# Theory Code_Binary_Nat

```(*  Title:      HOL/Library/Code_Binary_Nat.thy
Author:     Florian Haftmann, TU Muenchen
*)

section ‹Implementation of natural numbers as binary numerals›

theory Code_Binary_Nat
imports Code_Abstract_Nat
begin

text ‹
When generating code for functions on natural numbers, the
canonical representation using \<^term>‹0::nat› and
\<^term>‹Suc› is unsuitable for computations involving large
numbers.  This theory refines the representation of
natural numbers for code generation to use binary
numerals, which do not grow linear in size but logarithmic.
›

subsection ‹Representation›

code_datatype "0::nat" nat_of_num

lemma [code]:
"num_of_nat 0 = Num.One"
"num_of_nat (nat_of_num k) = k"

lemma [code]:
"(1::nat) = Numeral1"
by simp

lemma [code_abbrev]: "Numeral1 = (1::nat)"
by simp

lemma [code]:
"Suc n = n + 1"
by simp

subsection ‹Basic arithmetic›

context
begin

declare [[code drop: "plus :: nat ⇒ _"]]

lemma plus_nat_code [code]:
"nat_of_num k + nat_of_num l = nat_of_num (k + l)"
"m + 0 = (m::nat)"
"0 + n = (n::nat)"

text ‹Bounded subtraction needs some auxiliary›

qualified definition dup :: "nat ⇒ nat" where
"dup n = n + n"

lemma dup_code [code]:
"dup 0 = 0"
"dup (nat_of_num k) = nat_of_num (Num.Bit0 k)"

qualified definition sub :: "num ⇒ num ⇒ nat option" where
"sub k l = (if k ≥ l then Some (numeral k - numeral l) else None)"

lemma sub_code [code]:
"sub Num.One Num.One = Some 0"
"sub (Num.Bit0 m) Num.One = Some (nat_of_num (Num.BitM m))"
"sub (Num.Bit1 m) Num.One = Some (nat_of_num (Num.Bit0 m))"
"sub Num.One (Num.Bit0 n) = None"
"sub Num.One (Num.Bit1 n) = None"
"sub (Num.Bit0 m) (Num.Bit0 n) = map_option dup (sub m n)"
"sub (Num.Bit1 m) (Num.Bit1 n) = map_option dup (sub m n)"
"sub (Num.Bit1 m) (Num.Bit0 n) = map_option (λq. dup q + 1) (sub m n)"
"sub (Num.Bit0 m) (Num.Bit1 n) = (case sub m n of None ⇒ None
| Some q ⇒ if q = 0 then None else Some (dup q - 1))"
Num.dbl_def Num.dbl_inc_def Num.dbl_dec_def
apply (simp_all add: sub_non_negative [symmetric, where ?'a = int])
done

declare [[code drop: "minus :: nat ⇒ _"]]

lemma minus_nat_code [code]:
"nat_of_num k - nat_of_num l = (case sub k l of None ⇒ 0 | Some j ⇒ j)"
"m - 0 = (m::nat)"
"0 - n = (0::nat)"
by (simp_all add: nat_of_num_numeral sub_non_positive sub_def)

declare [[code drop: "times :: nat ⇒ _"]]

lemma times_nat_code [code]:
"nat_of_num k * nat_of_num l = nat_of_num (k * l)"
"m * 0 = (0::nat)"
"0 * n = (0::nat)"

declare [[code drop: "HOL.equal :: nat ⇒ _"]]

lemma equal_nat_code [code]:
"HOL.equal 0 (0::nat) ⟷ True"
"HOL.equal 0 (nat_of_num l) ⟷ False"
"HOL.equal (nat_of_num k) 0 ⟷ False"
"HOL.equal (nat_of_num k) (nat_of_num l) ⟷ HOL.equal k l"

lemma equal_nat_refl [code nbe]:
"HOL.equal (n::nat) n ⟷ True"
by (rule equal_refl)

declare [[code drop: "less_eq :: nat ⇒ _"]]

lemma less_eq_nat_code [code]:
"0 ≤ (n::nat) ⟷ True"
"nat_of_num k ≤ 0 ⟷ False"
"nat_of_num k ≤ nat_of_num l ⟷ k ≤ l"

declare [[code drop: "less :: nat ⇒ _"]]

lemma less_nat_code [code]:
"(m::nat) < 0 ⟷ False"
"0 < nat_of_num l ⟷ True"
"nat_of_num k < nat_of_num l ⟷ k < l"

declare [[code drop: Euclidean_Rings.divmod_nat]]

lemma divmod_nat_code [code]:
"Euclidean_Rings.divmod_nat (nat_of_num k) (nat_of_num l) = divmod k l"
"Euclidean_Rings.divmod_nat m 0 = (0, m)"
"Euclidean_Rings.divmod_nat 0 n = (0, 0)"

end

subsection ‹Conversions›

declare [[code drop: of_nat]]

lemma of_nat_code [code]:
"of_nat 0 = 0"
"of_nat (nat_of_num k) = numeral k"