# Theory Numeral_Type

theory Numeral_Type
imports Cardinality
```(*  Title:      HOL/Library/Numeral_Type.thy
Author:     Brian Huffman
*)

section ‹Numeral Syntax for Types›

theory Numeral_Type
imports Cardinality
begin

subsection ‹Numeral Types›

typedef num0 = "UNIV :: nat set" ..
typedef num1 = "UNIV :: unit set" ..

typedef 'a bit0 = "{0 ..< 2 * int CARD('a::finite)}"
proof
show "0 ∈ {0 ..< 2 * int CARD('a)}"
by simp
qed

typedef 'a bit1 = "{0 ..< 1 + 2 * int CARD('a::finite)}"
proof
show "0 ∈ {0 ..< 1 + 2 * int CARD('a)}"
by simp
qed

lemma card_num0 [simp]: "CARD (num0) = 0"
unfolding type_definition.card [OF type_definition_num0]
by simp

lemma infinite_num0: "¬ finite (UNIV :: num0 set)"
using card_num0[unfolded card_eq_0_iff]
by simp

lemma card_num1 [simp]: "CARD(num1) = 1"
unfolding type_definition.card [OF type_definition_num1]
by (simp only: card_UNIV_unit)

lemma card_bit0 [simp]: "CARD('a bit0) = 2 * CARD('a::finite)"
unfolding type_definition.card [OF type_definition_bit0]
by simp

lemma card_bit1 [simp]: "CARD('a bit1) = Suc (2 * CARD('a::finite))"
unfolding type_definition.card [OF type_definition_bit1]
by simp

instance num1 :: finite
proof
show "finite (UNIV::num1 set)"
unfolding type_definition.univ [OF type_definition_num1]
using finite by (rule finite_imageI)
qed

instance bit0 :: (finite) card2
proof
show "finite (UNIV::'a bit0 set)"
unfolding type_definition.univ [OF type_definition_bit0]
by simp
show "2 ≤ CARD('a bit0)"
by simp
qed

instance bit1 :: (finite) card2
proof
show "finite (UNIV::'a bit1 set)"
unfolding type_definition.univ [OF type_definition_bit1]
by simp
show "2 ≤ CARD('a bit1)"
by simp
qed

subsection ‹Locales for for modular arithmetic subtypes›

locale mod_type =
fixes n :: int
and Rep :: "'a::{zero,one,plus,times,uminus,minus} ⇒ int"
and Abs :: "int ⇒ 'a::{zero,one,plus,times,uminus,minus}"
assumes type: "type_definition Rep Abs {0..<n}"
and size1: "1 < n"
and zero_def: "0 = Abs 0"
and one_def:  "1 = Abs 1"
and add_def:  "x + y = Abs ((Rep x + Rep y) mod n)"
and mult_def: "x * y = Abs ((Rep x * Rep y) mod n)"
and diff_def: "x - y = Abs ((Rep x - Rep y) mod n)"
and minus_def: "- x = Abs ((- Rep x) mod n)"
begin

lemma size0: "0 < n"
using size1 by simp

lemmas definitions =
zero_def one_def add_def mult_def minus_def diff_def

lemma Rep_less_n: "Rep x < n"
by (rule type_definition.Rep [OF type, simplified, THEN conjunct2])

lemma Rep_le_n: "Rep x ≤ n"
by (rule Rep_less_n [THEN order_less_imp_le])

lemma Rep_inject_sym: "x = y ⟷ Rep x = Rep y"
by (rule type_definition.Rep_inject [OF type, symmetric])

lemma Rep_inverse: "Abs (Rep x) = x"
by (rule type_definition.Rep_inverse [OF type])

