diff -r b04cc5e4e84c -r 7743d2ad71d1 Theories/Myhill_2.thy
--- a/Theories/Myhill_2.thy	Tue May 31 20:32:49 2011 +0000
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,470 +0,0 @@
-theory Myhill_2
-  imports Myhill_1 Prefix_subtract
-          "~~/src/HOL/Library/List_Prefix"
-begin
-
-section {* Direction @{text "regular language \<Rightarrow>finite partition"} *}
-
-definition
-  str_eq :: "string \<Rightarrow> lang \<Rightarrow> string \<Rightarrow> bool" ("_ \<approx>_ _")
-where
-  "x \<approx>A y \<equiv> (x, y) \<in> (\<approx>A)"
-
-definition 
-   tag_eq_rel :: "(string \<Rightarrow> 'b) \<Rightarrow> (string \<times> string) set" ("=_=")
-where
-   "=tag= \<equiv> {(x, y) | x y. tag x = tag y}"
-
-lemma finite_eq_tag_rel:
-  assumes rng_fnt: "finite (range tag)"
-  shows "finite (UNIV // =tag=)"
-proof -
-  let "?f" =  "\<lambda>X. tag ` X" and ?A = "(UNIV // =tag=)"
-  have "finite (?f ` ?A)" 
-  proof -
-    have "range ?f \<subseteq> (Pow (range tag))" unfolding Pow_def by auto
-    moreover 
-    have "finite (Pow (range tag))" using rng_fnt by simp
-    ultimately 
-    have "finite (range ?f)" unfolding image_def by (blast intro: finite_subset)
-    moreover
-    have "?f ` ?A \<subseteq> range ?f" by auto
-    ultimately show "finite (?f ` ?A)" by (rule rev_finite_subset) 
-  qed
-  moreover
-  have "inj_on ?f ?A"
-  proof -
-    { fix X Y
-      assume X_in: "X \<in> ?A"
-        and  Y_in: "Y \<in> ?A"
-        and  tag_eq: "?f X = ?f Y"
-      then obtain x y 
-        where "x \<in> X" "y \<in> Y" "tag x = tag y"
-        unfolding quotient_def Image_def image_def tag_eq_rel_def
-        by (simp) (blast)
-      with X_in Y_in 
-      have "X = Y"
-	unfolding quotient_def tag_eq_rel_def by auto
-    } 
-    then show "inj_on ?f ?A" unfolding inj_on_def by auto
-  qed
-  ultimately show "finite (UNIV // =tag=)" by (rule finite_imageD)
-qed
-
-lemma refined_partition_finite:
-  assumes fnt: "finite (UNIV // R1)"
-  and refined: "R1 \<subseteq> R2"
-  and eq1: "equiv UNIV R1" and eq2: "equiv UNIV R2"
-  shows "finite (UNIV // R2)"
-proof -
-  let ?f = "\<lambda>X. {R1 `` {x} | x. x \<in> X}" 
-    and ?A = "UNIV // R2" and ?B = "UNIV // R1"
-  have "?f ` ?A \<subseteq> Pow ?B"
-    unfolding image_def Pow_def quotient_def by auto
-  moreover
-  have "finite (Pow ?B)" using fnt by simp
-  ultimately  
-  have "finite (?f ` ?A)" by (rule finite_subset)
-  moreover
-  have "inj_on ?f ?A"
-  proof -
-    { fix X Y
-      assume X_in: "X \<in> ?A" and Y_in: "Y \<in> ?A" and eq_f: "?f X = ?f Y"
-      from quotientE [OF X_in]
-      obtain x where "X = R2 `` {x}" by blast
-      with equiv_class_self[OF eq2] have x_in: "x \<in> X" by simp
-      then have "R1 ``{x} \<in> ?f X" by auto
-      with eq_f have "R1 `` {x} \<in> ?f Y" by simp
-      then obtain y 
-        where y_in: "y \<in> Y" and eq_r1_xy: "R1 `` {x} = R1 `` {y}" by auto
-      with eq_equiv_class[OF _ eq1] 
-      have "(x, y) \<in> R1" by blast
-      with refined have "(x, y) \<in> R2" by auto
-      with quotient_eqI [OF eq2 X_in Y_in x_in y_in]
-      have "X = Y" .
