# Theory Countable

theory Countable
imports Old_Datatype Rat Nat_Bijection
(*  Title:      HOL/Library/Countable.thy
Author:     Alexander Krauss, TU Muenchen
Author:     Brian Huffman, Portland State University
Author:     Jasmin Blanchette, TU Muenchen
*)

section ‹Encoding (almost) everything into natural numbers›

theory Countable
imports Old_Datatype "~~/src/HOL/Rat" Nat_Bijection
begin

subsection ‹The class of countable types›

class countable =
assumes ex_inj: "∃to_nat :: 'a ⇒ nat. inj to_nat"

lemma countable_classI:
fixes f :: "'a ⇒ nat"
assumes "⋀x y. f x = f y ⟹ 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::countable ⇒ nat" where
"to_nat = (SOME f. inj f)"

definition from_nat :: "nat ⇒ 'a::countable" where
"from_nat = inv (to_nat :: 'a ⇒ 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 ⟷ x = y"
using injD [OF inj_to_nat] by auto

lemma from_nat_to_nat [simp]:
"from_nat (to_nat x) = x"

subsection ‹Finite types are countable›

subclass (in finite) countable
proof
have "finite (UNIV::'a set)" by (rule finite_UNIV)
with finite_conv_nat_seg_image [of "UNIV::'a set"]
obtain n and f :: "nat ⇒ '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 "∃to_nat :: 'a ⇒ nat. inj to_nat" by (rule exI[of inj])
qed

subsection ‹Automatically proving countability of old-style datatypes›

context
begin

qualified inductive finite_item :: "'a Old_Datatype.item ⇒ bool" where
undefined: "finite_item undefined"
| In0: "finite_item x ⟹ finite_item (Old_Datatype.In0 x)"
| In1: "finite_item x ⟹ finite_item (Old_Datatype.In1 x)"
| Leaf: "finite_item (Old_Datatype.Leaf a)"
| Scons: "⟦finite_item x; finite_item y⟧ ⟹ finite_item (Old_Datatype.Scons x y)"

qualified function nth_item :: "nat ⇒ ('a::countable) Old_Datatype.item"
where
"nth_item 0 = undefined"
| "nth_item (Suc n) =
(case sum_decode n of
Inl i ⇒
(case sum_decode i of
Inl j ⇒ Old_Datatype.In0 (nth_item j)
| Inr j ⇒ Old_Datatype.In1 (nth_item j))
| Inr i ⇒
(case sum_decode i of
Inl j ⇒ Old_Datatype.Leaf (from_nat j)
| Inr j ⇒
(case prod_decode j of
(a, b) ⇒ Old_Datatype.Scons (nth_item a) (nth_item b))))"
by pat_completeness auto

lemma le_sum_encode_Inl: "x ≤ y ⟹ x ≤ sum_encode (Inl y)"
unfolding sum_encode_def by simp

lemma le_sum_encode_Inr: "x ≤ y ⟹ x ≤ sum_encode (Inr y)"
unfolding sum_encode_def by simp

qualified 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 ⟹ ∃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))))) = Old_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))))) = Old_Datatype.In1 x" by simp
thus ?case ..
next
case (Leaf a)
have "nth_item (Suc (sum_encode (Inr (sum_encode (Inl (to_nat a)))))) = Old_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))))))) = Old_Datatype.Scons x y"
by simp
thus ?case ..
qed

theorem countable_datatype:
fixes Rep :: "'b ⇒ ('a::countable) Old_Datatype.item"
fixes Abs :: "('a::countable) Old_Datatype.item ⇒ 'b"
fixes rep_set :: "('a::countable) Old_Datatype.item ⇒ bool"
assumes type: "type_definition Rep Abs (Collect rep_set)"
assumes finite_item: "⋀x. rep_set x ⟹ finite_item x"
shows "OFCLASS('b, countable_class)"
proof
def f  "λy. 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 "∃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 "∃f::'b ⇒ nat. inj f"
by - (rule exI)
qed

ML ‹
fun old_countable_datatype_tac ctxt =
SUBGOAL (fn (goal, _) =>
let
val ty_name =
(case goal of
(_ \$ Const (@{const_name Pure.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 (Thm.concl_of typedef_thm) of
(_ \$ _ \$ _ \$ (_ \$ 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 insts = vars |> map (fn (_, T) => try (Thm.cterm_of ctxt)
(Const (@{const_name Countable.finite_item}, T)))
val induct_thm' = Thm.instantiate' [] insts induct_thm
val rules = @{thms finite_item.intros}
in
SOLVED' (fn i => EVERY
[resolve_tac ctxt @{thms countable_datatype} i,
resolve_tac ctxt [typedef_thm] i,
eresolve_tac ctxt [induct_thm'] i,
REPEAT (resolve_tac ctxt rules i ORELSE assume_tac ctxt i)]) 1
end)
›

end

subsection ‹Automatically proving countability of datatypes›

ML_file "bnf_lfp_countable.ML"

ML ‹
fun countable_datatype_tac ctxt st =
(case try (fn () => HEADGOAL (old_countable_datatype_tac ctxt) st) () of
SOME res => res
| NONE => BNF_LFP_Countable.countable_datatype_tac ctxt st);

(* compatibility *)
fun countable_tac ctxt =
SELECT_GOAL (countable_datatype_tac ctxt);
›

method_setup countable_datatype = ‹
Scan.succeed (SIMPLE_METHOD o countable_datatype_tac)
› "prove countable class instances for datatypes"

subsection ‹More Countable types›

text ‹Naturals›

instance nat :: countable
by (rule countable_classI [of "id"]) simp

text ‹Pairs›

instance prod :: (countable, countable) countable
by (rule countable_classI [of "λ(x, y). prod_encode (to_nat x, to_nat y)"])

text ‹Sums›

instance sum :: (countable, countable) countable
by (rule countable_classI [of "(λx. case x of Inl a ⇒ to_nat (False, to_nat a)
| Inr b ⇒ 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 countable_datatype

text ‹Lists›

instance list :: (countable) countable
by countable_datatype

text ‹String literals›

instance String.literal :: countable
by (rule countable_classI [of "to_nat ∘ String.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 "∃to_nat::('a ⇒ 'b) ⇒ nat. inj to_nat"
proof
show "inj (λf. to_nat (map f xs))"
by (rule injI, simp add: xs fun_eq_iff)
qed
qed

text ‹Typereps›

instance typerep :: countable
by countable_datatype

subsection ‹The rationals are countably infinite›

definition nat_to_rat_surj :: "nat ⇒ 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 "∃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)))"
thus "∃n. r = nat_to_rat_surj n" by(auto simp: Let_def)
qed
qed

lemma Rats_eq_range_nat_to_rat_surj: "ℚ = range nat_to_rat_surj"

context field_char_0
begin

lemma Rats_eq_range_of_rat_o_nat_to_rat_surj:
"ℚ = range (of_rat ∘ 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 ∈ ℚ ⟹ ∃n. r = of_rat (nat_to_rat_surj n)"

end

instance rat :: countable
proof
show "∃to_nat::rat ⇒ 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

end