lemma Abs_inverse: "m ∈ {0..<n} ⟹ Rep (Abs m) = m"
by (rule type_definition.Abs_inverse [OF type])

lemma Rep_Abs_mod: "Rep (Abs (m mod n)) = m mod n"
by (simp add: Abs_inverse pos_mod_conj [OF size0])

lemma Rep_Abs_0: "Rep (Abs 0) = 0"

lemma Rep_0: "Rep 0 = 0"

lemma Rep_Abs_1: "Rep (Abs 1) = 1"

lemma Rep_1: "Rep 1 = 1"

lemma Rep_mod: "Rep x mod n = Rep x"
apply (rule_tac x=x in type_definition.Abs_cases [OF type])
apply (simp add: type_definition.Abs_inverse [OF type])
done

lemmas Rep_simps =
Rep_inject_sym Rep_inverse Rep_Abs_mod Rep_mod Rep_Abs_0 Rep_Abs_1

lemma comm_ring_1: "OFCLASS('a, comm_ring_1_class)"
apply (intro_classes, unfold definitions)
apply (simp_all add: Rep_simps mod_simps field_simps)
done

end

locale mod_ring = mod_type n Rep Abs
for n :: int
and Rep :: "'a::{comm_ring_1} ⇒ int"
and Abs :: "int ⇒ 'a::{comm_ring_1}"
begin

lemma of_nat_eq: "of_nat k = Abs (int k mod n)"
apply (induct k)
done

lemma of_int_eq: "of_int z = Abs (z mod n)"
apply (cases z rule: int_diff_cases)
apply (simp add: Rep_simps of_nat_eq diff_def mod_simps)
done

lemma Rep_numeral:
"Rep (numeral w) = numeral w mod n"
using of_int_eq [of "numeral w"]

lemma iszero_numeral:
"iszero (numeral w::'a) ⟷ numeral w mod n = 0"
by (simp add: Rep_inject_sym Rep_numeral Rep_0 iszero_def)

lemma cases:
assumes 1: "⋀z. ⟦(x::'a) = of_int z; 0 ≤ z; z < n⟧ ⟹ P"
shows "P"
apply (cases x rule: type_definition.Abs_cases [OF type])
apply (rule_tac z="y" in 1)
done

lemma induct:
"(⋀z. ⟦0 ≤ z; z < n⟧ ⟹ P (of_int z)) ⟹ P (x::'a)"
by (cases x rule: cases) simp

end

subsection ‹Ring class instances›

text ‹
Unfortunately ‹ring_1› instance is not possible for
@{typ num1}, since 0 and 1 are not distinct.
›

instantiation num1 :: "{comm_ring,comm_monoid_mult,numeral}"
begin

lemma num1_eq_iff: "(x::num1) = (y::num1) ⟷ True"
by (induct x, induct y) simp

instance

end

instantiation
bit0 and bit1 :: (finite) "{zero,one,plus,times,uminus,minus}"
begin

definition Abs_bit0' :: "int ⇒ 'a bit0" where
"Abs_bit0' x = Abs_bit0 (x mod int CARD('a bit0))"

definition Abs_bit1' :: "int ⇒ 'a bit1" where
"Abs_bit1' x = Abs_bit1 (x mod int CARD('a bit1))"

definition "0 = Abs_bit0 0"
definition "1 = Abs_bit0 1"
definition "x + y = Abs_bit0' (Rep_bit0 x + Rep_bit0 y)"
definition "x * y = Abs_bit0' (Rep_bit0 x * Rep_bit0 y)"
definition "x - y = Abs_bit0' (Rep_bit0 x - Rep_bit0 y)"
definition "- x = Abs_bit0' (- Rep_bit0 x)"

definition "0 = Abs_bit1 0"
definition "1 = Abs_bit1 1"
definition "x + y = Abs_bit1' (Rep_bit1 x + Rep_bit1 y)"
definition "x * y = Abs_bit1' (Rep_bit1 x * Rep_bit1 y)"
definition "x - y = Abs_bit1' (Rep_bit1 x - Rep_bit1 y)"
definition "- x = Abs_bit1' (- Rep_bit1 x)"

instance ..

