--- a/AFP-Submission/Derivatives.thy Tue Jun 14 12:37:46 2016 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,370 +0,0 @@
-section "Derivatives of regular expressions"
-
-(* Author: Christian Urban *)
-
-theory Derivatives
-imports Regular_Exp
-begin
-
-text{* This theory is based on work by Brozowski \cite{Brzozowski64} and Antimirov \cite{Antimirov95}. *}
-
-subsection {* Brzozowski's derivatives of regular expressions *}
-
-primrec
- deriv :: "'a \<Rightarrow> 'a rexp \<Rightarrow> 'a rexp"
-where
- "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)"
-
-primrec
- derivs :: "'a list \<Rightarrow> 'a rexp \<Rightarrow> 'a rexp"
-where
- "derivs [] r = r"
-| "derivs (c # s) r = derivs s (deriv c r)"
-
-
-lemma atoms_deriv_subset: "atoms (deriv x r) \<subseteq> atoms r"
-by (induction r) (auto)
-
-lemma atoms_derivs_subset: "atoms (derivs w r) \<subseteq> atoms r"
-by (induction w arbitrary: r) (auto dest: atoms_deriv_subset[THEN subsetD])
-
-lemma lang_deriv: "lang (deriv c r) = Deriv c (lang r)"
-by (induct r) (simp_all add: nullable_iff)
-
-lemma lang_derivs: "lang (derivs s r) = Derivs s (lang r)"
-by (induct s arbitrary: r) (simp_all add: lang_deriv)
-
-text {* A regular expression matcher: *}
-
-definition matcher :: "'a rexp \<Rightarrow> 'a list \<Rightarrow> bool" where
-"matcher r s = nullable (derivs s r)"
-
-lemma matcher_correctness: "matcher r s \<longleftrightarrow> s \<in> lang r"
-by (induct s arbitrary: r)
- (simp_all add: nullable_iff lang_deriv matcher_def Deriv_def)
-
-
-subsection {* Antimirov's partial derivatives *}
-
-abbreviation
- "Timess rs r \<equiv> (\<Union>r' \<in> rs. {Times r' r})"
-
-primrec
- pderiv :: "'a \<Rightarrow> 'a rexp \<Rightarrow> 'a rexp set"
-where
- "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)"
-
-primrec
- pderivs :: "'a list \<Rightarrow> 'a rexp \<Rightarrow> ('a rexp) set"
-where
- "pderivs [] r = {r}"
-| "pderivs (c # s) r = \<Union> (pderivs s ` pderiv c r)"
-
-abbreviation
- 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)"
-by (induct s1 arbitrary: r) (simp_all)
-
-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))"
-by (induct s) (simp_all)
-
-lemma pderivs_Atom:
- shows "pderivs s (Atom c) \<subseteq> {Atom c, One}"
-by (induct s) (simp_all)
-
-subsection {* Relating left-quotients and partial derivatives *}
-
-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)"
-proof (induct s arbitrary: r)
- case (Cons c s)
- 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)))"
- using ih by auto
- 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 derivatives and partial derivatives *}
-
-lemma deriv_pderiv:
- shows "\<Union> (lang ` (pderiv c r)) = lang (deriv c r)"
-unfolding lang_deriv Deriv_pderiv by simp
-
-lemma derivs_pderivs:
- shows "\<Union> (lang ` (pderivs s r)) = lang (derivs s r)"
-unfolding lang_derivs Derivs_pderivs by simp
-
-
-subsection {* Finiteness property of partial derivatives *}
-
-definition
- 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)
-
-definition
- "UNIV1 \<equiv> UNIV - {[]}"
-
-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(auto)
-apply(frule rev_subsetD)
-apply(rule pderivs_Atom)
-apply(simp)
-apply(case_tac xa)
-apply(auto split: if_splits)
-done
-
-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
-
-
-text {* Non-empty suffixes of a string (needed for the cases of @{const Times} and @{const Star} below) *}
-
-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 conc_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: conc_def)
-
-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)"
-proof (induct s rule: rev_induct)
- case (snoc c s)
- have ih: "pderivs s (Times r1 r2) \<subseteq> Timess (pderivs s r1) r2 \<union> (pderivs_lang (PSuf s) r2)"
- by fact
- 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))"
- using ih by fast
- also have "\<dots> = pderiv_set c (Timess (pderivs s r1) r2) \<union> pderiv_set c (pderivs_lang (PSuf s) r2)"
- by (simp)
- 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"
- by auto
- also
- have "\<dots> \<subseteq> Timess (pderiv_set c (pderivs s r1)) r2 \<union> pderiv c r2 \<union> pderivs_lang (PSuf s @@ {[c]}) r2"
- by (auto simp add: if_splits)
- 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)
- finally show ?case .
-qed (simp)
-
-lemma pderivs_lang_Times_aux1:
- assumes a: "s \<in> UNIV1"
- 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:
- assumes a: "s \<in> UNIV1"
- 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 subset_trans)
-apply(rule pderivs_Times)
-using pderivs_lang_Times_aux1 pderivs_lang_Times_aux2
-apply(blast)
-done
-
-lemma pderivs_Star:
- assumes a: "s \<noteq> []"
- shows "pderivs s (Star r) \<subseteq> Timess (pderivs_lang (PSuf s) r) (Star r)"
-using a
-proof (induct s rule: rev_induct)
- case (snoc c s)
- have ih: "s \<noteq> [] \<Longrightarrow> pderivs s (Star r) \<subseteq> Timess (pderivs_lang (PSuf s) r) (Star r)" by fact
- { assume asm: "s \<noteq> []"
- 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))"
- using ih[OF asm] by fast
- also have "\<dots> \<subseteq> Timess (pderiv_set c (pderivs_lang (PSuf s) r)) (Star r) \<union> pderiv c (Star r)"
- by (auto split: if_splits)
- 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)
- finally have ?case .
