src/HOL/Enum.thy

 
 lemma exists_code [code]: "(\<exists>x. P x) \<longleftrightarrow> enum_ex P"
   by (simp add: enum_ex)
 
 lemma exists1_code[code]: "(\<exists>!x. P x) \<longleftrightarrow> list_ex1 P enum"
-unfolding list_ex1_iff enum_UNIV by auto
+  by (auto simp add: enum_UNIV list_ex1_iff)
 
 
 subsection {* Default instances *}
-
-primrec n_lists :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list list" where
-  "n_lists 0 xs = [[]]"
-  | "n_lists (Suc n) xs = concat (map (\<lambda>ys. map (\<lambda>y. y # ys) xs) (n_lists n xs))"
-
-lemma n_lists_Nil [simp]: "n_lists n [] = (if n = 0 then [[]] else [])"
-  by (induct n) simp_all
-
-lemma length_n_lists: "length (n_lists n xs) = length xs ^ n"
-  by (induct n) (auto simp add: length_concat o_def listsum_triv)
-
-lemma length_n_lists_elem: "ys \<in> set (n_lists n xs) \<Longrightarrow> length ys = n"
-  by (induct n arbitrary: ys) auto
-
-lemma set_n_lists: "set (n_lists n xs) = {ys. length ys = n \<and> set ys \<subseteq> set xs}"
-proof (rule set_eqI)
-  fix ys :: "'a list"
-  show "ys \<in> set (n_lists n xs) \<longleftrightarrow> ys \<in> {ys. length ys = n \<and> set ys \<subseteq> set xs}"
-  proof -
-    have "ys \<in> set (n_lists n xs) \<Longrightarrow> length ys = n"
-      by (induct n arbitrary: ys) auto
-    moreover have "\<And>x. ys \<in> set (n_lists n xs) \<Longrightarrow> x \<in> set ys \<Longrightarrow> x \<in> set xs"
-      by (induct n arbitrary: ys) auto
-    moreover have "set ys \<subseteq> set xs \<Longrightarrow> ys \<in> set (n_lists (length ys) xs)"
-      by (induct ys) auto
-    ultimately show ?thesis by auto
-  qed
-qed
-
-lemma distinct_n_lists:
-  assumes "distinct xs"
-  shows "distinct (n_lists n xs)"
-proof (rule card_distinct)
-  from assms have card_length: "card (set xs) = length xs" by (rule distinct_card)
-  have "card (set (n_lists n xs)) = card (set xs) ^ n"
-  proof (induct n)
-    case 0 then show ?case by simp
-  next
-    case (Suc n)
-    moreover have "card (\<Union>ys\<in>set (n_lists n xs). (\<lambda>y. y # ys) ` set xs)
-      = (\<Sum>ys\<in>set (n_lists n xs). card ((\<lambda>y. y # ys) ` set xs))"
-      by (rule card_UN_disjoint) auto
-    moreover have "\<And>ys. card ((\<lambda>y. y # ys) ` set xs) = card (set xs)"
-      by (rule card_image) (simp add: inj_on_def)
-    ultimately show ?case by auto
-  qed
-  also have "\<dots> = length xs ^ n" by (simp add: card_length)
-  finally show "card (set (n_lists n xs)) = length (n_lists n xs)"
-    by (simp add: length_n_lists)
-qed
 
 lemma map_of_zip_enum_is_Some:
   assumes "length ys = length (enum \<Colon> 'a\<Colon>enum list)"
   shows "\<exists>y. map_of (zip (enum \<Colon> 'a\<Colon>enum list) ys) x = Some y"
 proof -

 qed
 
 definition
   all_n_lists :: "(('a :: enum) list \<Rightarrow> bool) \<Rightarrow> nat \<Rightarrow> bool"
 where
-  "all_n_lists P n = (\<forall>xs \<in> set (n_lists n enum). P xs)"
+  "all_n_lists P n = (\<forall>xs \<in> set (List.n_lists n enum). P xs)"
 
