theory Myhill_1imports Main Folds "~~/src/HOL/Library/While_Combinator"beginsection {* 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 \<equiv> 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)"unfolding folds_defapply(rule set_eqI)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 `` {s} | s . s \<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_def quotient_defby autosection {* Equational systems *}text {* The two kinds of terms in the rhs of equations. *}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)"endlemma L_rhs_union_distrib: fixes A B::"rhs_item set" shows "L A \<union> L B = L (A \<union> B)"by simptext {* Transitions between equivalence classes *}definition transition :: "lang \<Rightarrow> char \<Rightarrow> lang \<Rightarrow> bool" ("_ \<Turnstile>_\<Rightarrow>_" [100,100,100] 100)where "Y \<Turnstile>c\<Rightarrow> X \<equiv> Y ;; {[c]} \<subseteq> X"text {* Initial equational system *}definition "Init_rhs CS X \<equiv> if ([] \<in> X) then {Lam EMPTY} \<union> {Trn Y (CHAR c) | Y c. Y \<in> CS \<and> Y \<Turnstile>c\<Rightarrow> X} else {Trn Y (CHAR c)| Y c. Y \<in> CS \<and> Y \<Turnstile>c\<Rightarrow> X}"definition "Init CS \<equiv> {(X, Init_rhs CS X) | X. X \<in> CS}"section {* Arden Operation on equations *}text {* The function @{text "attach_rexp r item"} SEQ-composes @{text r} to the right of every rhs-item.*}fun append_rexp :: "rexp \<Rightarrow> rhs_item \<Rightarrow> rhs_item"where "append_rexp r (Lam rexp) = Lam (SEQ rexp r)"| "append_rexp r (Trn X rexp) = Trn X (SEQ rexp r)"definition "append_rhs_rexp rhs rexp \<equiv> (append_rexp rexp) ` rhs"definition "Arden X rhs \<equiv> append_rhs_rexp (rhs - {Trn X r | r. Trn X r \<in> rhs}) (STAR (\<Uplus> {r. Trn X r \<in> rhs}))"section {* Substitution Operation on equations *}text {* Suppose and equation @{text "X = xrhs"}, @{text "Subst"} substitutes all occurences of @{text "X"} in @{text "rhs"} by @{text "xrhs"}.*}definition "Subst rhs X xrhs \<equiv> (rhs - {Trn X r | r. Trn X r \<in> rhs}) \<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}.*}types esystem = "(lang \<times> rhs_item set) set"definition Subst_all :: "esystem \<Rightarrow> lang \<Rightarrow> rhs_item set \<Rightarrow> esystem"where "Subst_all ES X xrhs \<equiv> {(Y, Subst yrhs X xrhs) | Y yrhs. (Y, yrhs) \<in> ES}"text {* The following term @{text "remove ES Y yrhs"} removes the equation @{text "Y = yrhs"} from equational system @{text "ES"} by replacing all occurences of @{text "Y"} by its definition (using @{text "eqs_subst"}). The @{text "Y"}-definition is made non-recursive using Arden's transformation @{text "arden_variate Y yrhs"}. *}definition "Remove ES X xrhs \<equiv> Subst_all (ES - {(X, xrhs)}) X (Arden X xrhs)"section {* While-combinator *}text {* The following term @{text "Iter X ES"} represents one iteration in the while loop. It arbitrarily chooses a @{text "Y"} different from @{text "X"} to remove.