author | nipkow |
Thu, 12 Jun 2014 18:47:16 +0200 | |
changeset 57247 | 8191ccf6a1bd |
parent 55088 | 57c82e01022b |
child 57818 | 51aa30c9ee4e |
permissions | -rw-r--r-- |
31596 | 1 |
(* 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 |
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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_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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-- {* tailored towards simple instantiation *} |
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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 {* Implementations using @{class enum} *} |
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subsubsection {* Unbounded operations and quantifiers *} |
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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 {* An executable choice operator *} |
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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 `P x` in_enum[of x, unfolded enum_eq] `x \<noteq> a` 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 {* 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 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\<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: fun_eq_iff le_fun_def order_less_le) |
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subsubsection {* Operations on relations *} |
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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 {* Bounded accessible part *} |
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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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190 |
|
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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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|
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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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205 |
|
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206 |
lemma acc_subseteq_bacc: |
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207 |
assumes "finite r" |
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208 |
shows "Wellfounded.acc r \<subseteq> (\<Union>n. bacc r n)" |
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209 |
proof |
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210 |
fix x |
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211 |
assume "x : Wellfounded.acc r" |
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212 |
then have "\<exists> n. x : bacc r n" |
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213 |
proof (induct x arbitrary: rule: acc.induct) |
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214 |
case (accI x) |
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215 |
then have "\<forall>y. \<exists> n. (y, x) \<in> r --> y : bacc r n" by simp |
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216 |
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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217 |
obtain n where "\<And>y. (y, x) : r \<Longrightarrow> y : bacc r n" |
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|
218 |
proof |
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|
219 |
fix y assume y: "(y, x) : r" |
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220 |
with n have "y : bacc r (n y)" by auto |
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221 |
moreover have "n y <= Max ((%(y, x). n y) ` r)" |
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|
222 |
using y `finite r` by (auto intro!: Max_ge) |
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|
223 |
note bacc_mono[OF this, of r] |
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224 |
ultimately show "y : bacc r (Max ((%(y, x). n y) ` r))" by auto |
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225 |
qed |
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226 |
then show ?case |
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227 |
by (auto simp add: Let_def intro!: exI[of _ "Suc n"]) |
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228 |
qed |
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|
229 |
then show "x : (UN n. bacc r n)" by auto |
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230 |
qed |
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|
231 |
|
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|
232 |
lemma acc_bacc_eq: |
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233 |
fixes A :: "('a :: finite \<times> 'a) set" |
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|
234 |
assumes "finite A" |
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|
235 |
shows "Wellfounded.acc A = bacc A (card (UNIV :: 'a set))" |
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|
236 |
using assms by (metis acc_subseteq_bacc bacc_subseteq_acc bacc_upper_bound order_eq_iff) |
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|
237 |
|
49949 | 238 |
lemma [code]: |
239 |
fixes xs :: "('a::finite \<times> 'a) list" |
|
54295 | 240 |
shows "Wellfounded.acc (set xs) = bacc (set xs) (card_UNIV TYPE('a))" |
49949 | 241 |
by (simp add: card_UNIV_def acc_bacc_eq) |
242 |
||
26348 | 243 |
|
49949 | 244 |
subsection {* Default instances for @{class enum} *} |
26348 | 245 |
|
26444 | 246 |
lemma map_of_zip_enum_is_Some: |
247 |
assumes "length ys = length (enum \<Colon> 'a\<Colon>enum list)" |
|
248 |
shows "\<exists>y. map_of (zip (enum \<Colon> 'a\<Colon>enum list) ys) x = Some y" |
|
249 |
proof - |
|
250 |
from assms have "x \<in> set (enum \<Colon> 'a\<Colon>enum list) \<longleftrightarrow> |
|
251 |
(\<exists>y. map_of (zip (enum \<Colon> 'a\<Colon>enum list) ys) x = Some y)" |
|
252 |
by (auto intro!: map_of_zip_is_Some) |