 lemma [code]:
   "all_n_lists P n = (if n = 0 then P [] else enum_all (%x. all_n_lists (%xs. P (x # xs)) (n - 1)))"
 unfolding all_n_lists_def enum_all
 by (cases n) (auto simp add: enum_UNIV)
 
 definition
   ex_n_lists :: "(('a :: enum) list \<Rightarrow> bool) \<Rightarrow> nat \<Rightarrow> bool"
 where
-  "ex_n_lists P n = (\<exists>xs \<in> set (n_lists n enum). P xs)"
+  "ex_n_lists P n = (\<exists>xs \<in> set (List.n_lists n enum). P xs)"
 
 lemma [code]:
   "ex_n_lists P n = (if n = 0 then P [] else enum_ex (%x. ex_n_lists (%xs. P (x # xs)) (n - 1)))"
 unfolding ex_n_lists_def enum_ex
 by (cases n) (auto simp add: enum_UNIV)

 
 instantiation "fun" :: (enum, enum) enum
 begin
 
 definition
-  "enum = map (\<lambda>ys. the o map_of (zip (enum\<Colon>'a list) ys)) (n_lists (length (enum\<Colon>'a\<Colon>enum list)) enum)"
+  "enum = map (\<lambda>ys. the o map_of (zip (enum\<Colon>'a list) ys)) (List.n_lists (length (enum\<Colon>'a\<Colon>enum list)) enum)"
 
 definition
   "enum_all P = all_n_lists (\<lambda>bs. P (the o map_of (zip enum bs))) (length (enum :: 'a list))"
 
 definition

 qed
 
 end
 
 lemma enum_fun_code [code]: "enum = (let enum_a = (enum \<Colon> 'a\<Colon>{enum, equal} list)
-  in map (\<lambda>ys. the o map_of (zip enum_a ys)) (n_lists (length enum_a) enum))"
+  in map (\<lambda>ys. the o map_of (zip enum_a ys)) (List.n_lists (length enum_a) enum))"
   by (simp add: enum_fun_def Let_def)
 
 lemma enum_all_fun_code [code]:
   "enum_all P = (let enum_a = (enum :: 'a::{enum, equal} list)
    in all_n_lists (\<lambda>bs. P (the o map_of (zip enum_a bs))) (length enum_a))"

     unfolding enum_ex_bool_def by (auto, case_tac x) auto
 qed (auto simp add: enum_bool_def UNIV_bool)
 
 end
 
-primrec product :: "'a list \<Rightarrow> 'b list \<Rightarrow> ('a \<times> 'b) list" where
-  "product [] _ = []"
-  | "product (x#xs) ys = map (Pair x) ys @ product xs ys"
-
-lemma product_list_set:
-  "set (product xs ys) = set xs \<times> set ys"
-  by (induct xs) auto
-
-lemma distinct_product:
-  assumes "distinct xs" and "distinct ys"
-  shows "distinct (product xs ys)"
-  using assms by (induct xs)
-    (auto intro: inj_onI simp add: product_list_set distinct_map)
-
 instantiation prod :: (enum, enum) enum
 begin
 
 definition
-  "enum = product enum enum"
+  "enum = List.product enum enum"
 
 definition
   "enum_all P = enum_all (%x. enum_all (%y. P (x, y)))"
 
 definition

 
 instantiation char :: enum
 begin
 
 definition
-  "enum = map (split Char) (product enum enum)"
+  "enum = map (split Char) (List.product enum enum)"
 
 lemma enum_chars [code]:
   "enum = chars"
   unfolding enum_char_def chars_def enum_nibble_def by simp
 

