(* Title: HOL/Library/Countable.thy
Author: Alexander Krauss, TU Muenchen
Author: Brian Huffman, Portland State University
*)
header {* Encoding (almost) everything into natural numbers *}
theory Countable
imports Main Rat Nat_Bijection
begin
subsection {* The class of countable types *}
class countable =
assumes ex_inj: "\<exists>to_nat \<Colon> 'a \<Rightarrow> nat. inj to_nat"
lemma countable_classI:
fixes f :: "'a \<Rightarrow> nat"
assumes "\<And>x y. f x = f y \<Longrightarrow> x = y"
shows "OFCLASS('a, countable_class)"
proof (intro_classes, rule exI)
show "inj f"
by (rule injI [OF assms]) assumption
qed
subsection {* Conversion functions *}
definition to_nat :: "'a\<Colon>countable \<Rightarrow> nat" where
"to_nat = (SOME f. inj f)"
definition from_nat :: "nat \<Rightarrow> 'a\<Colon>countable" where
"from_nat = inv (to_nat \<Colon> 'a \<Rightarrow> nat)"
lemma inj_to_nat [simp]: "inj to_nat"
by (rule exE_some [OF ex_inj]) (simp add: to_nat_def)
lemma inj_on_to_nat[simp, intro]: "inj_on to_nat S"
using inj_to_nat by (auto simp: inj_on_def)
lemma surj_from_nat [simp]: "surj from_nat"
unfolding from_nat_def by (simp add: inj_imp_surj_inv)
lemma to_nat_split [simp]: "to_nat x = to_nat y \<longleftrightarrow> x = y"
using injD [OF inj_to_nat] by auto
lemma from_nat_to_nat [simp]:
"from_nat (to_nat x) = x"
by (simp add: from_nat_def)
subsection {* Countable types *}
instance nat :: countable
by (rule countable_classI [of "id"]) simp
subclass (in finite) countable
proof
have "finite (UNIV\<Colon>'a set)" by (rule finite_UNIV)
with finite_conv_nat_seg_image [of "UNIV::'a set"]
obtain n and f :: "nat \<Rightarrow> 'a"
where "UNIV = f ` {i. i < n}" by auto
then have "surj f" unfolding surj_def by auto
then have "inj (inv f)" by (rule surj_imp_inj_inv)
then show "\<exists>to_nat \<Colon> 'a \<Rightarrow> nat. inj to_nat" by (rule exI[of inj])
qed
text {* Pairs *}
instance prod :: (countable, countable) countable
by (rule countable_classI [of "\<lambda>(x, y). prod_encode (to_nat x, to_nat y)"])
(auto simp add: prod_encode_eq)
text {* Sums *}
instance sum :: (countable, countable) countable
by (rule countable_classI [of "(\<lambda>x. case x of Inl a \<Rightarrow> to_nat (False, to_nat a)
| Inr b \<Rightarrow> to_nat (True, to_nat b))"])
(simp split: sum.split_asm)
text {* Integers *}
instance int :: countable
by (rule countable_classI [of "int_encode"])
(simp add: int_encode_eq)
text {* Options *}
instance option :: (countable) countable
by (rule countable_classI [of "option_case 0 (Suc \<circ> to_nat)"])
(simp split: option.split_asm)
text {* Lists *}
instance list :: (countable) countable
by (rule countable_classI [of "list_encode \<circ> map to_nat"])
(simp add: list_encode_eq)
text {* Further *}
instance String.literal :: countable
by (rule countable_classI [of "to_nat o explode"])
(auto simp add: explode_inject)
text {* Functions *}
instance "fun" :: (finite, countable) countable
proof
obtain xs :: "'a list" where xs: "set xs = UNIV"
using finite_list [OF finite_UNIV] ..
