author | wenzelm |
Tue, 09 Apr 2013 21:39:55 +0200 | |
changeset 51667 | 354c23ef2784 |
parent 51188 | 9b5bf1a9a710 |
child 52143 | 36ffe23b25f8 |
permissions | -rw-r--r-- |
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(* Title: HOL/Library/Cardinality.thy |
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Author: Brian Huffman, Andreas Lochbihler |
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*) |
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header {* Cardinality of types *} |
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theory Cardinality |
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imports Phantom_Type |
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begin |
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|
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subsection {* Preliminary lemmas *} |
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(* These should be moved elsewhere *) |
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|
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lemma (in type_definition) univ: |
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"UNIV = Abs ` A" |
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proof |
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show "Abs ` A \<subseteq> UNIV" by (rule subset_UNIV) |
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show "UNIV \<subseteq> Abs ` A" |
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proof |
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fix x :: 'b |
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have "x = Abs (Rep x)" by (rule Rep_inverse [symmetric]) |
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moreover have "Rep x \<in> A" by (rule Rep) |
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ultimately show "x \<in> Abs ` A" by (rule image_eqI) |
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qed |
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qed |
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|
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lemma (in type_definition) card: "card (UNIV :: 'b set) = card A" |
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by (simp add: univ card_image inj_on_def Abs_inject) |
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lemma finite_range_Some: "finite (range (Some :: 'a \<Rightarrow> 'a option)) = finite (UNIV :: 'a set)" |
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by(auto dest: finite_imageD intro: inj_Some) |
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||
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lemma infinite_literal: "\<not> finite (UNIV :: String.literal set)" |
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proof - |
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have "inj STR" by(auto intro: injI) |
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thus ?thesis |
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by(auto simp add: type_definition.univ[OF type_definition_literal] infinite_UNIV_listI dest: finite_imageD) |
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qed |
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|
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subsection {* Cardinalities of types *} |
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|
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syntax "_type_card" :: "type => nat" ("(1CARD/(1'(_')))") |
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translations "CARD('t)" => "CONST card (CONST UNIV \<Colon> 't set)" |
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typed_print_translation (advanced) {* |
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let |
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fun card_univ_tr' ctxt _ [Const (@{const_syntax UNIV}, Type (_, [T]))] = |
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Syntax.const @{syntax_const "_type_card"} $ Syntax_Phases.term_of_typ ctxt T |
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in [(@{const_syntax card}, card_univ_tr')] end |
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*} |
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lemma card_prod [simp]: "CARD('a \<times> 'b) = CARD('a) * CARD('b)" |
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unfolding UNIV_Times_UNIV [symmetric] by (simp only: card_cartesian_product) |
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lemma card_UNIV_sum: "CARD('a + 'b) = (if CARD('a) \<noteq> 0 \<and> CARD('b) \<noteq> 0 then CARD('a) + CARD('b) else 0)" |
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unfolding UNIV_Plus_UNIV[symmetric] |
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by(auto simp add: card_eq_0_iff card_Plus simp del: UNIV_Plus_UNIV) |
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lemma card_sum [simp]: "CARD('a + 'b) = CARD('a::finite) + CARD('b::finite)" |
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by(simp add: card_UNIV_sum) |
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lemma card_UNIV_option: "CARD('a option) = (if CARD('a) = 0 then 0 else CARD('a) + 1)" |
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proof - |
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have "(None :: 'a option) \<notin> range Some" by clarsimp |
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thus ?thesis |
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by(simp add: UNIV_option_conv card_eq_0_iff finite_range_Some card_insert_disjoint card_image) |
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qed |
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lemma card_option [simp]: "CARD('a option) = Suc CARD('a::finite)" |
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by(simp add: card_UNIV_option) |
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lemma card_UNIV_set: "CARD('a set) = (if CARD('a) = 0 then 0 else 2 ^ CARD('a))" |
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by(simp add: Pow_UNIV[symmetric] card_eq_0_iff card_Pow del: Pow_UNIV) |
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lemma card_set [simp]: "CARD('a set) = 2 ^ CARD('a::finite)" |
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by(simp add: card_UNIV_set) |
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lemma card_nat [simp]: "CARD(nat) = 0" |
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by (simp add: card_eq_0_iff) |
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lemma card_fun: "CARD('a \<Rightarrow> 'b) = (if CARD('a) \<noteq> 0 \<and> CARD('b) \<noteq> 0 \<or> CARD('b) = 1 then CARD('b) ^ CARD('a) else 0)" |
