--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Theories/Regular.thy Wed Mar 23 12:17:30 2011 +0000
@@ -0,0 +1,311 @@
+theory Regular
+imports Main Folds
+begin
+
+section {* Preliminary definitions *}
+
+type_synonym lang = "string set"
+
+
+text {* Sequential composition of two languages *}
+
+definition
+ Seq :: "lang \<Rightarrow> lang \<Rightarrow> lang" (infixr ";;" 100)
+where
+ "A ;; B = {s\<^isub>1 @ s\<^isub>2 | s\<^isub>1 s\<^isub>2. s\<^isub>1 \<in> A \<and> s\<^isub>2 \<in> B}"
+
+
+text {* Some properties of operator @{text ";;"}. *}
+
+lemma seq_add_left:
+ assumes a: "A = B"
+ shows "C ;; A = C ;; B"
+using a by simp
+
+lemma seq_union_distrib_right:
+ shows "(A \<union> B) ;; C = (A ;; C) \<union> (B ;; C)"
+unfolding Seq_def by auto
+
+lemma seq_union_distrib_left:
+ shows "C ;; (A \<union> B) = (C ;; A) \<union> (C ;; B)"
+unfolding Seq_def by auto
+
+lemma seq_intro:
+ assumes a: "x \<in> A" "y \<in> B"
+ shows "x @ y \<in> A ;; B "
+using a by (auto simp: Seq_def)
+
+lemma seq_assoc:
+ shows "(A ;; B) ;; C = A ;; (B ;; C)"
+unfolding Seq_def
+apply(auto)
+apply(blast)
+by (metis append_assoc)
+
+lemma seq_empty [simp]:
+ shows "A ;; {[]} = A"
+ and "{[]} ;; A = A"
+by (simp_all add: Seq_def)
+
+lemma seq_null [simp]:
+ shows "A ;; {} = {}"
+ and "{} ;; A = {}"
+by (simp_all add: Seq_def)
+
+
+text {* Power and Star of a language *}
+
+fun
+ pow :: "lang \<Rightarrow> nat \<Rightarrow> lang" (infixl "\<up>" 100)
+where
+ "A \<up> 0 = {[]}"
+| "A \<up> (Suc n) = A ;; (A \<up> n)"
+
+definition
+ Star :: "lang \<Rightarrow> lang" ("_\<star>" [101] 102)
+where
+ "A\<star> \<equiv> (\<Union>n. A \<up> n)"
+
+
+lemma star_start[intro]:
+ shows "[] \<in> A\<star>"
+proof -
+ have "[] \<in> A \<up> 0" by auto
+ then show "[] \<in> A\<star>" unfolding Star_def by blast
+qed
+
+lemma star_step [intro]:
+ assumes a: "s1 \<in> A"
+ and b: "s2 \<in> A\<star>"
+ shows "s1 @ s2 \<in> A\<star>"
+proof -
+ from b obtain n where "s2 \<in> A \<up> n" unfolding Star_def by auto
+ then have "s1 @ s2 \<in> A \<up> (Suc n)" using a by (auto simp add: Seq_def)
+ then show "s1 @ s2 \<in> A\<star>" unfolding Star_def by blast
+qed
+
+lemma star_induct[consumes 1, case_names start step]:
+ assumes a: "x \<in> A\<star>"
+ and b: "P []"
+ and c: "\<And>s1 s2. \<lbrakk>s1 \<in> A; s2 \<in> A\<star>; P s2\<rbrakk> \<Longrightarrow> P (s1 @ s2)"
+ shows "P x"
+proof -
+ from a obtain n where "x \<in> A \<up> n" unfolding Star_def by auto
+ then show "P x"
+ by (induct n arbitrary: x)
+ (auto intro!: b c simp add: Seq_def Star_def)
+qed
+
+lemma star_intro1:
+ assumes a: "x \<in> A\<star>"
+ and b: "y \<in> A\<star>"
+ shows "x @ y \<in> A\<star>"
+using a b
+by (induct rule: star_induct) (auto)
+
+lemma star_intro2:
+ assumes a: "y \<in> A"
+ shows "y \<in> A\<star>"
+proof -
+ from a have "y @ [] \<in> A\<star>" by blast
+ then show "y \<in> A\<star>" by simp
+qed
+
+lemma star_intro3:
+ assumes a: "x \<in> A\<star>"
+ and b: "y \<in> A"
+ shows "x @ y \<in> A\<star>"
+using a b by (blast intro: star_intro1 star_intro2)
+
+lemma star_cases:
+ shows "A\<star> = {[]} \<union> A ;; A\<star>"
+proof
+ { fix x
