src/HOL/Enum.thy
author wenzelm
Tue Sep 01 22:32:58 2015 +0200 (2015-09-01)
changeset 61076 bdc1e2f0a86a
parent 60758 d8d85a8172b5
child 61169 4de9ff3ea29a
permissions -rw-r--r--
eliminated \<Colon>;
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(* Author: Florian Haftmann, TU Muenchen *)
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section \<open>Finite types as explicit enumerations\<close>
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theory Enum
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imports Map Groups_List
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begin
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subsection \<open>Class @{text enum}\<close>
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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_UNIV: "enum_all P \<longleftrightarrow> Ball UNIV P"
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  assumes enum_ex_UNIV: "enum_ex P \<longleftrightarrow> Bex UNIV P" 
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   -- \<open>tailored towards simple instantiation\<close>
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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:
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  "set enum = UNIV"
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  by (simp only: UNIV_enum)
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lemma in_enum: "x \<in> set enum"
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  by (simp add: enum_UNIV)
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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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lemma card_UNIV_length_enum:
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  "card (UNIV :: 'a set) = length enum"
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  by (simp add: UNIV_enum distinct_card enum_distinct)
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lemma enum_all [simp]:
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  "enum_all = HOL.All"
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  by (simp add: fun_eq_iff enum_all_UNIV)
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lemma enum_ex [simp]:
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  "enum_ex = HOL.Ex" 
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  by (simp add: fun_eq_iff enum_ex_UNIV)
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end
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subsection \<open>Implementations using @{class enum}\<close>
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subsubsection \<open>Unbounded operations and quantifiers\<close>
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lemma Collect_code [code]:
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  "Collect P = set (filter P enum)"
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  by (simp add: enum_UNIV)
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lemma vimage_code [code]:
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  "f -` B = set (filter (%x. f x : B) enum_class.enum)"
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  unfolding vimage_def Collect_code ..
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definition card_UNIV :: "'a itself \<Rightarrow> nat"
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where
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  [code del]: "card_UNIV TYPE('a) = card (UNIV :: 'a set)"
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lemma [code]:
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  "card_UNIV TYPE('a :: enum) = card (set (Enum.enum :: 'a list))"
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  by (simp only: card_UNIV_def enum_UNIV)
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lemma all_code [code]: "(\<forall>x. P x) \<longleftrightarrow> enum_all P"
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  by simp
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lemma exists_code [code]: "(\<exists>x. P x) \<longleftrightarrow> enum_ex P"
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  by simp
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lemma exists1_code [code]: "(\<exists>!x. P x) \<longleftrightarrow> list_ex1 P enum"
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  by (auto simp add: list_ex1_iff enum_UNIV)
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subsubsection \<open>An executable choice operator\<close>
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definition
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  [code del]: "enum_the = The"
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lemma [code]:
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  "The P = (case filter P enum of [x] => x | _ => enum_the P)"
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proof -
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  {
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    fix a
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    assume filter_enum: "filter P enum = [a]"
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    have "The P = a"
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    proof (rule the_equality)
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      fix x
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      assume "P x"
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      show "x = a"
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      proof (rule ccontr)
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        assume "x \<noteq> a"
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        from filter_enum obtain us vs
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          where enum_eq: "enum = us @ [a] @ vs"
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          and "\<forall> x \<in> set us. \<not> P x"
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          and "\<forall> x \<in> set vs. \<not> P x"
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          and "P a"
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          by (auto simp add: filter_eq_Cons_iff) (simp only: filter_empty_conv[symmetric])
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        with \<open>P x\<close> in_enum[of x, unfolded enum_eq] \<open>x \<noteq> a\<close> show "False" by auto
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      qed
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    next
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      from filter_enum show "P a" by (auto simp add: filter_eq_Cons_iff)
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    qed
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  }
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  from this show ?thesis
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    unfolding enum_the_def by (auto split: list.split)
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qed
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declare [[code abort: enum_the]]
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code_printing
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  constant enum_the \<rightharpoonup> (Eval) "(fn '_ => raise Match)"
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subsubsection \<open>Equality and order on functions\<close>
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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 fun_eq_iff enum_UNIV)
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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 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::enum \<Rightarrow> 'b::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: fun_eq_iff le_fun_def order_less_le)
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subsubsection \<open>Operations on relations\<close>
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lemma [code]:
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  "Id = image (\<lambda>x. (x, x)) (set Enum.enum)"
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  by (auto intro: imageI in_enum)
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lemma tranclp_unfold [code]:
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  "tranclp r a b \<longleftrightarrow> (a, b) \<in> trancl {(x, y). r x y}"
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  by (simp add: trancl_def)
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lemma rtranclp_rtrancl_eq [code]:
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  "rtranclp r x y \<longleftrightarrow> (x, y) \<in> rtrancl {(x, y). r x y}"
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  by (simp add: rtrancl_def)
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lemma max_ext_eq [code]:
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  "max_ext R = {(X, Y). finite X \<and> finite Y \<and> Y \<noteq> {} \<and> (\<forall>x. x \<in> X \<longrightarrow> (\<exists>xa \<in> Y. (x, xa) \<in> R))}"
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  by (auto simp add: max_ext.simps)
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lemma max_extp_eq [code]:
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  "max_extp r x y \<longleftrightarrow> (x, y) \<in> max_ext {(x, y). r x y}"
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  by (simp add: max_ext_def)
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lemma mlex_eq [code]:
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  "f <*mlex*> R = {(x, y). f x < f y \<or> (f x \<le> f y \<and> (x, y) \<in> R)}"
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  by (auto simp add: mlex_prod_def)
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subsubsection \<open>Bounded accessible part\<close>
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primrec bacc :: "('a \<times> 'a) set \<Rightarrow> nat \<Rightarrow> 'a set" 
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where
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  "bacc r 0 = {x. \<forall> y. (y, x) \<notin> r}"
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| "bacc r (Suc n) = (bacc r n \<union> {x. \<forall>y. (y, x) \<in> r \<longrightarrow> y \<in> bacc r n})"
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lemma bacc_subseteq_acc:
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  "bacc r n \<subseteq> Wellfounded.acc r"
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  by (induct n) (auto intro: acc.intros)
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lemma bacc_mono:
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  "n \<le> m \<Longrightarrow> bacc r n \<subseteq> bacc r m"
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  by (induct rule: dec_induct) auto
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lemma bacc_upper_bound:
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  "bacc (r :: ('a \<times> 'a) set)  (card (UNIV :: 'a::finite set)) = (\<Union>n. bacc r n)"
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proof -
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  have "mono (bacc r)" unfolding mono_def by (simp add: bacc_mono)
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  moreover have "\<forall>n. bacc r n = bacc r (Suc n) \<longrightarrow> bacc r (Suc n) = bacc r (Suc (Suc n))" by auto
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  moreover have "finite (range (bacc r))" by auto
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  ultimately show ?thesis
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   by (intro finite_mono_strict_prefix_implies_finite_fixpoint)
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     (auto intro: finite_mono_remains_stable_implies_strict_prefix)
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qed
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lemma acc_subseteq_bacc:
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  assumes "finite r"
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  shows "Wellfounded.acc r \<subseteq> (\<Union>n. bacc r n)"
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proof
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  fix x
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  assume "x : Wellfounded.acc r"
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  then have "\<exists> n. x : bacc r n"
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  proof (induct x arbitrary: rule: acc.induct)
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    case (accI x)
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    then have "\<forall>y. \<exists> n. (y, x) \<in> r --> y : bacc r n" by simp
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    from choice[OF this] obtain n where n: "\<forall>y. (y, x) \<in> r \<longrightarrow> y \<in> bacc r (n y)" ..
