imports Regular_Exp

section {* Leftquotients, Derivatives and Partial Derivatives *}

text{* This theory is based on work by Brozowski \cite{Brzozowski64} and Antimirov \cite{Antimirov95}. *}

subsection {* Left-Quotients of languages *}

definition Deriv :: "'a \<Rightarrow> 'a lang \<Rightarrow> 'a lang"

where "Deriv x A = { xs. x#xs \<in> A }"

definition Derivs :: "'a list \<Rightarrow> 'a lang \<Rightarrow> 'a lang"

where "Derivs xs A = { ys. xs @ ys \<in> A }"

  Derivss :: "'a list \<Rightarrow> 'a lang set \<Rightarrow> 'a lang"

where

  "Derivss s As \<equiv> \<Union> (Derivs s) ` As"

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)

qed

  shows "Deriv c (A @@ B) = (Deriv c A) @@ B \<union> (if [] \<in> A then Deriv c B else {})"

unfolding Deriv_def conc_def

lemma Deriv_star [simp]:

  shows "Deriv c (star A) = (Deriv c A) @@ star A"

  have incl: "[] \<in> A \<Longrightarrow> Deriv c (star A) \<subseteq> (Deriv c A) @@ star A"

    unfolding Deriv_def conc_def 

    apply(auto simp add: Cons_eq_append_conv)

  have "Deriv c (star A) = Deriv c (A @@ star A \<union> {[]})"

  also have "... = Deriv c (A @@ star A)"

    by (simp only: Deriv_union) (simp)

  also have "... =  (Deriv c A) @@ (star A) \<union> (if [] \<in> A then Deriv c (star A) else {})"

   also have "... =  (Deriv c A) @@ star A"

  finally show "Deriv c (star A) = (Deriv c A) @@ star A" . 

lemma Derivs_simps [simp]:

  shows "Derivs [] A = A"

  and   "Derivs (c # s) A = Derivs s (Deriv c A)"

  and   "Derivs (s1 @ s2) A = Derivs s2 (Derivs s1 A)"

unfolding Derivs_def Deriv_def by auto

(*

lemma Deriv_insert_eps[simp]: 

"Deriv a (insert [] A) = Deriv a A"

by (auto simp: Deriv_def)

*)

  deriv :: "'a \<Rightarrow> 'a rexp \<Rightarrow> 'a rexp"

  "deriv c (Zero) = Zero"

| "deriv c (One) = Zero"

| "deriv c (Atom c') = (if c = c' then One else Zero)"

| "deriv c (Plus r1 r2) = Plus (deriv c r1) (deriv c r2)"

| "deriv c (Times r1 r2) = 

    (if nullable r1 then Plus (Times (deriv c r1) r2) (deriv c r2) else Times (deriv c r1) r2)"

| "deriv c (Star r) = Times (deriv c r) (Star r)"

  derivs :: "'a list \<Rightarrow> 'a rexp \<Rightarrow> 'a rexp"

  "derivs [] r = r"

| "derivs (c # s) r = derivs s (deriv c r)"

lemma nullable_iff:

  shows "nullable r \<longleftrightarrow> [] \<in> lang r"

lemma Deriv_deriv:

  shows "Deriv c (lang r) = lang (deriv c r)"

by (induct r) (simp_all add: nullable_iff)

lemma Derivs_derivs:

  shows "Derivs s (lang r) = lang (derivs s r)"

by (induct s arbitrary: r) (simp_all add: Deriv_deriv)

subsection {* Antimirov's partial derivivatives *}

  pderiv :: "'a \<Rightarrow> 'a rexp \<Rightarrow> 'a rexp set"

  "pderiv c Zero = {}"

| "pderiv c One = {}"

| "pderiv c (Atom c') = (if c = c' then {One} else {})"

| "pderiv c (Plus r1 r2) = (pderiv c r1) \<union> (pderiv c r2)"

| "pderiv c (Times r1 r2) = 

    (if nullable r1 then Timess (pderiv c r1) r2 \<union> pderiv c r2 else Timess (pderiv c r1) r2)"

| "pderiv c (Star r) = Timess (pderiv c r) (Star r)"

  pderivs :: "'a list \<Rightarrow> 'a rexp \<Rightarrow> ('a rexp) set"

  "pderivs [] r = {r}"

| "pderivs (c # s) r = \<Union> (pderivs s) ` (pderiv c r)"

 pderiv_set :: "'a \<Rightarrow> 'a rexp set \<Rightarrow> 'a rexp set"

where

  "pderiv_set c rs \<equiv> \<Union> pderiv c ` rs"

abbreviation

  pderivs_set :: "'a list \<Rightarrow> 'a rexp set \<Rightarrow> 'a rexp set"

where

  "pderivs_set s rs \<equiv> \<Union> (pderivs s) ` rs"

lemma pderivs_append:

