(* Author: Florian Haftmann, TU Muenchen *)
section \<open>Finite types as explicit enumerations\<close>
theory Enum
imports Map Groups_List
begin
subsection \<open>Class \<open>enum\<close>\<close>
class enum =
fixes enum :: "'a list"
fixes enum_all :: "('a \<Rightarrow> bool) \<Rightarrow> bool"
fixes enum_ex :: "('a \<Rightarrow> bool) \<Rightarrow> bool"
assumes UNIV_enum: "UNIV = set enum"
and enum_distinct: "distinct enum"
assumes enum_all_UNIV: "enum_all P \<longleftrightarrow> Ball UNIV P"
assumes enum_ex_UNIV: "enum_ex P \<longleftrightarrow> Bex UNIV P"
\<comment> \<open>tailored towards simple instantiation\<close>
begin
subclass finite proof
qed (simp add: UNIV_enum)
lemma enum_UNIV:
"set enum = UNIV"
by (simp only: UNIV_enum)
lemma in_enum: "x \<in> set enum"
by (simp add: enum_UNIV)
lemma enum_eq_I:
assumes "\<And>x. x \<in> set xs"
shows "set enum = set xs"
proof -
from assms UNIV_eq_I have "UNIV = set xs" by auto
with enum_UNIV show ?thesis by simp
qed
lemma card_UNIV_length_enum:
"card (UNIV :: 'a set) = length enum"
by (simp add: UNIV_enum distinct_card enum_distinct)
lemma enum_all [simp]:
"enum_all = HOL.All"
by (simp add: fun_eq_iff enum_all_UNIV)
lemma enum_ex [simp]:
"enum_ex = HOL.Ex"
by (simp add: fun_eq_iff enum_ex_UNIV)
end
subsection \<open>Implementations using @{class enum}\<close>
subsubsection \<open>Unbounded operations and quantifiers\<close>
lemma Collect_code [code]:
"Collect P = set (filter P enum)"
by (simp add: enum_UNIV)
lemma vimage_code [code]:
"f -` B = set (filter (%x. f x : B) enum_class.enum)"
unfolding vimage_def Collect_code ..
definition card_UNIV :: "'a itself \<Rightarrow> nat"
where
[code del]: "card_UNIV TYPE('a) = card (UNIV :: 'a set)"
lemma [code]:
"card_UNIV TYPE('a :: enum) = card (set (Enum.enum :: 'a list))"
by (simp only: card_UNIV_def enum_UNIV)
lemma all_code [code]: "(\<forall>x. P x) \<longleftrightarrow> enum_all P"
by simp
lemma exists_code [code]: "(\<exists>x. P x) \<longleftrightarrow> enum_ex P"
by simp
lemma exists1_code [code]: "(\<exists>!x. P x) \<longleftrightarrow> list_ex1 P enum"
by (auto simp add: list_ex1_iff enum_UNIV)
subsubsection \<open>An executable choice operator\<close>
definition
[code del]: "enum_the = The"
lemma [code]:
"The P = (case filter P enum of [x] => x | _ => enum_the P)"
proof -
{
fix a
assume filter_enum: "filter P enum = [a]"
have "The P = a"
proof (rule the_equality)
fix x
assume "P x"
show "x = a"
proof (rule ccontr)
assume "x \<noteq> a"
from filter_enum obtain us vs
where enum_eq: "enum = us @ [a] @ vs"
and "\<forall> x \<in> set us. \<not> P x"
and "\<forall> x \<in> set vs. \<not> P x"
and "P a"
by (auto simp add: filter_eq_Cons_iff) (simp only: filter_empty_conv[symmetric])
with \<open>P x\<close> in_enum[of x, unfolded enum_eq] \<open>x \<noteq> a\<close> show "False" by auto
qed
next
from filter_enum show "P a" by (auto simp add: filter_eq_Cons_iff)
qed
}
from this show ?thesis
unfolding enum_the_def by (auto split: list.split)
qed
declare [[code abort: enum_the]]
code_printing
constant enum_the \<rightharpoonup> (Eval) "(fn '_ => raise Match)"
subsubsection \<open>Equality and order on functions\<close>
instantiation "fun" :: (enum, equal) equal
begin
definition
"HOL.equal f g \<longleftrightarrow> (\<forall>x \<in> set enum. f x = g x)"
instance proof
qed (simp_all add: equal_fun_def fun_eq_iff enum_UNIV)
end
lemma [code]:
"HOL.equal f g \<longleftrightarrow> enum_all (%x. f x = g x)"
by (auto simp add: equal fun_eq_iff)
lemma [code nbe]:
"HOL.equal (f :: _ \<Rightarrow> _) f \<longleftrightarrow> True"
by (fact equal_refl)
lemma order_fun [code]:
fixes f g :: "'a::enum \<Rightarrow> 'b::order"
shows "f \<le> g \<longleftrightarrow> enum_all (\<lambda>x. f x \<le> g x)"
and "f < g \<longleftrightarrow> f \<le> g \<and> enum_ex (\<lambda>x. f x \<noteq> g x)"
by (simp_all add: fun_eq_iff le_fun_def order_less_le)
subsubsection \<open>Operations on relations\<close>
lemma [code]:
"Id = image (\<lambda>x. (x, x)) (set Enum.enum)"
by (auto intro: imageI in_enum)
lemma tranclp_unfold [code]:
"tranclp r a b \<longleftrightarrow> (a, b) \<in> trancl {(x, y). r x y}"
by (simp add: trancl_def)
lemma rtranclp_rtrancl_eq [code]:
"rtranclp r x y \<longleftrightarrow> (x, y) \<in> rtrancl {(x, y). r x y}"
by (simp add: rtrancl_def)
lemma max_ext_eq [code]:
"max_ext R = {(X, Y). finite X \<and> finite Y \<and> Y \<noteq> {} \<and> (\<forall>x. x \<in> X \<longrightarrow> (\<exists>xa \<in> Y. (x, xa) \<in> R))}"
by (auto simp add: max_ext.simps)
lemma max_extp_eq [code]:
"max_extp r x y \<longleftrightarrow> (x, y) \<in> max_ext {(x, y). r x y}"
by (simp add: max_ext_def)
lemma mlex_eq [code]:
"f <*mlex*> R = {(x, y). f x < f y \<or> (f x \<le> f y \<and> (x, y) \<in> R)}"
by (auto simp add: mlex_prod_def)
subsubsection \<open>Bounded accessible part\<close>
primrec bacc :: "('a \<times> 'a) set \<Rightarrow> nat \<Rightarrow> 'a set"
where
"bacc r 0 = {x. \<forall> y. (y, x) \<notin> r}"
