theory Derivatives

imports Myhill_2

begin

section {* Left-Quotients and Derivatives *}

subsection {* Left-Quotients *}

definition

  Delta :: "'a lang \<Rightarrow> 'a lang"

where

  "Delta A = (if [] \<in> A then {[]} else {})"

definition

  Der :: "'a \<Rightarrow> 'a lang \<Rightarrow> 'a lang"

where

  "Der c A \<equiv> {s. [c] @ s \<in> A}"

definition

  Ders :: "'a list \<Rightarrow> 'a lang \<Rightarrow> 'a lang"

where

  "Ders s A \<equiv> {s'. s @ s' \<in> A}"

definition

  Ders_set :: "'a lang \<Rightarrow> 'a lang \<Rightarrow> 'a lang"

where

  "Ders_set A B \<equiv> {s' | s s'. s @ s' \<in> B \<and> s \<in> A}"

lemma Ders_set_Ders:

  shows "Ders_set A B = (\<Union>s \<in> A. Ders s B)"

unfolding Ders_set_def Ders_def

by auto

lemma Der_zero [simp]:

  shows "Der c {} = {}"

unfolding Der_def

by auto

lemma Der_one [simp]:

  shows "Der c {[]} = {}"

unfolding Der_def

by auto

lemma Der_atom [simp]:

  shows "Der c {[d]} = (if c = d then {[]} else {})"

unfolding Der_def

by auto

lemma Der_union [simp]:

  shows "Der c (A \<union> B) = Der c A \<union> Der c B"

unfolding Der_def

by auto

lemma Der_conc [simp]:

  shows "Der c (A \<cdot> B) = (Der c A) \<cdot> B \<union> (Delta A \<cdot> Der c B)"

unfolding Der_def Delta_def conc_def

by (auto simp add: Cons_eq_append_conv)

lemma Der_star [simp]:

  shows "Der c (A\<star>) = (Der c A) \<cdot> A\<star>"

proof -

  have incl: "Delta A \<cdot> Der c (A\<star>) \<subseteq> (Der c A) \<cdot> A\<star>"

    unfolding Der_def Delta_def 

    apply(auto)

    apply(drule star_decom)

    apply(auto simp add: Cons_eq_append_conv)

    done

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

    by (simp only: star_cases[symmetric])

  also have "... = Der c (A \<cdot> A\<star>)"

    by (simp only: Der_union Der_one) (simp)

  also have "... = (Der c A) \<cdot> A\<star> \<union> (Delta A \<cdot> Der c (A\<star>))"

    by simp

  also have "... =  (Der c A) \<cdot> A\<star>"

    using incl by auto

  finally show "Der c (A\<star>) = (Der c A) \<cdot> A\<star>" . 

qed

lemma Ders_singleton:

  shows "Ders [c] A = Der c A"

unfolding Der_def Ders_def

by simp

lemma Ders_append:

  shows "Ders (s1 @ s2) A = Ders s2 (Ders s1 A)"

unfolding Ders_def by simp 

text {* Relating the Myhill-Nerode relation with left-quotients. *}

lemma MN_Rel_Ders:

  shows "x \<approx>A y \<longleftrightarrow> Ders x A = Ders y A"

unfolding Ders_def str_eq_def str_eq_rel_def

by auto

subsection {* Brozowsky's derivatives of regular expressions *}

fun

  nullable :: "'a rexp \<Rightarrow> bool"

where

  "nullable (Zero) = False"

| "nullable (One) = True"

| "nullable (Atom c) = False"

| "nullable (Plus r1 r2) = (nullable r1 \<or> nullable r2)"

| "nullable (Times r1 r2) = (nullable r1 \<and> nullable r2)"

| "nullable (Star r) = True"

fun

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

where

  "der c (Zero) = Zero"

| "der c (One) = Zero"

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

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

| "der c (Times r1 r2) = Plus (Times (der c r1) r2) (if nullable r1 then der c r2 else Zero)"

