--- a/src/HOL/Library/Cardinality.thy Wed May 30 16:05:21 2012 +0200
+++ b/src/HOL/Library/Cardinality.thy Thu May 31 16:58:38 2012 +0200
@@ -1,5 +1,5 @@
(* Title: HOL/Library/Cardinality.thy
- Author: Brian Huffman
+ Author: Brian Huffman, Andreas Lochbihler
*)
header {* Cardinality of types *}
@@ -86,4 +86,326 @@
lemma one_less_int_card: "1 < int CARD('a::card2)"
using one_less_card [where 'a='a] by simp
+subsection {* A type class for computing the cardinality of types *}
+
+class card_UNIV =
+ fixes card_UNIV :: "'a itself \<Rightarrow> nat"
+ assumes card_UNIV: "card_UNIV x = card (UNIV :: 'a set)"
+begin
+
+lemma card_UNIV_neq_0_finite_UNIV:
+ "card_UNIV x \<noteq> 0 \<longleftrightarrow> finite (UNIV :: 'a set)"
+by(simp add: card_UNIV card_eq_0_iff)
+
+lemma card_UNIV_ge_0_finite_UNIV:
+ "card_UNIV x > 0 \<longleftrightarrow> finite (UNIV :: 'a set)"
+by(auto simp add: card_UNIV intro: card_ge_0_finite finite_UNIV_card_ge_0)
+
+lemma card_UNIV_eq_0_infinite_UNIV:
+ "card_UNIV x = 0 \<longleftrightarrow> \<not> finite (UNIV :: 'a set)"
+by(simp add: card_UNIV card_eq_0_iff)
+
+definition is_list_UNIV :: "'a list \<Rightarrow> bool"
+where "is_list_UNIV xs = (let c = card_UNIV (TYPE('a)) in if c = 0 then False else size (remdups xs) = c)"
+
+lemma is_list_UNIV_iff: fixes xs :: "'a list"
+ shows "is_list_UNIV xs \<longleftrightarrow> set xs = UNIV"
+proof
+ assume "is_list_UNIV xs"
+ hence c: "card_UNIV (TYPE('a)) > 0" and xs: "size (remdups xs) = card_UNIV (TYPE('a))"
+ unfolding is_list_UNIV_def by(simp_all add: Let_def split: split_if_asm)
+ from c have fin: "finite (UNIV :: 'a set)" by(auto simp add: card_UNIV_ge_0_finite_UNIV)
+ have "card (set (remdups xs)) = size (remdups xs)" by(subst distinct_card) auto
+ also note set_remdups
+ finally show "set xs = UNIV" using fin unfolding xs card_UNIV by-(rule card_eq_UNIV_imp_eq_UNIV)
+next
+ assume xs: "set xs = UNIV"
+ from finite_set[of xs] have fin: "finite (UNIV :: 'a set)" unfolding xs .
+ hence "card_UNIV (TYPE ('a)) \<noteq> 0" unfolding card_UNIV_neq_0_finite_UNIV .
