--- a/src/HOL/Library/Cardinality.thy Fri Jun 01 14:34:46 2012 +0200
+++ b/src/HOL/Library/Cardinality.thy Fri Jun 01 15:33:31 2012 +0200
@@ -27,6 +27,9 @@
lemma (in type_definition) card: "card (UNIV :: 'b set) = card A"
by (simp add: univ card_image inj_on_def Abs_inject)
+lemma finite_range_Some: "finite (range (Some :: 'a \<Rightarrow> 'a option)) = finite (UNIV :: 'a set)"
+by(auto dest: finite_imageD intro: inj_Some)
+
subsection {* Cardinalities of types *}
@@ -47,23 +50,104 @@
lemma card_prod [simp]: "CARD('a \<times> 'b) = CARD('a) * CARD('b)"
unfolding UNIV_Times_UNIV [symmetric] by (simp only: card_cartesian_product)
+lemma card_UNIV_sum: "CARD('a + 'b) = (if CARD('a) \<noteq> 0 \<and> CARD('b) \<noteq> 0 then CARD('a) + CARD('b) else 0)"
+unfolding UNIV_Plus_UNIV[symmetric]
+by(auto simp add: card_eq_0_iff card_Plus simp del: UNIV_Plus_UNIV)
+
lemma card_sum [simp]: "CARD('a + 'b) = CARD('a::finite) + CARD('b::finite)"
- unfolding UNIV_Plus_UNIV [symmetric] by (simp only: finite card_Plus)
+by(simp add: card_UNIV_sum)
+
+lemma card_UNIV_option: "CARD('a option) = (if CARD('a) = 0 then 0 else CARD('a) + 1)"
+proof -
+ have "(None :: 'a option) \<notin> range Some" by clarsimp
+ thus ?thesis
+ by(simp add: UNIV_option_conv card_eq_0_iff finite_range_Some card_insert_disjoint card_image)
+qed
lemma card_option [simp]: "CARD('a option) = Suc CARD('a::finite)"
- unfolding UNIV_option_conv
- apply (subgoal_tac "(None::'a option) \<notin> range Some")
- apply (simp add: card_image)
- apply fast
- done
+by(simp add: card_UNIV_option)
+
+lemma card_UNIV_set: "CARD('a set) = (if CARD('a) = 0 then 0 else 2 ^ CARD('a))"
+by(simp add: Pow_UNIV[symmetric] card_eq_0_iff card_Pow del: Pow_UNIV)
lemma card_set [simp]: "CARD('a set) = 2 ^ CARD('a::finite)"
- unfolding Pow_UNIV [symmetric]
- by (simp only: card_Pow finite)
+by(simp add: card_UNIV_set)
lemma card_nat [simp]: "CARD(nat) = 0"
by (simp add: card_eq_0_iff)
+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)"
+proof -
+ { assume "0 < CARD('a)" and "0 < CARD('b)"
+ 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('b) = length bs"
+ unfolding bs[symmetric] distinct_card[OF distb] ..
