theory Derivatives+
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imports Regular_Exp+
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begin+
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+
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section {* Leftquotients, Derivatives and Partial Derivatives *}+
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+
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text{* This theory is based on work by Brozowski \cite{Brzozowski64} and Antimirov \cite{Antimirov95}. *}+
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+
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subsection {* Left-Quotients of languages *}+
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+
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definition Deriv :: "'a \<Rightarrow> 'a lang \<Rightarrow> 'a lang"+
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where "Deriv x A = { xs. x#xs \<in> A }"+
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+
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definition Derivs :: "'a list \<Rightarrow> 'a lang \<Rightarrow> 'a lang"+
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where "Derivs xs A = { ys. xs @ ys \<in> A }"+
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+
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abbreviation +
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  Derivss :: "'a list \<Rightarrow> 'a lang set \<Rightarrow> 'a lang"+
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where+
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  "Derivss s As \<equiv> \<Union> (Derivs s) ` As"+
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+
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+
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lemma Deriv_empty[simp]:   "Deriv a {} = {}"+
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  and Deriv_epsilon[simp]: "Deriv a {[]} = {}"+
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  and Deriv_char[simp]:    "Deriv a {[b]} = (if a = b then {[]} else {})"+
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  and Deriv_union[simp]:   "Deriv a (A \<union> B) = Deriv a A \<union> Deriv a B"+
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by (auto simp: Deriv_def)+
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+
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lemma Deriv_conc_subset:+
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"Deriv a A @@ B \<subseteq> Deriv a (A @@ B)" (is "?L \<subseteq> ?R")+
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proof +
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  fix w assume "w \<in> ?L"+
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  then obtain u v where "w = u @ v" "a # u \<in> A" "v \<in> B"+
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    by (auto simp: Deriv_def)+
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  then have "a # w \<in> A @@ B"+
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    by (auto intro: concI[of "a # u", simplified])+
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  thus "w \<in> ?R" by (auto simp: Deriv_def)+
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qed+
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+
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lemma Der_conc [simp]:+
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  shows "Deriv c (A @@ B) = (Deriv c A) @@ B \<union> (if [] \<in> A then Deriv c B else {})"+
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unfolding Deriv_def conc_def+
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by (auto simp add: Cons_eq_append_conv)+
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+
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lemma Deriv_star [simp]:+
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  shows "Deriv c (star A) = (Deriv c A) @@ star A"+
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proof -+
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  have incl: "[] \<in> A \<Longrightarrow> Deriv c (star A) \<subseteq> (Deriv c A) @@ star A"+
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    unfolding Deriv_def conc_def +
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    apply(auto simp add: Cons_eq_append_conv)+
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    apply(drule star_decom)+
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    apply(auto simp add: Cons_eq_append_conv)+
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    done+
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+
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  have "Deriv c (star A) = Deriv c (A @@ star A \<union> {[]})"+
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    by (simp only: star_unfold_left[symmetric])+
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  also have "... = Deriv c (A @@ star A)"+
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    by (simp only: Deriv_union) (simp)+
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  also have "... =  (Deriv c A) @@ (star A) \<union> (if [] \<in> A then Deriv c (star A) else {})"+
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    by simp+
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   also have "... =  (Deriv c A) @@ star A"+
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    using incl by auto+
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  finally show "Deriv c (star A) = (Deriv c A) @@ star A" . +
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qed+
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+
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lemma Derivs_simps [simp]:+
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  shows "Derivs [] A = A"+
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  and   "Derivs (c # s) A = Derivs s (Deriv c A)"+
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  and   "Derivs (s1 @ s2) A = Derivs s2 (Derivs s1 A)"+
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unfolding Derivs_def Deriv_def by auto+
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+
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(*+
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lemma Deriv_insert_eps[simp]: +
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"Deriv a (insert [] A) = Deriv a A"+
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by (auto simp: Deriv_def)+
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*)+
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+
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+
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+
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subsection {* Brozowsky's derivatives of regular expressions *}+