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    obtain n where "\<And>y. (y, x) : r \<Longrightarrow> y : bacc r n"
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    proof
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      fix y assume y: "(y, x) : r"
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      with n have "y : bacc r (n y)" by auto
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      moreover have "n y <= Max ((%(y, x). n y) ` r)"
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        using y \<open>finite r\<close> by (auto intro!: Max_ge)
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      note bacc_mono[OF this, of r]
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      ultimately show "y : bacc r (Max ((%(y, x). n y) ` r))" by auto
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    qed
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    then show ?case
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      by (auto simp add: Let_def intro!: exI[of _ "Suc n"])
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  qed
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  then show "x : (UN n. bacc r n)" by auto
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qed
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lemma acc_bacc_eq:
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  fixes A :: "('a :: finite \<times> 'a) set"
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  assumes "finite A"
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  shows "Wellfounded.acc A = bacc A (card (UNIV :: 'a set))"
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  using assms by (metis acc_subseteq_bacc bacc_subseteq_acc bacc_upper_bound order_eq_iff)
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lemma [code]:
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  fixes xs :: "('a::finite \<times> 'a) list"
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  shows "Wellfounded.acc (set xs) = bacc (set xs) (card_UNIV TYPE('a))"
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  by (simp add: card_UNIV_def acc_bacc_eq)
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subsection \<open>Default instances for @{class enum}\<close>
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lemma map_of_zip_enum_is_Some:
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  assumes "length ys = length (enum :: 'a::enum list)"
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  shows "\<exists>y. map_of (zip (enum :: 'a::enum list) ys) x = Some y"
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proof -
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  from assms have "x \<in> set (enum :: 'a::enum list) \<longleftrightarrow>
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    (\<exists>y. map_of (zip (enum :: 'a::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::enum list"
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  assumes length: "length xs = length (enum :: 'a::enum list)"
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      "length ys = length (enum :: 'a::enum list)"
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    and map_of: "the \<circ> map_of (zip (enum :: 'a::enum list) xs) = the \<circ> map_of (zip (enum :: 'a::enum list) ys)"
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  shows "xs = ys"
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proof -
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  have "map_of (zip (enum :: 'a list) xs) = map_of (zip (enum :: '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 :: 'a list) xs) x = Some y1"
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        and "map_of (zip (enum :: 'a list) ys) x = Some y2" by blast
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    moreover from map_of
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      have "the (map_of (zip (enum :: 'a::enum list) xs) x) = the (map_of (zip (enum :: 'a::enum list) ys) x)"
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      by (auto dest: fun_cong)
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    ultimately show "map_of (zip (enum :: 'a::enum list) xs) x = map_of (zip (enum :: 'a::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 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 \<longleftrightarrow> (\<forall>xs \<in> set (List.n_lists n enum). P xs)"
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lemma [code]:
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  "all_n_lists P n \<longleftrightarrow> (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 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 \<longleftrightarrow> (\<exists>xs \<in> set (List.n_lists n enum). P xs)"
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lemma [code]:
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  "ex_n_lists P n \<longleftrightarrow> (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::'a list) ys)) (List.n_lists (length (enum::'a::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 :: ('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 :: 'a::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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   316
  qed
haftmann@26444
   317
next
haftmann@26444
   318
  from map_of_zip_enum_inject
wenzelm@61076
   319
  show "distinct (enum :: ('a \<Rightarrow> 'b) list)"
haftmann@26444
   320
    by (auto intro!: inj_onI simp add: enum_fun_def
haftmann@49950
   321
      distinct_map distinct_n_lists enum_distinct set_n_lists)
bulwahn@41078
   322
next
bulwahn@41078
   323
  fix P
haftmann@49950
   324
  show "enum_all (P :: ('a \<Rightarrow> 'b) \<Rightarrow> bool) = Ball UNIV P"
bulwahn@41078
   325
  proof
bulwahn@41078
   326
    assume "enum_all P"
haftmann@49950
   327
    show "Ball UNIV P"
bulwahn@41078
   328
    proof
bulwahn@41078
   329
      fix f :: "'a \<Rightarrow> 'b"
wenzelm@61076
   330
      have f: "f = the \<circ> map_of (zip (enum :: 'a::enum list) (map f enum))"
bulwahn@41078
   331
        by (auto simp add: map_of_zip_map fun_eq_iff intro: in_enum)
wenzelm@60758
   332
      from \<open>enum_all P\<close> have "P (the \<circ> map_of (zip enum (map f enum)))"
bulwahn@41078
   333
        unfolding enum_all_fun_def all_n_lists_def
bulwahn@41078
   334
        apply (simp add: set_n_lists)
bulwahn@41078
   335
        apply (erule_tac x="map f enum" in allE)
bulwahn@41078
   336
        apply (auto intro!: in_enum)
bulwahn@41078
   337
        done
bulwahn@41078
   338
      from this f show "P f" by auto
bulwahn@41078
   339
    qed
bulwahn@41078
   340
  next
haftmann@49950
   341
    assume "Ball UNIV P"
bulwahn@41078
   342
    from this show "enum_all P"
bulwahn@41078
   343
      unfolding enum_all_fun_def all_n_lists_def by auto
bulwahn@41078
   344
  qed
bulwahn@41078
   345
next
bulwahn@41078
   346
  fix P
haftmann@49950
   347
  show "enum_ex (P :: ('a \<Rightarrow> 'b) \<Rightarrow> bool) = Bex UNIV P"
bulwahn@41078
   348
  proof
bulwahn@41078
   349
    assume "enum_ex P"
haftmann@49950
   350
    from this show "Bex UNIV P"
bulwahn@41078
   351
      unfolding enum_ex_fun_def ex_n_lists_def by auto
bulwahn@41078
   352
  next
haftmann@49950
   353
    assume "Bex UNIV P"
bulwahn@41078
   354
    from this obtain f where "P f" ..
wenzelm@61076
   355
    have f: "f = the \<circ> map_of (zip (enum :: 'a::enum list) (map f enum))"
bulwahn@41078
   356
      by (auto simp add: map_of_zip_map fun_eq_iff intro: in_enum) 
wenzelm@61076
   357
    from \<open>P f\<close> this have "P (the \<circ> map_of (zip (enum :: 'a::enum list) (map f enum)))"
bulwahn@41078
   358
      by auto
bulwahn@41078
   359
    from  this show "enum_ex P"
bulwahn@41078
   360
      unfolding enum_ex_fun_def ex_n_lists_def
bulwahn@41078
   361
      apply (auto simp add: set_n_lists)
bulwahn@41078
   362
      apply (rule_tac x="map f enum" in exI)
bulwahn@41078
   363
      apply (auto intro!: in_enum)
bulwahn@41078
   364
      done
bulwahn@41078
   365
  qed
haftmann@26444
   366
qed
haftmann@26444
   367
haftmann@26444
   368
end
haftmann@26444
   369
wenzelm@61076
   370
lemma enum_fun_code [code]: "enum = (let enum_a = (enum :: 'a::{enum, equal} list)
haftmann@49948
   371
  in map (\<lambda>ys. the o map_of (zip enum_a ys)) (List.n_lists (length enum_a) enum))"
haftmann@28245
   372
