# Theory Cardinality

```(*  Title:      HOL/Library/Cardinality.thy
Author:     Brian Huffman, Andreas Lochbihler
*)

section ‹Cardinality of types›

theory Cardinality
imports Phantom_Type
begin

subsection ‹Preliminary lemmas›
(* These should be moved elsewhere *)

lemma (in type_definition) univ:
"UNIV = Abs ` A"
proof
show "Abs ` A ⊆ UNIV" by (rule subset_UNIV)
show "UNIV ⊆ Abs ` A"
proof
fix x :: 'b
have "x = Abs (Rep x)" by (rule Rep_inverse [symmetric])
moreover have "Rep x ∈ A" by (rule Rep)
ultimately show "x ∈ 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)

subsection ‹Cardinalities of types›

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

translations "CARD('t)" => "CONST card (CONST UNIV :: 't set)"

print_translation ‹
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_prod [simp]: "CARD('a × '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) ≠ 0 ∧ CARD('b) ≠ 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) ∉ range Some" by clarsimp
thus ?thesis
by (simp add: UNIV_option_conv card_eq_0_iff finite_range_Some 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: card_eq_0_iff card_Pow flip: 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 ⇒ 'b) = (if CARD('a) ≠ 0 ∧ CARD('b) ≠ 0 ∨ 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 (λys. the ∘ map_of (zip as ys)) (List.n_lists (length as) bs)"
have "UNIV = set ?xs"
proof(rule UNIV_eq_I)
fix f :: "'a ⇒ 'b"
from as have "f = the ∘ map_of (zip as (map f as))"
thus "f ∈ 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 ∈ set (List.n_lists (length as) bs)"
and ys: "ys ∈ set (List.n_lists (length as) bs)"
and eq: "the ∘ map_of (zip as xs) = the ∘ 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 "∃y. map_of (zip as xs) x = Some y" "∃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 ⇒ '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 = {λx :: 'a. b ::'b}"
proof(rule UNIV_eq_I)
fix x :: "'a ⇒ 'b"
{ fix y
have "x y ∈ UNIV" ..
hence "x y = b" unfolding b by simp }
thus "x ∈ {λx. b}" by(auto)
qed
have "CARD('a ⇒ '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

corollary finite_UNIV_fun:
"finite (UNIV :: ('a ⇒ 'b) set) ⟷
finite (UNIV :: 'a set) ∧ finite (UNIV :: 'b set) ∨ CARD('b) = 1"
(is "?lhs ⟷ ?rhs")
proof -
have "?lhs ⟷ CARD('a ⇒ 'b) > 0" by(simp add: card_gt_0_iff)
also have "… ⟷ CARD('a) > 0 ∧ CARD('b) > 0 ∨ CARD('b) = 1"
also have "… = ?rhs" by(simp add: card_gt_0_iff)
finally show ?thesis .
qed

lemma card_literal: "CARD(String.literal) = 0"

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 ≤ CARD('a::finite)"

class CARD_1 =
assumes CARD_1: "CARD ('a) = 1"
begin

subclass finite
proof
from CARD_1 show "finite (UNIV :: 'a set)"
using finite_UNIV_fun by fastforce
qed

end

text ‹Class for cardinality "at least 2"›

class card2 = finite +
assumes two_le_card: "2 ≤ 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 deciding finiteness of types›

type_synonym 'a finite_UNIV = "('a, bool) phantom"

class finite_UNIV =
fixes finite_UNIV :: "('a, bool) phantom"
assumes finite_UNIV: "finite_UNIV = Phantom('a) (finite (UNIV :: 'a set))"

lemma finite_UNIV_code [code_unfold]:
"finite (UNIV :: 'a :: finite_UNIV set)
⟷ of_phantom (finite_UNIV :: 'a finite_UNIV)"

subsection ‹A type class for computing the cardinality of types›

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

