src/HOL/Enum.thy
author bulwahn
Mon, 30 Jan 2012 13:55:24 +0100
changeset 46361 87d5d36a9005
parent 46359 9bc43dc49d0a
child 47221 7205eb4a0a05
child 47230 6584098d5378
permissions -rw-r--r--
adding code equations for max_extp and mlex
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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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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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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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    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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diff changeset
   313
0f8e23edd357 added theory Library/Enum.thy
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parents:
diff changeset
   314
primrec product :: "'a list \<Rightarrow> 'b list \<Rightarrow> ('a \<times> 'b) list" where
0f8e23edd357 added theory Library/Enum.thy
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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
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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
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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
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bulwahn
parents: 40900
diff changeset
   353
definition
051251fde456 adding more efficient implementations for quantifiers in Enum
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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
instantiation nibble :: enum
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   374
begin
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   375
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   376
definition
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   377
  "enum = [Nibble0, Nibble1, Nibble2, Nibble3, Nibble4, Nibble5, Nibble6, Nibble7,
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   378
    Nibble8, Nibble9, NibbleA, NibbleB, NibbleC, NibbleD, NibbleE, NibbleF]"
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   379
41078
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   380
definition
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   381
  "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
   382
     \<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
   383
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   384
definition
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   385
  "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
   386
     \<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
   387
31464
b2aca38301c4 tuned proofs
haftmann
parents: 31193
diff changeset
   388
instance proof
41078
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   389
  fix P
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   390
  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
   391
    unfolding enum_all_nibble_def
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   392
    by (auto, case_tac x) auto
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   393
next
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   394
  fix P
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   395
  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
   396
    unfolding enum_ex_nibble_def
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   397
    by (auto, case_tac x) auto
31464
b2aca38301c4 tuned proofs
haftmann
parents: 31193
diff changeset
   398
qed (simp_all add: enum_nibble_def UNIV_nibble)
26348
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   399
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   400
end
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   401
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   402
instantiation char :: enum
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   403
begin
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   404
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   405
definition
37765
26bdfb7b680b dropped superfluous [code del]s
haftmann
parents: 37678
diff changeset
   406
  "enum = map (split Char) (product enum enum)"
26444
6a5faa5bcf19 instance for functions, explicit characters
haftmann
parents: 26348
diff changeset
   407
31482
7288382fd549 using constant "chars"
haftmann
parents: 31464
diff changeset
   408
lemma enum_chars [code]:
7288382fd549 using constant "chars"
haftmann
parents: 31464
diff changeset
   409
  "enum = chars"
7288382fd549 using constant "chars"
haftmann
parents: 31464
diff changeset
   410
  unfolding enum_char_def chars_def enum_nibble_def by simp
26348
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   411
41078
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   412
definition
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   413
  "enum_all P = list_all P chars"
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 = list_ex P chars"
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   417
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   418
lemma set_enum_char: "set (enum :: char list) = UNIV"
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   419
    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
   420
31464
b2aca38301c4 tuned proofs
haftmann
parents: 31193
diff changeset
   421
instance proof
41078
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   422
  fix P
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   423
  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
   424
    unfolding enum_all_char_def enum_chars[symmetric]
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   425
    by (auto simp add: list_all_iff set_enum_char)
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   426
next
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   427
  fix P
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   428
  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
   429
    unfolding enum_ex_char_def enum_chars[symmetric]
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   430
    by (auto simp add: list_ex_iff set_enum_char)
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   431
next
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   432
  show "distinct (enum :: char list)"
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   433
    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
   434
qed (auto simp add: set_enum_char)
26348
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   435
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   436
end
0f8e23edd357 added theory Library/Enum.thy
haftmann
parents:
diff changeset
   437
29024
6cfa380af73b instantiation option :: (enum) enum
huffman
parents: 28684
diff changeset
   438
instantiation option :: (enum) enum
6cfa380af73b instantiation option :: (enum) enum
huffman
parents: 28684
diff changeset
   439
begin
6cfa380af73b instantiation option :: (enum) enum
huffman
parents: 28684
diff changeset
   440
6cfa380af73b instantiation option :: (enum) enum
huffman
parents: 28684
diff changeset
   441
definition
6cfa380af73b instantiation option :: (enum) enum
huffman
parents: 28684
diff changeset
   442
  "enum = None # map Some enum"
6cfa380af73b instantiation option :: (enum) enum
huffman
parents: 28684
diff changeset
   443
41078
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   444
definition
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   445
  "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
   446
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   447
definition
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   448
  "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
   449
31464
b2aca38301c4 tuned proofs
haftmann
parents: 31193
diff changeset
   450
instance proof
41078
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   451
