src/HOL/Enum.thy
author bulwahn
Mon, 13 Dec 2010 08:51:52 +0100
changeset 41115 2c362ff5daf4
parent 41085 a549ff1d4070
child 45117 3911cf09899a
permissions -rw-r--r--
adding an executable THE operator on finite types
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(* Author: Florian Haftmann, TU Muenchen *)
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header {* Finite types as explicit enumerations *}
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theory Enum
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imports Map String
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begin
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subsection {* Class @{text enum} *}
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class enum =
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  fixes enum :: "'a list"
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  fixes enum_all :: "('a \<Rightarrow> bool) \<Rightarrow> bool"
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  fixes enum_ex  :: "('a \<Rightarrow> bool) \<Rightarrow> bool"
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  assumes UNIV_enum: "UNIV = set enum"
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    and enum_distinct: "distinct enum"
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  assumes enum_all : "enum_all P = (\<forall> x. P x)"
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  assumes enum_ex  : "enum_ex P = (\<exists> x. P x)" 
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begin
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subclass finite proof
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qed (simp add: UNIV_enum)
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lemma enum_UNIV: "set enum = UNIV" unfolding UNIV_enum ..
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lemma in_enum: "x \<in> set enum"
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  unfolding enum_UNIV by auto
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lemma enum_eq_I:
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  assumes "\<And>x. x \<in> set xs"
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  shows "set enum = set xs"
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proof -
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  from assms UNIV_eq_I have "UNIV = set xs" by auto
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  with enum_UNIV show ?thesis by simp
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qed
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end
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subsection {* Equality and order on functions *}
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instantiation "fun" :: (enum, equal) equal
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begin
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definition
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  "HOL.equal f g \<longleftrightarrow> (\<forall>x \<in> set enum. f x = g x)"
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instance proof
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qed (simp_all add: equal_fun_def enum_UNIV fun_eq_iff)
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end
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lemma [code]:
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  "HOL.equal f g \<longleftrightarrow> enum_all (%x. f x = g x)"
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by (auto simp add: equal enum_all fun_eq_iff)
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lemma [code nbe]:
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  "HOL.equal (f :: _ \<Rightarrow> _) f \<longleftrightarrow> True"
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  by (fact equal_refl)
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lemma order_fun [code]:
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  fixes f g :: "'a\<Colon>enum \<Rightarrow> 'b\<Colon>order"
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  shows "f \<le> g \<longleftrightarrow> enum_all (\<lambda>x. f x \<le> g x)"
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    and "f < g \<longleftrightarrow> f \<le> g \<and> enum_ex (\<lambda>x. f x \<noteq> g x)"
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  by (simp_all add: enum_all enum_ex fun_eq_iff le_fun_def order_less_le)
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subsection {* Quantifiers *}
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lemma all_code [code]: "(\<forall>x. P x) \<longleftrightarrow> enum_all P"
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  by (simp add: enum_all)
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lemma exists_code [code]: "(\<exists>x. P x) \<longleftrightarrow> enum_ex P"
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  by (simp add: enum_ex)
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lemma exists1_code[code]: "(\<exists>!x. P x) \<longleftrightarrow> list_ex1 P enum"
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unfolding list_ex1_iff enum_UNIV by auto
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subsection {* Default instances *}
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primrec n_lists :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list list" where
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  "n_lists 0 xs = [[]]"
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  | "n_lists (Suc n) xs = concat (map (\<lambda>ys. map (\<lambda>y. y # ys) xs) (n_lists n xs))"
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lemma n_lists_Nil [simp]: "n_lists n [] = (if n = 0 then [[]] else [])"
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  by (induct n) simp_all
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lemma length_n_lists: "length (n_lists n xs) = length xs ^ n"
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  by (induct n) (auto simp add: length_concat o_def listsum_triv)
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lemma length_n_lists_elem: "ys \<in> set (n_lists n xs) \<Longrightarrow> length ys = n"
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  by (induct n arbitrary: ys) auto
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lemma set_n_lists: "set (n_lists n xs) = {ys. length ys = n \<and> set ys \<subseteq> set xs}"
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proof (rule set_eqI)
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  fix ys :: "'a list"
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  show "ys \<in> set (n_lists n xs) \<longleftrightarrow> ys \<in> {ys. length ys = n \<and> set ys \<subseteq> set xs}"
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  proof -
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    have "ys \<in> set (n_lists n xs) \<Longrightarrow> length ys = n"
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      by (induct n arbitrary: ys) auto
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    moreover have "\<And>x. ys \<in> set (n_lists n xs) \<Longrightarrow> x \<in> set ys \<Longrightarrow> x \<in> set xs"
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      by (induct n arbitrary: ys) auto
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    moreover have "set ys \<subseteq> set xs \<Longrightarrow> ys \<in> set (n_lists (length ys) xs)"
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      by (induct ys) auto
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    ultimately show ?thesis by auto
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  qed
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qed
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lemma distinct_n_lists:
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  assumes "distinct xs"
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  shows "distinct (n_lists n xs)"
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proof (rule card_distinct)
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  from assms have card_length: "card (set xs) = length xs" by (rule distinct_card)
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  have "card (set (n_lists n xs)) = card (set xs) ^ n"
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  proof (induct n)
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    case 0 then show ?case by simp
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  next
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    case (Suc n)
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    moreover have "card (\<Union>ys\<in>set (n_lists n xs). (\<lambda>y. y # ys) ` set xs)
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      = (\<Sum>ys\<in>set (n_lists n xs). card ((\<lambda>y. y # ys) ` set xs))"
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      by (rule card_UN_disjoint) auto
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    moreover have "\<And>ys. card ((\<lambda>y. y # ys) ` set xs) = card (set xs)"
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      by (rule card_image) (simp add: inj_on_def)
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    ultimately show ?case by auto
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  qed
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  also have "\<dots> = length xs ^ n" by (simp add: card_length)
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  finally show "card (set (n_lists n xs)) = length (n_lists n xs)"
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    by (simp add: length_n_lists)
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qed
