac x = s2a in exI,auto simp:Seq_def)
+
+lemma lang_of_append_rhs:
+ "L (append_rhs_rexp rhs r) = L rhs ;; L r"
+apply (auto simp:append_rhs_rexp_def image_def)
+apply (auto simp:Seq_def)
+apply (rule_tac x = "L xb ;; L r" in exI, auto simp add:Seq_def)
+by (rule_tac x = "attach_rexp r xb" in exI, auto simp:Seq_def)
+
+lemma classes_of_union_distrib:
+ "classes_of A \<union> classes_of B = classes_of (A \<union> B)"
+by (auto simp add:classes_of_def)
+
+lemma lefts_of_union_distrib:
+ "lefts_of A \<union> lefts_of B = lefts_of (A \<union> B)"
+by (auto simp:lefts_of_def)
+
+
+subsubsection {* Intialization *}
+
+text {*
+ The following several lemmas until @{text "init_ES_satisfy_Inv"} shows that
+ the initial equational system satisfies invariant @{text "Inv"}.
+ *}
+
+lemma defined_by_str:
+ "\<lbrakk>s \<in> X; X \<in> UNIV // (\<approx>Lang)\<rbrakk> \<Longrightarrow> X = (\<approx>Lang) `` {s}"
+by (auto simp:quotient_def Image_def str_eq_rel_def)
+
+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 clarsimp
+ moreover
+ have "s \<in> Y" unfolding Y_def
+ unfolding Image_def str_eq_rel_def by simp
+ ultimately show thesis by (blast intro: that)
+qed
+
+lemma l_eq_r_in_eqs:
+ assumes X_in_eqs: "(X, xrhs) \<in> (eqs (UNIV // (\<approx>Lang)))"
+ shows "X = L xrhs"
+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 = []"
+ thus ?thesis using X_in_eqs "(1)"
+ by (auto simp:eqs_def init_rhs_def)
+ next
+ assume not_empty: "x \<noteq> []"
+ then obtain clist c where decom: "x = clist @ [c]"
+ by (case_tac x rule:rev_cases, auto)
+ have "X \<in> UNIV // (\<approx>Lang)" using X_in_eqs by (auto simp:eqs_def)
+ then obtain Y
+ where "Y \<in> UNIV // (\<approx>Lang)"
+ and "Y ;; {[c]} \<subseteq> X"
+ and "clist \<in> Y"
+ using decom "(1)" every_eqclass_has_transition by blast
+ hence
+ "x \<in> L {Trn Y (CHAR c)| Y c. Y \<in> UNIV // (\<approx>Lang) \<and> Y ;; {[c]} \<subseteq> X}"
+ using "(1)" decom
+ by (simp, rule_tac x = "Trn Y (CHAR c)" in exI, simp add:Seq_def)
+ thus ?thesis using X_in_eqs "(1)"
+ by (simp add:eqs_def init_rhs_def)
+ qed
+ qed
+next
+ show "L xrhs \<subseteq> X" using X_in_eqs
+ by (auto simp:eqs_def init_rhs_def)
+qed
+
+lemma finite_init_rhs:
+ assumes finite: "finite CS"
+ shows "finite (init_rhs CS X)"
+proof-
+ have "finite {Trn Y (CHAR c) |Y c. Y \<in> CS \<and> Y ;; {[c]} \<subseteq> X}" (is "finite ?A")
+ proof -
+ def S \<equiv> "{(Y, c)| Y c. Y \<in> CS \<and> Y ;; {[c]} \<subseteq> X}"
+ def h \<equiv> "\<lambda> (Y, c). Trn Y (CHAR c)"
+ have "finite (CS \<times> (UNIV::char set))" using finite by auto
