diff -r a8a442ba0dbf -r e93760534354 Derivs.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Derivs.thy	Wed May 18 19:54:43 2011 +0000
@@ -0,0 +1,521 @@
+theory Derivs
+imports Closure
+begin
+
+section {* Experiments with Derivatives -- independent of Myhill-Nerode *}
+
+subsection {* Left-Quotients *}
+
+definition
+  Delta :: "lang \<Rightarrow> lang"
+where
+  "Delta A = (if [] \<in> A then {[]} else {})"
+
+definition
+  Der :: "char \<Rightarrow> lang \<Rightarrow> lang"
+where
+  "Der c A \<equiv> {s. [c] @ s \<in> A}"
+
+definition
+  Ders :: "string \<Rightarrow> lang \<Rightarrow> lang"
+where
+  "Ders s A \<equiv> {s'. s @ s' \<in> A}"
+
+definition
+  Ders_set :: "lang \<Rightarrow> lang \<Rightarrow> lang"
+where
+  "Ders_set A B \<equiv> {s' | s s'. s @ s' \<in> B \<and> s \<in> A}"
+
+lemma Ders_set_Ders:
+  shows "Ders_set A B = (\<Union>s \<in> A. Ders s B)"
+unfolding Ders_set_def Ders_def
+by auto
+
+lemma Der_null [simp]:
+  shows "Der c {} = {}"
+unfolding Der_def
+by auto
+
+lemma Der_empty [simp]:
+  shows "Der c {[]} = {}"
+unfolding Der_def
+by auto
+
+lemma Der_char [simp]:
+  shows "Der c {[d]} = (if c = d then {[]} else {})"
+unfolding Der_def
+by auto
+
+lemma Der_union [simp]:
+  shows "Der c (A \<union> B) = Der c A \<union> Der c B"
+unfolding Der_def
+by auto
+
+lemma Der_seq [simp]:
+  shows "Der c (A ;; B) = (Der c A) ;; B \<union> (Delta A ;; Der c B)"
+unfolding Der_def Delta_def
+unfolding Seq_def
+by (auto simp add: Cons_eq_append_conv)
+
+lemma Der_star [simp]:
+  shows "Der c (A\<star>) = (Der c A) ;; A\<star>"
+proof -
+  have incl: "Delta A ;; Der c (A\<star>) \<subseteq> (Der c A) ;; A\<star>"
+    unfolding Der_def Delta_def Seq_def
+    apply(auto)
+    apply(drule star_decom)
+    apply(auto simp add: Cons_eq_append_conv)
+    done
+    
+  have "Der c (A\<star>) = Der c ({[]} \<union> A ;; A\<star>)"
+    by (simp only: star_cases[symmetric])
+  also have "... = Der c (A ;; A\<star>)"
+    by (simp only: Der_union Der_empty) (simp)
+  also have "... = (Der c A) ;; A\<star> \<union> (Delta A ;; Der c (A\<star>))"
+    by simp
+  also have "... =  (Der c A) ;; A\<star>"
+    using incl by auto
+  finally show "Der c (A\<star>) = (Der c A) ;; A\<star>" . 