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proof - |
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{ assume "0 < CARD('a)" and "0 < CARD('b)" |
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hence fina: "finite (UNIV :: 'a set)" and finb: "finite (UNIV :: 'b set)" |
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by(simp_all only: card_ge_0_finite) |
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from finite_distinct_list[OF finb] obtain bs |
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where bs: "set bs = (UNIV :: 'b set)" and distb: "distinct bs" by blast |
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from finite_distinct_list[OF fina] obtain as |
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where as: "set as = (UNIV :: 'a set)" and dista: "distinct as" by blast |
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have cb: "CARD('b) = length bs" |
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unfolding bs[symmetric] distinct_card[OF distb] .. |
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have ca: "CARD('a) = length as" |
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unfolding as[symmetric] distinct_card[OF dista] .. |
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let ?xs = "map (\<lambda>ys. the o map_of (zip as ys)) (List.n_lists (length as) bs)" |
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have "UNIV = set ?xs" |
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proof(rule UNIV_eq_I) |
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fix f :: "'a \<Rightarrow> 'b" |
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from as have "f = the \<circ> map_of (zip as (map f as))" |
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by(auto simp add: map_of_zip_map) |
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thus "f \<in> set ?xs" using bs by(auto simp add: set_n_lists) |
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qed |
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moreover have "distinct ?xs" unfolding distinct_map |
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proof(intro conjI distinct_n_lists distb inj_onI) |
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fix xs ys :: "'b list" |
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assume xs: "xs \<in> set (List.n_lists (length as) bs)" |
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and ys: "ys \<in> set (List.n_lists (length as) bs)" |
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and eq: "the \<circ> map_of (zip as xs) = the \<circ> map_of (zip as ys)" |
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from xs ys have [simp]: "length xs = length as" "length ys = length as" |
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by(simp_all add: length_n_lists_elem) |
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have "map_of (zip as xs) = map_of (zip as ys)" |
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proof |
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fix x |
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from as bs have "\<exists>y. map_of (zip as xs) x = Some y" "\<exists>y. map_of (zip as ys) x = Some y" |
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by(simp_all add: map_of_zip_is_Some[symmetric]) |
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with eq show "map_of (zip as xs) x = map_of (zip as ys) x" |
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by(auto dest: fun_cong[where x=x]) |
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qed |
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with dista show "xs = ys" by(simp add: map_of_zip_inject) |
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qed |
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hence "card (set ?xs) = length ?xs" by(simp only: distinct_card) |
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moreover have "length ?xs = length bs ^ length as" by(simp add: length_n_lists) |
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ultimately have "CARD('a \<Rightarrow> 'b) = CARD('b) ^ CARD('a)" using cb ca by simp } |
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moreover { |
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assume cb: "CARD('b) = 1" |
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then obtain b where b: "UNIV = {b :: 'b}" by(auto simp add: card_Suc_eq) |
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have eq: "UNIV = {\<lambda>x :: 'a. b ::'b}" |
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proof(rule UNIV_eq_I) |
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fix x :: "'a \<Rightarrow> 'b" |
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{ fix y |
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have "x y \<in> UNIV" .. |
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hence "x y = b" unfolding b by simp } |
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thus "x \<in> {\<lambda>x. b}" by(auto) |
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qed |
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have "CARD('a \<Rightarrow> 'b) = 1" unfolding eq by simp } |
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ultimately show ?thesis |
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by(auto simp del: One_nat_def)(auto simp add: card_eq_0_iff dest: finite_fun_UNIVD2 finite_fun_UNIVD1) |
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qed |
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||
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corollary finite_UNIV_fun: |
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"finite (UNIV :: ('a \<Rightarrow> 'b) set) \<longleftrightarrow> |
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finite (UNIV :: 'a set) \<and> finite (UNIV :: 'b set) \<or> CARD('b) = 1" |
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(is "?lhs \<longleftrightarrow> ?rhs") |
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proof - |
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have "?lhs \<longleftrightarrow> CARD('a \<Rightarrow> 'b) > 0" by(simp add: card_gt_0_iff) |
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also have "\<dots> \<longleftrightarrow> CARD('a) > 0 \<and> CARD('b) > 0 \<or> CARD('b) = 1" |
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by(simp add: card_fun) |
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also have "\<dots> = ?rhs" by(simp add: card_gt_0_iff) |
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finally show ?thesis . |
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qed |