  by (simp add: enum_fun_def Let_def)
haftmann@26444
   373
bulwahn@41078
   374
lemma enum_all_fun_code [code]:
bulwahn@41078
   375
  "enum_all P = (let enum_a = (enum :: 'a::{enum, equal} list)
bulwahn@41078
   376
   in all_n_lists (\<lambda>bs. P (the o map_of (zip enum_a bs))) (length enum_a))"
haftmann@49950
   377
  by (simp only: enum_all_fun_def Let_def)
bulwahn@41078
   378
bulwahn@41078
   379
lemma enum_ex_fun_code [code]:
bulwahn@41078
   380
  "enum_ex P = (let enum_a = (enum :: 'a::{enum, equal} list)
bulwahn@41078
   381
   in ex_n_lists (\<lambda>bs. P (the o map_of (zip enum_a bs))) (length enum_a))"
haftmann@49950
   382
  by (simp only: enum_ex_fun_def Let_def)
haftmann@45963
   383
haftmann@45963
   384
instantiation set :: (enum) enum
haftmann@45963
   385
begin
haftmann@45963
   386
haftmann@45963
   387
definition
haftmann@45963
   388
  "enum = map set (sublists enum)"
haftmann@45963
   389
haftmann@45963
   390
definition
haftmann@45963
   391
  "enum_all P \<longleftrightarrow> (\<forall>A\<in>set enum. P (A::'a set))"
haftmann@45963
   392
haftmann@45963
   393
definition
haftmann@45963
   394
  "enum_ex P \<longleftrightarrow> (\<exists>A\<in>set enum. P (A::'a set))"
haftmann@45963
   395
haftmann@45963
   396
instance proof
haftmann@45963
   397
qed (simp_all add: enum_set_def enum_all_set_def enum_ex_set_def sublists_powset distinct_set_sublists
haftmann@45963
   398
  enum_distinct enum_UNIV)
huffman@29024
   399
huffman@29024
   400
end
huffman@29024
   401
haftmann@49950
   402
instantiation unit :: enum
haftmann@49950
   403
begin
haftmann@49950
   404
haftmann@49950
   405
definition
haftmann@49950
   406
  "enum = [()]"
haftmann@49950
   407
haftmann@49950
   408
definition
haftmann@49950
   409
  "enum_all P = P ()"
haftmann@49950
   410
haftmann@49950
   411
definition
haftmann@49950
   412
  "enum_ex P = P ()"
haftmann@49950
   413
haftmann@49950
   414
instance proof
haftmann@49950
   415
qed (auto simp add: enum_unit_def enum_all_unit_def enum_ex_unit_def)
haftmann@49950
   416
haftmann@49950
   417
end
haftmann@49950
   418
haftmann@49950
   419
instantiation bool :: enum
haftmann@49950
   420
begin
haftmann@49950
   421
haftmann@49950
   422
definition
haftmann@49950
   423
  "enum = [False, True]"
haftmann@49950
   424
haftmann@49950
   425
definition
haftmann@49950
   426
  "enum_all P \<longleftrightarrow> P False \<and> P True"
haftmann@49950
   427
haftmann@49950
   428
definition
haftmann@49950
   429
  "enum_ex P \<longleftrightarrow> P False \<or> P True"
haftmann@49950
   430
haftmann@49950
   431
instance proof
haftmann@49950
   432
qed (simp_all only: enum_bool_def enum_all_bool_def enum_ex_bool_def UNIV_bool, simp_all)
haftmann@49950
   433
haftmann@49950
   434
end
haftmann@49950
   435
haftmann@49950
   436
instantiation prod :: (enum, enum) enum
haftmann@49950
   437
begin
haftmann@49950
   438
haftmann@49950
   439
definition
haftmann@49950
   440
  "enum = List.product enum enum"
haftmann@49950
   441
haftmann@49950
   442
definition
haftmann@49950
   443
  "enum_all P = enum_all (%x. enum_all (%y. P (x, y)))"
haftmann@49950
   444
haftmann@49950
   445
definition
haftmann@49950
   446
  "enum_ex P = enum_ex (%x. enum_ex (%y. P (x, y)))"
haftmann@49950
   447
haftmann@49950
   448
 
haftmann@49950
   449
instance by default
nipkow@57247
   450
  (simp_all add: enum_prod_def distinct_product
haftmann@49950
   451
    enum_UNIV enum_distinct enum_all_prod_def enum_ex_prod_def)
haftmann@49950
   452
haftmann@49950
   453
end
haftmann@49950
   454
haftmann@49950
   455
instantiation sum :: (enum, enum) enum
haftmann@49950
   456
begin
haftmann@49950
   457
haftmann@49950
   458
definition
haftmann@49950
   459
  "enum = map Inl enum @ map Inr enum"
haftmann@49950
   460
haftmann@49950
   461
definition
haftmann@49950
   462
  "enum_all P \<longleftrightarrow> enum_all (\<lambda>x. P (Inl x)) \<and> enum_all (\<lambda>x. P (Inr x))"
haftmann@49950
   463
haftmann@49950
   464
definition
haftmann@49950
   465
  "enum_ex P \<longleftrightarrow> enum_ex (\<lambda>x. P (Inl x)) \<or> enum_ex (\<lambda>x. P (Inr x))"
haftmann@49950
   466
haftmann@49950
   467
instance proof
haftmann@49950
   468
qed (simp_all only: enum_sum_def enum_all_sum_def enum_ex_sum_def UNIV_sum,
haftmann@49950
   469
  auto simp add: enum_UNIV distinct_map enum_distinct)
haftmann@49950
   470
haftmann@49950
   471
end
haftmann@49950
   472
haftmann@49950
   473
instantiation option :: (enum) enum
haftmann@49950
   474
begin
haftmann@49950
   475
haftmann@49950
   476
definition
haftmann@49950
   477
  "enum = None # map Some enum"
haftmann@49950
   478
haftmann@49950
   479
definition
haftmann@49950
   480
  "enum_all P \<longleftrightarrow> P None \<and> enum_all (\<lambda>x. P (Some x))"
haftmann@49950
   481
haftmann@49950
   482
definition
haftmann@49950
   483
  "enum_ex P \<longleftrightarrow> P None \<or> enum_ex (\<lambda>x. P (Some x))"
haftmann@49950
   484
haftmann@49950
   485
instance proof
haftmann@49950
   486
qed (simp_all only: enum_option_def enum_all_option_def enum_ex_option_def UNIV_option_conv,
haftmann@49950
   487
  auto simp add: distinct_map enum_UNIV enum_distinct)
haftmann@49950
   488
haftmann@49950
   489
end
haftmann@49950
   490
haftmann@45963
   491
wenzelm@60758
   492
subsection \<open>Small finite types\<close>
bulwahn@40647
   493
wenzelm@60758
   494
text \<open>We define small finite types for use in Quickcheck\<close>
bulwahn@40647
   495
wenzelm@58659
   496
datatype (plugins only: code "quickcheck" extraction) finite_1 =
blanchet@58350
   497
  a\<^sub>1
bulwahn@40647
   498
wenzelm@53015
   499
notation (output) a\<^sub>1  ("a\<^sub>1")
bulwahn@40900
   500
haftmann@49950
   501
lemma UNIV_finite_1:
wenzelm@53015
   502
  "UNIV = {a\<^sub>1}"
haftmann@49950
   503
  by (auto intro: finite_1.exhaust)
haftmann@49950
   504
bulwahn@40647
   505
instantiation finite_1 :: enum
bulwahn@40647
   506
begin
bulwahn@40647
   507
bulwahn@40647
   508
definition
wenzelm@53015
   509
  "enum = [a\<^sub>1]"
bulwahn@40647
   510
bulwahn@41078
   511
definition
wenzelm@53015
   512
  "enum_all P = P a\<^sub>1"
bulwahn@41078
   513
bulwahn@41078
   514
definition
wenzelm@53015
   515
  "enum_ex P = P a\<^sub>1"
bulwahn@41078
   516
bulwahn@40647
   517
instance proof
haftmann@49950
   518
qed (simp_all only: enum_finite_1_def enum_all_finite_1_def enum_ex_finite_1_def UNIV_finite_1, simp_all)
bulwahn@40647
   519
huffman@29024
   520
end
bulwahn@40647
   521
bulwahn@40651
   522
instantiation finite_1 :: linorder
bulwahn@40651
   523
begin
bulwahn@40651
   524
haftmann@49950
   525
definition less_finite_1 :: "finite_1 \<Rightarrow> finite_1 \<Rightarrow> bool"
haftmann@49950
   526
where
haftmann@49950
   527
  "x < (y :: finite_1) \<longleftrightarrow> False"
haftmann@49950
   528
bulwahn@40651
   529
definition less_eq_finite_1 :: "finite_1 \<Rightarrow> finite_1 \<Rightarrow> bool"
bulwahn@40651
   530
where
haftmann@49950
   531
  "x \<le> (y :: finite_1) \<longleftrightarrow> True"
bulwahn@40651
   532
bulwahn@40651
   533
instance
bulwahn@40651
   534
apply (intro_classes)
bulwahn@40651
   535
apply (auto simp add: less_finite_1_def less_eq_finite_1_def)
bulwahn@40651
   536
apply (metis finite_1.exhaust)
bulwahn@40651
   537
done
bulwahn@40651
   538
bulwahn@40651
   539
end
bulwahn@40651
   540
Andreas@57922
   541
instance finite_1 :: "{dense_linorder, wellorder}"
Andreas@57922
   542
by intro_classes (simp_all add: less_finite_1_def)
Andreas@57922
   543
Andreas@57818
   544
instantiation finite_1 :: complete_lattice
Andreas@57818
   545
begin
Andreas@57818
   546
Andreas@57818
   547
definition [simp]: "Inf = (\<lambda>_. a\<^sub>1)"
Andreas@57818
   548
definition [simp]: "Sup = (\<lambda>_. a\<^sub>1)"
Andreas@57818
   549
definition [simp]: "bot = a\<^sub>1"
Andreas@57818
   550
definition [simp]: "top = a\<^sub>1"
Andreas@57818
   551
definition [simp]: "inf = (\<lambda>_ _. a\<^sub>1)"
Andreas@57818
   552
definition [simp]: "sup = (\<lambda>_ _. a\<^sub>1)"
Andreas@57818
   553
Andreas@57818
   554
instance by intro_classes(simp_all add: less_eq_finite_1_def)
Andreas@57818
   555
end
Andreas@57818
   556
Andreas@57818
   557
instance finite_1 :: complete_distrib_lattice
Andreas@57818
   558
by intro_classes(simp_all add: INF_def SUP_def)
Andreas@57818
   559
Andreas@57818
   560
instance finite_1 :: complete_linorder ..