end

interpretation bit0:
mod_type "int CARD('a::finite bit0)"
"Rep_bit0 :: 'a::finite bit0 ⇒ int"
"Abs_bit0 :: int ⇒ 'a::finite bit0"
apply (rule mod_type.intro)
apply (rule one_less_int_card)
apply (rule zero_bit0_def)
apply (rule one_bit0_def)
apply (rule plus_bit0_def [unfolded Abs_bit0'_def])
apply (rule times_bit0_def [unfolded Abs_bit0'_def])
apply (rule minus_bit0_def [unfolded Abs_bit0'_def])
apply (rule uminus_bit0_def [unfolded Abs_bit0'_def])
done

interpretation bit1:
mod_type "int CARD('a::finite bit1)"
"Rep_bit1 :: 'a::finite bit1 ⇒ int"
"Abs_bit1 :: int ⇒ 'a::finite bit1"
apply (rule mod_type.intro)
apply (rule one_less_int_card)
apply (rule zero_bit1_def)
apply (rule one_bit1_def)
apply (rule plus_bit1_def [unfolded Abs_bit1'_def])
apply (rule times_bit1_def [unfolded Abs_bit1'_def])
apply (rule minus_bit1_def [unfolded Abs_bit1'_def])
apply (rule uminus_bit1_def [unfolded Abs_bit1'_def])
done

instance bit0 :: (finite) comm_ring_1
by (rule bit0.comm_ring_1)

instance bit1 :: (finite) comm_ring_1
by (rule bit1.comm_ring_1)

interpretation bit0:
mod_ring "int CARD('a::finite bit0)"
"Rep_bit0 :: 'a::finite bit0 ⇒ int"
"Abs_bit0 :: int ⇒ 'a::finite bit0"
..

interpretation bit1:
mod_ring "int CARD('a::finite bit1)"
"Rep_bit1 :: 'a::finite bit1 ⇒ int"
"Abs_bit1 :: int ⇒ 'a::finite bit1"
..

text ‹Set up cases, induction, and arithmetic›

lemmas bit0_cases [case_names of_int, cases type: bit0] = bit0.cases
lemmas bit1_cases [case_names of_int, cases type: bit1] = bit1.cases

lemmas bit0_induct [case_names of_int, induct type: bit0] = bit0.induct
lemmas bit1_induct [case_names of_int, induct type: bit1] = bit1.induct

lemmas bit0_iszero_numeral [simp] = bit0.iszero_numeral
lemmas bit1_iszero_numeral [simp] = bit1.iszero_numeral

lemmas [simp] = eq_numeral_iff_iszero [where 'a="'a bit0"] for dummy :: "'a::finite"
lemmas [simp] = eq_numeral_iff_iszero [where 'a="'a bit1"] for dummy :: "'a::finite"

subsection ‹Order instances›

instantiation bit0 and bit1 :: (finite) linorder begin
definition "a < b ⟷ Rep_bit0 a < Rep_bit0 b"
definition "a ≤ b ⟷ Rep_bit0 a ≤ Rep_bit0 b"
definition "a < b ⟷ Rep_bit1 a < Rep_bit1 b"
definition "a ≤ b ⟷ Rep_bit1 a ≤ Rep_bit1 b"

instance
by(intro_classes)
(auto simp add: less_eq_bit0_def less_bit0_def less_eq_bit1_def less_bit1_def Rep_bit0_inject Rep_bit1_inject)
end

lemma (in preorder) tranclp_less: "op <⇧+⇧+ = op <"
by(auto simp add: fun_eq_iff intro: less_trans elim: tranclp.induct)

instance bit0 and bit1 :: (finite) wellorder
proof -
have "wf {(x :: 'a bit0, y). x < y}"
by(auto simp add: trancl_def tranclp_less intro!: finite_acyclic_wf acyclicI)
thus "OFCLASS('a bit0, wellorder_class)"
by(rule wf_wellorderI) intro_classes
next
have "wf {(x :: 'a bit1, y). x < y}"
by(auto simp add: trancl_def tranclp_less intro!: finite_acyclic_wf acyclicI)
thus "OFCLASS('a bit1, wellorder_class)"
by(rule wf_wellorderI) intro_classes
qed

subsection ‹Code setup and type classes for code generation›

text ‹Code setup for @{typ num0} and @{typ num1}›

definition Num0 :: num0 where "Num0 = Abs_num0 0"
code_datatype Num0

instantiation num0 :: equal begin
definition equal_num0 :: "num0 ⇒ num0 ⇒ bool"