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lemma map_of_zip_enum_is_Some:
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  assumes "length ys = length (enum \<Colon> 'a\<Colon>enum list)"
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  shows "\<exists>y. map_of (zip (enum \<Colon> 'a\<Colon>enum list) ys) x = Some y"
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proof -
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  from assms have "x \<in> set (enum \<Colon> 'a\<Colon>enum list) \<longleftrightarrow>
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    (\<exists>y. map_of (zip (enum \<Colon> 'a\<Colon>enum list) ys) x = Some y)"
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    by (auto intro!: map_of_zip_is_Some)
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  then show ?thesis using enum_UNIV by auto
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qed
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lemma map_of_zip_enum_inject:
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  fixes xs ys :: "'b\<Colon>enum list"
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  assumes length: "length xs = length (enum \<Colon> 'a\<Colon>enum list)"
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      "length ys = length (enum \<Colon> 'a\<Colon>enum list)"
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    and map_of: "the \<circ> map_of (zip (enum \<Colon> 'a\<Colon>enum list) xs) = the \<circ> map_of (zip (enum \<Colon> 'a\<Colon>enum list) ys)"
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  shows "xs = ys"
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proof -
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  have "map_of (zip (enum \<Colon> 'a list) xs) = map_of (zip (enum \<Colon> 'a list) ys)"
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  proof
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    fix x :: 'a
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    from length map_of_zip_enum_is_Some obtain y1 y2
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      where "map_of (zip (enum \<Colon> 'a list) xs) x = Some y1"
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        and "map_of (zip (enum \<Colon> 'a list) ys) x = Some y2" by blast
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    moreover from map_of have "the (map_of (zip (enum \<Colon> 'a\<Colon>enum list) xs) x) = the (map_of (zip (enum \<Colon> 'a\<Colon>enum list) ys) x)"
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      by (auto dest: fun_cong)
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    ultimately show "map_of (zip (enum \<Colon> 'a\<Colon>enum list) xs) x = map_of (zip (enum \<Colon> 'a\<Colon>enum list) ys) x"
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      by simp
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  qed
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  with length enum_distinct show "xs = ys" by (rule map_of_zip_inject)
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qed
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definition
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  all_n_lists :: "(('a :: enum) list \<Rightarrow> bool) \<Rightarrow> nat \<Rightarrow> bool"
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where
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  "all_n_lists P n = (\<forall>xs \<in> set (n_lists n enum). P xs)"
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lemma [code]:
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  "all_n_lists P n = (if n = 0 then P [] else enum_all (%x. all_n_lists (%xs. P (x # xs)) (n - 1)))"
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unfolding all_n_lists_def enum_all
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by (cases n) (auto simp add: enum_UNIV)
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definition
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  ex_n_lists :: "(('a :: enum) list \<Rightarrow> bool) \<Rightarrow> nat \<Rightarrow> bool"
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where
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  "ex_n_lists P n = (\<exists>xs \<in> set (n_lists n enum). P xs)"
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lemma [code]:
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  "ex_n_lists P n = (if n = 0 then P [] else enum_ex (%x. ex_n_lists (%xs. P (x # xs)) (n - 1)))"
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unfolding ex_n_lists_def enum_ex
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by (cases n) (auto simp add: enum_UNIV)
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instantiation "fun" :: (enum, enum) enum
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begin
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definition
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  "enum = map (\<lambda>ys. the o map_of (zip (enum\<Colon>'a list) ys)) (n_lists (length (enum\<Colon>'a\<Colon>enum list)) enum)"
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definition
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  "enum_all P = all_n_lists (\<lambda>bs. P (the o map_of (zip enum bs))) (length (enum :: 'a list))"
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definition
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  "enum_ex P = ex_n_lists (\<lambda>bs. P (the o map_of (zip enum bs))) (length (enum :: 'a list))"
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instance proof
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  show "UNIV = set (enum \<Colon> ('a \<Rightarrow> 'b) list)"
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  proof (rule UNIV_eq_I)
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    fix f :: "'a \<Rightarrow> 'b"
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    have "f = the \<circ> map_of (zip (enum \<Colon> 'a\<Colon>enum list) (map f enum))"
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      by (auto simp add: map_of_zip_map fun_eq_iff intro: in_enum)
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    then show "f \<in> set enum"
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      by (auto simp add: enum_fun_def set_n_lists intro: in_enum)
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  qed
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next
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  from map_of_zip_enum_inject
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  show "distinct (enum \<Colon> ('a \<Rightarrow> 'b) list)"
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    by (auto intro!: inj_onI simp add: enum_fun_def
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      distinct_map distinct_n_lists enum_distinct set_n_lists enum_all)
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next
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  fix P
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  show "enum_all (P :: ('a \<Rightarrow> 'b) \<Rightarrow> bool) = (\<forall>x. P x)"
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  proof
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    assume "enum_all P"
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    show "\<forall>x. P x"
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    proof
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      fix f :: "'a \<Rightarrow> 'b"
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      have f: "f = the \<circ> map_of (zip (enum \<Colon> 'a\<Colon>enum list) (map f enum))"
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        by (auto simp add: map_of_zip_map fun_eq_iff intro: in_enum)
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      from `enum_all P` have "P (the \<circ> map_of (zip enum (map f enum)))"
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        unfolding enum_all_fun_def all_n_lists_def
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        apply (simp add: set_n_lists)
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        apply (erule_tac x="map f enum" in allE)
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        apply (auto intro!: in_enum)
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        done
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      from this f show "P f" by auto
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    qed
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  next
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    assume "\<forall>x. P x"
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    from this show "enum_all P"
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      unfolding enum_all_fun_def all_n_lists_def by auto
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  qed
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next
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  fix P
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  show "enum_ex (P :: ('a \<Rightarrow> 'b) \<Rightarrow> bool) = (\<exists>x. P x)"
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  proof
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    assume "enum_ex P"
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    from this show "\<exists>x. P x"
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      unfolding enum_ex_fun_def ex_n_lists_def by auto
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  next
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    assume "\<exists>x. P x"
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    from this obtain f where "P f" ..