Andreas@57818
   561
Andreas@57922
   562
lemma finite_1_eq: "x = a\<^sub>1"
Andreas@57922
   563
by(cases x) simp
Andreas@57922
   564
wenzelm@60758
   565
simproc_setup finite_1_eq ("x::finite_1") = \<open>
wenzelm@59582
   566
  fn _ => fn _ => fn ct =>
wenzelm@59582
   567
    (case Thm.term_of ct of
wenzelm@59582
   568
      Const (@{const_name a\<^sub>1}, _) => NONE
wenzelm@59582
   569
    | _ => SOME (mk_meta_eq @{thm finite_1_eq}))
wenzelm@60758
   570
\<close>
Andreas@57922
   571
wenzelm@59582
   572
instantiation finite_1 :: complete_boolean_algebra
wenzelm@59582
   573
begin
Andreas@57922
   574
definition [simp]: "op - = (\<lambda>_ _. a\<^sub>1)"
Andreas@57922
   575
definition [simp]: "uminus = (\<lambda>_. a\<^sub>1)"
Andreas@57922
   576
instance by intro_classes simp_all
Andreas@57922
   577
end
Andreas@57922
   578
Andreas@57922
   579
instantiation finite_1 :: 
Andreas@57922
   580
  "{linordered_ring_strict, linordered_comm_semiring_strict, ordered_comm_ring,
Andreas@57922
   581
    ordered_cancel_comm_monoid_diff, comm_monoid_mult, ordered_ring_abs,
Andreas@57922
   582
    one, Divides.div, sgn_if, inverse}"
Andreas@57922
   583
begin
Andreas@57922
   584
definition [simp]: "Groups.zero = a\<^sub>1"
Andreas@57922
   585
definition [simp]: "Groups.one = a\<^sub>1"
Andreas@57922
   586
definition [simp]: "op + = (\<lambda>_ _. a\<^sub>1)"
Andreas@57922
   587
definition [simp]: "op * = (\<lambda>_ _. a\<^sub>1)"
Andreas@57922
   588
definition [simp]: "op mod = (\<lambda>_ _. a\<^sub>1)" 
Andreas@57922
   589
definition [simp]: "abs = (\<lambda>_. a\<^sub>1)"
Andreas@57922
   590
definition [simp]: "sgn = (\<lambda>_. a\<^sub>1)"
Andreas@57922
   591
definition [simp]: "inverse = (\<lambda>_. a\<^sub>1)"
haftmann@60352
   592
definition [simp]: "divide = (\<lambda>_ _. a\<^sub>1)"
Andreas@57922
   593
Andreas@57922
   594
instance by intro_classes(simp_all add: less_finite_1_def)
Andreas@57922
   595
end
Andreas@57922
   596
Andreas@57922
   597
declare [[simproc del: finite_1_eq]]
wenzelm@53015
   598
hide_const (open) a\<^sub>1
bulwahn@40657
   599
wenzelm@58659
   600
datatype (plugins only: code "quickcheck" extraction) finite_2 =
blanchet@58350
   601
  a\<^sub>1 | a\<^sub>2
bulwahn@40647
   602
wenzelm@53015
   603
notation (output) a\<^sub>1  ("a\<^sub>1")
wenzelm@53015
   604
notation (output) a\<^sub>2  ("a\<^sub>2")
bulwahn@40900
   605
haftmann@49950
   606
lemma UNIV_finite_2:
wenzelm@53015
   607
  "UNIV = {a\<^sub>1, a\<^sub>2}"
haftmann@49950
   608
  by (auto intro: finite_2.exhaust)
haftmann@49950
   609
bulwahn@40647
   610
instantiation finite_2 :: enum
bulwahn@40647
   611
begin
bulwahn@40647
   612
bulwahn@40647
   613
definition
wenzelm@53015
   614
  "enum = [a\<^sub>1, a\<^sub>2]"
bulwahn@40647
   615
bulwahn@41078
   616
definition
wenzelm@53015
   617
  "enum_all P \<longleftrightarrow> P a\<^sub>1 \<and> P a\<^sub>2"
bulwahn@41078
   618
bulwahn@41078
   619
definition
wenzelm@53015
   620
  "enum_ex P \<longleftrightarrow> P a\<^sub>1 \<or> P a\<^sub>2"
bulwahn@41078
   621
bulwahn@40647
   622
instance proof
haftmann@49950
   623
qed (simp_all only: enum_finite_2_def enum_all_finite_2_def enum_ex_finite_2_def UNIV_finite_2, simp_all)
bulwahn@40647
   624
bulwahn@40647
   625
end
bulwahn@40647
   626
bulwahn@40651
   627
instantiation finite_2 :: linorder
bulwahn@40651
   628
begin
bulwahn@40651
   629
bulwahn@40651
   630
definition less_finite_2 :: "finite_2 \<Rightarrow> finite_2 \<Rightarrow> bool"
bulwahn@40651
   631
where
wenzelm@53015
   632
  "x < y \<longleftrightarrow> x = a\<^sub>1 \<and> y = a\<^sub>2"
bulwahn@40651
   633
bulwahn@40651
   634
definition less_eq_finite_2 :: "finite_2 \<Rightarrow> finite_2 \<Rightarrow> bool"
bulwahn@40651
   635
where
haftmann@49950
   636
  "x \<le> y \<longleftrightarrow> x = y \<or> x < (y :: finite_2)"
bulwahn@40651
   637
bulwahn@40651
   638
instance
bulwahn@40651
   639
apply (intro_classes)
bulwahn@40651
   640
apply (auto simp add: less_finite_2_def less_eq_finite_2_def)
haftmann@49950
   641
apply (metis finite_2.nchotomy)+
bulwahn@40651
   642
done
bulwahn@40651
   643
bulwahn@40651
   644
end
bulwahn@40651
   645
Andreas@57922
   646
instance finite_2 :: wellorder
Andreas@57922
   647
by(rule wf_wellorderI)(simp add: less_finite_2_def, intro_classes)
Andreas@57922
   648
Andreas@57818
   649
instantiation finite_2 :: complete_lattice
Andreas@57818
   650
begin
Andreas@57818
   651
Andreas@57818
   652
definition "\<Sqinter>A = (if a\<^sub>1 \<in> A then a\<^sub>1 else a\<^sub>2)"
Andreas@57818
   653
definition "\<Squnion>A = (if a\<^sub>2 \<in> A then a\<^sub>2 else a\<^sub>1)"
Andreas@57818
   654
definition [simp]: "bot = a\<^sub>1"
Andreas@57818
   655
definition [simp]: "top = a\<^sub>2"
Andreas@57818
   656
definition "x \<sqinter> y = (if x = a\<^sub>1 \<or> y = a\<^sub>1 then a\<^sub>1 else a\<^sub>2)"
Andreas@57818
   657
definition "x \<squnion> y = (if x = a\<^sub>2 \<or> y = a\<^sub>2 then a\<^sub>2 else a\<^sub>1)"
Andreas@57818
   658
Andreas@57818
   659
lemma neq_finite_2_a\<^sub>1_iff [simp]: "x \<noteq> a\<^sub>1 \<longleftrightarrow> x = a\<^sub>2"
Andreas@57818
   660
by(cases x) simp_all
Andreas@57818
   661
Andreas@57818
   662
lemma neq_finite_2_a\<^sub>1_iff' [simp]: "a\<^sub>1 \<noteq> x \<longleftrightarrow> x = a\<^sub>2"
Andreas@57818
   663
by(cases x) simp_all
Andreas@57818
   664
Andreas@57818
   665
lemma neq_finite_2_a\<^sub>2_iff [simp]: "x \<noteq> a\<^sub>2 \<longleftrightarrow> x = a\<^sub>1"
Andreas@57818
   666
by(cases x) simp_all
Andreas@57818
   667
Andreas@57818
   668
lemma neq_finite_2_a\<^sub>2_iff' [simp]: "a\<^sub>2 \<noteq> x \<longleftrightarrow> x = a\<^sub>1"
Andreas@57818
   669
by(cases x) simp_all
Andreas@57818
   670
Andreas@57818
   671
instance
Andreas@57818
   672
proof
Andreas@57818
   673
  fix x :: finite_2 and A
Andreas@57818
   674
  assume "x \<in> A"
Andreas@57818
   675
  then show "\<Sqinter>A \<le> x" "x \<le> \<Squnion>A"
Andreas@57818
   676
    by(case_tac [!] x)(auto simp add: less_eq_finite_2_def less_finite_2_def Inf_finite_2_def Sup_finite_2_def)
Andreas@57818
   677
qed(auto simp add: less_eq_finite_2_def less_finite_2_def inf_finite_2_def sup_finite_2_def Inf_finite_2_def Sup_finite_2_def)
Andreas@57818
   678
end
Andreas@57818
   679
Andreas@57818
   680
instance finite_2 :: complete_distrib_lattice
Andreas@57818
   681
by(intro_classes)(auto simp add: INF_def SUP_def sup_finite_2_def inf_finite_2_def Inf_finite_2_def Sup_finite_2_def)
Andreas@57818
   682
Andreas@57818
   683
instance finite_2 :: complete_linorder ..