  fix P
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   452
  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
   453
    unfolding enum_all_option_def enum_all
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   454
    by (auto, case_tac x) auto
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   455
next
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   456
  fix P
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   457
  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
   458
    unfolding enum_ex_option_def enum_ex
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   459
    by (auto, case_tac x) auto
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   460
qed (auto simp add: enum_UNIV enum_option_def, rule option.exhaust, auto intro: simp add: distinct_map enum_distinct)
45963
1c7e6454883e enum type class instance for `set`; dropped misfitting code lemma for trancl
haftmann
parents: 45144
diff changeset
   461
end
1c7e6454883e enum type class instance for `set`; dropped misfitting code lemma for trancl
haftmann
parents: 45144
diff changeset
   462
1c7e6454883e enum type class instance for `set`; dropped misfitting code lemma for trancl
haftmann
parents: 45144
diff changeset
   463
primrec sublists :: "'a list \<Rightarrow> 'a list list" where
1c7e6454883e enum type class instance for `set`; dropped misfitting code lemma for trancl
haftmann
parents: 45144
diff changeset
   464
  "sublists [] = [[]]"
1c7e6454883e enum type class instance for `set`; dropped misfitting code lemma for trancl
haftmann
parents: 45144
diff changeset
   465
  | "sublists (x#xs) = (let xss = sublists xs in map (Cons x) xss @ xss)"
1c7e6454883e enum type class instance for `set`; dropped misfitting code lemma for trancl
haftmann
parents: 45144
diff changeset
   466
1c7e6454883e enum type class instance for `set`; dropped misfitting code lemma for trancl
haftmann
parents: 45144
diff changeset
   467
lemma length_sublists:
1c7e6454883e enum type class instance for `set`; dropped misfitting code lemma for trancl
haftmann
parents: 45144
diff changeset
   468
  "length (sublists xs) = Suc (Suc (0\<Colon>nat)) ^ length xs"
1c7e6454883e enum type class instance for `set`; dropped misfitting code lemma for trancl
haftmann
parents: 45144
diff changeset
   469
  by (induct xs) (simp_all add: Let_def)
1c7e6454883e enum type class instance for `set`; dropped misfitting code lemma for trancl
haftmann
parents: 45144
diff changeset
   470
1c7e6454883e enum type class instance for `set`; dropped misfitting code lemma for trancl
haftmann
parents: 45144
diff changeset
   471
lemma sublists_powset:
1c7e6454883e enum type class instance for `set`; dropped misfitting code lemma for trancl
haftmann
parents: 45144
diff changeset
   472
  "set ` set (sublists xs) = Pow (set xs)"
1c7e6454883e enum type class instance for `set`; dropped misfitting code lemma for trancl
haftmann
parents: 45144
diff changeset
   473
proof -
1c7e6454883e enum type class instance for `set`; dropped misfitting code lemma for trancl
haftmann
parents: 45144
diff changeset
   474
  have aux: "\<And>x A. set ` Cons x ` A = insert x ` set ` A"
1c7e6454883e enum type class instance for `set`; dropped misfitting code lemma for trancl
haftmann
parents: 45144
diff changeset
   475
    by (auto simp add: image_def)
1c7e6454883e enum type class instance for `set`; dropped misfitting code lemma for trancl
haftmann
parents: 45144
diff changeset
   476
  have "set (map set (sublists xs)) = Pow (set xs)"
1c7e6454883e enum type class instance for `set`; dropped misfitting code lemma for trancl
haftmann
parents: 45144
diff changeset
   477
    by (induct xs)
1c7e6454883e enum type class instance for `set`; dropped misfitting code lemma for trancl
haftmann
parents: 45144
diff changeset
   478
      (simp_all add: aux Let_def Pow_insert Un_commute comp_def del: map_map)
1c7e6454883e enum type class instance for `set`; dropped misfitting code lemma for trancl
haftmann
parents: 45144
diff changeset
   479
  then show ?thesis by simp
1c7e6454883e enum type class instance for `set`; dropped misfitting code lemma for trancl
haftmann
parents: 45144
diff changeset
   480
qed
1c7e6454883e enum type class instance for `set`; dropped misfitting code lemma for trancl
haftmann
parents: 45144
diff changeset
   481
1c7e6454883e enum type class instance for `set`; dropped misfitting code lemma for trancl
haftmann
parents: 45144
diff changeset
   482
lemma distinct_set_sublists:
1c7e6454883e enum type class instance for `set`; dropped misfitting code lemma for trancl
haftmann
parents: 45144
diff changeset
   483
  assumes "distinct xs"
1c7e6454883e enum type class instance for `set`; dropped misfitting code lemma for trancl
haftmann
parents: 45144
diff changeset
   484
  shows "distinct (map set (sublists xs))"
1c7e6454883e enum type class instance for `set`; dropped misfitting code lemma for trancl
haftmann
parents: 45144
diff changeset
   485
proof (rule card_distinct)
1c7e6454883e enum type class instance for `set`; dropped misfitting code lemma for trancl
haftmann
parents: 45144
diff changeset
   486
  have "finite (set xs)" by rule
1c7e6454883e enum type class instance for `set`; dropped misfitting code lemma for trancl
haftmann
parents: 45144
diff changeset
   487
  then have "card (Pow (set xs)) = Suc (Suc 0) ^ card (set xs)" by (rule card_Pow)
1c7e6454883e enum type class instance for `set`; dropped misfitting code lemma for trancl
haftmann
parents: 45144
diff changeset
   488
  with assms distinct_card [of xs]
1c7e6454883e enum type class instance for `set`; dropped misfitting code lemma for trancl
haftmann
parents: 45144
diff changeset
   489
    have "card (Pow (set xs)) = Suc (Suc 0) ^ length xs" by simp
1c7e6454883e enum type class instance for `set`; dropped misfitting code lemma for trancl
haftmann
parents: 45144
diff changeset
   490
  then show "card (set (map set (sublists xs))) = length (map set (sublists xs))"
1c7e6454883e enum type class instance for `set`; dropped misfitting code lemma for trancl
haftmann
parents: 45144
diff changeset
   491
    by (simp add: sublists_powset length_sublists)
1c7e6454883e enum type class instance for `set`; dropped misfitting code lemma for trancl
haftmann
parents: 45144
diff changeset
   492
qed
1c7e6454883e enum type class instance for `set`; dropped misfitting code lemma for trancl
haftmann
parents: 45144
diff changeset
   493
1c7e6454883e enum type class instance for `set`; dropped misfitting code lemma for trancl
haftmann
parents: 45144
diff changeset
   494
instantiation set :: (enum) enum
1c7e6454883e enum type class instance for `set`; dropped misfitting code lemma for trancl
haftmann
parents: 45144
diff changeset
   495
begin
1c7e6454883e enum type class instance for `set`; dropped misfitting code lemma for trancl
haftmann
parents: 45144
diff changeset
   496
1c7e6454883e enum type class instance for `set`; dropped misfitting code lemma for trancl
haftmann
parents: 45144
diff changeset
   497
definition
1c7e6454883e enum type class instance for `set`; dropped misfitting code lemma for trancl
haftmann
parents: 45144
diff changeset
   498
  "enum = map set (sublists enum)"
1c7e6454883e enum type class instance for `set`; dropped misfitting code lemma for trancl
haftmann
parents: 45144
diff changeset
   499
1c7e6454883e enum type class instance for `set`; dropped misfitting code lemma for trancl
haftmann
parents: 45144
diff changeset
   500
definition
1c7e6454883e enum type class instance for `set`; dropped misfitting code lemma for trancl
haftmann
parents: 45144
diff changeset
   501
  "enum_all P \<longleftrightarrow> (\<forall>A\<in>set enum. P (A::'a set))"
1c7e6454883e enum type class instance for `set`; dropped misfitting code lemma for trancl
haftmann
parents: 45144
diff changeset
   502