where "equal_num0 = op ="
instance by intro_classes (simp add: equal_num0_def)
end

lemma equal_num0_code [code]:
"equal_class.equal Num0 Num0 = True"
by(rule equal_refl)

code_datatype "1 :: num1"

instantiation num1 :: equal begin
definition equal_num1 :: "num1 ⇒ num1 ⇒ bool"
where "equal_num1 = op ="
instance by intro_classes (simp add: equal_num1_def)
end

lemma equal_num1_code [code]:
"equal_class.equal (1 :: num1) 1 = True"
by(rule equal_refl)

instantiation num1 :: enum begin
definition "enum_class.enum = [1 :: num1]"
definition "enum_class.enum_all P = P (1 :: num1)"
definition "enum_class.enum_ex P = P (1 :: num1)"
instance
by intro_classes
(auto simp add: enum_num1_def enum_all_num1_def enum_ex_num1_def num1_eq_iff Ball_def,
(metis (full_types) num1_eq_iff)+)
end

instantiation num0 and num1 :: card_UNIV begin
definition "finite_UNIV = Phantom(num0) False"
definition "card_UNIV = Phantom(num0) 0"
definition "finite_UNIV = Phantom(num1) True"
definition "card_UNIV = Phantom(num1) 1"
instance
by intro_classes
(simp_all add: finite_UNIV_num0_def card_UNIV_num0_def infinite_num0 finite_UNIV_num1_def card_UNIV_num1_def)
end

text ‹Code setup for @{typ "'a bit0"} and @{typ "'a bit1"}›

declare
bit0.Rep_inverse[code abstype]
bit0.Rep_0[code abstract]
bit0.Rep_1[code abstract]

lemma Abs_bit0'_code [code abstract]:
"Rep_bit0 (Abs_bit0' x :: 'a :: finite bit0) = x mod int (CARD('a bit0))"
by(auto simp add: Abs_bit0'_def intro!: Abs_bit0_inverse)

lemma inj_on_Abs_bit0:
"inj_on (Abs_bit0 :: int ⇒ 'a bit0) {0..<2 * int CARD('a :: finite)}"
by(auto intro: inj_onI simp add: Abs_bit0_inject)

declare
bit1.Rep_inverse[code abstype]
bit1.Rep_0[code abstract]
bit1.Rep_1[code abstract]

lemma Abs_bit1'_code [code abstract]:
"Rep_bit1 (Abs_bit1' x :: 'a :: finite bit1) = x mod int (CARD('a bit1))"
by(auto simp add: Abs_bit1'_def intro!: Abs_bit1_inverse)

lemma inj_on_Abs_bit1:
"inj_on (Abs_bit1 :: int ⇒ 'a bit1) {0..<1 + 2 * int CARD('a :: finite)}"
by(auto intro: inj_onI simp add: Abs_bit1_inject)

instantiation bit0 and bit1 :: (finite) equal begin

definition "equal_class.equal x y ⟷ Rep_bit0 x = Rep_bit0 y"
definition "equal_class.equal x y ⟷ Rep_bit1 x = Rep_bit1 y"

instance
by intro_classes (simp_all add: equal_bit0_def equal_bit1_def Rep_bit0_inject Rep_bit1_inject)

end

instantiation bit0 :: (finite) enum begin
definition "(enum_class.enum :: 'a bit0 list) = map (Abs_bit0' ∘ int) (upt 0 (CARD('a bit0)))"
definition "enum_class.enum_all P = (∀b :: 'a bit0 ∈ set enum_class.enum. P b)"
definition "enum_class.enum_ex P = (∃b :: 'a bit0 ∈ set enum_class.enum. P b)"

instance
proof(intro_classes)
show "distinct (enum_class.enum :: 'a bit0 list)"
by (simp add: enum_bit0_def distinct_map inj_on_def Abs_bit0'_def Abs_bit0_inject mod_pos_pos_trivial)

show univ_eq: "(UNIV :: 'a bit0 set) = set enum_class.enum"
unfolding enum_bit0_def type_definition.Abs_image[OF type_definition_bit0, symmetric]
by(simp add: image_comp [symmetric] inj_on_Abs_bit0 card_image image_int_atLeastLessThan)
(auto intro!: image_cong[OF refl] simp add: Abs_bit0'_def mod_pos_pos_trivial)

fix P :: "'a bit0 ⇒ bool"
show "enum_class.enum_all P = Ball UNIV P"