*}definition "Iter X ES \<equiv> (let (Y, yrhs) = SOME (Y, yrhs). (Y, yrhs) \<in> ES \<and> X \<noteq> Y in Remove ES Y yrhs)"lemma IterI2: assumes "(Y, yrhs) \<in> ES" and "X \<noteq> Y" and "\<And>Y yrhs. \<lbrakk>(Y, yrhs) \<in> ES; X \<noteq> Y\<rbrakk> \<Longrightarrow> Q (Remove ES Y yrhs)" shows "Q (Iter X ES)"unfolding Iter_def using assmsby (rule_tac a="(Y, yrhs)" in someI2) (auto)text {* The following term @{text "Reduce X ES"} repeatedly removes characteriztion equations for unknowns other than @{text "X"} until one is left.*}abbreviation "Cond ES \<equiv> card ES \<noteq> 1"definition "Solve X ES \<equiv> while Cond (Iter X) ES"text {* Since the @{text "while"} combinator from HOL library is used to implement @{text "Solve X ES"}, the induction principle @{thm [source] while_rule} is used to proved the desired properties of @{text "Solve X ES"}. For this purpose, an invariant predicate @{text "invariant"} is defined in terms of a series of auxilliary predicates:*}section {* Invariants *}text {* Every variable is defined at most once 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 "sound_eqs ES \<equiv> \<forall>(X, rhs) \<in> ES. X = L rhs"text {* @{text "ardenable 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 "ardenable rhs \<equiv> (\<forall> Y r. Trn Y r \<in> rhs \<longrightarrow> [] \<notin> L r)"text {* The following @{text "ardenable_all ES"} requires that Arden's transformation is applicable to every equation of equational system @{text "ES"}.*}definition "ardenable_all ES \<equiv> \<forall>(X, rhs) \<in> ES. ardenable rhs"text {* @{text "finite_rhs ES"} requires every equation in @{text "rhs"} be finite.*}definition "finite_rhs ES \<equiv> \<forall>(X, rhs) \<in> ES. finite rhs"lemma finite_rhs_def2: "finite_rhs ES = (\<forall> X rhs. (X, rhs) \<in> ES \<longrightarrow> finite rhs)"unfolding finite_rhs_def by autotext {* @{text "classes_of rhs"} returns all variables (or equivalent classes) occuring in @{text "rhs"}. *}definition "rhss rhs \<equiv> {X | X r. Trn X r \<in> rhs}"text {* @{text "lefts_of ES"} returns all variables defined by an equational system @{text "ES"}.*}definition "lhss ES \<equiv> {Y | Y yrhs. (Y, yrhs) \<in> ES}"text {* The following @{text "valid_eqs ES"} requires that every variable occuring on the right hand side of equations is already defined by some equation in @{text "ES"}.*}definition "valid_eqs ES \<equiv> \<forall>(X, rhs) \<in> ES. rhss rhs \<subseteq> lhss ES"text {* The invariant @{text "invariant(ES)"} is a conjunction of all the previously defined constaints. *}definition "invariant ES \<equiv> finite ES \<and> finite_rhs ES \<and> sound_eqs ES \<and> distinct_equas ES \<and> ardenable_all ES \<and> valid_eqs ES"lemma invariantI: assumes "sound_eqs ES" "finite ES" "distinct_equas ES" "ardenable_all ES" "finite_rhs ES" "valid_eqs ES" shows "invariant ES"using assms by (simp add: invariant_def)subsection {* The proof of this direction *}subsubsection {* Basic properties *}text {* The following are some basic properties of the above definitions.