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+
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fun+
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  nullable :: "'a rexp \<Rightarrow> bool"+
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where+
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  "nullable (Zero) = False"+
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| "nullable (One) = True"+
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| "nullable (Atom c) = False"+
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| "nullable (Plus r1 r2) = (nullable r1 \<or> nullable r2)"+
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| "nullable (Times r1 r2) = (nullable r1 \<and> nullable r2)"+
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| "nullable (Star r) = True"+
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+
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fun+
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  deriv :: "'a \<Rightarrow> 'a rexp \<Rightarrow> 'a rexp"+
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where+
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  "deriv c (Zero) = Zero"+
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| "deriv c (One) = Zero"+
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| "deriv c (Atom c') = (if c = c' then One else Zero)"+
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| "deriv c (Plus r1 r2) = Plus (deriv c r1) (deriv c r2)"+
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| "deriv c (Times r1 r2) = +
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    (if nullable r1 then Plus (Times (deriv c r1) r2) (deriv c r2) else Times (deriv c r1) r2)"+
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| "deriv c (Star r) = Times (deriv c r) (Star r)"+
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+
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fun +
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  derivs :: "'a list \<Rightarrow> 'a rexp \<Rightarrow> 'a rexp"+
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where+
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  "derivs [] r = r"+
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| "derivs (c # s) r = derivs s (deriv c r)"+
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+
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+
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lemma nullable_iff:+
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  shows "nullable r \<longleftrightarrow> [] \<in> lang r"+
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by (induct r) (auto simp add: conc_def split: if_splits)+
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+
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lemma Deriv_deriv:+
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  shows "Deriv c (lang r) = lang (deriv c r)"+
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by (induct r) (simp_all add: nullable_iff)+
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+
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lemma Derivs_derivs:+
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  shows "Derivs s (lang r) = lang (derivs s r)"+
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by (induct s arbitrary: r) (simp_all add: Deriv_deriv)+
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+
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+
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subsection {* Antimirov's partial derivivatives *}+
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+
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abbreviation+
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  "Timess rs r \<equiv> {Times r' r | r'. r' \<in> rs}"+
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+
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fun+
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  pderiv :: "'a \<Rightarrow> 'a rexp \<Rightarrow> 'a rexp set"+
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where+
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  "pderiv c Zero = {}"+
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| "pderiv c One = {}"+
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| "pderiv c (Atom c') = (if c = c' then {One} else {})"+
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| "pderiv c (Plus r1 r2) = (pderiv c r1) \<union> (pderiv c r2)"+
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| "pderiv c (Times r1 r2) = +
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    (if nullable r1 then Timess (pderiv c r1) r2 \<union> pderiv c r2 else Timess (pderiv c r1) r2)"+
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| "pderiv c (Star r) = Timess (pderiv c r) (Star r)"+
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+
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fun+
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  pderivs :: "'a list \<Rightarrow> 'a rexp \<Rightarrow> ('a rexp) set"+
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where+
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  "pderivs [] r = {r}"+
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| "pderivs (c # s) r = \<Union> (pderivs s) ` (pderiv c r)"+
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+
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abbreviation+
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 pderiv_set :: "'a \<Rightarrow> 'a rexp set \<Rightarrow> 'a rexp set"+
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where+
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  "pderiv_set c rs \<equiv> \<Union> pderiv c ` rs"+
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+
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abbreviation+
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  pderivs_set :: "'a list \<Rightarrow> 'a rexp set \<Rightarrow> 'a rexp set"+
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where+
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  "pderivs_set s rs \<equiv> \<Union> (pderivs s) ` rs"+
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+
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lemma pderivs_append:+
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  "pderivs (s1 @ s2) r = \<Union> (pderivs s2) ` (pderivs s1 r)"+
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by (induct s1 arbitrary: r) (simp_all)+
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+
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lemma pderivs_snoc:+
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  shows "pderivs (s @ [c]) r = pderiv_set c (pderivs s r)"+
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by (simp add: pderivs_append)+