+ have "x \<in> A\<star> \<Longrightarrow> x \<in> {[]} \<union> A ;; A\<star>"
+ unfolding Seq_def
+ by (induct rule: star_induct) (auto)
+ }
+ then show "A\<star> \<subseteq> {[]} \<union> A ;; A\<star>" by auto
+next
+ show "{[]} \<union> A ;; A\<star> \<subseteq> A\<star>"
+ unfolding Seq_def by auto
+qed
+
+lemma star_decom:
+ assumes a: "x \<in> A\<star>" "x \<noteq> []"
+ shows "\<exists>a b. x = a @ b \<and> a \<noteq> [] \<and> a \<in> A \<and> b \<in> A\<star>"
+using a
+by (induct rule: star_induct) (blast)+
+
+lemma
+ shows seq_Union_left: "B ;; (\<Union>n. A \<up> n) = (\<Union>n. B ;; (A \<up> n))"
+ and seq_Union_right: "(\<Union>n. A \<up> n) ;; B = (\<Union>n. (A \<up> n) ;; B)"
+unfolding Seq_def by auto
+
+lemma seq_pow_comm:
+ shows "A ;; (A \<up> n) = (A \<up> n) ;; A"
+by (induct n) (simp_all add: seq_assoc[symmetric])
+
+lemma seq_star_comm:
+ shows "A ;; A\<star> = A\<star> ;; A"
+unfolding Star_def seq_Union_left
+unfolding seq_pow_comm seq_Union_right
+by simp
+
+
+text {* Two lemmas about the length of strings in @{text "A \<up> n"} *}
+
+lemma pow_length:
+ assumes a: "[] \<notin> A"
+ and b: "s \<in> A \<up> Suc n"
+ shows "n < length s"
+using b
+proof (induct n arbitrary: s)
+ case 0
+ have "s \<in> A \<up> Suc 0" by fact
+ with a have "s \<noteq> []" by auto
+ then show "0 < length s" by auto
+next
+ case (Suc n)
+ have ih: "\<And>s. s \<in> A \<up> Suc n \<Longrightarrow> n < length s" by fact
+ have "s \<in> A \<up> Suc (Suc n)" by fact
+ then obtain s1 s2 where eq: "s = s1 @ s2" and *: "s1 \<in> A" and **: "s2 \<in> A \<up> Suc n"
+ by (auto simp add: Seq_def)
+ from ih ** have "n < length s2" by simp
+ moreover have "0 < length s1" using * a by auto
+ ultimately show "Suc n < length s" unfolding eq
+ by (simp only: length_append)
+qed
+
+lemma seq_pow_length:
+ assumes a: "[] \<notin> A"
+ and b: "s \<in> B ;; (A \<up> Suc n)"
+ shows "n < length s"
+proof -
+ from b obtain s1 s2 where eq: "s = s1 @ s2" and *: "s2 \<in> A \<up> Suc n"
+ unfolding Seq_def by auto
+ from * have " n < length s2" by (rule pow_length[OF a])
+ then show "n < length s" using eq by simp
+qed
+
+
+section {* A modified version of Arden's lemma *}
+
+text {* A helper lemma for Arden *}
+
+lemma arden_helper:
+ assumes eq: "X = X ;; A \<union> B"
+ shows "X = X ;; (A \<up> Suc n) \<union> (\<Union>m\<in>{0..n}. B ;; (A \<up> m))"
+proof (induct n)
+ case 0
+ show "X = X ;; (A \<up> Suc 0) \<union> (\<Union>(m::nat)\<in>{0..0}. B ;; (A \<up> m))"
+ using eq by simp
+next
+ case (Suc n)
+ have ih: "X = X ;; (A \<up> Suc n) \<union> (\<Union>m\<in>{0..n}. B ;; (A \<up> m))" by fact
+ also have "\<dots> = (X ;; A \<union> B) ;; (A \<up> Suc n) \<union> (\<Union>m\<in>{0..n}. B ;; (A \<up> m))" using eq by simp
+ also have "\<dots> = X ;; (A \<up> Suc (Suc n)) \<union> (B ;; (A \<up> Suc n)) \<union> (\<Union>m\<in>{0..n}. B ;; (A \<up> m))"
+ by (simp add: seq_union_distrib_right seq_assoc)
+ also have "\<dots> = X ;; (A \<up> Suc (Suc n)) \<union> (\<Union>m\<in>{0..Suc n}. B ;; (A \<up> m))"
+ by (auto simp add: le_Suc_eq)
+ finally show "X = X ;; (A \<up> Suc (Suc n)) \<union> (\<Union>m\<in>{0..Suc n}. B ;; (A \<up> m))" .