| "bacc r (Suc n) = (bacc r n \<union> {x. \<forall>y. (y, x) \<in> r \<longrightarrow> y \<in> bacc r n})"
lemma bacc_subseteq_acc:
"bacc r n \<subseteq> Wellfounded.acc r"
by (induct n) (auto intro: acc.intros)
lemma bacc_mono:
"n \<le> m \<Longrightarrow> bacc r n \<subseteq> bacc r m"
by (induct rule: dec_induct) auto
lemma bacc_upper_bound:
"bacc (r :: ('a \<times> 'a) set) (card (UNIV :: 'a::finite set)) = (\<Union>n. bacc r n)"
proof -
have "mono (bacc r)" unfolding mono_def by (simp add: bacc_mono)
moreover have "\<forall>n. bacc r n = bacc r (Suc n) \<longrightarrow> bacc r (Suc n) = bacc r (Suc (Suc n))" by auto
moreover have "finite (range (bacc r))" by auto
ultimately show ?thesis
by (intro finite_mono_strict_prefix_implies_finite_fixpoint)
(auto intro: finite_mono_remains_stable_implies_strict_prefix)
qed
lemma acc_subseteq_bacc:
assumes "finite r"
shows "Wellfounded.acc r \<subseteq> (\<Union>n. bacc r n)"
proof
fix x
assume "x : Wellfounded.acc r"
then have "\<exists> n. x : bacc r n"
proof (induct x arbitrary: rule: acc.induct)
case (accI x)
then have "\<forall>y. \<exists> n. (y, x) \<in> r --> y : bacc r n" by simp
from choice[OF this] obtain n where n: "\<forall>y. (y, x) \<in> r \<longrightarrow> y \<in> bacc r (n y)" ..
obtain n where "\<And>y. (y, x) : r \<Longrightarrow> y : bacc r n"
proof
fix y assume y: "(y, x) : r"
with n have "y : bacc r (n y)" by auto
moreover have "n y <= Max ((%(y, x). n y) ` r)"
using y \<open>finite r\<close> by (auto intro!: Max_ge)
note bacc_mono[OF this, of r]
ultimately show "y : bacc r (Max ((%(y, x). n y) ` r))" by auto
qed
then show ?case
by (auto simp add: Let_def intro!: exI[of _ "Suc n"])
qed
then show "x : (UN n. bacc r n)" by auto
qed
lemma acc_bacc_eq:
fixes A :: "('a :: finite \<times> 'a) set"
assumes "finite A"
shows "Wellfounded.acc A = bacc A (card (UNIV :: 'a set))"
using assms by (metis acc_subseteq_bacc bacc_subseteq_acc bacc_upper_bound order_eq_iff)
lemma [code]:
fixes xs :: "('a::finite \<times> 'a) list"
shows "Wellfounded.acc (set xs) = bacc (set xs) (card_UNIV TYPE('a))"
by (simp add: card_UNIV_def acc_bacc_eq)
subsection \<open>Default instances for @{class enum}\<close>
lemma map_of_zip_enum_is_Some:
assumes "length ys = length (enum :: 'a::enum list)"
shows "\<exists>y. map_of (zip (enum :: 'a::enum list) ys) x = Some y"
proof -
from assms have "x \<in> set (enum :: 'a::enum list) \<longleftrightarrow>
(\<exists>y. map_of (zip (enum :: 'a::enum list) ys) x = Some y)"
by (auto intro!: map_of_zip_is_Some)
then show ?thesis using enum_UNIV by auto
qed
lemma map_of_zip_enum_inject:
fixes xs ys :: "'b::enum list"
assumes length: "length xs = length (enum :: 'a::enum list)"
"length ys = length (enum :: 'a::enum list)"
and map_of: "the \<circ> map_of (zip (enum :: 'a::enum list) xs) = the \<circ> map_of (zip (enum :: 'a::enum list) ys)"
shows "xs = ys"
proof -
have "map_of (zip (enum :: 'a list) xs) = map_of (zip (enum :: 'a list) ys)"
proof
fix x :: 'a
from length map_of_zip_enum_is_Some obtain y1 y2
where "map_of (zip (enum :: 'a list) xs) x = Some y1"
and "map_of (zip (enum :: 'a list) ys) x = Some y2" by blast
moreover from map_of
have "the (map_of (zip (enum :: 'a::enum list) xs) x) = the (map_of (zip (enum :: 'a::enum list) ys) x)"
by (auto dest: fun_cong)
ultimately show "map_of (zip (enum :: 'a::enum list) xs) x = map_of (zip (enum :: 'a::enum list) ys) x"
by simp
qed
with length enum_distinct show "xs = ys" by (rule map_of_zip_inject)
qed
definition all_n_lists :: "(('a :: enum) list \<Rightarrow> bool) \<Rightarrow> nat \<Rightarrow> bool"
where
"all_n_lists P n \<longleftrightarrow> (\<forall>xs \<in> set (List.n_lists n enum). P xs)"
lemma [code]:
"all_n_lists P n \<longleftrightarrow> (if n = 0 then P [] else enum_all (%x. all_n_lists (%xs. P (x # xs)) (n - 1)))"
unfolding all_n_lists_def enum_all
by (cases n) (auto simp add: enum_UNIV)
definition ex_n_lists :: "(('a :: enum) list \<Rightarrow> bool) \<Rightarrow> nat \<Rightarrow> bool"
where
"ex_n_lists P n \<longleftrightarrow> (\<exists>xs \<in> set (List.n_lists n enum). P xs)"
lemma [code]:
"ex_n_lists P n \<longleftrightarrow> (if n = 0 then P [] else enum_ex (%x. ex_n_lists (%xs. P (x # xs)) (n - 1)))"
unfolding ex_n_lists_def enum_ex
by (cases n) (auto simp add: enum_UNIV)
instantiation "fun" :: (enum, enum) enum
begin
definition
"enum = map (\<lambda>ys. the o map_of (zip (enum::'a list) ys)) (List.n_lists (length (enum::'a::enum list)) enum)"
definition
"enum_all P = all_n_lists (\<lambda>bs. P (the o map_of (zip enum bs))) (length (enum :: 'a list))"
definition
"enum_ex P = ex_n_lists (\<lambda>bs. P (the o map_of (zip enum bs))) (length (enum :: 'a list))"
instance proof
show "UNIV = set (enum :: ('a \<Rightarrow> 'b) list)"
proof (rule UNIV_eq_I)
fix f :: "'a \<Rightarrow> 'b"
have "f = the \<circ> map_of (zip (enum :: 'a::enum list) (map f enum))"
by (auto simp add: map_of_zip_map fun_eq_iff intro: in_enum)
then show "f \<in> set enum"
by (auto simp add: enum_fun_def set_n_lists intro: in_enum)
qed