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

function 

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

where

  "ders [] r = r"

| "ders (s @ [c]) r = der c (ders s r)"

by (auto) (metis rev_cases)

termination

  by (relation "measure (length o fst)") (auto)

lemma Delta_nullable:

  shows "Delta (lang r) = (if nullable r then {[]} else {})"

unfolding Delta_def

by (induct r) (auto simp add: conc_def split: if_splits)

lemma Der_der:

  shows "Der c (lang r) = lang (der c r)"

by (induct r) (simp_all add: Delta_nullable)

lemma Ders_ders:

  shows "Ders s (lang r) = lang (ders s r)"

apply(induct s rule: rev_induct)

apply(simp add: Ders_def)

apply(simp only: ders.simps)

apply(simp only: Ders_append)

apply(simp only: Ders_singleton)

apply(simp only: Der_der)

done

subsection {* Antimirov's Partial Derivatives *}

abbreviation

  "Times_set rs r \<equiv> {Times r' r | r'. r' \<in> rs}"

fun

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

where

  "pder c Zero = {Zero}"

| "pder c One = {Zero}"

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

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

| "pder c (Times r1 r2) = Times_set (pder c r1) r2 \<union> (if nullable r1 then pder c r2 else {})"

| "pder c (Star r) = Times_set (pder c r) (Star r)"

abbreviation

  "pder_set c rs \<equiv> \<Union>r \<in> rs. pder c r"

function 

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

where

  "pders [] r = {r}"

| "pders (s @ [c]) r = pder_set c (pders s r)"

by (auto) (metis rev_cases)

termination

  by (relation "measure (length o fst)") (auto)

abbreviation

  "pders_set A r \<equiv> \<Union>s \<in> A. pders s r"

lemma pders_append:

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

apply(induct s2 arbitrary: s1 r rule: rev_induct)

apply(simp)

apply(subst append_assoc[symmetric])

apply(simp only: pders.simps)

apply(auto)

done

lemma pders_singleton:

  "pders [c] r = pder c r"

apply(subst append_Nil[symmetric])

apply(simp only: pders.simps)

apply(simp)

done

lemma pders_set_lang:

  shows "(\<Union> (lang ` pder_set c rs)) = (\<Union>r \<in> rs. (\<Union>lang ` (pder c r)))"

unfolding image_def 

by auto

lemma pders_Zero [simp]:

  shows "pders s Zero = {Zero}"

by (induct s rule: rev_induct) (simp_all)

lemma pders_One [simp]:

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

by (induct s rule: rev_induct) (auto)

lemma pders_Atom [simp]:

  shows "pders s (Atom c) = (if s = [] then {Atom c} else (if s = [c] then {One} else {Zero}))"

by (induct s rule: rev_induct) (auto)

lemma pders_Plus [simp]:

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

by (induct s rule: rev_induct) (auto)

text {* Non-empty suffixes of a string *}

definition

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

lemma Suf:

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

unfolding Suf_def conc_def

by (auto simp add: append_eq_append_conv2 append_eq_Cons_conv)

lemma Suf_Union:

  shows "(\<Union>v \<in> Suf s \<cdot> {[c]}. P v) = (\<Union>v \<in> Suf s. P (v @ [c]))"

by (auto simp add: conc_def)

lemma pders_Times:

  shows "pders s (Times r1 r2) \<subseteq> Times_set (pders s r1) r2 \<union> (\<Union>v \<in> Suf s. pders v r2)"

proof (induct s rule: rev_induct)

  case (snoc c s)

  have ih: "pders s (Times r1 r2) \<subseteq> Times_set (pders s r1) r2 \<union> (\<Union>v \<in> Suf s. pders v r2)" 

    by fact

  have "pders (s @ [c]) (Times r1 r2) = pder_set c (pders s (Times r1 r2))" by simp

  also have "\<dots> \<subseteq> pder_set c (Times_set (pders s r1) r2 \<union> (\<Union>v \<in> Suf s. pders v r2))"

    using ih by (auto) (blast)

  also have "\<dots> = pder_set c (Times_set (pders s r1) r2) \<union> pder_set c (\<Union>v \<in> Suf s. pders v r2)"

    by (simp)

  also have "\<dots> = pder_set c (Times_set (pders s r1) r2) \<union> (\<Union>v \<in> Suf s. pder_set c (pders v r2))"

    by (simp)

  also have "\<dots> \<subseteq> pder_set c (Times_set (pders s r1) r2) \<union> (pder c r2) \<union> (\<Union>v \<in> Suf s. pders (v @ [c]) r2)"

    by (auto)

  also have "\<dots> \<subseteq> Times_set (pder_set c (pders s r1)) r2 \<union> (pder c r2) \<union> (\<Union>v \<in> Suf s. pders (v @ [c]) r2)"

    by (auto simp add: if_splits) (blast)

  also have "\<dots> = Times_set (pders (s @ [c]) r1) r2 \<union> (\<Union>v \<in> Suf (s @ [c]). pders v r2)"

    apply(subst (2) pders.simps)

    apply(simp only: Suf)

    apply(simp add: Suf_Union pders_singleton)

    apply(auto)

    done

  finally show ?case .