1c7e6454883e enum type class instance for `set`; dropped misfitting code lemma for trancl
haftmann
parents: 45144
diff changeset
   503
definition
1c7e6454883e enum type class instance for `set`; dropped misfitting code lemma for trancl
haftmann
parents: 45144
diff changeset
   504
  "enum_ex P \<longleftrightarrow> (\<exists>A\<in>set enum. P (A::'a set))"
1c7e6454883e enum type class instance for `set`; dropped misfitting code lemma for trancl
haftmann
parents: 45144
diff changeset
   505
1c7e6454883e enum type class instance for `set`; dropped misfitting code lemma for trancl
haftmann
parents: 45144
diff changeset
   506
instance proof
1c7e6454883e enum type class instance for `set`; dropped misfitting code lemma for trancl
haftmann
parents: 45144
diff changeset
   507
qed (simp_all add: enum_set_def enum_all_set_def enum_ex_set_def sublists_powset distinct_set_sublists
1c7e6454883e enum type class instance for `set`; dropped misfitting code lemma for trancl
haftmann
parents: 45144
diff changeset
   508
  enum_distinct enum_UNIV)
29024
6cfa380af73b instantiation option :: (enum) enum
huffman
parents: 28684
diff changeset
   509
6cfa380af73b instantiation option :: (enum) enum
huffman
parents: 28684
diff changeset
   510
end
6cfa380af73b instantiation option :: (enum) enum
huffman
parents: 28684
diff changeset
   511
45963
1c7e6454883e enum type class instance for `set`; dropped misfitting code lemma for trancl
haftmann
parents: 45144
diff changeset
   512
40647
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   513
subsection {* Small finite types *}
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   514
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   515
text {* We define small finite types for the use in Quickcheck *}
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   516
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   517
datatype finite_1 = a\<^isub>1
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   518
40900
1d5f76d79856 adding shorter output syntax for the finite types of quickcheck
bulwahn
parents: 40898
diff changeset
   519
notation (output) a\<^isub>1  ("a\<^isub>1")
1d5f76d79856 adding shorter output syntax for the finite types of quickcheck
bulwahn
parents: 40898
diff changeset
   520
40647
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   521
instantiation finite_1 :: enum
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   522
begin
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   523
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   524
definition
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   525
  "enum = [a\<^isub>1]"
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   526
41078
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   527
definition
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   528
  "enum_all P = P a\<^isub>1"
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   529
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   530
definition
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   531
  "enum_ex P = P a\<^isub>1"
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   532
40647
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   533
instance proof
41078
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   534
  fix P
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   535
  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
   536
    unfolding enum_all_finite_1_def
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   537
    by (auto, case_tac x) auto
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   538
next
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   539
  fix P
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   540
  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
   541
    unfolding enum_ex_finite_1_def
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   542
    by (auto, case_tac x) auto
40647
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   543
qed (auto simp add: enum_finite_1_def intro: finite_1.exhaust)
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   544
29024
6cfa380af73b instantiation option :: (enum) enum
huffman
parents: 28684
diff changeset
   545
end
40647
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   546
40651
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   547
instantiation finite_1 :: linorder
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   548
begin
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   549
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   550
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
   551
where
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   552
  "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
   553
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   554
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
   555
where
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   556
  "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
   557
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   558
instance
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   559
apply (intro_classes)
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   560
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
   561
apply (metis finite_1.exhaust)
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   562
done
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   563
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   564
end
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   565
41085
a549ff1d4070 adding a smarter enumeration scheme for finite functions
bulwahn
parents: 41078
diff changeset
   566
hide_const (open) a\<^isub>1
40657
58a6ba7ccfc5 hiding the constants
bulwahn
parents: 40652
diff changeset
   567
40647
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   568
datatype finite_2 = a\<^isub>1 | a\<^isub>2
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   569
40900
1d5f76d79856 adding shorter output syntax for the finite types of quickcheck
bulwahn
parents: 40898
diff changeset
   570
notation (output) a\<^isub>1  ("a\<^isub>1")
1d5f76d79856 adding shorter output syntax for the finite types of quickcheck
bulwahn
parents: 40898
diff changeset
   571
notation (output) a\<^isub>2  ("a\<^isub>2")
1d5f76d79856 adding shorter output syntax for the finite types of quickcheck
bulwahn
parents: 40898
diff changeset
   572
40647
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   573
instantiation finite_2 :: enum
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   574
begin
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   575
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   576
definition
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   577
  "enum = [a\<^isub>1, a\<^isub>2]"
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   578
41078
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   579
definition
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   580
  "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
   581
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   582
definition
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   583
  "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
   584
40647
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   585
instance proof
41078
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   586
  fix P
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   587
  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
   588
    unfolding enum_all_finite_2_def
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   589
    by (auto, case_tac x) auto
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   590
next
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   591
  fix P
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   592
  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