+ hence "finite S" using S_def
+ by (rule_tac B = "CS \<times> UNIV" in finite_subset, auto)
+ moreover have "?A = h ` S" by (auto simp: S_def h_def image_def)
+ ultimately show ?thesis
+ by auto
+ qed
+ thus ?thesis by (simp add:init_rhs_def)
+qed
+
+lemma init_ES_satisfy_Inv:
+ assumes finite_CS: "finite (UNIV // (\<approx>Lang))"
+ shows "Inv (eqs (UNIV // (\<approx>Lang)))"
+proof -
+ have "finite (eqs (UNIV // (\<approx>Lang)))" using finite_CS
+ by (simp add:eqs_def)
+ moreover have "distinct_equas (eqs (UNIV // (\<approx>Lang)))"
+ by (simp add:distinct_equas_def eqs_def)
+ moreover have "ardenable (eqs (UNIV // (\<approx>Lang)))"
+ by (auto simp add:ardenable_def eqs_def init_rhs_def rhs_nonempty_def del:L_rhs.simps)
+ moreover have "valid_eqns (eqs (UNIV // (\<approx>Lang)))"
+ using l_eq_r_in_eqs by (simp add:valid_eqns_def)
+ moreover have "non_empty (eqs (UNIV // (\<approx>Lang)))"
+ by (auto simp:non_empty_def eqs_def quotient_def Image_def str_eq_rel_def)
+ moreover have "finite_rhs (eqs (UNIV // (\<approx>Lang)))"
+ using finite_init_rhs[OF finite_CS]
+ by (auto simp:finite_rhs_def eqs_def)
+ moreover have "self_contained (eqs (UNIV // (\<approx>Lang)))"
+ by (auto simp:self_contained_def eqs_def init_rhs_def classes_of_def lefts_of_def)
+ ultimately show ?thesis by (simp add:Inv_def)
+qed
+
+subsubsection {*
+ Interation step
+ *}
+
+text {*
+ From this point until @{text "iteration_step"}, it is proved
+ that there exists iteration steps which keep @{text "Inv(ES)"} while
+ decreasing the size of @{text "ES"}.
+ *}
+lemma arden_variate_keeps_eq:
+ assumes l_eq_r: "X = L rhs"
+ and not_empty: "[] \<notin> L (rexp_of rhs X)"
+ and finite: "finite rhs"
+ shows "X = L (arden_variate X rhs)"
+proof -
+ def A \<equiv> "L (rexp_of rhs X)"
+ def b \<equiv> "rhs - items_of rhs X"
+ def B \<equiv> "L b"
+ have "X = B ;; A\<star>"
+ proof-
+ have "rhs = items_of rhs X \<union> b" by (auto simp:b_def items_of_def)
+ hence "L rhs = L(items_of rhs X \<union> b)" by simp
+ hence "L rhs = L(items_of rhs X) \<union> B" by (simp only:L_rhs_union_distrib B_def)
+ with lang_of_rexp_of
+ have "L rhs = X ;; A \<union> B " using finite by (simp only:B_def b_def A_def)
+ thus ?thesis
+ using l_eq_r not_empty
+ apply (drule_tac B = B and X = X in ardens_revised)
+ by (auto simp:A_def simp del:L_rhs.simps)
+ qed
+ moreover have "L (arden_variate X rhs) = (B ;; A\<star>)" (is "?L = ?R")
+ by (simp only:arden_variate_def L_rhs_union_distrib lang_of_append_rhs
+ B_def A_def b_def L_rexp.simps seq_union_distrib)