*}lemma finite_Trn: assumes fin: "finite rhs" shows "finite {r. Trn Y r \<in> rhs}"proof - have "finite {Trn Y r | Y r. Trn Y r \<in> rhs}" by (rule rev_finite_subset[OF fin]) (auto) then have "finite ((\<lambda>(Y, r). Trn Y r) ` {(Y, r) | Y r. Trn Y r \<in> rhs})" by (simp add: image_Collect) then have "finite {(Y, r) | Y r. Trn Y r \<in> rhs}" by (erule_tac finite_imageD) (simp add: inj_on_def) then show "finite {r. Trn Y r \<in> rhs}" by (erule_tac f="snd" in finite_surj) (auto simp add: image_def)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: "ardenable rhs" shows "[] \<notin> L (\<Uplus> {r. Trn X r \<in> rhs})"using finite nonempty ardenable_defusing finite_Trn[OF finite]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) apply(auto) apply(rule_tac x= "Trn X xa" in exI) apply(auto simp add: Seq_def) doneqedlemma lang_of_append: "L (append_rexp r rhs_item) = L rhs_item ;; L r"by (induct rule: append_rexp.induct) (auto simp add: seq_assoc)lemma lang_of_append_rhs: "L (append_rhs_rexp rhs r) = L rhs ;; L r"unfolding append_rhs_rexp_defby (auto simp add: Seq_def lang_of_append)lemma rhss_union_distrib: shows "rhss (A \<union> B) = rhss A \<union> rhss B"by (auto simp add: rhss_def)lemma lhss_union_distrib: shows "lhss (A \<union> B) = lhss A \<union> lhss B"by (auto simp add: lhss_def)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: assumes "s \<in> X" "X \<in> UNIV // \<approx>A" shows "X = \<approx>A `` {s}"using assmsunfolding quotient_def Image_def str_eq_rel_defby autolemma every_eqclass_has_transition: assumes has_str: "s @ [c] \<in> X" and in_CS: "X \<in> UNIV // \<approx>A" obtains Y where "Y \<in> UNIV // \<approx>A" and "Y ;; {[c]} \<subseteq> X" and "s \<in> Y"proof - def Y \<equiv> "\<approx>A `` {s}" have "Y \<in> UNIV // \<approx>A" unfolding Y_def quotient_def by auto moreover have "X = \<approx>A `` {s @ [c]}" using has_str in_CS defined_by_str by blast then have "Y ;; {[c]} \<subseteq> X" unfolding Y_def Image_def Seq_def unfolding str_eq_rel_def by clarsimp moreover have "s \<in> Y" unfolding Y_def unfolding Image_def str_eq_rel_def by simp ultimately show thesis using that by blastqedlemma l_eq_r_in_eqs: assumes X_in_eqs: "(X, rhs) \<in> Init (UNIV // \<approx>A)" shows "X = L rhs"proof show "X \<subseteq> L rhs" proof fix x assume "(1)": "x \<in> X" show "x \<in> L rhs" proof (cases "x = []") assume empty: "x = []" thus ?thesis using X_in_eqs "(1)" by (auto simp: Init_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>A" using X_in_eqs by (auto simp:Init_def) then obtain Y where "Y \<in> UNIV // \<approx>A" 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>A \<and> Y \<Turnstile>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: Init_def Init_rhs_def) qed qednext show "L rhs \<subseteq> X" using X_in_eqs by (auto simp:Init_def Init_rhs_def transition_def) qedlemma test: assumes X_in_eqs: "(X, rhs) \<in> Init (UNIV // \<approx>A)" shows "X = \<Union> (L ` rhs)"using assmsby (drule_tac l_eq_r_in_eqs) (simp)lemma finite_Init_rhs: assumes finite: "finite CS" shows "finite (Init_rhs CS X)"proof- def S \<equiv> "{(Y, c)| Y c. Y \<in> CS \<and> Y ;; {[c]} \<subseteq> X}" def h \<equiv> "\<lambda> (Y, c). Trn Y (CHAR c)" have "finite (CS \<times> (UNIV::char