   593
    unfolding enum_ex_finite_2_def
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   594
    by (auto, case_tac x) auto
40647
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   595
qed (auto simp add: enum_finite_2_def intro: finite_2.exhaust)
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   596
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   597
end
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   598
40651
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   599
instantiation finite_2 :: linorder
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   600
begin
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   601
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   602
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
   603
where
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   604
  "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
   605
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   606
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
   607
where
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   608
  "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
   609
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   610
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   611
instance
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   612
apply (intro_classes)
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   613
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
   614
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
   615
done
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   616
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   617
end
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   618
41085
a549ff1d4070 adding a smarter enumeration scheme for finite functions
bulwahn
parents: 41078
diff changeset
   619
hide_const (open) a\<^isub>1 a\<^isub>2
40657
58a6ba7ccfc5 hiding the constants
bulwahn
parents: 40652
diff changeset
   620
40651
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   621
40647
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   622
datatype finite_3 = a\<^isub>1 | a\<^isub>2 | a\<^isub>3
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   623
40900
1d5f76d79856 adding shorter output syntax for the finite types of quickcheck
bulwahn
parents: 40898
diff changeset
   624
notation (output) a\<^isub>1  ("a\<^isub>1")
1d5f76d79856 adding shorter output syntax for the finite types of quickcheck
bulwahn
parents: 40898
diff changeset
   625
notation (output) a\<^isub>2  ("a\<^isub>2")
1d5f76d79856 adding shorter output syntax for the finite types of quickcheck
bulwahn
parents: 40898
diff changeset
   626
notation (output) a\<^isub>3  ("a\<^isub>3")
1d5f76d79856 adding shorter output syntax for the finite types of quickcheck
bulwahn
parents: 40898
diff changeset
   627
40647
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   628
instantiation finite_3 :: enum
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   629
begin
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   630
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   631
definition
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   632
  "enum = [a\<^isub>1, a\<^isub>2, a\<^isub>3]"
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   633
41078
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   634
definition
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   635
  "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
   636
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   637
definition
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   638
  "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
   639
40647
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   640
instance proof
41078
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   641
  fix P
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   642
  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
   643
    unfolding enum_all_finite_3_def
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   644
    by (auto, case_tac x) auto
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   645
next
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   646
  fix P
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   647
  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
   648
    unfolding enum_ex_finite_3_def
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   649
    by (auto, case_tac x) auto
40647
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   650
qed (auto simp add: enum_finite_3_def intro: finite_3.exhaust)
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   651
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   652
end
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   653
40651
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   654
instantiation finite_3 :: linorder
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   655
begin
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   656
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   657
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
   658
where
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   659
  "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
   660
     | 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
   661
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   662
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
   663
where
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   664
  "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
   665
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   666
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   667
instance proof (intro_classes)
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   668
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
   669
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   670
end
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   671
41085
a549ff1d4070 adding a smarter enumeration scheme for finite functions
bulwahn
parents: 41078
diff changeset
   672
hide_const (open) a\<^isub>1 a\<^isub>2 a\<^isub>3
40657
58a6ba7ccfc5 hiding the constants
bulwahn
parents: 40652
diff changeset
   673
40651
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   674
40647
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   675
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
   676
40900
1d5f76d79856 adding shorter output syntax for the finite types of quickcheck
bulwahn
parents: 40898
diff changeset
   677
notation (output) a\<^isub>1  ("a\<^isub>1")
1d5f76d79856 adding shorter output syntax for the finite types of quickcheck
bulwahn
parents: 40898
diff changeset
   678
notation (output) a\<^isub>2  ("a\<^isub>2")
1d5f76d79856 adding shorter output syntax for the finite types of quickcheck
bulwahn
parents: 40898
diff changeset
   679
notation (output) a\<^isub>3  ("a\<^isub>3")
1d5f76d79856 adding shorter output syntax for the finite types of quickcheck
bulwahn
parents: 40898
diff changeset
   680
notation (output) a\<^isub>4  ("a\<^isub>4")
1d5f76d79856 adding shorter output syntax for the finite types of quickcheck
bulwahn
parents: 40898
diff changeset
   681
40647
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   682
instantiation finite_4 :: enum
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   683
begin
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   684
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   685
definition
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   686
  "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
   687
41078
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   688
definition
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   689
  "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
   690