+ moreover have "size (remdups xs) = card (set (remdups xs))"
+ by(subst distinct_card) auto
+ ultimately show "is_list_UNIV xs" using xs by(simp add: is_list_UNIV_def Let_def card_UNIV)
+qed
+
+lemma card_UNIV_eq_0_is_list_UNIV_False:
+ assumes cU0: "card_UNIV x = 0"
+ shows "is_list_UNIV = (\<lambda>xs. False)"
+proof(rule ext)
+ fix xs :: "'a list"
+ from cU0 have "\<not> finite (UNIV :: 'a set)"
+ by(auto simp only: card_UNIV_eq_0_infinite_UNIV)
+ moreover have "finite (set xs)" by(rule finite_set)
+ ultimately have "(UNIV :: 'a set) \<noteq> set xs" by(auto simp del: finite_set)
+ thus "is_list_UNIV xs = False" unfolding is_list_UNIV_iff by simp
+qed
+
end
+
+subsection {* Instantiations for @{text "card_UNIV"} *}
+
+subsubsection {* @{typ "nat"} *}
+
+instantiation nat :: card_UNIV begin
+
+definition "card_UNIV_class.card_UNIV = (\<lambda>a :: nat itself. 0)"
+
+instance proof
+ fix x :: "nat itself"
+ show "card_UNIV x = card (UNIV :: nat set)"
+ unfolding card_UNIV_nat_def by simp
+qed
+
+end
+
+subsubsection {* @{typ "int"} *}
+
+instantiation int :: card_UNIV begin
+
+definition "card_UNIV_class.card_UNIV = (\<lambda>a :: int itself. 0)"
+
+instance proof
+ fix x :: "int itself"
+ show "card_UNIV x = card (UNIV :: int set)"
+ unfolding card_UNIV_int_def by(simp add: infinite_UNIV_int)
+qed
+
+end
+
+subsubsection {* @{typ "'a list"} *}
+
+instantiation list :: (type) card_UNIV begin
+
+definition "card_UNIV_class.card_UNIV = (\<lambda>a :: 'a list itself. 0)"
+
+instance proof
+ fix x :: "'a list itself"
+ show "card_UNIV x = card (UNIV :: 'a list set)"
+ unfolding card_UNIV_list_def by(simp add: infinite_UNIV_listI)
+qed
+
+end
+
+subsubsection {* @{typ "unit"} *}
+
+lemma card_UNIV_unit: "card (UNIV :: unit set) = 1"
+ unfolding UNIV_unit by simp
+
+instantiation unit :: card_UNIV begin
+
+definition card_UNIV_unit_def:
+ "card_UNIV_class.card_UNIV = (\<lambda>a :: unit itself. 1)"
+
+instance proof
+ fix x :: "unit itself"
+ show "card_UNIV x = card (UNIV :: unit set)"
+ by(simp add: card_UNIV_unit_def card_UNIV_unit)
+qed
+
+end
+
+subsubsection {* @{typ "bool"} *}
+
+lemma card_UNIV_bool: "card (UNIV :: bool set) = 2"
+ unfolding UNIV_bool by simp
+
+instantiation bool :: card_UNIV begin
+
+definition card_UNIV_bool_def:
+ "card_UNIV_class.card_UNIV = (\<lambda>a :: bool itself. 2)"
+
+instance proof
+ fix x :: "bool itself"
+ show "card_UNIV x = card (UNIV :: bool set)"
+ by(simp add: card_UNIV_bool_def card_UNIV_bool)
+qed
+
+end
+
+subsubsection {* @{typ "char"} *}
+
+lemma card_UNIV_char: "card (UNIV :: char set) = 256"
+proof -
+ from enum_distinct
+ have "card (set (Enum.enum :: char list)) = length (Enum.enum :: char list)"
+ by (rule distinct_card)
+ also have "set Enum.enum = (UNIV :: char set)" by (auto intro: in_enum)
+ also note enum_chars
+ finally show ?thesis by (simp add: chars_def)
+qed
+
+instantiation char :: card_UNIV begin
+
+definition card_UNIV_char_def:
+ "card_UNIV_class.card_UNIV = (\<lambda>a :: char itself. 256)"
+
+instance proof
+ fix x :: "char itself"
+ show "card_UNIV x = card (UNIV :: char set)"
+ by(simp add: card_UNIV_char_def card_UNIV_char)
+qed
+
+end
+
+subsubsection {* @{typ "'a \<times> 'b"} *}
+
+instantiation prod :: (card_UNIV, card_UNIV) card_UNIV begin
+
+definition card_UNIV_product_def:
+ "card_UNIV_class.card_UNIV = (\<lambda>a :: ('a \<times> 'b) itself. card_UNIV (TYPE('a)) * card_UNIV (TYPE('b)))"
+
+instance proof
+ fix x :: "('a \<times> 'b) itself"
+ show "card_UNIV x = card (UNIV :: ('a \<times> 'b) set)"