next
from map_of_zip_enum_inject
show "distinct (enum :: ('a \<Rightarrow> 'b) list)"
by (auto intro!: inj_onI simp add: enum_fun_def
distinct_map distinct_n_lists enum_distinct set_n_lists)
next
fix P
show "enum_all (P :: ('a \<Rightarrow> 'b) \<Rightarrow> bool) = Ball UNIV P"
proof
assume "enum_all P"
show "Ball UNIV P"
proof
fix f :: "'a \<Rightarrow> 'b"
have f: "f = the \<circ> map_of (zip (enum :: 'a::enum list) (map f enum))"
by (auto simp add: map_of_zip_map fun_eq_iff intro: in_enum)
from \<open>enum_all P\<close> have "P (the \<circ> map_of (zip enum (map f enum)))"
unfolding enum_all_fun_def all_n_lists_def
apply (simp add: set_n_lists)
apply (erule_tac x="map f enum" in allE)
apply (auto intro!: in_enum)
done
from this f show "P f" by auto
qed
next
assume "Ball UNIV P"
from this show "enum_all P"
unfolding enum_all_fun_def all_n_lists_def by auto
qed
next
fix P
show "enum_ex (P :: ('a \<Rightarrow> 'b) \<Rightarrow> bool) = Bex UNIV P"
proof
assume "enum_ex P"
from this show "Bex UNIV P"
unfolding enum_ex_fun_def ex_n_lists_def by auto
next
assume "Bex UNIV P"
from this obtain f where "P f" ..
have f: "f = the \<circ> map_of (zip (enum :: 'a::enum list) (map f enum))"
by (auto simp add: map_of_zip_map fun_eq_iff intro: in_enum)
from \<open>P f\<close> this have "P (the \<circ> map_of (zip (enum :: 'a::enum list) (map f enum)))"
by auto
from this show "enum_ex P"
unfolding enum_ex_fun_def ex_n_lists_def
apply (auto simp add: set_n_lists)
apply (rule_tac x="map f enum" in exI)
apply (auto intro!: in_enum)
done
qed
qed
end
lemma enum_fun_code [code]: "enum = (let enum_a = (enum :: 'a::{enum, equal} list)
in map (\<lambda>ys. the o map_of (zip enum_a ys)) (List.n_lists (length enum_a) enum))"
by (simp add: enum_fun_def Let_def)
lemma enum_all_fun_code [code]:
"enum_all P = (let enum_a = (enum :: 'a::{enum, equal} list)
in all_n_lists (\<lambda>bs. P (the o map_of (zip enum_a bs))) (length enum_a))"
by (simp only: enum_all_fun_def Let_def)
lemma enum_ex_fun_code [code]:
"enum_ex P = (let enum_a = (enum :: 'a::{enum, equal} list)
in ex_n_lists (\<lambda>bs. P (the o map_of (zip enum_a bs))) (length enum_a))"
by (simp only: enum_ex_fun_def Let_def)
instantiation set :: (enum) enum
begin
definition
"enum = map set (sublists enum)"
definition
"enum_all P \<longleftrightarrow> (\<forall>A\<in>set enum. P (A::'a set))"
definition
"enum_ex P \<longleftrightarrow> (\<exists>A\<in>set enum. P (A::'a set))"
instance proof
qed (simp_all add: enum_set_def enum_all_set_def enum_ex_set_def sublists_powset distinct_set_sublists
enum_distinct enum_UNIV)
end
instantiation unit :: enum
begin
definition
"enum = [()]"
definition
"enum_all P = P ()"
definition
"enum_ex P = P ()"
instance proof
qed (auto simp add: enum_unit_def enum_all_unit_def enum_ex_unit_def)
end
instantiation bool :: enum
begin
definition
"enum = [False, True]"
definition
"enum_all P \<longleftrightarrow> P False \<and> P True"
definition
"enum_ex P \<longleftrightarrow> P False \<or> P True"
instance proof
qed (simp_all only: enum_bool_def enum_all_bool_def enum_ex_bool_def UNIV_bool, simp_all)
end
instantiation prod :: (enum, enum) enum
begin
definition
"enum = List.product enum enum"
definition
"enum_all P = enum_all (%x. enum_all (%y. P (x, y)))"
definition
"enum_ex P = enum_ex (%x. enum_ex (%y. P (x, y)))"
instance
by standard
(simp_all add: enum_prod_def distinct_product
enum_UNIV enum_distinct enum_all_prod_def enum_ex_prod_def)
end
instantiation sum :: (enum, enum) enum
begin
definition
"enum = map Inl enum @ map Inr enum"
definition
"enum_all P \<longleftrightarrow> enum_all (\<lambda>x. P (Inl x)) \<and> enum_all (\<lambda>x. P (Inr x))"
definition
"enum_ex P \<longleftrightarrow> enum_ex (\<lambda>x. P (Inl x)) \<or> enum_ex (\<lambda>x. P (Inr x))"
instance proof
qed (simp_all only: enum_sum_def enum_all_sum_def enum_ex_sum_def UNIV_sum,
auto simp add: enum_UNIV distinct_map enum_distinct)
end
instantiation option :: (enum) enum
begin
definition
"enum = None # map Some enum"
definition
"enum_all P \<longleftrightarrow> P None \<and> enum_all (\<lambda>x. P (Some x))"
definition
"enum_ex P \<longleftrightarrow> P None \<or> enum_ex (\<lambda>x. P (Some x))"
instance proof
qed (simp_all only: enum_option_def enum_all_option_def enum_ex_option_def UNIV_option_conv,
auto simp add: distinct_map enum_UNIV enum_distinct)
end
subsection \<open>Small finite types\<close>
text \<open>We define small finite types for use in Quickcheck\<close>
datatype (plugins only: code "quickcheck" extraction) finite_1 =
a\<^sub>1
notation (output) a\<^sub>1 ("a\<^sub>1")
lemma UNIV_finite_1:
"UNIV = {a\<^sub>1}"
by (auto intro: finite_1.exhaust)
instantiation finite_1 :: enum
begin
definition
"enum = [a\<^sub>1]"
definition
"enum_all P = P a\<^sub>1"
definition
"enum_ex P = P a\<^sub>1"
instance proof
qed (simp_all only: enum_finite_1_def enum_all_finite_1_def enum_ex_finite_1_def UNIV_finite_1, simp_all)
end
instantiation finite_1 :: linorder
begin