lemma is_list_UNIV_iff: "is_list_UNIV xs ⟷ 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)

type_synonym 'a card_UNIV = "('a, nat) phantom"

class card_UNIV = finite_UNIV +
fixes card_UNIV :: "'a card_UNIV"
assumes card_UNIV: "card_UNIV = Phantom('a) CARD('a)"

subsection ‹Instantiations for ‹card_UNIV››

instantiation nat :: card_UNIV begin
definition "finite_UNIV = Phantom(nat) False"
definition "card_UNIV = Phantom(nat) 0"
instance by intro_classes (simp_all add: finite_UNIV_nat_def card_UNIV_nat_def)
end

instantiation int :: card_UNIV begin
definition "finite_UNIV = Phantom(int) False"
definition "card_UNIV = Phantom(int) 0"
instance by intro_classes (simp_all add: card_UNIV_int_def finite_UNIV_int_def)
end

instantiation natural :: card_UNIV begin
definition "finite_UNIV = Phantom(natural) False"
definition "card_UNIV = Phantom(natural) 0"
instance
by standard
(auto simp add: finite_UNIV_natural_def card_UNIV_natural_def card_eq_0_iff
type_definition.univ [OF type_definition_natural] natural_eq_iff
dest!: finite_imageD intro: inj_onI)
end

instantiation integer :: card_UNIV begin
definition "finite_UNIV = Phantom(integer) False"
definition "card_UNIV = Phantom(integer) 0"
instance
by standard
(auto simp add: finite_UNIV_integer_def card_UNIV_integer_def card_eq_0_iff
type_definition.univ [OF type_definition_integer]
dest!: finite_imageD intro: inj_onI)
end

instantiation list :: (type) card_UNIV begin
definition "finite_UNIV = Phantom('a list) False"
definition "card_UNIV = Phantom('a list) 0"
instance by intro_classes (simp_all add: card_UNIV_list_def finite_UNIV_list_def infinite_UNIV_listI)
end

instantiation unit :: card_UNIV begin
definition "finite_UNIV = Phantom(unit) True"
definition "card_UNIV = Phantom(unit) 1"
instance by intro_classes (simp_all add: card_UNIV_unit_def finite_UNIV_unit_def)
end

instantiation bool :: card_UNIV begin
definition "finite_UNIV = Phantom(bool) True"
definition "card_UNIV = Phantom(bool) 2"
end

instantiation char :: card_UNIV begin
definition "finite_UNIV = Phantom(char) True"
definition "card_UNIV = Phantom(char) 256"
instance by intro_classes (simp_all add: card_UNIV_char_def card_UNIV_char finite_UNIV_char_def)
end

instantiation prod :: (finite_UNIV, finite_UNIV) finite_UNIV begin
definition "finite_UNIV = Phantom('a × 'b)
(of_phantom (finite_UNIV :: 'a finite_UNIV) ∧ of_phantom (finite_UNIV :: 'b finite_UNIV))"
instance by intro_classes (simp add: finite_UNIV_prod_def finite_UNIV finite_prod)
end

instantiation prod :: (card_UNIV, card_UNIV) card_UNIV begin
definition "card_UNIV = Phantom('a × 'b)
(of_phantom (card_UNIV :: 'a card_UNIV) * of_phantom (card_UNIV :: 'b card_UNIV))"
instance by intro_classes (simp add: card_UNIV_prod_def card_UNIV)
end

instantiation sum :: (finite_UNIV, finite_UNIV) finite_UNIV begin
definition "finite_UNIV = Phantom('a + 'b)
(of_phantom (finite_UNIV :: 'a finite_UNIV) ∧ of_phantom (finite_UNIV :: 'b finite_UNIV))"
instance
by intro_classes (simp add: finite_UNIV_sum_def finite_UNIV)
end

instantiation sum :: (card_UNIV, card_UNIV) card_UNIV begin
definition "card_UNIV = Phantom('a + 'b)
(let ca = of_phantom (card_UNIV :: 'a card_UNIV);
cb = of_phantom (card_UNIV :: 'b card_UNIV)
in if ca ≠ 0 ∧ cb ≠ 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" :: (finite_UNIV, card_UNIV) finite_UNIV begin
definition "finite_UNIV = Phantom('a ⇒ 'b)
