(* 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