+qed
+
+
+lemma Ders_singleton:
+  shows "Ders [c] A = Der c A"
+unfolding Der_def Ders_def
+by simp
+
+lemma Ders_append:
+  shows "Ders (s1 @ s2) A = Ders s2 (Ders s1 A)"
+unfolding Ders_def by simp 
+
+lemma MN_Rel_Ders:
+  shows "x \<approx>A y \<longleftrightarrow> Ders x A = Ders y A"
+unfolding Ders_def str_eq_def str_eq_rel_def
+by auto
+
+subsection {* Brozowsky's derivatives of regular expressions *}
+
+fun
+  nullable :: "rexp \<Rightarrow> bool"
+where
+  "nullable (NULL) = False"
+| "nullable (EMPTY) = True"
+| "nullable (CHAR c) = False"
+| "nullable (ALT r1 r2) = (nullable r1 \<or> nullable r2)"
+| "nullable (SEQ r1 r2) = (nullable r1 \<and> nullable r2)"
+| "nullable (STAR r) = True"
+
+fun
+  der :: "char \<Rightarrow> rexp \<Rightarrow> rexp"
+where
+  "der c (NULL) = NULL"
+| "der c (EMPTY) = NULL"
+| "der c (CHAR c') = (if c = c' then EMPTY else NULL)"
+| "der c (ALT r1 r2) = ALT (der c r1) (der c r2)"
+| "der c (SEQ r1 r2) = ALT (SEQ (der c r1) r2) (if nullable r1 then der c r2 else NULL)"
+| "der c (STAR r) = SEQ (der c r) (STAR r)"
+
+function 
+  ders :: "string \<Rightarrow> rexp \<Rightarrow> rexp"
+where
+  "ders [] r = r"
+| "ders (s @ [c]) r = der c (ders s r)"
+by (auto) (metis rev_cases)
+
+termination
+  by (relation "measure (length o fst)") (auto)
+
+lemma Delta_nullable:
+  shows "Delta (L r) = (if nullable r then {[]} else {})"
+unfolding Delta_def
+by (induct r) (auto simp add: Seq_def split: if_splits)
+
+lemma Der_der:
+  fixes r::rexp
+  shows "Der c (L r) = L (der c r)"
+by (induct r) (simp_all add: Delta_nullable)
+
+lemma Ders_ders:
+  fixes r::rexp
+  shows "Ders s (L r) = L (ders s r)"
+apply(induct s rule: rev_induct)
+apply(simp add: Ders_def)
+apply(simp only: ders.simps)
+apply(simp only: Ders_append)
+apply(simp only: Ders_singleton)
+apply(simp only: Der_der)
+done
+
+
+subsection {* Antimirov's Partial Derivatives *}
+
+abbreviation
+  "SEQS R r \<equiv> {SEQ r' r | r'. r' \<in> R}"
+
+fun
+  pder :: "char \<Rightarrow> rexp \<Rightarrow> rexp set"
+where
+  "pder c NULL = {NULL}"
+| "pder c EMPTY = {NULL}"
+| "pder c (CHAR c') = (if c = c' then {EMPTY} else {NULL})"
+| "pder c (ALT r1 r2) = (pder c r1) \<union> (pder c r2)"
+| "pder c (SEQ r1 r2) = SEQS (pder c r1) r2 \<union> (if nullable r1 then pder c r2 else {})"
+| "pder c (STAR r) = SEQS (pder c r) (STAR r)"
+
+abbreviation
+  "pder_set c R \<equiv> \<Union>r \<in> R. pder c r"
+
+function 
+  pders :: "string \<Rightarrow> rexp \<Rightarrow> rexp set"
+where
+  "pders [] r = {r}"
+| "pders (s @ [c]) r = pder_set c (pders s r)"
+by (auto) (metis rev_cases)
+
+termination
+  by (relation "measure (length o fst)") (auto)
+
+abbreviation
+  "pders_set A r \<equiv> \<Union>s \<in> A. pders s r"
+
+lemma pders_append:
+  "pders (s1 @ s2) r = \<Union> (pders s2) ` (pders s1 r)"
+apply(induct s2 arbitrary: s1 r rule: rev_induct)
+apply(simp)
+apply(subst append_assoc[symmetric])
+apply(simp only: pders.simps)
+apply(auto)
+done
+
+lemma pders_singleton:
+  "pders [c] r = pder c r"
+apply(subst append_Nil[symmetric])
+apply(simp only: pders.simps)
+apply(simp)
+done
+
+lemma pder_set_lang:
+  shows "(\<Union> (L ` pder_set c R)) = (\<Union>r \<in> R. (\<Union>L ` (pder c r)))"
+unfolding image_def 
+by auto
+
+lemma
+  shows seq_UNION_left:  "B ;; (\<Union>n\<in>C. A n) = (\<Union>n\<in>C. B ;; A n)"
+  and   seq_UNION_right: "(\<Union>n\<in>C. A n) ;; B = (\<Union>n\<in>C. A n ;; B)"
+unfolding Seq_def by auto
+
+lemma Der_pder:
+  fixes r::rexp
+  shows "Der c (L r) = \<Union> L ` (pder c r)"
+by (induct r) (auto simp add: Delta_nullable seq_UNION_right)
+
+lemma Ders_pders:
+  fixes r::rexp
+  shows "Ders s (L r) = \<Union> L ` (pders s r)"
+proof (induct s rule: rev_induct)
+  case (snoc c s)
+  have ih: "Ders s (L r) = \<Union> L ` (pders s r)" by fact
+  have "Ders (s @ [c]) (L r) = Ders [c] (Ders s (L r))"
+    by (simp add: Ders_append)
+  also have "\<dots> = Der c (\<Union> L ` (pders s r))" using ih
+    by (simp add: Ders_singleton)
+  also have "\<dots> = (\<Union>r\<in>pders s r. Der c (L r))" 
+    unfolding Der_def image_def by auto
+  also have "\<dots> = (\<Union>r\<in>pders s r. (\<Union> L `  (pder c r)))"
+    by (simp add: Der_pder)
+  also have "\<dots> = (\<Union>L ` (pder_set c (pders s r)))"
+    by (simp add: pder_set_lang)
+  also have "\<dots> = (\<Union>L ` (pders (s @ [c]) r))"
+    by simp
+  finally show "Ders (s @ [c]) (L r) = \<Union>L ` pders (s @ [c]) r" .