Andreas@57818
   684
haftmann@59867
   685
instantiation finite_2 :: "{field, abs_if, ring_div, sgn_if, semiring_div}" begin
Andreas@57922
   686
definition [simp]: "0 = a\<^sub>1"
Andreas@57922
   687
definition [simp]: "1 = a\<^sub>2"
Andreas@57922
   688
definition "x + y = (case (x, y) of (a\<^sub>1, a\<^sub>1) \<Rightarrow> a\<^sub>1 | (a\<^sub>2, a\<^sub>2) \<Rightarrow> a\<^sub>1 | _ \<Rightarrow> a\<^sub>2)"
Andreas@57922
   689
definition "uminus = (\<lambda>x :: finite_2. x)"
Andreas@57922
   690
definition "op - = (op + :: finite_2 \<Rightarrow> _)"
Andreas@57922
   691
definition "x * y = (case (x, y) of (a\<^sub>2, a\<^sub>2) \<Rightarrow> a\<^sub>2 | _ \<Rightarrow> a\<^sub>1)"
Andreas@57922
   692
definition "inverse = (\<lambda>x :: finite_2. x)"
haftmann@60352
   693
definition "divide = (op * :: finite_2 \<Rightarrow> _)"
Andreas@57922
   694
definition "abs = (\<lambda>x :: finite_2. x)"
Andreas@57922
   695
definition "x mod y = (case (x, y) of (a\<^sub>2, a\<^sub>1) \<Rightarrow> a\<^sub>2 | _ \<Rightarrow> a\<^sub>1)"
Andreas@57922
   696
definition "sgn = (\<lambda>x :: finite_2. x)"
Andreas@57922
   697
instance
Andreas@57922
   698
by intro_classes
Andreas@57922
   699
  (simp_all add: plus_finite_2_def uminus_finite_2_def minus_finite_2_def times_finite_2_def
haftmann@60352
   700
       inverse_finite_2_def divide_finite_2_def abs_finite_2_def mod_finite_2_def sgn_finite_2_def
Andreas@57922
   701
     split: finite_2.splits)
Andreas@57922
   702
end
Andreas@57922
   703
haftmann@58646
   704
lemma two_finite_2 [simp]:
haftmann@58646
   705
  "2 = a\<^sub>1"
haftmann@58646
   706
  by (simp add: numeral.simps plus_finite_2_def)
haftmann@58646
   707
  
wenzelm@53015
   708
hide_const (open) a\<^sub>1 a\<^sub>2
bulwahn@40657
   709
wenzelm@58659
   710
datatype (plugins only: code "quickcheck" extraction) finite_3 =
blanchet@58350
   711
  a\<^sub>1 | a\<^sub>2 | a\<^sub>3
bulwahn@40647
   712
wenzelm@53015
   713
notation (output) a\<^sub>1  ("a\<^sub>1")
wenzelm@53015
   714
notation (output) a\<^sub>2  ("a\<^sub>2")
wenzelm@53015
   715
notation (output) a\<^sub>3  ("a\<^sub>3")
bulwahn@40900
   716
haftmann@49950
   717
lemma UNIV_finite_3:
wenzelm@53015
   718
  "UNIV = {a\<^sub>1, a\<^sub>2, a\<^sub>3}"
haftmann@49950
   719
  by (auto intro: finite_3.exhaust)
haftmann@49950
   720
bulwahn@40647
   721
instantiation finite_3 :: enum
bulwahn@40647
   722
begin
bulwahn@40647
   723
bulwahn@40647
   724
definition
wenzelm@53015
   725
  "enum = [a\<^sub>1, a\<^sub>2, a\<^sub>3]"
bulwahn@40647
   726
bulwahn@41078
   727
definition
wenzelm@53015
   728
  "enum_all P \<longleftrightarrow> P a\<^sub>1 \<and> P a\<^sub>2 \<and> P a\<^sub>3"
bulwahn@41078
   729
bulwahn@41078
   730
definition
wenzelm@53015
   731
  "enum_ex P \<longleftrightarrow> P a\<^sub>1 \<or> P a\<^sub>2 \<or> P a\<^sub>3"
bulwahn@41078
   732
bulwahn@40647
   733
instance proof
haftmann@49950
   734
qed (simp_all only: enum_finite_3_def enum_all_finite_3_def enum_ex_finite_3_def UNIV_finite_3, simp_all)
bulwahn@40647
   735
bulwahn@40647
   736
end
bulwahn@40647
   737
bulwahn@40651
   738
instantiation finite_3 :: linorder
bulwahn@40651
   739
begin
bulwahn@40651
   740
bulwahn@40651
   741
definition less_finite_3 :: "finite_3 \<Rightarrow> finite_3 \<Rightarrow> bool"
bulwahn@40651
   742
where
wenzelm@53015
   743
  "x < y = (case x of a\<^sub>1 \<Rightarrow> y \<noteq> a\<^sub>1 | a\<^sub>2 \<Rightarrow> y = a\<^sub>3 | a\<^sub>3 \<Rightarrow> False)"
bulwahn@40651
   744
bulwahn@40651
   745
definition less_eq_finite_3 :: "finite_3 \<Rightarrow> finite_3 \<Rightarrow> bool"
bulwahn@40651
   746
where
haftmann@49950
   747
  "x \<le> y \<longleftrightarrow> x = y \<or> x < (y :: finite_3)"
bulwahn@40651
   748
bulwahn@40651
   749
instance proof (intro_classes)
bulwahn@40651
   750
qed (auto simp add: less_finite_3_def less_eq_finite_3_def split: finite_3.split_asm)
bulwahn@40651
   751
bulwahn@40651
   752
end
bulwahn@40651
   753
Andreas@57922
   754
instance finite_3 :: wellorder
Andreas@57922
   755
proof(rule wf_wellorderI)
Andreas@57922
   756
  have "inv_image less_than (case_finite_3 0 1 2) = {(x, y). x < y}"
Andreas@57922
   757
    by(auto simp add: less_finite_3_def split: finite_3.splits)
Andreas@57922
   758
  from this[symmetric] show "wf \<dots>" by simp
Andreas@57922
   759
qed intro_classes
Andreas@57922
   760
Andreas@57818
   761
instantiation finite_3 :: complete_lattice
Andreas@57818
   762
begin
Andreas@57818
   763
Andreas@57818
   764
definition "\<Sqinter>A = (if a\<^sub>1 \<in> A then a\<^sub>1 else if a\<^sub>2 \<in> A then a\<^sub>2 else a\<^sub>3)"
Andreas@57818
   765
definition "\<Squnion>A = (if a\<^sub>3 \<in> A then a\<^sub>3 else if a\<^sub>2 \<in> A then a\<^sub>2 else a\<^sub>1)"
Andreas@57818
   766
definition [simp]: "bot = a\<^sub>1"
Andreas@57818
   767
definition [simp]: "top = a\<^sub>3"
Andreas@57818
   768
definition [simp]: "inf = (min :: finite_3 \<Rightarrow> _)"
Andreas@57818
   769
definition [simp]: "sup = (max :: finite_3 \<Rightarrow> _)"
Andreas@57818
   770
Andreas@57818
   771
instance
Andreas@57818
   772
proof
Andreas@57818
   773
  fix x :: finite_3 and A
Andreas@57818
   774
  assume "x \<in> A"
Andreas@57818
   775
  then show "\<Sqinter>A \<le> x" "x \<le> \<Squnion>A"
Andreas@57818
   776
    by(case_tac [!] x)(auto simp add: Inf_finite_3_def Sup_finite_3_def less_eq_finite_3_def less_finite_3_def)
Andreas@57818
   777
next
Andreas@57818
   778
  fix A and z :: finite_3
Andreas@57818
   779
  assume "\<And>x. x \<in> A \<Longrightarrow> z \<le> x"
Andreas@57818
   780
  then show "z \<le> \<Sqinter>A"
Andreas@57818
   781
    by(cases z)(auto simp add: Inf_finite_3_def less_eq_finite_3_def less_finite_3_def)
Andreas@57818
   782
next
Andreas@57818
   783
  fix A and z :: finite_3
Andreas@57818
   784
  assume *: "\<And>x. x \<in> A \<Longrightarrow> x \<le> z"
Andreas@57818
   785
  show "\<Squnion>A \<le> z"
Andreas@57818
   786
    by(auto simp add: Sup_finite_3_def less_eq_finite_3_def less_finite_3_def dest: *)
Andreas@57818
   787
qed(auto simp add: Inf_finite_3_def Sup_finite_3_def)
Andreas@57818
   788
end
Andreas@57818
   789
Andreas@57818
   790
instance finite_3 :: complete_distrib_lattice
Andreas@57818
   791
proof
Andreas@57818
   792
  fix a :: finite_3 and B
Andreas@57818
   793
  show "a \<squnion> \<Sqinter>B = (\<Sqinter>b\<in>B. a \<squnion> b)"
Andreas@57818
   794
  proof(cases a "\<Sqinter>B" rule: finite_3.exhaust[case_product finite_3.exhaust])
Andreas@57818
   795
    case a\<^sub>2_a\<^sub>3
Andreas@57818
   796
    then have "\<And>x. x \<in> B \<Longrightarrow> x = a\<^sub>3"
Andreas@57818
   797
      by(case_tac x)(auto simp add: Inf_finite_3_def split: split_if_asm)
Andreas@57818
   798
    then show ?thesis using a\<^sub>2_a\<^sub>3
Andreas@57818
   799
      by(auto simp add: INF_def Inf_finite_3_def max_def less_eq_finite_3_def less_finite_3_def split: split_if_asm)
Andreas@57818
   800
  qed(auto simp add: INF_def Inf_finite_3_def max_def less_finite_3_def less_eq_finite_3_def split: split_if_asm)
Andreas@57818
   801
  show "a \<sqinter> \<Squnion>B = (\<Squnion>b\<in>B. a \<sqinter> b)"
Andreas@57818
   802
    by(cases a "\<Squnion>B" rule: finite_3.exhaust[case_product finite_3.exhaust])
Andreas@57818
   803
      (auto simp add: SUP_def Sup_finite_3_def min_def less_finite_3_def less_eq_finite_3_def split: split_if_asm)
Andreas@57818
   804
qed
Andreas@57818
   805
Andreas@57818
   806
instance finite_3 :: complete_linorder ..