set))" using finite by auto then have "finite S" using S_def by (rule_tac B = "CS \<times> UNIV" in finite_subset) (auto) moreover have "{Trn Y (CHAR c) |Y c. Y \<in> CS \<and> Y ;; {[c]} \<subseteq> X} = h ` S" unfolding S_def h_def image_def by auto ultimately have "finite {Trn Y (CHAR c) |Y c. Y \<in> CS \<and> Y ;; {[c]} \<subseteq> X}" by auto then show "finite (Init_rhs CS X)" unfolding Init_rhs_def transition_def by simpqedlemma Init_ES_satisfies_invariant: assumes finite_CS: "finite (UNIV // \<approx>A)" shows "invariant (Init (UNIV // \<approx>A))"proof (rule invariantI) show "sound_eqs (Init (UNIV // \<approx>A))" unfolding sound_eqs_def using l_eq_r_in_eqs by auto show "finite (Init (UNIV // \<approx>A))" using finite_CS unfolding Init_def by simp show "distinct_equas (Init (UNIV // \<approx>A))" unfolding distinct_equas_def Init_def by simp show "ardenable_all (Init (UNIV // \<approx>A))" unfolding ardenable_all_def Init_def Init_rhs_def ardenable_def by auto show "finite_rhs (Init (UNIV // \<approx>A))" using finite_Init_rhs[OF finite_CS] unfolding finite_rhs_def Init_def by auto show "valid_eqs (Init (UNIV // \<approx>A))" unfolding valid_eqs_def Init_def Init_rhs_def rhss_def lhss_def by autoqedsubsubsection {* Interation step *}text {* From this point until @{text "iteration_step"}, the correctness of the iteration step @{text "Iter X ES"} is proved.*}lemma Arden_keeps_eq: assumes l_eq_r: "X = L rhs" and not_empty: "ardenable rhs" and finite: "finite rhs" shows "X = L (Arden X rhs)"proof - def A \<equiv> "L (\<Uplus>{r. Trn X r \<in> rhs})" def b \<equiv> "rhs - {Trn X r | r. Trn X r \<in> rhs}" def B \<equiv> "L b" have "X = B ;; A\<star>" proof - have "L rhs = L({Trn X r | r. Trn X r \<in> rhs} \<union> b)" by (auto simp: b_def) also have "\<dots> = X ;; A \<union> B" unfolding L_rhs_union_distrib[symmetric] by (simp only: lang_of_rexp_of finite B_def A_def) finally show ?thesis apply(rule_tac arden[THEN iffD1]) apply(simp (no_asm) add: A_def) using finite_Trn[OF finite] not_empty apply(simp add: ardenable_def) using l_eq_r apply(simp) done qed moreover have "L (Arden X rhs) = B ;; A\<star>" by (simp only:Arden_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_keeps_finite: "finite rhs \<Longrightarrow> finite (Arden X rhs)"by (auto simp:Arden_def append_keeps_finite)lemma append_keeps_nonempty: "ardenable rhs \<Longrightarrow> ardenable (append_rhs_rexp rhs r)"apply (auto simp:ardenable_def append_rhs_rexp_def)by (case_tac x, auto simp:Seq_def)lemma nonempty_set_sub: "ardenable rhs \<Longrightarrow> ardenable (rhs - A)"by (auto simp:ardenable_def)lemma nonempty_set_union: "\<lbrakk>ardenable rhs; ardenable rhs'\<rbrakk> \<Longrightarrow> ardenable (rhs \<union> rhs')"by (auto simp:ardenable_def)lemma Arden_keeps_nonempty: "ardenable rhs \<Longrightarrow> ardenable (Arden X rhs)"by (simp only:Arden_def append_keeps_nonempty nonempty_set_sub)lemma Subst_keeps_nonempty: "\<lbrakk>ardenable rhs; ardenable xrhs\<rbrakk> \<Longrightarrow> ardenable (Subst rhs X xrhs)"by (simp only:Subst_def append_keeps_nonempty