+ ultimately show ?thesis by simp
+qed
+
+lemma append_keeps_finite:
+ "finite rhs \<Longrightarrow> finite (append_rhs_rexp rhs r)"
+by (auto simp:append_rhs_rexp_def)
+
+lemma arden_variate_keeps_finite:
+ "finite rhs \<Longrightarrow> finite (arden_variate X rhs)"
+by (auto simp:arden_variate_def append_keeps_finite)
+
+lemma append_keeps_nonempty:
+ "rhs_nonempty rhs \<Longrightarrow> rhs_nonempty (append_rhs_rexp rhs r)"
+apply (auto simp:rhs_nonempty_def append_rhs_rexp_def)
+by (case_tac x, auto simp:Seq_def)
+
+lemma nonempty_set_sub:
+ "rhs_nonempty rhs \<Longrightarrow> rhs_nonempty (rhs - A)"
+by (auto simp:rhs_nonempty_def)
+
+lemma nonempty_set_union:
+ "\<lbrakk>rhs_nonempty rhs; rhs_nonempty rhs'\<rbrakk> \<Longrightarrow> rhs_nonempty (rhs \<union> rhs')"
+by (auto simp:rhs_nonempty_def)
+
+lemma arden_variate_keeps_nonempty:
+ "rhs_nonempty rhs \<Longrightarrow> rhs_nonempty (arden_variate X rhs)"
+by (simp only:arden_variate_def append_keeps_nonempty nonempty_set_sub)
+
+
+lemma rhs_subst_keeps_nonempty:
+ "\<lbrakk>rhs_nonempty rhs; rhs_nonempty xrhs\<rbrakk> \<Longrightarrow> rhs_nonempty (rhs_subst rhs X xrhs)"
+by (simp only:rhs_subst_def append_keeps_nonempty nonempty_set_union nonempty_set_sub)
+
+lemma rhs_subst_keeps_eq:
+ assumes substor: "X = L xrhs"
+ and finite: "finite rhs"
+ shows "L (rhs_subst rhs X xrhs) = L rhs" (is "?Left = ?Right")
+proof-
+ def A \<equiv> "L (rhs - items_of rhs X)"
+ have "?Left = A \<union> L (append_rhs_rexp xrhs (rexp_of rhs X))"
+ by (simp only:rhs_subst_def L_rhs_union_distrib A_def)
+ moreover have "?Right = A \<union> L (items_of rhs X)"
+ proof-
+ have "rhs = (rhs - items_of rhs X) \<union> (items_of rhs X)" by (auto simp:items_of_def)
+ thus ?thesis by (simp only:L_rhs_union_distrib A_def)
+ qed
+ moreover have "L (append_rhs_rexp xrhs (rexp_of rhs X)) = L (items_of rhs X)"
+ using finite substor by (simp only:lang_of_append_rhs lang_of_rexp_of)
+ ultimately show ?thesis by simp
+qed
+
+lemma rhs_subst_keeps_finite_rhs:
+ "\<lbrakk>finite rhs; finite yrhs\<rbrakk> \<Longrightarrow> finite (rhs_subst rhs Y yrhs)"
+by (auto simp:rhs_subst_def append_keeps_finite)
+
+lemma eqs_subst_keeps_finite:
+ assumes finite:"finite (ES:: (string set \<times> rhs_item set) set)"
+ shows "finite (eqs_subst ES Y yrhs)"
+proof -
+ have "finite {(Ya, rhs_subst yrhsa Y yrhs) |Ya yrhsa. (Ya, yrhsa) \<in> ES}"
+ (is "finite ?A")
+ proof-
+ def eqns' \<equiv> "{((Ya::string set), yrhsa)| Ya yrhsa. (Ya, yrhsa) \<in> ES}"
