--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Regular.thy	Tue May 31 20:32:49 2011 +0000
@@ -0,0 +1,302 @@
+(* Author: Christian Urban, Xingyuan Zhang, Chunhan Wu *)
+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 "\<cdot>" 100)
+where 
+  "A \<cdot> 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 "\<cdot>"}. *}
+
+lemma seq_add_left:
+  assumes a: "A = B"
+  shows "C \<cdot> A = C \<cdot> B"
+using a by simp
+
+lemma seq_union_distrib_right:
+  shows "(A \<union> B) \<cdot> C = (A \<cdot> C) \<union> (B \<cdot> C)"
+unfolding Seq_def by auto
+
+lemma seq_union_distrib_left:
+  shows "C \<cdot> (A \<union> B) = (C \<cdot> A) \<union> (C \<cdot> B)"
+unfolding Seq_def by  auto
+
+lemma seq_intro:
+  assumes a: "x \<in> A" "y \<in> B"
+  shows "x @ y \<in> A \<cdot> B "
+using a by (auto simp: Seq_def)
+
+lemma seq_assoc:
+  shows "(A \<cdot> B) \<cdot> C = A \<cdot> (B \<cdot> C)"
+unfolding Seq_def
+apply(auto)
+apply(blast)
+by (metis append_assoc)
+
+lemma seq_empty [simp]:
+  shows "A \<cdot> {[]} = A"
+  and   "{[]} \<cdot> A = A"
+by (simp_all add: Seq_def)
+
+lemma seq_null [simp]:
+  shows "A \<cdot> {} = {}"
+  and   "{} \<cdot> 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 \<cdot> (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 \<cdot> A\<star>"
+proof
+  { fix x
+    have "x \<in> A\<star> \<Longrightarrow> x \<in> {[]} \<union> A \<cdot> A\<star>"
+      unfolding Seq_def
+    by (induct rule: star_induct) (auto)
+  }
+  then show "A\<star> \<subseteq> {[]} \<union> A \<cdot> A\<star>" by auto
+next
+  show "{[]} \<union> A \<cdot> 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 seq_Union_left: 
+  shows "B \<cdot> (\<Union>n. A \<up> n) = (\<Union>n. B \<cdot> (A \<up> n))"
+unfolding Seq_def by auto
+
+lemma seq_Union_right: 
+  shows "(\<Union>n. A \<up> n) \<cdot> B = (\<Union>n. (A \<up> n) \<cdot> B)"
+unfolding Seq_def by auto
+
+lemma seq_pow_comm:
+  shows "A \<cdot> (A \<up> n) = (A \<up> n) \<cdot> A"
+by (induct n) (simp_all add: seq_assoc[symmetric])
+
+lemma seq_star_comm:
+  shows "A \<cdot> A\<star> = A\<star> \<cdot> 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 \<cdot> (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 \<cdot> A \<union> B"
+  shows "X = X \<cdot> (A \<up> Suc n) \<union> (\<Union>m\<in>{0..n}. B \<cdot> (A \<up> m))"
+proof (induct n)
+  case 0 
+  show "X = X \<cdot> (A \<up> Suc 0) \<union> (\<Union>(m::nat)\<in>{0..0}. B \<cdot> (A \<up> m))"
+    using eq by simp
+next
+  case (Suc n)
+  have ih: "X = X \<cdot> (A \<up> Suc n) \<union> (\<Union>m\<in>{0..n}. B \<cdot> (A \<up> m))" by fact
+  also have "\<dots> = (X \<cdot> A \<union> B) \<cdot> (A \<up> Suc n) \<union> (\<Union>m\<in>{0..n}. B \<cdot> (A \<up> m))" using eq by simp
+  also have "\<dots> = X \<cdot> (A \<up> Suc (Suc n)) \<union> (B \<cdot> (A \<up> Suc n)) \<union> (\<Union>m\<in>{0..n}. B \<cdot> (A \<up> m))"
+    by (simp add: seq_union_distrib_right seq_assoc)
+  also have "\<dots> = X \<cdot> (A \<up> Suc (Suc n)) \<union> (\<Union>m\<in>{0..Suc n}. B \<cdot> (A \<up> m))"
+    by (auto simp add: le_Suc_eq)
+  finally show "X = X \<cdot> (A \<up> Suc (Suc n)) \<union> (\<Union>m\<in>{0..Suc n}. B \<cdot> (A \<up> m))" .
+qed
+
+theorem arden:
+  assumes nemp: "[] \<notin> A"
+  shows "X = X \<cdot> A \<union> B \<longleftrightarrow> X = B \<cdot> A\<star>"
+proof
+  assume eq: "X = B \<cdot> A\<star>"
+  have "A\<star> = {[]} \<union> A\<star> \<cdot> A" 
+    unfolding seq_star_comm[symmetric]
+    by (rule star_cases)
+  then have "B \<cdot> A\<star> = B \<cdot> ({[]} \<union> A\<star> \<cdot> A)"
+    by (rule seq_add_left)
+  also have "\<dots> = B \<union> B \<cdot> (A\<star> \<cdot> A)"
+    unfolding seq_union_distrib_left by simp
+  also have "\<dots> = B \<union> (B \<cdot> A\<star>) \<cdot> A" 
+    by (simp only: seq_assoc)
+  finally show "X = X \<cdot> A \<union> B" 
+    using eq by blast 
+next
+  assume eq: "X = X \<cdot> A \<union> B"
+  { fix n::nat
+    have "B \<cdot> (A \<up> n) \<subseteq> X" using arden_helper[OF eq, of "n"] by auto }
+  then have "B \<cdot> 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 \<cdot> (A \<up> Suc k)" 
+      using seq_pow_length[OF nemp] by blast
+    assume "s \<in> X"
+    then have "s \<in> X \<cdot> (A \<up> Suc k) \<union> (\<Union>m\<in>{0..k}. B \<cdot> (A \<up> m))"
+      using arden_helper[OF eq, of "k"] by auto
+    then have "s \<in> (\<Union>m\<in>{0..k}. B \<cdot> (A \<up> m))" using not_in by auto
+    moreover
+    have "(\<Union>m\<in>{0..k}. B \<cdot> (A \<up> m)) \<subseteq> (\<Union>n. B \<cdot> (A \<up> n))" by auto
+    ultimately 
+    have "s \<in> B \<cdot> A\<star>" 
+      unfolding seq_Union_left Star_def by auto }
+  then have "X \<subseteq> B \<cdot> A\<star>" by auto
+  ultimately 
+  show "X = B \<cdot> A\<star>" by simp
+qed
+
+
+section {* Regular Expressions *}
+
+datatype rexp =
+  NULL
+| EMPTY
+| CHAR char
+| SEQ rexp rexp
+| ALT rexp rexp
+| STAR rexp
+
+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) \<cdot> (L_rexp r2)"
+  | "L_rexp (ALT r1 r2) = (L_rexp r1) \<union> (L_rexp r2)"
+  | "L_rexp (STAR r) = (L_rexp r)\<star>"
+
+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_exp}.
+*}
+
+lemma folds_alt_simp [simp]:
+  fixes rs::"rexp set"
+  assumes a: "finite rs"
+  shows "L_rexp (\<Uplus>rs) = \<Union> (L_rexp ` 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