Andreas@57818
   807
haftmann@59867
   808
instantiation finite_3 :: "{field, abs_if, ring_div, semiring_div, sgn_if}" begin
Andreas@57922
   809
definition [simp]: "0 = a\<^sub>1"
Andreas@57922
   810
definition [simp]: "1 = a\<^sub>2"
Andreas@57922
   811
definition
Andreas@57922
   812
  "x + y = (case (x, y) of
Andreas@57922
   813
     (a\<^sub>1, a\<^sub>1) \<Rightarrow> a\<^sub>1 | (a\<^sub>2, a\<^sub>3) \<Rightarrow> a\<^sub>1 | (a\<^sub>3, a\<^sub>2) \<Rightarrow> a\<^sub>1
Andreas@57922
   814
   | (a\<^sub>1, a\<^sub>2) \<Rightarrow> a\<^sub>2 | (a\<^sub>2, a\<^sub>1) \<Rightarrow> a\<^sub>2 | (a\<^sub>3, a\<^sub>3) \<Rightarrow> a\<^sub>2
Andreas@57922
   815
   | _ \<Rightarrow> a\<^sub>3)"
Andreas@57922
   816
definition "- x = (case x of a\<^sub>1 \<Rightarrow> a\<^sub>1 | a\<^sub>2 \<Rightarrow> a\<^sub>3 | a\<^sub>3 \<Rightarrow> a\<^sub>2)"
Andreas@57922
   817
definition "x - y = x + (- y :: finite_3)"
Andreas@57922
   818
definition "x * y = (case (x, y) of (a\<^sub>2, a\<^sub>2) \<Rightarrow> a\<^sub>2 | (a\<^sub>3, a\<^sub>3) \<Rightarrow> a\<^sub>2 | (a\<^sub>2, a\<^sub>3) \<Rightarrow> a\<^sub>3 | (a\<^sub>3, a\<^sub>2) \<Rightarrow> a\<^sub>3 | _ \<Rightarrow> a\<^sub>1)"
Andreas@57922
   819
definition "inverse = (\<lambda>x :: finite_3. x)" 
haftmann@60429
   820
definition "x div y = x * inverse (y :: finite_3)"
Andreas@57922
   821
definition "abs = (\<lambda>x :: finite_3. x)"
Andreas@57922
   822
definition "x mod y = (case (x, y) of (a\<^sub>2, a\<^sub>1) \<Rightarrow> a\<^sub>2 | (a\<^sub>3, a\<^sub>1) \<Rightarrow> a\<^sub>3 | _ \<Rightarrow> a\<^sub>1)"
Andreas@57922
   823
definition "sgn = (\<lambda>x. case x of a\<^sub>1 \<Rightarrow> a\<^sub>1 | _ \<Rightarrow> a\<^sub>2)"
Andreas@57922
   824
instance
Andreas@57922
   825
by intro_classes
Andreas@57922
   826
  (simp_all add: plus_finite_3_def uminus_finite_3_def minus_finite_3_def times_finite_3_def
haftmann@60352
   827
       inverse_finite_3_def divide_finite_3_def abs_finite_3_def mod_finite_3_def sgn_finite_3_def
Andreas@57922
   828
       less_finite_3_def
Andreas@57922
   829
     split: finite_3.splits)
Andreas@57922
   830
end
Andreas@57922
   831
haftmann@58646
   832
haftmann@58646
   833
wenzelm@53015
   834
hide_const (open) a\<^sub>1 a\<^sub>2 a\<^sub>3
bulwahn@40657
   835
wenzelm@58659
   836
datatype (plugins only: code "quickcheck" extraction) finite_4 =
blanchet@58350
   837
  a\<^sub>1 | a\<^sub>2 | a\<^sub>3 | a\<^sub>4
bulwahn@40647
   838
wenzelm@53015
   839
notation (output) a\<^sub>1  ("a\<^sub>1")
wenzelm@53015
   840
notation (output) a\<^sub>2  ("a\<^sub>2")
wenzelm@53015
   841
notation (output) a\<^sub>3  ("a\<^sub>3")
wenzelm@53015
   842
notation (output) a\<^sub>4  ("a\<^sub>4")
bulwahn@40900
   843
haftmann@49950
   844
lemma UNIV_finite_4:
wenzelm@53015
   845
  "UNIV = {a\<^sub>1, a\<^sub>2, a\<^sub>3, a\<^sub>4}"
haftmann@49950
   846
  by (auto intro: finite_4.exhaust)
haftmann@49950
   847
bulwahn@40647
   848
instantiation finite_4 :: enum
bulwahn@40647
   849
begin
bulwahn@40647
   850
bulwahn@40647
   851
definition
wenzelm@53015
   852
  "enum = [a\<^sub>1, a\<^sub>2, a\<^sub>3, a\<^sub>4]"
bulwahn@40647
   853
bulwahn@41078
   854
definition
wenzelm@53015
   855
  "enum_all P \<longleftrightarrow> P a\<^sub>1 \<and> P a\<^sub>2 \<and> P a\<^sub>3 \<and> P a\<^sub>4"
bulwahn@41078
   856
bulwahn@41078
   857
definition
wenzelm@53015
   858
  "enum_ex P \<longleftrightarrow> P a\<^sub>1 \<or> P a\<^sub>2 \<or> P a\<^sub>3 \<or> P a\<^sub>4"
bulwahn@41078
   859
bulwahn@40647
   860
instance proof
haftmann@49950
   861
qed (simp_all only: enum_finite_4_def enum_all_finite_4_def enum_ex_finite_4_def UNIV_finite_4, simp_all)
bulwahn@40647
   862
bulwahn@40647
   863
end
bulwahn@40647
   864
Andreas@57818
   865
instantiation finite_4 :: complete_lattice begin
Andreas@57818
   866
wenzelm@60758
   867
text \<open>@{term a\<^sub>1} $<$ @{term a\<^sub>2},@{term a\<^sub>3} $<$ @{term a\<^sub>4},
wenzelm@60758
   868
  but @{term a\<^sub>2} and @{term a\<^sub>3} are incomparable.\<close>
Andreas@57818
   869
Andreas@57818
   870
definition
Andreas@57818
   871
  "x < y \<longleftrightarrow> (case (x, y) of
Andreas@57818
   872
     (a\<^sub>1, a\<^sub>1) \<Rightarrow> False | (a\<^sub>1, _) \<Rightarrow> True
Andreas@57818
   873
   |  (a\<^sub>2, a\<^sub>4) \<Rightarrow> True
Andreas@57818
   874
   |  (a\<^sub>3, a\<^sub>4) \<Rightarrow> True  | _ \<Rightarrow> False)"
Andreas@57818
   875
Andreas@57818
   876
definition 
Andreas@57818
   877
  "x \<le> y \<longleftrightarrow> (case (x, y) of
Andreas@57818
   878
     (a\<^sub>1, _) \<Rightarrow> True
Andreas@57818
   879
   | (a\<^sub>2, a\<^sub>2) \<Rightarrow> True | (a\<^sub>2, a\<^sub>4) \<Rightarrow> True
Andreas@57818
   880
   | (a\<^sub>3, a\<^sub>3) \<Rightarrow> True | (a\<^sub>3, a\<^sub>4) \<Rightarrow> True
Andreas@57818
   881
   | (a\<^sub>4, a\<^sub>4) \<Rightarrow> True | _ \<Rightarrow> False)"
Andreas@57818
   882
Andreas@57818
   883
definition
Andreas@57818
   884
  "\<Sqinter>A = (if a\<^sub>1 \<in> A \<or> a\<^sub>2 \<in> A \<and> a\<^sub>3 \<in> A then a\<^sub>1 else if a\<^sub>2 \<in> A then a\<^sub>2 else if a\<^sub>3 \<in> A then a\<^sub>3 else a\<^sub>4)"
Andreas@57818
   885
definition
Andreas@57818
   886