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lemma card_nibble: "CARD(nibble) = 16" |
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unfolding UNIV_nibble by simp |
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lemma card_UNIV_char: "CARD(char) = 256" |
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proof - |
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have "inj (\<lambda>(x, y). Char x y)" by(auto intro: injI) |
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thus ?thesis unfolding UNIV_char by(simp add: card_image card_nibble) |
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qed |
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lemma card_literal: "CARD(String.literal) = 0" |
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by(simp add: card_eq_0_iff infinite_literal) |
30001 | 163 |
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subsection {* Classes with at least 1 and 2 *} |
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text {* Class finite already captures "at least 1" *} |
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lemma zero_less_card_finite [simp]: "0 < CARD('a::finite)" |
|
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unfolding neq0_conv [symmetric] by simp |
170 |
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30001 | 171 |
lemma one_le_card_finite [simp]: "Suc 0 \<le> CARD('a::finite)" |
172 |
by (simp add: less_Suc_eq_le [symmetric]) |
|
173 |
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174 |
text {* Class for cardinality "at least 2" *} |
|
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176 |
class card2 = finite + |
|
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assumes two_le_card: "2 \<le> CARD('a)" |
|
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179 |
lemma one_less_card: "Suc 0 < CARD('a::card2)" |
|
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using two_le_card [where 'a='a] by simp |
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182 |
lemma one_less_int_card: "1 < int CARD('a::card2)" |
|
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using one_less_card [where 'a='a] by simp |
|
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||
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subsection {* A type class for deciding finiteness of types *} |
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type_synonym 'a finite_UNIV = "('a, bool) phantom" |
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class finite_UNIV = |
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fixes finite_UNIV :: "('a, bool) phantom" |
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assumes finite_UNIV: "finite_UNIV = Phantom('a) (finite (UNIV :: 'a set))" |
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lemma finite_UNIV_code [code_unfold]: |
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"finite (UNIV :: 'a :: finite_UNIV set) |
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\<longleftrightarrow> of_phantom (finite_UNIV :: 'a finite_UNIV)" |
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by(simp add: finite_UNIV) |
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subsection {* A type class for computing the cardinality of types *} |
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definition is_list_UNIV :: "'a list \<Rightarrow> bool" |
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where "is_list_UNIV xs = (let c = CARD('a) in if c = 0 then False else size (remdups xs) = c)" |
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lemma is_list_UNIV_iff: "is_list_UNIV xs \<longleftrightarrow> set xs = UNIV" |
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by(auto simp add: is_list_UNIV_def Let_def card_eq_0_iff List.card_set[symmetric] |
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dest: subst[where P="finite", OF _ finite_set] card_eq_UNIV_imp_eq_UNIV) |
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type_synonym 'a card_UNIV = "('a, nat) phantom" |
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class card_UNIV = finite_UNIV + |
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fixes card_UNIV :: "'a card_UNIV" |
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assumes card_UNIV: "card_UNIV = Phantom('a) CARD('a)" |
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subsection {* Instantiations for @{text "card_UNIV"} *} |
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instantiation nat :: card_UNIV begin |
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definition "finite_UNIV = Phantom(nat) False" |
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definition "card_UNIV = Phantom(nat) 0" |
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instance by intro_classes (simp_all add: finite_UNIV_nat_def card_UNIV_nat_def) |
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end |
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instantiation int :: card_UNIV begin |
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definition "finite_UNIV = Phantom(int) False" |
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definition "card_UNIV = Phantom(int) 0" |
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instance by intro_classes (simp_all add: card_UNIV_int_def finite_UNIV_int_def infinite_UNIV_int) |
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end |
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instantiation natural :: card_UNIV begin |
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definition "finite_UNIV = Phantom(natural) False" |
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definition "card_UNIV = Phantom(natural) 0" |
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instance proof |
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qed (auto simp add: finite_UNIV_natural_def card_UNIV_natural_def card_eq_0_iff |
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type_definition.univ [OF type_definition_natural] natural_eq_iff |
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dest!: finite_imageD intro: inj_onI) |
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end |
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instantiation integer :: card_UNIV begin |
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definition "finite_UNIV = Phantom(integer) False" |
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definition "card_UNIV = Phantom(integer) 0" |
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instance proof |
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qed (auto simp add: finite_UNIV_integer_def card_UNIV_integer_def card_eq_0_iff |