definition less_finite_1 :: "finite_1 \<Rightarrow> finite_1 \<Rightarrow> bool"
where
"x < (y :: finite_1) \<longleftrightarrow> False"
definition less_eq_finite_1 :: "finite_1 \<Rightarrow> finite_1 \<Rightarrow> bool"
where
"x \<le> (y :: finite_1) \<longleftrightarrow> True"
instance
apply (intro_classes)
apply (auto simp add: less_finite_1_def less_eq_finite_1_def)
apply (metis finite_1.exhaust)
done
end
instance finite_1 :: "{dense_linorder, wellorder}"
by intro_classes (simp_all add: less_finite_1_def)
instantiation finite_1 :: complete_lattice
begin
definition [simp]: "Inf = (\<lambda>_. a\<^sub>1)"
definition [simp]: "Sup = (\<lambda>_. a\<^sub>1)"
definition [simp]: "bot = a\<^sub>1"
definition [simp]: "top = a\<^sub>1"
definition [simp]: "inf = (\<lambda>_ _. a\<^sub>1)"
definition [simp]: "sup = (\<lambda>_ _. a\<^sub>1)"
instance by intro_classes(simp_all add: less_eq_finite_1_def)
end
instance finite_1 :: complete_distrib_lattice
by intro_classes(simp_all add: INF_def SUP_def)
instance finite_1 :: complete_linorder ..
lemma finite_1_eq: "x = a\<^sub>1"
by(cases x) simp
simproc_setup finite_1_eq ("x::finite_1") = \<open>
fn _ => fn _ => fn ct =>
(case Thm.term_of ct of
Const (@{const_name a\<^sub>1}, _) => NONE
| _ => SOME (mk_meta_eq @{thm finite_1_eq}))
\<close>
instantiation finite_1 :: complete_boolean_algebra
begin
definition [simp]: "op - = (\<lambda>_ _. a\<^sub>1)"
definition [simp]: "uminus = (\<lambda>_. a\<^sub>1)"
instance by intro_classes simp_all
end
instantiation finite_1 ::
"{linordered_ring_strict, linordered_comm_semiring_strict, ordered_comm_ring,
ordered_cancel_comm_monoid_diff, comm_monoid_mult, ordered_ring_abs,
one, Divides.div, sgn_if, inverse}"
begin
definition [simp]: "Groups.zero = a\<^sub>1"
definition [simp]: "Groups.one = a\<^sub>1"
definition [simp]: "op + = (\<lambda>_ _. a\<^sub>1)"
definition [simp]: "op * = (\<lambda>_ _. a\<^sub>1)"
definition [simp]: "op mod = (\<lambda>_ _. a\<^sub>1)"
definition [simp]: "abs = (\<lambda>_. a\<^sub>1)"
definition [simp]: "sgn = (\<lambda>_. a\<^sub>1)"
definition [simp]: "inverse = (\<lambda>_. a\<^sub>1)"
definition [simp]: "divide = (\<lambda>_ _. a\<^sub>1)"
instance by intro_classes(simp_all add: less_finite_1_def)
end
declare [[simproc del: finite_1_eq]]
hide_const (open) a\<^sub>1
datatype (plugins only: code "quickcheck" extraction) finite_2 =
a\<^sub>1 | a\<^sub>2
notation (output) a\<^sub>1 ("a\<^sub>1")
notation (output) a\<^sub>2 ("a\<^sub>2")
lemma UNIV_finite_2:
"UNIV = {a\<^sub>1, a\<^sub>2}"
by (auto intro: finite_2.exhaust)
instantiation finite_2 :: enum
begin
definition
"enum = [a\<^sub>1, a\<^sub>2]"
definition
"enum_all P \<longleftrightarrow> P a\<^sub>1 \<and> P a\<^sub>2"
definition
"enum_ex P \<longleftrightarrow> P a\<^sub>1 \<or> P a\<^sub>2"
instance proof
qed (simp_all only: enum_finite_2_def enum_all_finite_2_def enum_ex_finite_2_def UNIV_finite_2, simp_all)
end
instantiation finite_2 :: linorder
begin
definition less_finite_2 :: "finite_2 \<Rightarrow> finite_2 \<Rightarrow> bool"
where
"x < y \<longleftrightarrow> x = a\<^sub>1 \<and> y = a\<^sub>2"
definition less_eq_finite_2 :: "finite_2 \<Rightarrow> finite_2 \<Rightarrow> bool"
where
"x \<le> y \<longleftrightarrow> x = y \<or> x < (y :: finite_2)"
instance
apply (intro_classes)
apply (auto simp add: less_finite_2_def less_eq_finite_2_def)
apply (metis finite_2.nchotomy)+
done
end
instance finite_2 :: wellorder
by(rule wf_wellorderI)(simp add: less_finite_2_def, intro_classes)
instantiation finite_2 :: complete_lattice
begin
definition "\<Sqinter>A = (if a\<^sub>1 \<in> A then a\<^sub>1 else a\<^sub>2)"
definition "\<Squnion>A = (if a\<^sub>2 \<in> A then a\<^sub>2 else a\<^sub>1)"
definition [simp]: "bot = a\<^sub>1"
definition [simp]: "top = a\<^sub>2"
definition "x \<sqinter> y = (if x = a\<^sub>1 \<or> y = a\<^sub>1 then a\<^sub>1 else a\<^sub>2)"
definition "x \<squnion> y = (if x = a\<^sub>2 \<or> y = a\<^sub>2 then a\<^sub>2 else a\<^sub>1)"
lemma neq_finite_2_a\<^sub>1_iff [simp]: "x \<noteq> a\<^sub>1 \<longleftrightarrow> x = a\<^sub>2"
by(cases x) simp_all
lemma neq_finite_2_a\<^sub>1_iff' [simp]: "a\<^sub>1 \<noteq> x \<longleftrightarrow> x = a\<^sub>2"
by(cases x) simp_all
lemma neq_finite_2_a\<^sub>2_iff [simp]: "x \<noteq> a\<^sub>2 \<longleftrightarrow> x = a\<^sub>1"
by(cases x) simp_all
lemma neq_finite_2_a\<^sub>2_iff' [simp]: "a\<^sub>2 \<noteq> x \<longleftrightarrow> x = a\<^sub>1"
by(cases x) simp_all
instance
proof
fix x :: finite_2 and A
assume "x \<in> A"
then show "\<Sqinter>A \<le> x" "x \<le> \<Squnion>A"
by(case_tac [!] x)(auto simp add: less_eq_finite_2_def less_finite_2_def Inf_finite_2_def Sup_finite_2_def)
qed(auto simp add: less_eq_finite_2_def less_finite_2_def inf_finite_2_def sup_finite_2_def Inf_finite_2_def Sup_finite_2_def)
end
instance finite_2 :: complete_distrib_lattice
by(intro_classes)(auto simp add: INF_def SUP_def sup_finite_2_def inf_finite_2_def Inf_finite_2_def Sup_finite_2_def)
instance finite_2 :: complete_linorder ..
