theory Myhill_1imports Main Foldsbeginsection {* Preliminary definitions *}types lang = "string set"text {* Sequential composition of two languages *}definition Seq :: "lang \<Rightarrow> lang \<Rightarrow> lang" (infixr ";;" 100)where "A ;; B = {s\<^isub>1 @ s\<^isub>2 | s\<^isub>1 s\<^isub>2. s\<^isub>1 \<in> A \<and> s\<^isub>2 \<in> B}"text {* Some properties of operator @{text ";;"}. *}lemma seq_add_left: assumes a: "A = B" shows "C ;; A = C ;; B"using a by simplemma seq_union_distrib_right: shows "(A \<union> B) ;; C = (A ;; C) \<union> (B ;; C)"unfolding Seq_def by autolemma seq_union_distrib_left: shows "C ;; (A \<union> B) = (C ;; A) \<union> (C ;; B)"unfolding Seq_def by autolemma seq_intro: assumes a: "x \<in> A" "y \<in> B" shows "x @ y \<in> A ;; B "using a by (auto simp: Seq_def)lemma seq_assoc: shows "(A ;; B) ;; C = A ;; (B ;; C)"unfolding Seq_defapply(auto)apply(blast)by (metis append_assoc)lemma seq_empty [simp]: shows "A ;; {[]} = A" and "{[]} ;; A = A"by (simp_all add: Seq_def)text {* Power and Star of a language *}fun pow :: "lang \<Rightarrow> nat \<Rightarrow> lang" (infixl "\<up>" 100)where "A \<up> 0 = {[]}"| "A \<up> (Suc n) = A ;; (A \<up> n)" definition Star :: "lang \<Rightarrow> lang" ("_\<star>" [101] 102)where "A\<star> \<equiv> (\<Union>n. A \<up> n)"lemma star_start[intro]: shows "[] \<in> A\<star>"proof - have "[] \<in> A \<up> 0" by auto then show "[] \<in> A\<star>" unfolding Star_def by blastqedlemma 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 blastqedlemma 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)qedlemma star_intro1: assumes a: "x \<in> A\<star>" and b: "y \<in> A\<star>" shows "x @ y \<in> A\<star>"using a bby (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 simpqedlemma star_intro3: assumes a: "x \<in> A\<star>" and b: "y \<in> A" shows "x @ y \<in> A\<star>"using a b by (blast intro: star_intro1 star_intro2)lemma star_cases: shows "A\<star> = {[]} \<union> A ;; A\<star>"proof { fix x have "x \<in> A\<star> \<Longrightarrow> x \<in> {[]} \<union> A ;; A\<star>" unfolding Seq_def by (induct rule: star_induct) (auto) } then show "A\<star> \<subseteq> {[]} \<union> A ;; A\<star>" by autonext show "{[]} \<union> A ;; A\<star> \<subseteq> A\<star>" unfolding Seq_def by autoqedlemma 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 aby (induct rule: star_induct) (blast)+lemma shows seq_Union_left: "B ;; (\<Union>n. A \<up> n) = (\<Union>n. B ;; (A \<up> n))" and seq_Union_right: "(\<Union>n. A \<up> n) ;; B = (\<Union>n. (A \<up> n) ;; B)"unfolding Seq_def by autolemma seq_pow_comm: shows "A ;; (A \<up> n) = (A \<up> n) ;; A"by (induct n) (simp_all add: seq_assoc[symmetric])lemma seq_star_comm: shows "A ;; A\<star> = A\<star> ;; A"unfolding Star_def seq_Union_leftunfolding seq_pow_comm seq_Union_right by simptext {* 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 bproof (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 autonext 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)qedlemma seq_pow_length: assumes a: "[] \<notin> A" and b: "s \<in> B ;; (A \<up> Suc n)" shows "n < length s"proof - from b obtain s1 s2 where eq: "s = s1 @ s2" and *: "s2 \<in> A \<up> Suc n" unfolding Seq_def by auto from * have " n < length s2" by (rule pow_length[OF a]) then show "n < length s" using eq