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   691
definition
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   692
  "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
   693
40647
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   694
instance proof
41078
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   695
  fix P
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   696
  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
   697
    unfolding enum_all_finite_4_def
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   698
    by (auto, case_tac x) auto
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   699
next
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   700
  fix P
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   701
  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
   702
    unfolding enum_ex_finite_4_def
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   703
    by (auto, case_tac x) auto
40647
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   704
qed (auto simp add: enum_finite_4_def intro: finite_4.exhaust)
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   705
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   706
end
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   707
41085
a549ff1d4070 adding a smarter enumeration scheme for finite functions
bulwahn
parents: 41078
diff changeset
   708
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
   709
9752ba7348b5 adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents: 40650
diff changeset
   710
40647
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   711
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
   712
40900
1d5f76d79856 adding shorter output syntax for the finite types of quickcheck
bulwahn
parents: 40898
diff changeset
   713
notation (output) a\<^isub>1  ("a\<^isub>1")
1d5f76d79856 adding shorter output syntax for the finite types of quickcheck
bulwahn
parents: 40898
diff changeset
   714
notation (output) a\<^isub>2  ("a\<^isub>2")
1d5f76d79856 adding shorter output syntax for the finite types of quickcheck
bulwahn
parents: 40898
diff changeset
   715
notation (output) a\<^isub>3  ("a\<^isub>3")
1d5f76d79856 adding shorter output syntax for the finite types of quickcheck
bulwahn
parents: 40898
diff changeset
   716
notation (output) a\<^isub>4  ("a\<^isub>4")
1d5f76d79856 adding shorter output syntax for the finite types of quickcheck
bulwahn
parents: 40898
diff changeset
   717
notation (output) a\<^isub>5  ("a\<^isub>5")
1d5f76d79856 adding shorter output syntax for the finite types of quickcheck
bulwahn
parents: 40898
diff changeset
   718
40647
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   719
instantiation finite_5 :: enum
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   720
begin
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   721
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   722
definition
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   723
  "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
   724
41078
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   725
definition
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   726
  "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
   727
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   728
definition
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   729
  "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
   730
40647
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   731
instance proof
41078
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   732
  fix P
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   733
  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
   734
    unfolding enum_all_finite_5_def
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   735
    by (auto, case_tac x) auto
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   736
next
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   737
  fix P
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   738
  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
   739
    unfolding enum_ex_finite_5_def
051251fde456 adding more efficient implementations for quantifiers in Enum
bulwahn
parents: 40900
diff changeset
   740
    by (auto, case_tac x) auto
40647
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   741
qed (auto simp add: enum_finite_5_def intro: finite_5.exhaust)
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   742
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   743
end
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   744
46352
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   745
hide_const (open) a\<^isub>1 a\<^isub>2 a\<^isub>3 a\<^isub>4 a\<^isub>5
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   746
41115
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   747
subsection {* An executable THE operator on finite types *}
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   748
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   749
definition
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   750
  [code del]: "enum_the P = The P"
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   751
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   752
lemma [code]:
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   753
  "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
   754
proof -
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   755
  {
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   756
    fix a
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   757
    assume filter_enum: "filter P enum = [a]"
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   758
    have "The P = a"
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   759
    proof (rule the_equality)
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   760
      fix x
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   761
      assume "P x"
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   762
      show "x = a"
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   763
      proof (rule ccontr)
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   764
        assume "x \<noteq> a"
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   765
        from filter_enum obtain us vs
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   766
          where enum_eq: "enum = us @ [a] @ vs"
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   767
          and "\<forall> x \<in> set us. \<not> P x"
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   768
          and "\<forall> x \<in> set vs. \<not> P x"
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   769
          and "P a"
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   770
          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
   771
        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
   772
      qed
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   773
    next
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   774
      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
   775
    qed
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   776
  }
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   777
  from this show ?thesis
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   778
    unfolding enum_the_def by (auto split: list.split)
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   779
qed
2c362ff5daf4 adding an executable THE operator on finite types
bulwahn
parents: 41085
diff changeset
   780
46329
cf3b387ba667 adding code equation for Collect on finite types
bulwahn
parents: 45963
diff changeset
   781
code_abort enum_the
46336
39fe503602fb evaluation of THE with a non-singleton set raises a Match exception during the evaluation to yield a potential counterexample in quickcheck.