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+
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lemma pderivs_simps [simp]:+
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  shows "pderivs s Zero = (if s = [] then {Zero} else {})"+
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  and   "pderivs s One = (if s = [] then {One} else {})"+
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  and   "pderivs s (Plus r1 r2) = (if s = [] then {Plus r1 r2} else (pderivs s r1) \<union> (pderivs s r2))"+
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by (induct s) (simp_all)+
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+
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lemma pderivs_Atom:+
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  shows "pderivs s (Atom c) \<subseteq> {Atom c, One}"+
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by (induct s) (simp_all)+
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+
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subsection {* Relating left-quotients and partial derivivatives *}+
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+
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lemma Deriv_pderiv:+
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  shows "Deriv c (lang r) = \<Union> lang ` (pderiv c r)"+
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by (induct r) (auto simp add: nullable_iff conc_UNION_distrib)+
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+
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lemma Derivs_pderivs:+
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  shows "Derivs s (lang r) = \<Union> lang ` (pderivs s r)"+
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proof (induct s arbitrary: r)+
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  case (Cons c s)+
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  have ih: "\<And>r. Derivs s (lang r) = \<Union> lang ` (pderivs s r)" by fact+
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  have "Derivs (c # s) (lang r) = Derivs s (Deriv c (lang r))" by simp+
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  also have "\<dots> = Derivs s (\<Union> lang ` (pderiv c r))" by (simp add: Deriv_pderiv)+
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  also have "\<dots> = Derivss s (lang ` (pderiv c r))"+
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    by (auto simp add:  Derivs_def)+
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  also have "\<dots> = \<Union> lang ` (pderivs_set s (pderiv c r))"+
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    using ih by auto+
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  also have "\<dots> = \<Union> lang ` (pderivs (c # s) r)" by simp+
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  finally show "Derivs (c # s) (lang r) = \<Union> lang ` pderivs (c # s) r" .+
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qed (simp add: Derivs_def)+
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+
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subsection {* Relating derivivatives and partial derivivatives *}+
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+
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lemma deriv_pderiv:+
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  shows "(\<Union> lang ` (pderiv c r)) = lang (deriv c r)"+
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unfolding Deriv_deriv[symmetric] Deriv_pderiv by simp+
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+
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lemma derivs_pderivs:+
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  shows "(\<Union> lang ` (pderivs s r)) = lang (derivs s r)"+
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unfolding Derivs_derivs[symmetric] Derivs_pderivs by simp+
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+
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+
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subsection {* Finiteness property of partial derivivatives *}+
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+
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definition+
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  pderivs_lang :: "'a lang \<Rightarrow> 'a rexp \<Rightarrow> 'a rexp set"+
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where+
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  "pderivs_lang A r \<equiv> \<Union>x \<in> A. pderivs x r"+
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+
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lemma pderivs_lang_subsetI:+
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  assumes "\<And>s. s \<in> A \<Longrightarrow> pderivs s r \<subseteq> C"+
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  shows "pderivs_lang A r \<subseteq> C"+
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using assms unfolding pderivs_lang_def by (rule UN_least)+
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+
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lemma pderivs_lang_union:+
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  shows "pderivs_lang (A \<union> B) r = (pderivs_lang A r \<union> pderivs_lang B r)"+
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by (simp add: pderivs_lang_def)+
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+
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lemma pderivs_lang_subset:+
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  shows "A \<subseteq> B \<Longrightarrow> pderivs_lang A r \<subseteq> pderivs_lang B r"+
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by (auto simp add: pderivs_lang_def)+
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+
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definition+
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  "UNIV1 \<equiv> UNIV - {[]}"+
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+
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lemma pderivs_lang_Zero [simp]:+
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  shows "pderivs_lang UNIV1 Zero = {}"+
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unfolding UNIV1_def pderivs_lang_def by auto+
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+
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lemma pderivs_lang_One [simp]:+
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  shows "pderivs_lang UNIV1 One = {}"+
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unfolding UNIV1_def pderivs_lang_def by (auto split: if_splits)+
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+
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lemma pderivs_lang_Atom [simp]:+
