theory PDerivs

  imports Spec

begin

abbreviation

  "SEQs rs r \<equiv> (\<Union>r' \<in> rs. {SEQ r' r})"

lemma SEQs_eq_image:

  "SEQs rs r = (\<lambda>r'. SEQ r' r) ` rs"

  by auto

primrec

  pder :: "char \<Rightarrow> rexp \<Rightarrow> rexp set"

where

  "pder c ZERO = {}"

| "pder c ONE = {}"

| "pder c (CHAR d) = (if c = d then {ONE} else {})"

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

| "pder c (SEQ r1 r2) = 

    (if nullable r1 then SEQs (pder c r1) r2 \<union> pder c r2 else SEQs (pder c r1) r2)"

| "pder c (STAR r) = SEQs (pder c r) (STAR r)"

primrec

  pders :: "char list \<Rightarrow> rexp \<Rightarrow> rexp set"

where

  "pders [] r = {r}"

| "pders (c # s) r = \<Union> (pders s ` pder c r)"

abbreviation

 pder_set :: "char \<Rightarrow> rexp set \<Rightarrow> rexp set"

where

  "pder_set c rs \<equiv> \<Union> (pder c ` rs)"

abbreviation

  pders_set :: "char list \<Rightarrow> rexp set \<Rightarrow> rexp set"

where

  "pders_set s rs \<equiv> \<Union> (pders s ` rs)"

lemma pders_append:

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

by (induct s1 arbitrary: r) (simp_all)

lemma pders_snoc:

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

by (simp add: pders_append)

lemma pders_simps [simp]:

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

  and   "pders s ONE = (if s = [] then {ONE} else {})"

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

by (induct s) (simp_all)

lemma pders_CHAR:

  shows "pders s (CHAR c) \<subseteq> {CHAR c, ONE}"

by (induct s) (simp_all)

subsection \<open>Relating left-quotients and partial derivatives\<close>

lemma Sequ_UNION_distrib:

shows "A ;; \<Union>(M ` I) = \<Union>((\<lambda>i. A ;; M i) ` I)"

and   "\<Union>(M ` I) ;; A = \<Union>((\<lambda>i. M i ;; A) ` I)"

by (auto simp add: Sequ_def)

lemma Der_pder:

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

by (induct r) (simp_all add: nullable_correctness Sequ_UNION_distrib)

lemma Ders_pders:

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

proof (induct s arbitrary: r)

  case (Cons c s)

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

  have "Ders (c # s) (L r) = Ders s (Der c (L r))" by (simp add: Ders_def Der_def)

  also have "\<dots> = Ders s (\<Union> (L ` pder c r))" by (simp add: Der_pder)

  also have "\<dots> = (\<Union>A\<in>(L ` (pder c r)). (Ders s A))"

    by (auto simp add:  Ders_def)

  also have "\<dots> = \<Union> (L ` (pders_set s (pder c r)))"

    using ih by auto

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

  finally show "Ders (c # s) (L r) = \<Union> (L ` pders (c # s) r)" .

qed (simp add: Ders_def)

subsection \<open>Relating derivatives and partial derivatives\<close>

lemma der_pder:

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

unfolding der_correctness Der_pder by simp

lemma ders_pders:

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

unfolding der_correctness ders_correctness Ders_pders by simp

subsection \<open>Finiteness property of partial derivatives\<close>

definition

  pders_Set :: "string set \<Rightarrow> rexp \<Rightarrow> rexp set"

where

  "pders_Set A r \<equiv> \<Union>x \<in> A. pders x r"

lemma pders_Set_subsetI:

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

  shows "pders_Set A r \<subseteq> C"

using assms unfolding pders_Set_def by (rule UN_least)

lemma pders_Set_union:

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

by (simp add: pders_Set_def)

lemma pders_Set_subset:

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

by (auto simp add: pders_Set_def)

definition

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

lemma pders_Set_ZERO [simp]:

  shows "pders_Set UNIV1 ZERO = {}"

unfolding UNIV1_def pders_Set_def by auto

lemma pders_Set_ONE [simp]:

  shows "pders_Set UNIV1 ONE = {}"

unfolding UNIV1_def pders_Set_def by (auto split: if_splits)

lemma pders_Set_CHAR [simp]:

  shows "pders_Set UNIV1 (CHAR c) = {ONE}"

unfolding UNIV1_def pders_Set_def 

apply(auto)

apply(frule rev_subsetD)

apply(rule pders_CHAR)

apply(simp)

apply(case_tac xa)

apply(auto split: if_splits)

done

lemma pders_Set_ALT [simp]:

  shows "pders_Set UNIV1 (ALT r1 r2) = pders_Set UNIV1 r1 \<union> pders_Set UNIV1 r2"

unfolding UNIV1_def pders_Set_def by auto

text \<open>Non-empty suffixes of a string (needed for the cases of @{const SEQ} and @{const STAR} below)\<close>

definition

  "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 Sequ_def

by (auto simp add: append_eq_append_conv2 append_eq_Cons_conv)

lemma PSuf_Union:

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

by (auto simp add: Sequ_def)

lemma pders_Set_snoc:

  shows "pders_Set (PSuf s ;; {[c]}) r = (pder_set c (pders_Set (PSuf s) r))"

unfolding pders_Set_def

by (simp add: PSuf_Union pders_snoc)

lemma pderivs_SEQ:

  shows "pders s (SEQ r1 r2) \<subseteq> SEQs (pders s r1) r2 \<union> (pders_Set (PSuf s) r2)"

proof (induct s rule: rev_induct)

  case (snoc c s)

  have ih: "pders s (SEQ r1 r2) \<subseteq> SEQs (pders s r1) r2 \<union> (pders_Set (PSuf s) r2)" 

    by fact

  have "pders (s @ [c]) (SEQ r1 r2) = pder_set c (pders s (SEQ r1 r2))" 

    by (simp add: pders_snoc)

  also have "\<dots> \<subseteq> pder_set c (SEQs (pders s r1) r2 \<union> (pders_Set (PSuf s) r2))"

    using ih by fastforce

  also have "\<dots> = pder_set c (SEQs (pders s r1) r2) \<union> pder_set c (pders_Set (PSuf s) r2)"

    by (simp)

  also have "\<dots> = pder_set c (SEQs (pders s r1) r2) \<union> pders_Set (PSuf s ;; {[c]}) r2"

    by (simp add: pders_Set_snoc)

  also 

  have "\<dots> \<subseteq> pder_set c (SEQs (pders s r1) r2) \<union> pder c r2 \<union> pders_Set (PSuf s ;; {[c]}) r2"

    by auto

  also 

  have "\<dots> \<subseteq> SEQs (pder_set c (pders s r1)) r2 \<union> pder c r2 \<union> pders_Set (PSuf s ;; {[c]}) r2"

    by (auto simp add: if_splits)

  also have "\<dots> = SEQs (pders (s @ [c]) r1) r2 \<union> pder c r2 \<union> pders_Set (PSuf s ;; {[c]}) r2"

    by (simp add: pders_snoc)

  also have "\<dots> \<subseteq> SEQs (pders (s @ [c]) r1) r2 \<union> pders_Set (PSuf (s @ [c])) r2"

    unfolding pders_Set_def by (auto simp add: PSuf_snoc)  

  finally show ?case .

qed (simp) 

lemma pders_Set_SEQ_aux1:

  assumes a: "s \<in> UNIV1"

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

using a unfolding UNIV1_def PSuf_def pders_Set_def by auto

lemma pders_Set_SEQ_aux2:

  assumes a: "s \<in> UNIV1"

  shows "SEQs (pders s r1) r2 \<subseteq> SEQs (pders_Set UNIV1 r1) r2"

using a unfolding pders_Set_def by auto

lemma pders_Set_SEQ:

  shows "pders_Set UNIV1 (SEQ r1 r2) \<subseteq> SEQs (pders_Set UNIV1 r1) r2 \<union> pders_Set UNIV1 r2"

apply(rule pders_Set_subsetI)

apply(rule subset_trans)

apply(rule pderivs_SEQ)

using pders_Set_SEQ_aux1 pders_Set_SEQ_aux2

apply auto

apply blast

done

lemma pders_STAR:

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

  shows "pders s (STAR r) \<subseteq> SEQs (pders_Set (PSuf s) 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> SEQs (pders_Set (PSuf s) r) (STAR r)" by fact

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

    have "pders (s @ [c]) (STAR r) = pder_set c (pders s (STAR r))" by (simp add: pders_snoc)

    also have "\<dots> \<subseteq> pder_set c (SEQs (pders_Set (PSuf s) r) (STAR r))"

      using ih[OF asm] by fast

    also have "\<dots> \<subseteq> SEQs (pder_set c (pders_Set (PSuf s) r)) (STAR r) \<union> pder c (STAR r)"

      by (auto split: if_splits)

    also have "\<dots> \<subseteq> SEQs (pders_Set (PSuf (s @ [c])) r) (STAR r) \<union> (SEQs (pder c r) (STAR r))"

      by (simp only: PSuf_snoc pders_Set_snoc pders_Set_union)

         (auto simp add: pders_Set_def)

    also have "\<dots> = SEQs (pders_Set (PSuf (s @ [c])) r) (STAR r)"

      by (auto simp add: PSuf_snoc PSuf_Union pders_snoc pders_Set_def)

    finally have ?case .