+ have ca: "CARD('a) = 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)
+ 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('a \<Rightarrow> 'b) = CARD('b) ^ CARD('a)" using cb ca by simp }
+ moreover {
+ assume cb: "CARD('b) = 1"
+ 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)
+ qed
+ have "CARD('a \<Rightarrow> 'b) = 1" unfolding eq by simp }
+ ultimately show ?thesis
+ by(auto simp del: One_nat_def)(auto simp add: card_eq_0_iff dest: finite_fun_UNIVD2 finite_fun_UNIVD1)
+qed
+
+lemma card_nibble: "CARD(nibble) = 16"
+unfolding UNIV_nibble by simp
+
+lemma card_UNIV_char: "CARD(char) = 256"
+proof -
+ have "inj (\<lambda>(x, y). Char x y)" by(auto intro: injI)
+ thus ?thesis unfolding UNIV_char by(simp add: card_image card_nibble)
+qed
+
+lemma card_literal: "CARD(String.literal) = 0"
+proof -
+ have "inj STR" by(auto intro: injI)
+ thus ?thesis by(simp add: type_definition.univ[OF type_definition_literal] card_image infinite_UNIV_listI)
+qed
subsection {* Classes with at least 1 and 2 *}
@@ -97,10 +181,6 @@
by(auto simp add: is_list_UNIV_def Let_def card_eq_0_iff List.card_set[symmetric]
dest: subst[where P="finite", OF _ finite_set] card_eq_UNIV_imp_eq_UNIV)
-lemma card_UNIV_eq_0_is_list_UNIV_False:
- "CARD('a) = 0 \<Longrightarrow> is_list_UNIV = (\<lambda>xs :: 'a list. False)"
-by(simp add: is_list_UNIV_def[abs_def])
-
class card_UNIV =
fixes card_UNIV :: "'a itself \<Rightarrow> nat"
assumes card_UNIV: "card_UNIV x = CARD('a)"
@@ -108,164 +188,119 @@
lemma card_UNIV_code [code_unfold]: "CARD('a :: card_UNIV) = card_UNIV TYPE('a)"
by(simp add: card_UNIV)
-subsection {* Instantiations for @{text "card_UNIV"} *}
+lemma finite_UNIV_conv_card_UNIV [code_unfold]:
+ "finite (UNIV :: 'a :: card_UNIV set) \<longleftrightarrow> card_UNIV TYPE('a) > 0"
+by(simp add: card_UNIV card_gt_0_iff)
-subsubsection {* @{typ "nat"} *}
+subsection {* Instantiations for @{text "card_UNIV"} *}
instantiation nat :: card_UNIV begin
definition "card_UNIV_class.card_UNIV = (\<lambda>a :: nat itself. 0)"
instance by intro_classes (simp add: card_UNIV_nat_def)
end
-subsubsection {* @{typ "int"} *}
-
instantiation int :: card_UNIV begin
definition "card_UNIV = (\<lambda>a :: int itself. 0)"
instance by intro_classes (simp add: card_UNIV_int_def infinite_UNIV_int)
end
-subsubsection {* @{typ "'a list"} *}
-
+print_classes
instantiation list :: (type) card_UNIV begin
definition "card_UNIV = (\<lambda>a :: 'a list itself. 0)"
instance by intro_classes (simp add: card_UNIV_list_def infinite_UNIV_listI)
end
-subsubsection {* @{typ "unit"} *}
-
instantiation unit :: card_UNIV begin
definition "card_UNIV = (\<lambda>a :: unit itself. 1)"
instance by intro_classes (simp add: card_UNIV_unit_def card_UNIV_unit)
end
-subsubsection {* @{typ "bool"} *}
-
instantiation bool :: card_UNIV begin
definition "card_UNIV = (\<lambda>a :: bool itself. 2)"
instance by(intro_classes)(simp add: card_UNIV_bool_def card_UNIV_bool)
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_class.card_UNIV = (\<lambda>a :: char itself. 256)"
instance by intro_classes (simp add: card_UNIV_char_def card_UNIV_char)
end
-subsubsection {* @{typ "'a \<times> 'b"} *}
-
instantiation prod :: (card_UNIV, card_UNIV) card_UNIV begin
definition "card_UNIV = (\<lambda>a :: ('a \<times> 'b) itself. card_UNIV (TYPE('a)) * card_UNIV (TYPE('b)))"
-instance
- by intro_classes (simp add: card_UNIV_prod_def card_UNIV UNIV_Times_UNIV[symmetric] card_cartesian_product del: UNIV_Times_UNIV)
+instance by intro_classes (simp add: card_UNIV_prod_def card_UNIV)
end
-subsubsection {* @{typ "'a + 'b"} *}
-
instantiation sum :: (card_UNIV, card_UNIV) card_UNIV begin
-definition "card_UNIV_class.card_UNIV = (\<lambda>a :: ('a + 'b) itself.