-    } 
-    then show "inj_on ?f ?A" unfolding inj_on_def by blast 
-  qed
-  ultimately show "finite (UNIV // R2)" by (rule finite_imageD)
-qed
-
-lemma tag_finite_imageD:
-  assumes rng_fnt: "finite (range tag)" 
-  and same_tag_eqvt: "\<And>m n. tag m = tag n \<Longrightarrow> m \<approx>A n"
-  shows "finite (UNIV // \<approx>A)"
-proof (rule_tac refined_partition_finite [of "=tag="])
-  show "finite (UNIV // =tag=)" by (rule finite_eq_tag_rel[OF rng_fnt])
-next
-  from same_tag_eqvt
-  show "=tag= \<subseteq> \<approx>A" unfolding tag_eq_rel_def str_eq_def
-    by auto
-next
-  show "equiv UNIV =tag="
-    unfolding equiv_def tag_eq_rel_def refl_on_def sym_def trans_def
-    by auto
-next
-  show "equiv UNIV (\<approx>A)" 
-    unfolding equiv_def str_eq_rel_def sym_def refl_on_def trans_def
-    by blast
-qed
-
-
-subsection {* The proof *}
-
-subsubsection {* The base case for @{const "NULL"} *}
-
-lemma quot_null_eq:
-  shows "UNIV // \<approx>{} = {UNIV}"
-unfolding quotient_def Image_def str_eq_rel_def by auto
-
-lemma quot_null_finiteI [intro]:
-  shows "finite (UNIV // \<approx>{})"
-unfolding quot_null_eq by simp
-
-
-subsubsection {* The base case for @{const "EMPTY"} *}
-
-lemma quot_empty_subset:
-  shows "UNIV // \<approx>{[]} \<subseteq> {{[]}, UNIV - {[]}}"
-proof
-  fix x
-  assume "x \<in> UNIV // \<approx>{[]}"
-  then obtain y where h: "x = {z. (y, z) \<in> \<approx>{[]}}" 
-    unfolding quotient_def Image_def by blast
-  show "x \<in> {{[]}, UNIV - {[]}}"
-  proof (cases "y = []")
-    case True with h
-    have "x = {[]}" by (auto simp: str_eq_rel_def)
-    thus ?thesis by simp
-  next
-    case False with h
-    have "x = UNIV - {[]}" by (auto simp: str_eq_rel_def)
-    thus ?thesis by simp
-  qed
-qed
-
-lemma quot_empty_finiteI [intro]:
-  shows "finite (UNIV // \<approx>{[]})"
-by (rule finite_subset[OF quot_empty_subset]) (simp)
-
-
-subsubsection {* The base case for @{const "CHAR"} *}
-
-lemma quot_char_subset:
-  "UNIV // (\<approx>{[c]}) \<subseteq> {{[]},{[c]}, UNIV - {[], [c]}}"
-proof 
-  fix x 
-  assume "x \<in> UNIV // \<approx>{[c]}"
-  then obtain y where h: "x = {z. (y, z) \<in> \<approx>{[c]}}" 
-    unfolding quotient_def Image_def by blast
-  show "x \<in> {{[]},{[c]}, UNIV - {[], [c]}}"
-  proof -
-    { assume "y = []" hence "x = {[]}" using h 
-        by (auto simp:str_eq_rel_def) } 
-    moreover 
-    { assume "y = [c]" hence "x = {[c]}" using h 
-        by (auto dest!:spec[where x = "[]"] simp:str_eq_rel_def) } 
-    moreover 
-    { assume "y \<noteq> []" and "y \<noteq> [c]"
-      hence "\<forall> z. (y @ z) \<noteq> [c]" by (case_tac y, auto)
-      moreover have "\<And> p. (p \<noteq> [] \<and> p \<noteq> [c]) = (\<forall> q. p @ q \<noteq> [c])" 
-        by (case_tac p, auto)
-      ultimately have "x = UNIV - {[],[c]}" using h
-        by (auto simp add:str_eq_rel_def)
-    } 
-    ultimately show ?thesis by blast
-  qed
-qed
-
-lemma quot_char_finiteI [intro]:
-  shows "finite (UNIV // \<approx>{[c]})"
-by (rule finite_subset[OF quot_char_subset]) (simp)
-
-
-subsubsection {* The inductive case for @{const ALT} *}
-
-definition 
-  tag_str_ALT :: "lang \<Rightarrow> lang \<Rightarrow> string \<Rightarrow> (lang \<times> lang)"
-where
-  "tag_str_ALT A B \<equiv> (\<lambda>x. (\<approx>A `` {x}, \<approx>B `` {x}))"
-
-lemma quot_union_finiteI [intro]:
-  fixes L1 L2::"lang"
-  assumes finite1: "finite (UNIV // \<approx>A)"
-  and     finite2: "finite (UNIV // \<approx>B)"
-  shows "finite (UNIV // \<approx>(A \<union> B))"
-proof (rule_tac tag = "tag_str_ALT A B" in tag_finite_imageD)
-  have "finite ((UNIV // \<approx>A) \<times> (UNIV // \<approx>B))" 
-    using finite1 finite2 by auto
-  then show "finite (range (tag_str_ALT A B))"
-    unfolding tag_str_ALT_def quotient_def
-    by (rule rev_finite_subset) (auto)
-next
-  show "\<And>x y. tag_str_ALT A B x = tag_str_ALT A B y \<Longrightarrow> x \<approx>(A \<union> B) y"
-    unfolding tag_str_ALT_def
-    unfolding str_eq_def
-    unfolding str_eq_rel_def
-    by auto
-qed
-
-
-subsubsection {* The inductive case for @{text "SEQ"}*}
-
-definition 
-  tag_str_SEQ :: "lang \<Rightarrow> lang \<Rightarrow> string \<Rightarrow> (lang \<times> lang set)"
-where
-  "tag_str_SEQ L1 L2 \<equiv>
-     (\<lambda>x. (\<approx>L1 `` {x}, {(\<approx>L2 `` {x - xa}) | xa.  xa \<le> x \<and> xa \<in> L1}))"
-
-lemma Seq_in_cases:
-  assumes "x @ z \<in> A ;; B"
-  shows "(\<exists> x' \<le> x. x' \<in> A \<and> (x - x') @ z \<in> B) \<or> 
-         (\<exists> z' \<le> z. (x @ z') \<in> A \<and> (z - z') \<in> B)"
-using assms
-unfolding Seq_def prefix_def
-by (auto simp add: append_eq_append_conv2)
-
-lemma tag_str_SEQ_injI:
-  assumes eq_tag: "tag_str_SEQ A B x = tag_str_SEQ A B y" 
-  shows "x \<approx>(A ;; B) y"
-proof -
-  { fix x y z
-    assume xz_in_seq: "x @ z \<in> A ;; B"
-    and tag_xy: "tag_str_SEQ A B x = tag_str_SEQ A B y"
-    have"y @ z \<in> A ;; B" 
-    proof -
-      { (* first case with x' in A and (x - x') @ z in B *)
-        fix x'
-        assume h1: "x' \<le> x" and h2: "x' \<in> A" and h3: "(x - x') @ z \<in> B"
-        obtain y' 
-          where "y' \<le> y" 
-          and "y' \<in> A" 
-          and "(y - y') @ z \<in> B"
-        proof -
-          have "{\<approx>B `` {x - x'} |x'. x' \<le> x \<and> x' \<in> A} = 
-                {\<approx>B `` {y - y'} |y'. y' \<le> y \<and> y' \<in> A}" (is "?Left = ?Right")
-            using tag_xy unfolding tag_str_SEQ_def by simp
-          moreover 
-	  have "\<approx>B `` {x - x'} \<in> ?Left" using h1 h2 by auto
-          ultimately 
-	  have "\<approx>B `` {x - x'} \<in> ?Right" by simp
-          then obtain y' 
-            where eq_xy': "\<approx>B `` {x - x'} = \<approx>B `` {y - y'}" 
-            and pref_y': "y' \<le> y" and y'_in: "y' \<in> A"
-            by simp blast
-	  
-	  have "(x - x')  \<approx>B (y - y')" using eq_xy'