+qed (simp add: Ders_def)
+
+lemma Ders_set_pders_set:
+  fixes r::rexp
+  shows "Ders_set A (L r) = (\<Union> L ` (pders_set A r))"
+by (simp add: Ders_set_Ders Ders_pders)
+
+lemma pders_NULL [simp]:
+  shows "pders s NULL = {NULL}"
+by (induct s rule: rev_induct) (simp_all)
+
+lemma pders_EMPTY [simp]:
+  shows "pders s EMPTY = (if s = [] then {EMPTY} else {NULL})"
+by (induct s rule: rev_induct) (auto)
+
+lemma pders_CHAR [simp]:
+  shows "pders s (CHAR c) = (if s = [] then {CHAR c} else (if s = [c] then {EMPTY} else {NULL}))"
+by (induct s rule: rev_induct) (auto)
+
+lemma pders_ALT [simp]:
+  shows "pders s (ALT r1 r2) = (if s = [] then {ALT r1 r2} else (pders s r1) \<union> (pders s r2))"
+by (induct s rule: rev_induct) (auto)
+
+definition
+  "Suf s \<equiv> {v. v \<noteq> [] \<and> (\<exists>u. u @ v = s)}"
+
+lemma Suf:
+  shows "Suf (s @ [c]) = (Suf s) ;; {[c]} \<union> {[c]}"
+unfolding Suf_def Seq_def
+by (auto simp add: append_eq_append_conv2 append_eq_Cons_conv)
+
+lemma Suf_Union:
+  shows "(\<Union>v \<in> Suf s ;; {[c]}. P v) = (\<Union>v \<in> Suf s. P (v @ [c]))"
+by (auto simp add: Seq_def)
+
+lemma inclusion1:
+  shows "pder_set c (SEQS R r2) \<subseteq> SEQS (pder_set c R) r2 \<union> (pder c r2)"
+apply(auto simp add: if_splits)
+apply(blast)
+done
+
+lemma pders_SEQ:
+  shows "pders s (SEQ r1 r2) \<subseteq> SEQS (pders s r1) r2 \<union> (\<Union>v \<in> Suf s. pders v 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> (\<Union>v \<in> Suf s. pders v r2)" 
+    by fact
+  have "pders (s @ [c]) (SEQ r1 r2) = pder_set c (pders s (SEQ r1 r2))" by simp
+  also have "\<dots> \<subseteq> pder_set c (SEQS (pders s r1) r2 \<union> (\<Union>v \<in> Suf s. pders v r2))"
+    using ih by auto 
+  also have "\<dots> = pder_set c (SEQS (pders s r1) r2) \<union> pder_set c (\<Union>v \<in> Suf s. pders v r2)"
+    by (simp)
+  also have "\<dots> = pder_set c (SEQS (pders s r1) r2) \<union> (\<Union>v \<in> Suf s. pder_set c (pders v r2))"
+    by (simp)
+  also have "\<dots> \<subseteq> pder_set c (SEQS (pders s r1) r2) \<union> (pder c r2) \<union> (\<Union>v \<in> Suf s. pders (v @ [c]) r2)"
+    by (auto)
+  also have "\<dots> \<subseteq> SEQS (pder_set c (pders s r1)) r2 \<union> (pder c r2) \<union> (\<Union>v \<in> Suf s. pders (v @ [c]) r2)"
+    using inclusion1 by blast
+  also have "\<dots> = SEQS (pders (s @ [c]) r1) r2 \<union> (\<Union>v \<in> Suf (s @ [c]). pders v r2)"
+    apply(subst (2) pders.simps)
+    apply(simp only: Suf)
+    apply(simp add: Suf_Union pders_singleton)
+    apply(auto)
+    done
+  finally show ?case .