+ by(simp add: card_UNIV_product_def card_UNIV UNIV_Times_UNIV[symmetric] card_cartesian_product del: UNIV_Times_UNIV)
+qed
+
+end
+
+subsubsection {* @{typ "'a + 'b"} *}
+
+instantiation sum :: (card_UNIV, card_UNIV) card_UNIV begin
+
+definition card_UNIV_sum_def:
+ "card_UNIV_class.card_UNIV = (\<lambda>a :: ('a + 'b) itself. let ca = card_UNIV (TYPE('a)); cb = card_UNIV (TYPE('b))
+ in if ca \<noteq> 0 \<and> cb \<noteq> 0 then ca + cb else 0)"
+
+instance proof
+ fix x :: "('a + 'b) itself"
+ show "card_UNIV x = card (UNIV :: ('a + 'b) set)"
+ by (auto simp add: card_UNIV_sum_def card_UNIV card_eq_0_iff UNIV_Plus_UNIV[symmetric] finite_Plus_iff Let_def card_Plus simp del: UNIV_Plus_UNIV dest!: card_ge_0_finite)
+qed
+
+end
+
+subsubsection {* @{typ "'a \<Rightarrow> 'b"} *}
+
+instantiation "fun" :: (card_UNIV, card_UNIV) card_UNIV begin
+
+definition card_UNIV_fun_def:
+ "card_UNIV_class.card_UNIV = (\<lambda>a :: ('a \<Rightarrow> 'b) itself. let ca = card_UNIV (TYPE('a)); cb = card_UNIV (TYPE('b))
+ in if ca \<noteq> 0 \<and> cb \<noteq> 0 \<or> cb = 1 then cb ^ ca else 0)"
+
+instance proof
+ fix x :: "('a \<Rightarrow> 'b) itself"
+
+ { assume "0 < card (UNIV :: 'a set)"
+ and "0 < card (UNIV :: 'b set)"
+ hence fina: "finite (UNIV :: 'a set)" and finb: "finite (UNIV :: 'b set)"
+ by(simp_all only: card_ge_0_finite)
+ from finite_distinct_list[OF finb] obtain bs
+ where bs: "set bs = (UNIV :: 'b set)" and distb: "distinct bs" by blast
+ from finite_distinct_list[OF fina] obtain as
+ where as: "set as = (UNIV :: 'a set)" and dista: "distinct as" by blast
+ have cb: "card (UNIV :: 'b set) = length bs"
+ unfolding bs[symmetric] distinct_card[OF distb] ..
+ have ca: "card (UNIV :: 'a set) = length as"
+ unfolding as[symmetric] distinct_card[OF dista] ..
+ let ?xs = "map (\<lambda>ys. the o map_of (zip as ys)) (Enum.n_lists (length as) bs)"
+ have "UNIV = set ?xs"
+ proof(rule UNIV_eq_I)
+ fix f :: "'a \<Rightarrow> 'b"
+ from as have "f = the \<circ> map_of (zip as (map f as))"
+ by(auto simp add: map_of_zip_map intro: ext)
+ thus "f \<in> set ?xs" using bs by(auto simp add: set_n_lists)
+ qed
+ moreover have "distinct ?xs" unfolding distinct_map
+ proof(intro conjI distinct_n_lists distb inj_onI)
+ fix xs ys :: "'b list"
+ assume xs: "xs \<in> set (Enum.n_lists (length as) bs)"
+ and ys: "ys \<in> set (Enum.n_lists (length as) bs)"
+ and eq: "the \<circ> map_of (zip as xs) = the \<circ> map_of (zip as ys)"
+ from xs ys have [simp]: "length xs = length as" "length ys = length as"
+ by(simp_all add: length_n_lists_elem)
+ have "map_of (zip as xs) = map_of (zip as ys)"
+ proof
+ fix x
+ from as bs have "\<exists>y. map_of (zip as xs) x = Some y" "\<exists>y. map_of (zip as ys) x = Some y"
+ by(simp_all add: map_of_zip_is_Some[symmetric])
+ with eq show "map_of (zip as xs) x = map_of (zip as ys) x"
+ by(auto dest: fun_cong[where x=x])
+ qed
+ with dista show "xs = ys" by(simp add: map_of_zip_inject)
+ qed
+ hence "card (set ?xs) = length ?xs" by(simp only: distinct_card)
+ moreover have "length ?xs = length bs ^ length as" by(simp add: length_n_lists)
+ ultimately have "card (UNIV :: ('a \<Rightarrow> 'b) set) = card (UNIV :: 'b set) ^ card (UNIV :: 'a set)"
+ using cb ca by simp }
+ moreover {
+ assume cb: "card (UNIV :: 'b set) = Suc 0"
+ then obtain b where b: "UNIV = {b :: 'b}" by(auto simp add: card_Suc_eq)
+ have eq: "UNIV = {\<lambda>x :: 'a. b ::'b}"
+ proof(rule UNIV_eq_I)
+ fix x :: "'a \<Rightarrow> 'b"
+ { fix y
+ have "x y \<in> UNIV" ..