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    have f: "f = the \<circ> map_of (zip (enum \<Colon> 'a\<Colon>enum list) (map f enum))"
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      by (auto simp add: map_of_zip_map fun_eq_iff intro: in_enum) 
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    from `P f` this have "P (the \<circ> map_of (zip (enum \<Colon> 'a\<Colon>enum list) (map f enum)))"
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      by auto
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    from  this show "enum_ex P"
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      unfolding enum_ex_fun_def ex_n_lists_def
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      apply (auto simp add: set_n_lists)
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      apply (rule_tac x="map f enum" in exI)
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      apply (auto intro!: in_enum)
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      done
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  qed
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qed
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end
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lemma enum_fun_code [code]: "enum = (let enum_a = (enum \<Colon> 'a\<Colon>{enum, equal} list)
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  in map (\<lambda>ys. the o map_of (zip enum_a ys)) (n_lists (length enum_a) enum))"
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  by (simp add: enum_fun_def Let_def)
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lemma enum_all_fun_code [code]:
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  "enum_all P = (let enum_a = (enum :: 'a::{enum, equal} list)
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   in all_n_lists (\<lambda>bs. P (the o map_of (zip enum_a bs))) (length enum_a))"
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  by (simp add: enum_all_fun_def Let_def)
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lemma enum_ex_fun_code [code]:
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  "enum_ex P = (let enum_a = (enum :: 'a::{enum, equal} list)
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   in ex_n_lists (\<lambda>bs. P (the o map_of (zip enum_a bs))) (length enum_a))"
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  by (simp add: enum_ex_fun_def Let_def)
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instantiation unit :: enum
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begin
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definition
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  "enum = [()]"
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definition
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  "enum_all P = P ()"
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   281
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definition
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  "enum_ex P = P ()"
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   285
instance proof
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qed (auto simp add: enum_unit_def UNIV_unit enum_all_unit_def enum_ex_unit_def intro: unit.exhaust)
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end
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instantiation bool :: enum
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begin
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definition
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  "enum = [False, True]"
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definition
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  "enum_all P = (P False \<and> P True)"
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   298
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   299
definition
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  "enum_ex P = (P False \<or> P True)"
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instance proof
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  fix P
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  show "enum_all (P :: bool \<Rightarrow> bool) = (\<forall>x. P x)"
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   305
    unfolding enum_all_bool_def by (auto, case_tac x) auto
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next
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   307
  fix P
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   308
  show "enum_ex (P :: bool \<Rightarrow> bool) = (\<exists>x. P x)"
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   309
    unfolding enum_ex_bool_def by (auto, case_tac x) auto
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qed (auto simp add: enum_bool_def UNIV_bool)
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end
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   313
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   314
primrec product :: "'a list \<Rightarrow> 'b list \<Rightarrow> ('a \<times> 'b) list" where
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   315
  "product [] _ = []"
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   316
  | "product (x#xs) ys = map (Pair x) ys @ product xs ys"
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   317
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   318
lemma product_list_set:
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   319
  "set (product xs ys) = set xs \<times> set ys"
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   320
  by (induct xs) auto
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   321
26444
6a5faa5bcf19 instance for functions, explicit characters
haftmann
parents: 26348
diff changeset
   322
lemma distinct_product:
6a5faa5bcf19 instance for functions, explicit characters
haftmann
parents: 26348
diff changeset
   323
  assumes "distinct xs" and "distinct ys"
6a5faa5bcf19 instance for functions, explicit characters
haftmann
parents: 26348
diff changeset
   324
  shows "distinct (product xs ys)"
6a5faa5bcf19 instance for functions, explicit characters
haftmann
parents: 26348
diff changeset
   325
  using assms by (induct xs)
6a5faa5bcf19 instance for functions, explicit characters
haftmann
parents: 26348
diff changeset
   326
    (auto intro: inj_onI simp add: product_list_set distinct_map)
6a5faa5bcf19 instance for functions, explicit characters
haftmann
parents: 26348
diff changeset
   327
37678
0040bafffdef "prod" and "sum" replace "*" and "+" respectively
haftmann
parents: 37601
diff changeset
   328
instantiation prod :: (enum, enum) enum
26348
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   329
begin
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   330
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   331
definition
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   332
  "enum = product enum enum"
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   333
41078
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   334
definition
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   335
  "enum_all P = enum_all (%x. enum_all (%y. P (x, y)))"
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   336
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   337
definition
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   338
  "enum_ex P = enum_ex (%x. enum_ex (%y. P (x, y)))"
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   339
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   340
 
26348
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   341
instance by default
41078
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   342
  (simp_all add: enum_prod_def product_list_set distinct_product
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   343
    enum_UNIV enum_distinct enum_all_prod_def enum_all enum_ex_prod_def enum_ex)
26348
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   344
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   345
end
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   346
37678