instantiation finite_2 :: "{field, abs_if, ring_div, sgn_if, semiring_div}" begin
definition [simp]: "0 = a\<^sub>1"
definition [simp]: "1 = a\<^sub>2"
definition "x + y = (case (x, y) of (a\<^sub>1, a\<^sub>1) \<Rightarrow> a\<^sub>1 | (a\<^sub>2, a\<^sub>2) \<Rightarrow> a\<^sub>1 | _ \<Rightarrow> a\<^sub>2)"
definition "uminus = (\<lambda>x :: finite_2. x)"
definition "op - = (op + :: finite_2 \<Rightarrow> _)"
definition "x * y = (case (x, y) of (a\<^sub>2, a\<^sub>2) \<Rightarrow> a\<^sub>2 | _ \<Rightarrow> a\<^sub>1)"
definition "inverse = (\<lambda>x :: finite_2. x)"
definition "divide = (op * :: finite_2 \<Rightarrow> _)"
definition "abs = (\<lambda>x :: finite_2. x)"
definition "x mod y = (case (x, y) of (a\<^sub>2, a\<^sub>1) \<Rightarrow> a\<^sub>2 | _ \<Rightarrow> a\<^sub>1)"
definition "sgn = (\<lambda>x :: finite_2. x)"
instance
by intro_classes
(simp_all add: plus_finite_2_def uminus_finite_2_def minus_finite_2_def times_finite_2_def
inverse_finite_2_def divide_finite_2_def abs_finite_2_def mod_finite_2_def sgn_finite_2_def
split: finite_2.splits)
end
lemma two_finite_2 [simp]:
"2 = a\<^sub>1"
by (simp add: numeral.simps plus_finite_2_def)
hide_const (open) a\<^sub>1 a\<^sub>2
datatype (plugins only: code "quickcheck" extraction) finite_3 =
a\<^sub>1 | a\<^sub>2 | a\<^sub>3
notation (output) a\<^sub>1 ("a\<^sub>1")
notation (output) a\<^sub>2 ("a\<^sub>2")
notation (output) a\<^sub>3 ("a\<^sub>3")
lemma UNIV_finite_3:
"UNIV = {a\<^sub>1, a\<^sub>2, a\<^sub>3}"
by (auto intro: finite_3.exhaust)
instantiation finite_3 :: enum
begin
definition
"enum = [a\<^sub>1, a\<^sub>2, a\<^sub>3]"
definition
"enum_all P \<longleftrightarrow> P a\<^sub>1 \<and> P a\<^sub>2 \<and> P a\<^sub>3"
definition
"enum_ex P \<longleftrightarrow> P a\<^sub>1 \<or> P a\<^sub>2 \<or> P a\<^sub>3"
instance proof
qed (simp_all only: enum_finite_3_def enum_all_finite_3_def enum_ex_finite_3_def UNIV_finite_3, simp_all)
end
instantiation finite_3 :: linorder
begin
definition less_finite_3 :: "finite_3 \<Rightarrow> finite_3 \<Rightarrow> bool"
where
"x < y = (case x of a\<^sub>1 \<Rightarrow> y \<noteq> a\<^sub>1 | a\<^sub>2 \<Rightarrow> y = a\<^sub>3 | a\<^sub>3 \<Rightarrow> False)"
definition less_eq_finite_3 :: "finite_3 \<Rightarrow> finite_3 \<Rightarrow> bool"
where
"x \<le> y \<longleftrightarrow> x = y \<or> x < (y :: finite_3)"
instance proof (intro_classes)
qed (auto simp add: less_finite_3_def less_eq_finite_3_def split: finite_3.split_asm)
end
instance finite_3 :: wellorder
proof(rule wf_wellorderI)
have "inv_image less_than (case_finite_3 0 1 2) = {(x, y). x < y}"
by(auto simp add: less_finite_3_def split: finite_3.splits)
from this[symmetric] show "wf \<dots>" by simp
qed intro_classes
instantiation finite_3 :: complete_lattice
begin
definition "\<Sqinter>A = (if a\<^sub>1 \<in> A then a\<^sub>1 else if a\<^sub>2 \<in> A then a\<^sub>2 else a\<^sub>3)"
definition "\<Squnion>A = (if a\<^sub>3 \<in> A then a\<^sub>3 else if a\<^sub>2 \<in> A then a\<^sub>2 else a\<^sub>1)"
definition [simp]: "bot = a\<^sub>1"
definition [simp]: "top = a\<^sub>3"
definition [simp]: "inf = (min :: finite_3 \<Rightarrow> _)"
definition [simp]: "sup = (max :: finite_3 \<Rightarrow> _)"
instance
proof
fix x :: finite_3 and A
assume "x \<in> A"
then show "\<Sqinter>A \<le> x" "x \<le> \<Squnion>A"
by(case_tac [!] x)(auto simp add: Inf_finite_3_def Sup_finite_3_def less_eq_finite_3_def less_finite_3_def)
next
fix A and z :: finite_3
assume "\<And>x. x \<in> A \<Longrightarrow> z \<le> x"
then show "z \<le> \<Sqinter>A"
by(cases z)(auto simp add: Inf_finite_3_def less_eq_finite_3_def less_finite_3_def)
next
fix A and z :: finite_3
assume *: "\<And>x. x \<in> A \<Longrightarrow> x \<le> z"
show "\<Squnion>A \<le> z"
by(auto simp add: Sup_finite_3_def less_eq_finite_3_def less_finite_3_def dest: *)
qed(auto simp add: Inf_finite_3_def Sup_finite_3_def)
end
instance finite_3 :: complete_distrib_lattice
proof
fix a :: finite_3 and B
show "a \<squnion> \<Sqinter>B = (\<Sqinter>b\<in>B. a \<squnion> b)"
proof(cases a "\<Sqinter>B" rule: finite_3.exhaust[case_product finite_3.exhaust])
case a\<^sub>2_a\<^sub>3
then have "\<And>x. x \<in> B \<Longrightarrow> x = a\<^sub>3"
by(case_tac x)(auto simp add: Inf_finite_3_def split: split_if_asm)
then show ?thesis using a\<^sub>2_a\<^sub>3
by(auto simp add: INF_def Inf_finite_3_def max_def less_eq_finite_3_def less_finite_3_def split: split_if_asm)
qed(auto simp add: INF_def Inf_finite_3_def max_def less_finite_3_def less_eq_finite_3_def split: split_if_asm)
show "a \<sqinter> \<Squnion>B = (\<Squnion>b\<in>B. a \<sqinter> b)"
by(cases a "\<Squnion>B" rule: finite_3.exhaust[case_product finite_3.exhaust])
(auto simp add: SUP_def Sup_finite_3_def min_def less_finite_3_def less_eq_finite_3_def split: split_if_asm)
qed
instance finite_3 :: complete_linorder ..