by simpqedsection {* A modified version of Arden's lemma *}text {* A helper lemma for Arden *}lemma arden_helper: assumes eq: "X = X ;; A \<union> B" shows "X = X ;; (A \<up> Suc n) \<union> (\<Union>m\<in>{0..n}. B ;; (A \<up> m))"proof (induct n) case 0 show "X = X ;; (A \<up> Suc 0) \<union> (\<Union>(m::nat)\<in>{0..0}. B ;; (A \<up> m))" using eq by simpnext case (Suc n) have ih: "X = X ;; (A \<up> Suc n) \<union> (\<Union>m\<in>{0..n}. B ;; (A \<up> m))" by fact also have "\<dots> = (X ;; A \<union> B) ;; (A \<up> Suc n) \<union> (\<Union>m\<in>{0..n}. B ;; (A \<up> m))" using eq by simp also have "\<dots> = X ;; (A \<up> Suc (Suc n)) \<union> (B ;; (A \<up> Suc n)) \<union> (\<Union>m\<in>{0..n}. B ;; (A \<up> m))" by (simp add: seq_union_distrib_right seq_assoc) also have "\<dots> = X ;; (A \<up> Suc (Suc n)) \<union> (\<Union>m\<in>{0..Suc n}. B ;; (A \<up> m))" by (auto simp add: le_Suc_eq) finally show "X = X ;; (A \<up> Suc (Suc n)) \<union> (\<Union>m\<in>{0..Suc n}. B ;; (A \<up> m))" .qedtheorem arden: assumes nemp: "[] \<notin> A" shows "X = X ;; A \<union> B \<longleftrightarrow> X = B ;; A\<star>"proof assume eq: "X = B ;; A\<star>" have "A\<star> = {[]} \<union> A\<star> ;; A" unfolding seq_star_comm[symmetric] by (rule star_cases) then have "B ;; A\<star> = B ;; ({[]} \<union> A\<star> ;; A)" by (rule seq_add_left) also have "\<dots> = B \<union> B ;; (A\<star> ;; A)" unfolding seq_union_distrib_left by simp also have "\<dots> = B \<union> (B ;; A\<star>) ;; A" by (simp only: seq_assoc) finally show "X = X ;; A \<union> B" using eq by blast next assume eq: "X = X ;; A \<union> B" { fix n::nat have "B ;; (A \<up> n) \<subseteq> X" using arden_helper[OF eq, of "n"] by auto } then have "B ;; A\<star> \<subseteq> X" unfolding Seq_def Star_def UNION_def by auto moreover { fix s::string obtain k where "k = length s" by auto then have not_in: "s \<notin> X ;; (A \<up> Suc k)" using seq_pow_length[OF nemp] by blast assume "s \<in> X" then have "s \<in> X ;; (A \<up> Suc k) \<union> (\<Union>m\<in>{0..k}. B ;; (A \<up> m))" using arden_helper[OF eq, of "k"] by auto then have "s \<in> (\<Union>m\<in>{0..k}. B ;; (A \<up> m))" using not_in by auto moreover have "(\<Union>m\<in>{0..k}. B ;; (A \<up> m)) \<subseteq> (\<Union>n. B ;; (A \<up> n))" by auto ultimately have "s \<in> B ;; A\<star>" unfolding seq_Union_left Star_def by auto } then have "X \<subseteq> B ;; A\<star>" by auto ultimately show "X = B ;; A\<star>" by simpqedsection {* Regular Expressions *}datatype rexp = NULL| EMPTY| CHAR char| SEQ rexp rexp| ALT rexp rexp| STAR rexptext {* The function @{text L} is overloaded, with the idea that @{text "L x"} evaluates to the language represented by the object @{text x}.*}consts L:: "'a \<Rightarrow> lang"overloading L_rexp \<equiv> "L:: rexp \<Rightarrow> lang"beginfun 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) ;; (L_rexp r2)" | "L_rexp (ALT r1 r2) = (L_rexp r1) \<union> (L_rexp r2)" | "L_rexp (STAR r) = (L_rexp r)\<star>"endtext {* ALT-combination of a set or regulare expressions *}abbreviation Setalt ("\<Uplus>_" [1000] 999) where "\<Uplus>A == folds ALT NULL A"text {* For finite sets, @{term Setalt} is preserved under @{term L}.