+qed (simp)
+
+lemma pders_STAR:
+  assumes a: "s \<noteq> []"
+  shows "pders s (STAR r) \<subseteq> (\<Union>v \<in> Suf s. SEQS (pders v 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> (\<Union>v\<in>Suf s. SEQS (pders v r) (STAR r))" by fact
+  { assume asm: "s \<noteq> []"
+    have "pders (s @ [c]) (STAR r) = pder_set c (pders s (STAR r))" by simp
+    also have "\<dots> \<subseteq> (pder_set c (\<Union>v\<in>Suf s. SEQS (pders v r) (STAR r)))"
+      using ih[OF asm] by blast
+    also have "\<dots> = (\<Union>v\<in>Suf s. pder_set c (SEQS (pders v r) (STAR r)))"
+      by simp
+    also have "\<dots> \<subseteq> (\<Union>v\<in>Suf s. (SEQS (pder_set c (pders v r)) (STAR r) \<union> pder c (STAR r)))"
+      using inclusion1 by blast
+    also have "\<dots> = (\<Union>v\<in>Suf s. (SEQS (pder_set c (pders v r)) (STAR r))) \<union> pder c (STAR r)"
+      using asm by (auto simp add: Suf_def)
+    also have "\<dots> = (\<Union>v\<in>Suf s. (SEQS (pders (v @ [c]) r) (STAR r))) \<union> (SEQS (pder c r) (STAR r))"
+      by simp
+    also have "\<dots> = (\<Union>v\<in>Suf (s @ [c]). (SEQS (pders v r) (STAR r)))"
+      apply(simp only: Suf)
+      apply(simp add: Suf_Union pders_singleton)
+      apply(auto)
+      done
+    finally have ?case .
+  }
+  moreover
+  { assume asm: "s = []"
+    then have ?case
+      apply(simp add: pders_singleton Suf_def)
+      apply(auto)
+      apply(rule_tac x="[c]" in exI)
+      apply(simp add: pders_singleton)
+      done
+  }
+  ultimately show ?case by blast
+qed (simp)
+
+abbreviation 
+  "UNIV1 \<equiv> UNIV - {[]}"
+
+lemma pders_set_NULL:
+  shows "pders_set UNIV1 NULL = {NULL}"
+by auto
+
+lemma pders_set_EMPTY:
+  shows "pders_set UNIV1 EMPTY = {NULL}"
+by (auto split: if_splits)
+
+lemma pders_set_CHAR:
+  shows "pders_set UNIV1 (CHAR c) \<subseteq> {EMPTY, NULL}"
+by (auto split: if_splits)
+
+lemma pders_set_ALT:
+  shows "pders_set UNIV1 (ALT r1 r2) = pders_set UNIV1 r1 \<union> pders_set UNIV1 r2"
+by auto
+
+lemma pders_set_SEQ_aux:
+  assumes a: "s \<in> UNIV1"
+  shows "pders_set (Suf s) r2 \<subseteq> pders_set UNIV1 r2"
+using a by (auto simp add: Suf_def)
+
+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 UN_least)
+apply(rule subset_trans)
+apply(rule pders_SEQ)
+apply(simp)
+apply(rule conjI) 
+apply(auto)[1]
+apply(rule subset_trans)
+apply(rule pders_set_SEQ_aux)
+apply(auto)
+done
+
+lemma pders_set_STAR:
+  shows "pders_set UNIV1 (STAR r) \<subseteq> SEQS (pders_set UNIV1 r) (STAR r)"
+apply(rule UN_least)
+apply(rule subset_trans)
+apply(rule pders_STAR)
+apply(simp)
+apply(simp add: Suf_def)
+apply(auto)
+done
+
+lemma finite_SEQS:
+  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)
+apply(simp only: pders_set_EMPTY)
+apply(simp)
+apply(rule finite_subset)
+apply(rule pders_set_CHAR)
+apply(simp)
+apply(rule finite_subset)
+apply(rule pders_set_SEQ)
+apply(simp only: finite_SEQS finite_Un)
+apply(simp)
+apply(simp only: pders_set_ALT)
+apply(simp)
+apply(rule finite_subset)
+apply(rule pders_set_STAR)
+apply(simp only: finite_SEQS)
+done
+    
+lemma pders_set_UNIV_UNIV1:
+  shows "pders_set UNIV r = pders [] r \<union> pders_set UNIV1 r"
+apply(auto)
+apply(rule_tac x="[]" in exI)
+apply(simp)
+done
+
+lemma finite_pders_set_UNIV:
+  shows "finite (pders_set UNIV r)"
+unfolding pders_set_UNIV_UNIV1
+by (simp add: finite_pders_set_UNIV1)
+
+lemma finite_pders_set:
+  shows "finite (pders_set A r)"
+apply(rule rev_finite_subset)
+apply(rule_tac r="r" in finite_pders_set_UNIV)
+apply(auto)
+done
+
+lemma finite_pders:
+  shows "finite (pders s r)"
+using finite_pders_set[where A="{s}" and r="r"]
+by simp
+
+lemma finite_pders2:
+  shows "finite {pders s r | s. s \<in> A}"
+proof -
+  have "{pders s r | s. s \<in> A} \<subseteq> Pow (pders_set A r)" by auto
+  moreover
+  have "finite (Pow (pders_set A r))"
+    using finite_pders_set by simp
+  ultimately 
+  show "finite {pders s r | s. s \<in> A}"
+    by(rule finite_subset)
+qed
+
+
+lemma Myhill_Nerode3:
+  fixes r::"rexp"
+  shows "finite (UNIV // \<approx>(L r))"
+proof -
+  have "finite (UNIV // =(\<lambda>x. pders x r)=)"
+  proof - 
+    have "range (\<lambda>x. pders x r) \<subseteq> {pders s r | s. s \<in> UNIV}" by auto
+    moreover 
+    have "finite {pders s r | s. s \<in> UNIV}" by (rule finite_pders2)
+    ultimately
+    have "finite (range (\<lambda>x. pders x r))"
+      by (rule finite_subset)
+    then show "finite (UNIV // =(\<lambda>x. pders x r)=)" 
+      by (rule finite_eq_tag_rel)
+  qed
+  moreover 
+  have " =(\<lambda>x. pders x r)= \<subseteq> \<approx>(L r)"
+    unfolding tag_eq_rel_def
+    by (auto simp add: str_eq_def[symmetric] MN_Rel_Ders Ders_pders)
+  moreover 
+  have "equiv UNIV =(\<lambda>x. pders x r)="
+    unfolding equiv_def refl_on_def sym_def trans_def
+    unfolding tag_eq_rel_def
+    by auto
+  moreover
+  have "equiv UNIV (\<approx>(L r))"
+    unfolding equiv_def refl_on_def sym_def trans_def
+    unfolding str_eq_rel_def
+    by auto
+  ultimately show "finite (UNIV // \<approx>(L r))" 
+    by (rule refined_partition_finite)
+qed
+
+
+section {* Closure under Left-Quotients *}
+
+lemma closure_left_quotient:
+  assumes "regular A"
+  shows "regular (Ders_set B A)"
+proof -
+  from assms obtain r::rexp where eq: "L r = A" by auto
+  have fin: "finite (pders_set B r)" by (rule finite_pders_set)
+  
+  have "Ders_set B (L r) = (\<Union> L ` (pders_set B r))"
+    by (simp add: Ders_set_pders_set)
+  also have "\<dots> = L (\<Uplus>(pders_set B r))" using fin by simp
+  finally have "Ders_set B A = L (\<Uplus>(pders_set B r))" using eq
+    by simp
+  then show "regular (Ders_set B A)" by auto
+qed
+
+
+section {* Relating standard and partial derivations *}
+
+lemma
+  shows "(\<Union> L ` (pder c r)) = L (der c r)"
+unfolding Der_der[symmetric] Der_pder by simp
+
+lemma
+  shows "(\<Union> L ` (pders s r)) = L (ders s r)"
+unfolding Ders_ders[symmetric] Ders_pders by simp
+
+
+
+fun
+  width :: "rexp \<Rightarrow> nat"
+where
+  "width (NULL) = 0"
+| "width (EMPTY) = 0"
+| "width (CHAR c) = 1"
+| "width (ALT r1 r2) = width r1 + width r2"
+| "width (SEQ r1 r2) = width r1 + width r2"
+| "width (STAR r) = width r"
+
+
+ 
+end
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