- }
- moreover
- { assume asm: "s = []"
- then have ?case by (auto simp add: pderivs_lang_def pderivs_snoc PSuf_def)
- }
- ultimately show ?case by blast
-qed (simp)
-
-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 subset_trans)
-apply(rule pderivs_Star)
-apply(simp add: UNIV1_def)
-apply(simp add: UNIV1_def PSuf_def)
-apply(auto simp add: pderivs_lang_def)
-done
-
-lemma finite_Timess [simp]:
- assumes a: "finite A"
- shows "finite (Timess A r)"
-using a by auto
-
-lemma finite_pderivs_lang_UNIV1:
- shows "finite (pderivs_lang UNIV1 r)"
-apply(induct r)
-apply(simp_all add:
- finite_subset[OF pderivs_lang_Times]
- finite_subset[OF pderivs_lang_Star])
-done
-
-lemma pderivs_lang_UNIV:
- shows "pderivs_lang UNIV r = pderivs [] r \<union> pderivs_lang UNIV1 r"
-unfolding UNIV1_def pderivs_lang_def
-by blast
-
-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)
-
-
-text{* The following relationship between the alphabetic width of regular expressions
-(called @{text awidth} below) and the number of partial derivatives was proved
-by Antimirov~\cite{Antimirov95} and formalized by Max Haslbeck. *}
-
-fun awidth :: "'a rexp \<Rightarrow> nat" where
-"awidth Zero = 0" |
-"awidth One = 0" |
-"awidth (Atom a) = 1" |
-"awidth (Plus r1 r2) = awidth r1 + awidth r2" |
-"awidth (Times r1 r2) = awidth r1 + awidth r2" |
-"awidth (Star r1) = awidth r1"
-
-lemma card_Timess_pderivs_lang_le:
- "card (Timess (pderivs_lang A r) s) \<le> card (pderivs_lang A r)"
-by (metis card_image_le finite_pderivs_lang image_eq_UN)
-
-lemma card_pderivs_lang_UNIV1_le_awidth: "card (pderivs_lang UNIV1 r) \<le> awidth r"
-proof (induction r)
- case (Plus r1 r2)
- have "card (pderivs_lang UNIV1 (Plus r1 r2)) = card (pderivs_lang UNIV1 r1 \<union> pderivs_lang UNIV1 r2)" by simp
- also have "\<dots> \<le> card (pderivs_lang UNIV1 r1) + card (pderivs_lang UNIV1 r2)"
- by(simp add: card_Un_le)
- also have "\<dots> \<le> awidth (Plus r1 r2)" using Plus.IH by simp
- finally show ?case .
-next
- case (Times r1 r2)
- have "card (pderivs_lang UNIV1 (Times r1 r2)) \<le> card (Timess (pderivs_lang UNIV1 r1) r2 \<union> pderivs_lang UNIV1 r2)"
- by (simp add: card_mono finite_pderivs_lang pderivs_lang_Times)
- also have "\<dots> \<le> card (Timess (pderivs_lang UNIV1 r1) r2) + card (pderivs_lang UNIV1 r2)"
- by (simp add: card_Un_le)
- also have "\<dots> \<le> card (pderivs_lang UNIV1 r1) + card (pderivs_lang UNIV1 r2)"
- by (simp add: card_Timess_pderivs_lang_le)
- also have "\<dots> \<le> awidth (Times r1 r2)" using Times.IH by simp
- finally show ?case .
-next
- case (Star r)
- have "card (pderivs_lang UNIV1 (Star r)) \<le> card (Timess (pderivs_lang UNIV1 r) (Star r))"
- by (simp add: card_mono finite_pderivs_lang pderivs_lang_Star)
- also have "\<dots> \<le> card (pderivs_lang UNIV1 r)" by (rule card_Timess_pderivs_lang_le)
- also have "\<dots> \<le> awidth (Star r)" by (simp add: Star.IH)
- finally show ?case .
-qed (auto)
-
-text{* Antimirov's Theorem 3.4: *}
-theorem card_pderivs_lang_UNIV_le_awidth: "card (pderivs_lang UNIV r) \<le> awidth r + 1"
-proof -
- have "card (insert r (pderivs_lang UNIV1 r)) \<le> Suc (card (pderivs_lang UNIV1 r))"
- by(auto simp: card_insert_if[OF finite_pderivs_lang_UNIV1])
- also have "\<dots> \<le> Suc (awidth r)" by(simp add: card_pderivs_lang_UNIV1_le_awidth)
- finally show ?thesis by(simp add: pderivs_lang_UNIV)
-qed
-
-text{* Antimirov's Corollary 3.5: *}
-corollary card_pderivs_lang_le_awidth: "card (pderivs_lang A r) \<le> awidth r + 1"
-by(rule order_trans[OF
- card_mono[OF finite_pderivs_lang_UNIV pderivs_lang_subset[OF subset_UNIV]]
- card_pderivs_lang_UNIV_le_awidth])
-
-end
\ No newline at end of file