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type_definition.univ [OF type_definition_integer] infinite_UNIV_int |
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dest!: finite_imageD intro: inj_onI) |
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end |
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instantiation list :: (type) card_UNIV begin |
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definition "finite_UNIV = Phantom('a list) False" |
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definition "card_UNIV = Phantom('a list) 0" |
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instance by intro_classes (simp_all add: card_UNIV_list_def finite_UNIV_list_def infinite_UNIV_listI) |
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end |
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instantiation unit :: card_UNIV begin |
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definition "finite_UNIV = Phantom(unit) True" |
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definition "card_UNIV = Phantom(unit) 1" |
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instance by intro_classes (simp_all add: card_UNIV_unit_def finite_UNIV_unit_def) |
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end |
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instantiation bool :: card_UNIV begin |
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definition "finite_UNIV = Phantom(bool) True" |
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definition "card_UNIV = Phantom(bool) 2" |
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instance by(intro_classes)(simp_all add: card_UNIV_bool_def finite_UNIV_bool_def) |
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end |
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instantiation nibble :: card_UNIV begin |
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definition "finite_UNIV = Phantom(nibble) True" |
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definition "card_UNIV = Phantom(nibble) 16" |
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instance by(intro_classes)(simp_all add: card_UNIV_nibble_def card_nibble finite_UNIV_nibble_def) |
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end |
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instantiation char :: card_UNIV begin |
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definition "finite_UNIV = Phantom(char) True" |
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definition "card_UNIV = Phantom(char) 256" |
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instance by intro_classes (simp_all add: card_UNIV_char_def card_UNIV_char finite_UNIV_char_def) |
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end |
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instantiation prod :: (finite_UNIV, finite_UNIV) finite_UNIV begin |
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definition "finite_UNIV = Phantom('a \<times> 'b) |
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(of_phantom (finite_UNIV :: 'a finite_UNIV) \<and> of_phantom (finite_UNIV :: 'b finite_UNIV))" |
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instance by intro_classes (simp add: finite_UNIV_prod_def finite_UNIV finite_prod) |
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end |
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instantiation prod :: (card_UNIV, card_UNIV) card_UNIV begin |
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definition "card_UNIV = Phantom('a \<times> 'b) |
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(of_phantom (card_UNIV :: 'a card_UNIV) * of_phantom (card_UNIV :: 'b card_UNIV))" |
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instance by intro_classes (simp add: card_UNIV_prod_def card_UNIV) |
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end |
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instantiation sum :: (finite_UNIV, finite_UNIV) finite_UNIV begin |
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definition "finite_UNIV = Phantom('a + 'b) |
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(of_phantom (finite_UNIV :: 'a finite_UNIV) \<and> of_phantom (finite_UNIV :: 'b finite_UNIV))" |
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instance |
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by intro_classes (simp add: UNIV_Plus_UNIV[symmetric] finite_UNIV_sum_def finite_UNIV del: UNIV_Plus_UNIV) |
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end |
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instantiation sum :: (card_UNIV, card_UNIV) card_UNIV begin |
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definition "card_UNIV = Phantom('a + 'b) |
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(let ca = of_phantom (card_UNIV :: 'a card_UNIV); |
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cb = of_phantom (card_UNIV :: 'b card_UNIV) |
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in if ca \<noteq> 0 \<and> cb \<noteq> 0 then ca + cb else 0)" |
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instance by intro_classes (auto simp add: card_UNIV_sum_def card_UNIV card_UNIV_sum) |
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end |
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instantiation "fun" :: (finite_UNIV, card_UNIV) finite_UNIV begin |
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definition "finite_UNIV = Phantom('a \<Rightarrow> 'b) |
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(let cb = of_phantom (card_UNIV :: 'b card_UNIV) |
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in cb = 1 \<or> of_phantom (finite_UNIV :: 'a finite_UNIV) \<and> cb \<noteq> 0)" |
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instance |
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by intro_classes (auto simp add: finite_UNIV_fun_def Let_def card_UNIV finite_UNIV finite_UNIV_fun card_gt_0_iff) |
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end |
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instantiation "fun" :: (card_UNIV, card_UNIV) card_UNIV begin |
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definition "card_UNIV = Phantom('a \<Rightarrow> 'b) |
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(let ca = of_phantom (card_UNIV :: 'a card_UNIV); |
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cb = of_phantom (card_UNIV :: 'b card_UNIV) |
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in if ca \<noteq> 0 \<and> cb \<noteq> 0 \<or> cb = 1 then cb ^ ca else 0)" |
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instance by intro_classes (simp add: card_UNIV_fun_def card_UNIV Let_def card_fun) |