+ def h \<equiv> "\<lambda> ((Ya::string set), yrhsa). (Ya, rhs_subst yrhsa Y yrhs)"
+ have "finite (h ` eqns')" using finite h_def eqns'_def by auto
+ moreover have "?A = h ` eqns'" by (auto simp:h_def eqns'_def)
+ ultimately show ?thesis by auto
+ qed
+ thus ?thesis by (simp add:eqs_subst_def)
+qed
+
+lemma eqs_subst_keeps_finite_rhs:
+ "\<lbrakk>finite_rhs ES; finite yrhs\<rbrakk> \<Longrightarrow> finite_rhs (eqs_subst ES Y yrhs)"
+by (auto intro:rhs_subst_keeps_finite_rhs simp add:eqs_subst_def finite_rhs_def)
+
+lemma append_rhs_keeps_cls:
+ "classes_of (append_rhs_rexp rhs r) = classes_of rhs"
+apply (auto simp:classes_of_def append_rhs_rexp_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_variate_removes_cl:
+ "classes_of (arden_variate Y yrhs) = classes_of yrhs - {Y}"
+apply (simp add:arden_variate_def append_rhs_keeps_cls items_of_def)
+by (auto simp:classes_of_def)
+
+lemma lefts_of_keeps_cls:
+ "lefts_of (eqs_subst ES Y yrhs) = lefts_of ES"
+by (auto simp:lefts_of_def eqs_subst_def)
+
+lemma rhs_subst_updates_cls:
+ "X \<notin> classes_of xrhs \<Longrightarrow>
+ classes_of (rhs_subst rhs X xrhs) = classes_of rhs \<union> classes_of xrhs - {X}"
+apply (simp only:rhs_subst_def append_rhs_keeps_cls
+ classes_of_union_distrib[THEN sym])
+by (auto simp:classes_of_def items_of_def)
+
+lemma eqs_subst_keeps_self_contained:
+ fixes Y
+ assumes sc: "self_contained (ES \<union> {(Y, yrhs)})" (is "self_contained ?A")
+ shows "self_contained (eqs_subst ES Y (arden_variate Y yrhs))"
+ (is "self_contained ?B")
+proof-
+ { fix X xrhs'
+ assume "(X, xrhs') \<in> ?B"
+ then obtain xrhs
+ where xrhs_xrhs': "xrhs' = rhs_subst xrhs Y (arden_variate Y yrhs)"
+ and X_in: "(X, xrhs) \<in> ES" by (simp add:eqs_subst_def, blast)
+ have "classes_of xrhs' \<subseteq> lefts_of ?B"
+ proof-
+ have "lefts_of ?B = lefts_of ES" by (auto simp add:lefts_of_def eqs_subst_def)
+ moreover have "classes_of xrhs' \<subseteq> lefts_of ES"
+ proof-
+ have "classes_of xrhs' \<subseteq>
+ classes_of xrhs \<union> classes_of (arden_variate Y yrhs) - {Y}"
+ proof-
+ have "Y \<notin> classes_of (arden_variate Y yrhs)"
+ using arden_variate_removes_cl by simp
+ thus ?thesis using xrhs_xrhs' by (auto simp:rhs_subst_updates_cls)
+ qed
+ moreover have "classes_of xrhs \<subseteq> lefts_of ES \<union> {Y}" using X_in sc
+ apply (simp only:self_contained_def lefts_of_union_distrib[THEN sym])
+ by (drule_tac x = "(X, xrhs)" in bspec, auto simp:lefts_of_def)
+ moreover have "classes_of (arden_variate Y yrhs) \<subseteq> lefts_of ES \<union> {Y}"
+ using sc
