src/HOL/Library/Cardinality.thy
 author Andreas Lochbihler Fri, 01 Jun 2012 20:40:34 +0200 changeset 48062 9014e78ccde2 parent 48060 1f4d00a7f59f child 48067 9f458b3efb5c permissions -rw-r--r--
improved code setup for card, finite, subset
```
(*  Title:      HOL/Library/Cardinality.thy
Author:     Brian Huffman, Andreas Lochbihler
*)

header {* Cardinality of types *}

theory Cardinality
imports "~~/src/HOL/Main"
begin

subsection {* Preliminary lemmas *}
(* These should be moved elsewhere *)

lemma (in type_definition) univ:
"UNIV = Abs ` A"
proof
show "Abs ` A \<subseteq> UNIV" by (rule subset_UNIV)
show "UNIV \<subseteq> Abs ` A"
proof
fix x :: 'b
have "x = Abs (Rep x)" by (rule Rep_inverse [symmetric])
moreover have "Rep x \<in> A" by (rule Rep)
ultimately show "x \<in> Abs ` A" by (rule image_eqI)
qed
qed

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 *}

syntax "_type_card" :: "type => nat" ("(1CARD/(1'(_')))")

translations "CARD('t)" => "CONST card (CONST UNIV \<Colon> 't set)"

let
fun card_univ_tr' ctxt _ [Const (@{const_syntax UNIV}, Type (_, [T, _]))] =
Syntax.const @{syntax_const "_type_card"} \$ Syntax_Phases.term_of_typ ctxt T;
in [(@{const_syntax card}, card_univ_tr')] end
*}

lemma card_unit [simp]: "CARD(unit) = 1"
unfolding UNIV_unit by simp

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)"

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)"

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)"

lemma card_nat [simp]: "CARD(nat) = 0"

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))"
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"
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"
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  *}

text {* Class finite already captures "at least 1" *}

lemma zero_less_card_finite [simp]: "0 < CARD('a::finite)"
unfolding neq0_conv [symmetric] by simp

lemma one_le_card_finite [simp]: "Suc 0 \<le> CARD('a::finite)"

text {* Class for cardinality "at least 2" *}

class card2 = finite +
assumes two_le_card: "2 \<le> CARD('a)"

lemma one_less_card: "Suc 0 < CARD('a::card2)"
using two_le_card [where 'a='a] by simp

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 *}

definition is_list_UNIV :: "'a list \<Rightarrow> bool"
where "is_list_UNIV xs = (let c = CARD('a) in if c = 0 then False else size (remdups xs) = c)"

lemmas [code_unfold] = is_list_UNIV_def[abs_def]

lemma is_list_UNIV_iff: "is_list_UNIV xs \<longleftrightarrow> set xs = UNIV"
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)

class card_UNIV =
fixes card_UNIV :: "'a itself \<Rightarrow> nat"
assumes card_UNIV: "card_UNIV x = CARD('a)"

lemma card_UNIV_code [code_unfold]: "CARD('a :: card_UNIV) = card_UNIV TYPE('a)"

lemma finite_UNIV_conv_card_UNIV [code_unfold]:
"finite (UNIV :: 'a :: card_UNIV set) \<longleftrightarrow> card_UNIV TYPE('a) > 0"

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

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

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

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

instantiation bool :: card_UNIV begin
definition "card_UNIV = (\<lambda>a :: bool itself. 2)"
end

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

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)
end

instantiation sum :: (card_UNIV, card_UNIV) card_UNIV begin
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_UNIV_sum)
end

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 by intro_classes (simp add: card_UNIV_fun_def card_UNIV Let_def card_fun)
end

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

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)

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)

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)

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 sets *}

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)

context fixes xs :: "'a :: card_UNIV list"
begin

definition card' :: "'a set \<Rightarrow> nat"
where [simp, code del, code_abbrev]: "card' = card"

lemma card'_code [code]:
"card' (set xs) = length (remdups xs)"
"card' (List.coset xs) =  card_UNIV TYPE('a) - length (remdups xs)"

lemma card'_UNIV [code_unfold]: "card' (UNIV :: 'a :: card_UNIV set) = card_UNIV TYPE('a)"

definition finite' :: "'a set \<Rightarrow> bool"
where [simp, code del, code_abbrev]: "finite' = finite"

lemma finite'_code [code]:
"finite' (set xs) \<longleftrightarrow> True"
"finite' (List.coset xs) \<longleftrightarrow> CARD('a) > 0"

definition subset' :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"
where [simp, code del, code_abbrev]: "subset' = op \<subseteq>"

lemma subset'_code [code]:
"subset' A (List.coset ys) \<longleftrightarrow> (\<forall>y \<in> set ys. y \<notin> A)"
"subset' (set ys) B \<longleftrightarrow> (\<forall>y \<in> set ys. y \<in> B)"
"subset' (List.coset xs) (set ys) \<longleftrightarrow> (let n = CARD('a) in n > 0 \<and> card(set (xs @ ys)) = n)"
by(auto simp add: Let_def card_gt_0_iff dest: card_eq_UNIV_imp_eq_UNIV intro: arg_cong[where f=card])
(metis finite_compl finite_set rev_finite_subset)

definition eq_set :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"
where [simp, code del, code_abbrev]: "eq_set = op ="

lemma eq_set_code [code]:
fixes ys
defines "rhs \<equiv>
let n = CARD('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

end

notepad begin (* test code setup *)
have "List.coset [True] = set [False] \<and> List.coset [] \<subseteq> List.set [True, False] \<and> finite (List.coset [True])"
by eval
end

hide_const (open) card' finite' subset' eq_set

end
```