nonempty_set_union nonempty_set_sub)lemma Subst_keeps_eq: assumes substor: "X = L xrhs" and finite: "finite rhs" shows "L (Subst rhs X xrhs) = L rhs" (is "?Left = ?Right")proof- def A \<equiv> "L (rhs - {Trn X r | r. Trn X r \<in> rhs})" have "?Left = A \<union> L (append_rhs_rexp xrhs (\<Uplus>{r. Trn X r \<in> rhs}))" unfolding Subst_def unfolding L_rhs_union_distrib[symmetric] by (simp add: A_def) moreover have "?Right = A \<union> L ({Trn X r | r. Trn X r \<in> rhs})" proof- have "rhs = (rhs - {Trn X r | r. Trn X r \<in> rhs}) \<union> ({Trn X r | r. Trn X r \<in> rhs})" by auto thus ?thesis unfolding A_def unfolding L_rhs_union_distrib by simp qed moreover have "L (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_keeps_finite_rhs: "\<lbrakk>finite rhs; finite yrhs\<rbrakk> \<Longrightarrow> finite (Subst rhs Y yrhs)"by (auto simp:Subst_def append_keeps_finite)lemma Subst_all_keeps_finite: assumes finite: "finite ES" shows "finite (Subst_all ES Y yrhs)"proof - def eqns \<equiv> "{(X::lang, rhs) |X rhs. (X, rhs) \<in> ES}" def h \<equiv> "\<lambda>(X::lang, rhs). (X, Subst rhs Y yrhs)" have "finite (h ` eqns)" using finite h_def eqns_def by auto moreover have "Subst_all ES Y yrhs = h ` eqns" unfolding h_def eqns_def Subst_all_def by auto ultimately show "finite (Subst_all ES Y yrhs)" by simpqedlemma Subst_all_keeps_finite_rhs: "\<lbrakk>finite_rhs ES; finite yrhs\<rbrakk> \<Longrightarrow> finite_rhs (Subst_all ES Y yrhs)"by (auto intro:Subst_keeps_finite_rhs simp add:Subst_all_def finite_rhs_def)lemma append_rhs_keeps_cls: "rhss (append_rhs_rexp rhs r) = rhss rhs"apply (auto simp:rhss_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_removes_cl: "rhss (Arden Y yrhs) = rhss yrhs - {Y}"apply (simp add:Arden_def append_rhs_keeps_cls)by (auto simp:rhss_def)lemma lhss_keeps_cls: "lhss (Subst_all ES Y yrhs) = lhss ES"by (auto simp:lhss_def Subst_all_def)lemma Subst_updates_cls: "X \<notin> rhss xrhs \<Longrightarrow> rhss (Subst rhs X xrhs) = rhss rhs \<union> rhss xrhs - {X}"apply (simp only:Subst_def append_rhs_keeps_cls rhss_union_distrib)by (auto simp:rhss_def)lemma Subst_all_keeps_valid_eqs: assumes sc: "valid_eqs (ES \<union> {(Y, yrhs)})" (is "valid_eqs ?A") shows "valid_eqs (Subst_all ES Y (Arden Y yrhs))" (is "valid_eqs ?B")proof - { fix X xrhs' assume "(X, xrhs') \<in> ?B" then obtain xrhs where xrhs_xrhs': "xrhs' = Subst xrhs Y (Arden Y yrhs)" and X_in: "(X, xrhs) \<in> ES" by (simp add:Subst_all_def, blast) have "rhss xrhs' \<subseteq> lhss ?B" proof- have "lhss ?B = lhss ES" by (auto simp add:lhss_def Subst_all_def) moreover have "rhss xrhs' \<subseteq> lhss ES" proof- have "rhss xrhs' \<subseteq> rhss xrhs \<union> rhss (Arden Y yrhs) - {Y}" proof- have "Y \<notin> rhss (Arden Y yrhs)" using Arden_removes_cl by simp thus ?thesis using xrhs_xrhs' by (auto simp:Subst_updates_cls) qed moreover have "rhss xrhs \<subseteq> lhss ES \<union> {Y}" using X_in sc apply (simp only:valid_eqs_def lhss_union_distrib) by (drule_tac x = "(X, xrhs)" in bspec, auto simp:lhss_def) moreover have "rhss (Arden Y yrhs) \<subseteq> lhss ES \<union> {Y}" using sc by (auto simp add:Arden_removes_cl valid_eqs_def lhss_def) ultimately show ?thesis by auto qed ultimately show ?thesis by simp qed } thus ?thesis by (auto simp only:Subst_all_def valid_eqs_def)qedlemma Subst_all_satisfies_invariant: assumes invariant_ES: "invariant (ES \<union> {(Y, yrhs)})" shows "invariant (Subst_all ES Y (Arden Y yrhs))"proof (rule invariantI) have Y_eq_yrhs: "Y = L yrhs" using invariant_ES by (simp only:invariant_def sound_eqs_def, blast) have finite_yrhs: "finite yrhs" using invariant_ES by (auto simp:invariant_def finite_rhs_def) have nonempty_yrhs: "ardenable yrhs" using invariant_ES by (auto simp:invariant_def ardenable_all_def) show "sound_eqs (Subst_all ES Y (Arden Y yrhs))" proof - have "Y = L (Arden Y yrhs)" using Y_eq_yrhs invariant_ES finite_yrhs using finite_Trn[OF finite_yrhs] apply(rule_tac Arden_keeps_eq) apply(simp_all) unfolding invariant_def ardenable_all_def ardenable_def apply(auto) done thus ?thesis using invariant_ES unfolding invariant_def finite_rhs_def2 sound_eqs_def Subst_all_def by (auto simp add: Subst_keeps_eq simp del: L_rhs.simps) qed show "finite (Subst_all ES Y (Arden Y yrhs))" using invariant_ES by (simp add:invariant_def Subst_all_keeps_finite) show "distinct_equas (Subst_all ES Y (Arden Y yrhs))" using invariant_ES unfolding distinct_equas_def Subst_all_def invariant_def by auto show "ardenable_all (Subst_all ES Y (Arden Y yrhs))" proof - { fix X rhs assume "(X, rhs) \<in> ES" hence "ardenable rhs" using prems invariant_ES by (auto simp add:invariant_def ardenable_all_def) with nonempty_yrhs have "ardenable (Subst rhs Y (Arden Y yrhs))" by (simp add:nonempty_yrhs Subst_keeps_nonempty Arden_keeps_nonempty) } thus ?thesis by (auto simp add:ardenable_all_def Subst_all_def) qed show "finite_rhs (Subst_all ES Y (Arden Y yrhs))" proof- have "finite_rhs ES" using invariant_ES by (simp add:invariant_def finite_rhs_def) moreover have "finite (Arden Y yrhs)" proof - have "finite yrhs" using invariant_ES by (auto simp:invariant_def finite_rhs_def) thus ?thesis using Arden_keeps_finite by simp qed ultimately show ?thesis by (simp add:Subst_all_keeps_finite_rhs) qed show "valid_eqs (Subst_all ES Y (Arden Y yrhs))" using invariant_ES Subst_all_keeps_valid_eqs by (simp add:invariant_def)qedlemma Remove_in_card_measure: assumes finite: "finite ES" and in_ES: "(X, rhs) \<in> ES" shows "(Remove ES X rhs, ES) \<in> measure card"proof - def f \<equiv> "\<lambda> x. ((fst x)::lang, Subst (snd x) X (Arden X rhs))" def ES' \<equiv> "ES - {(X, rhs)}" have "Subst_all ES' X (Arden X rhs) = f ` ES'" apply (auto simp: Subst_all_def f_def image_def) by (rule_tac x = "(Y, yrhs)" in bexI, simp+) then have "card (Subst_all ES' X (Arden X rhs)) \<le> card ES'" unfolding ES'_def using finite by (auto intro: card_image_le) also have "\<dots> < card ES" unfolding ES'_def using in_ES finite by (rule_tac card_Diff1_less) finally show "(Remove ES X rhs, ES) \<in> measure card" unfolding Remove_def ES'_def by simpqedlemma Subst_all_cls_remains: "(X, xrhs) \<in> ES \<Longrightarrow> \<exists> xrhs'. (X, xrhs') \<in> (Subst_all