src/HOL/Library/Numeral_Type.thy

number_ring instances for numeral types

(*  Title:      HOL/Library/Numeral_Type.thy
    Author:     Brian Huffman
*)

header {* Numeral Syntax for Types *}

theory Numeral_Type
imports Plain "~~/src/HOL/Presburger"
begin

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

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 (CONST 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 [(@{const_syntax 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: 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)

lemma card_finite_pos [simp]: "0 < CARD('a::finite)"
  unfolding neq0_conv [symmetric] by simp


subsection {* Numeral Types *}

typedef (open) num0 = "UNIV :: nat set" ..
typedef (open) num1 = "UNIV :: unit set" ..

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

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

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]
    by simp
qed

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

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

lemma card_num0: "CARD (num0) = 0"
  by (simp add: infinite_UNIV_nat card_eq_0_iff 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 {* Locale for modular arithmetic subtypes *}

locale mod_type =
  fixes n :: int
  and Rep :: "'a::{zero,one,plus,times,uminus,minus,power} \<Rightarrow> int"
  and Abs :: "int \<Rightarrow> 'a::{zero,one,plus,times,uminus,minus,power}"
  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)"
  and power_def: "x ^ k = Abs (Rep x ^ k mod n)"
begin

lemma size0: "0 < n"
by (cut_tac size1, simp)

lemmas definitions =
  zero_def one_def add_def mult_def minus_def diff_def power_def

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

lemma Rep_le_n: "Rep x \<le> n"
by (rule Rep_less_n [THEN order_less_imp_le])

lemma Rep_inject_sym: "x = y \<longleftrightarrow> 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 \<in> {0..<n} \<Longrightarrow> 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 IntDiv.pos_mod_conj [OF size0])

lemma Rep_Abs_0: "Rep (Abs 0) = 0"
by (simp add: Abs_inverse size0)

lemma Rep_0: "Rep 0 = 0"
by (simp add: zero_def Rep_Abs_0)

lemma Rep_Abs_1: "Rep (Abs 1) = 1"
by (simp add: Abs_inverse size1)

lemma Rep_1: "Rep 1 = 1"
by (simp add: one_def Rep_Abs_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])
apply (simp add: mod_pos_pos_trivial)
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 zmod_simps ring_simps)
done

lemma recpower: "OFCLASS('a, recpower_class)"
apply (intro_classes, unfold definitions)
apply (simp_all add: Rep_simps zmod_simps add_ac mult_assoc
                     mod_pos_pos_trivial size1)
done

end

locale mod_ring = mod_type +
  constrains n :: int
  and Rep :: "'a::{number_ring,power} \<Rightarrow> int"
  and Abs :: "int \<Rightarrow> 'a::{number_ring,power}"
begin

lemma of_nat_eq: "of_nat k = Abs (int k mod n)"
apply (induct k)
apply (simp add: zero_def)
apply (simp add: Rep_simps add_def one_def zmod_simps add_ac)
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 zmod_simps)
done

lemma Rep_number_of:
  "Rep (number_of w) = number_of w mod n"
by (simp add: number_of_eq of_int_eq Rep_Abs_mod)

lemma iszero_number_of:
  "iszero (number_of w::'a) \<longleftrightarrow> number_of w mod n = 0"
by (simp add: Rep_simps number_of_eq of_int_eq iszero_def zero_def)

lemma cases:
  assumes 1: "\<And>z. \<lbrakk>(x::'a) = of_int z; 0 \<le> z; z < n\<rbrakk> \<Longrightarrow> P"
  shows "P"
apply (cases x rule: type_definition.Abs_cases [OF type])
apply (rule_tac z="y" in 1)
apply (simp_all add: of_int_eq mod_pos_pos_trivial)
done

lemma induct:
  "(\<And>z. \<lbrakk>0 \<le> z; z < n\<rbrakk> \<Longrightarrow> P (of_int z)) \<Longrightarrow> P (x::'a)"
by (cases x rule: cases) simp

end


subsection {* Number ring instances *}

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

definition Abs_bit0' :: "int \<Rightarrow> 'a bit0" where
  "Abs_bit0' x = Abs_bit0 (x mod (2 * int CARD('a)))"

definition Abs_bit1' :: "int \<Rightarrow> 'a bit1" where
  "Abs_bit1' x = Abs_bit1 (x mod (1 + 2 * int CARD('a)))"

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 "x ^ k = Abs_bit0' (Rep_bit0 x ^ k)"

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)"
definition "x ^ k = Abs_bit1' (Rep_bit1 x ^ k)"

instance ..

end

interpretation bit0!:
  mod_type "2 * int CARD('a::finite)"
           "Rep_bit0 :: 'a::finite bit0 \<Rightarrow> int"
           "Abs_bit0 :: int \<Rightarrow> 'a::finite bit0"
apply (rule mod_type.intro)
apply (rule type_definition_bit0)
using card_finite_pos [where ?'a='a] apply arith
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])
apply (rule power_bit0_def [unfolded Abs_bit0'_def])
done

interpretation bit1!:
  mod_type "1 + 2 * int CARD('a::finite)"
           "Rep_bit1 :: 'a::finite bit1 \<Rightarrow> int"
           "Abs_bit1 :: int \<Rightarrow> 'a::finite bit1"
apply (rule mod_type.intro)
apply (rule type_definition_bit1)
apply simp
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])
apply (rule power_bit1_def [unfolded Abs_bit1'_def])
done

instance bit0 :: (finite) "{comm_ring_1,recpower}"
  by (rule bit0.comm_ring_1 bit0.recpower)+

instance bit1 :: (finite) "{comm_ring_1,recpower}"
  by (rule bit1.comm_ring_1 bit1.recpower)+

instantiation bit0 and bit1 :: (finite) number_ring
begin

definition "(number_of w :: _ bit0) = of_int w"

definition "(number_of w :: _ bit1) = of_int w"

instance proof
qed (rule number_of_bit0_def number_of_bit1_def)+

end

interpretation bit0!:
  mod_ring "2 * int CARD('a::finite)"
           "Rep_bit0 :: 'a::finite bit0 \<Rightarrow> int"
           "Abs_bit0 :: int \<Rightarrow> 'a::finite bit0"
  ..

interpretation bit1!:
  mod_ring "1 + 2 * int CARD('a::finite)"
           "Rep_bit1 :: 'a::finite bit1 \<Rightarrow> int"
           "Abs_bit1 :: int \<Rightarrow> 'a::finite bit1"
  ..

text {* Set up cases, induction, and arithmetic *}

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

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

lemmas bit0_iszero_number_of [simp] = bit0.iszero_number_of
lemmas bit1_iszero_number_of [simp] = bit1.iszero_number_of

declare power_Suc [where ?'a="'a::finite bit0", standard, simp]
declare power_Suc [where ?'a="'a::finite bit1", standard, simp]


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

end