*}lemma folds_alt_simp [simp]: fixes rs::"rexp set" assumes a: "finite rs" shows "L (\<Uplus>rs) = \<Union> (L ` rs)"apply(rule set_eqI)apply(simp add: folds_def)apply(rule someI2_ex)apply(rule_tac finite_imp_fold_graph[OF a])apply(erule fold_graph.induct)apply(auto)donesection {* Direction @{text "finite partition \<Rightarrow> regular language"} *}text {* Just a technical lemma for collections and pairs *}lemma Pair_Collect[simp]: shows "(x, y) \<in> {(x, y). P x y} \<longleftrightarrow> P x y"by simptext {* 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)}"text {* Among the equivalence clases of @{text "\<approx>A"}, the set @{text "finals A"} singles out those which contains the strings from @{text A}.*}definition finals :: "lang \<Rightarrow> lang set"where "finals A \<equiv> {\<approx>A `` {x} | x . x \<in> A}"lemma lang_is_union_of_finals: shows "A = \<Union> finals A"unfolding finals_defunfolding Image_defunfolding str_eq_rel_defapply(auto)apply(drule_tac x = "[]" in spec)apply(auto)donelemma finals_in_partitions: shows "finals A \<subseteq> (UNIV // \<approx>A)"unfolding finals_defunfolding quotient_defby autosection {* Equational systems *}datatype rhs_item = Lam "rexp" (* Lambda-marker *) | Trn "lang" "rexp" (* Transition *)overloading L_rhs_item \<equiv> "L:: rhs_item \<Rightarrow> lang"begin fun L_rhs_item:: "rhs_item \<Rightarrow> lang" where "L_rhs_item (Lam r) = L r" | "L_rhs_item (Trn X r) = X ;; L r"endoverloading L_rhs \<equiv> "L:: rhs_item set \<Rightarrow> lang"begin fun L_rhs:: "rhs_item set \<Rightarrow> lang" where "L_rhs rhs = \<Union> (L ` rhs)"enddefinition "trns_of rhs X \<equiv> {Trn X r | r. Trn X r \<in> rhs}"text {* Transitions between equivalence classes *}definition transition :: "lang \<Rightarrow> rexp \<Rightarrow> lang \<Rightarrow> bool" ("_ \<Turnstile>_\<Rightarrow>_" [100,100,100] 100)where "Y \<Turnstile>r\<Rightarrow> X \<equiv> Y ;; (L r) \<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>(CHAR c)\<Rightarrow> X} else {Trn Y (CHAR c)| Y c. Y \<in> CS \<and> Y \<Turnstile>(CHAR c)\<Rightarrow> X}"definition "eqs CS \<equiv> {(X, init_rhs CS X) | X. X \<in> CS}"section {* Arden Operation on equations *}text {* The function @{text "attach_rexp r item"} SEQ-composes @{text r} to the right of every rhs-item.*}fun attach_rexp :: "rexp \<Rightarrow> rhs_item \<Rightarrow> rhs_item"where "attach_rexp r (Lam rexp) = Lam (SEQ rexp r)"| "attach_rexp r (Trn X rexp) = Trn X (SEQ rexp r)"definition "append_rhs_rexp rhs rexp \<equiv> (attach_rexp rexp) ` rhs"definition "arden_op X rhs \<equiv> append_rhs_rexp (rhs - trns_of rhs X) (STAR (\<Uplus> {r. Trn X r \<in> rhs}))"section {* Substitution Operation on equations *}text {* Suppose and equation @{text "X = xrhs"}, @{text "subst_op"} substitutes all occurences of @{text "X"} in @{text "rhs"} by @{text "xrhs"}.*}definition "subst_op rhs X xrhs \<equiv> (rhs - (trns_of rhs X)) \<union> (append_rhs_rexp xrhs (\<Uplus> {r. Trn X r \<in> rhs}))"text {* @{text "eqs_subst ES X xrhs"} substitutes @{text xrhs} into every equation of the equational system @{text ES}.*}definition "subst_op_all ES X xrhs \<equiv> {(Y, subst_op yrhs X xrhs) | Y yrhs. (Y, yrhs) \<in> ES}"section {* Well-founded iteration *}text {* The computation of regular expressions for equivalence 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 invariant 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"}.