   show "enum_ex (P :: 'a option \<Rightarrow> bool) = (\<exists>x. P x)"
     unfolding enum_ex_option_def enum_ex
     by (auto, case_tac x) auto
 qed (auto simp add: enum_UNIV enum_option_def, rule option.exhaust, auto intro: simp add: distinct_map enum_distinct)
 end
-
-primrec sublists :: "'a list \<Rightarrow> 'a list list" where
-  "sublists [] = [[]]"
-  | "sublists (x#xs) = (let xss = sublists xs in map (Cons x) xss @ xss)"
-
-lemma length_sublists:
-  "length (sublists xs) = 2 ^ length xs"
-  by (induct xs) (simp_all add: Let_def)
-
-lemma sublists_powset:
-  "set ` set (sublists xs) = Pow (set xs)"
-proof -
-  have aux: "\<And>x A. set ` Cons x ` A = insert x ` set ` A"
-    by (auto simp add: image_def)
-  have "set (map set (sublists xs)) = Pow (set xs)"
-    by (induct xs)
-      (simp_all add: aux Let_def Pow_insert Un_commute comp_def del: map_map)
-  then show ?thesis by simp
-qed
-
-lemma distinct_set_sublists:
-  assumes "distinct xs"
-  shows "distinct (map set (sublists xs))"
-proof (rule card_distinct)
-  have "finite (set xs)" by rule
-  then have "card (Pow (set xs)) = 2 ^ card (set xs)" by (rule card_Pow)
-  with assms distinct_card [of xs]
-    have "card (Pow (set xs)) = 2 ^ length xs" by simp
-  then show "card (set (map set (sublists xs))) = length (map set (sublists xs))"
-    by (simp add: sublists_powset length_sublists)
-qed
 
 instantiation set :: (enum) enum
 begin
 
 definition

 
 end
 
 hide_const (open) a\<^isub>1 a\<^isub>2 a\<^isub>3 a\<^isub>4 a\<^isub>5
 
+
 subsection {* An executable THE operator on finite types *}
 
 definition
-  [code del]: "enum_the P = The P"
+  [code del]: "enum_the = The"
 
 lemma [code]:
   "The P = (case filter P enum of [x] => x | _ => enum_the P)"
 proof -
   {

 qed
 
 code_abort enum_the
 code_const enum_the (Eval "(fn p => raise Match)")
 
+
 subsection {* Further operations on finite types *}
 
-lemma Collect_code[code]:
+lemma Collect_code [code]:
   "Collect P = set (filter P enum)"
 by (auto simp add: enum_UNIV)
 
 lemma [code]:
   "Id = image (%x. (x, x)) (set Enum.enum)"

 
 lemma tranclp_unfold [code, no_atp]:
   "tranclp r a b \<equiv> (a, b) \<in> trancl {(x, y). r x y}"
 by (simp add: trancl_def)
 
-lemma rtranclp_rtrancl_eq[code, no_atp]:
+lemma rtranclp_rtrancl_eq [code, no_atp]:
   "rtranclp r x y = ((x, y) : rtrancl {(x, y). r x y})"
 unfolding rtrancl_def by auto
 
-lemma max_ext_eq[code]:
+lemma max_ext_eq [code]:
   "max_ext R = {(X, Y). finite X & finite Y & Y ~={} & (ALL x. x : X --> (EX xa : Y. (x, xa) : R))}"
 by (auto simp add: max_ext.simps)
 
 lemma max_extp_eq[code]:
   "max_extp r x y = ((x, y) : max_ext {(x, y). r x y})"

 lemma mlex_eq[code]:
   "f <*mlex*> R = {(x, y). f x < f y \<or> (f x <= f y \<and> (x, y) : R)}"
 unfolding mlex_prod_def by auto
 
