Theory Code_Binary_Nat

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


header {* 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"
by (simp_all add: nat_of_num_inverse)

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 *}

lemma [code, code del]:
"(plus :: nat => _) = plus" ..

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)"
by (simp_all add: nat_of_num_numeral)

text {* Bounded subtraction needs some auxiliary *}

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)"
by (simp_all add: dup_def numeral_Bit0)

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) = Option.map dup (sub m n)"
"sub (Num.Bit1 m) (Num.Bit1 n) = Option.map dup (sub m n)"
"sub (Num.Bit1 m) (Num.Bit0 n) = Option.map (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))"

apply (auto simp add: nat_of_num_numeral
Num.dbl_def Num.dbl_inc_def Num.dbl_dec_def
Let_def le_imp_diff_is_add BitM_plus_one sub_def dup_def)
apply (simp_all add: sub_non_positive)
apply (simp_all add: sub_non_negative [symmetric, where ?'a = int])
done

lemma [code, code del]:
"(minus :: nat => _) = minus" ..

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)

lemma [code, code del]:
"(times :: nat => _) = times" ..

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)"
by (simp_all add: nat_of_num_numeral)

lemma [code, code del]:
"(HOL.equal :: nat => _) = HOL.equal" ..

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"
by (simp_all add: nat_of_num_numeral equal)

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

lemma [code, code del]:
"(less_eq :: nat => _) = less_eq" ..

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"
by (simp_all add: nat_of_num_numeral)

lemma [code, code del]:
"(less :: nat => _) = less" ..

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"
by (simp_all add: nat_of_num_numeral)

lemma [code, code del]:
"divmod_nat = divmod_nat" ..

lemma divmod_nat_code [code]:
"divmod_nat (nat_of_num k) (nat_of_num l) = divmod k l"
"divmod_nat m 0 = (0, m)"
"divmod_nat 0 n = (0, 0)"
by (simp_all add: prod_eq_iff nat_of_num_numeral del: div_nat_numeral mod_nat_numeral)


subsection {* Conversions *}

lemma [code, code del]:
"of_nat = of_nat" ..

lemma of_nat_code [code]:
"of_nat 0 = 0"
"of_nat (nat_of_num k) = numeral k"
by (simp_all add: nat_of_num_numeral)


code_identifier
code_module Code_Binary_Nat \<rightharpoonup>
(SML) Arith and (OCaml) Arith and (Haskell) Arith

hide_const (open) dup sub

end