|
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253 |
then show ?thesis using enum_UNIV by auto |
26444 | 254 |
qed |
255 |
||
256 |
lemma map_of_zip_enum_inject: |
|
257 |
fixes xs ys :: "'b\<Colon>enum list" |
|
258 |
assumes length: "length xs = length (enum \<Colon> 'a\<Colon>enum list)" |
|
259 |
"length ys = length (enum \<Colon> 'a\<Colon>enum list)" |
|
260 |
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)" |
|
261 |
shows "xs = ys" |
|
262 |
proof - |
|
263 |
have "map_of (zip (enum \<Colon> 'a list) xs) = map_of (zip (enum \<Colon> 'a list) ys)" |
|
264 |
proof |
|
265 |
fix x :: 'a |
|
266 |
from length map_of_zip_enum_is_Some obtain y1 y2 |
|
267 |
where "map_of (zip (enum \<Colon> 'a list) xs) x = Some y1" |
|
268 |
and "map_of (zip (enum \<Colon> 'a list) ys) x = Some y2" by blast |
|
47230 | 269 |
moreover from map_of |
270 |
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)" |
|
26444 | 271 |
by (auto dest: fun_cong) |
272 |
ultimately show "map_of (zip (enum \<Colon> 'a\<Colon>enum list) xs) x = map_of (zip (enum \<Colon> 'a\<Colon>enum list) ys) x" |
|
273 |
by simp |
|
274 |
qed |
|
275 |
with length enum_distinct show "xs = ys" by (rule map_of_zip_inject) |
|
276 |
qed |
|
277 |
||
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278 |
definition all_n_lists :: "(('a :: enum) list \<Rightarrow> bool) \<Rightarrow> nat \<Rightarrow> bool" |
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279 |
where |
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|
280 |
"all_n_lists P n \<longleftrightarrow> (\<forall>xs \<in> set (List.n_lists n enum). P xs)" |
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|
281 |
|
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|
282 |
lemma [code]: |
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283 |
"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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|
284 |
unfolding all_n_lists_def enum_all |
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|
285 |
by (cases n) (auto simp add: enum_UNIV) |
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|
286 |
|
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|
287 |
definition ex_n_lists :: "(('a :: enum) list \<Rightarrow> bool) \<Rightarrow> nat \<Rightarrow> bool" |
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|
288 |
where |
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|
289 |
"ex_n_lists P n \<longleftrightarrow> (\<exists>xs \<in> set (List.n_lists n enum). P xs)" |
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|
290 |
|
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|
291 |
lemma [code]: |
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|
292 |
"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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|
293 |
unfolding ex_n_lists_def enum_ex |
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|
294 |
by (cases n) (auto simp add: enum_UNIV) |
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|
295 |
|
26444 | 296 |
instantiation "fun" :: (enum, enum) enum |
297 |
begin |
|
298 |
||
299 |
definition |
|
49948
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moved quite generic material from theory Enum to more appropriate places
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|
300 |
"enum = map (\<lambda>ys. the o map_of (zip (enum\<Colon>'a list) ys)) (List.n_lists (length (enum\<Colon>'a\<Colon>enum list)) enum)" |
26444 | 301 |
|
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|
302 |
definition |
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|
303 |
"enum_all P = all_n_lists (\<lambda>bs. P (the o map_of (zip enum bs))) (length (enum :: 'a list))" |
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|
304 |
|
051251fde456
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|
305 |
definition |
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|
306 |
"enum_ex P = ex_n_lists (\<lambda>bs. P (the o map_of (zip enum bs))) (length (enum :: 'a list))" |
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|
307 |
|
26444 | 308 |
instance proof |
309 |
show "UNIV = set (enum \<Colon> ('a \<Rightarrow> 'b) list)" |
|
310 |
proof (rule UNIV_eq_I) |
|
311 |
fix f :: "'a \<Rightarrow> 'b" |
|
312 |
have "f = the \<circ> map_of (zip (enum \<Colon> 'a\<Colon>enum list) (map f enum))" |
|
40683 | 313 |
by (auto simp add: map_of_zip_map fun_eq_iff intro: in_enum) |
26444 | 314 |
then show "f \<in> set enum" |
40683 | 315 |
by (auto simp add: enum_fun_def set_n_lists intro: in_enum) |
26444 | 316 |
qed |
317 |
next |
|
318 |
from map_of_zip_enum_inject |
|
319 |
show "distinct (enum \<Colon> ('a \<Rightarrow> 'b) list)" |
|
320 |
by (auto intro!: inj_onI simp add: enum_fun_def |
|
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|
321 |
distinct_map distinct_n_lists enum_distinct set_n_lists) |
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|
322 |
next |
051251fde456
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changeset
|
323 |
fix P |
49950
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|
324 |
show "enum_all (P :: ('a \<Rightarrow> 'b) \<Rightarrow> bool) = Ball UNIV P" |
41078
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|
325 |
proof |
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|
326 |
assume "enum_all P" |
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changeset
|
327 |
show "Ball UNIV P" |
41078
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|
328 |
proof |
051251fde456
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changeset
|
329 |
fix f :: "'a \<Rightarrow> 'b" |
051251fde456
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changeset
|
330 |
have f: "f = the \<circ> map_of (zip (enum \<Colon> 'a\<Colon>enum list) (map f enum))" |
051251fde456
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changeset