 subsection {* Executable accessible part *}
-(* FIXME: should be moved somewhere else !? *)
-
-subsubsection {* Finite monotone eventually stable sequences *}
-
-lemma finite_mono_remains_stable_implies_strict_prefix:
-  fixes f :: "nat \<Rightarrow> 'a::order"
-  assumes S: "finite (range f)" "mono f" and eq: "\<forall>n. f n = f (Suc n) \<longrightarrow> f (Suc n) = f (Suc (Suc n))"
-  shows "\<exists>N. (\<forall>n\<le>N. \<forall>m\<le>N. m < n \<longrightarrow> f m < f n) \<and> (\<forall>n\<ge>N. f N = f n)"
-  using assms
-proof -
-  have "\<exists>n. f n = f (Suc n)"
-  proof (rule ccontr)
-    assume "\<not> ?thesis"
-    then have "\<And>n. f n \<noteq> f (Suc n)" by auto
-    then have "\<And>n. f n < f (Suc n)"
-      using  `mono f` by (auto simp: le_less mono_iff_le_Suc)
-    with lift_Suc_mono_less_iff[of f]
-    have "\<And>n m. n < m \<Longrightarrow> f n < f m" by auto
-    then have "inj f"
-      by (auto simp: inj_on_def) (metis linorder_less_linear order_less_imp_not_eq)
-    with `finite (range f)` have "finite (UNIV::nat set)"
-      by (rule finite_imageD)
-    then show False by simp
-  qed
-  then obtain n where n: "f n = f (Suc n)" ..
-  def N \<equiv> "LEAST n. f n = f (Suc n)"
-  have N: "f N = f (Suc N)"
-    unfolding N_def using n by (rule LeastI)
-  show ?thesis
-  proof (intro exI[of _ N] conjI allI impI)
-    fix n assume "N \<le> n"
-    then have "\<And>m. N \<le> m \<Longrightarrow> m \<le> n \<Longrightarrow> f m = f N"
-    proof (induct rule: dec_induct)
-      case (step n) then show ?case
-        using eq[rule_format, of "n - 1"] N
-        by (cases n) (auto simp add: le_Suc_eq)
-    qed simp
-    from this[of n] `N \<le> n` show "f N = f n" by auto
-  next
-    fix n m :: nat assume "m < n" "n \<le> N"
-    then show "f m < f n"
-    proof (induct rule: less_Suc_induct[consumes 1])
-      case (1 i)
-      then have "i < N" by simp
-      then have "f i \<noteq> f (Suc i)"
-        unfolding N_def by (rule not_less_Least)
-      with `mono f` show ?case by (simp add: mono_iff_le_Suc less_le)
-    qed auto
-  qed
-qed
-
-lemma finite_mono_strict_prefix_implies_finite_fixpoint:
-  fixes f :: "nat \<Rightarrow> 'a set"
-  assumes S: "\<And>i. f i \<subseteq> S" "finite S"
-    and inj: "\<exists>N. (\<forall>n\<le>N. \<forall>m\<le>N. m < n \<longrightarrow> f m \<subset> f n) \<and> (\<forall>n\<ge>N. f N = f n)"
-  shows "f (card S) = (\<Union>n. f n)"
-proof -
-  from inj obtain N where inj: "(\<forall>n\<le>N. \<forall>m\<le>N. m < n \<longrightarrow> f m \<subset> f n)" and eq: "(\<forall>n\<ge>N. f N = f n)" by auto
-
-  { fix i have "i \<le> N \<Longrightarrow> i \<le> card (f i)"
-    proof (induct i)
-      case 0 then show ?case by simp
-    next
-      case (Suc i)
-      with inj[rule_format, of "Suc i" i]
-      have "(f i) \<subset> (f (Suc i))" by auto
-      moreover have "finite (f (Suc i))" using S by (rule finite_subset)
-      ultimately have "card (f i) < card (f (Suc i))" by (intro psubset_card_mono)
-      with Suc show ?case using inj by auto
-    qed
-  }
-  then have "N \<le> card (f N)" by simp
-  also have "\<dots> \<le> card S" using S by (intro card_mono)
-  finally have "f (card S) = f N" using eq by auto
