theory Myhill_1 imports Main List_Prefix Prefix_subtract Preludebegin(*text {* \begin{figure} \centering \scalebox{0.95}{ \begin{tikzpicture}[->,>=latex,shorten >=1pt,auto,node distance=1.2cm, semithick] \node[state,initial] (n1) {$1$}; \node[state,accepting] (n2) [right = 10em of n1] {$2$}; \path (n1) edge [bend left] node {$0$} (n2) (n1) edge [loop above] node{$1$} (n1) (n2) edge [loop above] node{$0$} (n2) (n2) edge [bend left] node {$1$} (n1) ; \end{tikzpicture}} \caption{An example automaton (or partition)}\label{fig:example_automata} \end{figure}*}*)section {* Preliminary definitions *}types lang = "string set"text {* Sequential composition of two languages @{text "L1"} and @{text "L2"} *}definition Seq :: "lang \<Rightarrow> lang \<Rightarrow> lang" ("_ ;; _" [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 :: "lang \<Rightarrow> lang" ("_\<star>" [101] 102) for L 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 blastby (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 {* 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 rexptext {* 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"beginfun 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(* Just a technical lemma. *)lemma [simp]: shows "(x, y) \<in> {(x, y). P x y} \<longleftrightarrow> P x y"by simptext {* @{text "\<approx>L"} is an equivalent class defined by language @{text "Lang"}.*}definition str_eq_rel ("\<approx>_" [100] 100)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) qednext show "\<Union> (finals Lang) \<subseteq> Lang" apply (clarsimp simp:finals_def str_eq_rel_def) by (drule_tac x = "[]" in spec, auto)qedsection {* Direction @{text "finite partition \<Rightarrow> regular language"}*}subsection {* Ardens lemma *}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 qedqedsubsection {* Defintions peculiar to this direction *}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_eq_intro, simp add:folds_def)apply (rule someI2_ex, erule finite_imp_fold_graph)by (erule fold_graph.induct, auto)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"endtext {* 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)"endtext {* 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 qedqedtext {* 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 simplemma 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 simpqedlemma rexp_of_empty: assumes finite:"finite rhs" and nonempty:"rhs_nonempty rhs" shows "[] \<notin> L (rexp_of rhs X)"using finite nonempty rhs_nonempty_defby (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 autolemma 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)qedlemma 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)qedlemma [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)qedlemma 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 qednext show "L xrhs \<subseteq> X" using X_in_eqs by (auto simp:eqs_def init_rhs_def) qedlemma 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)qedlemma 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)qedsubsubsection {* 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 simpqed 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 simpqedlemma 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)qedlemma 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)qedlemma 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)qedlemma 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)qedlemma 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 autoqedlemma 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 blastqedsubsubsection {* 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 blastnext case False thus ?thesis using history iteration_step X_in_ES by (rule_tac f = card in wf_iter, auto)qedlemma 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 autoqedlemma 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)qedlemma 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 blastqed end