  "pderivs (s1 @ s2) r = \<Union> (pderivs s2) ` (pderivs s1 r)"

lemma pderivs_snoc:

  shows "pderivs (s @ [c]) r = pderiv_set c (pderivs s r)"

by (simp add: pderivs_append)

lemma pderivs_simps [simp]:

  shows "pderivs s Zero = (if s = [] then {Zero} else {})"

  and   "pderivs s One = (if s = [] then {One} else {})"

  and   "pderivs s (Plus r1 r2) = (if s = [] then {Plus r1 r2} else (pderivs s r1) \<union> (pderivs s r2))"

lemma pderivs_Atom:

  shows "pderivs s (Atom c) \<subseteq> {Atom c, One}"

subsection {* Relating left-quotients and partial derivivatives *}

lemma Deriv_pderiv:

  shows "Deriv c (lang r) = \<Union> lang ` (pderiv c r)"

by (induct r) (auto simp add: nullable_iff conc_UNION_distrib)

lemma Derivs_pderivs:

  shows "Derivs s (lang r) = \<Union> lang ` (pderivs s r)"

  have ih: "\<And>r. Derivs s (lang r) = \<Union> lang ` (pderivs s r)" by fact

  have "Derivs (c # s) (lang r) = Derivs s (Deriv c (lang r))" by simp

  also have "\<dots> = Derivs s (\<Union> lang ` (pderiv c r))" by (simp add: Deriv_pderiv)

  also have "\<dots> = Derivss s (lang ` (pderiv c r))"

    by (auto simp add:  Derivs_def)

  also have "\<dots> = \<Union> lang ` (pderivs_set s (pderiv c r))"

  also have "\<dots> = \<Union> lang ` (pderivs (c # s) r)" by simp

  finally show "Derivs (c # s) (lang r) = \<Union> lang ` pderivs (c # s) r" .

qed (simp add: Derivs_def)

subsection {* Relating derivivatives and partial derivivatives *}

lemma deriv_pderiv:

  shows "(\<Union> lang ` (pderiv c r)) = lang (deriv c r)"

unfolding Deriv_deriv[symmetric] Deriv_pderiv by simp

lemma derivs_pderivs:

  shows "(\<Union> lang ` (pderivs s r)) = lang (derivs s r)"

unfolding Derivs_derivs[symmetric] Derivs_pderivs by simp

subsection {* Finiteness property of partial derivivatives *}

  pderivs_lang :: "'a lang \<Rightarrow> 'a rexp \<Rightarrow> 'a rexp set"

where

  "pderivs_lang A r \<equiv> \<Union>x \<in> A. pderivs x r"

lemma pderivs_lang_subsetI:

  assumes "\<And>s. s \<in> A \<Longrightarrow> pderivs s r \<subseteq> C"

  shows "pderivs_lang A r \<subseteq> C"

using assms unfolding pderivs_lang_def by (rule UN_least)

lemma pderivs_lang_union:

  shows "pderivs_lang (A \<union> B) r = (pderivs_lang A r \<union> pderivs_lang B r)"

by (simp add: pderivs_lang_def)

lemma pderivs_lang_subset:

  shows "A \<subseteq> B \<Longrightarrow> pderivs_lang A r \<subseteq> pderivs_lang B r"

by (auto simp add: pderivs_lang_def)

lemma pderivs_lang_Zero [simp]:

  shows "pderivs_lang UNIV1 Zero = {}"

unfolding UNIV1_def pderivs_lang_def by auto

lemma pderivs_lang_One [simp]:

  shows "pderivs_lang UNIV1 One = {}"

unfolding UNIV1_def pderivs_lang_def by (auto split: if_splits)

lemma pderivs_lang_Atom [simp]:

  shows "pderivs_lang UNIV1 (Atom c) = {One}"

unfolding UNIV1_def pderivs_lang_def 

apply(rule pderivs_Atom)

lemma pderivs_lang_Plus [simp]:

  shows "pderivs_lang UNIV1 (Plus r1 r2) = pderivs_lang UNIV1 r1 \<union> pderivs_lang UNIV1 r2"

unfolding UNIV1_def pderivs_lang_def by auto

  "PSuf s \<equiv> {v. v \<noteq> [] \<and> (\<exists>u. u @ v = s)}"

lemma PSuf_snoc:

  shows "PSuf (s @ [c]) = (PSuf s) @@ {[c]} \<union> {[c]}"

unfolding PSuf_def conc_def

lemma PSuf_Union:

  shows "(\<Union>v \<in> PSuf s @@ {[c]}. f v) = (\<Union>v \<in> PSuf s. f (v @ [c]))"

lemma pderivs_lang_snoc:

  shows "pderivs_lang (PSuf s @@ {[c]}) r = (pderiv_set c (pderivs_lang (PSuf s) r))"

unfolding pderivs_lang_def

by (simp add: PSuf_Union pderivs_snoc)

lemma pderivs_Times:

  shows "pderivs s (Times r1 r2) \<subseteq> Timess (pderivs s r1) r2 \<union> (pderivs_lang (PSuf s) r2)"

  have ih: "pderivs s (Times r1 r2) \<subseteq> Timess (pderivs s r1) r2 \<union> (pderivs_lang (PSuf s) r2)" 