0040bafffdef "prod" and "sum" replace "*" and "+" respectively
haftmann
parents: 37601
diff changeset
   347
instantiation sum :: (enum, enum) enum
26348
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   348
begin
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   349
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   350
definition
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   351
  "enum = map Inl enum @ map Inr enum"
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   352
41078
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   353
definition
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   354
  "enum_all P = (enum_all (%x. P (Inl x)) \<and> enum_all (%x. P (Inr x)))"
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   355
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   356
definition
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   357
  "enum_ex P = (enum_ex (%x. P (Inl x)) \<or> enum_ex (%x. P (Inr x)))"
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   358
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   359
instance proof
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   360
  fix P
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   361
  show "enum_all (P :: ('a + 'b) \<Rightarrow> bool) = (\<forall>x. P x)"
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   362
    unfolding enum_all_sum_def enum_all
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   363
    by (auto, case_tac x) auto
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   364
next
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   365
  fix P
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   366
  show "enum_ex (P :: ('a + 'b) \<Rightarrow> bool) = (\<exists>x. P x)"
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   367
    unfolding enum_ex_sum_def enum_ex
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   368
    by (auto, case_tac x) auto
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   369
qed (auto simp add: enum_UNIV enum_sum_def, case_tac x, auto intro: inj_onI simp add: distinct_map enum_distinct)
26348
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   370
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   371
end
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   372
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   373
primrec sublists :: "'a list \<Rightarrow> 'a list list" where
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   374
  "sublists [] = [[]]"
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   375
  | "sublists (x#xs) = (let xss = sublists xs in map (Cons x) xss @ xss)"
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   376
26444
6a5faa5bcf19 instance for functions, explicit characters
haftmann
parents: 26348
diff changeset
   377
lemma length_sublists:
6a5faa5bcf19 instance for functions, explicit characters
haftmann
parents: 26348
diff changeset
   378
  "length (sublists xs) = Suc (Suc (0\<Colon>nat)) ^ length xs"
6a5faa5bcf19 instance for functions, explicit characters
haftmann
parents: 26348
diff changeset
   379
  by (induct xs) (simp_all add: Let_def)
6a5faa5bcf19 instance for functions, explicit characters
haftmann
parents: 26348
diff changeset
   380
26348
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   381
lemma sublists_powset:
26444
6a5faa5bcf19 instance for functions, explicit characters
haftmann
parents: 26348
diff changeset
   382
  "set ` set (sublists xs) = Pow (set xs)"
26348
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   383
proof -
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   384
  have aux: "\<And>x A. set ` Cons x ` A = insert x ` set ` A"
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   385
    by (auto simp add: image_def)
26444
6a5faa5bcf19 instance for functions, explicit characters
haftmann
parents: 26348
diff changeset
   386
  have "set (map set (sublists xs)) = Pow (set xs)"
26348
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   387
    by (induct xs)
33639
603320b93668 New list theorems; added map_map to simpset, this is the prefered direction; allow sorting by a key
hoelzl
parents: 33635
diff changeset
   388
      (simp_all add: aux Let_def Pow_insert Un_commute comp_def del: map_map)
26444
6a5faa5bcf19 instance for functions, explicit characters
haftmann
parents: 26348
diff changeset
   389
  then show ?thesis by simp
6a5faa5bcf19 instance for functions, explicit characters
haftmann
parents: 26348
diff changeset
   390
qed
6a5faa5bcf19 instance for functions, explicit characters
haftmann
parents: 26348
diff changeset
   391
6a5faa5bcf19 instance for functions, explicit characters
haftmann
parents: 26348
diff changeset
   392
lemma distinct_set_sublists:
6a5faa5bcf19 instance for functions, explicit characters
haftmann
parents: 26348
diff changeset
   393
  assumes "distinct xs"
6a5faa5bcf19 instance for functions, explicit characters
haftmann
parents: 26348
diff changeset
   394
  shows "distinct (map set (sublists xs))"
6a5faa5bcf19 instance for functions, explicit characters
haftmann
parents: 26348
diff changeset
   395
proof (rule card_distinct)
6a5faa5bcf19 instance for functions, explicit characters
haftmann
parents: 26348
diff changeset
   396
  have "finite (set xs)" by rule
6a5faa5bcf19 instance for functions, explicit characters
haftmann
parents: 26348
diff changeset
   397
  then have "card (Pow (set xs)) = Suc (Suc 0) ^ card (set xs)" by (rule card_Pow)
6a5faa5bcf19 instance for functions, explicit characters
haftmann
parents: 26348
diff changeset
   398
  with assms distinct_card [of xs]
6a5faa5bcf19 instance for functions, explicit characters
haftmann
parents: 26348
diff changeset
   399
    have "card (Pow (set xs)) = Suc (Suc 0) ^ length xs" by simp
6a5faa5bcf19 instance for functions, explicit characters
haftmann
parents: 26348
diff changeset
   400
  then show "card (set (map set (sublists xs))) = length (map set (sublists xs))"
6a5faa5bcf19 instance for functions, explicit characters
haftmann
parents: 26348
diff changeset
   401
    by (simp add: sublists_powset length_sublists)
26348
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   402
qed
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   403
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   404
instantiation nibble :: enum
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   405
begin
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   406
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   407
definition
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   408
  "enum = [Nibble0, Nibble1, Nibble2, Nibble3, Nibble4, Nibble5, Nibble6, Nibble7,
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   409
    Nibble8, Nibble9, NibbleA, NibbleB, NibbleC, NibbleD, NibbleE, NibbleF]"
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   410
41078
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   411
definition
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   412
  "enum_all P = (P Nibble0 \<and> P Nibble1 \<and> P Nibble2 \<and> P Nibble3 \<and> P Nibble4 \<and> P Nibble5 \<and> P Nibble6 \<and> P Nibble7
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   413
     \<and> P Nibble8 \<and> P Nibble9 \<and> P NibbleA \<and> P NibbleB \<and> P NibbleC \<and> P NibbleD \<and> P NibbleE \<and> P NibbleF)"
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   414
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   415
definition
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   416
  "enum_ex P = (P Nibble0 \<or> P Nibble1 \<or> P Nibble2 \<or> P Nibble3 \<or> P Nibble4 \<or> P Nibble5 \<or> P Nibble6 \<or> P Nibble7
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   417
     \<or> P Nibble8 \<or> P Nibble9 \<or> P NibbleA \<or> P NibbleB \<or> P NibbleC \<or> P NibbleD \<or> P NibbleE \<or> P NibbleF)"
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   418
31464
b2aca38301c4 tuned proofs
haftmann
parents: 31193
diff changeset
   419
instance proof
41078
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   420
  fix P
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   421