ES Y yrhs)"by (auto simp: Subst_all_def)lemma card_noteq_1_has_more: assumes card:"Cond ES" and e_in: "(X, xrhs) \<in> ES" and finite: "finite ES" shows "\<exists>(Y, yrhs) \<in> ES. (X, xrhs) \<noteq> (Y, yrhs)"proof- have "card ES > 1" using card e_in finite by (cases "card ES") (auto) then have "card (ES - {(X, xrhs)}) > 0" using finite e_in by auto then have "(ES - {(X, xrhs)}) \<noteq> {}" using finite by (rule_tac notI, simp) then show "\<exists>(Y, yrhs) \<in> ES. (X, xrhs) \<noteq> (Y, yrhs)" by autoqedlemma iteration_step_measure: assumes Inv_ES: "invariant ES" and X_in_ES: "(X, xrhs) \<in> ES" and Cnd: "Cond ES " shows "(Iter X ES, ES) \<in> measure card"proof - have fin: "finite ES" using Inv_ES unfolding invariant_def by simp then obtain Y yrhs where Y_in_ES: "(Y, yrhs) \<in> ES" and not_eq: "(X, xrhs) \<noteq> (Y, yrhs)" using Cnd X_in_ES by (drule_tac card_noteq_1_has_more) (auto) then have "(Y, yrhs) \<in> ES " "X \<noteq> Y" using X_in_ES Inv_ES unfolding invariant_def distinct_equas_def by auto then show "(Iter X ES, ES) \<in> measure card" apply(rule IterI2) apply(rule Remove_in_card_measure) apply(simp_all add: fin) doneqedlemma iteration_step_invariant: assumes Inv_ES: "invariant ES" and X_in_ES: "(X, xrhs) \<in> ES" and Cnd: "Cond ES" shows "invariant (Iter X ES)"proof - have finite_ES: "finite ES" using Inv_ES by (simp add: invariant_def) then obtain Y yrhs where Y_in_ES: "(Y, yrhs) \<in> ES" and not_eq: "(X, xrhs) \<noteq> (Y, yrhs)" using Cnd X_in_ES by (drule_tac card_noteq_1_has_more) (auto) then have "(Y, yrhs) \<in> ES" "X \<noteq> Y" using X_in_ES Inv_ES unfolding invariant_def distinct_equas_def by auto then show "invariant (Iter X ES)" proof(rule IterI2) fix Y yrhs assume h: "(Y, yrhs) \<in> ES" "X \<noteq> Y" then have "ES - {(Y, yrhs)} \<union> {(Y, yrhs)} = ES" by auto then show "invariant (Remove ES Y yrhs)" unfolding Remove_def using Inv_ES thm Subst_all_satisfies_invariant by (rule_tac Subst_all_satisfies_invariant) (simp) qedqedlemma iteration_step_ex: assumes Inv_ES: "invariant ES" and X_in_ES: "(X, xrhs) \<in> ES" and Cnd: "Cond ES" shows "\<exists>xrhs'. (X, xrhs') \<in> (Iter X ES)"proof - have finite_ES: "finite ES" using Inv_ES by (simp add: invariant_def) then obtain Y yrhs where Y_in_ES: "(Y, yrhs) \<in> ES" and not_eq: "(X, xrhs) \<noteq> (Y, yrhs)" using Cnd X_in_ES by (drule_tac card_noteq_1_has_more) (auto) then have "(Y, yrhs) \<in> ES " "X \<noteq> Y" using X_in_ES Inv_ES unfolding invariant_def distinct_equas_def by auto then show "\<exists>xrhs'. (X, xrhs') \<in> (Iter X ES)" apply(rule IterI2) unfolding Remove_def apply(rule Subst_all_cls_remains) using X_in_ES apply(auto) doneqedsubsubsection {* Conclusion of the proof *}lemma Solve: assumes fin: "finite (UNIV // \<approx>A)" and X_in: "X \<in> (UNIV // \<approx>A)" shows "\<exists>rhs. Solve X (Init (UNIV // \<approx>A)) = {(X, rhs)} \<and> invariant {(X, rhs)}"proof - def Inv \<equiv> "\<lambda>ES. invariant ES \<and> (\<exists>rhs. (X, rhs) \<in> ES)" have "Inv (Init (UNIV // \<approx>A))" unfolding Inv_def