-            unfolding Image_def str_eq_rel_def str_eq_def by auto
-          with h3 have "(y - y') @ z \<in> B" 
-	    unfolding str_eq_rel_def str_eq_def by simp
-          with pref_y' y'_in 
-          show ?thesis using that by blast
-        qed
-	then have "y @ z \<in> A ;; B" by (erule_tac prefixE) (auto simp: Seq_def)
-      } 
-      moreover 
-      { (* second case with x @ z' in A and z - z' in B *)
-        fix z'
-        assume h1: "z' \<le> z" and h2: "(x @ z') \<in> A" and h3: "z - z' \<in> B"
-	 have "\<approx>A `` {x} = \<approx>A `` {y}" 
-           using tag_xy unfolding tag_str_SEQ_def by simp
-         with h2 have "y @ z' \<in> A"
-            unfolding Image_def str_eq_rel_def str_eq_def by auto
-        with h1 h3 have "y @ z \<in> A ;; B"
-	  unfolding prefix_def Seq_def
-	  by (auto) (metis append_assoc)
-      }
-      ultimately show "y @ z \<in> A ;; B" 
-	using Seq_in_cases [OF xz_in_seq] by blast
-    qed
-  }
-  from this [OF _ eq_tag] and this [OF _ eq_tag [THEN sym]]
-    show "x \<approx>(A ;; B) y" unfolding str_eq_def str_eq_rel_def by blast
-qed 
-
-lemma quot_seq_finiteI [intro]:
-  fixes L1 L2::"lang"
-  assumes fin1: "finite (UNIV // \<approx>L1)" 
-  and     fin2: "finite (UNIV // \<approx>L2)" 
-  shows "finite (UNIV // \<approx>(L1 ;; L2))"
-proof (rule_tac tag = "tag_str_SEQ L1 L2" in tag_finite_imageD)
-  show "\<And>x y. tag_str_SEQ L1 L2 x = tag_str_SEQ L1 L2 y \<Longrightarrow> x \<approx>(L1 ;; L2) y"
-    by (rule tag_str_SEQ_injI)
-next
-  have *: "finite ((UNIV // \<approx>L1) \<times> (Pow (UNIV // \<approx>L2)))" 
-    using fin1 fin2 by auto
-  show "finite (range (tag_str_SEQ L1 L2))" 
-    unfolding tag_str_SEQ_def
-    apply(rule finite_subset[OF _ *])
-    unfolding quotient_def
-    by auto
-qed
-
-
-subsubsection {* The inductive case for @{const "STAR"} *}
-
-definition 
-  tag_str_STAR :: "lang \<Rightarrow> string \<Rightarrow> lang set"
-where
-  "tag_str_STAR L1 \<equiv> (\<lambda>x. {\<approx>L1 `` {x - xa} | xa. xa < x \<and> xa \<in> L1\<star>})"
-
-text {* A technical lemma. *}
-lemma finite_set_has_max: "\<lbrakk>finite A; A \<noteq> {}\<rbrakk> \<Longrightarrow> 
-           (\<exists> max \<in> A. \<forall> a \<in> A. f a <= (f max :: nat))"
-proof (induct rule:finite.induct)
-  case emptyI thus ?case by simp
-next
-  case (insertI A a)
-  show ?case
-  proof (cases "A = {}")
-    case True thus ?thesis by (rule_tac x = a in bexI, auto)
-  next
-    case False
-    with insertI.hyps and False  
-    obtain max 
-      where h1: "max \<in> A" 
-      and h2: "\<forall>a\<in>A. f a \<le> f max" by blast
-    show ?thesis
-    proof (cases "f a \<le> f max")
-      assume "f a \<le> f max"
-      with h1 h2 show ?thesis by (rule_tac x = max in bexI, auto)
-    next
-      assume "\<not> (f a \<le> f max)"
-      thus ?thesis using h2 by (rule_tac x = a in bexI, auto)
-    qed
-  qed
-qed
-
-
