(* Author: Andreas Lochbihler, KIT *)
header {* A type class for computing the cardinality of a type's universe *}
theory Card_Univ imports Main begin
subsection {* A type class for computing the cardinality of a type's universe *}
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_nat_def:
"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_int_def:
"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_list_def:
"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
end