  have "pderivs (s @ [c]) (Times r1 r2) = pderiv_set c (pderivs s (Times r1 r2))" 

    by (simp add: pderivs_snoc)

  also have "\<dots> \<subseteq> pderiv_set c (Timess (pderivs s r1) r2 \<union> (pderivs_lang (PSuf s) r2))"

  also have "\<dots> = pderiv_set c (Timess (pderivs s r1) r2) \<union> pderiv_set c (pderivs_lang (PSuf s) r2)"

  also have "\<dots> = pderiv_set c (Timess (pderivs s r1) r2) \<union> pderivs_lang (PSuf s @@ {[c]}) r2"

    by (simp add: pderivs_lang_snoc)

  also 

  have "\<dots> \<subseteq> pderiv_set c (Timess (pderivs s r1) r2) \<union> pderiv c r2 \<union> pderivs_lang (PSuf s @@ {[c]}) r2"

  also 

  have "\<dots> \<subseteq> Timess (pderiv_set c (pderivs s r1)) r2 \<union> pderiv c r2 \<union> pderivs_lang (PSuf s @@ {[c]}) r2"

  also have "\<dots> = Timess (pderivs (s @ [c]) r1) r2 \<union> pderiv c r2 \<union> pderivs_lang (PSuf s @@ {[c]}) r2"

    by (simp add: pderivs_snoc)

  also have "\<dots> \<subseteq> Timess (pderivs (s @ [c]) r1) r2 \<union> pderivs_lang (PSuf (s @ [c])) r2"

    unfolding pderivs_lang_def by (auto simp add: PSuf_snoc)  

lemma pderivs_lang_Times_aux1:

  shows "pderivs_lang (PSuf s) r \<subseteq> pderivs_lang UNIV1 r"

using a unfolding UNIV1_def PSuf_def pderivs_lang_def by auto

lemma pderivs_lang_Times_aux2:

  shows "Timess (pderivs s r1) r2 \<subseteq> Timess (pderivs_lang UNIV1 r1) r2"

using a unfolding pderivs_lang_def by auto

lemma pderivs_lang_Times:

  shows "pderivs_lang UNIV1 (Times r1 r2) \<subseteq> Timess (pderivs_lang UNIV1 r1) r2 \<union> pderivs_lang UNIV1 r2"

apply(rule pderivs_lang_subsetI)

apply(rule pderivs_Times)

using pderivs_lang_Times_aux1 pderivs_lang_Times_aux2

lemma pderivs_Star:

  shows "pderivs s (Star r) \<subseteq> Timess (pderivs_lang (PSuf s) r) (Star r)"

  have ih: "s \<noteq> [] \<Longrightarrow> pderivs s (Star r) \<subseteq> Timess (pderivs_lang (PSuf s) r) (Star r)" by fact

    have "pderivs (s @ [c]) (Star r) = pderiv_set c (pderivs s (Star r))" by (simp add: pderivs_snoc)

    also have "\<dots> \<subseteq> pderiv_set c (Timess (pderivs_lang (PSuf s) r) (Star r))"

    also have "\<dots> \<subseteq> Timess (pderiv_set c (pderivs_lang (PSuf s) r)) (Star r) \<union> pderiv c (Star r)"

    also have "\<dots> \<subseteq> Timess (pderivs_lang (PSuf (s @ [c])) r) (Star r) \<union> (Timess (pderiv c r) (Star r))"

      by (simp only: PSuf_snoc pderivs_lang_snoc pderivs_lang_union)

         (auto simp add: pderivs_lang_def)

    also have "\<dots> = Timess (pderivs_lang (PSuf (s @ [c])) r) (Star r)"

      by (auto simp add: PSuf_snoc PSuf_Union pderivs_snoc pderivs_lang_def)

      apply (auto simp add: pderivs_lang_def pderivs_snoc PSuf_def)

lemma pderivs_lang_Star:

  shows "pderivs_lang UNIV1 (Star r) \<subseteq> Timess (pderivs_lang UNIV1 r) (Star r)"

apply(rule pderivs_lang_subsetI)

apply(rule pderivs_Star)

apply(simp add: UNIV1_def PSuf_def)

apply(auto simp add: pderivs_lang_def)

lemma finite_pderivs_lang_UNIV1:

  shows "finite (pderivs_lang UNIV1 r)"

  finite_subset[OF pderivs_lang_Times]

  finite_subset[OF pderivs_lang_Star])

lemma pderivs_lang_UNIV:

  shows "pderivs_lang UNIV r = pderivs [] r \<union> pderivs_lang UNIV1 r"

unfolding UNIV1_def pderivs_lang_def

lemma finite_pderivs_lang_UNIV:

  shows "finite (pderivs_lang UNIV r)"

unfolding pderivs_lang_UNIV

by (simp add: finite_pderivs_lang_UNIV1)

lemma finite_pderivs_lang:

  shows "finite (pderivs_lang A r)"

by (metis finite_pderivs_lang_UNIV pderivs_lang_subset rev_finite_subset subset_UNIV)