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end |
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instantiation option :: (finite_UNIV) finite_UNIV begin |
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definition "finite_UNIV = Phantom('a option) (of_phantom (finite_UNIV :: 'a finite_UNIV))" |
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instance by intro_classes (simp add: finite_UNIV_option_def finite_UNIV) |
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end |
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instantiation option :: (card_UNIV) card_UNIV begin |
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definition "card_UNIV = Phantom('a option) |
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(let c = of_phantom (card_UNIV :: 'a card_UNIV) in if c \<noteq> 0 then Suc c else 0)" |
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instance by intro_classes (simp add: card_UNIV_option_def card_UNIV card_UNIV_option) |
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end |
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instantiation String.literal :: card_UNIV begin |
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definition "finite_UNIV = Phantom(String.literal) False" |
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definition "card_UNIV = Phantom(String.literal) 0" |
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instance |
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by intro_classes (simp_all add: card_UNIV_literal_def finite_UNIV_literal_def infinite_literal card_literal) |
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end |
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|
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instantiation set :: (finite_UNIV) finite_UNIV begin |
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definition "finite_UNIV = Phantom('a set) (of_phantom (finite_UNIV :: 'a finite_UNIV))" |
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instance by intro_classes (simp add: finite_UNIV_set_def finite_UNIV Finite_Set.finite_set) |
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end |
341 |
||
342 |
instantiation set :: (card_UNIV) card_UNIV begin |
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definition "card_UNIV = Phantom('a set) |
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(let c = of_phantom (card_UNIV :: 'a card_UNIV) in if c = 0 then 0 else 2 ^ c)" |
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instance by intro_classes (simp add: card_UNIV_set_def card_UNIV_set card_UNIV) |
48051 | 346 |
end |
347 |
||
48060 | 348 |
lemma UNIV_finite_1: "UNIV = set [finite_1.a\<^isub>1]" |
349 |
by(auto intro: finite_1.exhaust) |
|
350 |
||
351 |
lemma UNIV_finite_2: "UNIV = set [finite_2.a\<^isub>1, finite_2.a\<^isub>2]" |
|
352 |
by(auto intro: finite_2.exhaust) |
|
48051 | 353 |
|
48060 | 354 |
lemma UNIV_finite_3: "UNIV = set [finite_3.a\<^isub>1, finite_3.a\<^isub>2, finite_3.a\<^isub>3]" |
355 |
by(auto intro: finite_3.exhaust) |
|
48051 | 356 |
|
48060 | 357 |
lemma UNIV_finite_4: "UNIV = set [finite_4.a\<^isub>1, finite_4.a\<^isub>2, finite_4.a\<^isub>3, finite_4.a\<^isub>4]" |
358 |
by(auto intro: finite_4.exhaust) |
|
359 |
||
360 |
lemma UNIV_finite_5: |
|
361 |
"UNIV = set [finite_5.a\<^isub>1, finite_5.a\<^isub>2, finite_5.a\<^isub>3, finite_5.a\<^isub>4, finite_5.a\<^isub>5]" |
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362 |
by(auto intro: finite_5.exhaust) |
|
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48060 | 364 |
instantiation Enum.finite_1 :: card_UNIV begin |
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definition "finite_UNIV = Phantom(Enum.finite_1) True" |
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definition "card_UNIV = Phantom(Enum.finite_1) 1" |
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instance |
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by intro_classes (simp_all add: UNIV_finite_1 card_UNIV_finite_1_def finite_UNIV_finite_1_def) |
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end |
370 |
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371 |
instantiation Enum.finite_2 :: card_UNIV begin |
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definition "finite_UNIV = Phantom(Enum.finite_2) True" |
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definition "card_UNIV = Phantom(Enum.finite_2) 2" |
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instance |
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by intro_classes (simp_all add: UNIV_finite_2 card_UNIV_finite_2_def finite_UNIV_finite_2_def) |
48060 | 376 |
end |
48051 | 377 |
|
48060 | 378 |
instantiation Enum.finite_3 :: card_UNIV begin |
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definition "finite_UNIV = Phantom(Enum.finite_3) True" |
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definition "card_UNIV = Phantom(Enum.finite_3) 3" |
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instance |
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382 |
by intro_classes (simp_all add: UNIV_finite_3 card_UNIV_finite_3_def finite_UNIV_finite_3_def) |
48060 | 383 |
end |
384 |
||
385 |
instantiation Enum.finite_4 :: card_UNIV begin |
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definition "finite_UNIV = Phantom(Enum.finite_4) True" |
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definition "card_UNIV = Phantom(Enum.finite_4) 4" |
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388 |
instance |
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389 |
by intro_classes (simp_all add: UNIV_finite_4 card_UNIV_finite_4_def finite_UNIV_finite_4_def) |
48060 | 390 |
end |
391 |
||
392 |
instantiation Enum.finite_5 :: card_UNIV begin |
|
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definition "finite_UNIV = Phantom(Enum.finite_5) True" |
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definition "card_UNIV = Phantom(Enum.finite_5) 5" |
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instance |
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396 |
by intro_classes (simp_all add: UNIV_finite_5 card_UNIV_finite_5_def finite_UNIV_finite_5_def) |
48051 | 397 |
end |
398 |
||
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subsection {* Code setup for sets *} |
48051 | 400 |
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text {* |
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402 |
Implement @{term "CARD('a)"} via @{term card_UNIV} and provide |