  "\<Squnion>A = (if a\<^sub>4 \<in> A \<or> a\<^sub>2 \<in> A \<and> a\<^sub>3 \<in> A then a\<^sub>4 else if a\<^sub>2 \<in> A then a\<^sub>2 else if a\<^sub>3 \<in> A then a\<^sub>3 else a\<^sub>1)"
Andreas@57818
   887
definition [simp]: "bot = a\<^sub>1"
Andreas@57818
   888
definition [simp]: "top = a\<^sub>4"
Andreas@57818
   889
definition
Andreas@57818
   890
  "x \<sqinter> y = (case (x, y) of
Andreas@57818
   891
     (a\<^sub>1, _) \<Rightarrow> a\<^sub>1 | (_, a\<^sub>1) \<Rightarrow> a\<^sub>1 | (a\<^sub>2, a\<^sub>3) \<Rightarrow> a\<^sub>1 | (a\<^sub>3, a\<^sub>2) \<Rightarrow> a\<^sub>1
Andreas@57818
   892
   | (a\<^sub>2, _) \<Rightarrow> a\<^sub>2 | (_, a\<^sub>2) \<Rightarrow> a\<^sub>2
Andreas@57818
   893
   | (a\<^sub>3, _) \<Rightarrow> a\<^sub>3 | (_, a\<^sub>3) \<Rightarrow> a\<^sub>3
Andreas@57818
   894
   | _ \<Rightarrow> a\<^sub>4)"
Andreas@57818
   895
definition
Andreas@57818
   896
  "x \<squnion> y = (case (x, y) of
Andreas@57818
   897
     (a\<^sub>4, _) \<Rightarrow> a\<^sub>4 | (_, a\<^sub>4) \<Rightarrow> a\<^sub>4 | (a\<^sub>2, a\<^sub>3) \<Rightarrow> a\<^sub>4 | (a\<^sub>3, a\<^sub>2) \<Rightarrow> a\<^sub>4
Andreas@57818
   898
  | (a\<^sub>2, _) \<Rightarrow> a\<^sub>2 | (_, a\<^sub>2) \<Rightarrow> a\<^sub>2
Andreas@57818
   899
  | (a\<^sub>3, _) \<Rightarrow> a\<^sub>3 | (_, a\<^sub>3) \<Rightarrow> a\<^sub>3
Andreas@57818
   900
  | _ \<Rightarrow> a\<^sub>1)"
Andreas@57818
   901
Andreas@57818
   902
instance
Andreas@57818
   903
proof
Andreas@57818
   904
  fix A and z :: finite_4
Andreas@57818
   905
  assume *: "\<And>x. x \<in> A \<Longrightarrow> x \<le> z"
Andreas@57818
   906
  show "\<Squnion>A \<le> z"
Andreas@57818
   907
    by(auto simp add: Sup_finite_4_def less_eq_finite_4_def dest!: * split: finite_4.splits)
Andreas@57818
   908
next
Andreas@57818
   909
  fix A and z :: finite_4
Andreas@57818
   910
  assume *: "\<And>x. x \<in> A \<Longrightarrow> z \<le> x"
Andreas@57818
   911
  show "z \<le> \<Sqinter>A"
Andreas@57818
   912
    by(auto simp add: Inf_finite_4_def less_eq_finite_4_def dest!: * split: finite_4.splits)
Andreas@57818
   913
qed(auto simp add: less_finite_4_def less_eq_finite_4_def Inf_finite_4_def Sup_finite_4_def inf_finite_4_def sup_finite_4_def split: finite_4.splits)
Andreas@57818
   914
Andreas@57818
   915
end
Andreas@57818
   916
Andreas@57818
   917
instance finite_4 :: complete_distrib_lattice
Andreas@57818
   918
proof
Andreas@57818
   919
  fix a :: finite_4 and B
Andreas@57818
   920
  show "a \<squnion> \<Sqinter>B = (\<Sqinter>b\<in>B. a \<squnion> b)"
Andreas@57818
   921
    by(cases a "\<Sqinter>B" rule: finite_4.exhaust[case_product finite_4.exhaust])
Andreas@57818
   922
      (auto simp add: sup_finite_4_def Inf_finite_4_def INF_def split: finite_4.splits split_if_asm)
Andreas@57818
   923
  show "a \<sqinter> \<Squnion>B = (\<Squnion>b\<in>B. a \<sqinter> b)"
Andreas@57818
   924
    by(cases a "\<Squnion>B" rule: finite_4.exhaust[case_product finite_4.exhaust])
Andreas@57818
   925
      (auto simp add: inf_finite_4_def Sup_finite_4_def SUP_def split: finite_4.splits split_if_asm)
Andreas@57818
   926
qed
Andreas@57818
   927
Andreas@57922
   928
instantiation finite_4 :: complete_boolean_algebra begin
Andreas@57922
   929
definition "- x = (case x of a\<^sub>1 \<Rightarrow> a\<^sub>4 | a\<^sub>2 \<Rightarrow> a\<^sub>3 | a\<^sub>3 \<Rightarrow> a\<^sub>2 | a\<^sub>4 \<Rightarrow> a\<^sub>1)"
Andreas@57922
   930
definition "x - y = x \<sqinter> - (y :: finite_4)"
Andreas@57922
   931
instance
Andreas@57922
   932
by intro_classes
Andreas@57922
   933
  (simp_all add: inf_finite_4_def sup_finite_4_def uminus_finite_4_def minus_finite_4_def split: finite_4.splits)
Andreas@57922
   934
end
Andreas@57922
   935
wenzelm@53015
   936
hide_const (open) a\<^sub>1 a\<^sub>2 a\<^sub>3 a\<^sub>4
bulwahn@40651
   937
wenzelm@58659
   938
datatype (plugins only: code "quickcheck" extraction) finite_5 =
blanchet@58350
   939
  a\<^sub>1 | a\<^sub>2 | a\<^sub>3 | a\<^sub>4 | a\<^sub>5
bulwahn@40647
   940
wenzelm@53015
   941
notation (output) a\<^sub>1  ("a\<^sub>1")
wenzelm@53015
   942
notation (output) a\<^sub>2  ("a\<^sub>2")
wenzelm@53015
   943
notation (output) a\<^sub>3  ("a\<^sub>3")
wenzelm@53015
   944
notation (output) a\<^sub>4  ("a\<^sub>4")
wenzelm@53015
   945
notation (output) a\<^sub>5  ("a\<^sub>5")
bulwahn@40900
   946
haftmann@49950
   947
lemma UNIV_finite_5:
wenzelm@53015
   948
  "UNIV = {a\<^sub>1, a\<^sub>2, a\<^sub>3, a\<^sub>4, a\<^sub>5}"
haftmann@49950
   949
  by (auto intro: finite_5.exhaust)
haftmann@49950
   950
bulwahn@40647
   951
instantiation finite_5 :: enum
bulwahn@40647
   952
begin
bulwahn@40647
   953
bulwahn@40647
   954
definition
wenzelm@53015
   955
  "enum = [a\<^sub>1, a\<^sub>2, a\<^sub>3, a\<^sub>4, a\<^sub>5]"
bulwahn@40647
   956
bulwahn@41078
   957
definition
wenzelm@53015
   958
  "enum_all P \<longleftrightarrow> P a\<^sub>1 \<and> P a\<^sub>2 \<and> P a\<^sub>3 \<and> P a\<^sub>4 \<and> P a\<^sub>5"
bulwahn@41078
   959
bulwahn@41078
   960
definition
wenzelm@53015
   961
  "enum_ex P \<longleftrightarrow> P a\<^sub>1 \<or> P a\<^sub>2 \<or> P a\<^sub>3 \<or> P a\<^sub>4 \<or> P a\<^sub>5"
bulwahn@41078
   962
bulwahn@40647
   963
instance proof
haftmann@49950
   964
qed (simp_all only: enum_finite_5_def enum_all_finite_5_def enum_ex_finite_5_def UNIV_finite_5, simp_all)
bulwahn@40647