bulwahn
parents: 46329
diff changeset
   782
code_const enum_the (Eval "(fn p => raise Match)")
46329
cf3b387ba667 adding code equation for Collect on finite types
bulwahn
parents: 45963
diff changeset
   783
cf3b387ba667 adding code equation for Collect on finite types
bulwahn
parents: 45963
diff changeset
   784
subsection {* Further operations on finite types *}
cf3b387ba667 adding code equation for Collect on finite types
bulwahn
parents: 45963
diff changeset
   785
cf3b387ba667 adding code equation for Collect on finite types
bulwahn
parents: 45963
diff changeset
   786
lemma [code]:
cf3b387ba667 adding code equation for Collect on finite types
bulwahn
parents: 45963
diff changeset
   787
  "Collect P = set (filter P enum)"
cf3b387ba667 adding code equation for Collect on finite types
bulwahn
parents: 45963
diff changeset
   788
by (auto simp add: enum_UNIV)
45140
339a8b3c4791 bouned transitive closure
haftmann
parents: 45119
diff changeset
   789
46357
2795607a1f40 adding code equation for tranclp
bulwahn
parents: 46352
diff changeset
   790
lemma tranclp_unfold [code, no_atp]:
2795607a1f40 adding code equation for tranclp
bulwahn
parents: 46352
diff changeset
   791
  "tranclp r a b \<equiv> (a, b) \<in> trancl {(x, y). r x y}"
2795607a1f40 adding code equation for tranclp
bulwahn
parents: 46352
diff changeset
   792
by (simp add: trancl_def)
46352
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   793
46359
9bc43dc49d0a adding code equation for rtranclp in Enum
bulwahn
parents: 46358
diff changeset
   794
lemma rtranclp_rtrancl_eq[code, no_atp]:
9bc43dc49d0a adding code equation for rtranclp in Enum
bulwahn
parents: 46358
diff changeset
   795
  "rtranclp r x y = ((x, y) : rtrancl {(x, y). r x y})"
9bc43dc49d0a adding code equation for rtranclp in Enum
bulwahn
parents: 46358
diff changeset
   796
unfolding rtrancl_def by auto
9bc43dc49d0a adding code equation for rtranclp in Enum
bulwahn
parents: 46358
diff changeset
   797
46358
b2a936486685 adding code equation for max_ext
bulwahn
parents: 46357
diff changeset
   798
lemma max_ext_eq[code]:
b2a936486685 adding code equation for max_ext
bulwahn
parents: 46357
diff changeset
   799
  "max_ext R = {(X, Y). finite X & finite Y & Y ~={} & (ALL x. x : X --> (EX xa : Y. (x, xa) : R))}"
b2a936486685 adding code equation for max_ext
bulwahn
parents: 46357
diff changeset
   800
by (auto simp add: max_ext.simps)
b2a936486685 adding code equation for max_ext
bulwahn
parents: 46357
diff changeset
   801
46361
87d5d36a9005 adding code equations for max_extp and mlex
bulwahn
parents: 46359
diff changeset
   802
lemma max_extp_eq[code]:
87d5d36a9005 adding code equations for max_extp and mlex
bulwahn
parents: 46359
diff changeset
   803
  "max_extp r x y = ((x, y) : max_ext {(x, y). r x y})"
87d5d36a9005 adding code equations for max_extp and mlex
bulwahn
parents: 46359
diff changeset
   804
unfolding max_ext_def by auto
87d5d36a9005 adding code equations for max_extp and mlex
bulwahn
parents: 46359
diff changeset
   805
87d5d36a9005 adding code equations for max_extp and mlex
bulwahn
parents: 46359
diff changeset
   806
lemma mlex_eq[code]:
87d5d36a9005 adding code equations for max_extp and mlex
bulwahn
parents: 46359
diff changeset
   807
  "f <*mlex*> R = {(x, y). f x < f y \<or> (f x <= f y \<and> (x, y) : R)}"
87d5d36a9005 adding code equations for max_extp and mlex
bulwahn
parents: 46359
diff changeset
   808
unfolding mlex_prod_def by auto
87d5d36a9005 adding code equations for max_extp and mlex
bulwahn
parents: 46359
diff changeset
   809
46352
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   810
subsection {* Executable accessible part *}
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   811
(* FIXME: should be moved somewhere else !? *)
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   812
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   813
subsubsection {* Finite monotone eventually stable sequences *}
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   814
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   815
lemma finite_mono_remains_stable_implies_strict_prefix:
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   816
  fixes f :: "nat \<Rightarrow> 'a::order"
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   817
  assumes S: "finite (range f)" "mono f" and eq: "\<forall>n. f n = f (Suc n) \<longrightarrow> f (Suc n) = f (Suc (Suc n))"
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   818
  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)"
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   819
  using assms
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   820
proof -
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   821
  have "\<exists>n. f n = f (Suc n)"
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   822
  proof (rule ccontr)
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   823
    assume "\<not> ?thesis"
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   824
    then have "\<And>n. f n \<noteq> f (Suc n)" by auto
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   825
    then have "\<And>n. f n < f (Suc n)"
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   826
      using  `mono f` by (auto simp: le_less mono_iff_le_Suc)
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   827
    with lift_Suc_mono_less_iff[of f]
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   828
    have "\<And>n m. n < m \<Longrightarrow> f n < f m" by auto
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   829
    then have "inj f"
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   830
      by (auto simp: inj_on_def) (metis linorder_less_linear order_less_imp_not_eq)
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   831
    with `finite (range f)` have "finite (UNIV::nat set)"
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   832
      by (rule finite_imageD)
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   833
    then show False by simp
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   834
  qed
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   835
  then guess n .. note n = this
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   836
  def N \<equiv> "LEAST n. f n = f (Suc n)"