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implementations for @{term "finite"}, @{term "card"}, @{term "op \<subseteq>"}, |
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404 |
and @{term "op ="}if the calling context already provides @{class finite_UNIV} |
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and @{class card_UNIV} instances. If we implemented the latter |
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406 |
always via @{term card_UNIV}, we would require instances of essentially all |
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407 |
element types, i.e., a lot of instantiation proofs and -- at run time -- |
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408 |
possibly slow dictionary constructions. |
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409 |
*} |
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410 |
|
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411 |
definition card_UNIV' :: "'a card_UNIV" |
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412 |
where [code del]: "card_UNIV' = Phantom('a) CARD('a)" |
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413 |
|
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414 |
lemma CARD_code [code_unfold]: |
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"CARD('a) = of_phantom (card_UNIV' :: 'a card_UNIV)" |
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416 |
by(simp add: card_UNIV'_def) |
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417 |
|
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418 |
lemma card_UNIV'_code [code]: |
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419 |
"card_UNIV' = card_UNIV" |
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420 |
by(simp add: card_UNIV card_UNIV'_def) |
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421 |
|
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422 |
hide_const (open) card_UNIV' |
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423 |
|
48051 | 424 |
lemma card_Compl: |
425 |
"finite A \<Longrightarrow> card (- A) = card (UNIV :: 'a set) - card (A :: 'a set)" |
|
426 |
by (metis Compl_eq_Diff_UNIV card_Diff_subset top_greatest) |
|
427 |
||
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428 |
context fixes xs :: "'a :: finite_UNIV list" |
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429 |
begin |
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430 |
|
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431 |
definition finite' :: "'a set \<Rightarrow> bool" |
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432 |
where [simp, code del, code_abbrev]: "finite' = finite" |
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433 |
|
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434 |
lemma finite'_code [code]: |
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435 |
"finite' (set xs) \<longleftrightarrow> True" |
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436 |
"finite' (List.coset xs) \<longleftrightarrow> of_phantom (finite_UNIV :: 'a finite_UNIV)" |
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437 |
by(simp_all add: card_gt_0_iff finite_UNIV) |
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438 |
|
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439 |
end |
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440 |
|
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context fixes xs :: "'a :: card_UNIV list" |
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442 |
begin |
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443 |
|
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444 |
definition card' :: "'a set \<Rightarrow> nat" |
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where [simp, code del, code_abbrev]: "card' = card" |
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446 |
|
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447 |
lemma card'_code [code]: |
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"card' (set xs) = length (remdups xs)" |
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449 |
"card' (List.coset xs) = of_phantom (card_UNIV :: 'a card_UNIV) - length (remdups xs)" |
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450 |
by(simp_all add: List.card_set card_Compl card_UNIV) |
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451 |
|
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452 |
|
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453 |
definition subset' :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" |
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454 |
where [simp, code del, code_abbrev]: "subset' = op \<subseteq>" |
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455 |
|
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456 |
lemma subset'_code [code]: |
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457 |
"subset' A (List.coset ys) \<longleftrightarrow> (\<forall>y \<in> set ys. y \<notin> A)" |
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458 |
"subset' (set ys) B \<longleftrightarrow> (\<forall>y \<in> set ys. y \<in> B)" |
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459 |
"subset' (List.coset xs) (set ys) \<longleftrightarrow> (let n = CARD('a) in n > 0 \<and> card(set (xs @ ys)) = n)" |
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|
460 |
by(auto simp add: Let_def card_gt_0_iff dest: card_eq_UNIV_imp_eq_UNIV intro: arg_cong[where f=card]) |
9014e78ccde2
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changeset
|
461 |
(metis finite_compl finite_set rev_finite_subset) |
9014e78ccde2
improved code setup for card, finite, subset
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changeset
|
462 |
|
51139
c8e3cf3520b3
partially revert 0dac0158b8d4 as it too aggressively spreads card_UNIV type class whose dictionary constructions can slow down generated code;
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parents:
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|
463 |
definition eq_set :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" |
c8e3cf3520b3
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|
464 |
where [simp, code del, code_abbrev]: "eq_set = op =" |
c8e3cf3520b3
partially revert 0dac0158b8d4 as it too aggressively spreads card_UNIV type class whose dictionary constructions can slow down generated code;
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parents:
51116
diff
changeset
|
465 |
|
c8e3cf3520b3
partially revert 0dac0158b8d4 as it too aggressively spreads card_UNIV type class whose dictionary constructions can slow down generated code;
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|
466 |
lemma eq_set_code [code]: |
c8e3cf3520b3