+definition "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
- by intro_classes (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)
+instance by intro_classes (auto simp add: card_UNIV_sum_def card_UNIV card_UNIV_sum)
end
-subsubsection {* @{typ "'a \<Rightarrow> 'b"} *}
-
instantiation "fun" :: (card_UNIV, card_UNIV) card_UNIV begin
-
-definition "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"
+definition "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 by intro_classes (simp add: card_UNIV_fun_def card_UNIV Let_def card_fun)
+end
- { 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)
- 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)
- 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
+instantiation option :: (card_UNIV) card_UNIV begin
+definition "card_UNIV = (\<lambda>a :: 'a option itself.
+ let c = card_UNIV (TYPE('a)) in if c \<noteq> 0 then Suc c else 0)"
+instance by intro_classes (simp add: card_UNIV_option_def card_UNIV card_UNIV_option)
+end
+instantiation String.literal :: card_UNIV begin
+definition "card_UNIV = (\<lambda>a :: String.literal itself. 0)"
+instance by intro_classes (simp add: card_UNIV_literal_def card_literal)
+end
+
+instantiation set :: (card_UNIV) card_UNIV begin
+definition "card_UNIV = (\<lambda>a :: 'a set itself.
+ let c = card_UNIV (TYPE('a)) in if c = 0 then 0 else 2 ^ c)"
+instance by intro_classes (simp add: card_UNIV_set_def card_UNIV_set card_UNIV)
end
-subsubsection {* @{typ "'a option"} *}
+
+lemma UNIV_finite_1: "UNIV = set [finite_1.a\<^isub>1]"
+by(auto intro: finite_1.exhaust)
+
+lemma UNIV_finite_2: "UNIV = set [finite_2.a\<^isub>1, finite_2.a\<^isub>2]"
+by(auto intro: finite_2.exhaust)
-instantiation option :: (card_UNIV) card_UNIV
-begin
+lemma UNIV_finite_3: "UNIV = set [finite_3.a\<^isub>1, finite_3.a\<^isub>2, finite_3.a\<^isub>3]"
+by(auto intro: finite_3.exhaust)
-definition "card_UNIV = (\<lambda>a :: 'a option itself. let c = card_UNIV (TYPE('a)) in if c \<noteq> 0 then Suc c else 0)"
+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]"
+by(auto intro: finite_4.exhaust)
+
+lemma UNIV_finite_5:
+ "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]"
+by(auto intro: finite_5.exhaust)
-instance proof
- fix x :: "'a option itself"
- show "card_UNIV x = card (UNIV :: 'a option set)"
- by(auto simp add: UNIV_option_conv 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
+instantiation Enum.finite_1 :: card_UNIV begin
+definition "card_UNIV = (\<lambda>a :: Enum.finite_1 itself. 1)"
+instance by intro_classes (simp add: UNIV_finite_1 card_UNIV_finite_1_def)
+end
+
+instantiation Enum.finite_2 :: card_UNIV begin
+definition "card_UNIV = (\<lambda>a :: Enum.finite_2 itself. 2)"
+instance by intro_classes (simp add: UNIV_finite_2 card_UNIV_finite_2_def)
+end
+instantiation Enum.finite_3 :: card_UNIV begin
+definition "card_UNIV = (\<lambda>a :: Enum.finite_3 itself. 3)"
+instance by intro_classes (simp add: UNIV_finite_3 card_UNIV_finite_3_def)
+end
+
+instantiation Enum.finite_4 :: card_UNIV begin
+definition "card_UNIV = (\<lambda>a :: Enum.finite_4 itself. 4)"
+instance by intro_classes (simp add: UNIV_finite_4 card_UNIV_finite_4_def)
+end
+
+instantiation Enum.finite_5 :: card_UNIV begin
+definition "card_UNIV = (\<lambda>a :: Enum.finite_5 itself. 5)"
+instance by intro_classes (simp add: UNIV_finite_5 card_UNIV_finite_5_def)
end
subsection {* Code setup for equality on sets *}