-text {* The following is a technical lemma, which helps to show the range finiteness of tag function. *}
-
-lemma finite_strict_prefix_set: "finite {xa. xa < (x::string)}"
-apply (induct x rule:rev_induct, simp)
-apply (subgoal_tac "{xa. xa < xs @ [x]} = {xa. xa < xs} \<union> {xs}")
-by (auto simp:strict_prefix_def)
-
-
-lemma tag_str_STAR_injI:
-  assumes eq_tag: "tag_str_STAR L\<^isub>1 v = tag_str_STAR L\<^isub>1 w"
-  shows "v \<approx>(L\<^isub>1\<star>) w"
-proof-
-  { fix x y z
-    assume xz_in_star: "x @ z \<in> L\<^isub>1\<star>" 
-      and tag_xy: "tag_str_STAR L\<^isub>1 x = tag_str_STAR L\<^isub>1 y"
-    have "y @ z \<in> L\<^isub>1\<star>"
-    proof(cases "x = []")
-      case True
-      with tag_xy have "y = []" 
-        by (auto simp add: tag_str_STAR_def strict_prefix_def)
-      thus ?thesis using xz_in_star True by simp
-    next
-      case False
-      let ?S = "{xa. xa < x \<and> xa \<in> L\<^isub>1\<star> \<and> (x - xa) @ z \<in> L\<^isub>1\<star>}"
-      have "finite ?S"
-        by (rule_tac B = "{xa. xa < x}" in finite_subset, 
-          auto simp:finite_strict_prefix_set)
-      moreover have "?S \<noteq> {}" using False xz_in_star
-        by (simp, rule_tac x = "[]" in exI, auto simp:strict_prefix_def)
-      ultimately have "\<exists> xa_max \<in> ?S. \<forall> xa \<in> ?S. length xa \<le> length xa_max" 
-        using finite_set_has_max by blast
-      then obtain xa_max 
-        where h1: "xa_max < x" 
-        and h2: "xa_max \<in> L\<^isub>1\<star>" 
-        and h3: "(x - xa_max) @ z \<in> L\<^isub>1\<star>" 
-        and h4:"\<forall> xa < x. xa \<in> L\<^isub>1\<star> \<and> (x - xa) @ z \<in> L\<^isub>1\<star>  
-                                     \<longrightarrow> length xa \<le> length xa_max"
-        by blast
-      obtain ya 
-        where h5: "ya < y" and h6: "ya \<in> L\<^isub>1\<star>" 
-        and eq_xya: "(x - xa_max) \<approx>L\<^isub>1 (y - ya)"
-      proof-
-        from tag_xy have "{\<approx>L\<^isub>1 `` {x - xa} |xa. xa < x \<and> xa \<in> L\<^isub>1\<star>} = 
-          {\<approx>L\<^isub>1 `` {y - xa} |xa. xa < y \<and> xa \<in> L\<^isub>1\<star>}" (is "?left = ?right")
-          by (auto simp:tag_str_STAR_def)
-        moreover have "\<approx>L\<^isub>1 `` {x - xa_max} \<in> ?left" using h1 h2 by auto
-        ultimately have "\<approx>L\<^isub>1 `` {x - xa_max} \<in> ?right" by simp
-        thus ?thesis using that 
-          apply (simp add:Image_def str_eq_rel_def str_eq_def) by blast
-      qed 
-      have "(y - ya) @ z \<in> L\<^isub>1\<star>" 
-      proof-
-        obtain za zb where eq_zab: "z = za @ zb" 
-          and l_za: "(y - ya)@za \<in> L\<^isub>1" and ls_zb: "zb \<in> L\<^isub>1\<star>"
-        proof -
-          from h1 have "(x - xa_max) @ z \<noteq> []" 
-            by (auto simp:strict_prefix_def elim:prefixE)
-          from star_decom [OF h3 this]
-          obtain a b where a_in: "a \<in> L\<^isub>1" 
-            and a_neq: "a \<noteq> []" and b_in: "b \<in> L\<^isub>1\<star>" 