  show "enum_all (P :: nibble \<Rightarrow> bool) = (\<forall>x. P x)"
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   422
    unfolding enum_all_nibble_def
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   423
    by (auto, case_tac x) auto
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   424
next
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   425
  fix P
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   426
  show "enum_ex (P :: nibble \<Rightarrow> bool) = (\<exists>x. P x)"
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   427
    unfolding enum_ex_nibble_def
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   428
    by (auto, case_tac x) auto
31464
b2aca38301c4 tuned proofs
haftmann
parents: 31193
diff changeset
   429
qed (simp_all add: enum_nibble_def UNIV_nibble)
26348
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   430
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   431
end
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   432
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   433
instantiation char :: enum
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   434
begin
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   435
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   436
definition
37765
26bdfb7b680b dropped superfluous [code del]s
haftmann
parents: 37678
diff changeset
   437
  "enum = map (split Char) (product enum enum)"
26444
6a5faa5bcf19 instance for functions, explicit characters
haftmann
parents: 26348
diff changeset
   438
31482
7288382fd549 using constant "chars"
haftmann
parents: 31464
diff changeset
   439
lemma enum_chars [code]:
7288382fd549 using constant "chars"
haftmann
parents: 31464
diff changeset
   440
  "enum = chars"
7288382fd549 using constant "chars"
haftmann
parents: 31464
diff changeset
   441
  unfolding enum_char_def chars_def enum_nibble_def by simp
26348
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   442
41078
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   443
definition
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   444
  "enum_all P = list_all P chars"
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   445
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   446
definition
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   447
  "enum_ex P = list_ex P chars"
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   448
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   449
lemma set_enum_char: "set (enum :: char list) = UNIV"
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   450
    by (auto intro: char.exhaust simp add: enum_char_def product_list_set enum_UNIV full_SetCompr_eq [symmetric])
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   451
31464
b2aca38301c4 tuned proofs
haftmann
parents: 31193
diff changeset
   452
instance proof
41078
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   453
  fix P
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   454
  show "enum_all (P :: char \<Rightarrow> bool) = (\<forall>x. P x)"
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   455
    unfolding enum_all_char_def enum_chars[symmetric]
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   456
    by (auto simp add: list_all_iff set_enum_char)
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   457
next
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   458
  fix P
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   459
  show "enum_ex (P :: char \<Rightarrow> bool) = (\<exists>x. P x)"
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   460
    unfolding enum_ex_char_def enum_chars[symmetric]
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   461
    by (auto simp add: list_ex_iff set_enum_char)
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   462
next
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   463
  show "distinct (enum :: char list)"
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   464
    by (auto intro: inj_onI simp add: enum_char_def product_list_set distinct_map distinct_product enum_distinct)
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   465
qed (auto simp add: set_enum_char)
26348
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   466
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   467
end
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   468
29024
6cfa380af73b instantiation option :: (enum) enum
huffman
parents: 28684
diff changeset
   469
instantiation option :: (enum) enum
6cfa380af73b instantiation option :: (enum) enum
huffman
parents: 28684
diff changeset
   470
begin
6cfa380af73b instantiation option :: (enum) enum
huffman
parents: 28684
diff changeset
   471
6cfa380af73b instantiation option :: (enum) enum
huffman
parents: 28684
diff changeset
   472
definition
6cfa380af73b instantiation option :: (enum) enum
huffman
parents: 28684
diff changeset
   473
  "enum = None # map Some enum"
6cfa380af73b instantiation option :: (enum) enum
huffman
parents: 28684
diff changeset
   474
41078
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   475
definition
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   476
  "enum_all P = (P None \<and> enum_all (%x. P (Some x)))"
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   477
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   478
definition
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   479
  "enum_ex P = (P None \<or> enum_ex (%x. P (Some x)))"
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   480
31464
b2aca38301c4 tuned proofs
haftmann
parents: 31193
diff changeset
   481
instance proof
41078
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   482
  fix P
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   483
  show "enum_all (P :: 'a option \<Rightarrow> bool) = (\<forall>x. P x)"
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   484
    unfolding enum_all_option_def enum_all
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   485
    by (auto, case_tac x) auto
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   486
next
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   487
  fix P
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   488
  show "enum_ex (P :: 'a option \<Rightarrow> bool) = (\<exists>x. P x)"
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   489
    unfolding enum_ex_option_def enum_ex
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   490
    by (auto, case_tac x) auto
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   491
qed (auto simp add: enum_UNIV enum_option_def, rule option.exhaust, auto intro: simp add: distinct_map enum_distinct)
29024
6cfa380af73b instantiation option :: (enum) enum
huffman
parents: 28684
diff changeset
   492
6cfa380af73b instantiation option :: (enum) enum
huffman
parents: 28684
diff changeset
   493
end
6cfa380af73b instantiation option :: (enum) enum
huffman
parents: 28684
diff changeset
   494
40647
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   495
subsection {* Small finite types *}
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   496
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   497
text {* We define small finite types for the use in Quickcheck *}
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   498
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   499
datatype finite_1 = a\<^isub>1
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   500
40900
1d5f76d79856 adding shorter output syntax for the finite types of quickcheck
bulwahn
parents: 40898
diff changeset
   501
notation (output) a\<^isub>1  ("a\<^isub>1")
1d5f76d79856 adding shorter output syntax for the finite types of quickcheck
bulwahn
parents: 40898
diff changeset
   502
40647
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   503
instantiation finite_1 :: enum
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   504
begin
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   505
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   506
definition
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   507