qed (simp)

lemma pders_Star:

  assumes a: "s \<noteq> []"

  shows "pders s (Star r) \<subseteq> (\<Union>v \<in> Suf s. Times_set (pders v r) (Star r))"

using a

proof (induct s rule: rev_induct)

  case (snoc c s)

  have ih: "s \<noteq> [] \<Longrightarrow> pders s (Star r) \<subseteq> (\<Union>v\<in>Suf s. Times_set (pders v r) (Star r))" by fact

  { assume asm: "s \<noteq> []"

    have "pders (s @ [c]) (Star r) = pder_set c (pders s (Star r))" by simp

    also have "\<dots> \<subseteq> (pder_set c (\<Union>v\<in>Suf s. Times_set (pders v r) (Star r)))"

      using ih[OF asm] by blast

    also have "\<dots> = (\<Union>v\<in>Suf s. pder_set c (Times_set (pders v r) (Star r)))"

      by simp

    also have "\<dots> \<subseteq> (\<Union>v\<in>Suf s. (Times_set (pder_set c (pders v r)) (Star r) \<union> pder c (Star r)))"

      by (auto split: if_splits) 

    also have "\<dots> = (\<Union>v\<in>Suf s. (Times_set (pder_set c (pders v r)) (Star r))) \<union> pder c (Star r)"

      using asm by (auto simp add: Suf_def)

    also have "\<dots> = (\<Union>v\<in>Suf s. (Times_set (pders (v @ [c]) r) (Star r))) \<union> (Times_set (pder c r) (Star r))"

      by simp

    also have "\<dots> = (\<Union>v\<in>Suf (s @ [c]). (Times_set (pders v r) (Star r)))"

      apply(simp only: Suf)

      apply(simp add: Suf_Union pders_singleton)

      apply(auto)

      done

    finally have ?case .

  }

  moreover

  { assume asm: "s = []"

    then have ?case

      apply(simp add: pders_singleton Suf_def)

      apply(auto)

      apply(rule_tac x="[c]" in exI)

      apply(simp add: pders_singleton)

      done

  }

  ultimately show ?case by blast

qed (simp)

abbreviation 

  "UNIV1 \<equiv> UNIV - {[]}"

lemma pders_set_Zero:

  shows "pders_set UNIV1 Zero = {Zero}"

by auto

lemma pders_set_One:

  shows "pders_set UNIV1 One = {Zero}"

by (auto split: if_splits)

lemma pders_set_Atom:

  shows "pders_set UNIV1 (Atom c) \<subseteq> {One, Zero}"

by (auto split: if_splits)

lemma pders_set_Plus:

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

by auto

lemma pders_set_Times_aux:

  assumes a: "s \<in> UNIV1"

  shows "pders_set (Suf s) r2 \<subseteq> pders_set UNIV1 r2"

using a by (auto simp add: Suf_def)

lemma pders_set_Times:

  shows "pders_set UNIV1 (Times r1 r2) \<subseteq> Times_set (pders_set UNIV1 r1) r2 \<union> pders_set UNIV1 r2"

apply(rule UN_least)

apply(rule subset_trans)

apply(rule pders_Times)

apply(simp)

apply(rule conjI) 

apply(auto)[1]

apply(rule subset_trans)

apply(rule pders_set_Times_aux)

apply(auto)

done

lemma pders_set_Star:

  shows "pders_set UNIV1 (Star r) \<subseteq> Times_set (pders_set UNIV1 r) (Star r)"

apply(rule UN_least)

apply(rule subset_trans)

apply(rule pders_Star)

apply(simp)

apply(simp add: Suf_def)

apply(auto)

done

lemma finite_Times_set:

  assumes a: "finite A"

  shows "finite (Times_set A r)"

using a by (auto)

lemma finite_pders_set_UNIV1:

  shows "finite (pders_set UNIV1 r)"

apply(induct r)

apply(simp)

apply(simp only: pders_set_One)

apply(simp)

apply(rule finite_subset)

apply(rule pders_set_Atom)

apply(simp)

apply(simp only: pders_set_Plus)

apply(simp)

apply(rule finite_subset)

apply(rule pders_set_Times)

apply(simp only: finite_Times_set finite_Un)

apply(simp)

apply(rule finite_subset)

apply(rule pders_set_Star)