instantiation finite_3 :: "{field, abs_if, ring_div, semiring_div, sgn_if}" begin
definition [simp]: "0 = a\<^sub>1"
definition [simp]: "1 = a\<^sub>2"
definition
"x + y = (case (x, y) of
(a\<^sub>1, a\<^sub>1) \<Rightarrow> a\<^sub>1 | (a\<^sub>2, a\<^sub>3) \<Rightarrow> a\<^sub>1 | (a\<^sub>3, a\<^sub>2) \<Rightarrow> a\<^sub>1
| (a\<^sub>1, a\<^sub>2) \<Rightarrow> a\<^sub>2 | (a\<^sub>2, a\<^sub>1) \<Rightarrow> a\<^sub>2 | (a\<^sub>3, a\<^sub>3) \<Rightarrow> a\<^sub>2
| _ \<Rightarrow> a\<^sub>3)"
definition "- x = (case x of a\<^sub>1 \<Rightarrow> a\<^sub>1 | a\<^sub>2 \<Rightarrow> a\<^sub>3 | a\<^sub>3 \<Rightarrow> a\<^sub>2)"
definition "x - y = x + (- y :: finite_3)"
definition "x * y = (case (x, y) of (a\<^sub>2, a\<^sub>2) \<Rightarrow> a\<^sub>2 | (a\<^sub>3, a\<^sub>3) \<Rightarrow> a\<^sub>2 | (a\<^sub>2, a\<^sub>3) \<Rightarrow> a\<^sub>3 | (a\<^sub>3, a\<^sub>2) \<Rightarrow> a\<^sub>3 | _ \<Rightarrow> a\<^sub>1)"
definition "inverse = (\<lambda>x :: finite_3. x)"
definition "x div y = x * inverse (y :: finite_3)"
definition "abs = (\<lambda>x :: finite_3. x)"
definition "x mod y = (case (x, y) of (a\<^sub>2, a\<^sub>1) \<Rightarrow> a\<^sub>2 | (a\<^sub>3, a\<^sub>1) \<Rightarrow> a\<^sub>3 | _ \<Rightarrow> a\<^sub>1)"
definition "sgn = (\<lambda>x. case x of a\<^sub>1 \<Rightarrow> a\<^sub>1 | _ \<Rightarrow> a\<^sub>2)"
instance
by intro_classes
(simp_all add: plus_finite_3_def uminus_finite_3_def minus_finite_3_def times_finite_3_def
inverse_finite_3_def divide_finite_3_def abs_finite_3_def mod_finite_3_def sgn_finite_3_def
less_finite_3_def
split: finite_3.splits)
end
hide_const (open) a\<^sub>1 a\<^sub>2 a\<^sub>3
datatype (plugins only: code "quickcheck" extraction) finite_4 =
a\<^sub>1 | a\<^sub>2 | a\<^sub>3 | a\<^sub>4
notation (output) a\<^sub>1 ("a\<^sub>1")
notation (output) a\<^sub>2 ("a\<^sub>2")
notation (output) a\<^sub>3 ("a\<^sub>3")
notation (output) a\<^sub>4 ("a\<^sub>4")
lemma UNIV_finite_4:
"UNIV = {a\<^sub>1, a\<^sub>2, a\<^sub>3, a\<^sub>4}"
by (auto intro: finite_4.exhaust)
instantiation finite_4 :: enum
begin
definition
"enum = [a\<^sub>1, a\<^sub>2, a\<^sub>3, a\<^sub>4]"
definition
"enum_all P \<longleftrightarrow> P a\<^sub>1 \<and> P a\<^sub>2 \<and> P a\<^sub>3 \<and> P a\<^sub>4"
definition
"enum_ex P \<longleftrightarrow> P a\<^sub>1 \<or> P a\<^sub>2 \<or> P a\<^sub>3 \<or> P a\<^sub>4"
instance proof
qed (simp_all only: enum_finite_4_def enum_all_finite_4_def enum_ex_finite_4_def UNIV_finite_4, simp_all)
end
instantiation finite_4 :: complete_lattice begin
text \<open>@{term a\<^sub>1} $<$ @{term a\<^sub>2},@{term a\<^sub>3} $<$ @{term a\<^sub>4},
but @{term a\<^sub>2} and @{term a\<^sub>3} are incomparable.\<close>
definition
"x < y \<longleftrightarrow> (case (x, y) of
(a\<^sub>1, a\<^sub>1) \<Rightarrow> False | (a\<^sub>1, _) \<Rightarrow> True
| (a\<^sub>2, a\<^sub>4) \<Rightarrow> True
| (a\<^sub>3, a\<^sub>4) \<Rightarrow> True | _ \<Rightarrow> False)"
definition
"x \<le> y \<longleftrightarrow> (case (x, y) of
(a\<^sub>1, _) \<Rightarrow> True
| (a\<^sub>2, a\<^sub>2) \<Rightarrow> True | (a\<^sub>2, a\<^sub>4) \<Rightarrow> True
| (a\<^sub>3, a\<^sub>3) \<Rightarrow> True | (a\<^sub>3, a\<^sub>4) \<Rightarrow> True
| (a\<^sub>4, a\<^sub>4) \<Rightarrow> True | _ \<Rightarrow> False)"
definition
"\<Sqinter>A = (if a\<^sub>1 \<in> A \<or> a\<^sub>2 \<in> A \<and> a\<^sub>3 \<in> A then a\<^sub>1 else if a\<^sub>2 \<in> A then a\<^sub>2 else if a\<^sub>3 \<in> A then a\<^sub>3 else a\<^sub>4)"
definition
