made the theories compatible with the existing developments in the AFP; old theories are in the directory Attic
(* Author: Tobias Nipkow, Alex Krauss *)header "Regular sets"theory Regular_Setimports Mainbegintype_synonym 'a lang = "'a list set"definition conc :: "'a lang \<Rightarrow> 'a lang \<Rightarrow> 'a lang" (infixr "@@" 75) where"A @@ B = {xs@ys | xs ys. xs:A & ys:B}"overloading lang_pow == "compow :: nat \<Rightarrow> 'a lang \<Rightarrow> 'a lang"begin primrec lang_pow :: "nat \<Rightarrow> 'a lang \<Rightarrow> 'a lang" where "lang_pow 0 A = {[]}" | "lang_pow (Suc n) A = A @@ (lang_pow n A)"enddefinition star :: "'a lang \<Rightarrow> 'a lang" where"star A = (\<Union>n. A ^^ n)"definition deriv :: "'a \<Rightarrow> 'a lang \<Rightarrow> 'a lang"where "deriv x L = { xs. x#xs \<in> L }"coinductive bisimilar :: "'a list set \<Rightarrow> 'a list set \<Rightarrow> bool" where"([] \<in> K \<longleftrightarrow> [] \<in> L) \<Longrightarrow> (\<And>x. bisimilar (deriv x K) (deriv x L)) \<Longrightarrow> bisimilar K L"subsection{* @{term "op @@"} *}lemma concI[simp,intro]: "u : A \<Longrightarrow> v : B \<Longrightarrow> u@v : A @@ B"by (auto simp add: conc_def)lemma concE[elim]: assumes "w \<in> A @@ B"obtains u v where "u \<in> A" "v \<in> B" "w = u@v"using assms by (auto simp: conc_def)lemma conc_mono: "A \<subseteq> C \<Longrightarrow> B \<subseteq> D \<Longrightarrow> A @@ B \<subseteq> C @@ D"by (auto simp: conc_def) lemma conc_empty[simp]: shows "{} @@ A = {}" and "A @@ {} = {}"by autolemma conc_epsilon[simp]: shows "{[]} @@ A = A" and "A @@ {[]} = A"by (simp_all add:conc_def)lemma conc_assoc: "(A @@ B) @@ C = A @@ (B @@ C)"by (auto elim!: concE) (simp only: append_assoc[symmetric] concI)lemma conc_Un_distrib:shows "A @@ (B \<union> C) = A @@ B \<union> A @@ C"and "(A \<union> B) @@ C = A @@ C \<union> B @@ C"by autolemma conc_UNION_distrib:shows "A @@ UNION I M = UNION I (%i. A @@ M i)"and "UNION I M @@ A = UNION I (%i. M i @@ A)"by autosubsection{* @{term "A ^^ n"} *}lemma lang_pow_add: "A ^^ (n + m) = A ^^ n @@ A ^^ m"by (induct n) (auto simp: conc_assoc)lemma lang_pow_empty: "{} ^^ n = (if n = 0 then {[]} else {})"by (induct n) autolemma lang_pow_empty_Suc[simp]: "({}::'a lang) ^^ Suc n = {}"by (simp add: lang_pow_empty)lemma length_lang_pow_ub: "ALL w : A. length w \<le> k \<Longrightarrow> w : A^^n \<Longrightarrow> length w \<le> k*n"by(induct n arbitrary: w) (fastsimp simp: conc_def)+lemma length_lang_pow_lb: "ALL w : A. length w \<ge> k \<Longrightarrow> w : A^^n \<Longrightarrow> length w \<ge> k*n"by(induct n arbitrary: w) (fastsimp simp: conc_def)+subsection{* @{const star} *}lemma star_if_lang_pow[simp]: "w : A ^^ n \<Longrightarrow> w : star A"by (auto simp: star_def)lemma Nil_in_star[iff]: "[] : star A"proof (rule star_if_lang_pow) show "[] : A ^^ 0" by simpqedlemma star_if_lang[simp]: assumes "w : A" shows "w : star A"proof (rule star_if_lang_pow) show "w : A ^^ 1" using `w : A` by simpqedlemma append_in_starI[simp]:assumes "u : star A" and "v : star A" shows "u@v : star A"proof - from `u : star A` obtain m where "u : A ^^ m" by (auto simp: star_def) moreover from `v : star A` obtain n where "v : A ^^ n" by (auto simp: star_def) ultimately have "u@v : A ^^ (m+n)" by (simp add: lang_pow_add) thus ?thesis by simpqedlemma conc_star_star: "star A @@ star A = star A"by (auto simp: conc_def)lemma star_induct[consumes 1, case_names Nil append, induct