   965
bulwahn@40647
   966
end
bulwahn@40647
   967
Andreas@57818
   968
instantiation finite_5 :: complete_lattice
Andreas@57818
   969
begin
Andreas@57818
   970
wenzelm@60758
   971
text \<open>The non-distributive pentagon lattice $N_5$\<close>
Andreas@57818
   972
Andreas@57818
   973
definition
Andreas@57818
   974
  "x < y \<longleftrightarrow> (case (x, y) of
Andreas@57818
   975
     (a\<^sub>1, a\<^sub>1) \<Rightarrow> False | (a\<^sub>1, _) \<Rightarrow> True
Andreas@57818
   976
   | (a\<^sub>2, a\<^sub>3) \<Rightarrow> True  | (a\<^sub>2, a\<^sub>5) \<Rightarrow> True
Andreas@57818
   977
   | (a\<^sub>3, a\<^sub>5) \<Rightarrow> True
Andreas@57818
   978
   | (a\<^sub>4, a\<^sub>5) \<Rightarrow> True  | _ \<Rightarrow> False)"
Andreas@57818
   979
Andreas@57818
   980
definition
Andreas@57818
   981
  "x \<le> y \<longleftrightarrow> (case (x, y) of
Andreas@57818
   982
     (a\<^sub>1, _) \<Rightarrow> True
Andreas@57818
   983
   | (a\<^sub>2, a\<^sub>2) \<Rightarrow> True | (a\<^sub>2, a\<^sub>3) \<Rightarrow> True | (a\<^sub>2, a\<^sub>5) \<Rightarrow> True
Andreas@57818
   984
   | (a\<^sub>3, a\<^sub>3) \<Rightarrow> True | (a\<^sub>3, a\<^sub>5) \<Rightarrow> True
Andreas@57818
   985
   | (a\<^sub>4, a\<^sub>4) \<Rightarrow> True | (a\<^sub>4, a\<^sub>5) \<Rightarrow> True
Andreas@57818
   986
   | (a\<^sub>5, a\<^sub>5) \<Rightarrow> True | _ \<Rightarrow> False)"
Andreas@57818
   987
Andreas@57818
   988
definition
Andreas@57818
   989
  "\<Sqinter>A = 
Andreas@57818
   990
  (if a\<^sub>1 \<in> A \<or> a\<^sub>4 \<in> A \<and> (a\<^sub>2 \<in> A \<or> a\<^sub>3 \<in> A) then a\<^sub>1
Andreas@57818
   991
   else if a\<^sub>2 \<in> A then a\<^sub>2
Andreas@57818
   992
   else if a\<^sub>3 \<in> A then a\<^sub>3
Andreas@57818
   993
   else if a\<^sub>4 \<in> A then a\<^sub>4
Andreas@57818
   994
   else a\<^sub>5)"
Andreas@57818
   995
definition
Andreas@57818
   996
  "\<Squnion>A = 
Andreas@57818
   997
  (if a\<^sub>5 \<in> A \<or> a\<^sub>4 \<in> A \<and> (a\<^sub>2 \<in> A \<or> a\<^sub>3 \<in> A) then a\<^sub>5
Andreas@57818
   998
   else if a\<^sub>3 \<in> A then a\<^sub>3
Andreas@57818
   999
   else if a\<^sub>2 \<in> A then a\<^sub>2
Andreas@57818
  1000
   else if a\<^sub>4 \<in> A then a\<^sub>4
Andreas@57818
  1001
   else a\<^sub>1)"
Andreas@57818
  1002
definition [simp]: "bot = a\<^sub>1"
Andreas@57818
  1003
definition [simp]: "top = a\<^sub>5"
Andreas@57818
  1004
definition
Andreas@57818
  1005
  "x \<sqinter> y = (case (x, y) of
Andreas@57818
  1006
     (a\<^sub>1, _) \<Rightarrow> a\<^sub>1 | (_, a\<^sub>1) \<Rightarrow> a\<^sub>1 | (a\<^sub>2, a\<^sub>4) \<Rightarrow> a\<^sub>1 | (a\<^sub>4, a\<^sub>2) \<Rightarrow> a\<^sub>1 | (a\<^sub>3, a\<^sub>4) \<Rightarrow> a\<^sub>1 | (a\<^sub>4, a\<^sub>3) \<Rightarrow> a\<^sub>1
Andreas@57818
  1007
   | (a\<^sub>2, _) \<Rightarrow> a\<^sub>2 | (_, a\<^sub>2) \<Rightarrow> a\<^sub>2
Andreas@57818
  1008
   | (a\<^sub>3, _) \<Rightarrow> a\<^sub>3 | (_, a\<^sub>3) \<Rightarrow> a\<^sub>3
Andreas@57818
  1009
   | (a\<^sub>4, _) \<Rightarrow> a\<^sub>4 | (_, a\<^sub>4) \<Rightarrow> a\<^sub>4
Andreas@57818
  1010
   | _ \<Rightarrow> a\<^sub>5)"
Andreas@57818
  1011
definition
Andreas@57818
  1012
  "x \<squnion> y = (case (x, y) of
Andreas@57818
  1013
     (a\<^sub>5, _) \<Rightarrow> a\<^sub>5 | (_, a\<^sub>5) \<Rightarrow> a\<^sub>5 | (a\<^sub>2, a\<^sub>4) \<Rightarrow> a\<^sub>5 | (a\<^sub>4, a\<^sub>2) \<Rightarrow> a\<^sub>5 | (a\<^sub>3, a\<^sub>4) \<Rightarrow> a\<^sub>5 | (a\<^sub>4, a\<^sub>3) \<Rightarrow> a\<^sub>5
Andreas@57818
  1014
   | (a\<^sub>3, _) \<Rightarrow> a\<^sub>3 | (_, a\<^sub>3) \<Rightarrow> a\<^sub>3
Andreas@57818
  1015
   | (a\<^sub>2, _) \<Rightarrow> a\<^sub>2 | (_, a\<^sub>2) \<Rightarrow> a\<^sub>2
Andreas@57818
  1016
   | (a\<^sub>4, _) \<Rightarrow> a\<^sub>4 | (_, a\<^sub>4) \<Rightarrow> a\<^sub>4
Andreas@57818
  1017
   | _ \<Rightarrow> a\<^sub>1)"
Andreas@57818
  1018
Andreas@57818
  1019
instance 
Andreas@57818
  1020
proof intro_classes
Andreas@57818
  1021
  fix A and z :: finite_5
Andreas@57818
  1022
  assume *: "\<And>x. x \<in> A \<Longrightarrow> z \<le> x"
Andreas@57818
  1023
  show "z \<le> \<Sqinter>A"
Andreas@57818
  1024
    by(auto simp add: less_eq_finite_5_def Inf_finite_5_def split: finite_5.splits split_if_asm dest!: *)
Andreas@57818
  1025
next
Andreas@57818
  1026
  fix A and z :: finite_5
Andreas@57818
  1027
  assume *: "\<And>x. x \<in> A \<Longrightarrow> x \<le> z"
Andreas@57818
  1028
  show "\<Squnion>A \<le> z"
Andreas@57818
  1029
    by(auto simp add: less_eq_finite_5_def Sup_finite_5_def split: finite_5.splits split_if_asm dest!: *)
Andreas@57818
  1030
qed(auto simp add: less_eq_finite_5_def less_finite_5_def inf_finite_5_def sup_finite_5_def Inf_finite_5_def Sup_finite_5_def split: finite_5.splits split_if_asm)
Andreas@57818
  1031
Andreas@57818
  1032
end
Andreas@57818
  1033
wenzelm@53015
  1034
hide_const (open) a\<^sub>1 a\<^sub>2 a\<^sub>3 a\<^sub>4 a\<^sub>5
bulwahn@46352
  1035
haftmann@49948
  1036
wenzelm@60758
  1037
subsection \<open>Closing up\<close>
bulwahn@40657
  1038
bulwahn@41085
  1039
hide_type (open) finite_1 finite_2 finite_3 finite_4 finite_5
haftmann@49948
  1040
hide_const (open) enum enum_all enum_ex all_n_lists ex_n_lists ntrancl
bulwahn@40647
  1041
bulwahn@40647
  1042
end