-            and ab_max: "(x - xa_max) @ z = a @ b" by blast
-          let ?za = "a - (x - xa_max)" and ?zb = "b"
-          have pfx: "(x - xa_max) \<le> a" (is "?P1") 
-            and eq_z: "z = ?za @ ?zb" (is "?P2")
-          proof -
-            have "((x - xa_max) \<le> a \<and> (a - (x - xa_max)) @ b = z) \<or> 
-              (a < (x - xa_max) \<and> ((x - xa_max) - a) @ z = b)" 
-              using append_eq_dest[OF ab_max] by (auto simp:strict_prefix_def)
-            moreover {
-              assume np: "a < (x - xa_max)" 
-                and b_eqs: "((x - xa_max) - a) @ z = b"
-              have "False"
-              proof -
-                let ?xa_max' = "xa_max @ a"
-                have "?xa_max' < x" 
-                  using np h1 by (clarsimp simp:strict_prefix_def diff_prefix) 
-                moreover have "?xa_max' \<in> L\<^isub>1\<star>" 
-                  using a_in h2 by (simp add:star_intro3) 
-                moreover have "(x - ?xa_max') @ z \<in> L\<^isub>1\<star>" 
-                  using b_eqs b_in np h1 by (simp add:diff_diff_append)
-                moreover have "\<not> (length ?xa_max' \<le> length xa_max)" 
-                  using a_neq by simp
-                ultimately show ?thesis using h4 by blast
-              qed }
-            ultimately show ?P1 and ?P2 by auto
-          qed
-          hence "(x - xa_max)@?za \<in> L\<^isub>1" using a_in by (auto elim:prefixE)
-          with eq_xya have "(y - ya) @ ?za \<in> L\<^isub>1" 
-            by (auto simp:str_eq_def str_eq_rel_def)
-           with eq_z and b_in 
-          show ?thesis using that by blast
-        qed
-        have "((y - ya) @ za) @ zb \<in> L\<^isub>1\<star>" using  l_za ls_zb by blast
-        with eq_zab show ?thesis by simp
-      qed
-      with h5 h6 show ?thesis 
-        by (drule_tac star_intro1, auto simp:strict_prefix_def elim:prefixE)
-    qed
-  } 
-  from this [OF _ eq_tag] and this [OF _ eq_tag [THEN sym]]
-    show  ?thesis unfolding str_eq_def str_eq_rel_def by blast
-qed
-
-lemma quot_star_finiteI [intro]:
-  assumes finite1: "finite (UNIV // \<approx>L1)"
-  shows "finite (UNIV // \<approx>(L1\<star>))"
-proof (rule_tac tag = "tag_str_STAR L1" in tag_finite_imageD)
-  show "\<And>x y. tag_str_STAR L1 x = tag_str_STAR L1 y \<Longrightarrow> x \<approx>(L1\<star>) y"
-    by (rule tag_str_STAR_injI)
-next
-  have *: "finite (Pow (UNIV // \<approx>L1))" 
-    using finite1 by auto
-  show "finite (range (tag_str_STAR L1))"
-    unfolding tag_str_STAR_def
-    apply(rule finite_subset[OF _ *])
-    unfolding quotient_def
-    by auto
-qed
-
-subsubsection{* The conclusion *}
-
-lemma Myhill_Nerode2:
-  fixes r::"rexp"
-  shows "finite (UNIV // \<approx>(L r))"
-by (induct r) (auto)
-
-
-theorem Myhill_Nerode:
-  shows "(\<exists>r::rexp. A = L r) \<longleftrightarrow> finite (UNIV // \<approx>A)"
-using Myhill_Nerode1 Myhill_Nerode2 by auto
-
-end