|
331 |
by (auto simp add: map_of_zip_map fun_eq_iff intro: in_enum) |
051251fde456
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changeset
|
332 |
from `enum_all P` have "P (the \<circ> map_of (zip enum (map f enum)))" |
051251fde456
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changeset
|
333 |
unfolding enum_all_fun_def all_n_lists_def |
051251fde456
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changeset
|
334 |
apply (simp add: set_n_lists) |
051251fde456
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changeset
|
335 |
apply (erule_tac x="map f enum" in allE) |
051251fde456
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changeset
|
336 |
apply (auto intro!: in_enum) |
051251fde456
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changeset
|
337 |
done |
051251fde456
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changeset
|
338 |
from this f show "P f" by auto |
051251fde456
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|
339 |
qed |
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changeset
|
340 |
next |
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changeset
|
341 |
assume "Ball UNIV P" |
41078
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changeset
|
342 |
from this show "enum_all P" |
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changeset
|
343 |
unfolding enum_all_fun_def all_n_lists_def by auto |
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changeset
|
344 |
qed |
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changeset
|
345 |
next |
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changeset
|
346 |
fix P |
49950
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|
347 |
show "enum_ex (P :: ('a \<Rightarrow> 'b) \<Rightarrow> bool) = Bex UNIV P" |
41078
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|
348 |
proof |
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|
349 |
assume "enum_ex P" |
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|
350 |
from this show "Bex UNIV P" |
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|
351 |
unfolding enum_ex_fun_def ex_n_lists_def by auto |
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changeset
|
352 |
next |
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changeset
|
353 |
assume "Bex UNIV P" |
41078
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|
354 |
from this obtain f where "P f" .. |
051251fde456
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changeset
|
355 |
have f: "f = the \<circ> map_of (zip (enum \<Colon> 'a\<Colon>enum list) (map f enum))" |
051251fde456
adding more efficient implementations for quantifiers in Enum
bulwahn
parents:
40900
diff
changeset
|
356 |
by (auto simp add: map_of_zip_map fun_eq_iff intro: in_enum) |
051251fde456
adding more efficient implementations for quantifiers in Enum
bulwahn
parents:
40900
diff
changeset
|
357 |
from `P f` this have "P (the \<circ> map_of (zip (enum \<Colon> 'a\<Colon>enum list) (map f enum)))" |
051251fde456
adding more efficient implementations for quantifiers in Enum
bulwahn
parents:
40900
diff
changeset
|
358 |
by auto |
051251fde456
adding more efficient implementations for quantifiers in Enum
bulwahn
parents:
40900
diff
changeset
|
359 |
from this show "enum_ex P" |
051251fde456
adding more efficient implementations for quantifiers in Enum
bulwahn
parents:
40900
diff
changeset
|
360 |
unfolding enum_ex_fun_def ex_n_lists_def |
051251fde456
adding more efficient implementations for quantifiers in Enum
bulwahn
parents:
40900
diff
changeset
|
361 |
apply (auto simp add: set_n_lists) |
051251fde456
adding more efficient implementations for quantifiers in Enum
bulwahn
parents:
40900
diff
changeset
|
362 |
apply (rule_tac x="map f enum" in exI) |
051251fde456
adding more efficient implementations for quantifiers in Enum
bulwahn
parents:
40900
diff
changeset
|
363 |
apply (auto intro!: in_enum) |
051251fde456
adding more efficient implementations for quantifiers in Enum
bulwahn
parents:
40900
diff
changeset
|
364 |
done |
051251fde456
adding more efficient implementations for quantifiers in Enum
bulwahn
parents:
40900
diff
changeset
|
365 |
qed |
26444 | 366 |
qed |
367 |
||
368 |
end |
|
369 |
||
38857
97775f3e8722
renamed class/constant eq to equal; tuned some instantiations
haftmann
parents:
37765
diff
changeset
|
370 |
lemma enum_fun_code [code]: "enum = (let enum_a = (enum \<Colon> 'a\<Colon>{enum, equal} list) |
49948
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
48123
diff
changeset
|
371 |
in map (\<lambda>ys. the o map_of (zip enum_a ys)) (List.n_lists (length enum_a) enum))" |
28245
9767dd8e1e54
celver code lemma avoid type ambiguity problem with Haskell
haftmann
parents:
27487
diff
changeset
|
372 |
by (simp add: enum_fun_def Let_def) |
26444 | 373 |
|
41078
051251fde456
adding more efficient implementations for quantifiers in Enum
bulwahn
parents:
40900
diff
changeset
|
374 |
lemma enum_all_fun_code [code]: |
051251fde456
adding more efficient implementations for quantifiers in Enum
bulwahn
parents:
40900
diff
changeset
|
375 |
"enum_all P = (let enum_a = (enum :: 'a::{enum, equal} list) |
051251fde456
adding more efficient implementations for quantifiers in Enum
bulwahn
parents:
40900
diff
changeset
|
376 |
in all_n_lists (\<lambda>bs. P (the o map_of (zip enum_a bs))) (length enum_a))" |
49950
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
377 |
by (simp only: enum_all_fun_def Let_def) |
41078
051251fde456
adding more efficient implementations for quantifiers in Enum
bulwahn
parents:
40900
diff
changeset
|
378 |
|
051251fde456
adding more efficient implementations for quantifiers in Enum
bulwahn
parents:
40900
diff
changeset
|
379 |
lemma enum_ex_fun_code [code]: |
051251fde456
adding more efficient implementations for quantifiers in Enum
bulwahn
parents:
40900
diff
changeset
|
380 |