  "enum = [a\<^isub>1]"
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   508
41078
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   509
definition
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   510
  "enum_all P = P a\<^isub>1"
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   511
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   512
definition
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   513
  "enum_ex P = P a\<^isub>1"
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   514
40647
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   515
instance proof
41078
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   516
  fix P
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   517
  show "enum_all (P :: finite_1 \<Rightarrow> bool) = (\<forall>x. P x)"
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   518
    unfolding enum_all_finite_1_def
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   519
    by (auto, case_tac x) auto
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   520
next
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   521
  fix P
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   522
  show "enum_ex (P :: finite_1 \<Rightarrow> bool) = (\<exists>x. P x)"
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   523
    unfolding enum_ex_finite_1_def
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   524
    by (auto, case_tac x) auto
40647
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   525
qed (auto simp add: enum_finite_1_def intro: finite_1.exhaust)
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   526
29024
6cfa380af73b instantiation option :: (enum) enum
huffman
parents: 28684
diff changeset
   527
end
40647
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   528
40651
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   529
instantiation finite_1 :: linorder
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   530
begin
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   531
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   532
definition less_eq_finite_1 :: "finite_1 \<Rightarrow> finite_1 \<Rightarrow> bool"
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   533
where
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   534
  "less_eq_finite_1 x y = True"
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   535
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   536
definition less_finite_1 :: "finite_1 \<Rightarrow> finite_1 \<Rightarrow> bool"
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   537
where
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   538
  "less_finite_1 x y = False"
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   539
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   540
instance
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   541
apply (intro_classes)
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   542
apply (auto simp add: less_finite_1_def less_eq_finite_1_def)
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   543
apply (metis finite_1.exhaust)
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   544
done
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   545
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   546
end
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   547
41085
a549ff1d4070 adding a smarter enumeration scheme for finite functions
bulwahn
parents: 41078
diff changeset
   548
hide_const (open) a\<^isub>1
40657
58a6ba7ccfc5 hiding the constants
bulwahn
parents: 40652
diff changeset
   549
40647
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   550
datatype finite_2 = a\<^isub>1 | a\<^isub>2
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   551
40900
1d5f76d79856 adding shorter output syntax for the finite types of quickcheck
bulwahn
parents: 40898
diff changeset
   552
notation (output) a\<^isub>1  ("a\<^isub>1")
1d5f76d79856 adding shorter output syntax for the finite types of quickcheck
bulwahn
parents: 40898
diff changeset
   553
notation (output) a\<^isub>2  ("a\<^isub>2")
1d5f76d79856 adding shorter output syntax for the finite types of quickcheck
bulwahn
parents: 40898
diff changeset
   554
40647
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   555
instantiation finite_2 :: enum
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   556
begin
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   557
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   558
definition
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   559
  "enum = [a\<^isub>1, a\<^isub>2]"
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   560
41078
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   561
definition
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   562
  "enum_all P = (P a\<^isub>1 \<and> P a\<^isub>2)"
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   563
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   564
definition
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   565
  "enum_ex P = (P a\<^isub>1 \<or> P a\<^isub>2)"
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   566
40647
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   567
instance proof
41078
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   568
  fix P
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   569
  show "enum_all (P :: finite_2 \<Rightarrow> bool) = (\<forall>x. P x)"
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   570
    unfolding enum_all_finite_2_def
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   571
    by (auto, case_tac x) auto
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   572
next
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   573
  fix P
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   574
  show "enum_ex (P :: finite_2 \<Rightarrow> bool) = (\<exists>x. P x)"
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   575
    unfolding enum_ex_finite_2_def
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   576
    by (auto, case_tac x) auto
40647
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   577
qed (auto simp add: enum_finite_2_def intro: finite_2.exhaust)
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   578
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   579
end
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   580
40651
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   581
instantiation finite_2 :: linorder
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   582
begin
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   583
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   584
definition less_finite_2 :: "finite_2 \<Rightarrow> finite_2 \<Rightarrow> bool"
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   585
where
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   586
  "less_finite_2 x y = ((x = a\<^isub>1) & (y = a\<^isub>2))"
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   587
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   588
definition less_eq_finite_2 :: "finite_2 \<Rightarrow> finite_2 \<Rightarrow> bool"
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   589
where
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   590
  "less_eq_finite_2 x y = ((x = y) \<or> (x < y))"
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   591
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   592
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   593
instance
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   594
apply (intro_classes)
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   595
apply (auto simp add: less_finite_2_def less_eq_finite_2_def)
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   596
apply (metis finite_2.distinct finite_2.nchotomy)+
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   597
done
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   598
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   599
end
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   600
41085
a549ff1d4070 adding a smarter enumeration scheme for finite functions