show "\<exists>to_nat::('a \<Rightarrow> 'b) \<Rightarrow> nat. inj to_nat"
proof
show "inj (\<lambda>f. to_nat (map f xs))"
by (rule injI, simp add: xs fun_eq_iff)
qed
qed
subsection {* The Rationals are Countably Infinite *}
definition nat_to_rat_surj :: "nat \<Rightarrow> rat" where
"nat_to_rat_surj n = (let (a,b) = prod_decode n
in Fract (int_decode a) (int_decode b))"
lemma surj_nat_to_rat_surj: "surj nat_to_rat_surj"
unfolding surj_def
proof
fix r::rat
show "\<exists>n. r = nat_to_rat_surj n"
proof (cases r)
fix i j assume [simp]: "r = Fract i j" and "j > 0"
have "r = (let m = int_encode i; n = int_encode j
in nat_to_rat_surj(prod_encode (m,n)))"
by (simp add: Let_def nat_to_rat_surj_def)
thus "\<exists>n. r = nat_to_rat_surj n" by(auto simp:Let_def)
qed
qed
lemma Rats_eq_range_nat_to_rat_surj: "\<rat> = range nat_to_rat_surj"
by (simp add: Rats_def surj_nat_to_rat_surj)
context field_char_0
begin
lemma Rats_eq_range_of_rat_o_nat_to_rat_surj:
"\<rat> = range (of_rat o nat_to_rat_surj)"
using surj_nat_to_rat_surj
by (auto simp: Rats_def image_def surj_def)
(blast intro: arg_cong[where f = of_rat])
lemma surj_of_rat_nat_to_rat_surj:
"r\<in>\<rat> \<Longrightarrow> \<exists>n. r = of_rat(nat_to_rat_surj n)"
by(simp add: Rats_eq_range_of_rat_o_nat_to_rat_surj image_def)
end
instance rat :: countable
proof
show "\<exists>to_nat::rat \<Rightarrow> nat. inj to_nat"
proof
have "surj nat_to_rat_surj"
by (rule surj_nat_to_rat_surj)
then show "inj (inv nat_to_rat_surj)"
by (rule surj_imp_inj_inv)
qed
qed
subsection {* Automatically proving countability of datatypes *}
inductive finite_item :: "'a Datatype.item \<Rightarrow> bool" where
undefined: "finite_item undefined"
| In0: "finite_item x \<Longrightarrow> finite_item (Datatype.In0 x)"
| In1: "finite_item x \<Longrightarrow> finite_item (Datatype.In1 x)"
| Leaf: "finite_item (Datatype.Leaf a)"
| Scons: "\<lbrakk>finite_item x; finite_item y\<rbrakk> \<Longrightarrow> finite_item (Datatype.Scons x y)"
function
nth_item :: "nat \<Rightarrow> ('a::countable) Datatype.item"
where
"nth_item 0 = undefined"
| "nth_item (Suc n) =
(case sum_decode n of
Inl i \<Rightarrow>
(case sum_decode i of
Inl j \<Rightarrow> Datatype.In0 (nth_item j)
| Inr j \<Rightarrow> Datatype.In1 (nth_item j))
| Inr i \<Rightarrow>
(case sum_decode i of
Inl j \<Rightarrow> Datatype.Leaf (from_nat j)
| Inr j \<Rightarrow>
(case prod_decode j of
(a, b) \<Rightarrow> Datatype.Scons (nth_item a) (nth_item b))))"
by pat_completeness auto
lemma le_sum_encode_Inl: "x \<le> y \<Longrightarrow> x \<le> sum_encode (Inl y)"
unfolding sum_encode_def by simp
lemma le_sum_encode_Inr: "x \<le> y \<Longrightarrow> x \<le> sum_encode (Inr y)"
unfolding sum_encode_def by simp
termination
by (relation "measure id")
(auto simp add: sum_encode_eq [symmetric] prod_encode_eq [symmetric]
le_imp_less_Suc le_sum_encode_Inl le_sum_encode_Inr
le_prod_encode_1 le_prod_encode_2)
lemma nth_item_covers: "finite_item x \<Longrightarrow> \<exists>n. nth_item n = x"
proof (induct set: finite_item)
case undefined
have "nth_item 0 = undefined" by simp
thus ?case ..
