--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/AFP-Submission/Derivatives.thy	Tue May 24 11:36:21 2016 +0100
@@ -0,0 +1,370 @@
+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