and "enum_class.enum_ex P = Bex UNIV P"
qed

end

instantiation bit1 :: (finite) enum begin
definition "(enum_class.enum :: 'a bit1 list) = map (Abs_bit1' ∘ int) (upt 0 (CARD('a bit1)))"
definition "enum_class.enum_all P = (∀b :: 'a bit1 ∈ set enum_class.enum. P b)"
definition "enum_class.enum_ex P = (∃b :: 'a bit1 ∈ set enum_class.enum. P b)"

instance
proof(intro_classes)
show "distinct (enum_class.enum :: 'a bit1 list)"
by(simp only: Abs_bit1'_def zmod_int[symmetric] enum_bit1_def distinct_map Suc_eq_plus1 card_bit1 o_apply inj_on_def)

show univ_eq: "(UNIV :: 'a bit1 set) = set enum_class.enum"
unfolding enum_bit1_def type_definition.Abs_image[OF type_definition_bit1, symmetric]
by(simp add: image_comp [symmetric] inj_on_Abs_bit1 card_image image_int_atLeastLessThan)
(auto intro!: image_cong[OF refl] simp add: Abs_bit1'_def mod_pos_pos_trivial)

fix P :: "'a bit1 ⇒ bool"
show "enum_class.enum_all P = Ball UNIV P"
and "enum_class.enum_ex P = Bex UNIV P"
qed

end

instantiation bit0 and bit1 :: (finite) finite_UNIV begin
definition "finite_UNIV = Phantom('a bit0) True"
definition "finite_UNIV = Phantom('a bit1) True"
instance by intro_classes (simp_all add: finite_UNIV_bit0_def finite_UNIV_bit1_def)
end

instantiation bit0 and bit1 :: ("{finite,card_UNIV}") card_UNIV begin
definition "card_UNIV = Phantom('a bit0) (2 * of_phantom (card_UNIV :: 'a card_UNIV))"
definition "card_UNIV = Phantom('a bit1) (1 + 2 * of_phantom (card_UNIV :: 'a card_UNIV))"
instance by intro_classes (simp_all add: card_UNIV_bit0_def card_UNIV_bit1_def card_UNIV)
end

subsection ‹Syntax›

syntax
"_NumeralType" :: "num_token => type"  ("_")
"_NumeralType0" :: type ("0")
"_NumeralType1" :: type ("1")

translations
(type) "1" == (type) "num1"
(type) "0" == (type) "num0"

parse_translation ‹
let
fun mk_bintype n =
let
fun mk_bit 0 = Syntax.const @{type_syntax bit0}
| mk_bit 1 = Syntax.const @{type_syntax bit1};
fun bin_of n =
if n = 1 then Syntax.const @{type_syntax num1}
else if n = 0 then Syntax.const @{type_syntax num0}
else if n = ~1 then raise TERM ("negative type numeral", [])
else
let val (q, r) = Integer.div_mod n 2;
in mk_bit r \$ bin_of q end;
in bin_of n end;

fun numeral_tr [Free (str, _)] = mk_bintype (the (Int.fromString str))
| numeral_tr ts = raise TERM ("numeral_tr", ts);

in [(@{syntax_const "_NumeralType"}, K numeral_tr)] end;
›

print_translation ‹
let
fun int_of [] = 0
| int_of (b :: bs) = b + 2 * int_of bs;

fun bin_of (Const (@{type_syntax num0}, _)) = []
| bin_of (Const (@{type_syntax num1}, _)) = [1]
| bin_of (Const (@{type_syntax bit0}, _) \$ bs) = 0 :: bin_of bs
| bin_of (Const (@{type_syntax bit1}, _) \$ bs) = 1 :: bin_of bs
| bin_of t = raise TERM ("bin_of", [t]);

fun bit_tr' b [t] =
let
val rev_digs = b :: bin_of t handle TERM _ => raise Match
val i = int_of rev_digs;
val num = string_of_int (abs i);
in
Syntax.const @{syntax_const "_NumeralType"} \$ Syntax.free num
end
| bit_tr' b _ = raise Match;
in
[(@{type_syntax bit0}, K (bit_tr' 0)),
(@{type_syntax bit1}, K (bit_tr' 1))]
end;
›

subsection ‹Examples›

lemma "CARD(0) = 0" by simp
lemma "CARD(17) = 17" by simp
lemma "8 * 11 ^ 3 - 6 = (2::5)" by simp

end
```