partially revert 0dac0158b8d4 as it too aggressively spreads card_UNIV type class whose dictionary constructions can slow down generated code;
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|
467 |
fixes ys |
48051 | 468 |
defines "rhs \<equiv> |
48059 | 469 |
let n = CARD('a) |
48051 | 470 |
in if n = 0 then False else |
471 |
let xs' = remdups xs; ys' = remdups ys |
|
472 |
in length xs' + length ys' = n \<and> (\<forall>x \<in> set xs'. x \<notin> set ys') \<and> (\<forall>y \<in> set ys'. y \<notin> set xs')" |
|
51139
c8e3cf3520b3
partially revert 0dac0158b8d4 as it too aggressively spreads card_UNIV type class whose dictionary constructions can slow down generated code;
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parents:
51116
diff
changeset
|
473 |
shows "eq_set (List.coset xs) (set ys) \<longleftrightarrow> rhs" (is ?thesis1) |
c8e3cf3520b3
partially revert 0dac0158b8d4 as it too aggressively spreads card_UNIV type class whose dictionary constructions can slow down generated code;
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parents:
51116
diff
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|
474 |
and "eq_set (set ys) (List.coset xs) \<longleftrightarrow> rhs" (is ?thesis2) |
c8e3cf3520b3
partially revert 0dac0158b8d4 as it too aggressively spreads card_UNIV type class whose dictionary constructions can slow down generated code;
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parents:
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diff
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|
475 |
and "eq_set (set xs) (set ys) \<longleftrightarrow> (\<forall>x \<in> set xs. x \<in> set ys) \<and> (\<forall>y \<in> set ys. y \<in> set xs)" (is ?thesis3) |
c8e3cf3520b3
partially revert 0dac0158b8d4 as it too aggressively spreads card_UNIV type class whose dictionary constructions can slow down generated code;
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parents:
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diff
changeset
|
476 |
and "eq_set (List.coset xs) (List.coset ys) \<longleftrightarrow> (\<forall>x \<in> set xs. x \<in> set ys) \<and> (\<forall>y \<in> set ys. y \<in> set xs)" (is ?thesis4) |
48051 | 477 |
proof - |
478 |
show ?thesis1 (is "?lhs \<longleftrightarrow> ?rhs") |
|
479 |
proof |
|
480 |
assume ?lhs thus ?rhs |
|
51139
c8e3cf3520b3
partially revert 0dac0158b8d4 as it too aggressively spreads card_UNIV type class whose dictionary constructions can slow down generated code;
Andreas Lochbihler
parents:
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diff
changeset
|
481 |
by(auto simp add: rhs_def Let_def List.card_set[symmetric] card_Un_Int[where A="set xs" and B="- set xs"] card_UNIV Compl_partition card_gt_0_iff dest: sym)(metis finite_compl finite_set) |
48051 | 482 |
next |
483 |
assume ?rhs |
|
484 |
moreover have "\<lbrakk> \<forall>y\<in>set xs. y \<notin> set ys; \<forall>x\<in>set ys. x \<notin> set xs \<rbrakk> \<Longrightarrow> set xs \<inter> set ys = {}" by blast |
|
485 |
ultimately show ?lhs |
|
51139
c8e3cf3520b3
partially revert 0dac0158b8d4 as it too aggressively spreads card_UNIV type class whose dictionary constructions can slow down generated code;
Andreas Lochbihler
parents:
51116
diff
changeset
|
486 |
by(auto simp add: rhs_def Let_def List.card_set[symmetric] card_UNIV card_gt_0_iff card_Un_Int[where A="set xs" and B="set ys"] dest: card_eq_UNIV_imp_eq_UNIV split: split_if_asm) |
48051 | 487 |
qed |
51139
c8e3cf3520b3
partially revert 0dac0158b8d4 as it too aggressively spreads card_UNIV type class whose dictionary constructions can slow down generated code;
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parents:
51116
diff
changeset
|
488 |
thus ?thesis2 unfolding eq_set_def by blast |
c8e3cf3520b3
partially revert 0dac0158b8d4 as it too aggressively spreads card_UNIV type class whose dictionary constructions can slow down generated code;
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parents:
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diff
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|
489 |
show ?thesis3 ?thesis4 unfolding eq_set_def List.coset_def by blast+ |
48051 | 490 |
qed |
491 |
||
51139
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partially revert 0dac0158b8d4 as it too aggressively spreads card_UNIV type class whose dictionary constructions can slow down generated code;
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parents:
51116
diff
changeset
|
492 |
end |
c8e3cf3520b3
partially revert 0dac0158b8d4 as it too aggressively spreads card_UNIV type class whose dictionary constructions can slow down generated code;
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parents:
51116
diff
changeset
|
493 |
|
c8e3cf3520b3
partially revert 0dac0158b8d4 as it too aggressively spreads card_UNIV type class whose dictionary constructions can slow down generated code;
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changeset
|
494 |
text {* |
c8e3cf3520b3
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|
495 |
Provide more informative exceptions than Match for non-rewritten cases. |
c8e3cf3520b3
partially revert 0dac0158b8d4 as it too aggressively spreads card_UNIV type class whose dictionary constructions can slow down generated code;
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|
496 |
If generated code raises one these exceptions, then a code equation calls |
c8e3cf3520b3
partially revert 0dac0158b8d4 as it too aggressively spreads card_UNIV type class whose dictionary constructions can slow down generated code;
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diff
changeset
|
497 |
the mentioned operator for an element type that is not an instance of |
c8e3cf3520b3
partially revert 0dac0158b8d4 as it too aggressively spreads card_UNIV type class whose dictionary constructions can slow down generated code;
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parents:
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diff
changeset
|
498 |
@{class card_UNIV} and is therefore not implemented via @{term card_UNIV}. |
c8e3cf3520b3
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Andreas Lochbihler
parents:
51116
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changeset
|
499 |
Constrain the element type with sort @{class card_UNIV} to change this. |
c8e3cf3520b3
partially revert 0dac0158b8d4 as it too aggressively spreads card_UNIV type class whose dictionary constructions can slow down generated code;
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parents:
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diff
changeset
|
500 |
*} |
c8e3cf3520b3
