src/HOL/Library/Numeral_Type.thy
 author kleing Mon, 20 Aug 2007 00:22:18 +0200 changeset 24332 e3a2b75b1cf9 child 24406 d96eb21fc1bc permissions -rw-r--r--
boolean algebras as locales and numbers as types by Brian Huffman
```
(*
ID:     \$Id\$
Author: Brian Huffman

Numeral Syntax for Types
*)

theory Numeral_Type
imports Infinite_Set
begin

subsection {* Preliminary lemmas *}
(* These should be moved elsewhere *)

lemma inj_Inl [simp]: "inj_on Inl A"
by (rule inj_onI, simp)

lemma inj_Inr [simp]: "inj_on Inr A"
by (rule inj_onI, simp)

lemma inj_Some [simp]: "inj_on Some A"
by (rule inj_onI, simp)

lemma card_Plus:
"[| finite A; finite B |] ==> card (A <+> B) = card A + card B"
unfolding Plus_def
apply (subgoal_tac "Inl ` A \<inter> Inr ` B = {}")
apply fast
done

lemma (in type_definition) univ:
"UNIV = Abs ` A"
proof
show "Abs ` A \<subseteq> UNIV" by (rule subset_UNIV)
show "UNIV \<subseteq> Abs ` A"
proof
fix x :: 'b
have "x = Abs (Rep x)" by (rule Rep_inverse [symmetric])
moreover have "Rep x \<in> A" by (rule Rep)
ultimately show "x \<in> Abs ` A" by (rule image_eqI)
qed
qed

lemma (in type_definition) card: "card (UNIV :: 'b set) = card A"
by (simp add: univ card_image inj_on_def Abs_inject)

subsection {* Cardinalities of types *}

syntax "_type_card" :: "type => nat" ("(1CARD/(1'(_')))")

translations "CARD(t)" => "card (UNIV::t set)"

lemma card_unit: "CARD(unit) = 1"
unfolding univ_unit by simp

lemma card_bool: "CARD(bool) = 2"
unfolding univ_bool by simp

lemma card_prod: "CARD('a::finite \<times> 'b::finite) = CARD('a) * CARD('b)"
unfolding univ_prod by (simp only: card_cartesian_product)

lemma card_sum: "CARD('a::finite + 'b::finite) = CARD('a) + CARD('b)"
unfolding univ_sum by (simp only: finite card_Plus)

lemma card_option: "CARD('a::finite option) = Suc CARD('a)"
unfolding univ_option
apply (subgoal_tac "(None::'a option) \<notin> range Some")
apply fast
done

lemma card_set: "CARD('a::finite set) = 2 ^ CARD('a)"
unfolding univ_set
by (simp only: card_Pow finite numeral_2_eq_2)

subsection {* Numeral Types *}

typedef (open) pls = "UNIV :: nat set" ..
typedef (open) num1 = "UNIV :: unit set" ..
typedef (open) 'a bit0 = "UNIV :: (bool * 'a) set" ..
typedef (open) 'a bit1 = "UNIV :: (bool * 'a) option set" ..

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) finite
proof
show "finite (UNIV::'a bit0 set)"
unfolding type_definition.univ [OF type_definition_bit0]
using finite by (rule finite_imageI)
qed

instance bit1 :: (finite) finite
proof
show "finite (UNIV::'a bit1 set)"
unfolding type_definition.univ [OF type_definition_bit1]
using finite by (rule finite_imageI)
qed

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

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

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

lemma card_pls: "CARD (pls) = 0"
by (simp add: type_definition.card [OF type_definition_pls])

lemmas card_univ_simps [simp] =
card_unit
card_bool
card_prod
card_sum
card_option
card_set
card_num1
card_bit0
card_bit1
card_pls

subsection {* Syntax *}

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

translations
"_NumeralType1" == (type) "num1"
"_NumeralType0" == (type) "pls"

parse_translation {*
let

val num1_const = Syntax.const "Numeral_Type.num1";
val pls_const = Syntax.const "Numeral_Type.pls";
val B0_const = Syntax.const "Numeral_Type.bit0";
val B1_const = Syntax.const "Numeral_Type.bit1";

fun mk_bintype n =
let
fun mk_bit n = if n = 0 then B0_const else B1_const;
fun bin_of n =
if n = 1 then num1_const
else if n = 0 then pls_const
else if n = ~1 then raise TERM ("negative type numeral", [])
else
let val (q, r) = IntInf.divMod (n, 2);
in mk_bit r \$ bin_of q end;
in bin_of n end;

fun numeral_tr (*"_NumeralType"*) [Const (str, _)] =
mk_bintype (valOf (IntInf.fromString str))
| numeral_tr (*"_NumeralType"*) ts = raise TERM ("numeral_tr", ts);

in [("_NumeralType", numeral_tr)] end;
*}

print_translation {*
let
fun int_of [] = 0
| int_of (b :: bs) = IntInf.fromInt b + (2 * int_of bs);

fun bin_of (Const ("pls", _)) = []
| bin_of (Const ("num1", _)) = [1]
| bin_of (Const ("bit0", _) \$ bs) = 0 :: bin_of bs
| bin_of (Const ("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 = IntInf.toString (IntInf.abs i);
in
Syntax.const "_NumeralType" \$ Syntax.free num
end
| bit_tr' b _ = raise Match;

in [("bit0", bit_tr' 0), ("bit1", bit_tr' 1)] end;
*}

subsection {* Classes with at values least 1 and 2  *}

text {* Class finite already captures "at least 1" *}

lemma zero_less_card_finite:
"0 < CARD('a::finite)"
proof (cases "CARD('a::finite) = 0")
case False thus ?thesis by (simp del: card_0_eq)
next
case True
thus ?thesis by (simp add: finite)
qed

lemma one_le_card_finite:
"Suc 0 <= CARD('a::finite)"
by (simp add: less_Suc_eq_le [symmetric] zero_less_card_finite)

text {* Class for cardinality "at least 2" *}

class card2 = finite +
assumes two_le_card: "2 <= CARD('a)"

lemma one_less_card: "Suc 0 < CARD('a::card2)"
using two_le_card [where 'a='a] by simp

instance bit0 :: (finite) card2