(*
ID: $Id$
Author: Brian Huffman
Numeral Syntax for Types
*)
header "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 (simp add: card_Un_disjoint card_image)
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)" => "CONST card (UNIV \<Colon> t set)"
typed_print_translation {*
let
fun card_univ_tr' show_sorts _ [Const (@{const_name UNIV}, Type(_,[T]))] =
Syntax.const "_type_card" $ Syntax.term_of_typ show_sorts T;
in [("card", card_univ_tr')]
end
*}
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_Times_UNIV [symmetric] by (simp only: card_cartesian_product)
lemma card_sum: "CARD('a::finite + 'b::finite) = CARD('a) + CARD('b)"
unfolding UNIV_Plus_UNIV [symmetric] by (simp only: finite card_Plus)
lemma card_option: "CARD('a::finite option) = Suc CARD('a)"
unfolding insert_None_conv_UNIV [symmetric]
apply (subgoal_tac "(None::'a option) \<notin> range Some")
apply (simp add: finite card_image)
apply fast
done
lemma card_set: "CARD('a::finite set) = 2 ^ CARD('a)"
unfolding Pow_UNIV [symmetric]
by (simp only: card_Pow finite numeral_2_eq_2)
subsection {* Numeral Types *}
typedef (open) num0 = "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_num0: "CARD (num0) = 0"
by (simp add: type_definition.card [OF type_definition_num0])
lemmas card_univ_simps [simp] =
card_unit
card_bool
card_prod
card_sum
card_option
card_set
card_num1
card_bit0
card_bit1
card_num0
subsection {* Syntax *}
syntax
"_NumeralType" :: "num_const => type" ("_")
"_NumeralType0" :: type ("0")
"_NumeralType1" :: type ("1")
translations
"_NumeralType1" == (type) "num1"
"_NumeralType0" == (type) "num0"
parse_translation {*
let
val num1_const = Syntax.const "Numeral_Type.num1";
val num0_const = Syntax.const "Numeral_Type.num0";
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 num0_const
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 (*"_NumeralType"*) [Const (str, _)] =
mk_bintype (valOf (Int.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) = b + 2 * int_of bs;
fun bin_of (Const ("num0", _)) = []
| 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 = string_of_int (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 least 1 and 2 *}
text {* Class finite already captures "at least 1" *}
lemma zero_less_card_finite [simp]:
"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 [simp]:
"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
by intro_classes (simp add: one_le_card_finite)
instance bit1 :: (finite) card2
by intro_classes (simp add: one_le_card_finite)
subsection {* Examples *}
lemma "CARD(0) = 0" by simp
lemma "CARD(17) = 17" by simp
end