apply(simp only: finite_Times_set)

done

lemma pders_set_UNIV_UNIV1:

  shows "pders_set UNIV r = pders [] r \<union> pders_set UNIV1 r"

apply(auto)

apply(rule_tac x="[]" in exI)

apply(simp)

done

lemma finite_pders_set_UNIV:

  shows "finite (pders_set UNIV r)"

unfolding pders_set_UNIV_UNIV1

by (simp add: finite_pders_set_UNIV1)

lemma finite_pders_set:

  shows "finite (pders_set A r)"

apply(rule rev_finite_subset)

apply(rule_tac r="r" in finite_pders_set_UNIV)

apply(auto)

done

lemma finite_pders:

  shows "finite (pders s r)"

using finite_pders_set[where A="{s}" and r="r"]

by simp

lemma finite_pders2:

  shows "finite {pders s r | s. s \<in> A}"

proof -

  have "{pders s r | s. s \<in> A} \<subseteq> Pow (pders_set A r)" by auto

  moreover

  have "finite (Pow (pders_set A r))"

    using finite_pders_set by simp

  ultimately 

  show "finite {pders s r | s. s \<in> A}"

    by(rule finite_subset)

qed

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

lemma Der_pder:

  shows "Der c (lang r) = \<Union> lang ` (pder c r)"

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

lemma Ders_pders:

  shows "Ders s (lang r) = \<Union> lang ` (pders s r)"

proof (induct s rule: rev_induct)

  case (snoc c s)

  have ih: "Ders s (lang r) = \<Union> lang ` (pders s r)" by fact

  have "Ders (s @ [c]) (lang r) = Ders [c] (Ders s (lang r))"

    by (simp add: Ders_append)

  also have "\<dots> = Der c (\<Union> lang ` (pders s r))" using ih

    by (simp add: Ders_singleton)

  also have "\<dots> = (\<Union>r\<in>pders s r. Der c (lang r))" 

    unfolding Der_def image_def by auto

  also have "\<dots> = (\<Union>r\<in>pders s r. (\<Union> lang `  (pder c r)))"

    by (simp add: Der_pder)

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

    by (simp add: pders_set_lang)

  also have "\<dots> = (\<Union>lang ` (pders (s @ [c]) r))"

    by simp

  finally show "Ders (s @ [c]) (lang r) = \<Union> lang ` pders (s @ [c]) r" .

qed (simp add: Ders_def)

lemma Ders_set_pders_set:

  shows "Ders_set A (lang r) = (\<Union> lang ` (pders_set A r))"

by (simp add: Ders_set_Ders Ders_pders)

subsection {* Relating derivatives and partial derivatives *}

lemma

  shows "(\<Union> lang ` (pder c r)) = lang (der c r)"

unfolding Der_der[symmetric] Der_pder by simp

lemma

  shows "(\<Union> lang ` (pders s r)) = lang (ders s r)"

unfolding Ders_ders[symmetric] Ders_pders by simp

subsection {*

  The second direction of the Myhill-Nerode theorem using

  partial derivatives.

*}

lemma Myhill_Nerode3:

  fixes r::"'a rexp"

  shows "finite (UNIV // \<approx>(lang r))"

proof -

  have "finite (UNIV // =(\<lambda>x. pders x r)=)"

  proof - 

    have "range (\<lambda>x. pders x r) = {pders s r | s. s \<in> UNIV}" by auto

    moreover 

    have "finite {pders s r | s. s \<in> UNIV}" by (rule finite_pders2)

    ultimately

    have "finite (range (\<lambda>x. pders x r))"

      by simp

    then show "finite (UNIV // =(\<lambda>x. pders x r)=)" 

      by (rule finite_eq_tag_rel)

  qed

  moreover 

  have "=(\<lambda>x. pders x r)= \<subseteq> \<approx>(lang r)"

    unfolding tag_eq_rel_def

    unfolding str_eq_def2

    unfolding MN_Rel_Ders

    unfolding Ders_pders

    by auto

  moreover 

  have "equiv UNIV =(\<lambda>x. pders x r)="

    unfolding equiv_def refl_on_def sym_def trans_def

    unfolding tag_eq_rel_def

    by auto

  moreover

  have "equiv UNIV (\<approx>(lang r))"

    unfolding equiv_def refl_on_def sym_def trans_def

    unfolding str_eq_rel_def

    by auto

  ultimately show "finite (UNIV // \<approx>(lang r))" 

    by (rule refined_partition_finite)

qed

end