-  then show ?thesis using eq inj[rule_format, of N]
-    apply auto
-    apply (case_tac "n < N")
-    apply (auto simp: not_less)
-    done
-qed
-
-subsubsection {* Bounded accessible part *}
-
-fun bacc :: "('a * 'a) set => nat => 'a set" 
-where
-  "bacc r 0 = {x. \<forall> y. (y, x) \<notin> r}"
-| "bacc r (Suc n) = (bacc r n Un {x. \<forall> y. (y, x) : r --> y : bacc r n})"
-
-lemma bacc_subseteq_acc:
-  "bacc r n \<subseteq> acc r"
-by (induct n) (auto intro: acc.intros)
-
-lemma bacc_mono:
-  "n <= m ==> bacc r n \<subseteq> bacc r m"
-by (induct rule: dec_induct) auto
-  
-lemma bacc_upper_bound:
-  "bacc (r :: ('a * 'a) set)  (card (UNIV :: ('a :: enum) set)) = (UN n. bacc r n)"
-proof -
-  have "mono (bacc r)" unfolding mono_def by (simp add: bacc_mono)
-  moreover have "\<forall>n. bacc r n = bacc r (Suc n) \<longrightarrow> bacc r (Suc n) = bacc r (Suc (Suc n))" by auto
-  moreover have "finite (range (bacc r))" by auto
-  ultimately show ?thesis
-   by (intro finite_mono_strict_prefix_implies_finite_fixpoint)
-     (auto intro: finite_mono_remains_stable_implies_strict_prefix  simp add: enum_UNIV)
-qed
-
-lemma acc_subseteq_bacc:
-  assumes "finite r"
-  shows "acc r \<subseteq> (UN n. bacc r n)"
-proof
-  fix x
-  assume "x : acc r"
-  then have "\<exists> n. x : bacc r n"
-  proof (induct x arbitrary: rule: acc.induct)
-    case (accI x)
-    then have "\<forall>y. \<exists> n. (y, x) \<in> r --> y : bacc r n" by simp
-    from choice[OF this] obtain n where n: "\<forall>y. (y, x) \<in> r \<longrightarrow> y \<in> bacc r (n y)" ..
-    obtain n where "\<And>y. (y, x) : r \<Longrightarrow> y : bacc r n"
-    proof
-      fix y assume y: "(y, x) : r"
-      with n have "y : bacc r (n y)" by auto
-      moreover have "n y <= Max ((%(y, x). n y) ` r)"
-        using y `finite r` by (auto intro!: Max_ge)
-      note bacc_mono[OF this, of r]
-      ultimately show "y : bacc r (Max ((%(y, x). n y) ` r))" by auto
-    qed
-    then show ?case
-      by (auto simp add: Let_def intro!: exI[of _ "Suc n"])
-  qed
-  then show "x : (UN n. bacc r n)" by auto
-qed
-
-lemma acc_bacc_eq: "acc ((set xs) :: (('a :: enum) * 'a) set) = bacc (set xs) (card (UNIV :: 'a set))"
-by (metis acc_subseteq_bacc bacc_subseteq_acc bacc_upper_bound finite_set order_eq_iff)
 
 definition 
   [code del]: "card_UNIV = card UNIV"
 
 lemma [code]:
   "card_UNIV TYPE('a :: enum) = card (set (Enum.enum :: 'a list))"
-unfolding card_UNIV_def enum_UNIV ..
-
-declare acc_bacc_eq[folded card_UNIV_def, code]
-
-lemma [code_unfold]: "accp r = (%x. x : acc {(x, y). r x y})"
-unfolding acc_def by simp
+  unfolding card_UNIV_def enum_UNIV ..
+
+lemma [code]:
+  fixes xs :: "('a::finite \<times> 'a) list"
+  shows "acc (set xs) = bacc (set xs) (card_UNIV TYPE('a))"
+  by (simp add: card_UNIV_def acc_bacc_eq)
+
+lemma [code_unfold]: "accp r = (\<lambda>x. x \<in> acc {(x, y). r x y})"
+  unfolding acc_def by simp
 
 subsection {* Closing up *}
 
 hide_type (open) finite_1 finite_2 finite_3 finite_4 finite_5
-hide_const (open) enum enum_all enum_ex n_lists all_n_lists ex_n_lists product ntrancl
-
-end
+hide_const (open) enum enum_all enum_ex all_n_lists ex_n_lists ntrancl
+
+end
+