+ by (auto simp add:arden_variate_removes_cl self_contained_def lefts_of_def)
+ ultimately show ?thesis by auto
+ qed
+ ultimately show ?thesis by simp
+ qed
+ } thus ?thesis by (auto simp only:eqs_subst_def self_contained_def)
+qed
+
+lemma eqs_subst_satisfy_Inv:
+ assumes Inv_ES: "Inv (ES \<union> {(Y, yrhs)})"
+ shows "Inv (eqs_subst ES Y (arden_variate Y yrhs))"
+proof -
+ have finite_yrhs: "finite yrhs"
+ using Inv_ES by (auto simp:Inv_def finite_rhs_def)
+ have nonempty_yrhs: "rhs_nonempty yrhs"
+ using Inv_ES by (auto simp:Inv_def ardenable_def)
+ have Y_eq_yrhs: "Y = L yrhs"
+ using Inv_ES by (simp only:Inv_def valid_eqns_def, blast)
+ have "distinct_equas (eqs_subst ES Y (arden_variate Y yrhs))"
+ using Inv_ES
+ by (auto simp:distinct_equas_def eqs_subst_def Inv_def)
+ moreover have "finite (eqs_subst ES Y (arden_variate Y yrhs))"
+ using Inv_ES by (simp add:Inv_def eqs_subst_keeps_finite)
+ moreover have "finite_rhs (eqs_subst ES Y (arden_variate Y yrhs))"
+ proof-
+ have "finite_rhs ES" using Inv_ES
+ by (simp add:Inv_def finite_rhs_def)
+ moreover have "finite (arden_variate Y yrhs)"
+ proof -
+ have "finite yrhs" using Inv_ES
+ by (auto simp:Inv_def finite_rhs_def)
+ thus ?thesis using arden_variate_keeps_finite by simp
+ qed
+ ultimately show ?thesis
+ by (simp add:eqs_subst_keeps_finite_rhs)
+ qed
+ moreover have "ardenable (eqs_subst ES Y (arden_variate Y yrhs))"
+ proof -
+ { fix X rhs
+ assume "(X, rhs) \<in> ES"
+ hence "rhs_nonempty rhs" using prems Inv_ES
+ by (simp add:Inv_def ardenable_def)
+ with nonempty_yrhs
+ have "rhs_nonempty (rhs_subst rhs Y (arden_variate Y yrhs))"
+ by (simp add:nonempty_yrhs
+ rhs_subst_keeps_nonempty arden_variate_keeps_nonempty)
+ } thus ?thesis by (auto simp add:ardenable_def eqs_subst_def)
+ qed
+ moreover have "valid_eqns (eqs_subst ES Y (arden_variate Y yrhs))"
+ proof-
+ have "Y = L (arden_variate Y yrhs)"
+ using Y_eq_yrhs Inv_ES finite_yrhs nonempty_yrhs
+ by (rule_tac arden_variate_keeps_eq, (simp add:rexp_of_empty)+)
+ thus ?thesis using Inv_ES
+ by (clarsimp simp add:valid_eqns_def
+ eqs_subst_def rhs_subst_keeps_eq Inv_def finite_rhs_def
+ simp del:L_rhs.simps)
+ qed
+ moreover have
+ non_empty_subst: "non_empty (eqs_subst ES Y (arden_variate Y yrhs))"
+ using Inv_ES by (auto simp:Inv_def non_empty_def eqs_subst_def)
+ moreover
+ have self_subst: "self_contained (eqs_subst ES Y (arden_variate Y yrhs))"
+ using Inv_ES eqs_subst_keeps_self_contained by (simp add:Inv_def)
+ ultimately show ?thesis using Inv_ES by (simp add:Inv_def)
+qed
+
+lemma eqs_subst_card_le:
+ assumes finite: "finite (ES::(string set \<times> rhs_item set) set)"
+ shows "card (eqs_subst ES Y yrhs) <= card ES"
+proof-
+ def f \<equiv> "\<lambda> x. ((fst x)::string set, rhs_subst (snd x) Y yrhs)"
+ have "eqs_subst ES Y yrhs = f ` ES"
+ apply (auto simp:eqs_subst_def f_def image_def)
+ by (rule_tac x = "(Ya, yrhsa)" in bexI, simp+)
+ thus ?thesis using finite by (auto intro:card_image_le)
+qed
+
+lemma eqs_subst_cls_remains:
+ "(X, xrhs) \<in> ES \<Longrightarrow> \<exists> xrhs'. (X, xrhs') \<in> (eqs_subst ES Y yrhs)"
+by (auto simp:eqs_subst_def)
+
+lemma card_noteq_1_has_more:
+ assumes card:"card S \<noteq> 1"
+ and e_in: "e \<in> S"
+ and finite: "finite S"
+ obtains e' where "e' \<in> S \<and> e \<noteq> e'"
+proof-
+ have "card (S - {e}) > 0"
+ proof -
+ have "card S > 1" using card e_in finite
+ by (case_tac "card S", auto)
+ thus ?thesis using finite e_in by auto
+ qed
+ hence "S - {e} \<noteq> {}" using finite by (rule_tac notI, simp)
+ thus "(\<And>e'. e' \<in> S \<and> e \<noteq> e' \<Longrightarrow> thesis) \<Longrightarrow> thesis" by auto
+qed
+
+lemma iteration_step:
+ assumes Inv_ES: "Inv ES"
+ and X_in_ES: "(X, xrhs) \<in> ES"
+ and not_T: "card ES \<noteq> 1"
+ shows "\<exists> ES'. (Inv ES' \<and> (\<exists> xrhs'.(X, xrhs') \<in> ES')) \<and>
+ (card ES', card ES) \<in> less_than" (is "\<exists> ES'. ?P ES'")
+proof -
+ have finite_ES: "finite ES" using Inv_ES by (simp add:Inv_def)
+ then obtain Y yrhs
+ where Y_in_ES: "(Y, yrhs) \<in> ES" and not_eq: "(X, xrhs) \<noteq> (Y, yrhs)"
+ using not_T X_in_ES by (drule_tac card_noteq_1_has_more, auto)
+ def ES' == "ES - {(Y, yrhs)}"
+ let ?ES'' = "eqs_subst ES' Y (arden_variate Y yrhs)"
+ have "?P ?ES''"
+ proof -
+ have "Inv ?ES''" using Y_in_ES Inv_ES
+ by (rule_tac eqs_subst_satisfy_Inv, simp add:ES'_def insert_absorb)
+ moreover have "\<exists>xrhs'. (X, xrhs') \<in> ?ES''" using not_eq X_in_ES
+ by (rule_tac ES = ES' in eqs_subst_cls_remains, auto simp add:ES'_def)
+ moreover have "(card ?ES'', card ES) \<in> less_than"
+ proof -
+ have "finite ES'" using finite_ES ES'_def by auto
+ moreover have "card ES' < card ES" using finite_ES Y_in_ES
+ by (auto simp:ES'_def card_gt_0_iff intro:diff_Suc_less)
+ ultimately show ?thesis
+ by (auto dest:eqs_subst_card_le elim:le_less_trans)
+ qed
+ ultimately show ?thesis by simp
+ qed
+ thus ?thesis by blast
+qed
+
+subsubsection {*
+ Conclusion of the proof
+ *}
+
+text {*
+ From this point until @{text "hard_direction"}, the hard direction is proved
+ through a simple application of the iteration principle.