  }

  moreover

  { assume asm: "s = []"

    then have ?case by (auto simp add: pders_Set_def pders_snoc PSuf_def)

  }

  ultimately show ?case by blast

qed (simp)

lemma pders_Set_STAR:

  shows "pders_Set UNIV1 (STAR r) \<subseteq> SEQs (pders_Set UNIV1 r) (STAR r)"

apply(rule pders_Set_subsetI)

apply(rule subset_trans)

apply(rule pders_STAR)

apply(simp add: UNIV1_def)

apply(simp add: UNIV1_def PSuf_def)

apply(auto simp add: pders_Set_def)

done

lemma finite_SEQs [simp]:

  assumes a: "finite A"

  shows "finite (SEQs A r)"

using a by auto

lemma finite_pders_Set_UNIV1:

  shows "finite (pders_Set UNIV1 r)"

apply(induct r)

apply(simp_all add: 

  finite_subset[OF pders_Set_SEQ]

  finite_subset[OF pders_Set_STAR])

done

lemma pders_Set_UNIV:

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

unfolding UNIV1_def pders_Set_def

by blast

lemma finite_pders_Set_UNIV:

  shows "finite (pders_Set UNIV r)"

unfolding pders_Set_UNIV

by (simp add: finite_pders_Set_UNIV1)

lemma finite_pders_set:

  shows "finite (pders_Set A r)"

by (metis finite_pders_Set_UNIV pders_Set_subset rev_finite_subset subset_UNIV)

text \<open>The following relationship between the alphabetic width of regular expressions

(called \<open>awidth\<close> below) and the number of partial derivatives was proved

by Antimirov~\cite{Antimirov95} and formalized by Max Haslbeck.\<close>

fun awidth :: "rexp \<Rightarrow> nat" where

"awidth ZERO = 0" |

"awidth ONE = 0" |

"awidth (CHAR a) = 1" |

"awidth (ALT r1 r2) = awidth r1 + awidth r2" |

"awidth (SEQ r1 r2) = awidth r1 + awidth r2" |

"awidth (STAR r1) = awidth r1"

lemma card_SEQs_pders_Set_le:

  shows  "card (SEQs (pders_Set A r) s) \<le> card (pders_Set A r)"

  using finite_pders_set 

  unfolding SEQs_eq_image 

  by (rule card_image_le)

lemma card_pders_set_UNIV1_le_awidth: 

  shows "card (pders_Set UNIV1 r) \<le> awidth r"

proof (induction r)

  case (ALT r1 r2)

  have "card (pders_Set UNIV1 (ALT r1 r2)) = card (pders_Set UNIV1 r1 \<union> pders_Set UNIV1 r2)" by simp

  also have "\<dots> \<le> card (pders_Set UNIV1 r1) + card (pders_Set UNIV1 r2)"

    by(simp add: card_Un_le)

  also have "\<dots> \<le> awidth (ALT r1 r2)" using ALT.IH by simp

  finally show ?case .

next

  case (SEQ r1 r2)

  have "card (pders_Set UNIV1 (SEQ r1 r2)) \<le> card (SEQs (pders_Set UNIV1 r1) r2 \<union> pders_Set UNIV1 r2)"

    by (simp add: card_mono finite_pders_set pders_Set_SEQ)

  also have "\<dots> \<le> card (SEQs (pders_Set UNIV1 r1) r2) + card (pders_Set UNIV1 r2)"

    by (simp add: card_Un_le)

  also have "\<dots> \<le> card (pders_Set UNIV1 r1) + card (pders_Set UNIV1 r2)"

    by (simp add: card_SEQs_pders_Set_le)

  also have "\<dots> \<le> awidth (SEQ r1 r2)" using SEQ.IH by simp

  finally show ?case .

next

  case (STAR r)

  have "card (pders_Set UNIV1 (STAR r)) \<le> card (SEQs (pders_Set UNIV1 r) (STAR r))"

    by (simp add: card_mono finite_pders_set pders_Set_STAR)

  also have "\<dots> \<le> card (pders_Set UNIV1 r)" by (rule card_SEQs_pders_Set_le)

  also have "\<dots> \<le> awidth (STAR r)" by (simp add: STAR.IH)

  finally show ?case .

qed (auto)

text\<open>Antimirov's Theorem 3.4:\<close>

theorem card_pders_set_UNIV_le_awidth: 

  shows "card (pders_Set UNIV r) \<le> awidth r + 1"

proof -

  have "card (insert r (pders_Set UNIV1 r)) \<le> Suc (card (pders_Set UNIV1 r))"

    by(auto simp: card_insert_if[OF finite_pders_Set_UNIV1])

  also have "\<dots> \<le> Suc (awidth r)" by(simp add: card_pders_set_UNIV1_le_awidth)

  finally show ?thesis by(simp add: pders_Set_UNIV)

qed 

text\<open>Antimirov's Corollary 3.5:\<close>

corollary card_pders_set_le_awidth: 

  shows "card (pders_Set A r) \<le> awidth r + 1"

proof -

  have "card (pders_Set A r) \<le> card (pders_Set UNIV r)"

    by (simp add: card_mono finite_pders_set pders_Set_subset)

  also have "... \<le> awidth r + 1"

    by (rule card_pders_set_UNIV_le_awidth)

  finally show "card (pders_Set A r) \<le> awidth r + 1" by simp

qed

end