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   837
  have N: "f N = f (Suc N)"
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   838
    unfolding N_def using n by (rule LeastI)
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   839
  show ?thesis
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   840
  proof (intro exI[of _ N] conjI allI impI)
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   841
    fix n assume "N \<le> n"
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   842
    then have "\<And>m. N \<le> m \<Longrightarrow> m \<le> n \<Longrightarrow> f m = f N"
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   843
    proof (induct rule: dec_induct)
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   844
      case (step n) then show ?case
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   845
        using eq[rule_format, of "n - 1"] N
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   846
        by (cases n) (auto simp add: le_Suc_eq)
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   847
    qed simp
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   848
    from this[of n] `N \<le> n` show "f N = f n" by auto
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   849
  next
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   850
    fix n m :: nat assume "m < n" "n \<le> N"
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   851
    then show "f m < f n"
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   852
    proof (induct rule: less_Suc_induct[consumes 1])
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   853
      case (1 i)
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   854
      then have "i < N" by simp
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   855
      then have "f i \<noteq> f (Suc i)"
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   856
        unfolding N_def by (rule not_less_Least)
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   857
      with `mono f` show ?case by (simp add: mono_iff_le_Suc less_le)
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   858
    qed auto
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   859
  qed
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   860
qed
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   861
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   862
lemma finite_mono_strict_prefix_implies_finite_fixpoint:
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   863
  fixes f :: "nat \<Rightarrow> 'a set"
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   864
  assumes S: "\<And>i. f i \<subseteq> S" "finite S"
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   865
    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)"
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   866
  shows "f (card S) = (\<Union>n. f n)"
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   867
proof -
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   868
  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
45117
3911cf09899a adding code equations for cardinality and (reflexive) transitive closure on finite types
bulwahn
parents: 41115
diff changeset
   869
46352
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   870
  { fix i have "i \<le> N \<Longrightarrow> i \<le> card (f i)"
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   871
    proof (induct i)
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   872
      case 0 then show ?case by simp
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   873
    next
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   874
      case (Suc i)
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   875
      with inj[rule_format, of "Suc i" i]
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   876
      have "(f i) \<subset> (f (Suc i))" by auto
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   877
      moreover have "finite (f (Suc i))" using S by (rule finite_subset)
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   878
      ultimately have "card (f i) < card (f (Suc i))" by (intro psubset_card_mono)
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   879
      with Suc show ?case using inj by auto
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   880
    qed
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   881
  }
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   882
  then have "N \<le> card (f N)" by simp
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   883
  also have "\<dots> \<le> card S" using S by (intro card_mono)
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   884
  finally have "f (card S) = f N" using eq by auto
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   885
  then show ?thesis using eq inj[rule_format, of N]
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   886
    apply auto
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   887
    apply (case_tac "n < N")
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   888
    apply (auto simp: not_less)
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   889
    done
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   890
qed
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   891
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   892
subsubsection {* Bounded accessible part *}
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   893
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   894
fun bacc :: "('a * 'a) set => nat => 'a set" 
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   895
where
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   896
  "bacc r 0 = {x. \<forall> y. (y, x) \<notin> r}"
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   897
| "bacc r (Suc n) = (bacc r n Un {x. \<forall> y. (y, x) : r --> y : bacc r n})"
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   898
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   899
lemma bacc_subseteq_acc:
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   900
  "bacc r n \<subseteq> acc r"
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   901
by (induct n) (auto intro: acc.intros)
40657
58a6ba7ccfc5 hiding the constants
bulwahn
parents: 40652
diff changeset
   902
46352