"enum_ex P = (let enum_a = (enum :: 'a::{enum, equal} list) |
051251fde456
adding more efficient implementations for quantifiers in Enum
bulwahn
parents:
40900
diff
changeset
|
381 |
in ex_n_lists (\<lambda>bs. P (the o map_of (zip enum_a bs))) (length enum_a))" |
49950
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
382 |
by (simp only: enum_ex_fun_def Let_def) |
45963
1c7e6454883e
enum type class instance for `set`; dropped misfitting code lemma for trancl
haftmann
parents:
45144
diff
changeset
|
383 |
|
1c7e6454883e
enum type class instance for `set`; dropped misfitting code lemma for trancl
haftmann
parents:
45144
diff
changeset
|
384 |
instantiation set :: (enum) enum |
1c7e6454883e
enum type class instance for `set`; dropped misfitting code lemma for trancl
haftmann
parents:
45144
diff
changeset
|
385 |
begin |
1c7e6454883e
enum type class instance for `set`; dropped misfitting code lemma for trancl
haftmann
parents:
45144
diff
changeset
|
386 |
|
1c7e6454883e
enum type class instance for `set`; dropped misfitting code lemma for trancl
haftmann
parents:
45144
diff
changeset
|
387 |
definition |
1c7e6454883e
enum type class instance for `set`; dropped misfitting code lemma for trancl
haftmann
parents:
45144
diff
changeset
|
388 |
"enum = map set (sublists enum)" |
1c7e6454883e
enum type class instance for `set`; dropped misfitting code lemma for trancl
haftmann
parents:
45144
diff
changeset
|
389 |
|
1c7e6454883e
enum type class instance for `set`; dropped misfitting code lemma for trancl
haftmann
parents:
45144
diff
changeset
|
390 |
definition |
1c7e6454883e
enum type class instance for `set`; dropped misfitting code lemma for trancl
haftmann
parents:
45144
diff
changeset
|
391 |
"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
|
392 |
|
1c7e6454883e
enum type class instance for `set`; dropped misfitting code lemma for trancl
haftmann
parents:
45144
diff
changeset
|
393 |
definition |
1c7e6454883e
enum type class instance for `set`; dropped misfitting code lemma for trancl
haftmann
parents:
45144
diff
changeset
|
394 |
"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
|
395 |
|
1c7e6454883e
enum type class instance for `set`; dropped misfitting code lemma for trancl
haftmann
parents:
45144
diff
changeset
|
396 |
instance proof |
1c7e6454883e
enum type class instance for `set`; dropped misfitting code lemma for trancl
haftmann
parents:
45144
diff
changeset
|
397 |
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
|
398 |
enum_distinct enum_UNIV) |
29024 | 399 |
|
400 |
end |
|
401 |
||
49950
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
402 |
instantiation unit :: enum |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
403 |
begin |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
404 |
|
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
405 |
definition |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
406 |
"enum = [()]" |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
407 |
|
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
408 |
definition |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
409 |
"enum_all P = P ()" |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
410 |
|
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
411 |
definition |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
412 |
"enum_ex P = P ()" |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
413 |
|
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
414 |
instance proof |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
415 |
qed (auto simp add: enum_unit_def enum_all_unit_def enum_ex_unit_def) |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
416 |
|
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
417 |
end |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
418 |
|
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
419 |
instantiation bool :: enum |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
420 |
begin |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
421 |
|
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
422 |
definition |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
423 |
"enum = [False, True]" |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
424 |
|
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
425 |
definition |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
426 |
"enum_all P \<longleftrightarrow> P False \<and> P True" |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
427 |
|
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
428 |
definition |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
429 |
"enum_ex P \<longleftrightarrow> P False \<or> P True" |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
430 |
|
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
431 |
instance proof |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
432 |
qed (simp_all only: enum_bool_def enum_all_bool_def enum_ex_bool_def UNIV_bool, simp_all) |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
433 |
|
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
434 |
end |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
435 |
|
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
436 |
instantiation prod :: (enum, enum) enum |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
437 |
begin |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
438 |
|
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
439 |
definition |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
440 |
"enum = List.product enum enum" |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
441 |
|
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
442 |
definition |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
443 |
"enum_all P = enum_all (%x. enum_all (%y. P (x, y)))" |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
444 |
|
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