*}section {* Invariants *}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 {* @{text "rhs_nonempty rhs"} requires regular expressions occuring in transitional items of @{text "rhs"} do 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"text {* @{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 {* @{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 {* @{text "lefts_of ES"} returns all variables defined by an 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 "invariant(ES)"} is a conjunction of all the previously defined constaints. *}definition "invariant ES \<equiv> valid_eqns ES \<and> finite ES \<and> distinct_equas ES \<and> ardenable 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: fixes A B::"rhs_item set" shows "L A \<union> L B = L (A \<union> B)"by simplemma 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)qedlemma 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) doneqedlemma rexp_of_empty: assumes finite:"finite rhs" and nonempty:"rhs_nonempty rhs" shows "[] \<notin> L (\<Uplus> {r. Trn X r \<in> rhs})"using finite nonempty rhs_nonempty_defusing finite_Trn[OF finite]by (auto)lemma [intro!]: "P (Trn X r) \<Longrightarrow> (\<exists>a. (\<exists>r. a = Trn X r \<and> P a))" by autolemma lang_of_rexp_of: assumes finite:"finite rhs" shows "L ({Trn X r| r. Trn X r \<in> rhs}) = X ;; (L (\<Uplus>{r. Trn X r \<in> rhs}))"proof - have "finite {r. Trn X r \<in> rhs}" by (rule finite_Trn[OF finite]) then show ?thesis apply(auto simp add: Seq_def) apply(rule_tac x = "s\<^isub>1" in exI, rule_tac x = "s\<^isub>2" in exI, auto) apply(rule_tac x= "Trn X xa" in exI) apply(auto simp: Seq_def) doneqedlemma rexp_of_lam_eq_lam_set: assumes fin: "finite rhs" shows "L (\<Uplus>{r. Lam r \<in> rhs}) = L ({Lam r | r. Lam r \<in> rhs})"proof - have "finite ({r. Lam r \<in> rhs})" using fin by (rule finite_Lam) then show ?thesis by autoqedlemma [simp]: "L (attach_rexp r xb) = L xb ;; L r"apply (cases xb, auto simp: Seq_def)apply(rule_tac x = "s\<^isub>1 @ s\<^isub>1'" in exI, rule_tac x = "s\<^isub>2'" in exI)apply(auto simp: Seq_def)donelemma 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_invariant"} shows that the initial equational system satisfies invariant @{text "invariant"}.*}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 \<Turnstile>(CHAR c)\<Rightarrow> X}" unfolding transition_def 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 transition_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 transition_def)qedlemma init_ES_satisfy_invariant: assumes finite_CS: "finite (UNIV // (\<approx>Lang))" shows "invariant (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 "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:invariant_def)qedsubsubsection {* Interation step *}text {* From this point until @{text "iteration_step"}, it is proved that there exists iteration steps which keep @{text "invariant(ES)"} while decreasing the size of @{text "ES"}.