"\<Squnion>A = (if a\<^sub>4 \<in> A \<or> a\<^sub>2 \<in> A \<and> a\<^sub>3 \<in> A then a\<^sub>4 else if a\<^sub>2 \<in> A then a\<^sub>2 else if a\<^sub>3 \<in> A then a\<^sub>3 else a\<^sub>1)"
definition [simp]: "bot = a\<^sub>1"
definition [simp]: "top = a\<^sub>4"
definition
"x \<sqinter> y = (case (x, y) of
(a\<^sub>1, _) \<Rightarrow> a\<^sub>1 | (_, a\<^sub>1) \<Rightarrow> a\<^sub>1 | (a\<^sub>2, a\<^sub>3) \<Rightarrow> a\<^sub>1 | (a\<^sub>3, a\<^sub>2) \<Rightarrow> a\<^sub>1
| (a\<^sub>2, _) \<Rightarrow> a\<^sub>2 | (_, a\<^sub>2) \<Rightarrow> a\<^sub>2
| (a\<^sub>3, _) \<Rightarrow> a\<^sub>3 | (_, a\<^sub>3) \<Rightarrow> a\<^sub>3
| _ \<Rightarrow> a\<^sub>4)"
definition
"x \<squnion> y = (case (x, y) of
(a\<^sub>4, _) \<Rightarrow> a\<^sub>4 | (_, a\<^sub>4) \<Rightarrow> a\<^sub>4 | (a\<^sub>2, a\<^sub>3) \<Rightarrow> a\<^sub>4 | (a\<^sub>3, a\<^sub>2) \<Rightarrow> a\<^sub>4
| (a\<^sub>2, _) \<Rightarrow> a\<^sub>2 | (_, a\<^sub>2) \<Rightarrow> a\<^sub>2
| (a\<^sub>3, _) \<Rightarrow> a\<^sub>3 | (_, a\<^sub>3) \<Rightarrow> a\<^sub>3
| _ \<Rightarrow> a\<^sub>1)"
instance
proof
fix A and z :: finite_4
assume *: "\<And>x. x \<in> A \<Longrightarrow> x \<le> z"
show "\<Squnion>A \<le> z"
by(auto simp add: Sup_finite_4_def less_eq_finite_4_def dest!: * split: finite_4.splits)
next
fix A and z :: finite_4
assume *: "\<And>x. x \<in> A \<Longrightarrow> z \<le> x"
show "z \<le> \<Sqinter>A"
by(auto simp add: Inf_finite_4_def less_eq_finite_4_def dest!: * split: finite_4.splits)
qed(auto simp add: less_finite_4_def less_eq_finite_4_def Inf_finite_4_def Sup_finite_4_def inf_finite_4_def sup_finite_4_def split: finite_4.splits)
end
instance finite_4 :: complete_distrib_lattice
proof
fix a :: finite_4 and B
show "a \<squnion> \<Sqinter>B = (\<Sqinter>b\<in>B. a \<squnion> b)"
by(cases a "\<Sqinter>B" rule: finite_4.exhaust[case_product finite_4.exhaust])
(auto simp add: sup_finite_4_def Inf_finite_4_def INF_def split: finite_4.splits split_if_asm)
show "a \<sqinter> \<Squnion>B = (\<Squnion>b\<in>B. a \<sqinter> b)"
by(cases a "\<Squnion>B" rule: finite_4.exhaust[case_product finite_4.exhaust])
(auto simp add: inf_finite_4_def Sup_finite_4_def SUP_def split: finite_4.splits split_if_asm)
qed
instantiation finite_4 :: complete_boolean_algebra begin
definition "- x = (case x of a\<^sub>1 \<Rightarrow> a\<^sub>4 | a\<^sub>2 \<Rightarrow> a\<^sub>3 | a\<^sub>3 \<Rightarrow> a\<^sub>2 | a\<^sub>4 \<Rightarrow> a\<^sub>1)"
definition "x - y = x \<sqinter> - (y :: finite_4)"
instance
by intro_classes
(simp_all add: inf_finite_4_def sup_finite_4_def uminus_finite_4_def minus_finite_4_def split: finite_4.splits)
end
hide_const (open) a\<^sub>1 a\<^sub>2 a\<^sub>3 a\<^sub>4
datatype (plugins only: code "quickcheck" extraction) finite_5 =
a\<^sub>1 | a\<^sub>2 | a\<^sub>3 | a\<^sub>4 | a\<^sub>5
notation (output) a\<^sub>1 ("a\<^sub>1")
notation (output) a\<^sub>2 ("a\<^sub>2")
notation (output) a\<^sub>3 ("a\<^sub>3")
notation (output) a\<^sub>4 ("a\<^sub>4")
notation (output) a\<^sub>5 ("a\<^sub>5")
lemma UNIV_finite_5:
"UNIV = {a\<^sub>1, a\<^sub>2, a\<^sub>3, a\<^sub>4, a\<^sub>5}"
by (auto intro: finite_5.exhaust)
instantiation finite_5 :: enum
begin
definition
"enum = [a\<^sub>1, a\<^sub>2, a\<^sub>3, a\<^sub>4, a\<^sub>5]"
definition
"enum_all P \<longleftrightarrow> P a\<^sub>1 \<and> P a\<^sub>2 \<and> P a\<^sub>3 \<and> P a\<^sub>4 \<and> P a\<^sub>5"
definition
"enum_ex P \<longleftrightarrow> P a\<^sub>1 \<or> P a\<^sub>2 \<or> P a\<^sub>3 \<or> P a\<^sub>4 \<or> P a\<^sub>5"
instance proof
qed (simp_all only: enum_finite_5_def enum_all_finite_5_def enum_ex_finite_5_def UNIV_finite_5, simp_all)
end