(let cb = of_phantom (card_UNIV :: 'b card_UNIV)
in cb = 1 ∨ of_phantom (finite_UNIV :: 'a finite_UNIV) ∧ cb ≠ 0)"
instance
by intro_classes (auto simp add: finite_UNIV_fun_def Let_def card_UNIV finite_UNIV finite_UNIV_fun card_gt_0_iff)
end

instantiation "fun" :: (card_UNIV, card_UNIV) card_UNIV begin
definition "card_UNIV = Phantom('a ⇒ 'b)
(let ca = of_phantom (card_UNIV :: 'a card_UNIV);
cb = of_phantom (card_UNIV :: 'b card_UNIV)
in if ca ≠ 0 ∧ cb ≠ 0 ∨ 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 :: (finite_UNIV) finite_UNIV begin
definition "finite_UNIV = Phantom('a option) (of_phantom (finite_UNIV :: 'a finite_UNIV))"
instance by intro_classes (simp add: finite_UNIV_option_def finite_UNIV)
end

instantiation option :: (card_UNIV) card_UNIV begin
definition "card_UNIV = Phantom('a option)
(let c = of_phantom (card_UNIV :: 'a card_UNIV) in if c ≠ 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 "finite_UNIV = Phantom(String.literal) False"
definition "card_UNIV = Phantom(String.literal) 0"
instance
by intro_classes (simp_all add: card_UNIV_literal_def finite_UNIV_literal_def infinite_literal card_literal)
end

instantiation set :: (finite_UNIV) finite_UNIV begin
definition "finite_UNIV = Phantom('a set) (of_phantom (finite_UNIV :: 'a finite_UNIV))"
instance by intro_classes (simp add: finite_UNIV_set_def finite_UNIV Finite_Set.finite_set)
end

instantiation set :: (card_UNIV) card_UNIV begin
definition "card_UNIV = Phantom('a set)
(let c = of_phantom (card_UNIV :: 'a card_UNIV) 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⇩1]"
by(auto intro: finite_1.exhaust)

lemma UNIV_finite_2: "UNIV = set [finite_2.a⇩1, finite_2.a⇩2]"
by(auto intro: finite_2.exhaust)

lemma UNIV_finite_3: "UNIV = set [finite_3.a⇩1, finite_3.a⇩2, finite_3.a⇩3]"
by(auto intro: finite_3.exhaust)

lemma UNIV_finite_4: "UNIV = set [finite_4.a⇩1, finite_4.a⇩2, finite_4.a⇩3, finite_4.a⇩4]"
by(auto intro: finite_4.exhaust)

lemma UNIV_finite_5:
"UNIV = set [finite_5.a⇩1, finite_5.a⇩2, finite_5.a⇩3, finite_5.a⇩4, finite_5.a⇩5]"
by(auto intro: finite_5.exhaust)

instantiation Enum.finite_1 :: card_UNIV begin
definition "finite_UNIV = Phantom(Enum.finite_1) True"
definition "card_UNIV = Phantom(Enum.finite_1) 1"
instance
by intro_classes (simp_all add: UNIV_finite_1 card_UNIV_finite_1_def finite_UNIV_finite_1_def)
end

instantiation Enum.finite_2 :: card_UNIV begin
definition "finite_UNIV = Phantom(Enum.finite_2) True"
definition "card_UNIV = Phantom(Enum.finite_2) 2"
instance
by intro_classes (simp_all add: UNIV_finite_2 card_UNIV_finite_2_def finite_UNIV_finite_2_def)
end

instantiation Enum.finite_3 :: card_UNIV begin
definition "finite_UNIV = Phantom(Enum.finite_3) True"
definition "card_UNIV = Phantom(Enum.finite_3) 3"
instance
by intro_classes (simp_all add: UNIV_finite_3 card_UNIV_finite_3_def finite_UNIV_finite_3_def)
end

instantiation Enum.finite_4 :: card_UNIV begin
definition "finite_UNIV = Phantom(Enum.finite_4) True"
definition "card_UNIV = Phantom(Enum.finite_4) 4"
instance
by intro_classes (simp_all add: UNIV_finite_4 card_UNIV_finite_4_def finite_UNIV_finite_4_def)
end

instantiation Enum.finite_5 :: card_UNIV begin
definition "finite_UNIV = Phantom(Enum.finite_5) True"
definition "card_UNIV = Phantom(Enum.finite_5) 5"
instance
by intro_classes (simp_all add: UNIV_finite_5 card_UNIV_finite_5_def finite_UNIV_finite_5_def)
end

end
```