bulwahn
parents: 41078
diff changeset
   601
hide_const (open) a\<^isub>1 a\<^isub>2
40657
58a6ba7ccfc5 hiding the constants
bulwahn
parents: 40652
diff changeset
   602
40651
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   603
40647
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   604
datatype finite_3 = a\<^isub>1 | a\<^isub>2 | a\<^isub>3
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   605
40900
1d5f76d79856 adding shorter output syntax for the finite types of quickcheck
bulwahn
parents: 40898
diff changeset
   606
notation (output) a\<^isub>1  ("a\<^isub>1")
1d5f76d79856 adding shorter output syntax for the finite types of quickcheck
bulwahn
parents: 40898
diff changeset
   607
notation (output) a\<^isub>2  ("a\<^isub>2")
1d5f76d79856 adding shorter output syntax for the finite types of quickcheck
bulwahn
parents: 40898
diff changeset
   608
notation (output) a\<^isub>3  ("a\<^isub>3")
1d5f76d79856 adding shorter output syntax for the finite types of quickcheck
bulwahn
parents: 40898
diff changeset
   609
40647
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   610
instantiation finite_3 :: enum
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   611
begin
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   612
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   613
definition
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   614
  "enum = [a\<^isub>1, a\<^isub>2, a\<^isub>3]"
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   615
41078
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   616
definition
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   617
  "enum_all P = (P a\<^isub>1 \<and> P a\<^isub>2 \<and> P a\<^isub>3)"
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   618
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   619
definition
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   620
  "enum_ex P = (P a\<^isub>1 \<or> P a\<^isub>2 \<or> P a\<^isub>3)"
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   621
40647
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   622
instance proof
41078
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   623
  fix P
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   624
  show "enum_all (P :: finite_3 \<Rightarrow> bool) = (\<forall>x. P x)"
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   625
    unfolding enum_all_finite_3_def
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   626
    by (auto, case_tac x) auto
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   627
next
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   628
  fix P
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   629
  show "enum_ex (P :: finite_3 \<Rightarrow> bool) = (\<exists>x. P x)"
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   630
    unfolding enum_ex_finite_3_def
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   631
    by (auto, case_tac x) auto
40647
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   632
qed (auto simp add: enum_finite_3_def intro: finite_3.exhaust)
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   633
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   634
end
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   635
40651
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   636
instantiation finite_3 :: linorder
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   637
begin
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   638
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   639
definition less_finite_3 :: "finite_3 \<Rightarrow> finite_3 \<Rightarrow> bool"
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   640
where
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   641
  "less_finite_3 x y = (case x of a\<^isub>1 => (y \<noteq> a\<^isub>1)
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   642
     | a\<^isub>2 => (y = a\<^isub>3)| a\<^isub>3 => False)"
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   643
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   644
definition less_eq_finite_3 :: "finite_3 \<Rightarrow> finite_3 \<Rightarrow> bool"
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   645
where
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   646
  "less_eq_finite_3 x y = ((x = y) \<or> (x < y))"
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   647
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   648
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   649
instance proof (intro_classes)
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   650
qed (auto simp add: less_finite_3_def less_eq_finite_3_def split: finite_3.split_asm)
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   651
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   652
end
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   653
41085
a549ff1d4070 adding a smarter enumeration scheme for finite functions
bulwahn
parents: 41078
diff changeset
   654
hide_const (open) a\<^isub>1 a\<^isub>2 a\<^isub>3
40657
58a6ba7ccfc5 hiding the constants
bulwahn
parents: 40652
diff changeset
   655
40651
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   656
40647
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   657
datatype finite_4 = a\<^isub>1 | a\<^isub>2 | a\<^isub>3 | a\<^isub>4
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   658
40900
1d5f76d79856 adding shorter output syntax for the finite types of quickcheck
bulwahn
parents: 40898
diff changeset
   659
notation (output) a\<^isub>1  ("a\<^isub>1")
1d5f76d79856 adding shorter output syntax for the finite types of quickcheck
bulwahn
parents: 40898
diff changeset
   660
notation (output) a\<^isub>2  ("a\<^isub>2")
1d5f76d79856 adding shorter output syntax for the finite types of quickcheck
bulwahn
parents: 40898
diff changeset
   661
notation (output) a\<^isub>3  ("a\<^isub>3")
1d5f76d79856 adding shorter output syntax for the finite types of quickcheck
bulwahn
parents: 40898
diff changeset
   662
notation (output) a\<^isub>4  ("a\<^isub>4")
1d5f76d79856 adding shorter output syntax for the finite types of quickcheck
bulwahn
parents: 40898
diff changeset
   663
40647
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   664
instantiation finite_4 :: enum
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   665
begin
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   666
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   667
definition
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   668
  "enum = [a\<^isub>1, a\<^isub>2, a\<^isub>3, a\<^isub>4]"
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   669
41078
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   670
definition
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   671
  "enum_all P = (P a\<^isub>1 \<and> P a\<^isub>2 \<and> P a\<^isub>3 \<and> P a\<^isub>4)"
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   672
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   673
definition
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   674
  "enum_ex P = (P a\<^isub>1 \<or> P a\<^isub>2 \<or> P a\<^isub>3 \<or> P a\<^isub>4)"
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   675
40647
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   676
instance proof
41078
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   677
  fix P
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   678
  show "enum_all (P :: finite_4 \<Rightarrow> bool) = (\<forall>x. P x)"
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   679
    unfolding enum_all_finite_4_def
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   680
    by (auto, case_tac x) auto
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   681
next
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   682
  fix P
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   683
  show "enum_ex (P :: finite_4 \<Rightarrow> bool) = (\<exists>x. P x)"
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   684
    unfolding enum_ex_finite_4_def
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   685