445 |
definition |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
446 |
"enum_ex P = enum_ex (%x. enum_ex (%y. P (x, y)))" |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
447 |
|
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
448 |
|
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
449 |
instance by default |
57247 | 450 |
(simp_all add: enum_prod_def distinct_product |
49950
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
451 |
enum_UNIV enum_distinct enum_all_prod_def enum_ex_prod_def) |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
452 |
|
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
453 |
end |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
454 |
|
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
455 |
instantiation sum :: (enum, enum) enum |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
456 |
begin |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
457 |
|
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
458 |
definition |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
459 |
"enum = map Inl enum @ map Inr enum" |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
460 |
|
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
461 |
definition |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
462 |
"enum_all P \<longleftrightarrow> enum_all (\<lambda>x. P (Inl x)) \<and> enum_all (\<lambda>x. P (Inr x))" |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
463 |
|
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
464 |
definition |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
465 |
"enum_ex P \<longleftrightarrow> enum_ex (\<lambda>x. P (Inl x)) \<or> enum_ex (\<lambda>x. P (Inr x))" |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
466 |
|
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
467 |
instance proof |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
468 |
qed (simp_all only: enum_sum_def enum_all_sum_def enum_ex_sum_def UNIV_sum, |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
469 |
auto simp add: enum_UNIV distinct_map enum_distinct) |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
470 |
|
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
471 |
end |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
472 |
|
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
473 |
instantiation option :: (enum) enum |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
474 |
begin |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
475 |
|
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
476 |
definition |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
477 |
"enum = None # map Some enum" |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
478 |
|
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
479 |
definition |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
480 |
"enum_all P \<longleftrightarrow> P None \<and> enum_all (\<lambda>x. P (Some x))" |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
481 |
|
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
482 |
definition |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
483 |
"enum_ex P \<longleftrightarrow> P None \<or> enum_ex (\<lambda>x. P (Some x))" |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
484 |
|
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
485 |
instance proof |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
486 |
qed (simp_all only: enum_option_def enum_all_option_def enum_ex_option_def UNIV_option_conv, |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
487 |
auto simp add: distinct_map enum_UNIV enum_distinct) |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
488 |
|
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
489 |
end |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
490 |
|
45963
1c7e6454883e
enum type class instance for `set`; dropped misfitting code lemma for trancl
haftmann
parents:
45144
diff
changeset
|
491 |
|
40647 | 492 |
subsection {* Small finite types *} |
493 |
||
494 |
text {* We define small finite types for the use in Quickcheck *} |
|
495 |
||
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52435
diff
changeset
|
496 |
datatype finite_1 = a\<^sub>1 |
40647 | 497 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52435
diff
changeset
|
498 |
notation (output) a\<^sub>1 ("a\<^sub>1") |
40900
1d5f76d79856
adding shorter output syntax for the finite types of quickcheck
bulwahn
parents:
40898
diff
changeset
|
499 |
|
49950
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
500 |
lemma UNIV_finite_1: |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52435
diff
changeset
|
501 |
"UNIV = {a\<^sub>1}" |
49950
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
502 |
by (auto intro: finite_1.exhaust) |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
503 |
|
40647 | 504 |
instantiation finite_1 :: enum |
505 |
begin |
|
506 |
||
507 |
definition |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52435
diff
changeset
|
508 |
"enum = [a\<^sub>1]" |
40647 | 509 |
|
41078
051251fde456
adding more efficient implementations for quantifiers in Enum
bulwahn
parents:
40900
diff
changeset
|
510 |
definition |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52435
diff
changeset
|
511 |
"enum_all P = P a\<^sub>1" |
41078
051251fde456
adding more efficient implementations for quantifiers in Enum
bulwahn
parents:
40900
diff
changeset
|
512 |
|
051251fde456
adding more efficient implementations for quantifiers in Enum
bulwahn
parents:
40900
diff
changeset
|
513 |
definition |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52435
diff
changeset
|
514 |
"enum_ex P = P a\<^sub>1" |
41078
051251fde456
adding more efficient implementations for quantifiers in Enum
bulwahn
parents:
40900
diff
changeset
|
515 |
|
40647 | 516 |
instance proof |
49950