using fin X_in by (simp add: Init_ES_satisfies_invariant, simp add: Init_def) moreover { fix ES assume inv: "Inv ES" and crd: "Cond ES" then have "Inv (Iter X ES)" unfolding Inv_def by (auto simp add: iteration_step_invariant iteration_step_ex) } moreover { fix ES assume inv: "Inv ES" and not_crd: "\<not>Cond ES" from inv obtain rhs where "(X, rhs) \<in> ES" unfolding Inv_def by auto moreover from not_crd have "card ES = 1" by simp ultimately have "ES = {(X, rhs)}" by (auto simp add: card_Suc_eq) then have "\<exists>rhs'. ES = {(X, rhs')} \<and> invariant {(X, rhs')}" using inv unfolding Inv_def by auto } moreover have "wf (measure card)" by simp moreover { fix ES assume inv: "Inv ES" and crd: "Cond ES" then have "(Iter X ES, ES) \<in> measure card" unfolding Inv_def apply(clarify) apply(rule_tac iteration_step_measure) apply(auto) done } ultimately show "\<exists>rhs. Solve X (Init (UNIV // \<approx>A)) = {(X, rhs)} \<and> invariant {(X, rhs)}" unfolding Solve_def by (rule while_rule)qedlemma every_eqcl_has_reg: assumes finite_CS: "finite (UNIV // \<approx>A)" and X_in_CS: "X \<in> (UNIV // \<approx>A)" shows "\<exists>r::rexp. X = L r" proof - from finite_CS X_in_CS obtain xrhs where Inv_ES: "invariant {(X, xrhs)}" using Solve by metis def A \<equiv> "Arden X xrhs" have "rhss xrhs \<subseteq> {X}" using Inv_ES unfolding valid_eqs_def invariant_def rhss_def lhss_def by auto then have "rhss A = {}" unfolding A_def by (simp add: Arden_removes_cl) then have eq: "{Lam r | r. Lam r \<in> A} = A" unfolding rhss_def by (auto, case_tac x, auto) have "finite A" using Inv_ES unfolding A_def invariant_def finite_rhs_def using Arden_keeps_finite by auto then have fin: "finite {r. Lam r \<in> A}" by (rule finite_Lam) have "X = L xrhs" using Inv_ES unfolding invariant_def sound_eqs_def by simp then have "X = L A" using Inv_ES unfolding A_def invariant_def ardenable_all_def finite_rhs_def by (rule_tac Arden_keeps_eq) (simp_all add: finite_Trn) then have "X = L {Lam r | r. Lam r \<in> A}" using eq by simp then have "X = L (\<Uplus>{r. Lam r \<in> A})" using fin by auto then show "\<exists>r::rexp. X = L r" by blastqedlemma bchoice_finite_set: assumes a: "\<forall>x \<in> S. \<exists>y. x = f y" and b: "finite S" shows "\<exists>ys. (\<Union> S) = \<Union>(f ` ys) \<and> finite ys"using bchoice[OF a] bapply(erule_tac exE)apply(rule_tac x="fa ` S" in exI)apply(auto)donetheorem Myhill_Nerode1: assumes finite_CS: "finite (UNIV // \<approx>A)" shows "\<exists>r::rexp. A = L r"proof - have fin: "finite (finals A)" using finals_in_partitions finite_CS by (rule finite_subset) have "\<forall>X \<in> (UNIV // \<approx>A). \<exists>r::rexp. X = L r" using finite_CS every_eqcl_has_reg by blast then have a: "\<forall>X \<in> finals A. \<exists>r::rexp. X = L r" using finals_in_partitions by auto then obtain rs::"rexp set" where "\<Union> (finals A) = \<Union>(L ` rs)" "finite rs" using fin by (auto dest: bchoice_finite_set) then have "A = L (\<Uplus>rs)" unfolding lang_is_union_of_finals[symmetric] by simp then show "\<exists>r::rexp. A = L r" by blastqed end