partially revert 0dac0158b8d4 as it too aggressively spreads card_UNIV type class whose dictionary constructions can slow down generated code;
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parents:
51116
diff
changeset
|
501 |
|
c8e3cf3520b3
partially revert 0dac0158b8d4 as it too aggressively spreads card_UNIV type class whose dictionary constructions can slow down generated code;
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parents:
51116
diff
changeset
|
502 |
definition card_coset_requires_card_UNIV :: "'a list \<Rightarrow> nat" |
c8e3cf3520b3
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changeset
|
503 |
where [code del, simp]: "card_coset_requires_card_UNIV xs = card (List.coset xs)" |
c8e3cf3520b3
partially revert 0dac0158b8d4 as it too aggressively spreads card_UNIV type class whose dictionary constructions can slow down generated code;
Andreas Lochbihler
parents:
51116
diff
changeset
|
504 |
|
c8e3cf3520b3
partially revert 0dac0158b8d4 as it too aggressively spreads card_UNIV type class whose dictionary constructions can slow down generated code;
Andreas Lochbihler
parents:
51116
diff
changeset
|
505 |
code_abort card_coset_requires_card_UNIV |
c8e3cf3520b3
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Andreas Lochbihler
parents:
51116
diff
changeset
|
506 |
|
c8e3cf3520b3
partially revert 0dac0158b8d4 as it too aggressively spreads card_UNIV type class whose dictionary constructions can slow down generated code;
Andreas Lochbihler
parents:
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diff
changeset
|
507 |
lemma card_coset_error [code]: |
c8e3cf3520b3
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|
508 |
"card (List.coset xs) = card_coset_requires_card_UNIV xs" |
c8e3cf3520b3
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Andreas Lochbihler
parents:
51116
diff
changeset
|
509 |
by(simp) |
c8e3cf3520b3
partially revert 0dac0158b8d4 as it too aggressively spreads card_UNIV type class whose dictionary constructions can slow down generated code;
Andreas Lochbihler
parents:
51116
diff
changeset
|
510 |
|
c8e3cf3520b3
partially revert 0dac0158b8d4 as it too aggressively spreads card_UNIV type class whose dictionary constructions can slow down generated code;
Andreas Lochbihler
parents:
51116
diff
changeset
|
511 |
definition coset_subseteq_set_requires_card_UNIV :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" |
c8e3cf3520b3
partially revert 0dac0158b8d4 as it too aggressively spreads card_UNIV type class whose dictionary constructions can slow down generated code;
Andreas Lochbihler
parents:
51116
diff
changeset
|
512 |
where [code del, simp]: "coset_subseteq_set_requires_card_UNIV xs ys \<longleftrightarrow> List.coset xs \<subseteq> set ys" |
c8e3cf3520b3
partially revert 0dac0158b8d4 as it too aggressively spreads card_UNIV type class whose dictionary constructions can slow down generated code;
Andreas Lochbihler
parents:
51116
diff
changeset
|
513 |
|
c8e3cf3520b3
partially revert 0dac0158b8d4 as it too aggressively spreads card_UNIV type class whose dictionary constructions can slow down generated code;
Andreas Lochbihler
parents:
51116
diff
changeset
|
514 |
code_abort coset_subseteq_set_requires_card_UNIV |
c8e3cf3520b3
partially revert 0dac0158b8d4 as it too aggressively spreads card_UNIV type class whose dictionary constructions can slow down generated code;
Andreas Lochbihler
parents:
51116
diff
changeset
|
515 |
|
c8e3cf3520b3
partially revert 0dac0158b8d4 as it too aggressively spreads card_UNIV type class whose dictionary constructions can slow down generated code;
Andreas Lochbihler
parents:
51116
diff
changeset
|
516 |
lemma coset_subseteq_set_code [code]: |
c8e3cf3520b3
partially revert 0dac0158b8d4 as it too aggressively spreads card_UNIV type class whose dictionary constructions can slow down generated code;
Andreas Lochbihler
parents:
51116
diff
changeset
|
517 |
"List.coset xs \<subseteq> set ys \<longleftrightarrow> |
c8e3cf3520b3
partially revert 0dac0158b8d4 as it too aggressively spreads card_UNIV type class whose dictionary constructions can slow down generated code;
Andreas Lochbihler
parents:
51116
diff
changeset
|
518 |
(if xs = [] \<and> ys = [] then False else coset_subseteq_set_requires_card_UNIV xs ys)" |
c8e3cf3520b3
partially revert 0dac0158b8d4 as it too aggressively spreads card_UNIV type class whose dictionary constructions can slow down generated code;
Andreas Lochbihler
parents:
51116
diff
changeset
|
519 |
by simp |
c8e3cf3520b3
partially revert 0dac0158b8d4 as it too aggressively spreads card_UNIV type class whose dictionary constructions can slow down generated code;
Andreas Lochbihler
parents:
51116
diff
changeset
|
520 |
|
c8e3cf3520b3
partially revert 0dac0158b8d4 as it too aggressively spreads card_UNIV type class whose dictionary constructions can slow down generated code;
Andreas Lochbihler
parents:
51116
diff
changeset
|
521 |
notepad begin -- "test code setup" |
c8e3cf3520b3
partially revert 0dac0158b8d4 as it too aggressively spreads card_UNIV type class whose dictionary constructions can slow down generated code;
Andreas Lochbihler
parents:
51116
diff
changeset
|
522 |
have "List.coset [True] = set [False] \<and> |
c8e3cf3520b3
partially revert 0dac0158b8d4 as it too aggressively spreads card_UNIV type class whose dictionary constructions can slow down generated code;
Andreas Lochbihler
parents:
51116
diff
changeset
|
523 |
List.coset [] \<subseteq> List.set [True, False] \<and> |
c8e3cf3520b3
partially revert 0dac0158b8d4 as it too aggressively spreads card_UNIV type class whose dictionary constructions can slow down generated code;
Andreas Lochbihler
parents:
51116
diff
changeset
|
524 |
finite (List.coset [True])" |
48062
9014e78ccde2
improved code setup for card, finite, subset
Andreas Lochbihler
parents:
48060
diff
changeset
|
525 |
by eval |
9014e78ccde2
improved code setup for card, finite, subset
Andreas Lochbihler
parents:
48060
diff
changeset
|
526 |
end |
9014e78ccde2
improved code setup for card, finite, subset
Andreas Lochbihler
parents:
48060
diff
changeset
|
527 |
|
51139
c8e3cf3520b3
partially revert 0dac0158b8d4 as it too aggressively spreads card_UNIV type class whose dictionary constructions can slow down generated code;
Andreas Lochbihler
parents:
51116
diff
changeset
|
528 |
hide_const (open) card' finite' subset' eq_set |
c8e3cf3520b3
partially revert 0dac0158b8d4 as it too aggressively spreads card_UNIV type class whose dictionary constructions can slow down generated code;
Andreas Lochbihler
parents:
51116
diff
changeset
|
529 |
|
48051 | 530 |
end |
49948
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49689
diff
changeset
|
531 |