+qed
+
+theorem arden:
+ assumes nemp: "[] \<notin> A"
+ shows "X = X ;; A \<union> B \<longleftrightarrow> X = B ;; A\<star>"
+proof
+ assume eq: "X = B ;; A\<star>"
+ have "A\<star> = {[]} \<union> A\<star> ;; A"
+ unfolding seq_star_comm[symmetric]
+ by (rule star_cases)
+ then have "B ;; A\<star> = B ;; ({[]} \<union> A\<star> ;; A)"
+ by (rule seq_add_left)
+ also have "\<dots> = B \<union> B ;; (A\<star> ;; A)"
+ unfolding seq_union_distrib_left by simp
+ also have "\<dots> = B \<union> (B ;; A\<star>) ;; A"
+ by (simp only: seq_assoc)
+ finally show "X = X ;; A \<union> B"
+ using eq by blast
+next
+ assume eq: "X = X ;; A \<union> B"
+ { fix n::nat
+ have "B ;; (A \<up> n) \<subseteq> X" using arden_helper[OF eq, of "n"] by auto }
+ then have "B ;; A\<star> \<subseteq> X"
+ unfolding Seq_def Star_def UNION_def by auto
+ moreover
+ { fix s::string
+ obtain k where "k = length s" by auto
+ then have not_in: "s \<notin> X ;; (A \<up> Suc k)"
+ using seq_pow_length[OF nemp] by blast
+ assume "s \<in> X"
+ then have "s \<in> X ;; (A \<up> Suc k) \<union> (\<Union>m\<in>{0..k}. B ;; (A \<up> m))"
+ using arden_helper[OF eq, of "k"] by auto
+ then have "s \<in> (\<Union>m\<in>{0..k}. B ;; (A \<up> m))" using not_in by auto
+ moreover
+ have "(\<Union>m\<in>{0..k}. B ;; (A \<up> m)) \<subseteq> (\<Union>n. B ;; (A \<up> n))" by auto
+ ultimately
+ have "s \<in> B ;; A\<star>"
+ unfolding seq_Union_left Star_def by auto }
+ then have "X \<subseteq> B ;; A\<star>" by auto
+ ultimately
+ show "X = B ;; A\<star>" by simp
+qed
+
+
+section {* Regular Expressions *}
+
+datatype rexp =
+ NULL
+| EMPTY
+| CHAR char
+| SEQ rexp rexp
+| ALT rexp rexp
+| STAR rexp
+
+
+text {*
+ The function @{text L} is overloaded, with the idea that @{text "L x"}
+ evaluates to the language represented by the object @{text x}.
+*}
+
+consts L:: "'a \<Rightarrow> lang"
+
+overloading L_rexp \<equiv> "L:: rexp \<Rightarrow> lang"
+begin
+fun
+ L_rexp :: "rexp \<Rightarrow> lang"
+where
+ "L_rexp (NULL) = {}"
+ | "L_rexp (EMPTY) = {[]}"
+ | "L_rexp (CHAR c) = {[c]}"
+ | "L_rexp (SEQ r1 r2) = (L_rexp r1) ;; (L_rexp r2)"
+ | "L_rexp (ALT r1 r2) = (L_rexp r1) \<union> (L_rexp r2)"
+ | "L_rexp (STAR r) = (L_rexp r)\<star>"
+end
+
+
+text {* ALT-combination for a set of regular expressions *}
+
+abbreviation
+ Setalt ("\<Uplus>_" [1000] 999)
+where
+ "\<Uplus>A \<equiv> folds ALT NULL A"
+
+text {*
+ For finite sets, @{term Setalt} is preserved under @{term L}.
+*}
+
+lemma folds_alt_simp [simp]:
+ fixes rs::"rexp set"
+ assumes a: "finite rs"
+ shows "L (\<Uplus>rs) = \<Union> (L ` rs)"
+unfolding folds_def
+apply(rule set_eqI)
+apply(rule someI2_ex)
+apply(rule_tac finite_imp_fold_graph[OF a])
+apply(erule fold_graph.induct)
+apply(auto)
+done
+
+end
\ No newline at end of file