src/HOL/Library/Countable.thy
 author wenzelm Wed, 08 Mar 2017 10:50:59 +0100 changeset 65151 a7394aa4d21c parent 63040 eb4ddd18d635 child 65411 3c628937899d permissions -rw-r--r--
tuned proofs;
```
(*  Title:      HOL/Library/Countable.thy
Author:     Alexander Krauss, TU Muenchen
Author:     Brian Huffman, Portland State University
Author:     Jasmin Blanchette, TU Muenchen
*)

section \<open>Encoding (almost) everything into natural numbers\<close>

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

subsection \<open>The class of countable types\<close>

class countable =
assumes ex_inj: "\<exists>to_nat :: '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 \<open>Conversion functions\<close>

definition to_nat :: "'a::countable \<Rightarrow> nat" where
"to_nat = (SOME f. inj f)"

definition from_nat :: "nat \<Rightarrow> 'a::countable" where
"from_nat = inv (to_nat :: '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"

subsection \<open>Finite types are countable\<close>

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 \<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 :: 'a \<Rightarrow> nat. inj to_nat" by (rule exI[of inj])
qed

subsection \<open>Automatically proving countability of old-style datatypes\<close>

context
begin

qualified inductive finite_item :: "'a Old_Datatype.item \<Rightarrow> bool" where
undefined: "finite_item undefined"
| In0: "finite_item x \<Longrightarrow> finite_item (Old_Datatype.In0 x)"
| In1: "finite_item x \<Longrightarrow> finite_item (Old_Datatype.In1 x)"
| Leaf: "finite_item (Old_Datatype.Leaf a)"
| Scons: "\<lbrakk>finite_item x; finite_item y\<rbrakk> \<Longrightarrow> finite_item (Old_Datatype.Scons x y)"

qualified function nth_item :: "nat \<Rightarrow> ('a::countable) Old_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> Old_Datatype.In0 (nth_item j)
| Inr j \<Rightarrow> Old_Datatype.In1 (nth_item j))
| Inr i \<Rightarrow>
(case sum_decode i of
Inl j \<Rightarrow> Old_Datatype.Leaf (from_nat j)
| Inr j \<Rightarrow>
(case prod_decode j of
(a, b) \<Rightarrow> Old_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

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 \<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))))) = 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 \<Rightarrow> ('a::countable) Old_Datatype.item"
fixes Abs :: "('a::countable) Old_Datatype.item \<Rightarrow> 'b"
fixes rep_set :: "('a::countable) Old_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
define f where "f y = (LEAST n. nth_item n = Rep y)" for 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 \<open>
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)
\<close>

end

subsection \<open>Automatically proving countability of datatypes\<close>

ML_file "../Tools/BNF/bnf_lfp_countable.ML"

ML \<open>
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);
\<close>

method_setup countable_datatype = \<open>
Scan.succeed (SIMPLE_METHOD o countable_datatype_tac)
\<close> "prove countable class instances for datatypes"

subsection \<open>More Countable types\<close>

text \<open>Naturals\<close>

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

text \<open>Pairs\<close>

instance prod :: (countable, countable) countable
by (rule countable_classI [of "\<lambda>(x, y). prod_encode (to_nat x, to_nat y)"])

text \<open>Sums\<close>

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 \<open>Integers\<close>

instance int :: countable
by (rule countable_classI [of int_encode]) (simp add: int_encode_eq)

text \<open>Options\<close>

instance option :: (countable) countable
by countable_datatype

text \<open>Lists\<close>

instance list :: (countable) countable
by countable_datatype

text \<open>String literals\<close>

instance String.literal :: countable
by (rule countable_classI [of "to_nat \<circ> String.explode"]) (auto simp add: explode_inject)

text \<open>Functions\<close>

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

text \<open>Typereps\<close>

instance typerep :: countable
by countable_datatype

subsection \<open>The rationals are countably infinite\<close>

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)))"
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"

context field_char_0
begin

lemma Rats_eq_range_of_rat_o_nat_to_rat_surj:
"\<rat> = range (of_rat \<circ> 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)"

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

end
```