    by (auto, case_tac x) auto
40647
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   686
qed (auto simp add: enum_finite_4_def intro: finite_4.exhaust)
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   687
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   688
end
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   689
41085
a549ff1d4070 adding a smarter enumeration scheme for finite functions
bulwahn
parents: 41078
diff changeset
   690
hide_const (open) a\<^isub>1 a\<^isub>2 a\<^isub>3 a\<^isub>4
40651
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   691
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   692
40647
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   693
datatype finite_5 = a\<^isub>1 | a\<^isub>2 | a\<^isub>3 | a\<^isub>4 | a\<^isub>5
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   694
40900
1d5f76d79856 adding shorter output syntax for the finite types of quickcheck
bulwahn
parents: 40898
diff changeset
   695
notation (output) a\<^isub>1  ("a\<^isub>1")
1d5f76d79856 adding shorter output syntax for the finite types of quickcheck
bulwahn
parents: 40898
diff changeset
   696
notation (output) a\<^isub>2  ("a\<^isub>2")
1d5f76d79856 adding shorter output syntax for the finite types of quickcheck
bulwahn
parents: 40898
diff changeset
   697
notation (output) a\<^isub>3  ("a\<^isub>3")
1d5f76d79856 adding shorter output syntax for the finite types of quickcheck
bulwahn
parents: 40898
diff changeset
   698
notation (output) a\<^isub>4  ("a\<^isub>4")
1d5f76d79856 adding shorter output syntax for the finite types of quickcheck
bulwahn
parents: 40898
diff changeset
   699
notation (output) a\<^isub>5  ("a\<^isub>5")
1d5f76d79856 adding shorter output syntax for the finite types of quickcheck
bulwahn
parents: 40898
diff changeset
   700
40647
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   701
instantiation finite_5 :: enum
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   702
begin
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   703
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   704
definition
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   705
  "enum = [a\<^isub>1, a\<^isub>2, a\<^isub>3, a\<^isub>4, a\<^isub>5]"
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   706
41078
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   707
definition
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   708
  "enum_all P = (P a\<^isub>1 \<and> P a\<^isub>2 \<and> P a\<^isub>3 \<and> P a\<^isub>4 \<and> P a\<^isub>5)"
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   709
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   710
definition
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   711
  "enum_ex P = (P a\<^isub>1 \<or> P a\<^isub>2 \<or> P a\<^isub>3 \<or> P a\<^isub>4 \<or> P a\<^isub>5)"
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   712
40647
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   713
instance proof
41078
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   714
  fix P
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   715
  show "enum_all (P :: finite_5 \<Rightarrow> bool) = (\<forall>x. P x)"
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   716
    unfolding enum_all_finite_5_def
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   717
    by (auto, case_tac x) auto
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   718
next
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   719
  fix P
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   720
  show "enum_ex (P :: finite_5 \<Rightarrow> bool) = (\<exists>x. P x)"
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   721
    unfolding enum_ex_finite_5_def
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   722
    by (auto, case_tac x) auto
40647
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   723
qed (auto simp add: enum_finite_5_def intro: finite_5.exhaust)
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   724
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   725
end
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   726
41115
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   727
subsection {* An executable THE operator on finite types *}
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   728
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   729
definition
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   730
  [code del]: "enum_the P = The P"
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   731
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   732
lemma [code]:
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   733
  "The P = (case filter P enum of [x] => x | _ => enum_the P)"
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   734
proof -
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   735
  {
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   736
    fix a
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   737
    assume filter_enum: "filter P enum = [a]"
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   738
    have "The P = a"
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   739
    proof (rule the_equality)
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   740
      fix x
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   741
      assume "P x"
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   742
      show "x = a"
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   743
      proof (rule ccontr)
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   744
        assume "x \<noteq> a"
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   745
        from filter_enum obtain us vs
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   746
          where enum_eq: "enum = us @ [a] @ vs"
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   747
          and "\<forall> x \<in> set us. \<not> P x"
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   748
          and "\<forall> x \<in> set vs. \<not> P x"
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   749
          and "P a"
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   750
          by (auto simp add: filter_eq_Cons_iff) (simp only: filter_empty_conv[symmetric])
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   751
        with `P x` in_enum[of x, unfolded enum_eq] `x \<noteq> a` show "False" by auto
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   752
      qed
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   753
    next
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   754
      from filter_enum show "P a" by (auto simp add: filter_eq_Cons_iff)
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   755
    qed
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   756
  }
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   757
  from this show ?thesis
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   758
    unfolding enum_the_def by (auto split: list.split)
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   759
qed
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   760
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   761
code_abort enum_the
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   762
41085
a549ff1d4070 adding a smarter enumeration scheme for finite functions
bulwahn
parents: 41078
diff changeset
   763
hide_const (open) a\<^isub>1 a\<^isub>2 a\<^isub>3 a\<^isub>4 a\<^isub>5
40657
58a6ba7ccfc5 hiding the constants
bulwahn
parents: 40652
diff changeset
   764
58a6ba7ccfc5 hiding the constants
bulwahn
parents: 40652
diff changeset
   765
41085
a549ff1d4070 adding a smarter enumeration scheme for finite functions
bulwahn
parents: 41078
diff changeset
   766
hide_type (open) finite_1 finite_2 finite_3 finite_4 finite_5
41078
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   767
hide_const (open) enum enum_all enum_ex n_lists all_n_lists ex_n_lists product
40647
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   768
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   769
end