instantiation finite_5 :: complete_lattice
begin
text \<open>The non-distributive pentagon lattice $N_5$\<close>
definition
"x < y \<longleftrightarrow> (case (x, y) of
(a\<^sub>1, a\<^sub>1) \<Rightarrow> False | (a\<^sub>1, _) \<Rightarrow> True
| (a\<^sub>2, a\<^sub>3) \<Rightarrow> True | (a\<^sub>2, a\<^sub>5) \<Rightarrow> True
| (a\<^sub>3, a\<^sub>5) \<Rightarrow> True
| (a\<^sub>4, a\<^sub>5) \<Rightarrow> True | _ \<Rightarrow> False)"
definition
"x \<le> y \<longleftrightarrow> (case (x, y) of
(a\<^sub>1, _) \<Rightarrow> True
| (a\<^sub>2, a\<^sub>2) \<Rightarrow> True | (a\<^sub>2, a\<^sub>3) \<Rightarrow> True | (a\<^sub>2, a\<^sub>5) \<Rightarrow> True
| (a\<^sub>3, a\<^sub>3) \<Rightarrow> True | (a\<^sub>3, a\<^sub>5) \<Rightarrow> True
| (a\<^sub>4, a\<^sub>4) \<Rightarrow> True | (a\<^sub>4, a\<^sub>5) \<Rightarrow> True
| (a\<^sub>5, a\<^sub>5) \<Rightarrow> True | _ \<Rightarrow> False)"
definition
"\<Sqinter>A =
(if a\<^sub>1 \<in> A \<or> a\<^sub>4 \<in> A \<and> (a\<^sub>2 \<in> A \<or> a\<^sub>3 \<in> A) then a\<^sub>1
else if a\<^sub>2 \<in> A then a\<^sub>2
else if a\<^sub>3 \<in> A then a\<^sub>3
else if a\<^sub>4 \<in> A then a\<^sub>4
else a\<^sub>5)"
definition
"\<Squnion>A =
(if a\<^sub>5 \<in> A \<or> a\<^sub>4 \<in> A \<and> (a\<^sub>2 \<in> A \<or> a\<^sub>3 \<in> A) then a\<^sub>5
else if a\<^sub>3 \<in> A then a\<^sub>3
else if a\<^sub>2 \<in> A then a\<^sub>2
else if a\<^sub>4 \<in> A then a\<^sub>4
else a\<^sub>1)"
definition [simp]: "bot = a\<^sub>1"
definition [simp]: "top = a\<^sub>5"
definition
"x \<sqinter> y = (case (x, y) of
(a\<^sub>1, _) \<Rightarrow> a\<^sub>1 | (_, a\<^sub>1) \<Rightarrow> a\<^sub>1 | (a\<^sub>2, a\<^sub>4) \<Rightarrow> a\<^sub>1 | (a\<^sub>4, a\<^sub>2) \<Rightarrow> a\<^sub>1 | (a\<^sub>3, a\<^sub>4) \<Rightarrow> a\<^sub>1 | (a\<^sub>4, a\<^sub>3) \<Rightarrow> a\<^sub>1
| (a\<^sub>2, _) \<Rightarrow> a\<^sub>2 | (_, a\<^sub>2) \<Rightarrow> a\<^sub>2
| (a\<^sub>3, _) \<Rightarrow> a\<^sub>3 | (_, a\<^sub>3) \<Rightarrow> a\<^sub>3
| (a\<^sub>4, _) \<Rightarrow> a\<^sub>4 | (_, a\<^sub>4) \<Rightarrow> a\<^sub>4
| _ \<Rightarrow> a\<^sub>5)"
definition
"x \<squnion> y = (case (x, y) of
(a\<^sub>5, _) \<Rightarrow> a\<^sub>5 | (_, a\<^sub>5) \<Rightarrow> a\<^sub>5 | (a\<^sub>2, a\<^sub>4) \<Rightarrow> a\<^sub>5 | (a\<^sub>4, a\<^sub>2) \<Rightarrow> a\<^sub>5 | (a\<^sub>3, a\<^sub>4) \<Rightarrow> a\<^sub>5 | (a\<^sub>4, a\<^sub>3) \<Rightarrow> a\<^sub>5
| (a\<^sub>3, _) \<Rightarrow> a\<^sub>3 | (_, a\<^sub>3) \<Rightarrow> a\<^sub>3
| (a\<^sub>2, _) \<Rightarrow> a\<^sub>2 | (_, a\<^sub>2) \<Rightarrow> a\<^sub>2
| (a\<^sub>4, _) \<Rightarrow> a\<^sub>4 | (_, a\<^sub>4) \<Rightarrow> a\<^sub>4
| _ \<Rightarrow> a\<^sub>1)"
instance
proof intro_classes
fix A and z :: finite_5
assume *: "\<And>x. x \<in> A \<Longrightarrow> z \<le> x"
show "z \<le> \<Sqinter>A"
by(auto simp add: less_eq_finite_5_def Inf_finite_5_def split: finite_5.splits split_if_asm dest!: *)
next
fix A and z :: finite_5
assume *: "\<And>x. x \<in> A \<Longrightarrow> x \<le> z"
show "\<Squnion>A \<le> z"
by(auto simp add: less_eq_finite_5_def Sup_finite_5_def split: finite_5.splits split_if_asm dest!: *)
qed(auto simp add: less_eq_finite_5_def less_finite_5_def inf_finite_5_def sup_finite_5_def Inf_finite_5_def Sup_finite_5_def split: finite_5.splits split_if_asm)
end
hide_const (open) a\<^sub>1 a\<^sub>2 a\<^sub>3 a\<^sub>4 a\<^sub>5
subsection \<open>Closing up\<close>
hide_type (open) finite_1 finite_2 finite_3 finite_4 finite_5
hide_const (open) enum enum_all enum_ex all_n_lists ex_n_lists ntrancl
end