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
517 |
qed (simp_all only: enum_finite_1_def enum_all_finite_1_def enum_ex_finite_1_def UNIV_finite_1, simp_all) |
40647 | 518 |
|
29024 | 519 |
end |
40647 | 520 |
|
40651
9752ba7348b5
adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents:
40650
diff
changeset
|
521 |
instantiation finite_1 :: linorder |
9752ba7348b5
adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents:
40650
diff
changeset
|
522 |
begin |
9752ba7348b5
adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents:
40650
diff
changeset
|
523 |
|
49950
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
524 |
definition less_finite_1 :: "finite_1 \<Rightarrow> finite_1 \<Rightarrow> bool" |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
525 |
where |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
526 |
"x < (y :: finite_1) \<longleftrightarrow> False" |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
527 |
|
40651
9752ba7348b5
adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents:
40650
diff
changeset
|
528 |
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
|
529 |
where |
49950
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
530 |
"x \<le> (y :: finite_1) \<longleftrightarrow> True" |
40651
9752ba7348b5
adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents:
40650
diff
changeset
|
531 |
|
9752ba7348b5
adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents:
40650
diff
changeset
|
532 |
instance |
9752ba7348b5
adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents:
40650
diff
changeset
|
533 |
apply (intro_classes) |
9752ba7348b5
adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents:
40650
diff
changeset
|
534 |
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
|
535 |
apply (metis finite_1.exhaust) |
9752ba7348b5
adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents:
40650
diff
changeset
|
536 |
done |
9752ba7348b5
adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents:
40650
diff
changeset
|
537 |
|
9752ba7348b5
adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents:
40650
diff
changeset
|
538 |
end |
9752ba7348b5
adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents:
40650
diff
changeset
|
539 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52435
diff
changeset
|
540 |
hide_const (open) a\<^sub>1 |
40657 | 541 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52435
diff
changeset
|
542 |
datatype finite_2 = a\<^sub>1 | a\<^sub>2 |
40647 | 543 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52435
diff
changeset
|
544 |
notation (output) a\<^sub>1 ("a\<^sub>1") |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52435
diff
changeset
|
545 |
notation (output) a\<^sub>2 ("a\<^sub>2") |
40900
1d5f76d79856
adding shorter output syntax for the finite types of quickcheck
bulwahn
parents:
40898
diff
changeset
|
546 |
|
49950
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
547 |
lemma UNIV_finite_2: |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52435
diff
changeset
|
548 |
"UNIV = {a\<^sub>1, a\<^sub>2}" |
49950
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
549 |
by (auto intro: finite_2.exhaust) |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
550 |
|
40647 | 551 |
instantiation finite_2 :: enum |
552 |
begin |
|
553 |
||
554 |
definition |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52435
diff
changeset
|
555 |
"enum = [a\<^sub>1, a\<^sub>2]" |
40647 | 556 |
|
41078
051251fde456
adding more efficient implementations for quantifiers in Enum
bulwahn
parents:
40900
diff
changeset
|
557 |
definition |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52435
diff
changeset
|
558 |
"enum_all P \<longleftrightarrow> P a\<^sub>1 \<and> P a\<^sub>2" |
41078
051251fde456
adding more efficient implementations for quantifiers in Enum
bulwahn
parents:
40900
diff
changeset
|
559 |
|
051251fde456
adding more efficient implementations for quantifiers in Enum
bulwahn
parents:
40900
diff
changeset
|
560 |
definition |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52435
diff
changeset
|
561 |
"enum_ex P \<longleftrightarrow> P a\<^sub>1 \<or> P a\<^sub>2" |
41078
051251fde456
adding more efficient implementations for quantifiers in Enum
bulwahn
parents:
40900
diff
changeset
|
562 |
|
40647 | 563 |
instance proof |
49950
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
564 |
qed (simp_all only: enum_finite_2_def enum_all_finite_2_def enum_ex_finite_2_def UNIV_finite_2, simp_all) |
40647 | 565 |
|
566 |
end |
|
567 |
||
40651
9752ba7348b5
adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents:
40650
diff
changeset
|
568 |
instantiation finite_2 :: linorder |
9752ba7348b5
adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents:
40650
diff
changeset
|
569 |
begin |
9752ba7348b5
adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents:
40650
diff
changeset
|
570 |
|
9752ba7348b5
adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents:
40650
diff
changeset
|
571 |
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
|
572 |
where |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52435
diff
changeset
|
573 |
"x < y \<longleftrightarrow> x = a\<^sub>1 \<and> y = a\<^sub>2" |
40651
9752ba7348b5
adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents:
40650
diff
changeset
|
574 |
|
9752ba7348b5
adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents:
40650
diff
changeset
|
575 |
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
|
576 |
where |
49950
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
577 |
"x \<le> y \<longleftrightarrow> x = y \<or> x < (y :: finite_2)" |
40651
9752ba7348b5
adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents:
40650
diff
changeset
|
578 |
|
9752ba7348b5
adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents:
40650
diff
changeset
|
579 |
instance |
9752ba7348b5
adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents:
40650
diff
changeset
|
580 |
apply (intro_classes) |
9752ba7348b5
adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents:
40650
diff
changeset
|
581 |
apply (auto simp add: less_finite_2_def less_eq_finite_2_def) |
49950
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
582 |
apply (metis finite_2.nchotomy)+ |
40651
9752ba7348b5
adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents:
40650
diff
changeset
|
583 |
done |
9752ba7348b5
adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents:
40650
diff
changeset
|
584 |
|
9752ba7348b5
adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents:
40650
diff
changeset
|
585 |
end |
9752ba7348b5
adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents:
40650
diff
changeset
|
586 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52435
diff
changeset
|
587 |
hide_const (open) a\<^sub>1 a\<^sub>2 |
40657 | 588 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52435
diff
changeset
|
589 |
datatype finite_3 = a\<^sub>1 | a\<^sub>2 | a\<^sub>3 |
40647 | 590 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52435
diff
changeset
|
591 |
notation (output) a\<^sub>1 ("a\<^sub>1") |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52435
diff
changeset
|
592 |
notation (output) a\<^sub>2 ("a\<^sub>2") |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52435
diff
changeset
|
593 |
notation (output) a\<^sub>3 ("a\<^sub>3") |
40900
1d5f76d79856
adding shorter output syntax for the finite types of quickcheck
bulwahn
parents:
40898
diff
changeset
|
594 |
|
49950
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
595 |
lemma UNIV_finite_3: |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52435
diff
changeset
|
596 |
"UNIV = {a\<^sub>1, a\<^sub>2, a\<^sub>3}" |
49950
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
597 |
by (auto intro: finite_3.exhaust) |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
598 |
|
40647 | 599 |
instantiation finite_3 :: enum |
600 |
begin |
|
601 |
||
602 |
definition |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52435
diff
changeset
|
603 |
"enum = [a\<^sub>1, a\<^sub>2, a\<^sub>3]" |
40647 | 604 |
|
41078
051251fde456
adding more efficient implementations for quantifiers in Enum
bulwahn
parents:
40900
diff
changeset
|
605 |
definition |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52435
diff
changeset
|
606 |
"enum_all P \<longleftrightarrow> P a\<^sub>1 \<and> P a\<^sub>2 \<and> P a\<^sub>3" |
41078
051251fde456
adding more efficient implementations for quantifiers in Enum
bulwahn
parents:
40900
diff
changeset
|
607 |
|
051251fde456
adding more efficient implementations for quantifiers in Enum
bulwahn
parents:
40900
diff
changeset
|
608 |
definition |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52435
diff
changeset
|
609 |
"enum_ex P \<longleftrightarrow> P a\<^sub>1 \<or> P a\<^sub>2 \<or> P a\<^sub>3" |
41078
051251fde456
adding more efficient implementations for quantifiers in Enum
bulwahn
parents:
40900
diff
changeset
|
610 |
|
40647 | 611 |
instance proof |
49950
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
612 |
qed (simp_all only: enum_finite_3_def enum_all_finite_3_def enum_ex_finite_3_def UNIV_finite_3, simp_all) |
40647 | 613 |
|
614 |
end |
|
615 |
||
40651
9752ba7348b5
adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents:
40650
diff
changeset
|
616 |
instantiation finite_3 :: linorder |
9752ba7348b5
adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents:
40650
diff
changeset
|
617 |
begin |
9752ba7348b5
adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents:
40650
diff
changeset
|
618 |
|
9752ba7348b5
adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents:
40650
diff
changeset
|
619 |
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
|
620 |
where |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52435
diff
changeset
|
621 |
"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)" |
40651
9752ba7348b5
adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents:
40650
diff
changeset
|
622 |
|
9752ba7348b5
adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents:
40650
diff
changeset
|
623 |
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
|
624 |
where |
49950
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
625 |
"x \<le> y \<longleftrightarrow> x = y \<or> x < (y :: finite_3)" |
40651
9752ba7348b5
adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents:
40650
diff
changeset
|
626 |
|
9752ba7348b5
adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents:
40650
diff
changeset
|
627 |
instance proof (intro_classes) |
9752ba7348b5
adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents:
40650
diff
changeset
|
628 |
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
|
629 |
|
9752ba7348b5
adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents:
40650
diff
changeset
|
630 |
end |
9752ba7348b5
adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents:
40650
diff
changeset
|
631 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52435
diff
changeset
|
632 |
hide_const (open) a\<^sub>1 a\<^sub>2 a\<^sub>3 |
40657 | 633 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52435
diff
changeset
|
634 |
datatype finite_4 = a\<^sub>1 | a\<^sub>2 | a\<^sub>3 | a\<^sub>4 |
40647 | 635 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52435
diff
changeset
|
636 |
notation (output) a\<^sub>1 ("a\<^sub>1") |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52435
diff
changeset
|
637 |
notation (output) a\<^sub>2 ("a\<^sub>2") |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52435
diff
changeset
|
638 |
notation (output) a\<^sub>3 ("a\<^sub>3") |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52435
diff
changeset
|
639 |
notation (output) a\<^sub>4 ("a\<^sub>4") |
40900
1d5f76d79856
adding shorter output syntax for the finite types of quickcheck