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  shows "pderivs_lang UNIV1 (Atom c) = {One}"+
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unfolding UNIV1_def pderivs_lang_def +
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apply(auto)+
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apply(frule rev_subsetD)+
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apply(rule pderivs_Atom)+
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apply(simp)+
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apply(case_tac xa)+
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apply(auto split: if_splits)+
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done+
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+
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lemma pderivs_lang_Plus [simp]:+
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  shows "pderivs_lang UNIV1 (Plus r1 r2) = pderivs_lang UNIV1 r1 \<union> pderivs_lang UNIV1 r2"+
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unfolding UNIV1_def pderivs_lang_def by auto+
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+
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+
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text {* Non-empty suffixes of a string (needed for teh cases of @{const Times} and @{const Star} *}+
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+
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definition+
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  "PSuf s \<equiv> {v. v \<noteq> [] \<and> (\<exists>u. u @ v = s)}"+
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+
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lemma PSuf_snoc:+
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  shows "PSuf (s @ [c]) = (PSuf s) @@ {[c]} \<union> {[c]}"+
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unfolding PSuf_def conc_def+
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by (auto simp add: append_eq_append_conv2 append_eq_Cons_conv)+
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+
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lemma PSuf_Union:+
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  shows "(\<Union>v \<in> PSuf s @@ {[c]}. f v) = (\<Union>v \<in> PSuf s. f (v @ [c]))"+
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by (auto simp add: conc_def)+
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+
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lemma pderivs_lang_snoc:+
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  shows "pderivs_lang (PSuf s @@ {[c]}) r = (pderiv_set c (pderivs_lang (PSuf s) r))"+
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unfolding pderivs_lang_def+
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by (simp add: PSuf_Union pderivs_snoc)+
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+
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lemma pderivs_Times:+
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  shows "pderivs s (Times r1 r2) \<subseteq> Timess (pderivs s r1) r2 \<union> (pderivs_lang (PSuf s) r2)"+
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proof (induct s rule: rev_induct)+
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  case (snoc c s)+
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  have ih: "pderivs s (Times r1 r2) \<subseteq> Timess (pderivs s r1) r2 \<union> (pderivs_lang (PSuf s) r2)" +
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    by fact+
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  have "pderivs (s @ [c]) (Times r1 r2) = pderiv_set c (pderivs s (Times r1 r2))" +
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    by (simp add: pderivs_snoc)+
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  also have "\<dots> \<subseteq> pderiv_set c (Timess (pderivs s r1) r2 \<union> (pderivs_lang (PSuf s) r2))"+
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    using ih by (auto) (blast)+
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  also have "\<dots> = pderiv_set c (Timess (pderivs s r1) r2) \<union> pderiv_set c (pderivs_lang (PSuf s) r2)"+
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    by (simp)+
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  also have "\<dots> = pderiv_set c (Timess (pderivs s r1) r2) \<union> pderivs_lang (PSuf s @@ {[c]}) r2"+
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    by (simp add: pderivs_lang_snoc)+
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  also +
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  have "\<dots> \<subseteq> pderiv_set c (Timess (pderivs s r1) r2) \<union> pderiv c r2 \<union> pderivs_lang (PSuf s @@ {[c]}) r2"+
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    by auto+
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  also +
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  have "\<dots> \<subseteq> Timess (pderiv_set c (pderivs s r1)) r2 \<union> pderiv c r2 \<union> pderivs_lang (PSuf s @@ {[c]}) r2"+
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    by (auto simp add: if_splits) (blast)+
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  also have "\<dots> = Timess (pderivs (s @ [c]) r1) r2 \<union> pderiv c r2 \<union> pderivs_lang (PSuf s @@ {[c]}) r2"+
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    by (simp add: pderivs_snoc)+
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  also have "\<dots> \<subseteq> Timess (pderivs (s @ [c]) r1) r2 \<union> pderivs_lang (PSuf (s @ [c])) r2"+
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    unfolding pderivs_lang_def by (auto simp add: PSuf_snoc)  +
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  finally show ?case .+
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qed (simp) +
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+
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lemma pderivs_lang_Times_aux1:+
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  assumes a: "s \<in> UNIV1"+
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  shows "pderivs_lang (PSuf s) r \<subseteq> pderivs_lang UNIV1 r"+
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using a unfolding UNIV1_def PSuf_def pderivs_lang_def by auto+
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+
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lemma pderivs_lang_Times_aux2:+
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  assumes a: "s \<in> UNIV1"+
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  shows "Timess (pderivs s r1) r2 \<subseteq> Timess (pderivs_lang UNIV1 r1) r2"+
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using a unfolding pderivs_lang_def by auto+
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+
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lemma pderivs_lang_Times:+