+*}
+
+lemma iteration_conc:
+ assumes history: "Inv ES"
+ and X_in_ES: "\<exists> xrhs. (X, xrhs) \<in> ES"
+ shows
+ "\<exists> ES'. (Inv ES' \<and> (\<exists> xrhs'. (X, xrhs') \<in> ES')) \<and> card ES' = 1"
+ (is "\<exists> ES'. ?P ES'")
+proof (cases "card ES = 1")
+ case True
+ thus ?thesis using history X_in_ES
+ by blast
+next
+ case False
+ thus ?thesis using history iteration_step X_in_ES
+ by (rule_tac f = card in wf_iter, auto)
+qed
+
+lemma last_cl_exists_rexp:
+ assumes ES_single: "ES = {(X, xrhs)}"
+ and Inv_ES: "Inv ES"
+ shows "\<exists> (r::rexp). L r = X" (is "\<exists> r. ?P r")
+proof-
+ let ?A = "arden_variate X xrhs"
+ have "?P (rexp_of_lam ?A)"
+ proof -
+ have "L (rexp_of_lam ?A) = L (lam_of ?A)"
+ proof(rule rexp_of_lam_eq_lam_set)
+ show "finite (arden_variate X xrhs)" using Inv_ES ES_single
+ by (rule_tac arden_variate_keeps_finite,
+ auto simp add:Inv_def finite_rhs_def)
+ qed
+ also have "\<dots> = L ?A"
+ proof-
+ have "lam_of ?A = ?A"
+ proof-
+ have "classes_of ?A = {}" using Inv_ES ES_single
+ by (simp add:arden_variate_removes_cl
+ self_contained_def Inv_def lefts_of_def)
+ thus ?thesis
+ by (auto simp only:lam_of_def classes_of_def, case_tac x, auto)
+ qed
+ thus ?thesis by simp
+ qed
+ also have "\<dots> = X"
+ proof(rule arden_variate_keeps_eq [THEN sym])
+ show "X = L xrhs" using Inv_ES ES_single
+ by (auto simp only:Inv_def valid_eqns_def)
+ next
+ from Inv_ES ES_single show "[] \<notin> L (rexp_of xrhs X)"
+ by(simp add:Inv_def ardenable_def rexp_of_empty finite_rhs_def)
+ next
+ from Inv_ES ES_single show "finite xrhs"
+ by (simp add:Inv_def finite_rhs_def)
+ qed
+ finally show ?thesis by simp
+ qed
+ thus ?thesis by auto
+qed
+
+lemma every_eqcl_has_reg:
+ assumes finite_CS: "finite (UNIV // (\<approx>Lang))"
+ and X_in_CS: "X \<in> (UNIV // (\<approx>Lang))"
+ shows "\<exists> (reg::rexp). L reg = X" (is "\<exists> r. ?E r")
+proof -
+ from X_in_CS have "\<exists> xrhs. (X, xrhs) \<in> (eqs (UNIV // (\<approx>Lang)))"
+ by (auto simp:eqs_def init_rhs_def)
+ then obtain ES xrhs where Inv_ES: "Inv ES"
+ and X_in_ES: "(X, xrhs) \<in> ES"
+ and card_ES: "card ES = 1"
+ using finite_CS X_in_CS init_ES_satisfy_Inv iteration_conc
+ by blast
+ hence ES_single_equa: "ES = {(X, xrhs)}"
+ by (auto simp:Inv_def dest!:card_Suc_Diff1 simp:card_eq_0_iff)
+ thus ?thesis using Inv_ES
+ by (rule last_cl_exists_rexp)
+qed
+
+lemma finals_in_partitions:
+ "finals Lang \<subseteq> (UNIV // (\<approx>Lang))"
+ by (auto simp:finals_def quotient_def)
+
+theorem hard_direction:
+ assumes finite_CS: "finite (UNIV // (\<approx>Lang))"
+ shows "\<exists> (reg::rexp). Lang = L reg"
+proof -
+ have "\<forall> X \<in> (UNIV // (\<approx>Lang)). \<exists> (reg::rexp). X = L reg"
+ using finite_CS every_eqcl_has_reg by blast
+ then obtain f
+ where f_prop: "\<forall> X \<in> (UNIV // (\<approx>Lang)). X = L ((f X)::rexp)"
+ by (auto dest:bchoice)
+ def rs \<equiv> "f ` (finals Lang)"
+ have "Lang = \<Union> (finals Lang)" using lang_is_union_of_finals by auto
+ also have "\<dots> = L (folds ALT NULL rs)"
+ proof -
+ have "finite rs"
+ proof -
+ have "finite (finals Lang)"
+ using finite_CS finals_in_partitions[of "Lang"]
+ by (erule_tac finite_subset, simp)
+ thus ?thesis using rs_def by auto
+ qed
+ thus ?thesis
+ using f_prop rs_def finals_in_partitions[of "Lang"] by auto
+ qed
+ finally show ?thesis by blast
+qed
+
+end
\ No newline at end of file