bulwahn
parents:
40898
diff
changeset
|
640 |
|
49950
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
641 |
lemma UNIV_finite_4: |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52435
diff
changeset
|
642 |
"UNIV = {a\<^sub>1, a\<^sub>2, a\<^sub>3, a\<^sub>4}" |
49950
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
643 |
by (auto intro: finite_4.exhaust) |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
644 |
|
40647 | 645 |
instantiation finite_4 :: enum |
646 |
begin |
|
647 |
||
648 |
definition |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52435
diff
changeset
|
649 |
"enum = [a\<^sub>1, a\<^sub>2, a\<^sub>3, a\<^sub>4]" |
40647 | 650 |
|
41078
051251fde456
adding more efficient implementations for quantifiers in Enum
bulwahn
parents:
40900
diff
changeset
|
651 |
definition |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52435
diff
changeset
|
652 |
"enum_all P \<longleftrightarrow> P a\<^sub>1 \<and> P a\<^sub>2 \<and> P a\<^sub>3 \<and> P a\<^sub>4" |
41078
051251fde456
adding more efficient implementations for quantifiers in Enum
bulwahn
parents:
40900
diff
changeset
|
653 |
|
051251fde456
adding more efficient implementations for quantifiers in Enum
bulwahn
parents:
40900
diff
changeset
|
654 |
definition |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52435
diff
changeset
|
655 |
"enum_ex P \<longleftrightarrow> P a\<^sub>1 \<or> P a\<^sub>2 \<or> P a\<^sub>3 \<or> P a\<^sub>4" |
41078
051251fde456
adding more efficient implementations for quantifiers in Enum
bulwahn
parents:
40900
diff
changeset
|
656 |
|
40647 | 657 |
instance proof |
49950
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
658 |
qed (simp_all only: enum_finite_4_def enum_all_finite_4_def enum_ex_finite_4_def UNIV_finite_4, simp_all) |
40647 | 659 |
|
660 |
end |
|
661 |
||
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52435
diff
changeset
|
662 |
hide_const (open) a\<^sub>1 a\<^sub>2 a\<^sub>3 a\<^sub>4 |
40651
9752ba7348b5
adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents:
40650
diff
changeset
|
663 |
|
9752ba7348b5
adding code equation for function equality; adding some instantiations for the finite types
bulwahn
parents:
40650
diff
changeset
|
664 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52435
diff
changeset
|
665 |
datatype finite_5 = a\<^sub>1 | a\<^sub>2 | a\<^sub>3 | a\<^sub>4 | a\<^sub>5 |
40647 | 666 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52435
diff
changeset
|
667 |
notation (output) a\<^sub>1 ("a\<^sub>1") |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52435
diff
changeset
|
668 |
notation (output) a\<^sub>2 ("a\<^sub>2") |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52435
diff
changeset
|
669 |
notation (output) a\<^sub>3 ("a\<^sub>3") |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52435
diff
changeset
|
670 |
notation (output) a\<^sub>4 ("a\<^sub>4") |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52435
diff
changeset
|
671 |
notation (output) a\<^sub>5 ("a\<^sub>5") |
40900
1d5f76d79856
adding shorter output syntax for the finite types of quickcheck
bulwahn
parents:
40898
diff
changeset
|
672 |
|
49950
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
673 |
lemma UNIV_finite_5: |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52435
diff
changeset
|
674 |
"UNIV = {a\<^sub>1, a\<^sub>2, a\<^sub>3, a\<^sub>4, a\<^sub>5}" |
49950
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
675 |
by (auto intro: finite_5.exhaust) |
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
676 |
|
40647 | 677 |
instantiation finite_5 :: enum |
678 |
begin |
|
679 |
||
680 |
definition |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52435
diff
changeset
|
681 |
"enum = [a\<^sub>1, a\<^sub>2, a\<^sub>3, a\<^sub>4, a\<^sub>5]" |
40647 | 682 |
|
41078
051251fde456
adding more efficient implementations for quantifiers in Enum
bulwahn
parents:
40900
diff
changeset
|
683 |
definition |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52435
diff
changeset
|
684 |
"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" |
41078
051251fde456
adding more efficient implementations for quantifiers in Enum
bulwahn
parents:
40900
diff
changeset
|
685 |
|
051251fde456
adding more efficient implementations for quantifiers in Enum
bulwahn
parents:
40900
diff
changeset
|
686 |
definition |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52435
diff
changeset
|
687 |
"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" |
41078
051251fde456
adding more efficient implementations for quantifiers in Enum
bulwahn
parents:
40900
diff
changeset
|
688 |
|
40647 | 689 |
instance proof |
49950
cd882d53ba6b
tailored enum specification towards simple instantiation;
haftmann
parents:
49949
diff
changeset
|
690 |
qed (simp_all only: enum_finite_5_def enum_all_finite_5_def enum_ex_finite_5_def UNIV_finite_5, simp_all) |
40647 | 691 |
|
692 |
end |
|
693 |
||
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52435
diff
changeset
|
694 |
hide_const (open) a\<^sub>1 a\<^sub>2 a\<^sub>3 a\<^sub>4 a\<^sub>5 |
46352
73b03235799a
an executable version of accessible part (only for finite types yet)
bulwahn
parents:
46336
diff
changeset
|
695 |
|
49948
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
48123
diff
changeset
|
696 |
|
46352
73b03235799a
an executable version of accessible part (only for finite types yet)
bulwahn
parents:
46336
diff
changeset
|
697 |
subsection {* Closing up *} |
40657 | 698 |
|
41085
a549ff1d4070
adding a smarter enumeration scheme for finite functions
bulwahn
parents:
41078
diff
changeset
|
699 |
hide_type (open) finite_1 finite_2 finite_3 finite_4 finite_5 |
49948
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
48123
diff
changeset
|
700 |
hide_const (open) enum enum_all enum_ex all_n_lists ex_n_lists ntrancl |
40647 | 701 |
|
702 |
end |