next
case (In0 x)
then obtain n where "nth_item n = x" by fast
hence "nth_item (Suc (sum_encode (Inl (sum_encode (Inl n)))))
= Datatype.In0 x" by simp
thus ?case ..
next
case (In1 x)
then obtain n where "nth_item n = x" by fast
hence "nth_item (Suc (sum_encode (Inl (sum_encode (Inr n)))))
= Datatype.In1 x" by simp
thus ?case ..
next
case (Leaf a)
have "nth_item (Suc (sum_encode (Inr (sum_encode (Inl (to_nat a))))))
= Datatype.Leaf a" by simp
thus ?case ..
next
case (Scons x y)
then obtain i j where "nth_item i = x" and "nth_item j = y" by fast
hence "nth_item
(Suc (sum_encode (Inr (sum_encode (Inr (prod_encode (i, j)))))))
= Datatype.Scons x y" by simp
thus ?case ..
qed
theorem countable_datatype:
fixes Rep :: "'b \<Rightarrow> ('a::countable) Datatype.item"
fixes Abs :: "('a::countable) Datatype.item \<Rightarrow> 'b"
fixes rep_set :: "('a::countable) Datatype.item \<Rightarrow> bool"
assumes type: "type_definition Rep Abs (Collect rep_set)"
assumes finite_item: "\<And>x. rep_set x \<Longrightarrow> finite_item x"
shows "OFCLASS('b, countable_class)"
proof
def f \<equiv> "\<lambda>y. LEAST n. nth_item n = Rep y"
{
fix y :: 'b
have "rep_set (Rep y)"
using type_definition.Rep [OF type] by simp
hence "finite_item (Rep y)"
by (rule finite_item)
hence "\<exists>n. nth_item n = Rep y"
by (rule nth_item_covers)
hence "nth_item (f y) = Rep y"
unfolding f_def by (rule LeastI_ex)
hence "Abs (nth_item (f y)) = y"
using type_definition.Rep_inverse [OF type] by simp
}
hence "inj f"
by (rule inj_on_inverseI)
thus "\<exists>f::'b \<Rightarrow> nat. inj f"
by - (rule exI)
qed
ML {*
fun countable_tac ctxt =
SUBGOAL (fn (goal, i) =>
let
val ty_name =
(case goal of
(_ $ Const (@{const_name TYPE}, Type (@{type_name itself}, [Type (n, _)]))) => n
| _ => raise Match)
val typedef_info = hd (Typedef.get_info ctxt ty_name)
val typedef_thm = #type_definition (snd typedef_info)
val pred_name =
(case HOLogic.dest_Trueprop (concl_of typedef_thm) of
(typedef $ rep $ abs $ (collect $ Const (n, _))) => n
| _ => raise Match)
val induct_info = Inductive.the_inductive ctxt pred_name
val pred_names = #names (fst induct_info)
val induct_thms = #inducts (snd induct_info)
val alist = pred_names ~~ induct_thms
val induct_thm = the (AList.lookup (op =) alist pred_name)
val vars = rev (Term.add_vars (Thm.prop_of induct_thm) [])
val thy = Proof_Context.theory_of ctxt
val insts = vars |> map (fn (_, T) => try (Thm.cterm_of thy)
(Const (@{const_name Countable.finite_item}, T)))
val induct_thm' = Drule.instantiate' [] insts induct_thm
val rules = @{thms finite_item.intros}
in
SOLVED' (fn i => EVERY
[rtac @{thm countable_datatype} i,
rtac typedef_thm i,
etac induct_thm' i,
REPEAT (resolve_tac rules i ORELSE atac i)]) 1
end)
*}
method_setup countable_datatype = {*
Scan.succeed (fn ctxt => SIMPLE_METHOD' (countable_tac ctxt))
*} "prove countable class instances for datatypes"
hide_const (open) finite_item nth_item
subsection {* Countable datatypes *}
instance typerep :: countable
by countable_datatype
end