+ hence "x y = b" unfolding b by simp }
+ thus "x \<in> {\<lambda>x. b}" by(auto intro: ext)
+ qed
+ have "card (UNIV :: ('a \<Rightarrow> 'b) set) = Suc 0" unfolding eq by simp }
+ ultimately show "card_UNIV x = card (UNIV :: ('a \<Rightarrow> 'b) set)"
+ unfolding card_UNIV_fun_def card_UNIV Let_def
+ by(auto simp del: One_nat_def)(auto simp add: card_eq_0_iff dest: finite_fun_UNIVD2 finite_fun_UNIVD1)
+qed
+
+end
+
+subsubsection {* @{typ "'a option"} *}
+
+instantiation option :: (card_UNIV) card_UNIV
+begin
+
+definition card_UNIV_option_def:
+ "card_UNIV_class.card_UNIV = (\<lambda>a :: 'a option itself. let c = card_UNIV (TYPE('a))
+ in if c \<noteq> 0 then Suc c else 0)"
+
+instance proof
+ fix x :: "'a option itself"
+ show "card_UNIV x = card (UNIV :: 'a option set)"
+ unfolding UNIV_option_conv
+ by(auto simp add: card_UNIV_option_def card_UNIV card_eq_0_iff Let_def intro: inj_Some dest: finite_imageD)
+ (subst card_insert_disjoint, auto simp add: card_eq_0_iff card_image inj_Some intro: finite_imageI card_ge_0_finite)
+qed
+
+end
+
+subsection {* Code setup for equality on sets *}
+
+definition eq_set :: "'a :: card_UNIV set \<Rightarrow> 'a :: card_UNIV set \<Rightarrow> bool"
+where [simp, code del]: "eq_set = op ="
+
+lemmas [code_unfold] = eq_set_def[symmetric]
+
+lemma card_Compl:
+ "finite A \<Longrightarrow> card (- A) = card (UNIV :: 'a set) - card (A :: 'a set)"
+by (metis Compl_eq_Diff_UNIV card_Diff_subset top_greatest)
+
+lemma eq_set_code [code]:
+ fixes xs ys :: "'a :: card_UNIV list"
+ defines "rhs \<equiv>
+ let n = card_UNIV TYPE('a)
+ in if n = 0 then False else
+ let xs' = remdups xs; ys' = remdups ys
+ 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')"
+ shows "eq_set (List.coset xs) (set ys) \<longleftrightarrow> rhs" (is ?thesis1)
+ and "eq_set (set ys) (List.coset xs) \<longleftrightarrow> rhs" (is ?thesis2)
+ 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)
+ 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)
+proof -
+ show ?thesis1 (is "?lhs \<longleftrightarrow> ?rhs")
+ proof
+ assume ?lhs thus ?rhs
+ 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)
+ next
+ assume ?rhs
+ 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
+ ultimately show ?lhs
+ 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)
+ qed
+ thus ?thesis2 unfolding eq_set_def by blast
+ show ?thesis3 ?thesis4 unfolding eq_set_def List.coset_def by blast+
+qed
+
+(* test code setup *)
+value [code] "List.coset [True] = set [False] \<and> set [] = List.coset [True, False]"
+
+end