*}lemma arden_op_keeps_eq: assumes l_eq_r: "X = L rhs" and not_empty: "[] \<notin> L (\<Uplus>{r. Trn X r \<in> rhs})" and finite: "finite rhs" shows "X = L (arden_op X rhs)"proof - def A \<equiv> "L (\<Uplus>{r. Trn X r \<in> rhs})" def b \<equiv> "rhs - trns_of rhs X" def B \<equiv> "L b" have "X = B ;; A\<star>" proof- have "L rhs = L(trns_of rhs X \<union> b)" by (auto simp: b_def trns_of_def) also have "\<dots> = X ;; A \<union> B" unfolding trns_of_def unfolding L_rhs_union_distrib[symmetric] by (simp only: lang_of_rexp_of finite B_def A_def) finally show ?thesis using l_eq_r not_empty apply(rule_tac arden[THEN iffD1]) apply(simp add: A_def) apply(simp) done qed moreover have "L (arden_op X rhs) = (B ;; A\<star>)" by (simp only:arden_op_def L_rhs_union_distrib lang_of_append_rhs B_def A_def b_def L_rexp.simps seq_union_distrib_left) 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_op_keeps_finite: "finite rhs \<Longrightarrow> finite (arden_op X rhs)"by (auto simp:arden_op_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_op_keeps_nonempty: "rhs_nonempty rhs \<Longrightarrow> rhs_nonempty (arden_op X rhs)"by (simp only:arden_op_def append_keeps_nonempty nonempty_set_sub)lemma subst_op_keeps_nonempty: "\<lbrakk>rhs_nonempty rhs; rhs_nonempty xrhs\<rbrakk> \<Longrightarrow> rhs_nonempty (subst_op rhs X xrhs)"by (simp only:subst_op_def append_keeps_nonempty nonempty_set_union nonempty_set_sub)lemma subst_op_keeps_eq: assumes substor: "X = L xrhs" and finite: "finite rhs" shows "L (subst_op rhs X xrhs) = L rhs" (is "?Left = ?Right")proof- def A \<equiv> "L (rhs - trns_of rhs X)" have "?Left = A \<union> L (append_rhs_rexp xrhs (\<Uplus>{r. Trn X r \<in> rhs}))" unfolding subst_op_def unfolding L_rhs_union_distrib[symmetric] by (simp add: A_def) moreover have "?Right = A \<union> L ({Trn X r | r. Trn X r \<in> rhs})" proof- have "rhs = (rhs - trns_of rhs X) \<union> (trns_of rhs X)" by (auto simp add: trns_of_def) thus ?thesis unfolding A_def unfolding L_rhs_union_distrib unfolding trns_of_def by simp qed moreover have "L (append_rhs_rexp xrhs (\<Uplus>{r. Trn X r \<in> rhs})) = L ({Trn X r | r. Trn X r \<in> rhs})" using finite substor by (simp only:lang_of_append_rhs lang_of_rexp_of) ultimately show ?thesis by simpqedlemma subst_op_keeps_finite_rhs: "\<lbrakk>finite rhs; finite yrhs\<rbrakk> \<Longrightarrow> finite (subst_op rhs Y yrhs)"by (auto simp:subst_op_def append_keeps_finite)lemma subst_op_all_keeps_finite: assumes finite:"finite (ES:: (string set \<times> rhs_item set) set)" shows "finite (subst_op_all ES Y yrhs)"proof - have "finite {(Ya, subst_op 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, subst_op 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:subst_op_all_def)qedlemma subst_op_all_keeps_finite_rhs: "\<lbrakk>finite_rhs ES; finite yrhs\<rbrakk> \<Longrightarrow> finite_rhs (subst_op_all ES Y yrhs)"by (auto intro:subst_op_keeps_finite_rhs simp add:subst_op_all_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_op_removes_cl: "classes_of (arden_op Y yrhs) = classes_of yrhs - {Y}"apply (simp add:arden_op_def append_rhs_keeps_cls trns_of_def)by (auto simp:classes_of_def)lemma lefts_of_keeps_cls: "lefts_of (subst_op_all ES Y yrhs) = lefts_of ES"by (auto simp:lefts_of_def subst_op_all_def)lemma subst_op_updates_cls: "X \<notin> classes_of xrhs \<Longrightarrow> classes_of (subst_op rhs X xrhs) = classes_of rhs \<union> classes_of xrhs - {X}"apply (simp only:subst_op_def append_rhs_keeps_cls classes_of_union_distrib[THEN sym])by (auto simp:classes_of_def trns_of_def)lemma subst_op_all_keeps_self_contained: fixes Y assumes sc: "self_contained (ES \<union> {(Y, yrhs)})" (is "self_contained ?A") shows "self_contained (subst_op_all ES Y (arden_op Y yrhs))" (is "self_contained ?B")proof- { fix X xrhs' assume "(X, xrhs') \<in> ?B" then obtain xrhs where xrhs_xrhs': "xrhs' = subst_op xrhs Y (arden_op Y yrhs)" and X_in: "(X, xrhs) \<in> ES" by (simp add:subst_op_all_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 subst_op_all_def) moreover have "classes_of xrhs' \<subseteq> lefts_of ES" proof- have "classes_of xrhs' \<subseteq> classes_of xrhs \<union> classes_of (arden_op Y yrhs) - {Y}" proof- have "Y \<notin> classes_of (arden_op Y yrhs)" using arden_op_removes_cl by simp thus ?thesis using xrhs_xrhs' by (auto simp:subst_op_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_op Y yrhs) \<subseteq> lefts_of ES \<union> {Y}" using sc by (auto simp add:arden_op_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:subst_op_all_def self_contained_def)qedlemma subst_op_all_satisfy_invariant: assumes invariant_ES: "invariant (ES \<union> {(Y, yrhs)})" shows "invariant (subst_op_all ES Y (arden_op Y yrhs))"proof - have finite_yrhs: "finite yrhs" using invariant_ES by (auto simp:invariant_def finite_rhs_def) have nonempty_yrhs: "rhs_nonempty yrhs" using invariant_ES by (auto simp:invariant_def ardenable_def) have Y_eq_yrhs: "Y = L yrhs" using invariant_ES by (simp only:invariant_def valid_eqns_def, blast) have "distinct_equas (subst_op_all ES Y (arden_op Y yrhs))" using invariant_ES by (auto simp:distinct_equas_def subst_op_all_def invariant_def) moreover have "finite (subst_op_all ES Y (arden_op Y yrhs))" using invariant_ES by (simp add:invariant_def subst_op_all_keeps_finite) moreover have "finite_rhs (subst_op_all ES Y (arden_op Y yrhs))" proof- have "finite_rhs ES" using invariant_ES by (simp add:invariant_def finite_rhs_def) moreover have "finite (arden_op Y yrhs)" proof - have "finite yrhs" using invariant_ES by (auto simp:invariant_def finite_rhs_def) thus ?thesis using arden_op_keeps_finite by simp qed ultimately show ?thesis by (simp add:subst_op_all_keeps_finite_rhs) qed moreover have "ardenable (subst_op_all ES Y (arden_op Y yrhs))" proof - { fix X rhs assume "(X, rhs) \<in> ES" hence "rhs_nonempty rhs" using prems invariant_ES by (simp add:invariant_def ardenable_def) with nonempty_yrhs have "rhs_nonempty (subst_op rhs Y (arden_op Y yrhs))" by (simp add:nonempty_yrhs subst_op_keeps_nonempty arden_op_keeps_nonempty) } thus ?thesis by (auto simp add:ardenable_def subst_op_all_def) qed moreover have "valid_eqns (subst_op_all ES Y (arden_op Y yrhs))" proof- have "Y = L (arden_op Y yrhs)" using Y_eq_yrhs invariant_ES finite_yrhs nonempty_yrhs by (rule_tac arden_op_keeps_eq, (simp add:rexp_of_empty)+) thus ?thesis using invariant_ES by (clarsimp simp add:valid_eqns_def subst_op_all_def subst_op_keeps_eq invariant_def finite_rhs_def simp del:L_rhs.simps) qed moreover have self_subst: "self_contained (subst_op_all ES Y (arden_op Y yrhs))" using invariant_ES subst_op_all_keeps_self_contained by (simp add:invariant_def) ultimately show ?thesis using invariant_ES by (simp add:invariant_def)qedlemma subst_op_all_card_le: assumes finite: "finite (ES::(string set \<times> rhs_item set) set)" shows "card (subst_op_all ES Y yrhs) <= card ES"proof- def f \<equiv> "\<lambda> x. ((fst x)::string set, subst_op (snd x) Y yrhs)" have "subst_op_all ES Y yrhs = f ` ES" apply (auto simp:subst_op_all_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 subst_op_all_cls_remains: "(X, xrhs) \<in> ES \<Longrightarrow> \<exists> xrhs'. (X, xrhs') \<in> (subst_op_all ES Y yrhs)"by (auto simp:subst_op_all_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 invariant_ES: "invariant ES" and X_in_ES: "(X, xrhs) \<in> ES" and not_T: "card ES \<noteq> 1" shows "\<exists> ES'. (invariant 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 invariant_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 not_T X_in_ES by (drule_tac card_noteq_1_has_more, auto) def ES' == "ES - {(Y, yrhs)}" let ?ES'' = "subst_op_all ES' Y (arden_op Y yrhs)" have "?P ?ES''" proof - have "invariant ?ES''" using Y_in_ES invariant_ES by (rule_tac subst_op_all_satisfy_invariant, 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 subst_op_all_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:subst_op_all_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: "invariant ES" and X_in_ES: "\<exists> xrhs. (X, xrhs) \<in> ES" shows "\<exists> ES'. (invariant 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 invariant_ES: "invariant ES" shows "\<exists> (r::rexp). L r = X" (is "\<exists> r. ?P r")proof- def A \<equiv> "arden_op X xrhs" have "?P (\<Uplus>{r. Lam r \<in> A})" proof - have "L (\<Uplus>{r. Lam r \<in> A}) = L ({Lam r | r. Lam r \<in> A})" proof(rule rexp_of_lam_eq_lam_set) show "finite A" unfolding A_def using invariant_ES ES_single by (rule_tac arden_op_keeps_finite) (auto simp add: invariant_def finite_rhs_def) qed also have "\<dots> = L A" proof- have "{Lam r | r. Lam r \<in> A} = A" proof- have "classes_of A = {}" using invariant_ES ES_single unfolding A_def by (simp add:arden_op_removes_cl self_contained_def invariant_def lefts_of_def) thus ?thesis unfolding A_def by (auto simp only: classes_of_def, case_tac x, auto) qed thus ?thesis by simp qed also have "\<dots> = X" unfolding A_def proof(rule arden_op_keeps_eq [THEN sym]) show "X = L xrhs" using invariant_ES ES_single by (auto simp only:invariant_def valid_eqns_def) next from invariant_ES ES_single show "[] \<notin> L (\<Uplus>{r. Trn X r \<in> xrhs})" by(simp add:invariant_def ardenable_def rexp_of_empty finite_rhs_def) next from invariant_ES ES_single show "finite xrhs" by (simp add:invariant_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 invariant_ES: "invariant ES" and X_in_ES: "(X, xrhs) \<in> ES" and card_ES: "card ES = 1" using finite_CS X_in_CS init_ES_satisfy_invariant iteration_conc by blast hence ES_single_equa: "ES = {(X, xrhs)}" by (auto simp:invariant_def dest!:card_Suc_Diff1 simp:card_eq_0_iff) thus ?thesis using invariant_ES by (rule last_cl_exists_rexp)qedtheorem hard_direction: assumes finite_CS: "finite (UNIV // \<approx>A)" shows "\<exists>r::rexp. A = L r"proof - have "\<forall> X \<in> (UNIV // \<approx>A). \<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>A). X = L ((f X)::rexp)" by (auto dest: bchoice) def rs \<equiv> "f ` (finals A)" have "A = \<Union> (finals A)" using lang_is_union_of_finals by auto also have "\<dots> = L (\<Uplus>rs)" proof - have "finite rs" proof - have "finite (finals A)" using finite_CS finals_in_partitions[of "A"] 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 "A"] by auto qed finally show ?thesis by blastqed end