set: star]:assumes "w : star A" and "P []" and step: "!!u v. u : A \<Longrightarrow> v : star A \<Longrightarrow> P v \<Longrightarrow> P (u@v)"shows "P w"proof - { fix n have "w : A ^^ n \<Longrightarrow> P w" by (induct n arbitrary: w) (auto intro: `P []` step star_if_lang_pow) } with `w : star A` show "P w" by (auto simp: star_def)qedlemma star_empty[simp]: "star {} = {[]}"by (auto elim: star_induct)lemma star_epsilon[simp]: "star {[]} = {[]}"by (auto elim: star_induct)lemma star_idemp[simp]: "star (star A) = star A"by (auto elim: star_induct)lemma star_unfold_left: "star A = A @@ star A \<union> {[]}" (is "?L = ?R")proof show "?L \<subseteq> ?R" by (rule, erule star_induct) autoqed autolemma concat_in_star: "set ws \<subseteq> A \<Longrightarrow> concat ws : star A"by (induct ws) simp_alllemma in_star_iff_concat: "w : star A = (EX ws. set ws \<subseteq> A & w = concat ws)" (is "_ = (EX ws. ?R w ws)")proof assume "w : star A" thus "EX ws. ?R w ws" proof induct case Nil have "?R [] []" by simp thus ?case .. next case (append u v) moreover then obtain ws where "set ws \<subseteq> A \<and> v = concat ws" by blast ultimately have "?R (u@v) (u#ws)" by auto thus ?case .. qednext assume "EX us. ?R w us" thus "w : star A" by (auto simp: concat_in_star)qedlemma star_conv_concat: "star A = {concat ws|ws. set ws \<subseteq> A}"by (fastsimp simp: in_star_iff_concat)lemma star_insert_eps[simp]: "star (insert [] A) = star(A)"proof- { fix us have "set us \<subseteq> insert [] A \<Longrightarrow> EX vs. concat us = concat vs \<and> set vs \<subseteq> A" (is "?P \<Longrightarrow> EX vs. ?Q vs") proof let ?vs = "filter (%u. u \<noteq> []) us" show "?P \<Longrightarrow> ?Q ?vs" by (induct us) auto qed } thus ?thesis by (auto simp: star_conv_concat)qedlemma Arden:assumes "[] \<notin> A" and "X = A @@ X \<union> B"shows "X = star A @@ B"proof - { fix n have "X = A^^(n+1)@@X \<union> (\<Union>i\<le>n. A^^i@@B)" proof(induct n) case 0 show ?case using `X = A @@ X \<union> B` by simp next case (Suc n) have "X = A@@X Un B" by(rule assms(2)) also have "\<dots> = A@@(A^^(n+1)@@X \<union> (\<Union>i\<le>n. A^^i@@B)) Un B" by(subst Suc)(rule refl) also have "\<dots> = A^^(n+2)@@X \<union> (\<Union>i\<le>n. A^^(i+1)@@B) Un B" by(simp add:conc_UNION_distrib conc_assoc conc_Un_distrib) also have "\<dots> = A^^(n+2)@@X \<union> (UN i : {1..n+1}. A^^i@@B) \<union> B" by(subst UN_le_add_shift)(rule refl) also have "\<dots> = A^^(n+2)@@X \<union> (UN i : {1..n+1}. A^^i@@B) \<union> A^^0@@B" by(simp) also have "\<dots> = A^^(n+2)@@X \<union> (\<Union>i\<le>n+1. A^^i@@B)" by(auto simp: UN_le_eq_Un0) finally show ?case by simp qed } note 1 = this { fix w assume "w : X" let ?n = "size w" from `[] \<notin> A` have "ALL u : A. length u \<ge> 1" by (metis Suc_eq_plus1 add_leD2 le_0_eq length_0_conv not_less_eq_eq) hence "ALL u : A^^(?n+1). length u \<ge> ?n+1" by (metis length_lang_pow_lb nat_mult_1) hence "ALL u : A^^(?n+1)@@X. length u \<ge> ?n+1" by(auto simp only: conc_def length_append) hence "w \<notin> A^^(?n+1)@@X" by auto hence "w : star A @@ B" using `w : X` 1[of ?n] by(auto simp add: star_def conc_UNION_distrib) } moreover { fix w assume "w : star A @@ B" hence "EX n. w : A^^n @@ B" by(auto simp: conc_def star_def) hence "w : X" using 1 by blast } ultimately show ?thesis by blastqedsubsection{* @{const deriv} *}lemma deriv_empty[simp]: "deriv a {} = {}"and deriv_epsilon[simp]: "deriv a {[]} = {}"and deriv_char[simp]: "deriv a {[b]} = (if a = b then {[]} else {})"and deriv_union[simp]: "deriv a (A \<union> B) = deriv a A \<union> deriv a B"by (auto simp: deriv_def)lemma deriv_conc_subset:"deriv a A @@ B \<subseteq> deriv a (A @@ B)" (is "?L \<subseteq> ?R")proof fix w assume "w \<in> ?L" then obtain u v where "w = u @ v" "a # u \<in> A" "v \<in> B" by (auto simp: deriv_def) then have "a # w \<in> A @@ B" by (auto intro: concI[of "a # u", simplified]) thus "w \<in> ?R" by (auto simp: deriv_def)qedlemma deriv_conc1:assumes "[] \<notin> A"shows "deriv a (A @@ B) = deriv a A @@ B" (is "?L = ?R")proof show "?L \<subseteq> ?R" proof fix w assume "w \<in> ?L" then have "a # w \<in> A @@ B" by (simp add: deriv_def) then obtain x y where aw: "a # w = x @ y" "x \<in> A" "y \<in> B" by auto with `[] \<notin> A` obtain x' where "x = a # x'" by (cases x) auto with aw have "w = x' @ y" "x' \<in> deriv a A" by (auto simp: deriv_def) with `y \<in> B` show "w \<in> ?R" by simp qed show "?R \<subseteq> ?L" by (fact deriv_conc_subset)qedlemma deriv_conc2:assumes "[] \<in> A"shows "deriv a (A @@ B) = deriv a A @@ B \<union> deriv a B"(is "?L = ?R")proof show "?L \<subseteq> ?R" proof fix w assume "w \<in> ?L" then have "a # w \<in> A @@ B" by (simp add: deriv_def) then obtain x y where aw: "a # w = x @ y" "x \<in> A" "y \<in> B" by auto show "w \<in> ?R" proof (cases x) case Nil with aw have "w \<in> deriv a B" by (auto simp: deriv_def) thus ?thesis .. next case (Cons b x') with aw have "w = x' @ y" "x' \<in> deriv a A" by (auto simp: deriv_def) with `y \<in> B` show "w \<in> ?R" by simp qed qed from concI[OF `[] \<in> A`, simplified] have "B \<subseteq> A @@ B" .. then have "deriv a B \<subseteq> deriv a (A @@ B)" by (auto simp: deriv_def) with deriv_conc_subset[of a A B] show "?R \<subseteq> ?L" by autoqedlemma wlog_no_eps: assumes PB: "\<And>B. [] \<notin> B \<Longrightarrow> P B"assumes preserved: "\<And>A. P A \<Longrightarrow> P (insert [] A)"shows "P A"proof - let ?B = "A - {[]}" have "P ?B" by (rule PB) auto thus "P A" proof cases assume "[] \<in> A" then have "(insert [] ?B) = A" by auto with preserved[OF `P ?B`] show ?thesis by simp qed autoqedlemma deriv_insert_eps[simp]: "deriv a (insert [] A) = deriv a A"by (auto simp: deriv_def)lemma deriv_star[simp]: "deriv a (star A) = deriv a A @@ star A"proof (induct A rule: wlog_no_eps) fix B :: "'a list set" assume "[] \<notin> B" thus "deriv a (star B) = deriv a B @@ star B" by (subst star_unfold_left) (simp add: deriv_conc1)qed autosubsection{* @{const bisimilar} *}lemma equal_if_bisimilar:assumes "bisimilar K L" shows "K = L"proof (rule set_eqI) fix w from `bisimilar K L` show "w \<in> K \<longleftrightarrow> w \<in> L" proof (induct w arbitrary: K L) case Nil thus ?case by (auto elim: bisimilar.cases) next case (Cons a w K L) from `bisimilar K L` have "bisimilar (deriv a K) (deriv a L)" by (auto elim: bisimilar.cases) then have "w \<in> deriv a K \<longleftrightarrow> w \<in> deriv a L" by (rule Cons(1)) thus ?case by (auto simp: deriv_def) qedqedlemma language_coinduct:fixes R (infixl "\<sim>" 50)assumes "K \<sim> L"assumes "\<And>K L. K \<sim> L \<Longrightarrow> ([] \<in> K \<longleftrightarrow> [] \<in> L)"assumes "\<And>K L x. K \<sim> L \<Longrightarrow> deriv x K \<sim> deriv x L"shows "K = L"apply (rule equal_if_bisimilar)apply (rule bisimilar.coinduct[of R, OF `K \<sim> L`])apply (auto simp: assms)doneend