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   903
lemma bacc_mono:
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   904
  "n <= m ==> bacc r n \<subseteq> bacc r m"
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   905
by (induct rule: dec_induct) auto
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   906
  
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   907
lemma bacc_upper_bound:
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   908
  "bacc (r :: ('a * 'a) set)  (card (UNIV :: ('a :: enum) set)) = (UN n. bacc r n)"
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   909
proof -
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   910
  have "mono (bacc r)" unfolding mono_def by (simp add: bacc_mono)
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   911
  moreover have "\<forall>n. bacc r n = bacc r (Suc n) \<longrightarrow> bacc r (Suc n) = bacc r (Suc (Suc n))" by auto
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   912
  moreover have "finite (range (bacc r))" by auto
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   913
  ultimately show ?thesis
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   914
   by (intro finite_mono_strict_prefix_implies_finite_fixpoint)
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   915
     (auto intro: finite_mono_remains_stable_implies_strict_prefix  simp add: enum_UNIV)
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   916
qed
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   917
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   918
lemma acc_subseteq_bacc:
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   919
  assumes "finite r"
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   920
  shows "acc r \<subseteq> (UN n. bacc r n)"
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   921
proof
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   922
  fix x
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   923
  assume "x : acc r"
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   924
  from this have "\<exists> n. x : bacc r n"
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   925
  proof (induct x arbitrary: n rule: acc.induct)
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   926
    case (accI x)
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   927
    from accI have "\<forall> y. \<exists> n. (y, x) \<in> r --> y : bacc r n" by simp
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   928
    from choice[OF this] guess n .. note n = this
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   929
    have "\<exists> n. \<forall>y. (y, x) : r --> y : bacc r n"
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   930
    proof (safe intro!: exI[of _ "Max ((%(y,x). n y)`r)"])
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   931
      fix y assume y: "(y, x) : r"
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   932
      with n have "y : bacc r (n y)" by auto
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   933
      moreover have "n y <= Max ((%(y, x). n y) ` r)"
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   934
        using y `finite r` by (auto intro!: Max_ge)
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   935
      note bacc_mono[OF this, of r]
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   936
      ultimately show "y : bacc r (Max ((%(y, x). n y) ` r))" by auto
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   937
    qed
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   938
    from this guess n ..
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   939
    from this show ?case
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   940
      by (auto simp add: Let_def intro!: exI[of _ "Suc n"])
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   941
  qed
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   942
  thus "x : (UN n. bacc r n)" by auto
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   943
qed
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   944
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   945
lemma acc_bacc_eq: "acc ((set xs) :: (('a :: enum) * 'a) set) = bacc (set xs) (card (UNIV :: 'a set))"
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   946
by (metis acc_subseteq_bacc bacc_subseteq_acc bacc_upper_bound finite_set order_eq_iff)
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   947
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   948
definition 
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   949
  [code del]: "card_UNIV = card UNIV"
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   950
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   951
lemma [code]:
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   952
  "card_UNIV TYPE('a :: enum) = card (set (Enum.enum :: 'a list))"
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   953
unfolding card_UNIV_def enum_UNIV ..
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   954
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   955
declare acc_bacc_eq[folded card_UNIV_def, code]
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   956
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   957
lemma [code_unfold]: "accp r = (%x. x : acc {(x, y). r x y})"
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   958
unfolding acc_def by simp
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   959
73b03235799a an executable version of accessible part (only for finite types yet)
bulwahn
parents: 46336
diff changeset
   960
subsection {* Closing up *}
40657
58a6ba7ccfc5 hiding the constants
bulwahn
parents: 40652
diff changeset
   961
41085
a549ff1d4070 adding a smarter enumeration scheme for finite functions
bulwahn
parents: 41078
diff changeset
   962
hide_type (open) finite_1 finite_2 finite_3 finite_4 finite_5
45117
3911cf09899a adding code equations for cardinality and (reflexive) transitive closure on finite types
bulwahn
parents: 41115
diff changeset
   963
hide_const (open) enum enum_all enum_ex n_lists all_n_lists ex_n_lists product ntrancl
40647
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   964
6e92ca8e981b adding prototype for finite_type instantiations
bulwahn
parents: 39302
diff changeset
   965
end