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  shows "pderivs_lang UNIV1 (Times r1 r2) \<subseteq> Timess (pderivs_lang UNIV1 r1) r2 \<union> pderivs_lang UNIV1 r2"+
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apply(rule pderivs_lang_subsetI)+
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apply(rule subset_trans)+
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apply(rule pderivs_Times)+
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using pderivs_lang_Times_aux1 pderivs_lang_Times_aux2+
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apply(blast)+
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done+
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+
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lemma pderivs_Star:+
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  assumes a: "s \<noteq> []"+
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  shows "pderivs s (Star r) \<subseteq> Timess (pderivs_lang (PSuf s) r) (Star r)"+
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using a+
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proof (induct s rule: rev_induct)+
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  case (snoc c s)+
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  have ih: "s \<noteq> [] \<Longrightarrow> pderivs s (Star r) \<subseteq> Timess (pderivs_lang (PSuf s) r) (Star r)" by fact+
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  { assume asm: "s \<noteq> []"+
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    have "pderivs (s @ [c]) (Star r) = pderiv_set c (pderivs s (Star r))" by (simp add: pderivs_snoc)+
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    also have "\<dots> \<subseteq> pderiv_set c (Timess (pderivs_lang (PSuf s) r) (Star r))"+
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      using ih[OF asm] by (auto) (blast)+
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    also have "\<dots> \<subseteq> Timess (pderiv_set c (pderivs_lang (PSuf s) r)) (Star r) \<union> pderiv c (Star r)"+
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      by (auto split: if_splits) (blast)++
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    also have "\<dots> \<subseteq> Timess (pderivs_lang (PSuf (s @ [c])) r) (Star r) \<union> (Timess (pderiv c r) (Star r))"+
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      by (simp only: PSuf_snoc pderivs_lang_snoc pderivs_lang_union)+
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         (auto simp add: pderivs_lang_def)+
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    also have "\<dots> = Timess (pderivs_lang (PSuf (s @ [c])) r) (Star r)"+
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      by (auto simp add: PSuf_snoc PSuf_Union pderivs_snoc pderivs_lang_def)+
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    finally have ?case .+
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  }+
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  moreover+
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  { assume asm: "s = []"+
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    then have ?case+
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      apply (auto simp add: pderivs_lang_def pderivs_snoc PSuf_def)+
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      apply(rule_tac x = "[c]" in exI)+
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      apply(auto)+
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      done+
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  }+
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  ultimately show ?case by blast+
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qed (simp)+
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+
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lemma pderivs_lang_Star:+
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  shows "pderivs_lang UNIV1 (Star r) \<subseteq> Timess (pderivs_lang UNIV1 r) (Star r)"+
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apply(rule pderivs_lang_subsetI)+
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apply(rule subset_trans)+
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apply(rule pderivs_Star)+
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apply(simp add: UNIV1_def)+
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apply(simp add: UNIV1_def PSuf_def)+
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apply(auto simp add: pderivs_lang_def)+
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done+
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+
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lemma finite_Timess [simp]:+
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  assumes a: "finite A"+
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  shows "finite (Timess A r)"+
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using a by auto+
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+
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lemma finite_pderivs_lang_UNIV1:+
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  shows "finite (pderivs_lang UNIV1 r)"+
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apply(induct r)+
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apply(simp_all add: +
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  finite_subset[OF pderivs_lang_Times]+
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  finite_subset[OF pderivs_lang_Star])+
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done+
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    +
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lemma pderivs_lang_UNIV:+
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  shows "pderivs_lang UNIV r = pderivs [] r \<union> pderivs_lang UNIV1 r"+
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unfolding UNIV1_def pderivs_lang_def+
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by blast+
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+
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lemma finite_pderivs_lang_UNIV:+
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  shows "finite (pderivs_lang UNIV r)"+
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unfolding pderivs_lang_UNIV+
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by (simp add: finite_pderivs_lang_UNIV1)+
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+
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lemma finite_pderivs_lang:+
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  shows "finite (pderivs_lang A r)"+
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by (metis finite_pderivs_lang_UNIV pderivs_lang_subset rev_finite_subset subset_UNIV)+
−
+
−
+
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end+
−